Advances in the Analysis and Design of Marine Structures 1032506369, 9781032506364

Table of contents :
cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Preface
Organisation
Acknowledgements
Methods and tools for loads and load effects Hull girder loads
Verification of onboard loading computer by laser measurements
Assessment of loads on a ship hull during a sideways launching process based on model tests
Hull structural loads during a sideways launching process using fluid-structure interaction
Study on fatigue design loads based on actual encountered loads
Wave loads
Update of wave statistics standards for classification rules
Different strategies to improve isochrone voyage optimization algorithm
Estimation of nonlinear forces acting on floating bodies using machine learning
Comparative study of ship design load on various environmental conditions
On nonlinear wave loads of a mega-scale container ship using an elastic backbone model
Encounter spectra computation of heave motion based on full-scale measurements using ANN
Data-driven ship fatigue assessment based on pitch and heave motions
Numerical simulation of ship motion and non-linear sea loads of a modern frigate in regular waves
Slamming
Abnormal wave slamming impact of stiffened cylinders
Comparison between deforming meshes and overset meshes for water entry of a wedge
Experimental investigation on oblique entry of trimaran cross deck structure
Fluid-structure interaction analysis for bow-shaped structure subjected to slamming pressure
Dynamics and vibration
Motion analysis of a floating horizontal set of interconnected plates based on computer vision target tracking technique
Wind-induced vibration characteristics of typical guide rail frame structure in open area of large cruise ships
Wind-induced vibration characteristics of typical guide rail frame structure in open area of large cruise ships
Model test of a dual-spar floating wind farm in regular waves
Acoustic black holes: The new frontier for soundproofing on board ships
Wave energy
Experimental study on inflatable circular diaphragms used in the oscillating water column wave energy converter
Power capture performance analysis of a 2DoF nonlinear wave energy converter in regular waves
Fatigue of mooring lines in wave energy parks
Two-body, time domain model for a heaving point absorber
Wind turbine
Implied safety level in design codes for monopile offshore wind turbines - parametric study on the effect of partial safety factors
Experimental study of long-term scour damage for protected offshore wind foundations
Design of substructure of 10MW floating offshore wind turbine system based on dominant load parameters
Mooring system optimization of 12MW FOWT in environment condition in the southwest sea of Korea
Methods and tools for strength assessment Structural analysis
Hydroelastic response of a moored floating flexible structure based on Timoshenko-Mindlin beam theory
Hull deflection estimation model for marine shaft alignment applications
A study on the development of a FPSO standard model (KRISO-FPSO) I: Focusing on structural safety evaluation
Numerical analysis of the Ro-Ro vehicle ramp structure under moving load
Finite element analysis for ship production solutions
A mathematical model for variable cross-section hull girder withtime-varying mass characteristics
A method for fast calculation of vertical bending moment and stress distribution of hull structure in HMS system
Structure analysis of lightweight sail structures for wind-assisted ship propulsion
Ultimate strength
Rule formulation updates on buckling strength requirements in Common Structural Rules
Equivalent single layer approach for predicting ultimate strength of stiffened panel under different load combinations
Study on the modification and application of RTCL criterion considering the influence of mesh size
The ultimate strength of sandwich plates with joints under in-plane compression
Ultimate strength – hull girder
Hull girder ultimate strength assessment according to rules requirements
Hull girder collapse analysis in extreme wave by direct coupled simulation between CFD and FEM
Predicting ship’s ultimate longitudinal strength considering the lateral pressure loading
Ultimate strength – cyclic load
Evaluation of residual plastic strain on a stiffened panel subjected to compression and tension-compression cyclic load
FEM numerical strategies for the evaluation of the accumulated plastic strain due to a cyclic load condition
Response of plates and stiffened panels under pressure and cyclic biaxial loading
Collapse response analysis of a ship’s hull girder in cyclic focused waves using a hydro-elasto-plastic beam model
Fatigue
Consequence and uncertainty-informed fatigue life prediction of ships
Fatigue performance of aluminum alloy EN-AW 5083 cut edges producedby abrasive water jet and laser cutting
Structural life expectancy assessment of an aging combatant ship using spectral fatigue analyses
Fatigue testing of multiaxial non-proportionally loaded T-nodes
A study on the development of a FPSO standard model (KRISO-FPSO) II:Focusing on fatigue assessment
Predicting failure of dynamic subsea cables by electrical insulation breakdown due to water treeing
On the application of Engineering Critical Assessment specified in BS 7910 to ship structures
Numerical analysis of pit-to-crack transition under corrosion fatigue using a stochastic pit generation algorithm
Wind turbine – fatigue
The effects of surge underprediction on fatigue damage of floating offshore wind turbine dynamic cables
Tension angle assessment of mooring chain of a 10MW floating offshore wind turbine
Fatigue assessment of offshore wind turbine support structures subjected to seawater
Impact of preload loss on fatigue strength of blade root bolts of a Floating Offshore Wind Turbine (FOWT)
Offshore structures
Predicting long-term extreme responses using two approaches with a case study of a jacket structure
Experimental study on the lateral bending capacity of steel pipes with initialdents
Collapse and burst analysis for a subsea sandwich pipe
Stress distribution in uniplanar KT joints reinforced with fibre reinforcedpolymer subjected to the axial loadings
Ice
Application of the CASFOP taxonomy to analyse an ice incursion incident with a FPSO
Fluid-structure interaction simulations using a hydrodynamic plug-in HydroQus
A rate and pressure dependent elastoplastic material model for glacial ice colliding with marine structures
Study on the annual changing trend of Arctic Sea ice melting for merchant shipping
Impact
Experimental investigation of the repeated impact behaviour of rectangular plates
Crashworthiness analysis of semi-submersible platform column subjected to ship impact loads
Shape characterization and impact on the structural behavior of initially distorted, 4-mm thick ship-deck stiffened panels
Analysis of how the conditions in a collision scenario affect the size of a struck vessel’s damage opening and ultimate strength
Blast and explosion
Prediction of damage extents due to in-compartment explosions in warships
Anti-blast performance optimization design of corrugated sandwich structures based on BP-GA method
Time delay effect in the dynamic response of submerged cylinder subjected to an underwater explosion
Recent studies on the saturated impulse for ship structures under pressurepulse loading
Experimental analysis of structures Composites
Finite element analysis of filament wound composite materials split disk tests
Experimental and numerical study of composite materials drive shafts
Thermal and rheological study of structural adhesives for naval construction
Experimental analysis of ballistic impact on carbon, glass and hybrid composite under hydrostatic prestrain
Materials and fabrication of structures Corrosion
The integrity of corrosion protection systems of welded maritime structures under cyclic loading
Microstructural analysis of intergranular stress corrosion cracking in 5xxxseries aluminum reinforced with a composite patch
Uncertainty quantification within strain-based SHM schemes used for detecting thickness loss in ship hulls
Corrosion degradation impact on mechanical properties of structural steel
Methods and tools for structural design and optimization Ship design
Integration of a tank storage solution for alternative fuels on a RoRo ship
On the safety of offshore mooring systems
Experimental study on structure responses of triple wing sails to turbulence flows at multiple apparent wind angles
Numerical and experimental analysis of fire resistance for bulkhead and deck structures of ships and offshore installations
Ship design and optimization
Comparison of metaheuristic algorithms and constraint handling approaches for multi-objective optimization of a tanker
Structural optimization of cell guides for container securing as a retrofit solution on barges
Ship design optimization considering probabilistic compliance of decarbonization regulations
Lightweight design and topology optimization of marine structures using peridynamics
Structural reliability, safety and environmental protection Reliability
Review of the structural configuration and strength of metallic sandwich panels
Analysis of pontoon primary structure failure
Pontoon design and construction methodology of gangway
Significance of laser weld stiffness in vibration and buckling optimization of laser-welded web-core sandwich panels
A Bayesian approach for the quantification of strength model uncertainty factor in ultimate limit state
Structural reliability analysis of secondary hull detail
Concept of the probability-based ship operability analysis
Bayesian inference for the parameters of probabilistic corrosion model adopted by IACS
Author Index

Citation preview

Marine Technology and Ocean Engineering Series Volume 11

Advances in the Analysis and Design of Marine Structures Editors Gued es Soares Gar batov

Progress in the Analysis and Design of Marine Structures Editors: J.W. Ringsberg C. Guedes Soares

ADVANCES IN THE ANALYSIS AND DESIGN OF MARINE STRUCTURES

Advances in the Analysis and Design of Marine Structures is a collection of papers presented at MARSTRUCT 2023, the 9th International Conference on Marine Structures, held in Gothenburg, Sweden, 3-5 April 2023. The conference was organised by the Division of Marine Technology, Depart­ ment of Mechanics and Maritime Sciences at Chalmers University of Technology, in Gothenburg, Sweden. The MARSTRUCT Conference series deals with Ship and Offshore Structures, addressing topics in the fields of: • Methods and tools for loads and load effects • Methods and tools for strength assessment • Experimental analysis of structures • Materials and fabrication of structures • Methods and tools for structural design and optimization • Structural reliability, safety, and environmental protection The MARSTRUCT conferences series of started in Glasgow, UK in 2007, the second event of the series took place in Lisbon, Portugal in March 2009, the third in Hamburg, Germany in March 2011, the fourth in Espoo, Finland in March 2013, the fifth in Southampton, UK in March 2015, the sixth in Lisbon, Portugal in May 2017, the seventh in Dubrovnik, Croatia in May 2019, and the eighth event in Trondheim, Norway in June 2021. Advances in the Analysis and Design of Marine Structures is essential reading for academics, engineers and all professionals involved in the design of marine and offshore structures. The Proceedings in Marine Technology and Ocean Engineering series is dedicated to the publication of proceedings of peer-reviewed international conferences dealing with various aspects of ‘Marine Technol­ ogy and Ocean Engineering’. The Series includes the proceedings of the following conferences: the International Maritime Association of the Mediterranean (IMAM) conferences, the Marine Structures (MARSTRUCT) conferences, the Renewable Energies Offshore (RENEW) conferences and the Mari­ time Technology (MARTECH) conferences. The ‘Marine Technology and Ocean Engineering’ series is also open to new conferences that cover topics on the sustainable exploration and exploitation of marine resources in various fields, such as maritime transport and ports, usage of the ocean including coastal areas, nautical activities, the exploration and exploitation of mineral resources, the protection of the marine environment and its resources, and risk analysis, safety and reliability. The aim of the series is to stimulate advanced education and training through the wide dissemination of the results of scientific research.

Proceedings in Marine Technology and Ocean Engineering BOOK SERIES EDITOR Carlos Guedes Soares EDITORIAL BOARD MEMBERS R. Ajit Shenoi, Enrico Rizzuto, Fenando Lopez-Peña, Jani Romanov, Jorgen Amdahl, Joško Parunov ABOUT THE SERIES The ‘Proceedings in Marine Technology and Ocean Engineering’ series is devoted to the publica­ tion of proceedings of peer-reviewed international conferences dealing with various aspects of ‘Marine Technology and Ocean Engineering’. The Series includes the proceedings of the following conferences: the International Maritime Association of the Mediterranean (IMAM) Conferences, the Marine Structures (MARSTRUCT) Conferences, the Renewable Energies Offshore (RENEW) Conferences and the Maritime Technology (MARTECH) Conferences. The ‘Marine Technology and Ocean Engineering’ series is also open to new conferences that cover topics on the sustainable exploration and exploitation of marine resources in various fields, such as maritime transport and ports, usage of the ocean including coastal areas, nautical activities, the exploration and exploitation of mineral resources, the protection of the marine environment and its resources, and risk analysis, safety and reliability. The aim of the series is to stimulate advanced education and training through the wide dissemination of the results of scientific research. BOOKS IN THE SERIES Volume 1: Advances in Renewable Energies Offshore, 2019, C. Guedes Soares (Ed.). Volume 2: Trends in the Analysis and Design of Marine Structures, 2019, J. Parunov and C. Guedes Soares (Eds.). Volume 3: Sustainable Development and Innovations in Marine Technologies, 2020, P. Georgiev and C. Guedes Soares (Eds.). Volume 4: Developments in the Collision and Grounding of Ships and Offshore Structures, 2020, C. Guedes Soares (Ed.). Volume 5: Developments in Renewable Energies Offshore, 2021, C. Guedes Soares (Ed.). Volume 6: Developments in Maritime Technology and Engineering, 2021, C. Guedes Soares and T.A. Santos (Eds.) Volume 7: Developments in the Analysis and Design of Marine Structures, 2021, J. Amdahl and C. Guedes Soares (Eds.) Volume 8: Trends in Maritime Technology and Engineering, 2022, C. Guedes Soares and T.A. Santos (Eds.) Volume 9: Sustainable Development and Innovations in Marine Technologies, 2023, Selma Ergin and C. Guedes Soares (Eds.) Volume 10: Trends in Renewable Energies Offshore, 2023, C. Guedes Soares (Ed.) Volume 11: Advances in the Analysis and Design of Marine Structures, 2023, J.W. Ringsberg and C. Guedes Soares (Eds.) Proceedings in Marine Technology and Ocean Engineering (Print): ISSN: 2638-647X Proceedings in Marine Technology and Ocean Engineering (Online): eISSN: 2638-6461

PROCEEDINGS OF THE 9th INTERNATIONAL CONFERENCE ON MARINE STRUCTURES (MARSTRUCT 2023), GOTHENBURG, SWEDEN, 3-5 APRIL 2023

Advances in the Analysis and Design of Marine Structures

Edited by J.W. Ringsberg Chalmers University of Technology, Gothenburg, Sweden

C. Guedes Soares Técnico Lisboa, Universidade de Lisboa, Portugal

Front Cover Image: Stena Line Back cover image: Chalmers University of Technology, Gothenburg, Sweden First published 2023 by CRC Press/Balkema 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN e-mail: [email protected] www.routledge.com – www.taylorandfrancis.com CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2023 selection and editorial matter, J.W. Ringsberg & C. Guedes Soares; individual chap­ ters, the contributors Typeset by Integra Software Services Pvt. Ltd., Pondicherry, India The right of J.W. Ringsberg & C. Guedes Soares to be identified as the authors of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Library of Congress Cataloguing-in-Publication Data A catalogue record has been requested for this book

ISBN: 978-1-032-50636-4 (hbk) ISBN: 978-1-032-50810-8 (pbk) ISBN: 978-1-003-39975-9 (ebk) DOI: 10.1201/9781003399759

Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Editor(s), ISBN 978-1-032-50636-4

Table of Contents

Preface

xiii

Organisation

xv

Acknowledgements

xvii

Methods and tools for loads and load effects Hull girder loads Verification of onboard loading computer by laser measurements D. Greening & G. Storhaug

3

Assessment of loads on a ship hull during a sideways launching process based on model tests A. Ulbertus, M. Schöttelndreyer & S. Ehlers

13

Hull structural loads during a sideways launching process using fluid-structure interaction A. Ulbertus, M. Schöttelndreyer & S. Ehlers

23

Study on fatigue design loads based on actual encountered loads N. Yamamoto, T. Sugimoto & K. Ishibashi

33

Wave loads Update of wave statistics standards for classification rules H.N. Austefjord, G. de Hauteclocque, M.C. Johnson & T.Y. Zhu

43

Different strategies to improve isochrone voyage optimization algorithm Y. Chen, W. Mao & C. Zhang

53

Estimation of nonlinear forces acting on floating bodies using machine learning C. Eskilsson, S. Pashami, A. Holst & J. Palm

63

Comparative study of ship design load on various environmental conditions T. Firmandha & S. Anggara

73

On nonlinear wave loads of a mega-scale container ship using an elastic backbone model A. Hanaoka

77

Encounter spectra computation of heave motion based on full-scale measurements using ANN M. Katalinić, P. Matić, N. Assani & J. Parunov Data-driven ship fatigue assessment based on pitch and heave motions X. Lang, J.W. Ringsberg, W. Mao, D. Wu & C. Zhang

v

85 95

Numerical simulation of ship motion and non-linear sea loads of a modern frigate in regular waves Z. Zhang, N. Ma & Q. Shi

105

Slamming Abnormal wave slamming impact of stiffened cylinders S. Bjørgo Fimreite, Z.L. Yu & J. Amdahl

117

Comparison between deforming meshes and overset meshes for water entry of a wedge M.F. Silveira, S. Wang & C. Guedes Soares

125

Experimental investigation on oblique entry of trimaran cross deck structure S.-Q. Tang, S.-L. Sun, H.-L. Ren & X.-Q. Zhou

133

Fluid-structure interaction analysis for bow-shaped structure subjected to slamming pressure K. Toh, D. Yanagihara & K. Nagayama

141

Dynamics and vibration Motion analysis of a floating horizontal set of interconnected plates based on computer vision target tracking technique I.B.S. Bispo, P. Amouzadrad, S.C. Mohapatra & C. Guedes Soares

153

Wind-induced vibration characteristics of typical guide rail frame structure in open area of large cruise ships X.L. Feng, J. Gan, Y. Zhu, Z.H. Chen & W.G. Wu

161

Model test of a dual-spar floating wind farm in regular waves Z. Jiang, G. Liang, T. Lopez-Olocco, A. Medina-Manuel, L.A. Saavedra-Ynocente & A. Souto-Iglesias

171

Acoustic black holes: The new frontier for soundproofing on board ships G. Kyaw Oo D’Amore, G. Rognoni, M. Biot & F. Mauro

179

Wave energy Experimental study on inflatable circular diaphragms used in the oscillating water column wave energy converter F. Abad, S. Lotfian, S. Dai, G. Zhao, G. Alarcon, L. Yang, Y. Huang, Q. Xiao & F. Brennan

187

Power capture performance analysis of a 2DoF nonlinear wave energy converter in regular waves Y. Gao, K. Liu, Z.G. Gao, J. Wang & W. Jiang

195

Fatigue of mooring lines in wave energy parks X. Shao, J.W. Ringsberg, H.-D. Yao, Z. Li & E. Johnson

205

Two-body, time domain model for a heaving point absorber C. Stavropoulou, J. Engström & M. Göteman

213

vi

Wind turbine Implied safety level in design codes for monopile offshore wind turbines - parametric study on the effect of partial safety factors H. Amlashi, M. Karimirad & D. Barreto Experimental study of long-term scour damage for protected offshore wind foundations J. Chambel, T. Fazeres-Ferradosa, A.M. Bento, F. Taveira-Pinto & P. Lomónaco

223 235

Design of substructure of 10MW floating offshore wind turbine system based on dominant load parameters S. Park, H. Lee & J. Choung

245

Mooring system optimization of 12MW FOWT in environment condition in the southwest sea of Korea C. Shim, M.S. Kim, K. Kim & D. Jeong

255

Methods and tools for strength assessment Structural analysis Hydroelastic response of a moored floating flexible structure based on Timoshenko-Mindlin beam theory P. Amouzadrad, S.C. Mohapatra & C. Guedes Soares Hull deflection estimation model for marine shaft alignment applications A. Dardamanis, G.N. Rossopoulos & C.I. Papadopoulos A study on the development of a FPSO standard model (KRISO-FPSO) I: Focusing on structural safety evaluation K. Sim, M.S. Ki & K. Lee

265 275

285

Numerical analysis of the Ro-Ro vehicle ramp structure under moving load Y. Yang, L. Zhu, Q. Liang & Y. Chen

293

Finite element analysis for ship production solutions A. Zamarin, D. Bolf, M. Hadjina, N. Vukas, D. Klanjac & T. Matulja

299

A mathematical model for variable cross-section hull girder with time-varying mass characteristics Y. Zhang & Z. Hu

309

A method for fast calculation of vertical bending moment and stress distribution of hull structure in HMS system X.-Q. Zhou, Y. Liu, M. Hernandez Ramos & H.-L. Ren

319

Structure analysis of lightweight sail structures for wind-assisted ship propulsion H. Zhu, S. Bikkireddy, J.W. Ringsberg, H.-D. Yao & B. Ramne

327

Ultimate strength Rule formulation updates on buckling strength requirements in Common Structural Rules L. Brubak, A. Bøe, Y. Lv, K. Ishibashi & A. Bollero

vii

339

Equivalent single layer approach for predicting ultimate strength of stiffened panel under different load combinations M. Kõrgesaar, T. Putranto & J. Jelovica

347

Study on the modification and application of RTCL criterion considering the influence of mesh size T.Q. Yu, K. Liu, X.F. Wang, H.W. Liu & G. Wang

355

The ultimate strength of sandwich plates with joints under in-plane compression M.J. Zhao & D.Y. Wang

365

Ultimate strength – hull girder Hull girder ultimate strength assessment according to rules requirements M. Aguiari, C.M. Rizzo, M.P. Salio, E. García Sánchez, S. Lazaro Rey, M. Safta & A. Nedaei Hull girder collapse analysis in extreme wave by direct coupled simulation between CFD and FEM S.K. Pal, B. Xie, K. Iijima, A. Tatsumi & M. Fujikubo Predicting ship’s ultimate longitudinal strength considering the lateral pressure loading S.-H. Park & S.-R. Cho

375

385 393

Ultimate strength – cyclic load Evaluation of residual plastic strain on a stiffened panel subjected to compression and tension-compression cyclic load B. Barsotti & M. Gaiotti

403

FEM numerical strategies for the evaluation of the accumulated plastic strain due to a cyclic load condition B. Barsotti & M. Gaiotti

411

Response of plates and stiffened panels under pressure and cyclic biaxial loading S. Fanourgakis & E.S. Samuelides Collapse response analysis of a ship’s hull girder in cyclic focused waves using a hydro-elasto-plastic beam model A. Tatsumi, S. Li & S. Benson

419

429

Fatigue Consequence and uncertainty-informed fatigue life prediction of ships M.L. Deul, C.H.H. van Battum, M. Hoogeland & J.W. van Bergen Fatigue performance of aluminum alloy EN-AW 5083 cut edges produced by abrasive water jet and laser cutting J-H. Grimm, N. Lange, M. Braun, F. von Bock und Polach, J. Diniz e Castro, M. Köhler & K. Dilger Structural life expectancy assessment of an aging combatant ship using spectral fatigue analyses K.P. Hernández, B. Verma, D.A. Carvajal & H.A. Barrios

viii

437

445

453

Fatigue testing of multiaxial non-proportionally loaded T-nodes J.K. Kamau, M.L. Larsen, V. Arora, T. Holm-Jensen & M. Jepsen

461

A study on the development of a FPSO standard model (KRISO-FPSO) II: Focusing on fatigue assessment M.S. Ki, K. Sim & K. Lee

469

Predicting failure of dynamic subsea cables by electrical insulation breakdown due to water treeing Z. Li, J.W. Ringsberg, Y.V. Serdyuk, D. Svensson, E. Johnson & C. Andersson

477

On the application of Engineering Critical Assessment specified in BS 7910 to ship structures E. McCaig & Y. Wang Numerical analysis of pit-to-crack transition under corrosion fatigue using a stochastic pit generation algorithm M. Mokhtari, X. Wang & J. Amdahl

485

493

Wind turbine – fatigue The effects of surge underprediction on fatigue damage of floating offshore wind turbine dynamic cables E. Land, Z. Hu, N. Haley, W. Brindley & C. Ng

503

Tension angle assessment of mooring chain of a 10MW floating offshore wind turbine H. Lee, J. Choung & J.-B. Lee

511

Fatigue assessment of offshore wind turbine support structures subjected to seawater C. Woitzik, M. Braun, F. von Bock und Polach, S. Ehlers, S. Shojai & P. Schaumann

521

Impact of preload loss on fatigue strength of blade root bolts of a Floating Offshore Wind Turbine (FOWT) T. Zheng & N.Z. Chen

529

Offshore structures Predicting long-term extreme responses using two approaches with a case study of a jacket structure L. Li, S. Haver & A. Eltervaag

539

Experimental study on the lateral bending capacity of steel pipes with initial dents R. Li, B.Q. Chen & C. Guedes Soares

551

Collapse and burst analysis for a subsea sandwich pipe Y. Qu & N.Z. Chen

559

Stress distribution in uniplanar KT joints reinforced with fibre reinforced polymer subjected to the axial loadings E. Zavvar & C. Guedes Soares

565

Ice Application of the CASFOP taxonomy to analyse an ice incursion incident with a FPSO U. Bhardwaj, A.P. Teixeira & C. Guedes Soares

ix

579

Fluid-structure interaction simulations using a hydrodynamic plug-in HydroQus J. Choung, D.H. Yoon & J. Kim A rate and pressure dependent elastoplastic material model for glacial ice colliding with marine structures M. Mokhtari, E. Kim & J. Amdahl Study on the annual changing trend of Arctic Sea ice melting for merchant shipping Z. Wang, J. Zhang, D. Wu, W. Tian, X. Lang & W. Mao

589

597 605

Impact Experimental investigation of the repeated impact behaviour of rectangular plates X. He & C. Guedes Soares

615

Crashworthiness analysis of semi-submersible platform column subjected to ship impact loads F. Liu, R.-H. Li, C.-M. Liu, X.-Q. Zhou & G.-Q. Feng

621

Shape characterization and impact on the structural behavior of initially distorted, 4-mm thick ship-deck stiffened panels F. Mancini, H. Remes, J. Romanoff, P. Lehto, M. Rautiainen, A. Niraula & A. Niemelä

629

Analysis of how the conditions in a collision scenario affect the size of a struck vessel’s damage opening and ultimate strength J.W. Ringsberg, A. Kuznecovs & E. Johnson

639

Blast and explosion Prediction of damage extents due to in-compartment explosions in warships W. Chang, B.C. Cerik & J. Choung

651

Anti-blast performance optimization design of corrugated sandwich structures based on BP-GA method W. Qiu, K. Liu, J. Wang & Z.G. Gao

659

Time delay effect in the dynamic response of submerged cylinder subjected to an underwater explosion Y.P. Sone Oo, H. Le Sourne & K. Brunellière

669

Recent studies on the saturated impulse for ship structures under pressure pulse loading L. Zhu, L. Tian & T.X. Yu

679

Experimental analysis of structures Composites Finite element analysis of filament wound composite materials split disk tests E.P. Bilalis & N.G. Tsouvalis

691

Experimental and numerical study of composite materials drive shafts E.P. Bilalis & N.G. Tsouvalis

699

x

Thermal and rheological study of structural adhesives for naval construction F.J. Rodríguez-Dopico, B. Sánchez Silva, A. Álvarez García, J. López-Beceiro & R. Artiaga Experimental analysis of ballistic impact on carbon, glass and hybrid composite under hydrostatic prestrain V. Vijay Kumar, S. Rajendran & S. Ramakrishna

709

717

Materials and fabrication of structures Corrosion The integrity of corrosion protection systems of welded maritime structures under cyclic loading G. Andresen-Paulsen, M. Braun, F. von Bock und Polach, T. Marquardt, A. Momber & S. Ehlers

725

Microstructural analysis of intergranular stress corrosion cracking in 5xxx series aluminum reinforced with a composite patch X. Ma & S. TerMaath

733

Uncertainty quantification within strain-based SHM schemes used for detecting thickness loss in ship hulls N.E. Silionis & K.N. Anyfantis

743

Corrosion degradation impact on mechanical properties of structural steel K. Woloszyk & Y. Garbatov

751

Methods and tools for structural design and optimization Ship design Integration of a tank storage solution for alternative fuels on a RoRo ship C. Ait Aider, L. Roß, S. Ehlers, P. Kaeding & T. Lindemann

761

On the safety of offshore mooring systems R. Yttervik, G. Ersdal & N. Oma

773

Experimental study on structure responses of triple wing sails to turbulence flows at multiple apparent wind angles H. Zhu, V. Chernoray, H.-D. Yao, J.W. Ringsberg & B. Ramne

781

Numerical and experimental analysis of fire resistance for bulkhead and deck structures of ships and offshore installations S. Zong, K. Liu, J.X. Wang, Z.G. Gao & Q. Sun

789

Ship design and optimization Comparison of metaheuristic algorithms and constraint handling approaches for multiobjective optimization of a tanker Y. Cai & J. Jelovica

xi

801

Structural optimization of cell guides for container securing as a retrofit solution on barges S. Haberl, A.S. Milaković, F. von Bock und Polach, O. Detlefsen, M. Abdel-Maksoud, S. Ehlers, L. Horstmann & C. Ahlers Ship design optimization considering probabilistic compliance of decarbonization regulations J. Huang, Q. Wei & Y. Liu Lightweight design and topology optimization of marine structures using peridynamics A. Kendibilir & A. Kefal

809

819 827

Structural components Review of the structural configuration and strength of metallic sandwich panels M. Elsaka & C. Guedes Soares

837

Analysis of pontoon primary structure failure S. Komariyah, T. Firmandha & S. Anggara

849

Pontoon design and construction methodology of gangway R. Sundaravadivelu, R. Natarajan, R.S. Sakthivel, L. Sony & S. Muthuraman

857

Significance of laser weld stiffness in vibration and buckling optimization of laser-welded web-core sandwich panels S. Yan & J. Jelovica

863

Structural reliability, safety and environmental protection Reliability A Bayesian approach for the quantification of strength model uncertainty factor in ultimate limit state D.G. Georgiadis & E.S. Samuelides

871

Structural reliability analysis of secondary hull detail S.K. Kleivane & B.J. Leira

877

Concept of the probability-based ship operability analysis T. Petranović, J. Parunov & C. Guedes Soares

885

Bayesian inference for the parameters of probabilistic corrosion model adopted by IACS T. Takeuchi, N. Osawa & N. Yamamoto

891

Author index

899

xii

Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Editor(s), ISBN 978-1-032-50636-4

Preface

This book contains the papers presented at the 9th International Conference on Marine Structures, MARSTRUCT 2023, held on 3 to 5 April 2023, and organized by Chalmers University of Technology in Gothenburg, Sweden. This is the ninth event in the MARSTRUCT Conference series and follows on from previous events held in Glasgow – Scotland, Lisbon – Portugal, Hamburg – Germany, Espoo – Finland, Southampton – UK, Lisbon – Portugal, Dubrovnik – Croatia, and Trondheim – Norway, in 2007, 2009, 2011, 2013, 2015, 2017, 2019 and 2021 respectively. The main objective of the MARSTRUCT Conferences is to provide a specialised forum for academics, researchers and industrial participants to discuss progress in their research directly related to structural analysis and design of marine structures. It was the intention that the MARSTRUCT Conferences be specifically dedicated to marine structures, which comple­ ments other conferences on general aspects of ships and offshore structures already available. This series of Conferences is one of the main activities of the MARSTRUCT Virtual Institute (http://www.marstruct-vi.com), an association of research groups interested in cooperating in the field of marine structures, which was created in 2010 after the end of the Network of Excellence on Marine Structures (MARSTRUCT), which was funded by the European Union. The MARSTRUCT Virtual Institute was created with the same members as the EU project, but with the aim to extend that membership to other interested groups in the future. The 2023 Conference reflects the work conducted in the current trends in the analysis and design of marine structures, including the full range of methods, modelling procedures and experimental results. The aim is to promote knowledge that enables marine structures to be more efficient, environmentally friendly, reliable and safe using the latest methods and proced­ ures for their design and optimisation. The book contains 99 papers that are categorized into the following themes and areas of research: • • • • • •

Methods and tools for loads and load effects Methods and tools for strength assessment Experimental analysis of structures Materials and fabrication of structures Methods and tools for structural design and optimization Structural reliability, safety and environmental protection

We hope that the overview coming from experts of different sectors will be an important con­ tribution to the audience. The articles in this book were accepted after a peer-review process based on the full text of the papers. Thanks are due to the Technical Programme and Advisory Committees, who had most of the responsibility for reviewing the papers. We are also grateful to the additional

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anonymous reviewers who helped the authors deliver better papers by providing them with constructive comments. We hope this process contributed to a consistently good level of the papers included in the book. J.W. Ringsberg & C. Guedes Soares

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Editor(s), ISBN 978-1-032-50636-4

Organisation

Conference Chairman J.W. Ringsberg, Chalmers University of Technology, Sweden Technical Programme Committee C. Guedes Soares, Técnico Lisboa, University of Lisbon, Portugal (Chair) J. Amdahl, NTNU, Norway J. Andric, University of Zagreb, Croatia E. Begović, University of Naples-Frederico II, Italy S. Benson, Newcastle University, UK D. Dessi, INSEAN, Italy L. Domnisoru, University “Dunarea de Jos” at Galati, Romania S. Ehlers, DLR, Germany Y. Garbatov, Técnico Lisboa, University of Lisbon, Portugal A.M. Horn, DNV, Norway Z.Q. Hu, Newcastle University, UK P. Kujala, Aalto University, Finland H. Le Sourne, ICAM, School of Engineering, France E. Oterkus, University of Strathclyde, Glasgow, UK J. Parunov, University of Zagreb, Croatia J. Prpić-Oršić, University of Rijeka, Croatia P. Rigo, University of Liège, Belgium J.W. Ringsberg, Chalmers University of Technology, Sweden M. Gaiotti, University of Genova, Italy J. Romanoff, Aalto University, Finland M. Samuelides, National Technical University of Athens, Greece K. Tabri, Tallinn University of Technology, Estonia M. Taczala, West Pomeranian University of Technology, Poland F. von Bock and Polach, Hamburg University of Technology, Germany Advisory Committee S. Aksu, Defence Science & Technology Organisation, Australia S.R. Cho, UlsanLab Inc., Korea M. Collette, University of Michigan, USA W.C. Cui, Shanghai Ocean University, China S. Estefen, Federal University of Rio de Janeiro, Brazil B.-S. Jang, Seoul National University, Korea J. Jelovica, University of British Columbia, Canada Y.M. Low, National University Singapore, Singapore

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N.R. Mandal, Indian Institute of Technology, Kharagpur, India A. Morandi, Gusto MSC, USA L. Moro, Memorial University, Canada P.T. Pedersen, DTU, Denmark J.J. Jensen, DTU, Denmark C. Tian, CSSRC, China D.-Y. Wang, Shanghai Jiao Tong University, China G. Wang, Jiangsu University Technology, China X. Wang, ABS, USA Y. Yamada, National Maritime Research Institute, Japan N. Yamamoto, MIJAC from ClassNK, Japan X. Zhou, Harbin Engineering University, China L. Zhu, Wuhan University of Technology, China Local Organizing Committee J.W. Ringsberg, Chalmers University of Technology, Sweden R. Bensow, Chalmers University of Technology, Sweden Z. Li, Chalmers University of Technology, Sweden W. Mao, Chalmers University of Technology, Sweden H.-D. Yao, Chalmers University of Technology, Sweden Technical Programme Secretariat Maria de Fátima Pina, Técnico Lisboa, University of Lisbon, Portugal

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Editor(s), ISBN 978-1-032-50636-4

Acknowledgements

The MARSTRUCT 2023 conference organizers appreciate the sponsorship and financial sup­ port from Chalmers University of Technology, Chalmers University of Technology Founda­ tion, The Swedish Maritime Competence Centre LIGHTHOUSE, The Swedish Foundation for International Cooperation in Research and Higher Education (STINT), SIGMA Energy & Marine, as well as CENTEC, Técnico Lisboa.

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Methods and tools for loads and load effects Hull girder loads

Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Verification of onboard loading computer by laser measurements D. Greening Seaspan Ship Management Limited, Vancouver, Canada

G. Storhaug DNV, Oslo, Norway

ABSTRACT: Safety should be ensured both during design and operation. For cargo ships a loading computer is required onboard to check that the still water wave bending does not exceed design limits. This is associated with uncertainties. Additional means should be used to check the loading computer. Conventionally this is done by draft survey which is also associated with uncertainties. In this paper laser measurements have been used to check the loading computer. Recordings from two container ships are presented and discussed. The main result is that the likelihood of exceeding the design still water bending moment is low. Hence, IACS design require­ ments for container ships has an additional safety factor which consequently may imply additional steel, fuel consumption and CO2 emission. Through transparent sharing of data like in this paper design rules may be further improved to balance risk and costs, and laser measurements may reduce the operational risk.

(2015) and includes collapse check of the hull girder where the SWBM is essential. Several class societies applied such collapse strength requirements also before 2015 when IACS provided these unified requirements. The LC used to predict the SWBM may rely on sensors and manual input, and the cargo weight taken onboard may also be associated with uncer­ tainties. It is therefore a good practice to perform a second check of the SWBM predicted by the LC. Draft survey is one alternative to compare hull deflection measurements with the LC predictions. Another alternative is hull monitoring with strain gauges, but in this case a practical and cost-effective solution by using laser measurements has been applied.

1 INTRODUCTION 1.1

Still Water Bending Moment (SWBM)

About 85% of all goods is transported onboard cargo ships. Safety is essential, and ship design and operation must handle uncertainties. The hull girder loads shall not exceed the hull girder strength to avoid collapse. This is a fundamental requirement in design and oper­ ation. For cargo ships, the still water bending moment (SWBM) is of similar importance as the wave bending moment in extreme storms. For this reason, loading computer (LC) is required onboard to predict the SWBM and confirm that it is below acceptable design levels. This was required by IMO (1966) stating: “The master of every new ship shall be supplied with suffi­ cient information, in an approved form, to enable him to arrange for the loading and ballasting of his ship in such a way as to avoid the creation of any unaccept­ able stresses in the ship’s structure, provided that this requirement need not apply to any particular length, design or class of ship where the Administration con­ siders it to be unnecessary”. This has then been inter­ preted by IACS first in 1971 and latest by IACS (2010) stating that loading instruments are required “for all ships of category I of 100 m in length and above”. Category I refers to ships with large deck openings, but also to ships in general with uneven cargo and ballast loading as well as any gas carrier and chemical tanker. It therefore applies to container ships with large deck openings. Updated longitudinal strength rules for container ships was issued by IACS

1.2

Theory of laser measurements

The theory behind laser measurements is explained by Storhaug et al. (2017) who also presented prelim­ inary results. The theory in brief (in more layman terms this time) is explained in a few short steps: • The hull girder will deflect under uneven loading of upwards buoyancy forces and downwards bal­ last, cargo and light ship weights, and summariz­ ing these forces with their respective arm along the hull girder will define the SWBM distribution along the hull girder • When the SWBM is substantial then the deflec­ tion of the hull girder will be measurable, and the

DOI: 10.1201/9781003399759-1

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largest SWBM and the maximum curvature of the hull girder will be close to midship • In such cases the laser should be located at one side of the maximum curvature (e.g. stern) and the bullseye should be located on the other side of the maximum curvature (e.g. bow), and the laser and bullseye should be located as far apart from each other as possible to ensure that the measurable offset on the bullseye becomes sig­ nificant to reduce the measurement uncertainties • The offset on the bullseye is converted to a deflection of the hull girder which again is con­ verted to a SWBM. This is done through a structural model which has a representative lon­ gitudinal distribution of the bending stiffness along the hull, and where the deflection can be calculated based on the SWBM distribution. The size of the bullseye must be larger than the max­ imum offset from the design SWBM. • Proper calibration of the sensor and bullseye loca­ tion should be done to ensure that the offset is zero when the SWBM is zero, like in the docked position during construction or renewal survey when the hull girder is straight.

The laser is located at frame 206, 168.7 m forward of AP. The distance between the laser and the bulls­ eye is only 75.8 m, which implies that the offset on the screen is less than the hull girder deflection. This low distance compared to the ship length is because the passageway has some equipment which block the free line of sight. This low distance also increases the uncertainties in the conversion from offset to SWBM midship. The laser gun and arrangement for the ship two is improved and has both vertical and horizontal possibilities for calibration. This was then also later installed on the ship one. The laser gun with bracket is illustrated in Figure 2.

Figure 1. Illustration of bullseye in the passageway.

Some more details are provided of the actual measurement system on the two ships in the next section. Now a new and more robust laser setup system is developed and applied to a second sister ship, and several years of data is now also available for both ships. Results will be presented and compared with LC, and draft survey readings for ship two. The results may also be used in development of more refined partial safety factors being part of the ultim­ ate capacity (collapse) check. This will be further discussed in a later section.

Figure 2. Laser gun with adjustment possibilities for cali­ bration and mounted on a bracket.

2 MEASUREMENT SYSTEM 2.1

Calibration should be done in drydock where the laser beam should be adjusted to hit the zero offset on the bullseye. 100% of the design SWBM was cal­ culated to correspond to an offset on the bullseye of 12.3 cm for this arrangement (locations). Measurements for the ship one has been taken since 15th of June 2016 and for the ship two since 3rd of June 2018. For ship one, the calibration was suddenly off because of rotation of the mounting bracket, so Figure 2 also represents an improvement of the mounting bracket arrangement to be fitted to the ship’s frame system. For ship two, the calibration was first properly done 19th of June 2019.

Ships

The ships are 10 000 TEU container ships with length of 320 m between perpendiculars and 48.2 m beam. Further details are given by Storhaug et al. (2017). They operate in the North Pacific and North Atlantic and may also go through Suez Canal and the new Panama Canal and to South America. The ships may occasionally encounter storms above 6 m significant wave height although not frequently. 2.2

Laser measurements

The purpose of the measurement system is to pro­ vide the ship’s staff with a quick method to deter­ mine the safety of the departure condition. The laser measurement system is mounted in the passageway, which is indoor, accessible, and convenient. The bullseye is located on frame 300, 244.5 m forward of aft perpendicular (AP) as illustrated in Figure 1.

2.3

Draft survey

From Storhaug et al. (2017) it was concluded that the SWBM based on the draft survey was most uncertain and had a bias compared to the LC and the laser meas­ urement. For that purpose, also the draft survey pro­ cedure was improved by using measurement tapes

4

located on the deck corner to determine draft a forward, aft and midship draught marks. This is illus­ trated in Figure 3. The measurement tape measures the distance down to the water line while the distance from the deck corner to the keel is known, so the draft is the difference. At the tip of the measurement tape there is a sensor that gives a light when in contact with water. The measurement is affected by waves while for conventional draft survey the average wave effect may be judged easier than based on a top view from about 10 m distance.

The declared versus the actual cargo is shown in Figure 5 for each voyage. The average actual cargo is 61093 tons with a 95% confidence interval of ± 28804 tons. The average difference between the declared and actual cargo weights is 0.51% (actual more than declared) and with a 95% confidence interval of 2.4%. From this, the uncertainties to the cargo weights are small.

Figure 3. Measurement tape for draft survey.

3 MEASUREMENT RESULTS FOR SHIP TWO 3.1

Cargo results

The first results are the number of declared contain­ ers. For ship two the average is 6463 TEU containers (the real number of containers may be less as 40 feet container counts as two TEU) with a 95 % confi­ dence interval of ± 3425 containers (2 standard devi­ ations) meaning a variation between 3038 and 9888 in 95% of the cases. For container ships the situation is that the ships normally transport containers on all legs but on some legs containers are more full and on others more empty, so the number of container may have limited value. From Figure 4 it is observed that in periods the number of containers transported is less. This may be due to the covid period. There are also significant changes on coastal voyages, but most containers are onboard during long voyages. The days represent the reporting day of the voyage relative to 3rd of June 2018.

Figure 5. Tons of cargo per voyage for ship two.

3.2

Loading Computer (LC) results

The ship length is 320 m between perpendiculars (Lpp) and midship is then 160 m from AP. From Figure 6, the location of the maximum SWBM from the loading computer (LC) (type MACS3) is in aver­ age 127.1 m from AP with a 95% confidence interval of ±24.8 m. This implies that the maximum is located between 0.319 and 0.475Lpp so generally aft of midship and with the average of 0.397Lpp. The locations of the laser gun and the bullseye were at 0.527 and 0.764Lpp, respectively. It is then noticed that maximum curvature of the hull girder then occurs aft of the measurement area for the laser arrangement.

Figure 6. Location of maximum SWBM from LC for ship two.

Figure 4. Number of TEU per voyage for ship two.

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less than Lpp. It gives an idea where the ship strength needs to be maximum. The probability distribution for the SWBM utiliza­ tion at midship and at the maximum location are shown in Figures 9 and 10, respectively. Utilization is defined as the moment from the recording divided by the design moment. 70 to 80% is most probably at midship while 80 to 90% is most probable at max­ imum location. Sagging SWBM does not occur, and the minimum hogging, as required by IACS (2015), is well above 0, which occasionally is used in design.

Based on the maximum SWBM being located aft of midship it is obvious that the midship moment is generally less. The maximum SWBM is 674722 tons meter and the maximum midship SWBM is 626552 tons meter. Given that the maximum design SWBM is 6500 MNm (662589 tons meter), the maximum utilization is 1.018 and 0.946 for the maximum loca­ tion and midship location, respectively. The record­ ings are illustrated in Figure 7 where the maximum SWBM is mainly larger than the midship SWBM.

Figure 9. Probability distribution of midship SWBM util­ ization from LC for ship two.

Figure 7. Midship vs. maximum SWBM from LC for ship two.

The maximum moment versus the location of the maximum moment is shown in Figure 8. Compared to Figures 6 and 8 show at which location the utiliza­ tion is close to design. The average utilization is 0.82 with a 95% confidence interval of ±0.24. The maximum has a relatively long longitudinal extent where it may be close to design limits from about 115 m to 145 m forward of AP, so about between 0.35 and 0.45 Lpp. This overlaps the location of the maximum wave bending moment given by IACS (2015), with maximum extending from 0.35 to 0.55 L for hogging. L is the rule length being about 3-4%

Figure 10. Probability distribution of maximum SWBM utilization from LC for ship two.

3.3

Laser measurements

The laser measurements are shown in Figure 11 where the laser offset on the bullseye is converted to a midship SWBM by a structural beam model of the hull girder based on theory outlined by Storhaug et al. (2017) depending on the location of the laser gun and bullseye. The time instants for the laser cor­ respond to the time instants for the LC recordings. The average moment is 493050 ton meter corres­ ponding to an average utilization of 0.744 which is

Figure 8. Maximum SWBM vs. location for ship two from LC.

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significantly below the average utilization at the maximum location of 0.821 but closer to the average utilization at the midship of 0.715 from the LC. The laser measurements however show a higher variation of the 95% confidence interval of ±0.416 while the spread at midship for the LC was ±0.267.

Figure 13. Probability distribution of difference in midship SWBM utilization from laser and LC for ship two.

3.4

Draft measurement tape

The draft measurements have been improved a lot but there are a few (about 11 of about 240 recordings) which are by engineering judgement wrong, for instance when the draft midship is larger than the maximum scantling draft or the deflection is negative suggesting sagging condition (more draft midship than in the ends). Hence, there are some errors in some of the records based on writing errors or reading errors. The negative recordings are replaced first with average recordings. Then a correction has been used based on a statistical method where “wild cards” more than 2 standard deviations is removed and set to previous average. There are of course other ways to do this, but this results in the deflection readings shown in Figure 14. The average is 28.1 cm with a 95% confidence interval of ±14.5 cm. 0.5 m deflection is a rule of thumb value for a 300 m ship and corresponds well to the maximum.

Figure 11. Laser estimated SWBM at midship for ship two.

The probability distribution for the laser moment is shown in Figure 12 and clearly shows some higher values exceeding 100% but also some lower values down to 15%. This will be discussed later.

Figure 12. Probability distribution of midship SWBM util­ ization from laser measurements for ship two.

The probability distribution of the laser measure­ ments being larger than the midship SWBM from the LC is shown in Figure 13. The laser is within ±10% of the LC with more than 50% of the time but it is biased of being a bit larger with an average of 2.95% but with a 95% confidence interval of ±34%. Consid­ ering only recordings of midship SWBM from the LC above 70% utilization, the laser is in average 1.88% larger but still with a 95% confidence interval of ±32.4%. Considering the voyages with high load­ ing therefore does not make much difference.

Figure 14. Deflection estimates from draft measurement tape readings for ship two.

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The maximum deflection from Storhaug et al. (2017) is however stated as 0.425 m but this is assumed to fit to the maximum hogging from the loading manual said to correspond to 95% of max­ imum permissible hogging, so this is corrected to a design deflection of 0.447 m. The average utilization is thereby 62.8% with a 95% confidence interval of ±32.4%. The average is significantly less than from the LC and the laser, and the spread is larger than from the LC but less than the laser. It may also be a bias to the estimated deflection which would be more accurately estimated based on a global finite element model rather than beam theory with cross sections (done according to IACS, 2015), however the difference in average utilization is still regarded high.

observation above 110% utilization. The probability distribution for the utilization of midship SWBM of ship one is given in Figure 16. This also resembles the distribution for ship two in Figure 9, but ship one has one observation with utilization above 100%.

4 MEASUREMENT RESULTS FOR SHIP ONE 4.1

Cargo and Loading Computer (LC) results

Similar results are produced for ship one as for ship two. The average number of TEU is 6433 ± 3049. This is similar to ship two, but ship one does not seem to have been affected by a reduction during the covid period. It has however three voyages with zero containers due to yard visit for inspection. The actual cargo is 54456 tons ± 30218 tons. This is less in amount but with a slightly larger variation than ship two. For two cases the actual cargo was missing and set to the declared cargo. The difference between the actual and declared weights are regarded as miss declared weights and is within 0.04±2.7% (in the few cases with zero actual cargo a value had to be assigned to avoid overflow). The average is smaller than for ship two, but the variation is larger with five observations above 4%. The longitudinal location of the maximum moment is 128.8 m ±26.0 m from AP. This is similar but about 1.7 m forward of ship two and with 1.2 m larger 95% confidence interval. In Storhaug et al. (2017) it was mentioned that the midship moment and the maximum moment appeared to converge for larger values, basically suggesting that the maximum moment appears at midship when it is high. This was based on few observations, and now with many observations, this is no longer true. The width from the diagonal is still high at high values for ship one. This is also seen in Figure 7 for ship two. This is also supported by Figure 8 showing location of the maximum for ship two and this also resembles results from ship one. The utilization of the maximum SWBM for ship one is 0.823±0.232. This is similar to ship two. The utilization at midship is 0.721±0.249 which is also similar to ship two. The probability distribution for utilization of the maximum SWBM of ship one is given in Figure 15. This is similar to the probability distribution for ship two in Figure 10, however ship one has one

Figure 15. Probability distribution of maximum SWBM utilization from LC for ship one.

Figure 16. Probability distribution of midship SWBM util­ ization from LC for ship one.

4.2

Laser measurements

The laser measurements have been carried out since 15th of June 2016, however the measurement period for ship one may be divided into four periods: 1. 15th of June 2016 to 5th of October 2016: Voyage 1 to 19 with initial calibration. This is reported as preliminary results by Storhaug et al. (2017) and includes draft survey data. The remaining data sets do not include draft survey data. 2. 29th of October 2016 to 14th of December 2016: Voyage 20 to 29 with suspect recordings. It is

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and 0.45 utilization from Figure 18). The likelihood of midship SWBM being larger than 75% is high, and then less difference is required for the laser to suggest overloading, so the likelihood of LC saying that it is less than 100% utilization while the laser gun is saying more than 100% utilization is significant. This happens however not more than 10% of the time as suggested by Figure 17 when the laser suggest that hull girder loading above 100%. Within this 10% there may be false readings, meaning that the hull is not overloaded even though the laser gun suggests this.

believed that either ship vibrations or crew has caused the laser gun to shift out of calibrated pos­ ition on the voyage between 5th of October and 29th of October, since the difference with the LC is suddenly much larger. 3. 17th of January 2017 to 26th of March 2021: Voyage 30 to 290. A laser was recalibrated, but apparently not successfully. 4. 20th of April 2021 to 26th of October 2022: Voyage 291 to 400. The laser system was upgraded, similar to ship one, and recalibrated, but the successfulness of the calibration is questioned. The laser measurements for the three last periods need to be treated. The average of the midship moment from the LC is within 1% for the three first periods but increased with 6% for the last period. From the laser the average goes significantly down in measurement period 2, then significantly up in period 3 and then down again in period 4. At the time of the calibration for period 1 the laser suggests about 2.5% higher utilization than the LC. For period 3 it was about 11% higher while for period 2 about 25% lower and for period 4 also about 25% lower. Consider­ ing that for ship two the laser had in average about 3% higher utilization, similar to period 1, the mean of the laser measurements for ship one is adjusted. For period 2 and 3 the mean is adjusted to be the same as for period 1 while for period 4 it is set 6% higher than for period 1. This is by no means accurate, but we may be more interested in the variation than in the mean value and the correction is then also consistent with the change in the LC values for the midship SWBM. The variation of the laser measurements should then be well represented even though it is noted that the laser recordings for ship one is rounded off also to the nearest cm from the bulls­ eye reading. This is a bit inaccurate but based on many observations it is statistically OK. The distribution of the adjusted laser measure­ ments is shown in Figure 17. This resembles ship one where utilization above 100% was also observed, but ship one has no observations at 1525%. The laser suggests that the hull girder is over­ loaded about 10% of the time, which is a concerning high number, suggesting that 1 of 10 voyages gets a warning based on the laser gun alone. The difference between the laser and the LC reading for the midship SWBM is displayed in Figure 18 for ship one. The difference is 3.4% ±35.4%. This is simi­ lar to ship two. The laser is generally showing larger values, and the difference is mainly within -5% and +15% (60% of the time), but also quite probable with -15% and + 25% as well (27.5% of the time). The larger difference is a concern. For instance, the most probable midship moment from the LC is 75% (see Figure 16) leading to likelihood of LC suggesting OK values while likelihood of laser being more than 25% higher (100% utilization) is about 5% (sum of 0.35

Figure 17. Probability distribution of midship SWBM util­ ization from laser measurements for ship one.

Figure 18. Probability distribution of difference in midship SWBM utilization from laser and LC for ship one.

5 DISCUSSION The laser arrangement is similar for ship one and two. The distance between the laser gun and bullseye is 0.24Lpp. The location of the laser gun and the bullseye is forward of midship. The maximum moment of the ship is aft of midship around 0.4Lpp forward of AP. The laser arrangement with limited

9

distance and location is not ideal for capturing max­ imum curvature (=bending) of the hull girder. The difference between the laser measurements and the LC is often more than 10%. Still the average midship SWBM from the laser measurements is in line with the LC. The absolute midship SWBM from the laser may be unreliable for this arrangement, but still the laser arrangement may be regarded useful. The per­ centage time the laser arrangement is indicating overloading is low and about 10% for ship one (6.5% for ship two). The midship moment from the LC is only about 6% of the time above 90% utiliza­ tion. In practice this could be arranged as follows: If the laser suggests more than 100% utilization and the LC suggest more than 90% utilization, then take action to ensure hull girder loading within permis­ sible levels. This could be supported by draft survey, and if this also supports high deflection above limits, then ballasting or rearrangement of cargo may be necessary. This is relevant only a fraction of the time, e.g., less than 6% based on the LC but less than 6% · 0.4 = 2.4% based on the difference between the laser and LC being larger than 10% given that the LC is larger than 90%, so 2-3 of 100 voyages gets an action. This is based on the midship SWBM. The maximum SWBM from the LC should also be considered but if this indicate more than 100% loading, then action is needed anyway. Another aspect of the results is how this may affect design requirements for container ships. IACS (2015) requires an ultimate capacity check (=crosssectional collapse check with bottom buckling). Results herein suggest only hogging SWBM, so buckling of deck due to sagging wave moment with whipping combined with a small hogging moment is less likely, but both sagging and hogging checks are required by IACS (2015) by Equation 1:

shedding (load moving from weaker structure to stron­ ger structure). The cross-sectional analysis is however a pragmatic approach also used in common structural rules for oil tankers and bulk carriers (IACS 2022) although with different partial safety factors. For con­ tainer ships γDB = 1.15 is a substantial reduction of the capacity in hogging, while γM = 1.05 represents a small reduction to the collapse strength estimate. The γM and γDB may however be seen together as a substantial uncertainty to the simplified buckling approach. If more accurate nonlinear FEA is applied, then both γDB and γM should be reconsidered. γW = 1.2 suggest that the uncertainty to the wave moment is sig­ nificant and includes whipping. That may not always be true, so IACS (2015) requires that for post Panamax container ships with beam larger than 32.26 meters whipping shall be considered. How this whipping should be handled should be described by IACS soon. The left-hand side of Equation 1 represents the load which may be said to have a return period of 25 years. The right-hand side represents the capacity or strength. The partial safety factors intend to address the uncertainties documented through a structural reliability study, and they should be affected by the target probability of failure (collapse). Collapse like on MSC Napoli, MSC Carla, Mol Comfort and Neptun Sapphire are not acceptable and should be assigned with low target probability of failure. It is a question how this should be represented. It could be represented by a separate partial safety factor on the right-hand side, or it could be included within the γM. It could be convenient to have the target probability for failure as a separate partial safety factor, so it is understood when something is regarded totally unacceptable, has moderate or minor consequence. It becomes less transparent if it is mixed into the partial safety factors on the load side. So, if we assume that the partial safety factors on the load side represents the uncertainty to the respective SWBM and wave bending moment and that their combined return period shall be 25 years, then we have an issue. IACS (2015) defines γS to be 1.0. This means that from a probabilistic perspective the actual hogging moment is considered to always be the design hogging moment. This is not the case from the results herein and considering the max­ imum moment the γS may be set to something slightly larger than 0.82 due to 82% utilization being most probable (from the LC) but with a likelihood of also being larger. For midship γS could be something slightly larger than 0.72. This could be further stud­ ied by probabilistic analysis. No proper literature on this has been seen for container ships, but similar studies was used as background for the common structural rules (IACS, 2022). The main point here is that it appears to be a significant additional safety factor hidden in the plus sign between the SWBM and the wave bending moment. This results in a return period, which currently are much higher than 25 years and the safety margin is not transparent.

Where γS is partial safety factor for the SWBM, MS, γW is partial safety factor for the vertical wave moment, MW, γM is partial safety factor for various uncertainties related to the collapse capacity, MU, and γDB is partial safety factor for the double bottom effect. The latter is caused by external sea pressure pushing the double bottom inwards amidships in hogging due to a wave crest amidships. This causes additional compressive stresses in the bottom plating between the transverse bulkheads, so double bottom bending contributes to longitudinal compressive stresses which may cause buckling. This is a well understood physical effect but represented by a separate partial safety factor because the collapse capacity MU is based on cross-sectional buckling analysis rather than nonlinear finite element analysis (FEA) which is an acceptable alternative. Such non­ linear FEA may capture both the double bottom bending and progressive buckling behavior with load

10

Container ships have been designed for many years based on a collapse check with γW = 1.2 or larger (1.5 was used by DNV, 2010) and γS = 1.0. The additional safety margin hidden in the plus sign was then quite OK to have as whipping was not handled explicitly, but with whipping now being introduced explicitly it may be difficult to maintain γS = 1.0 without ending up with stricter design rules. The offshore industry may use two collapse checks; one with high partial safety factor on the dynamic loads (γW > 1.0) and reduced par­ tial safety factor on the static loads (γS < 1.0), and secondly a collapse check with high partial safety factor on the static loads (γS > 1.0) and reduced on the dynamic loads (γW < 1.0). A main reason is that the maximum static and dynamic load is considered not to occur at the same time. This may also be relevant for container ships.

with filling level of ballast tanks. Ballast may be redistributed, or containers moved. The laser arrangement may be improved prefer­ ably by moving the laser gun aft of the maximum hogging area to capture better the area where the hull girder curvature is maximum but also to cover a longer distance between the laser gun and bullseye. Another aspect is to preferably use the finite element design model which is required nowadays for con­ tainer ships above 290 m in length to establish both the conversion from offset on bullseye to wave bend­ ing moment and to confirm design deflection for comparison with draft surveys. Another important future work is to collect LC data based on many more container ships. Con­ tainer ships are being “onboarded” to the shore much more frequently nowadays to provide a data stream of sensor data continuously for different purposes. Relevant data may be shared with class societies through available data platforms like Veracity (www.veracity.com) to quickly establish probability distributions like in this paper for many container ships, for different sizes and dif­ ferent trades. This would be much improved basis for transparent and more reliable definition of the partial safety factor γS for SWBM of container ships which would benefit the container ship industry both when it comes to costs in design and operation while still ensuring safety and improving sustainability. Results in this paper suggests that there is a hidden safety factor for container ship designs since SWBM is most likely well below design limits and with low probability of exceeding design limits.

6 CONCLUSIONS AND FURTHER WORK A laser arrangement has been installed on two con­ tainer ships and has been used to predict the still water bending moment (SWBM) at midship. Several years of data have been collected and the laser pre­ dicted moment has been compared with results from the loading computer (LC) and a draft survey. The LC results show that the maximum moment is located aft of midship and most likely around 0.4Lpp forward of aft perpendicular. Most probable maximum utilization compared to design is about 82% and with a low probability of exceeding 100%. Midship the most probable utilization is 0.72. The laser measurements have slightly higher aver­ age utilization, but also higher variability. In that respect the laser measurements appear as more unreli­ able, but the installation of the laser gun and bullseye on these ships are far from ideal. Laser gun is located forward of midship with the bullseye about 0.24Lpp forward of the laser gun. The distance is short, and they are not located where the hull girder has the lar­ gest moment and curvature. Still the laser arrange­ ment is regarded as useful provided good calibration. The draft tape measurement is regarded as a more convenient alternative to the draft survey but has sig­ nificant less average utilization but a variability which is in between the LC and the laser measurements in this case. A practical way to ensure hull girder safety is to see if the laser suggests more than 100% utilization when the midship SWBM from the LC exceed 90%. Then additional measures should be taken. This may be relevant on only about 2-3 percent of the voyages. One way is to further check the draft tape measure­ ment, but also to see if something could be wrong

REFERENCES DNV, 2010, Longitudinal and buckling strength, Container carriers, Rules for ships Pt. 5 Ch.2 Sec.6 B205, July 2010. IACS, 2010, Requirements for loading conditions, loading manuals and loading instruments, IACS Unified require­ ments, Requirements concerning strength of ships, UR S1, May 2010. IACS, 2015, Longitudinal strength standard for container ships, IACS Unified requirements, Requirements con­ cerning strength of ships, UR S11A, June 2015. IMO, 1966, Information to be supplied to the Master, Regulation 10(1) International Convention on load lines, 1966. IACS, 2022, Common Structural Rules for Bulk Carriers and Oil Tankers, 01 Jan. 2022. Storhaug, G., Fredriksen, O., Greening, D. & Robinson, I., 2017, Practical verification of loading computer by laser measurements, Progress in the Analysis and Design of Marine Structures, Guedes Soares, C. & Garbatov Y., (Eds.), Taylor & Francis, London, UK, pp. 51–57.

11

Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Assessment of loads on a ship hull during a sideways launching process based on model tests A. Ulbertus & M. Schöttelndreyer Thyssenkrupp Marine Systems, Hamburg, Germany

S. Ehlers German Aerospace Center (DLR), Geesthacht, Germany

ABSTRACT: Before a sideways launching process of a vessel, the motion and stability of the vessel as well as structural loads are to be checked. High loads on the ship’s hull structure at the impact with the water surface can be observed. In this paper the loads resulting on the ship hull of a special purpose vessel are investigated based on model tests of a sideways launching process conducted at a ship model basin. The model tests considered the slipway and did enable a free 6-DOF movement of the ship model. During model tests the pressure were meas­ ured at different points of the highly loaded area of the ship hull. Different parameters of the sideways launching process were varied during model tests. This includes the loading condition of the ship, the height of the water level inside the launch basin, the height of the palls and the coefficient of friction between palls and slipway. The influence of each of these parameters on the resulting loads is assessed. In the end, a load model based on these results is derived in the form of a pressure-time function, which is suitable for the use in simulations based on the finite element method (FEM). This load model accounts for the high loads right after the impact of the ship hull with the water surface.

1 INTRODUCTION

2 MODEL TESTS

To ensure a safe sideways launching process of a vessel, different aspects are to be assessed before launching. Besides the ship motion, the resulting loads on the ship hull are of importance. However, data and information available in literature are scarce. A comprehensive overview of the process of sideways launching is given by Leavitt (1980). Besides the pro­ cess itself (preparation and execution), an analytical approach for assessing the ship motion based on the equations of motion is provided by Leavitt (1980). But no data/approach is provided to assess the loads result­ ing on the ship during a sideways launching process. During model tests of a sideways launching pro­ cess of a special purpose vessel conducted at a ship model basin in 2020, pressure-time-signals were measured for verification purposes. The data was used by Ulbertus et al. (2021), to verify the applic­ ability of an Arbitrary-Lagrangian-Eulerian approach using fluid-structure interaction. As a parametric study was carried out during model tests, the obtained data is able to provide an in-depth knowledge of the loads resulting on ship hull during a sideways launching process as well as the crucial parameters influencing these loads.

2.1

Ship model

Model tests of for a special purpose vessel were con­ ducted at a ship model basin. For this purpose, a ship model in the scale of 1:32 was used. The model con­ tained the complete watertight volume of the ship. The hull was built using natural fiber reinforced composites with an average thickness of 5mm. In addition, the superstructure of the ship was considered, which was built with ABS plastic using 3D-printing technology. Appendages installed during the sideways launching process were considered during model tests, too. This included two rudders, a pair of fin stabilizers, two pro­ pulsion shafts with bossing, V-brackets, hub of propel­ lers and dummy propeller blades as well as one pair of bilge keels. 2.2

Loading conditions

As the model tests were used as the basis for a parametric study, different loading conditions of the vessel with different positions of the center of gravity (COG) were considered. In total, four differ­ ent loading conditions were investigated:

DOI: 10.1201/9781003399759-2

13

– LC01: tanks in bottom structure filled to lower COG of ship (COG 8% lower and displacement 11% higher than LC01) – LC02: weight distribution as designated for the sideways launching process – LC03: COG 5% higher than LC02 – LC04: COG 10% higher than LC02 The loading conditions were realized by trim­ ming weights inside the ship model. These loading conditions were investigated to assess a variety of different weight distributions feasible for the side­ ways launching process. Limits regarding a safe position of the COG were investigated. A shift of the COG during the design phase is possible due to various reasons. 2.3

Figure 1. Setup of model tests for sideways launching pro­ cess at ship model basin.

2.4

Test matrix

One goal of the model tests was assessing the influ­ ence of different parameters regarding the sideways launching process of the special purpose vessel. Especially parameters regarding the ship, which can change during the design process, and external parameters, which are relevant for the sideways launching process, were investigated. This included the following four aspects:

Test setup at ship model basin

An illustration of the test setup is given in Figure 1. The tests at the ship model basin were conducted as closely to the conditions at a shipyard using such as a launching process as possible. This was achieved by a close cooperation with a potential shipyard for the construction of the investigated special purpose vessel. The launch basin including the pier and slip­ way is built in the scale of 1:32 using wooden plates and aluminum supports. This ensures that the condi­ tions on-site were met, e.g. reflections of waves within the launch basin. The slipway of the model tests was equipped with two low-friction roller conveyors to simulate the sliding process of the ship. Two palls were mounted to the ship, on which it moved down on the roller conveyors. These palls were built with wood and fixed to the hull during model tests. Using roller conveyors, the friction between the palls and the slipway during a full scale sideways launching process was not modelled directly. The fric­ tion between the palls and the slipway was considered by controlling the sliding speed during the model tests. This was realized by a winch system located at the top of the slipway. This winch system is con­ nected with strings to the palls of the ship model. At the beginning of the sideways launching process the winch was accelerated to the defined sliding speed, at which the rate of rotation of the winch was fixed. Shortly before the ship model started to rotate around the edge of the pier, the connection between the palls and winch system was released by a little ramp located near the end of the slipway. After the release the ship model was able to rotate freely around the edge of the pier. Inside the water – after the palls left the slipway/edge of pier – the ship was able to do a free six DOF movement. In addition, the water level within the ship model basin was raised and lowered by pumps. This allowed investigating the influence of the water level inside the launch basin (tidal range).

– change of loading condition (displacement and positon of COG) – height of palls hP – water level hWL inside launch basin (influence of tidal range) – variation of coefficient of friction μSW between palls and slipways (different speed of ship at the end of slipway) The test matrix used during model tests is given in Table 1. For the coefficient of friction between the palls and the slipway a design value μSW 0 = 0.03 was used, which was provided by the shipyard. This value is corresponding to common values as found in literature, e.g. Leavitt (1980). The height of palls is defined as the distance between the slipway and the keel of the ship. A design value of hP 0 = 1.00m was used. This is the lowest height, where clearance between the appendages of the ship hull and the edge of pier is given. The water level inside the launch basin is defined as the distance from the edge of the pier to the water surface (negative value = water level below edge of pier). The design value for the water level hWL 0 = -0.50m is corresponding to the water level observed during average high tide. Within this frame of the parametric study only one parameter was changed at a time between differ­ ent test setups. Parameters were varied to assess the possible design space for a safe sideways launching process of the investigated special purpose vessel. In addition, different “worst case” scenarios were investigated. In contrast to the parametric study, for each “worst case” scenario several parameters were changed simultaneously.

14

Table 1.

the pressure measurement/pressure-time-signal due to it’s dynamic nature. For this purpose differ­ ent conditions of the water surface inside the launching basin have been considered as well (glass-smooth vs. slightly choppy water surface).

Text matrix.

case

LC

μSW -

hWL m

hP m

LC_01* LC_02* LC_03 LC_04

LC01 LC02 LC03 LC04

μSW 0 μSW 0 μSW 0 μSW 0

hWL 0 hWL 0 hWL 0 hWL 0

hP 0 hP 0 hP 0 hP 0

HP_01 HP_02**

LC01 LC01

μSW 0 μSW 0

hWL 0 hWL 0

hP 0 hP 0 + 0.50

WL_01** WL_02 WL_03 WL_04

LC01 LC01 LC01 LC01

μSW 0 μSW 0 μSW 0 μSW 0

hWL 0 hWL 0 – 0.50 hWL 0 – 1.00 hWL 0 – 0.66

hP 0 hP 0 hP 0 hP 0

FC_01 FC_02 FC_03 FC_04** FC_05 FC_06 FC_07

LC01 LC01 LC01 LC01 LC01 LC01 LC01

0.25·μSW 0 0.50·μSW 0 0.75·μSW 0 μSW 0 1.25·μSW 0 1.50·μSW 0 2.00·μSW 0

hWL 0 hWL 0 hWL 0 hWL 0 hWL 0 hWL 0 hWL 0

hP 0 hP 0 hP 0 hP 0 hP 0 hP 0 hP 0

WC_01 WC_02 WC_03 WC_04 WC_05

LC01 LC02 LC03 LC04 LC01

0.25·μSW 0 0.25·μSW 0 0.25·μSW 0 0.25·μSW 0 0.25·μSW 0

hWL 0 – 0.50 hWL 0 – 0.50 hWL 0 – 0.50 hWL 0 – 0.50 hWL 0 – 0.66

hP 0 + 0.50 hP 0 + 0.50 hP 0 + 0.50 hP 0 + 0.50 hP 0 + 0.50

Figure 2. Visualization of test matrix – conditions at COG at moment of first contact with water surface.

2.5

Measurement equipment

During model tests, the ship motion as well as pres­ sure-time-signals at different points of the ship hull was measured. The ship motion was measured using an optical tracking system called Qualisys. The ship motion is measured for all six DOF. For this purpose the ship model was equipped with markers at different positions, which were captured by the optical tracking system. In addition, an iner­ tial measurement unit inside the ship model was used to capture the accelerations during the side­ ways launching process. The sampling rate of the optical tracking system and the inertial measure­ ment device used for the model tests was 200Hz (35.4Hz for full scale). The resulting pressure-time-signal at four differ­ ent points on the ship hull was investigated using pressure transducers. The pressure transducers were arranged around the area, where the ship hull did have the first contact with the water surface. This was done, to capture the presumably highest pres­ sures at the impact of the ship hull with the water surface. The pressure transducers were offset in lon­ gitudinal as well as well as horizontal direction (compare Figure 3). This arrangement allows for an investigation of a decay behavior of the peaks of pressure. The pressure-time-signals were measured using miniature pressure transducers XTM-190-series from Kulite. The measurement area of the pressure transducers are circular in shape with a diameter of 3.8mm, corresponding to 0.122m in full scale. The

* conducted 4x for replicating pressure measurement ** case equal to LC_01

The intention of the “worst case” scenarios was to obtain a better understanding of the limits for a safe sideways launching process. A visualization of the different conditions defined by the test matrix is presented in Figure 2. This figure visualizes the different conditions at the impact of the ship hull with the water surface. The x-axis of Figure 2 plots the translational velocity vyz impact of the ship in the yz-plane, while the y-axis represents the rotational velocity ωx impact about the x-axis of the ship at the moment of impact. During the planning of the model tests it was expected, that a higher translational velocity vyz impact would be more critical regarding the resulting loads on hull structure (higher kinetic energy), while a higher rota­ tional speed ωx impact would be more critical regard­ ing the stability of the ship (higher rotational energy). As Figure 2 indicates, the “worst case” scenarios are all in the lower right corner. This results in more critical conditions for the sideways launching process. In total 19 different test setups with a total of 26 measurement series were investigated. The two test setups LC_01 and LC_02 were conducted four times each, in order to evaluate the sensitivity of

15

pressure transducers used a sampling frequency of 4,800Hz (848.5Hz in full scale). Pressure trans­ ducers with a high sampling frequency were chosen to capture the highly dynamic peak of pressure right after the impact of the ship’s hull with the water sur­ face sufficiently. If not stated otherwise, all results presented within this paper are given for the full scale sideways launch­ ing process. An overview of the different data meas­ ured during the model tests as well as the applied scaling laws is provided in Table 2. The expression of λ in Table 2 is the model scale, while the subscript ms is corresponding to model scale and fs to full scale. The model tests were carried out based on Froude similarity. Table 2.

Figure 3. Conditions observed at moment of first contact of ship hull with water surface.

Measurement data and applied scaling law.

measurement data

measuring device

translational motion translational velocity translational acceleration rotational motion

optical tracking system inertial measurement unit optical tracking system inertial measurement unit pressure transducer

rotational velocity pressure-time-signal

scaling law λ λ0.5 1 1 λ-0.5 ρfs/ρms · λ3 Figure 4. Visualization of test matrix – conditions at ship hull at point of first contact with water surface.

3 EXPERIMENTAL RESULTS 3.1

will start to rotate around the pier and the bottom struc­ ture hits the water surface nearly perpendicular (see Figure 3). Measurement of the ship motion showed very consistent results. In case of LC_01 and LC_02 a coefficient of variation (mean value) of 2.1% for the heeling angle at the moment of impact with the water surface was observed. The coefficient of variation for the maximal heeling angle was 1.2%.

Ship motion

As the focus of this paper is the loads resulting on a ship hull during a sideways launching process, the experimental results regarding the ship motion are only discussed briefly. One important observation made during model tests is, that the deadrise angle βimpact (angle of attack between ship hull the water surface at the moment of impact) is nearly constant for all model tests. The condition at the moment of the first contact of the ship hull with the water surface is visualized in Figure 3. As Figure 3 illustrates, a value for βimpact of nearly 0° was present during all model tests. In Figure 4 βimpact is plotted over the resulting velocity vimpact normal to the ship hull at the moment of first contact with the water surface. As Figure 4 shows, βimpact only differed about 2° during all model tests, while vimpact is mostly dominated by the height of the drop around the edge of the pier. This is achieved either by an increased height of palls or a lower water level inside the launch basin. This condition is achieved due to the combination of the hull form of the special purpose vessel and the geometry of the launch basin at the shipyard. As the bottom of the ship is inclined by approx. 12°, the ship

3.2

Pressure-time-signal

In Figure 5 exemplary pressure-time-signals meas­ ured during model tests are shown for LC_01 (run 02). The pressure-time-signals consist of the hydrodynamic as well as hydrostatic pressure. The pressure-time-signals are calibrated to the standard atmosphere p0 of 101,325Pa. As βimpact ≈ 0°, high loads on the bottom of the vessel could be observed. In the area of first contact high and harsh peak pressures were measured. In Figure 5 this can be seen for pressure transducer P2 and P4. These peak pressures are only of very short dur­ ation. The peak pressure can only be observed between one and two measurement intervals, which is equal to a time of around 2.3ms. After the

16

peak pressure, the pressure decays rather quickly to the level of the dynamic pressure. Oscillations of the pressure-time-signals around the dynamic pressure can be observed. For pressure transducers, which were later in contact with the water surface (e.g. P1 in Figure 5), the pressure-time-signals are not as harsh. The peak pressure is lower and the decay of the peak pressure is of a longer time period. As P1 and P3 are the outer pressure transducers as shown in Figure 3, a spray root forming beneath the hull could be observed. This spray root has a significant influence on the resulting pressure-time signals. The influence of the spray root on the resulting pressure during slam­ ming events is discussed in more detail in literature. One example is the work of Javaherian et al. (2022). All in all, the pressure-time-signals observed during model tests of the sideways launching process show great resemblance with the results of slamming experiments. In the past century a lot of research regarding slamming events of ships and the resulting loads was conducted. An extensive overview of slamming loads and the assessment of the structural response is given by Wang & Guedes Soares (2017). A good overview of different slamming experiments conducted in the past is provided in the work of Lewis et al. (2010). 3.3

Peak pressure

At the moment of impact, a harsh peak pressure can observed for all pressure transducers comparable to slamming events. For LC_02 (design loading condi­ tion) a pressure peak of 1,069kPa (mean value) at pressure transducer P2 could be observed. The high­ est peak pressure was 1,812kPa observed during WC_01 at pressure transducer P2. Test setups LC_01 and LC_02 were conducted four times each to assess uncertainties regarding pressure measurements. The mean value of the peak pressure and the standard deviation of each pressure transducer are given in Table 3 and Figure 6. For pressure transducer P2 and P4 high coefficients of variation cv i max regarding the meas­ ured peak pressure of up to 22.5% could be observed. In the case of P1 and P3 the uncertainties are lower, as the pressure peak is not as harsh com­ pared to P2 and P4. Investigations regarding uncertainties related to pres­ sure measurements during slamming experiments can be found in literature. Lewis et al. (2010) investigated the general uncertainties regarding pressure measure­ ment during slamming events based on a symmetric wedge with an deadrise angle of 25°. A sophisticated test setup designed for best possible repeatability was

Figure 5. Pressure-time-signals measured during LC_01.

17

used by Lewis et al. (2010). The uncertainties due to systematic error during measurements using such a test setup were determined with 1.1%. As Tödter et al. (2020) showed, the uncertainties during slamming experiments with a flat bottom structure (βimpact = 0°) are significantly higher. At higher values for vimpact, coefficients of variation (mean value) between 6.5% for a rigid and 7.4% for a flexible bottom structure were observed. These values are based on measurement series con­ taining 30 runs of each test case. Using high speed cameras Tödter et al. (2020) visualized the complex flow conditions right after the impact. Due to the βimpact = 0° an cushion of air was trapped between the water surface and structure, which is subjected to oscillations and thus influenced the resulting pressure.

asymmetric wedge with horizontal as well as well vertical speed at impact. The configuration is com­ parable to the one observed during the model tests of the sideways launching process (compare Figure 3). For small deadrise angles, flow separ­ ation and ventilation did occur during experiments conducted by Judge et al. (2004). This ventilation lead to complex flow conditions around the inves­ tigated asymmetric wedge increasing uncertainties during pressure measurements compared to a symmetric wedge/slamming event. In addition, the motion of the ship during model tests of the sideways launching process is not restricted to one DOF. During the impact with the water surface the ship is able to move freely. The motion at this point consists of three DOF (vertical and horizontal translation, rotation about x-axis of the ship).

Figure 6. Mean peak pressure for LC_01 and LC_02.

Figure 7. Impulse transferred onto ship hull in a 50ms time period (mean value) for LC_01 and LC_02.

Table 3.

Table 4.

case

LC_01

LC_02

Peak pressure measured during model tests. pi max mean kPa

σi max kPa

cv i max %

P1

438

21

P2 P3 P4

1,015 538 927

228 42 122

P1 P2 P3 P4

475 1,069 538 927

6 184 103 39

1.26 17.22 17.10 4.99

transducer

Impulse on ship hull in a 50ms time period.

transducer

Ji 50ms kPa·s

σi 50ms kPa·s

cv i 50ms %

4.85

P1

9.61

0.23

2.48

22.50 7.79 13.12

P2 P3 P4

10.15 8.77 8.69

0.65 0.43 0.20

6.40 4.92 2.26

P1 P2 P3 P4

9.81 9.79 8.87 9.31

0.06 0.45 0.12 0.20

0.63 4.57 1.35 2.10

case

LC_01

LC_02

3.4

The coefficient of variation observed for P2 during model tests of the sideways launching process is between two and half and three times higher as the values obtained by Tödter et al. (2020) There are different explanations feasible for the higher uncertainties. Judge et al. (2004) investi­ gated the behavior and flow conditions around an

Impulse on ship hull

Due to the high coefficient of variation of the peak pressure, a parametric study based on these values is not reliable. This is especially true, as during the parametric study of the model tests only one run of the different test setups was conducted.

18

Therefore, a different approach based on the impulse is used to evaluate the influence of different parameters on the loads on the hull structure. The impulse Ji 50ms per unit area is calculated as follows:

results in an increase of the impulse transferred onto the ship hull after impact of approx. 15.2%. 4.3

In Equation (1) the impulse based on the pressure-time -signal pi of a pressure transducer is calculated for a time period of 50ms after the peak pressure occurs at the time tpi max. The impulse is illustrated in Figure 5. A 50ms time period is chosen, as at this point the peak pressure is decayed for all pressure transducers (compare Figure 5). The resulting impulse (mean value) for setup LC_01 as well as LC_02 and the corresponding stand­ ard deviation are given in Table 4 and Figure 7. The maximal coefficient of variation of the impulse cv i 50ms for P2 in case of LC_01 is reduced to 6.4% compared to 22.5% for the peak pressure. A comparison between Figure 6 with Figure 7 shows, that the impulse is more consistent/reduces uncertainties.

4.4

As the impulse Ji 50ms is used as the basis for the paramet­ ric study, only the influence of different parameters on the loads right after of the ship hull impact with the water sur­ face is assessed. The results presented within this section are based on pressure transducer P2, as this transducer shows the highest impulse. All correlations presented are valid for the other three pressure transducer as well. Loading condition

No significant influence of the loading condition on the impulse can be identified. For all cases the impulse is nearly constant, as vimpact is nearly constant (see Figure 4). The loading condition is however the most rele­ vant parameter regarding the ship motion during the sideways launching process. As expected, an increase of COG has the biggest influence, especially on the maximal heeling angle of the ship. 4.2

Speed of hull at impact

As the variation of the water level and height of palls shows, the biggest influence on the impulse on the ship hull has the height of the drop around the edge of the pier. As Figure 4 shows, the height of the drop correlates directly with vimpact. With an increased drop height, the ship is able to rotate for a longer time period around the edge of the pier gaining more rotational speed (compare Figure 2). Due to the geometry of the special purpose vessel and the slipway, this rotational speed correlates dir­ ectly with vimpact as a βimpact ≈ 0° at the moment of impact was observed for all model tests (see Figure 3). In Figure 8 the resulting impulse at P2 for all test setups is plotted as a function of vimpact. As Figure 8 shows, a linear correlation between the JP2 50ms and vimpact can be observed. The slope of this approximation is about 1.7kPa·s/m. Based on the design point, an increase of vimpact of 1m/s is equal to an increase of the impulse resulting on the ship hull of about 17.8%.

4 PARAMETRIC STUDY OF RESULTING LOADS ON SHIP HULL

4.1

Coefficient of friction

A decrease of the coefficient of friction between the ship and palls will result in higher speed of the ship while sliding down the slipway as well as at the moment of impact with the water. A decrease of the coefficient of friction to 0.25µ0 (= 12.9% increase of ship speed at the end of the slip­ way) results in an increase of the impulse by 10.1%. An increase of the coefficient of friction to 2.00µ0 (= 20.0% decrease of ship speed at the end of the slip­ way) correlates to a decrease of 13.8% of the impulse. In total, the influence of the coefficient of friction is minor. During the full scale sideways launching process, a maximal variation of the coefficient fric­ tion between FC_03 and FC_05 is expected. Within this range the impulse changes about ± 3.7%.

Water level and height of palls

A decrease of the water level inside the launch basin like the increase of the height of palls results in the same effect regarding Ji 50ms: the height of the drop around the edge of the pier is increased resulting in higher speed at the moment of impact. An increase of the height of the palls of 0.50m will result in an increase of J2 50ms of 7.6%, while a decrease of the water level inside the launch basin of 0.50m will correlate with an increase of 7.7%. Based on a linear approximation, an increase of the drop height around the edge of the pier of 1.00m

Figure 8. Influence of speed at impact on resulting impulse on ship hull for pressure transducer P2.

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5 LOAD MODEL 5.1

Derivation

As described in section 3.2, the pressure-time-signal observed during model tests, shows a harsh peak pressure followed by quick decay towards the dynamic pressure. The same behavior can be observed in other slamming experiments as well, e.g. in the work of Lewis et al. (2010), Tödter et al. (2020) or Javaherian et al. (2022). Such a behavior can be approximated by the use of an exponential function. The following approach is proposed to assess the loads resulting on the ship hull overserved during model tests of the sideways launching process:

Figure 9. Load model for sideways launching based on model tests.

The proposed load model based on Equation (2) can be implemented/used for structural dynamic ana­ lysis using the finite element method (FEM). An example for the application of this load model for FEM simulations is given by Ulbertus et al. (2023). A visualization of this load model is provided in Figure 9. In Equation (2) the pressure pmt resulting on the ship hull is described by three stages. In the first stage, no contact with the water is given (pmt = 0). After impact of the ship hull with the water surface at t = timpact the pressure rises instantly to the maximal peak pressure pmt max. Afterwards a decay of pressure is given by the exponential function, where the decay­ ing behavior is defined by the decay constant kmt. The time of the decay to the sum of the dynamic pressure pdyn and hydrostatic pressure pstat (third phase) is given by the expression tr mt. The factor kv in Equation (2) is a scaling factor, which considers the change of the impulse as a function of vimpact as shown in Figure 8. In the case of pressure transducer P2, kv can be defined as a function of vimpact in [m/s] as follows:

5.2

Limitations

The proposed load model can be seen as a conservative approach for assessing the loads resulting on the ship hull during a sideways launch­ ing process. Due to the conditions as shown in Figure 3 during model tests, very high peak pres­ sures were observed as βimpact ≈ 0°. As illustrated by Chuang (1967), the peak pressure resulting on a structure decreases significantly if the deadrise angle of the structure is increased. Applying the proposed load model to different geometric config­ uration – like a ship with a flat bottom impacting the water surface in the area of the bilge –will result in too high loads. As all configurations of the model tests lead to βimpact ≈ 0°, no investigations regarding the influence of the deadrise angle were possible. In addition, as the ship model used for model tests was quite rigid, effects of elasticity of the ship’s hull structure are not accounted for in the proposed load model. As shown by Tödter et al. (2020), the peak pressure is reduced by up to 30% for an elastic bottom structure compared to a rigid one.

Values for kmt and tr mt in Equation (2) can be derived by using the following conditions and solving the system of equations for those two parameters: – Ji 50ms of the load model is equal to value obtained from model tests – pmt = pdyn + pstat at t = timpact + tr mt

6 SUMMARY AND CONCLUSIONS

In Figure 9 values for Equation (2) obtained for P2 at LC_01 are given. A comparison of the proposed load model with measured pressure-time-signal of P2 at LC_01 is given in Figure 10. The proposed load model is able to fit the measured data quite well.

Model tests of a sideways launching process of a special purpose vessel were conducted. During these model tests the pressure-time-signals at differ­ ent point of the ship hull were measured.

20

In the end, a load model based on the model tests is proposed. This load model allows for the implant­ ation inside of FEM simulations, to investigate the structural response of the ship’s hull structure due to the loads of the sideways launching process.

REFERENCES Chuang, S.-L. 1967. Experiments on Slamming of Wedge-Shaped Bodies. Journal of Ship Research 11(3): 190–198. Javaherian, M.J., Ren Z. & Gilbert, C. 2022. Flow Visual­ ization, Hydrodynamics, and Structural Response of a Flexible Wedge in Water Entry Experiments. Journal of Ship Research: 1–13. Judge, C., Troesch, A. & Perlin, M. 2004. Initial water impact of a wedge at vertical and oblique angles. Jour­ nal of Engineering Mathematics 48(3): 279–303. Leavitt, C. M. 1980. Launching. In Taggart, R. (ed.), Ship Design and Construction: 657–698. New York: Society of Naval Architects and Marine Engineers. Lewis, S.G., Hudson, D.A., Turnock, S.R. & Taunton, D. J. 2010. Impact of a free-falling wedge with water: synchron­ ized visualization, pressure and acceleration measurements. Fluid Dynamics Research 42(3): 035509 (30pp). Tödter, S., el Moctar, O., Neugebauer, J. & Schellin, T. E. 2020. Experimentally Measured Hydroelastic Effects on Impact-Induced Loads During Flat Water Entry and Related Uncertainties. Journal of Offshore Mechanics and Arctic Engineering 142 (1): 011604–1-9. Ulbertus, A., Schöttelndreyer, M. & Ehlers, S. 2021. Side­ ways launching process of a ship using the Arbitrary-Lagrangian-Eulerian approach. 13th European LS-DYNA Conference. Ulm. Ulbertus, A., Schöttelndreyer, M. & Ehlers, S. 2023. Hull structural loads during a sideways launching process using fluid-structure interaction. 9th International Con­ ference on Marine Structures (MARSTRUCT 2023). Gothenburg. Wang, S. & Guedes Soares, C. 2017. Review of ship slam­ ming loads and responses. Journal of Marine Science and Application 16(4): 427–445.

Figure 10. Comparison of load model with pressure-timesignal obtained from model tests.

Due to geometric conditions right at impact (deadrise angle of approx. 0°), high peak pressures at the impact with water surface could be observed resembling the behavior during slamming events. During model tests different parameters relevant for the sideways launching process were varied, including the loading condition of the ship, the coef­ ficient of friction between the palls and the slipway, the height of the palls as well as the water level inside the launch basin (tidal effects). Based on the impulse resulting right after the impact with the water surface, the influence of these parameters on the loads acting upon the ship hull is assessed. For the investigated special purpose vessel inves­ tigated, the height of the drop around the edge of the pier before the impact with the water has the most significant influence on the resulting loads, as this leads to a higher velocity at the ship hull at the impact with the water surface. An increase of this height of 1m correlates with an increase of about 15.2% of the loads observed on the ship hull.

21

Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Hull structural loads during a sideways launching process using fluid-structure interaction A. Ulbertus & M. Schöttelndreyer Thyssenkrupp Marine Systems, Hamburg, Germany

S. Ehlers German Aerospace Center (DLR), Geesthacht, Germany

ABSTRACT: Within this paper the hull structural loads resulting during a sideways launching process of a special purpose vessel are investigated. Different approaches for modelling the hull structural loads are used. One approach is based on a load model derived from model tests of the sideways launching process, which is used for simulations with the finite element method (FEM). The second approach is simulating the sideways launching process using an Arbitrary Lagrangian-Eulerian (ALE) approach, where the loads are accounted for by the means of fluid-structure interaction (FSI). Two different load mechanisms are observed based on the simulations with the ALE-approach: loads due to the impact with the water surface and loads due to the deceleration of the hull struc­ ture. The influence of the two different loading mechanisms on the resulting stresses is investigated and conse­ quences regarding the design of the hull structure of the investigated special purpose vessel are discussed.

Lagrangian approach. For simulations carried out using FSI, the surrounding media of the ship hull are modelled using an ALE-approach within the FEM software LS-DYNA. A detailed description of the con­ cept and aspects regarding the numerical implementa­ tion of ALE-approaches is provided by Donea et al. (2004). The brief summary of the ALE-approach within this section is based on the work by Donea et al. (2004). In Lagrangian algorithms, which are mainly used for structural mechanics, the nodes of the computa­ tional mesh follow the displacement of the associ­ ated material during the simulation. Such algorithms are good for tracking interfaces between different materials (e.g. contact surfaces), but are generally less robust for high distortion. Eulerian algorithms are commonly used for fluid dynamics. In the Euler­ ian approach, the mesh is fixed and the fluid is moving with respect to the grid. Large movements or distortion like vortices can be simulated well, but interfaces or free surfaces can only be handled suffi­ ciently with a high mesh resolution at those areas. ALE-methods are combining the Lagrangian and Eulerian approach and are able to overcome the dis­ advantages of both approaches to some extent as explained by Donea et al. (2004). Using an ALE approach the nodes can move with a continuum (Lagrangian), be fixed in space (Eulerian) or move arbitrarily in a specified way to enable continuous rezoning. For this purpose, two calculation steps are

1 INTRODUCTION To avoid damages or even a failure of the ship’s hull structure during a sideways launching process, the result­ ing hull structural loads are to be assessed carefully. However, data and information available in literature are scarce. A comprehensive overview of the process of sideways launching is given by Leavitt (1980). Besides the process itself (preparation and execution), an analyt­ ical approach for assessing the ship motion based on the according ordinary differential equations of motion (ODE) is provided by Leavitt (1980). But no data or approach is available to assess the loads resulting on the vessel during a sideways launching process. Therefore, a variety of simulations using FEM were carried out during the design phase of a special purpose vessel. These simulations included different approaches for modelling the hull structure of the special purpose vessel as well the resulting hull structural loads, including one approach based on FSI. Thus an in-depth knowledge of the resulting hull structural loads as well as different load mechanism acting during a sideways launching pro­ cess is obtained and presented within this paper. 2 THEORETICAL BACKGROUND 2.1

ALE-approach

In this paper the FEM model of the hull structure of the special purpose vessel is described with a classical

DOI: 10.1201/9781003399759-3

23

needed for each time step of the simulation. The first step is a classical Lagrangian step. The second step – which is also referred to as an advection step by LSTC (2019) – performs a rezone of the computa­ tional mesh. In LS-DYNA, this rezone is done incre­ mentally, which means, that the nodes of the mesh are only moved by a small fraction of the element size of surrounding elements as explained in LSTC (2019). Due to this freedom by rezoning the computa­ tional mesh, an ALE-approach allows for greater dis­ tortions than a classic Lagrangian algorithm while simultaneously providing a higher resolution than an Eulerian algorithm. The drawback is higher compu­ tational costs for each time step. As the ALE-approach is able to track contact sur­ faces (ship hull ↔ water) and free surfaces (spray and waves after impact of ship with water) while allowing simultaneously for big distortions of the computational mesh (ship motion, free surfaces), it is a well suited choice for the simulation of a sideways launching process. The applicability and a verification of the ALE-approach for the use case of a sideways launching process is given by Ulbertus et al. (2021). Using the ALE-approach, it is possible to assess the loads resulting on the hull structure as well as the ship motion during the sideways launch­ ing process quite well. In naval architecture the ALE-approach is used for similar physical phenomena. Especially in the recent years publications using the ALE-approach are more common due to the development of computational methods/resources. One example is the investigation of slamming events. The ALE-approach was used by Yu et al. (2019), Truong et al. (2021), Wang et al. (2021) or Truong et al. (2022) among others to inves­ tigate the resulting loads during slamming experi­ ments. Even more complex use cases like the slamming loads on the bow of a tanker in irregular waves can be simulated using the ALE-approach as shown by Wang & Guedes Soares (2016). 2.2

the deceleration of the ship are accounted for as well. In addition, added masses around the ship hull are considered. A parametric study of the different settings of the FSI algorithm used for the simulation of the sideways launching process with the ALEapproach was carried out by Ulbertus et al. (2021) Fluid and gasses modelled with an ALE-approach can be defined by equation of states (EOS). An EOS is a thermodynamic equation linking different state variables such as the pressure p, volume V, tempera­ ture T or internal energy Ei to describe the state of a material under a given set of physical conditions. The EOS used for the simulation of the sideways launching process are provided in section 2.3. As the two parts of the model (hull structure and surrounding media) are set up independently of each other, this modelling approach provides a good level of flexibility during the ship design process. A modular approach for modelling can be used, allowing for an easy setup of simulations at different stages of the design of the ship’s hull structure. An example for such a modular modelling approach during the ship design process based on the ALEapproach is given by Ulbertus & Schöttelndreyer (2016). 2.3

Material models

2.3.1 Steel The hull structure of the special purpose vessel investigated is designed mainly with the high-tensile steel alloy 1.0584 (VL D-36). In the event of the impact of the hull structure with the water surface during the sideways launching process, plastic deformations of the ship’s hull structure could poten­ tially occur in a very short time period. Therefore, the material model used for the ship’s hull structure has to consider the following aspects: – adequate assessment of plastic deformation and hardening – strain rate dependency

FSI algorithm

To account for these two aspects effective stress strain curves are used for the FEM simulations, which are considering strain rate effects. The curves were obtained by Kubiczek et al. (2017) based on uniaxial tensile tests of VL D-36 according to DIN EN ISO 6892-1. These tensile tests were carried out for differ­ ent stain rates. Based on optical measurements the engineering stress strain curves were determined. The resulting engineering stress strain curves are shown in Figure 1. The effective stress strain curves were derived by an optimization routine in combination with a FEM model of the uniaxial tensile tests in LSDYNA. The effective stress strain curves used in the FEM model were fitted by the optimization algorithm, so that the results of the FEM simulations match the experimental data from the uniaxial tensile tests. More details regarding the used approach are given in Kubic­ zek et al. (2017).

The hull structure of the special purpose vessel (Lagrangian formulation) and the surrounding media (ALE formulation) are modelled independently from each other. The Lagrangian elements are coupled with ALE elements in LS-DYNA by a contact algo­ rithm based on the penalty method. Details regarding the used contact algorithm are given in LSTC (2019). A general overview of different contact algo­ rithms and their implementation is given by Wrig­ gers (2006). Using a contact algorithm, FSI is accounted for. Loads resulting from the impact of the ship hull with the water surface are transferred into the ship hull. As an explicit time integration scheme is used, a sufficient sampling frequency for assessing the peak pressure during impact is ensured. As the water is pushed away by the motion of the ship, loads from

24

As the engineering stress strain curves in Figure 1 show, the material behavior of VL D-36 is influ­ enced by the strain rate. To account for the strain effects the Cowper-Symonds equation as derived by Cowper & Symonds (1957) is used. The CowperSymonds equation describes the ratio of the yield strength under dynamic load σYD to the yield strength under static loads σY as a function of the strain rate ε̇:

In Equation (2) the pressure p inside the air is described as a function of the internal energy Ei, the specific heat ratio κ as well as the expression µ. The expression µ in Equation (2) is defined as follows:

The expression ρ/ρ0 in Equation (3) is the ratio of the current density ρ to the reference density ρ0. Using the gamma-law EOS it is possible, to describe isentropic processes. Compression and expansion processes of the air can be covered using this EOS, which are needed to account for the free surface of the water. The material parameters of air used for simulations are given in Table 2.

The coefficients CCS as well as qCS in Equation (1) are material parameters unique for each steel alloy. The corresponding coefficients CCS and qCS derived for VL D-36 by Kubiczek et al. (2017) are provided among other material properties in Table 1. For the simulations of the sideways launching process carried out in LS-DYNA, the effective stress strains curves are considered using the material model *MAT_MODIFIED_PIECEWISE_LINEAR _PLASTICITY, whereas the strain rate effects are considered using the Cowper-Symonds Equation (1).

Table 1.

Table 2.

Material properties used for VL D-36.

ρ0 kg/m³

E GPa

ν -

σY MPa

CCS s-1

qCS -

7,850

206

0.30

355*

2,225

3.33

Material properties used for air.

ρ0 kg/m³

κ -

1.20

1.40

2.3.3 Water Water is modelled using the Mie-Grüneisen EOS. As explained by Hiermaier (2008), the Mie-Grüneisen EOS was originally derived for calculating shock waves in solids at high pressures and is often used for the assessment of seismologic phenomena. According to Hiermaier (2008), the Mie-Grüneisen EOS can be formulated as follows:

* set to minimal yield strength for VL D-36 for simulations

In the Mie-Grüneisen EOS (4) the term c0 is the speed of sound at a reference state, ΓMG is the socalled Grüneisen parameter or Grüneisen gamma and SMG is a material coefficient. EOS (4) is only valid for compression. To account for expansion and tension waves inside a material, a second EOS has to be introduced. According to LSTC (2019), the fol­ lowing EOS can be used for this purpose: Figure 1. Engineering stress strain curves for VL D-36 by Kubiczek et al. (2017).

2.3.2 Air The air is modelled as an ideal gas. For this purpose the following EOS is used, which is also referred to as the gamma-law EOS by LSTC (2019):

Due to the high bulk modulus of water, it is pos­ sible to use the Mie-Grüneisen EOS to describe fluids like water under shock or impact loading.

25

Examples for this approach are given by Steinberg (1987), Shin et al. (1998) or Hamashima et al. (2004). The material coefficients for the MieGrüneisen EOS used for water in this paper are obtained from Hamashima et al. (2004). These values are slightly modified to meet the conditions at a shipyard using such as a launching process. Cor­ responding values are given in Table 3.

Table 4.

Table 3.

primary members of the hull structure. An average element between 400x400mm and 300x300mm is used for the global FEM model. This element size is a good comprise between assessing the structural dynamic behavior of the hull structure on a global level as well as the local behavior of bigger struc­ tural components like deck panels. Inside the global FEM model all relevant members of the hull structure are modelled explicitly. Panel structures are modelled using shell elements. The same is true for the web of frames and girders. A homogenous mesh is used whenever possible. Stiff­ eners, flanges and pillars are modelled with beam elements. A matching offset for each beam section is defined, in order to model frames/girders as well as panels correctly by considering the corresponding moment of inertia of the beam elements accordingly. The properties of the global FEM model are given in Table 4. The global FEM of the special purpose vessel is quite detailed for a global FEM model. The level of detail of this FEM model is equal to the recommendations for partial ship structural analysis according to the guideline for FEM analysis of DNV (2021).

FEM model

c0 m/s

SMG -

ΓMG -

1,010

1,485

1.79

1.65

3 FEM MODEL OF HULL STRUCTURE For the investigation of the hull structural loads during the sideways launching, different FEM models of the hull structure of the special purpose vessel are set up. Three different FEM models of the hull structure are used: – global FEM model – detailed FEM model of highly loaded area – combined FEM model (combination of global and detailed FEM model) An overview of the three different FEM models based on a side view of the ship is given in Figure 2. General properties of the different FEM models are provided in Table 4.

3.2

Beam elements

Mass elements

nodes

167,000 29,000 150,700

203,900 1,914,300 2,386,500

Detailed FEM model of highly loaded area

Different parts of the hull structure are highly loaded during the sideways launching process. This is espe­ cially true for parts of the hull structure, that will impact with the water surface during the sideways launching process. The resulting loads at impact are comparable to slamming events. These impact loads are especially critical for the shell of the hull struc­ ture and its supporting structural members (web frames and bottom structure). To assess the stresses inside these highly loaded areas of the hull structure, a detailed FEM model of these areas is set up. This area of the detailed FEM model is visualized in Figure 2. Modelling is done in accordance to the require­ ments for local structure strength analysis as described by the guideline for FEM analysis of DNV (2021). For the detailed FEM model a very fine mesh size of 50x50mm is used. Properties of the detailed FEM model are given in Table 4. The very fine mesh and extent of the model result in a high number of elements. All structural members are modelled as shell elements. Stiffeners subjected to bending loads are

Figure 2. Overview of FEM models of the hull structure of the investigated special purpose vessel.

3.1

Shell elements

global 197,600 110,900 detailed 2,236,800 155,900 combined 2,452,100 248,000

Material properties used for water.

ρ0 kg/m³

Properties of FEM models of the hull structure.

Global FEM model

A global FEM model of the hull structure of the spe­ cial purpose vessel is set up, that considers all

26

– masses relevant for structural dynamic behavior (e.g. paint, insulation, flooring): an additional mass of 40kg/m2 for decks and 25kg/m² for walls and bulkheads is used – residual mass: distributed on stiffer nodes of hull structure

meshed with at least three elements along the height of the web, to account for a resulting bending deformation. Only the bulbs of the bulb flat profiles are modelled using beam elements. The very fine mesh size of 50x50mm enable a detailed analysis of plating subjected to lateral loads and the resulting bending deformation. This is import­ ant for the shell plating of the hull structure subjected to impact high loads at the contact with the water sur­ face. Structural members subjected to compressive or shear loads, where buckling could potentially occur, can be assessed directly w/o the need of a separate check of the buckling strength, e.g. according to the rules/guidelines of classification societies. 3.3

In Table 4, the number of mass elements used to account for the loading condition of the special pur­ pose vessel during the sideways launching process is given. The correct consideration of the mass, the position of the center of gravity and moments of inertia is especially crucial for the simulations based on FSI using the ALE-approach, as the inertia of the ship is important to assess the motion/trajectory of the ship accurately.

Combined FEM model

Especially for the use with the ALE-approach, a combined FEM model is set up. In the combined FEM model the detailed FEM model of the highly loaded areas is integrated into the global FEM model of the hull structure. This is illustrated in Figure 2. By integrating the detailed FEM model into the global FEM model of the hull structure, the stiffness as well as the inertia of the complete hull structure is considered during simulations. By using the com­ bined FEM model all aspects relevant during the sideways launching process can be assessed by one simulation using FSI with the ALE-approach. This includes the complete motion of the ship (e.g. max­ imal heeling angle) as well as the structural dynamic response of the hull structure. In addition, there is no need to set additional boundary conditions (BC) during simulations. For the integration of the detailed into the global FEM model, the elements adjacent to the detailed FEM model are gradually refined (marked as “area of transition” in Figure 2), to fit the mesh size of the detailed FEM model at the cutting edges. The longi­ tudinal stiffeners of the global FEM model are con­ nected to the longitudinal stiffeners of the detailed FEM model by overlapping the beam elements of the global FEM model with the shell elements of the detailed FEM model. This allows for an adequate transfer of bending moments. 3.4

3.5

Added masses

For simulations with the ALE-approach the added masses are accounted for by using FSI. For the simula­ tions w/o FSI the added mass around the ship hull are to be considered separately. Korotkin (2009) provides approaches for considering added masses around vibrating hull structures. In the event of the sideways launching process, a lateral bending deformation of the shell plating is to be expected. This mode of deformation is comparable to the vibration of multispan plates as presented by Korotkin (2009). The added mass λplate plate for that mode of deformation can be calculated based on the following equation:

In Equation (6) aplate is the distance between the stiffeners, while bplate is the length of the panel. The coefficient µplate is function of the aspect ratio of the panel (apanel/bpanel) and the number of oscillating plate fields nplate. Values for µplate are provided by Korotkin (2009) for different BC. The added mass based on Equation (6) is considered for simulations w/o FSI for the shell plating. The ratio of the added mass to the weight of plate panel is about 3.3 for an average panel of the shell plating of the special pur­ pose vessel investigated.

Weight distribution

As for the ship motion as well as the structural dynamic behavior of the ship’s hull structure, the mass of the special purpose vessel including the moments of inertia has to be considered during FEM simulations. The modelled mass inside the FEmodels is split into four different parts:

4 MODELLING OF LOADS 4.1

Initial conditions for simulations

The initial conditions used for both simulation approaches are calculated using an analytical approach. Within this approach the sliding of the ship down the slipway as well as the tipping around the edge of the pier up to the first contact with the water surface is considered. By solving the corres­ ponding ODE of motion with the 4TH order Runge-

– hull structure: modelled directly with shell and beam elements – bigger systems: modelled as rigid bodies with corresponding mass and moment of inertia and connected to hull structure (e.g. engines, gear boxes, gensets)

27

Kutta method, the position as well as speed of the special purpose vessel relative to the water surface at the moment of impact is calculated. This analytical approach is based on the work by Leavitt (1980). More details regarding the forces acting upon the ship as well as corresponding ODE used for the analytical approach are given in Leavitt (1980). By using this analytical approach the influ­ ence of different parameters regarding the initial conditions can be assessed. This includes the loading condition of the ship, the coefficient of friction between palls and slipway, the height of the palls used during launching as well as the height of the water inside the launch basin (tidal effects). 4.2

The load model according to Equation (7) is implemented as a load function using the card *DEFINE_FUNCTION inside LS-DYNA. One function is defined for each panel of the shell plating (PS) of the FEM model (approx. 670 functions). The according time of impact timpact and speed at impact vimpact are estimated for each panel individually based on the analytical approach as described in sec­ tion 4.1. By calculating these values separately, the delay of impact at different points of the hull is considered. Compared to design loads of classification soci­ eties, the load model in Equation (7) based on model test does result in higher loads on the hull structure. E.g. if compared to the design loads for bottom slamming of the Rules of Lloyds Register (2019), the proposed load model of Ulbertus et al. (2023) does result in approx. 84% higher loads at the point of impact with the water surface.

Simulations w/o FSI

In 2020 model tests of the sideways launching pro­ cess of the special purpose vessel were conducted. During these model tests, the pressure-time signals at different points of the hull were measured. Based on these results a load model is derived considering the loads at impact of the hull with the water surface. Details are given by Ulbertus et al. (2023). The fol­ lowing approach is proposed to assess the pressure pmt resulting on the ship hull after the moment of impact at timpact:

4.3

Simulations with FSI

As explained in section 2.2, the hull structure and the ALE-domain are modelled independently from each other. The ALE-domain is modelled using regular 3D hexahedrons with an element size of 500mm. In the area, where the ship immerges into the water, local refinement of the ALE-domain is used. This results in an element size of 250mm. A more detailed description of the setup of the ALE-domain as well the approach used for the local refinement is given by Ulbertus et al. (2021). The ALE-domain is modelled, so that the condi­ tions of the launch basin – like the geometry of the pier – are accounted for. Due to the size of the launch basin around 7.3 million ALE elements are used for simulations with FSI. For comparison: if the ALE-domain is meshed w/o the local refine­ ment and a consistent element size of 250mm, about 12.6 million ALE elements would be necessary. For setting up the simulations with FSI based on the ALE-approach, only gravitational loads and the initial conditions are to be set. This includes the pos­ ition of the ship relative to the water surface and ini­ tializing the corresponding speed of the ship at the moment of impact. Other aspects like loads acting upon the hull structure or added masses are con­ sidered by the use of the FSI algorithm.

An overview of the load model and different coeffi­ cients is provided in Figure 3. As Figure 3 shows, a harsh peak pressure was observed during model tests comparable to slamming events. The term kv in Equa­ tion (7) is a scaling factor defined as a function of vimpact in [m/s] as follows:

5 RESULTS OF FEM SIMULATIONS 5.1

Overview of highly loaded areas

Based on the FEM simulations, different highly loaded areas of the hull structure of the special pur­ pose vessel can be identified. Some areas/parts of the following structural components are highly loaded during the sideways launching process:

Figure 3. Load model for sideways launching based on model tests.

28

– – – –

shell plating web frames of shell floor plating plating of transversal bulkheads

As model tests conducted by Ulbertus et al. (2023) show, the loads during the impact of the special purpose vessel with the water surface are mainly influenced by the speed of the hull at the moment of impact. Especially an increase of the height of the drop around the edge of the pier is the most relevant parameter. This increase can be either by using higher palls during the launching process or a decreased water level inside the launch basin (tidal effects). The second load mechanism is transverse forces due to deceleration of the ship. After contact with the water surface, the transverse motion of the vessel from sliding down the slipway is stopped rather quickly. As illustrated in Figure 3, espe­ cially the bottom structure of the vessel is deceler­ ated, while the upper part wants to remain its transversal motion due to the inertia of the hull structure. This behavior results in high shear stres­ ses inside transversal members of the hull struc­ ture. Widespread shear stresses in the plating of bulkheads in engine rooms between the bottom structure and the next higher, continuous deck are observed for the special purpose vessel as shown in Figure 6. Too high shear stresses could result in shear buckling of the bulkhead plating. As the bulkheads are the main load bearing components for the second load mechanism, this failure mode could potentially result in progressive collapse behavior. A loss of shear stiffness of the bulkheads could result in great deformations of the shell and its supporting members equaling a total loss of the affected part of the hull structure in a worst case scenario. As simulations with the ALE-approach show, the second load mechanism is dominated by the weight of the vessel during the sideways launch­ ing process. E.g. a decrease of the weight of approx. 10% during launching, offsets the result­ ing shear stress in Figure 6 by roughly 0.25·τperm.

In the following, the resulting hull structural loads, underlying load mechanism as well as the influence of different FEM and load models are discussed. 5.2

Load mechanism acting on hull structure during sideways launching process

During the sideways launching process, two differ­ ent load mechanisms can be observed. These are illustrated in Figure 4. The first load mechanism is hull structural loads resulting from the impact with the water surface. These loads are acting upon the shell of the hull structure and are transferred into the supporting structure of the shell (web frames and bottom struc­ ture). In the case of the special purpose vessel, a deadrise angle of nearly 0° can be observed (com­ pare Figure 3). As model tests conducted by Ulber­ tus et al. (2023) show, this configuration leads to very high peak pressures right after impact with the water surface. However, these loads are limited to the area of first contact of the ship hull. Potential damages or failure of the hull structure would be limited locally to individual structural components. Potential failure modes are permanent plastic deformation of the shell plating due to lateral bend­ ing and/or buckling of supporting members (bottom structure/web frames) due to the resulting compres­ sion loads.

Figure 4. Different load mechanism acting upon hull struc­ ture during sideways launching process of special purpose vessel.

Figure 5. Stresses inside highly loaded shell plating based on simulations w/o FSI.

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5.3

Influence of FEM models of hull structure

The combined FEM model is able to overcome the disadvantages of the global and detailed FEM model. The combined FEM model allows for a detailed assessment of the resulting stresses. Both the stiffness and inertia of the complete hull structure are considered using the combined FEM model, so that no additional BC are necessary thus enabling the use for simulations with FSI.

One major disadvantage of the global FEM model is that no meaningful evaluation of stresses inside the shell plating due to lateral bending is possible. As Figure 5 illustrates, a factor of three between the global and detailed FEM model can be observed. Damages due to the first load mechanism (loads at impact) cannot be assessed adequately. In addition, buckling of structural components under compression/ shear loads is to be checked separately resulting in very high post-processing effort. As a coarser mesh is used, the resulting stresses are lower throughout the hull structure as compared to the detailed FEM model. An advantage of the global FEM model is the consid­ eration of the complete stiffness and inertia of the hull structure, which is especially important for simulations using FSI. One of the major problems regarding the detailed FEM model is setting adequate BC. As Figure 5 shows, the influence of BC for simula­ tions w/o FSI is negligible. However, the influence for simulations with the ALE-approach is quite significant. Two different BC are checked for simulations with FSI. The first BC checked is dropping the detailed FEM model freely onto the water surface with the corresponding initial vel­ ocity. As the inertia of the rest of the hull structure is missing, the detailed FEM model is not able to immerse properly into the water. The process can be compared to skipping a stone above a water sur­ face resulting in too low stresses. In the second BC checked, the model is forced into the water based on the estimated trajectory of the vessel during the sideways launching pro­ cess. This is done by prescribing the motion at the cutting edges, where the rest of the hull structure would connect with the detailed FEM model. This induces significant stresses around the area of the cutting edges, resulting in wide­ spread shear buckling of one of the bulkheads. Stresses are overestimated.

5.4

Influence of load modelling

The hull structural loads resulting from the first load mechanism (loads due to impact) show a good agree­ ment between simulations w/o FSI and the ALEapproach (compare first 50ms of Figure 7). However, simulations w/o FSI show that the load model as given in section 4.2 is not able to account for the second load mechanism (deceleration of ship). As Figure 6 shows, the shear stresses result­ ing in the plating of the bulkhead are underesti­ mated by roughly a factor of factor two compared to simulations based on the ALE-approach. In Figure 7 a switch of loading direction inside the web frames due to the deceleration of the ship is observed based on the ALE-approach. This switch of loading direction cannot be accounted for by simulations w/o FSI. As explained above, the second load mechanism could potentially result in bigger damages of the hull structure due to the risk of progressive collapse behavior. If no simulations using FSI based on the ALE-approach were carried out, this load mechan­ ism could potentially be overlooked. 5.5

Comparison of computational effort

One drawback of the simulations with FSI using the ALE-approach is the higher computational effort compared to simulations w/o FSI. The following computational time is necessary to simulate 100ms of real time:

Figure 7. Stresses in corner of web frame plating based on combined FEM model.

Figure 6. Stresses inside plating of transversal bulkhead based on combined FEM model.

30

sideways launching process of the special purpose vessel. The second load mechanism can only be assessed using FSI with the ALE-approach. Simula­ tions w/o FSI based on the load model derived from model tests do not account for this load mechanism. For the use of FSI with the ALE-approach, a FEM model of the complete hull structure is neces­ sary to account for the stiffness and inertia of the whole vessel. A combined FEM model (integration of detailed FEM model of highly loaded area into global FEM model) is the best-suited option for this purpose. Such a combined FEM model allows for the assessment of both load mechanism and potential failure modes within one simulation using the ALEapproach. In addition, the ship motion during the sideways launching process (e.g. maximum heeling) can be assessed using this approach as well.

– w/o FSI + global FEM model: 0h49min – w/o FSI + combined FEM model: 18h07min – with FSI + combined FEM model: 35h03min Approx. a factor of two between simulations w/o FSI and FSI can be observed. This difference is not bigger, as the numerical aspects of the ALE-approach fot he use case of a sideways launching process were investigated in more detail and optimized regarding the resulting com­ putational effort (local refinement for ALEdomain, adjustment of number between advection cycles, use of MPP- instead of SMPimplementation of LS-DYNA). Details are pro­ vided in Ulbertus et al. (2021). Due to the significantly less number of elements as well as bigger time step due to bigger element size of the global FEM model, a factor of approx. 22 between the global and combined FEM model during simulations w/o FSI is given. No noteworthy difference between the detailed and combined FEMmodel regarding the computational effort can be noticed. All values are provided for simulations using eight cores on an Intel Xeon Gold 6134 processor @ 3.70GHz (boost speed) and LS-DYNA release R13.1.

REFERENCES Cowper, G.R. & Symonds, P.S. 1957. Strain-hardening and strain-rate effects in the impact loading of cantilever beams. Providence (Rhode Island): Division of Applied Mathematics, Brown University. DNV (ed.). 2021. Class Guideline: Finite Element Analysis (DNV-CG-0127). Edition August 2021. Bærum: DNV. Donea, J., Huerta, A., Ponthot, J.-P. & Rodríguez-Ferran, A. 2004. Arbitrary Lagrangian-Eulerian Methods. In Stein, E., de Borst, R. & Hughes, T.J.R. (eds.), Encyclo­ pedia of Computational Mechanics (1): 413–437, Hobo­ ken (New Jersey): John Wiley & Sons. Hamashima, H., Kato, Y. & Itoh, S. 2004. Determination of JWL Parameters for Non-Ideal Explosive. AIP Confer­ ence Proceedings 706(1): 331–334. Hiermaier, S. 2008. Structures Under Crash and Impact: Continuum Mechanics, Discretization and Experimental Characterization. New York: Springer Science+Busi­ ness Media. Leavitt, C. M. 1980. Launching. In Taggart, R. (ed.), Ship Design and Construction: 657–698. New York: Society of Naval Architects and Marine Engineers. Lloyd’s Register (ed.). 2019. Rules and Regulations for the Classification of Naval Ships. Edition January 2019. London: Lloyd’s Register. LSTC (ed.). 2019. LS-DYNA® Theory Manual. Edition 02/ 20/19 (r:10859). Livermore (California): LSTC. Korotkin, A. 2009. Added Masses of Ship Structures. New York: Springer Science+Business Media. Kubiczek, J., Burchard, K., Ehlers, S. & Schöttelndreyer, M. 2017. Material relationship identifi­ cation for finite element analysis at intermediate strain rates using optical measurements. In Guedes Soares, C. & Garbatov, Y. (eds.), Progress in the Analysis and Design of Marine Structures: 459–468. Boca Raton (Florida): CRC Press. Shin, Y.S., Lee, M., Lam, K.Y. & Yeo, K.S. 1998. Model­ ing Mitigation Effects of Watershield on Shock Waves. Shock and Vibration 5: 225–234. Steinberg, D.J. 1987. Spherical Explosions and the Equa­ tion of State of Water. Report No. UCID-20974. Liver­ more (California): Lawrence Livermore National Laboratory, University of California. Truong, D.D., Jang, B.-S., Janson, C.-E., Ringsberg, J. W., Yamada, Y., Takamoto, K., Kawamura, Y. &

6 SUMMARY AND CONCLUSIONS Systematic investigations of the hull structural loads during the sideways launching process of a special purpose vessel are carried out based on FEM simula­ tions. Different FEM models of the hull structure as well as different approaches to account for the result­ ing loads are used for this purpose. One approach is based on a load model derived on the results of model tests of the sideways launch­ ing process. The other approach is considering loads by the means of FSI based on an ALE-approach. Based on the simulations with the ALEapproach, two load mechanisms during the side­ ways launching process can be observed: loads due the impact with the water surface and loads due to the deceleration of the hull structure. This first load mechanism (loads due to impact) could potentially result in local damages of highly loaded areas of the shell plating due to lateral bending and/or buckling of supporting members (bottom structure/web frames) due to the resulting compression loads. The second load mechanism (loads due to deceleration) results in high shear stresses inside transversal members of the hull structure, especially in the bulkhead plating. Widespread shear buckling of bulkhead plating could result from these high shear stresses. The risk of subsequent progressive collapse behavior with greater extent of damages of the hull struc­ ture is potentially given by this load mechanism. Both load mechanisms are to be accounted for during the design of hull structure, to ensure a safe

31

launching process based on model tests. 9th Inter­ national Conference on Marine Structures (MAR­ STRUCT 2023). Gothenburg. Wang, S. & Guedes Soares, C. 2016. Experimental and numerical study of the slamming load on the bow of a chemical tanker in irregular waves. Ocean Engineer­ ing 111, 369–383. Wang, S., Islam, H. & Guedes Soares, C. 2021. Uncertainty due to discretization on the ALE algorithm for predict­ ing water slamming loads. Marine Structures (80): 103086. Wriggers, P. 2006. Computational Contact Mechanics. Berlin/Heidelberg: Springer-Verlag. Yu, Z., Amdahl, J., Greco, M. & Xu, H. 2019. Hydroplastic response of beams and stiffened panels subjected to extreme water slamming at small impact angles, part II: Numerical verification and analysis. Marine Struc­ tures 65: 114–133.

Ju, H.-B. 2021. Benchmark study on slamming response of flat-stiffened plates considering fluid-structure interaction. Marine Structures 79: 103040. Truong, D.D., Jang, B.-S., Ju, H.-B. & Han, S. W. 2022. Prediction of slamming pressure considering fluid-structure interaction. Part I: numerical simulations. Ships and Offshore Structures 17(1): 7–28. Ulbertus, A. & Schöttelndreyer, M. 2016. The hidden potential of global strength calculations during the design phase [translated from German language]. Jahr­ buch der Schiffbautechnischen Gesellschaft 110: 160–174 [in German]. Ulbertus, A. & Schöttelndreyer, M. & Ehlers, S. 2021. Sideways launching process of a ship using the Arbitrary-Lagrangian-Eulerian approach. 13th European LS-DYNA Conference. Ulm. Ulbertus, A., Schöttelndreyer, M. & Ehlers, S. 2023. Assessment of loads on a ship hull during a sideways

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Study on fatigue design loads based on actual encountered loads N. Yamamoto, T. Sugimoto & K. Ishibashi Nippon Kaiji Kyokai, Tokyo, Japan

ABSTRACT: Fatigue design is conventionally based on the cumulative fatigue damage for the design loads. However, how much safe has not been known since the actual loads encountered by the ships are unknown. Recently, it has become possible to determine the loads encountered by ships in operation from AIS and hind­ cast data. In this study, the wave loads encountered by the ships of various types were investigated, and the severe encountered loads independent to the ship type were obtained by applying a three Gaussian mixture model. The distribution of severe loads extracted and compiled from the frequency components of each ship type is a normal distribution and is used to evaluate the fatigue reliability of ships operating in severe sea conditions. Based on the relationship between the probability distribution of both load and fatigue strength, the fatigue design loads required to achieve the target reliability even for ships operating in severe sea condi­ tions were examined.

design loads required to achieve the target reliability (allowable damage probability) were examined, assuming that the capacity of fatigue strength in the service environment was appropriately obtained. In ship structural design rules of classification societies (NK 2021), strength design other than fatigue strength is generally carried out using NA loads, and the results of the stress analysis are also used in the fatigue assessment. An efficient design is possible when the fatigue design load is given based on the NA load. Therefore, in exam­ ining the fatigue design load, the magnitudes of the load and the fatigue damage are considered by using the ratio to the values calculated for the NA load is used.

1 INTRODUCTION Strength design is based on the relationship between load (demand) and strength (capacity). The capacity of fatigue strength depends largely on the environ­ mental conditions in which the structure is used, but the fatigue strength under basic conditions is repre­ sented by the design S-N curve (UK-HSE 1995, Hobbacher 2009). Since the demand of fatigue strength largely depends on the loads acting on the structure, it is important to know the actual working loads to examine the fatigue design loads. The North Atlantic (hereafter, NA) loads (IACS 2014) are sometimes used as design loads, but they include a large safety margin. Since it was not known what kind of seas merchant ships sail in and what kind of loads they encounter, it was not known how much safety the fatigue evaluation using the design loads gives to the actual ships. In recent years, the SOLAS Convention requires that ships be equipped with an Automatic Identifica­ tion System (hereafter, AIS) (IMO 2017), which reveals the actual route of individual ships. The development of re-forecasting technology has also made it possible to determine the wave conditions on a global scale from time to time. By using the time and position data from AIS and the correspond­ ing time and position wave data from hindcast (Hers­ bach 2018, Ardhuin 2014), it is now possible to accurately determine the actual sea conditions encountered by ships (Yamamoto 2020). In this paper, a study on the setting of fatigue design loads based on the actual sea condition data of a ship was conducted. In the study, the fatigue

2 FATIGUE LOAD 2.1

Wave loads and fatigue assessment

The fatigue strength assessment is usually based on the cumulative fatigue damage, which is calculated using S-N curve and frequency distribution of stress range. The S-N curve is expressed as follows in usual.

where C = intercept of S-N curve in log-log relation; m = inverse slope of S-N curve in log-log relation, m = 3 for welded joints and m = 4 for base material.

DOI: 10.1201/9781003399759-4

33

In general, long-term cumulative fatigue damage is calculated by using the long-term distribution of stress range. Then, the fatigue damage is expressed in the form using the long-term response value of the fluctuating stress range as the characteristic value. On the other hand, it is understood that the longterm cumulative fatigue damage is the accumulation of the short-term fatigue damages. The short-term fatigue damage is obtained in the form using the short-term stress response which is proportional to the significant wave height in the short-term sea state as the characteristic value. If the frequency of occurrence of short-term seas over a long-term period is known, then the long-term cumulative fatigue damage is proportional to the expected value of the m-power of the significant wave height of short sea states.

for individual ship for welded joints (m=3) of Pana­ max Bulk Carriers (65,000 ≤ DWT < 130,000) and Oil Tankers of VLCC (170,000 ≤ DWT). The magni­ tude of encountered loads by individual ships varies considerably due to differences in routes, ship oper­ ations, and other factors.

where β = characteristic value of the long-term distribution of the fluctuating stress range; HS = significant wave height of the short-term sea state; and E[ ] = expected value. Using the long-term fatigue damage and the accu­ mulated short-term fatigue damages based on the loads in North Atlantic (IACS 2001), the ratio of the long-term cumulative fatigue damage is equal to the ratio of the accumulated short-term fatigue damages. The m-squared root of the ratio will represent the magnitude of the load relative to the North Atlantic load. According to the relationship in equation (3), the long-term design loads can be examined based on the short-term oceanographic data obtained from AIS and hindcast.

Figure 1. Frequency distribution of the ratio of encountered fatigue loads to that in NA for welded joints.

2.3

where βNA = characteristic value of the long-term dis­ tribution of the stress range in NA; HS,NA = significant wave height of the short-term sea state in NA. 2.2

Severe fatigue loads

When examining fatigue design loads, it is necessary to establish the severe loads that a ship is likely to encounter from a safety side consideration. Since the ship’s encounter loads vary widely as shown in Figure 1, fatigue design loads are examined based on the severe encountered loads among them. Since the magnitude of the encountered loads is mainly depending on the navigation routes, it is assumed that the frequency components of the encountered loads as shown in Figure 1 are a mixture of the loads encountered in (1) routes of calm wave environments, (2) routes of average wave environ­ ments, and (3) routes of severe wave environments. The value defined on the right -hand side of equa­ tion (3) represents the characteristic value of the loads that the ship encounters during the navigation, so the distribution of the characteristic value in each classi­ fied route is considered to be normally distributed according to the central limit theorem when the vari­ ation in routes is narrowed down. If the distribution

Encountered fatigue loads

By integrating AIS data and hindcast data, it is pos­ sible to determine the actual time and location at which individual ships sailed and the sea states along the routes. In this study, the encountered wave loads by each ship from AIS data and IOWAGA’s hindcast data (Ardhuin 2014) for the period from January 2015 to October 2017 (34 months) for a total of 24,323 ships of various types were obtained (Yamamoto 2020). Figure 1 shows an example of the frequency dis­ tribution of the ratio defined in equation (3) obtained

34

of the respective encountered loads can be represented by a normal distribution, the distribution of the encountered loads can be represented by a three Gaussian mixture model as below.

updated according to the posterior distribution of the latent variables in equation (7). Then the rest unknown parameters are estimated by the most likeli­ hood estimates. These estimated parameters are used to the next updating and repeated until the likelihood of the model is the maximum (Dempster 1977). The frequency distribution of encountered loads shown in Figure 1 is distinguished as the mixed three distributions as shown in Figure 2. Loads classified as severe encountered loads which is distribution 3 are indicated by red color in Figure 2. The severe loads distribution for Panamax Bulk Carriers is the normal distribution with μ3 = 0.587 and σ3 = 0.041, the frac­ tion of severe loads’ distribution, χ3 , is 0.12. That for VLCC is the normal distribution with μ3 = 0.553 and σ3 = 0.044, the fraction of severe loads’ distribution, χ3 , is 0.07.

where χk = fraction of the distribution k; ’k = normal probability function of the distribution k; μk = mean of distribution k; and σk = standard deviation of dis­ tribution k. To identify the three Gaussian mixture model, latent variables representing which distri­ bution each ships’ encounter load belongs to are introduced.

Then, the probability of observing the characteristic value of encountered loads, xi is expressed as follows.

Posterior distribution of the latent variable after obtain­ ing the encountered load {xi } is obtained as follows.

Giving the frequencies N ¼ fni g for X ¼ fxi g, log-likelihood of N and γ is expressed as follows.

Figure 2. Divided frequency distributions of encountered loads for welded joints according to the three Gaussian mixture model.

Although the Panamax bulk carrier encounters a higher percentage of severe loads, there is not much difference in the severe encountered loads. Similarly, the distribution of severe encountered loads for each type of ship was obtained. Although the distributions of severe encountered loads for con­ tainer carriers of TEU < 4,000, general cargo ships, and oil tankers of DWT < 60,000 are relatively small, the similar distributions are obtained for the other type of ships. The main reason of relatively lower severe encountered loads for the three types of ships may be that they operate in coastal areas and do not sail in severe sea conditions very often.

Fraction of the distribution {χk } is estimated as follows.

By giving the initial values of unknown param­ eters, log-likelihood of the model is evaluated, and the fraction of the distribution in equation (9) is

35

The frequency components of the severe encoun­ tered loads identified according to the three Gaussian mixture model for each type of ship were compiled as follows. Subscript 3 refers to the load encountered in routes of severe wave environments. where μs = mean of the logarithmic fatigue life in the stress range S; σ = standard deviation of the logarith­ mic fatigue life. Fatigue damage used for strength assessment in design is defined as the ratio of the number of stress cycles to the reference fatigue life, Nref , in the design S-N curve, which is a deterministic value. Therefore, the distribution of fatigue strength is expressed by the fatigue damage by using the relation μs lnNref ¼ 2σ as follows.

where N SL = integrated frequencies of severe encoun­ tered loads; N SL;j = frequencies of severe encountered loads for ship type j; nj;i = frequency of i-th load for ship type j, n3 j;i = frequency of i-th severe load for ship type j,; ’j;3 = probability distribution of severe encountered load for ship type j. The obtained frequency distribution is shown in Figure 3. The distribution shows very good fitness to a normal distribution. Mean and standard deviation of the severe encountered loads, μSL and σSL , are 0.527 and 0.064 respectively. This distribution repre­ sents the characteristics of severe loads that general merchant ships, regardless of type, encounter in service.

When a member is designed for fatigue design load with criterion of fatigue damage, Dcr ratio of cal­ culated fatigue damage, DDL , to the criterion corres­ ponds to the fatigue strength of 1.0 in the fatigue strength distribution of equation (12). Therefore, the probability distribution of fatigue strength of fatiguedesigned members can be rewritten as the distribution of the ratio of fatigue damage to fatigue damage with NA load, DNA , as follows. This distribution represents the distribution of Capacity of fatigue damage.

where DDL = fatigue damage with fatigue design load; DNA = fatigue damage with NA load. 3.2 Figure 3. Frequency distribution of severe encountered loads regardless of ship type for welded joints obtained by compiling frequency components of severe load of each type of ship, and fitted normal distribution.

Since the distribution of the ratio of the severe encountered load to NA load is obtained by normal distribution, ’SL ðrÞ, shown in Figure 3, the distribu­ tion of the ratio of the fatigue damage to the fatigue damage with NA load, is obtained as follows accord­ ing to the relation of y ¼ rm . This is the distribution of Demand of fatigue damage.

3 PROBABILITY OF FATIGUE DAMAGE 3.1

Safety (failure probability) against encounter loads

Distribution of fatigue strength

In general, fatigue life is a random variable which follows a log-normal distribution in equation (6), and the standard deviation of the logarithmic life is taking a constant value regardless of the stress range. The S-N curve used in design shows the rela­ tionship between the stress range and the fatigue life with 97.7% probability of survival (taking a safety margin of twice the standard deviation of the logarithmic life to the average of the logarith­ mic life).

where ’SL = probability distribution of the ratio � of severe encountered load to NA load, N μSL ; σ2SL . The relationship between Demand and Capacity by means of fatigue damage is conceptually

36

illustrated as shown in Figure 4. Both Demand and Capacity are obtained by the relative values to the fatigue damage with NA load as given by equations (13) and (14). The fatigue damage probability can be evaluated as follows. The fatigue damage probability calculated by the following equation is used to estab­ lish the fatigue design load on the conservative side.

Table 1. Target reliability in terms of allowable annual failure probability to the considered limit state according to the safety class. Safety Classes Limit State

C

Serviceability

10-1 - 10-2

Ultimate Fatigue Accidental

where DD = demand of fatigue damage; DC = capacity of fatigue damage.

-2

10 - 10 10

-3

10

-4

-3

A

10-2 - 10-3 -3

10 - 10

-4

10-3 - 10-4 10-4 - 10-5

10

-4

10-5

10

-5

10-6

corresponds to the safety class ‘B’, and the fatigue strength of ordinary hull structural members corres­ ponds to safety the class ‘C’ (IACS 2019). The allowable damage probability for 25 years design life in the Fatigue limit state (hereafter, FLS) of Safety Class ‘C’, pf ;all , is calculated from the annual allowable damage probability Δpf ;all = 10-3, as follows.

4 FATIGUE DESIGN LOAD 4.1

B

Target reliability

Fatigue crack damage does not directly affect struc­ tural strength or immediately develop into major damage. Even if crack damage occurs, the structural integrity can be maintained by detecting and repair­ ing the damage before it progresses to major damage. On the other hand, a penetrating crack in a compartment barrier can impair the functionality of the compartment. The occurrence of multiple cracks can also affect the ship’s unavailability due to repairs. When considering the target reliability in fatigue design, the target reliability concept for pipelines and offshore structures shown in Table 1 (Bai, 2014) can be referred to. Here, safety class ‘A’ is the case where the occurrence of damage directly leads to loss of human life or a major disaster, safety class ‘B’ is the case where the occurrence of damage may cause loss of human life or a major disaster, and safety class ‘C’ is the case where the occurrence of damage does not affect the loss of life or a major disaster directly. The ultimate strength of a hull in vertical bending

This value is almost equal to the damage probabil­ ity, pf ¼ Fð 2Þ = 0.0228, which is given which is given by the usual design S-N curve. This means that the conventional fatigue design was correspond­ ing to the fatigue design by Safety Class ‘C’ in FLS. Thus, this value is used for Safety Class ‘C’ in FLS. 4.2

Fatigue design loads

The fatigue design load is set as a ratio to the NA load since NA load is used for the other strength design in general, and this ratio is used as the Design Load Factor (hereafter, DLF). When fatigue strength is determined with allowable fatigue damage, Dcr =1 to the fatigue damage calculated by the fatigue

Figure 4. Relationship between Demand and Capacity of fatigue strength of the member which is determined according to the fatigue design load with design criterion.

37

design load, the relationship between the magnitude of used fatigue design load and the damage probabil­ ities evaluated for each severe encountered loads are shown in Figure 5 for the case of welded joints with m = 3. The thin lines in the figure show the damage prob­ ability for each ship type when Demand’s distribu­ tion of fatigue damage is obtained from the encountered load distribution approximated by the three Gaussian mixture model in equation (4), and the red line shows the damage probability when Demand’s distribution of fatigue damage is obtained from the severe encountered loads independent of ship type shown in Figure 3. The red dashed line in the figure indicates the allowable damage probability for the design life of 25 years, pf ;all ¼ Fð 2Þ, which corresponds to the target value for Safety Class ‘C’ of FLS. The blue dashed line indicates the allowable damage probability for Safety Class ‘B’ of FLS, pf,all,B= 2.5×10-3. From the relationship between damage probability for the fatigue design load and the allowable damage prob­ ability, the fatigue design load to achieve the target reliability can be determined.

the fatigue design of liquefied gas containment sys­ tems in the IMO IGC Codes (IMO 1986), which uses different fatigue damage criteria. The fatigue design loads evaluated above are used for fatigue design independent to ship type to achieve at least the target reliability over the 25-year fatigue design life. If the design load is to be changed according to the ship type, the design load can be determined from the relationship between the damage probability for the encountered load and the target fatigue damage probability for each ship type shown by the thin lines in Figure 5. If the fatigue design life is to be longer than 25 years, the allow­ able damage probability should be set by changing 25 in equation (10) to the fatigue design life, and the design load should be the load that can achieve the individual target values. 5 CONCLUSIONS By using hindcast data associated with AIS data, it is possible to determine the sea conditions encountered by general merchant ships during their service. In studying the fatigue strength of the hull structure, since the characteristic value of the long-term fatigue load is proportional to the m-square root of the m-square expected value of the significant wave height in the sea condition along the service routes, the encountered fatigue loads are obtained for each individual ship. It was found that the magnitude of the encoun­ tered fatigue loads varied depending on the route taken by each ship, and that the frequency distribu­ tion exhibited a form of distribution with multiple peaks relating the magnitude of the encountered loads. Using the three Gaussian mixture model, the distribution of severe encountered loads for each ship type could be extracted and found that the dis­ tribution of severe encountered loads was distributed at approximately the same level of magnitude regardless of the ship type. By compiling the severe encountered loads for each ship type, a generalized distribution of severe encountered loads could be obtained and found that the distribution is normal distribution. This fatigue load distribution can be used to evaluate realistic fatigue reliability of a ship for loads encountered in realistic severe wave envir­ onments, which has not been known previously. A formulation of fatigue damage probability was developed for the case where the fatigue strength of a member is determined so that the cumulative fatigue damage at the fatigue design life for a given design load satisfies the allowable fatigue damage. In fatigue design, the magnitude of the design load required to achieve the setting target reliability can be determined specifically. This enables appropriate fatigue strength design. In addition, it is possible to flexibly perform fatigue design to ensure the required fatigue reliability according to the importance of the

Figure 5. Relation between fatigue design load and fatigue damage probability of a welded component designed with fatigue damage criteria of 1.

According to the relationship shown in Figure 5, the fatigue design of welded joints can be made which achieves the target reliability for severe loads regard­ less of the ship type by using the fatigue design load­ with a DLF of 0.56. Although the design load determined in this way may be set on the safe side for general operation, depending on the type of ship, the target reliability is achieved even when the ship is operating in the severe sea conditions. To satisfy the allowable damage probability for Safety Class ‘B’ of FLS, it is necessary to use the design load with a DLF of 0.66 for welded joints. In order to satisfy the allowable damage probability for Safety Class ‘B’ using the design load determined for Safety Class ‘C’, fatigue design can be satisfied by setting the criterion of fatigue damage, Dcr=0.63. These fatigue damage criteria generally correspond to

38

members to be evaluated and the scale of the mem­ bers constituting the structure. However, the fatigue strength and fatigue damage probabilities evaluated in this study are the values for the basic S-N relation of design S-N curve. Uncertainty in stress evaluation used for fatigue assessment is not considered. Since the actual fatigue strength depends on the history of static load changes and the actual working environment, it differs from the actual fatigue reliability. However, it is useful for the study of the design loads necessary to achieve the design goals.

Hersbach, H. et al. 2018. ERA5 hourly data on single levels from 1979 to present. Copernicus Climate Change Service (C3S) Climate Data Store (CDS). Hobbacher, A. 2009. IIW Recommendations for Fatigue Design of Welded Joints and Components. WRC Bul­ letin 520. The Welding Research Council. New York. IACS. 2001. Recommendation No.34. Standard Wave Data. IACS. 2014. Common Structural Rules for Bulk Carriers and Oil Tanker. IACS. 2019. Fatigue Assessment -Uncertainties, Failure Consequences and Benchmark. TB Report, IMO. 1986. International Code for the Construction and Equipment of Ships Carrying Liquefied Gases in Bulk (IGC Code). IMO. 2017. The International Convention for the Safety of Life at Sea. Nippon Kaiji Kyokai (NK). 2021. Rules for the Survey and Construction of Steel Ships. Part C: Hull Construction and Equipment. UK-HSE. 1995. 4th Edition Guidance Notes, Offshore Installations. Guidance on Design, Construction and Certification, Consolidated Edition with Amendment 3. Yamamoto, N. et al. 2020. Study on Fatigue Assessment taking into account Realistic Fatigue Loads. OMAE2020-18248

REFERENCES Ardhuin, F. et al. 2014. IOWAGA - WW3 - HINDCAST Global grid – ECMWF. Bai, Y. et al. 2014. Subsea Pipeline Integrity and Risk Man­ agement. Elsevier Dempster, A. P. et al. 1977. Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistics Society. Series B 39 (1). pp.1–38.

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Wave loads

Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Update of wave statistics standards for classification rules H.N. Austefjord DNV, Oslo, Norway

G. de Hauteclocque Bureau Veritas, Paris, France

M.C. Johnson Lloyds Register, Southampton, UK

T.Y. Zhu Nippon Kaiji Kyokai (ClassNK), Tokyo, Japan

ABSTRACT: Modern classification rules, that are used to assess the safety of hull structure of ships, are heavily based on direct calculations, i.e. numerical simulations. The range of waves that a ship should with­ stand, together with the operational profile (speed and heading), is a crucial input to those calculations and is currently provided in the IACS “Rec. No. 34”. This is used as a basis for the designs of almost all of the world’s commercial shipping. IACS has recently undertaken significant work to update this standard to reflect technical advances and knowledge accumulated over the last decades. An updated wave scatter diagram of wave height and period is now proposed, together with a slightly narrower spectral shape and directional spreading. The recommended heading and speed profiles remain mostly the same. This paper presents the technical justification for those changes.

1 INTRODUCTION

(IMO) has made the public broadcast of ship posi­ tions (Automatic Identification System, AIS) manda­ tory; this was intended as an aid to navigation local to any particular ship, but the aggregation of such records globally has provided an enormously valu­ able dataset. These two developments led IACS to propose a project team in 2018 to consider updating Rec. No. 34, with the idea that the AIS tracking data could be combined with co-located wave model data for the North Atlantic to produce an unbiased wave scatter diagram. The project team has submitted its work to IACS “Rec. No. 34 rev2”; this paper makes public much of the technical work performed in order to establish this updated recommendation. In this paper, the geographical area of review and the sources of ship track and wave model data are first presented. The method to calculate a ‘raw’ scat­ ter diagram is given, together with work underlying the recommendation for spectra shape and spreading. This is followed by a description of a smoothing pro­ cess employed to sanitise the raw scatter diagram. Results are then presented, showing how, for a testing database of 70 different ship types, ship responses (ship motion, accelerations, and wave loads) are expected to change when comparing the new “Rec. No. 34 rev2” with the old “Rec. No. 34

The world’s commercial shipping in global service is designed to structurally withstand a severe wave environment defined by the International Association of Classification Societies (IACS) - namely IACS “Rec. No. 34 rev1” (2001), which provides details of the North Atlantic Ocean, principally in the form of a scatter diagram giving the occurrence statistics of combinations of significant wave height (Hs) and average zero up-crossing wave period (Tz). IACS has faced some criticism of Rec. No. 34 rev1 (2001) because the underlying statistical data ori­ ginates in historical ‘eyeball’ observations from ships. Whilst these data were the best available at the time, studies have shown inaccuracies in human estimates. The effect of weather avoidance is embedded in the data but unquantifiable, and any bias, for example due to fixed shipping routes or ship types, cannot be identi­ fied either. Furthermore, the last observations included date back to 1984, so there is also concern that long term changes since that time are missing. In recent years numerical wave modelling has improved greatly in quality and has also become more readily available to the engineering community. Furthermore, the International Maritime Organization

DOI: 10.1201/9781003399759-5

43

rev1 (2001)”. Finally, known limitations of the approach are identified and discussed. 2 DATA SOURCE 2.1

Wave hindcast

The sources underlying the Rec. No. 34 rev1 scatterdiagram are visual observations from ships, last pub­ lished in 1986 (Hogben 1986). Whilst some correc­ tions were applied, those visual observations have been reported to have limited accuracy, especially concerning wave period (Bitner-Gregersen et al. 1995). Since Rec. No. 34 rev1, significant progress has taken place. Numerical hindcast analyses are nowadays common practice, and several reliable global datasets are publicly available. Based on the analysis of different datasets (de Hauteclocque et al. 2020), the IOWAGA (Integrated Ocean Waves for Geophysical and other Applications) dataset from Ifremer (Institut Français de Recherche pour l’Ex­ ploitation de la Mer) is used in this work (Ardhuin et al. 2011). As the IOWAGA dataset does not store full spectra, it is complemented with ERA5 (ECMWF Reanalysis v5) dataset (Hersbach et al. 2019) from ECMWF (European Centre for MediumRange Weather Forecasts) in Section 5. 2.2

Figure 1. Definition of North-Atlantic area for this work.

the way to this new wave standard. To tackle this, a database of 3D linear seakeeping responses was used. The vessels included are presented in Table 1. For each vessel, 3D BEM linear calculation (Chen 2004) is performed. The transfer functions are output for the quantities listed in Table 2. The RAOs are calcu­ lated with a 5° heading resolution, and are available at four speeds: 0 knots, 5 knots, Froude number of 0.1 and 75% of the service speed.

Table 1. Ship database. Number of ships investigated for each type and loading condition.

Ship position

As Rec. No. 34 is supposed to reflect waves encoun­ tered by ships, it is important to consider realistic combinations of routes and wave data (Eisinger, Bloch & Storhaug 2016) (Miratsu et al. 2019) (Mir­ atsu et al. 2020). The best way to do this is to com­ bine millions of in-voyage locations with individually co-located wave data. This naturally gives a full rep­ resentation of the routing effect in a ‘routed’ scatterdiagram. Voyages of over twenty thousand vessels were established by cleaning and resampling AIS data to the same temporal resolution as the hindcast wave data. The fleet is limited to cargo and passenger vessels longer than 90m. This means most commercial sea­ going ships are included. Excluded are many fishing vessels, offshore vessels, naval ships, and ships oper­ ating at fixed locations e.g. FPSOs. The time period analysed ranged from 2013 to 2020 (seven full years). The collection of voyages was made in the North-Atlantic, as defined by Figure 1. The choice of this area is further discussed in Section 7. Coastal traffic near islands was dis­ carded (~50 nautical miles). 2.3

Ship type

Full

Ballast

Tanker Bulk and Cargo Container-vessel LNG LPG RoRo Passenger-ship Total

16 19 21 5 5 3 5 74

11 16 10 0 0 0 0 37

Table 2.

Type of ship responses included in the dataset.

RAO label

Description

VBM HBM VSF Pitch Acc. Surge Acc. Sway Pressure wl Roll

Vertical Bending Moment amidship Horizontal Bending Moment amidship Vertical Shear Force, aft quarter Pitch motion Surge acceleration Sway acceleration Waterline pressure amidship Roll motion

Database of ship responses

The final objective of the new wave statistics standard is a better long-term prediction of ship responses. It is thus important to have this in mind when evaluating the different assumptions and compromises made on

Furthermore, for each response, a characteristic period Tc is calculated thanks to the available regres­ sion in (de Hauteclocque et al. 2016); the characteris­ tic length L=α from (de Hauteclocque et al. 2016) is

44

converted to period: Tc2 ¼ ð2 � π � L Þ= ðg � αÞ. Note that in this work, the characteristic period is not used for any quantitative derivation; but only to display results in a scale allowing for physical interpretation. Figures 2 and 3 illustrate this responses dataset, showing two RAOs, with very different characteris­ tic periods: the vertical bending moment of a long ship with large Tc , and the horizontal bending moment of a short ship (low Tc ).

period (s), where mn is the spectral moment of order n. 4 STATISTICAL MODEL The previous section introduced the process fol­ lowed by the IACS working group to derive the Rec. No. 34 rev2 scatter-diagram from the combination of vessel tracks and hindcast wave data. The empirical scatter-diagram obtained from AIS and hindcast data is fitted with a statistical model. It smooths out some of the sampling uncertainties, allows the possibility to extrapolate to unobserved wave periods and provides the scatter-diagram in a compact form (the scatter-diagram can be reconstructed at any desired resolution from a few coefficients). The statistical model underlying Table 1 of the Rec. No. 34 rev2 is written as:

Where pH ðHs Þ is the marginal distribution of the significant wave height, and pT0m1 ðT0m1 jHs Þ the con­ ditional distribution of the mean wave period.

Figure 2. VBM RAOs, long ship ship, L = 370m, Tc ¼18s.

4.1

Marginal distribution of Hs

A mixture of Weibull distributions is used to model the marginal distribution. The coefficients (given in Table 3) are determined by MLE (maximum likelyhood estimate) based on the discrete scatter diagram.

Figure 3. HBM RAOs short ship, L = 90.m, Tc =6s. Table 3.

Hs distribution coefficients. Coefficient

3 DISCRETE SCATTER-DIAGRAM α1 ε λ1 α2 λ2 χ

The approach for deriving discrete scatter diagrams with bad weather avoidance is summarised as follows: – Download AIS and IOWAGA hindcast data within the North-Atlantic area for a period of 7 years (from June 1st 2013 to June 30th 2020). – Clean and resample AIS data to 3-hour resolution, including outlier removal and interpolation to fill gaps in the records. – Match each AIS data point to the nearest hindcast data point. – Place encountered significant wave heights (Hs ) and mean wave periods (T0m1 � � ) in 0.1m and 0.1s bins. Here, T0m1 ¼ 2π mm01 is a mean wave

4.2

1.4230 0.9360 1.8150 1.3940 2.8050 0.9499

Conditional model

The conditional mean wave period distribution is modelled as a split generalized normal distribution:

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With

JONSWAP spectrum with gamma = 1.5 and a directional spreading of cos3 . From Figure 4, the Hs of the sea-states contribut­ ing the most to the extreme are, roughly speaking, between, 7.5m and 16.5m. The characteristic period largely explains the variations observed: small Tc are associated with small Hs, large Tc with large Hs. As a simplification, hereafter, sea-states with Hs > 10m are considered as extreme sea-states. The range of sea-states contributing to the fatigue damage are, on the other hand, lower; in the range [3m, 7m] (Figure 5).

The parameters σu , σd and x0 are fitted, for each Hs bin, by MLE. The dependency of those param­ eters with Hs are then fitted with the shapes described in Equation (4), using least-square. Table 4 provides the coefficients thus obtained.

Figure 4. Contribution coefficients for extreme. Table 4. model.

Coefficients for conditional Coefficient

l0 l1 su0 su1 su2 su3 sl0 sl1

5.427251 -0.085340 2.549443 2.435955 0.705177 0.133225 0.018557 1.005918 Figure 5. Contribution coefficients for fatigue loads.

Finally, discretisation is performed into 1m and 1s bins to get the final scatter-diagram. Values in each bin are calculated using midpoints, except for Hs= [0.0m, 1.0m] where exact integration is used. The obtained discretized scatter-diagram is given in Appendix (Figure 19). 4.3

5 SPECTRUM SHAPE In Rec. No. 34 rev1, the spectral shape is a two parameters Pierson-Moskowitz spectrum (equivalent to JONSWAP withγ ¼ 1:0), associated with cos² spreading. In the present studies, analysis of full dir­ ectional spectra from hindcast data has shown that JONSWAP spectrum with γ ¼ 1:5 and a cos3 spread­ ing was more appropriate to represent extreme seastates. Furthermore, this spectral shape provides accurate results for fatigue loads as well. This sec­ tion gives some background justifications. The full spectra data here analysed are from the model ERA5, at a single point located in the North Atlantic, over the period of 25 years (1990-2014).

Contribution coefficients

Using the newly defined scatter-diagram, and work­ ing with the ship database introduced in 2.3, contri­ bution coefficients can be calculated to show the sensitivity to the seas state of both extreme and fatigue loads. The knowledge of those contribution coefficients allows prioritisation of the relevant seastates when simplified assessments are needed. The contribution coefficients are calculated using the response dataset from Section 2.3, assuming a

46

Figure 6 shows the shape of 306 sea-state spectra contributing the most to the 25-years extreme (~Hs > 10m), normalised according to T0m1. The extreme sea states have remarkably constant shape and seem to be well represented by a JONSWAP spectrum with peak­ edness factor γ= 1.5. This value of 1.5 has been obtained by a least-square minimisation. It is also observed that matching T0m1 or Tp provides much better results than Tz (de Hauteclocque & Lasbleis 2022). A slight trend of gamma increasing with Hs was observed; however, it was found that setting gamma as a function of Hs did not significantly improve the overall accuracy of ship responses. For simplicity and practicality, a fixed value gamma=1.5 is then recommended.

Finally, to evaluate the accuracy loss induced by this simple parametrization, a validation is per­ formed with the ship RAO dataset introduced Sec­ tion 2.3 and the ERA5 directional wave dataset introduced earlier in this section. The 25 years extreme values are calculated for all ship responses: - Using full spectra (reference) - Using γ ¼ 1:0 and n = 2 (Rec. No. 34 rev1) - Using γ ¼ 1:5 and n = 3 (Rec. No. 34 rev2) For extreme loads, the Rec. No. 34 rev1 shape results in a 7% quadratic error compared with the benchmark full spectrum case, which is reduced to 5% using Rec. No. 34 rev2 parameters. Fatigue loads (at 10-2 probability as the reference value) are less sensitive to spectrum shape. With the same testing data set, Rec. No. 34 rev1 and Rec. No. 34 rev2 show a quadratic error of 2.7% and 3.2% compared with the benchmark case, respect­ ively. Those errors are considered comparable and acceptable. These findings are confirmed by a similar analysis conducted at several locations (de Hauteclocque & Lasbleis 2022). 6 OPERATIONAL PROFILE The Rec. No. 34 rev1 includes recommendations for how ships are assumed to operate in different sea conditions. Equal probability for all ship headings is specified, and zero speed is assumed when evaluat­ ing extreme wave loads (strength assessment). In this section, using results from the combined AIS-hindcast dataset described in Section, Sec­ tion 2 the probability distributions of ship speeds and relative wave headings are estimated. All types of merchant ships over 90m navigating in the North Atlantic Area shown in Figure 1 are considered. The correlation between speed and heading with significant wave height is complex. In the following, it is assumed that the speed and heading can be investigated separately.

Figure 6. Shapes of contributing spectra (Hs > 10m) and the chosen parameterized spectrum (JONSWAP, γ ¼ 1:5), based on 25 years of hindcast data.

6.1

Heading

Accounting for the AIS-IOWAGA data including the entire range of Hs, we observe that the heading pro­ file is equiprobable, as currently assumed in Rec No. 34 rev1 (Figure 8). Uniform distribution is thus perfectly suited to fatigue calculations. However, looking only at extreme sea-states, the picture is different: beam seas are less likely, as shown in Figure 9. This figure presents the data in North-Atlantic only; it has been checked that using world-wide data which provides a similar picture. Two factors can explain this observation:

Figure 7. Shapes of contributing spectra (Hs>10m) as function of headings, together with the parametrized shape (cos3).

Similarly, Figure 7 shows the directional shape of sea-states contributing to the extreme. As with the frequency shape, the directional spreading is very similar among the different sea-states and well approximated by a cosn formulation with n=3.

– Ship’s captains avoid beam seas in harsh weather, to limit roll motion and increase stability.

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6.2

Speed

In the same fashion, the relationship between ship speed and heading is investigated. From Figure 11 it is observed that speed in head seas reduces signifi­ cantly with wave-height. The two most plausible reasons are: – Voluntary speed reduction to limit ship motions – Involuntary speed reduction due to added resist­ ance in waves. Looking slightly deeper, it appears that the speed reduction is strongly dependent on the relative wave heading. Figure 12 shows the speed reduction for each heading. It appears that the reduction is larger in head sea than in following sea.

Figure 8. Heading histogram, all data.

– Harsh weather happens in locations where routes are mostly east-west, with dominant wave direc­ tion from west. While the first explanation is considered as the main one, the below data cannot distinguish between the two effects. Whatever the cause, the practical effects are the same and evaluated on the ship response database.

Figure 11. Speed versus wave height in head sea, all ships. Figure 9. Heading histogram, Hs > 10m.

Figure 10 shows the relative differences between extreme responses considering the headings equi­ probable, or with the same distribution as Figure 9. A constant speed 5 knots has been assumed for sim­ plicity. The effect is small, and keeping a constant probability for headings thus appears to be a good compromise between simplicity and accuracy.

Figure 12. Average ship speed as function of Hs and rela­ tive wave heading, 0 being following sea and 180 head sea.

To assess the sensitivity of long-term results to assumptions made on vessel speed, long-term calcu­ lations of extreme responses are performed using data from Figure 12, simplified as follows to be com­ patible with the RAO dataset: - 0.75 Vs for 0°, 30° and 330°. - Froude number = 0.1 for 60°, 90°, 270° and 300°. - 5 knots for 120°, 150°, 180°, 210° and 240°.

Figure 10. Response sensitivity to heading distribution.

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The results are then compared with the less refined assumption from the Common Structural Rules for bulk carriers and oil tankers (IACS 2014): fixed 5 knots. The headings are considered equiprob­ able in both cases.

Filtering out ships having low GM with much larger roll motion at forward speed, we obtain Figure 14, which shows that the 5 knots assump­ tion is acceptable. For vessels with low metacentric height and oper­ ating without reduced speed in stern quartering seas appropriate speed and viscous damping need to be applied when evaluating roll related responses by numerical simulations. Furthermore, it is assumed that these effects can be considered in the develop­ ment of rule formulae of roll motions by individual classification society. 6.3

Summary on operational profile

The uniform heading distribution currently used in the IACS Common Structural Rules for bulk carriers and oil tankers (CSR) and in Rec. No. 34 rev1 is thus confirmed and the continued use is justified in Rec. No. 34 rev2. Using 5 knots for vertical shear force and bending moment and for loads in general for oil tankers and bulk carriers covered by CSR rules is mostly conser­ vative. At most 1% non-conservatism is observed for roll motion of bulk carriers and tankers. It is therefore considered acceptable to use 5 knots as basis for vertical shear and bending in unified requirements S11 and S11A as well as loads in gen­ eral for the CSR rules.

Figure 13. Effect of fixed 5 knots assumption.

From Figure 13, it is observed that assuming a 5 knots speed for extreme has small effect on most of the responses. However, large difference may arise for roll motion and related quantities such as pres­ sure on waterline. This is linked to the roll motion which can be large for low GM vessels in sternquartering seas at high speed (IMO 2007). While this is relevant, the discrepancies are overestimated in our calculations, for three reasons:

7 GEOGRAPHICAL AREA AND LOAD LEVEL

– The GMs used in the ship response dataset are a lower bound (full, scantling GM). However, ships tend to operate on average at larger GM. – In the ship response dataset, the roll damping is linear (6%) and does not vary with speed. In real­ ity, lift damping as well as the quadratic effect would attenuate large roll angles at large speed. – The fact that large roll angles are likely for low GM vessels in stern quartering seas is known and oper­ ational guidance are given to avoid those conditions (IMO 2007). The speed/heading/Hs statistics observed from AIS and hindcast do not include GM data, and do not allow this to be accounted for.

IACS Rec. No. 34 is based on wave data obtained from North Atlantic trade as this represents the most severe area ships tend to operate in. The basic idea of using a harsh design wave environment is that ships should not need to have geographical limita­ tions on their operation. Deciding on the exact polygon defining the North Atlantic is a subject for discussion. The wave char­ acteristics are not uniform across the whole basin. Selecting a small area with harsher weather will result in a stricter scatter diagram than if the polygon is expanded to include less severe areas. To under­ stand the consequence of the area selection an assessment of wave load level is made. Hydrodynamic strip theory analyses were per­ formed for 1500 vessels of different ship types and sizes. Each vessel was evaluated at multiple speeds between zero and full forward speed for relative wave directions with 30 degrees spacing. AIS data for the world fleet is, in this study, limited to merchant vessels longer than 90 meters with a minimum of one year of data. The resulting 44000 vessels are matched with the closest hydrodynamic model in terms of vessel type, length, breadth and ser­ vice speed. By matching the AIS data with IOWAGA hindcast data each vessel has a known series of wave height, period, relative direction and speed. This is used in long term response evaluation of the midship

Figure 14. Effect removing ship with high stern quartering roll.

49

vertical bending moment at 1 year return period as well as the 10-2 exceedance probability. Figure 15 presents the ratio between 1-year moment from actual operation normalised by the 25year design moment obtained using the Rec. No. 34 rev2. Figure 16 shows the ratio of 10-2 moment between experienced and design load used for fatigue assessment in the IACS Common Structural Rules for bulk carriers and oil tankers (CSR). Two variants of geographical areas are evaluated, one being somewhat smaller than the one finally selected. The smaller harsher area would slightly increase the 25-year extreme design wave loads, result­ ing in fewer vessels exceeding the 25-year design point. For fatigue the smaller area would easily result in no vessels exceeding the design value, meaning an over conservativism on fatigue loads. The effect of including the highly trafficked Bay of Biscay was con­ sidered but found not to make any difference to the loads, typically less than +/- 0.3% on extreme loads. It is also observed that the different vessel types do not strictly encounter the same sea-states. The option to provide different scatter-diagrams for differ­ ent types of vessels was considered and quickly dis­ carded and considered not practical. On the other hand, the knowledge from Figures 15 and 16 leaves the door open to further development of partial safety factors that would account for this observation.

8 CONSEQUENCE ASSESSMENT

Table 5.

Summary of changes. Rec. No. 34 rev1

Scatter-diagram Spectrum

Visual observation PiersonMoskowitz Cos² P = 10-8

Spreading Extreme definition Heading Uniform distribution Fatigue reference NA

Rec. No. 34 rev2 Hindcast JONSWAP γ= 1.5 Cos3 Return period = 25 years Uniform p=10-2

The changes from Rec. No. 34 rev1 to rev2 are sum­ marized in Table 5. The most significant change is the scatter-diagram itself, which induces relative lower loads. Then, the narrower spectrum and sharp spread­ ing tend to slightly increase the loads respectively. Finally, the new definition of extremes (RP=25 year, vs p ¼ 10 8 ) introduces a tiny reduction of the loads. The combined consequences of those updates are evaluated on the ship response dataset presented in Section 2.3, for extreme, and for fatigue loads. Figure 17 shows the consequences of extreme loads for both Rec. No. 34 rev1 and rev2 on all ship responses. Depending on the vessel and response type, the extreme loads are reduced from 10% to 30%. The characteristic period alone explains most of the variation: the reduction is relatively higher for responses with low characteristic period. Hence, for extreme loads, the new recommendation is – rela­ tively – more favorable to short vessels.

Figure 15. Exceedance rate of 25-year extreme design moment per year.

Figure 16. Exceedance rate of 10-2 fatigue design moment. Figure 17. Consequences on extreme loads.

This study gives an early idea of the level of the design load compared to what the world fleet experi­ ences and may act as input when the final safety level including both load and capacity will be calibrated by IACS.

Figure 18 shows the consequence of fatigue loads, evaluated at p = 10-2 for both Rec. No. 34 rev1 and rev2 on all ship responses. Compared to Rec. No. 34

50

IACS Rec. No. 34 rev2 scatter diagram does include some future-proofing.

rev1, the fatigue loads are significantly reduced in average, with reduction from -5% to -50%. As for extreme loads, the characteristic period of the response explains for most of the variation. On the other hand, this time, the Rec. No. 34 rev2 is, relatively, more favorable to long responses (i.e. long vessels).

9.3

9.4

9 LIMITATIONS Whilst the studies, techniques and data used by IACS to contribute to the up-issue of Rec. No. 34 are, at the moment of writing, considered state-ofthe-art, there are known limitations. These are high­ lighted here. Wave models

IACS Rec. No. 34 rev2 relies heavily on numerical hindcast data. Although those have been validated through comparison with buoy data and satellite altimeters some uncertainties remain. Wave model­ ling is an active academic field and the accuracy of the global wave models is expected to continue to improve year on year. 9.2

Statistics

Synchronised weather data with ship position was limited to only 7 years. This was compen­ sated by the fact that a huge number of ship positions was used, roughly 4500 ship-years, and that these later years were among the roughest recorded. The amount of data used is considered sufficient to correctly assess the 25 years ship responses, though this limitation is to be kept in mind when using the proposed scatter-diagram to estimate ship response at very low probabilities (i.e. very large return period). Even so, the new scatter diagrams are considered a huge improve­ ment on Rec. No. 34 rev1 derived from eyeball observations. Finally, the industry standard design approach that uses scatter-diagram is itself an approximation. By grouping time-series data into Hs-T0m1 bins, the serial correlation of sea-states is lost and can result in an overestimation bias (Mackay et al. 2021). Up to +5% conservatism on VBM is possible for large vessels.

Figure 18. Consequences on fatigue loads.

9.1

Bad weather avoidance

The bad-weather avoidance embedded within this work represents the current performance level of global shipping. The technical quality, availability and take-up of routing services is increasing under current industry drive towards digitalisation. Therefore, the new recommendation might be regarded as including a slightly conservative bias as time goes on and those improvements become more definite.

10 CONCLUSIONS A new wave standard is defined using state-of-the-art wave data sources combined with a ship position dataset. The wave scatter-diagram is significantly modified and includes the effect of bad-weather avoidance. Furthermore, the spectrum and spreading shapes are slightly narrower than in Rec. No. 34 rev1. The change of wave loads is not homogenous: it depends on the type of loads and the type and size of the vessels. It thus, theoretically, optimises how the steel is distributed on a vessel, and across the fleet. While Rec. No. 34 is an important document, it is only one piece acting as input to rule development. The average wave load reduction observed here will not necessarily translate directly into a reduction of the scantling. For instance, the current IACS unified requirement S11A for container vessels considers a routing factor to correct for the fact that Rec. No. 34 rev1 does not account properly for bad wea­ ther avoidance; this factor shall thus be adjusted when accounting for Rec. No. 34 rev2.

Climate change

The updated wave environment recommendations proposed by IACS are a present day snapshot and do not include any climate forecast change effects. The working group reviewed the work of the Intergov­ ernmental Panel on Climate Change (IPCC), and found that there was a great deal of uncertainty about the effects relevant to shipping. However, even changes at the highest end of IPCC projections of +/- 0.5m (positive or negative) in extreme wave heights for the North Atlantic would be expected to have negligible effect on the IACS Rec. No. 34 rev2 scatter diagram due to the robustness of the deriv­ ation procedure. Furthermore, ships in service will continue to avoid rough weather at the levels encap­ sulated in the new scatter diagram. In effect the

51

Further work is thus ongoing within IACS to update downstream documents, such as IACS URS11, IACS URS11A and the Common Structural Rules for bulk carriers and oil tankers (IACS 2014).

Mackay, E, de Hauteclocque, G, Vanem, E & Jonathan, P 2021, ‘The Effect of Serial Correlation in Environmental Conditions on Estimates of Extreme Events’, Ocean Engin­ eering, vol 242, p. 110092, viewed 8 November 2021. Miratsu, R, Fukui, T, Matsumoto, T & Zhu, T 2019, Quan­ titative Evaluation of Ship Operational Effect in Actu­ ally Encountered Sea States. https://doi.org/10.1115/ OMAE2019-95121 Miratsu, R, Fukui, T, Matsumoto, T & Zhu, T 2020, Study on Ship Operational Effect for Defining Design Values on Ship Motion and Loads in North Atlantic, https://doi. org/10.1115/OMAE2020-18193

ACKNOWLEDGEMENTS The work reported here was sponsored by the Inter­ national Association of Classification Societies. The authors also wish to acknowledge the contributions to this work of Dr. Norio Yamamoto, ClassNK, Quentin Derbanne, Marine Lasbleis, BV, Dr. Eivind Ruth, Dr. Elzbieta Maria Bitner-Gregersen, Dr. Tormod R. Landet, all three of DNV, Dr. Zhenhong Wang, LR.

APPENDIX

REFERENCES Ardhuin, F, Hanafin, J, Quilfen, Y, Chapron, B, Queffeulou, P & Obrebski, M 2011, ‘Calibration of the IOWAGA Global Wave Hindcast (1991–2011) Using ECMWF and CFSR Winds.’, 12th International Work­ shop on Wave Hindcasting and Forecasting, Kohala Coast, Hawai’i, HI, 2011., viewed 15 December 2020. Bitner-Gregersen, EM, Cramer, EH & Korbijn, F 1995, ‘Environmental Description For Long-Term Load Response of Ship Structures’, The Fifth International Offshore and Polar Engineering Conference, OnePetro, viewed 13 January 2021. Chen, XB 2004, ‘Hydrodynamics in Offshore and Naval Applications - Part I’, Keynote lecture of 6th Intl. Conf. HydroDynamics, Perth (Australia). de Hauteclocque, G & Lasbleis, M 2022, ‘Extreme Seast­ ate Parametrization and Its Consequences on Ship Responses.’, Proceedings of PRADS2022, Dubrovnik, Croatia. de Hauteclocque, G, Monroy, C, Bigot, F & Derbanne, Q 2016, ‘New Rules for Container-Ships - Simplified For­ mulae for Wave Loads.’, 13th International Symposium on Practical Design of Ships and Other Floating Struc­ tures (PRADS). de Hauteclocque, G, Zhu, T, Johnson, M, Austefjord, H & Bitner-Gregersen, E 2020, ‘Assessment of Global Wave Datasets for Long Term Response of Ships’, OMAE2020, Volume 2A: Structures, Safety, and Reli­ ability, viewed 16 April 2021. Eisinger, E, Bloch, H & Storhaug, G 2016, ‘A Method for Describing Ocean Environments for Ship Assessment’, Proc. 6th International Maritime Conference on Design for Safety., Hamburg. Hersbach, H, Bell, B, Berrisford, P, Horányi, A, Sabater, JMTN, Nicolas, J, Radu, R, Schepers, D, Simmons, A, Soci, C & Dee, D 2019, ‘Global Reanaly­ sis: Goodbye ERA-Interim, Hello ERA5ʹ, ECMWF Newsletter No. 159, pp. 17–24. Hogben, N 1986, ‘Global Wave Statistics’, British Mari­ time Technology. IACS 2014, ‘Common Structural Rules for Bulk Carriers and Oil Tankers’, Manual. IMO 2007, ‘Revised Guidance to the Master for Avoiding Dangerous Situations in Adverse Weather and Sea Conditions.’

Figure 19. Rec 34 rev2 scatter diagram.

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Different strategies to improve isochrone voyage optimization algorithm Yuhan Chen & Wengang Mao Department of Mechanics and Maritime Sciences, Chalmers University of Technology, Gothenburg, Sweden

Chi Zhang National Center for Waterborne Transport Safety, Wuhan University of Technology, Wuhan Hubei, China

ABSTRACT: Voyage optimization can be essential in ship routing for autonomous and intelligent operations. Various optimization algorithms were proposed to search for optimal ship routing with minimized fuel con­ sumption and accurate arrival time. In this paper, the Isochrone method, which is well-known for its robustness and efficiency, is further improved by different strategies to inherit the original capability of computation effi­ ciency, and overcome its incompetence of multi-objective optimization, reliable convergence for approaching destination and feasibility for practical navigation. Especially an innovative way of searching at a late stage is proposed, as well as a flexible evaluation of cost functions according to different optimization objectives. The improved Isochrone method can fast optimize a shipping route with multi-objective purposes, such as accurate expected time of arrival and minimum fuel consumption. The effectiveness and efficiency of different improved Isochrone methods are verified by several North Atlantic voyages collected by a chemical tanker.

as in Chen (1978), de Wit (1990) and Calvert et al. (1991) to optimize a ship’s route/path with fixed speed or power setting. In the static grid methods, if a ship’s speed/power is configured to vary along the voyage, it is recognized as 3D voyage optimization method. For example, the 2D dynamic programming method was further developed by Shao et al. (2012), Zaccone et al. (2017 & 2018), and Wang et al. (2019) to allow vol­ untary speed reduction to solve three-dimensional voyage optimization problems. However, voyage optimization should consider the balance between computation efficiency and the effectiveness of optimization algorithms for practical voyage planning. Sophisticated methods such as 3D deterministic and stochastic optimization algorithms are generally too complicated to solve voyage opti­ mization problems in a reasonable time for arbitrary voyages because of many variables and their ambiguous dependencies. According to the market survey by Simonsen et al. (2015), shipping compan­ ies would like to complete a voyage optimization in less than 1 minute, preferably within 15 seconds. Concerning the fact that large amounts of computa­ tion efforts are spent in extracting sea environmental data for all discretized waypoints (spatial and tem­ poral) during the optimization process, in this paper, the original Isochrone voyage optimization method is further developed to not only overcome the cons in the original Isochrone method but also provide more robust optimization results in a computation

1 INTRODUCTION A voyage planning system as an essential solution for e-navigation can be implemented to reduce air emis­ sions from shipping (DNV 2014) and assist autono­ mous ship navigation. A proper optimization algorithm is a crucial component of a ship’s voyage planning to achieve specific predefined objectives, such as minimum fuel cost, accurate time of arrival, etc. (Wang et al. 2019). The voyage optimization algo­ rithms can be categorized into deterministic algorithms and stochastic algorithms (Wang et al. 2021). For the voyage planning systems available in the shipping market, deterministic algorithms are widely used due to their efficiency of optimization. In those methods, a ship’s voyage is divided into several time stages. Dependent on how the searching area along the voyage in time is discretized, the optimization algo­ rithms can be divided into dynamic and static gridbased voyage optimizations. The isochrone ideas first proposed by James (1957) and implemented by Hagi­ wara (1989) for minimum time voyage and the Divid­ ing RECTangles algorithm, seen Larsson et al. (2015), are two typical dynamic grid searching methods. The static grid-based methods have predefined searching waypoints to choose from. If a ship’s speed is fixed along the voyage, the methods are recognized as 2D voyage optimization methods, such as the dynamic programming method proposed by Bellsman (1952) and utilized to develop voyage optimization systems

DOI: 10.1201/9781003399759-6

53

efficient manner. The effectiveness and efficiency of the improved method will be verified by full-scale measurement data from a chemical tanker.

(2) Simulate a ship departing from the departure loca­ tion X0=[x0, y0] at start time T0 to the destination location Xf=[xf, yf] at a constant speed, where x and y represent the longitude and latitude of the waypoints of the sailing area. The ship can follow headings Cref ± j·∆C (j= 0, 1, …, m) navigating distance ∆t·v in the first step, where Cref is the course of reference great cycle route connecting X0 to Xf, v is the navigation speed, and ∆C is the increment of heading. The first Isochrone is defined. The potential arrival waypoints at time T0 + ∆t is represented by X1(i)(i = 1, 2, …, 2k). The waypoint set {X1(i)} defines the first Isochrone. The engine power and fuel consumption for arriv­ ing at waypoints in {X1(i)} can be calculated and stored according to environmental data and ship performance model based on different operational and environmental conditions encountered at dif­ ferent waypoints associated with each time stage. (3) Define a set of sub-sectors by plotting great circle lines from departure location X0 to Xf, fol­ lowing Cref ± s·∆S1 (s= 0, 1, …, k), as shown in Figure 1, and the definition of ∆S1 is given in Section 3.2.1. In each sub-sector, only the way­ point X1(i) with minimum cost is preserved as the potential waypoint for the next Isochrone. In this paper, the function of cost evaluation is designed differently in different improvement approaches. The number of potential arrival way­ points in the Isochrone can be a constant con­ trolled by a specific parameter k instead of exponentially as the isochrone development advances. (4) Repeating procedures (2) and (3) until the min­ imum geographical distance between waypoints in the current Isochrone {Xn(i)} to the destination is larger than half the geographical distance between X0 and Xf. Regenerate the sub-sectors following Cinv ± s·∆Sn (s= 0, 1, …, k), where Cinv is the back azimuth course angle of great cycle route connecting Xf to X0. The reason for this regeneration is defining boundaries that routes generated inside do not have sharp turn­ ings. Connect the waypoints {Xn(i)} in the cur­ rent Isochrone to the destination point Xf using the great cycle lines. The back azimuth course angle of these great cycle lines is the new refer­ ence heading Cni for every waypoint {Xn(i)} in the current Isochrone. Thus, the ship in the ith waypoint of the nth Isochrone can follow head­ ings Cni ± j·∆C (j= 0, 1, …, m). The potential arrival waypoints at time T0 +n*∆t is Xn(i) (i = 1, 2, …, 2k). Improvements from different methods are all made within this part of the voyage, which will be presented in Section 3.2. (5) When the geographical distance between any waypoint in the Isochrone {Xn(i)} and destination Xf is less than a certain number, which in this paper is chosen to be sailing time left less than 24 hours, ∆C, the increment of heading during

2 OVERVIEW OF ORIGINAL ISOCHRONE ALGORITHM FOR VOYAGE OPTIMIZATION For a ship voyage planning system, some sailing constraints, e.g., land avoidance, no-go zones, traffic separation scheme, etc., should be defined first. Then, the core elements of such a system are optimization algorithms. This study aims to develop original Iso­ chrone method further to allow for fast and practical voyage planning. The Isochrones mean lines that the ship can reach with equal sailing time. The Isochrone voyage optimization method was initially proposed by James (1957) for manual use by navigators to help ships reach a destination as soon as possible (or on the exact estimated time of arrival (ETA)). For each time stage in the method, if all generated way­ points are kept on the next stage of Isochrone, the total number of potential waypoints will grow expo­ nentially. Overcoming the “curse of dimension” has been the motivation for researchers to improve the Isochrone method for more practical use. 2.1

Modified isochrone voyage optimization

The modified Isochrone algorithm introduced the sub-sector to only choose the good waypoints on the current stage for each sub-sector ΔD (Hagiwara 1989). The principal theory of this improved Iso­ chrone method is shown in Figure 1. To meet the requirement of fast routing planning, ship speed is assumed constant during the voyage unless encoun­ tering harsh weather conditions. The Isochrone method can be conducted as follows:

Figure 1. Definition of sub-sectors along the voyage.

(1) Determine the start point X0 and destination Xf as illustrated in Figure 1, and plot the great cycle line as a reference route. Determine time stage interval Δt; MetOcean data encountered in every time stage could be easily extracted. It can facili­ tate very fast estimation during the voyage opti­ mization process.

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voyage would be decreased to 10%~50%. This is because when approaching the surrounding des­ tination area, the width of reversed sub-sectors also narrows down significantly. Maintaining the normal increment of the heading angle will easily make successors of one node domain and others all in every sub-sector. (6) When the geographical distance between any waypoint in the Isochrone {Xn(i)} and destination Xf is less than ∆t·v, connect the waypoints in this Isochrone to destination location Xf using a great circle line. 2.2

•

Parameter sensitivity in isochrone methods •

An isochrone can be defined as a set of connected points that a ship can reach within constant sailing time, starting from one point and going in all pos­ sible headings. These isochrones are generated from each point of the previous set. The length of the line between the previous point and each point on the Isochrone depends on ship speed since sailing time is fixed. Furthermore, the shipping speed is depend­ ent on weather factors and engine setup, etc. In this method, there will be five essential parameters to be well specified and presented in Figure 2.

•

•

sections. Too large values of ∆C may lead to locally optimized predecessor waypoints. Many other candidate waypoints in nearby sub-sectors may be ruled out, leading to partly repeatable paths. m: number of successor waypoints for each of the current waypoints in the current stage, corres­ ponding to the number of spreading headings. It influences the expanding speed of searching way­ points in earlier time stages and helps to generate more feasible candidate paths by using a larger value of m. But if m becomes too large, the searching performance might be significantly reduced because a great computation effort is needed to search big areas/waypoints. ∆t: traveling time between two adjacent time stages. It controls the looseness of the searching grid along the direction toward the destination. Large ∆t may lead to slow converging of optimal searching towards the destination, and avoid too many overlapped candidate routes. However, it may cause sharp path turning in the candidate routes. ∆D: width of the searching limit within each local sub-sector, as shown in Figure 2. The width limit defines the searching grid along the voyage. Same as ∆t, small ∆D defines a narrow searching range perpendicular to the reference route. Small ∆D may cause overlapped sub-paths, while large ∆D leads to a sharp turning of the candidate route toward the destination. k: number of sub-sectors. Since each sub-sector preserves one optimal waypoint for each time stage, increasing k means that the number of way­ points in each stage of Isochrone will increase too. Similar to m, large k may improve perform­ ance and may also cause sharp turning and increase computational effort during the opti­ mization process.

All those parameters should be chosen within a reasonable range to allow for efficient voyage opti­ mization. They also depend on the length of the voyage and weather dynamics. Increasing the values of those parameters may have a positive impact on optimization results, but a trade-off between per­ formance and computational efficiency should be defined. In the original Isochrone method, a reference path is chosen for the discretization, and the reference path can be either the shortest route (the great circle route) or a typical sailing route. The optimization algorithm searches the next sub-routes with a series of headings C±j·∆C around the reference route as in Figure 2. After setting the basic configuration to discretize a ship’s sailing area in both space and time, some searching criteria should be defined to limit the number of candidate waypoints to proceed within the voyage optimization process. Otherwise, the number of possible waypoints (to form sub-routes) will increase exponentially as the evolution of time stages.

Figure 2. The graphic interpretation of Isochrone method.

They have a significant impact on voyage opti­ mization results because they control the generation of waypoints in the search grid. The algorithm should be able to generate a certain number of candi­ date paths that cover potential sailing areas. The fol­ lowing parameters are used to define search area and waypoints by the Isochrone method, • ∆C: the increment of heading angles between two adjacent sub-path from each of the current “opti­ mal” waypoints at each time stage. It can influ­ ence the search area ahead of the current waypoint. For example, if it is set to larger, gener­ ated searching waypoint in the next time stage will expand much faster in width and reach the width limit sooner. In addition, generated way­ points can easily spread among more sub-

55

To reduce the point number, Hagiwara (1989) devel­ oped k parallel lines of equal spacing ΔD on both sides of the reference ship route to form 2k sub-sectors. A ship is supposed to move within the area with a width of 2k � ΔD. At each time stage, only the waypoint closest in the distance to the destination is selected on each sub-sector to compose the next Iso­ chrone/time stage.

Figure 4. Concepts for different improvement strategies of Isochrone method for voyage optimization.

Figure 3. Potential routes using original Isochrone method proposed by Hagiwara (1989).

where dn is the expected travel distance in nth stages of isochrone waypoints set {Xn(i) (i = 1, 2, …, 2k)} after n�Δt hours:

The modified Isochrone method (Hagiwara 1989) has apparent shortcomings for ship voyage optimiza­ tion. For example, sub-routes in the last time stage have a very wide searching range, as seen in Figure 3. Most of those candidate routes with sharp turning in are not realistic for practical voyage plan­ ning. Several strategies to improve the original Iso­ chrone algorithms are investigated in this study and presented in the following sections.

Table 1. Methods used in different stages of a voyage in strategies. Second half voyage Name of First half improvement voyage Reversed Sub-Sec More Nodes

3 STRATEGIES FOR IMPROVEMENT Two methods to improve the original Isochrone method are investigated in this study, i.e., by modify­ ing the generation of sub-sectors, and by better defining the optimization criteria and cost function to choose optimal waypoints in each individual subsector. A flowchart for different approaches to improvement of Isochrone method studied in this paper is presented in Figure 4. For the improvement of the modified Isochrone method, the optimization process is divided into two parts from the departure to the destination. In the first half voyage, the ori­ ginal Isochrone method, as in Hagiwara (1989), is implemented. Then five strategies are used to opti­ mize the results for the second half of the voyage, and their concepts are presented in Figure 4. The improvement strategies and used methods are briefly summarized in Table 1, which will be presented in detail in the following subsections. 3.1

Subsector

Opti Power Original Isochrone A* Isochrone Half Dijkstra

Reversed SubSector

Searching method Original Isochrone Reserve more nodes Optimal power greedy search Additional heuristic function Dijkstra

Final Stage

∆C = ∆C *10% ~50%

in which Vs is the ship service speed fixed during the voyage optimization; ΔD is the local sub-sector width (i.e., resolution of the Isochrone). Note that ΔSn represents the heading angle range in the nth stage for each sub-sector, while ΔD, the parameter predefined as the width of the local subsector, indicates the width limit for each sub-sector in the distance. This equation indicates that the

Reversed subsector

In original Isochrone by Hagiwara (1989), the subsector is defined as:

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width ∆Sn is a monotonically increasing function of traveled distance dn. Therefore, in the last half of the voyage, if replacing traveled distance dn with distance from the destination (DNS) to calculate the width and plot from destination Xf, a reversed and symmetric subsector set can be generated, which gradually shrinks its range when approaching the end:

that establishes an evaluation function that involves both forward and backward cost estimation along the path:

where g(n) is the cost accumulated from departure, and h(n) is the heuristic function which is problemspecific and estimates the cost from the current node to the destination. In the original isochrone method, the cost function used in evaluating a node’s cost is the fuel consump­ tion from departure or current distance to the destin­ ation. In this approach, referring to A*, the cost function is extended with a heuristic term h(n), and f(n) accordingly becomes the overall fuel consumption for the whole voyage, g(n) is the fuel consumption from departure, and h(n) is the fuel consumption needed to reach the destination by estimation through ship per­ formance model, basing on the weather forecast.

where dtotal is the travel distance from X0 to Xf. In this method, the cost function is chosen to be the shortest distance to the destination. 3.2

More nodes preserved

One problem caused by the reversed sub-sector is, as the sub-sector width shrinks towards the destination, some nodes in the current stage with the lowest cost will quickly rule out other candidates and domain all the nearby sub-sectors; therefore, paths later devel­ oped will all derive from its successors. In the result­ ing path set, too many paths are overlapped, especially in the first half of the voyage, which can be considered as some early stages are trapped in a local minimization. This problem was addressed by the following procedures:

3.5

In the original isochrone algorithm, all nodes are con­ nected with a directional grid, which means the edges between nodes are a one-way route, pointing from the previous generation to the next. If a node is removed from the grid, all its predecessors will be automatically deleted since they will no longer form a complete path from departure to destination. This is one of the reasons the resulting path set easily contains over­ lapped paths. Dijkstra is the graph searching algorithm where A* extends from, which is also widely used because of its efficiency and little complexity. Different from Iso­ chrone, its network is undirected, and there is no bind­ ing between nodes. Therefore, in this approach, to avoid too many overlapped results, in the latter half of the voyage, a grid is developed based on the middle stage isochrone waypoints, as illustrated in Figure 5 below:

(1) The number of nodes/waypoints preserved in each sub-sector is increased and controlled by an extra parameter to prevent the waypoints from growing exponentially. (2) Every predecessor node is only allowed to keep a limited number of its successors, avoiding its domination. (3) This gives other candidate nodes chances to sur­ vive for a longer time and, to some extent, pre­ vent being denied because of temporary suboptimality. 3.3

Half Dijkstra isochrone

Optimal power search

Another way to generate more candidate paths is, in the latter half of the voyage, node searching is replaced by a greedy algorithm: remove the restric­ tion of sub-sector, every waypoint expands following the heading Cni ± j·∆C (j= 0, 1, …m), keeps the node with lowest fuel consumption and proceeds until reaching the destination. 3.4

A* isochrone

Alternatively, A* algorithm can be used to optimize the voyage’s second half. A* is a graph-searching algorithm used widely in many fields which has a competitive performance in optimality and compu­ tational efficiency. It is an informed search algorithm

Figure 5. Node grid generated in the latter half of voyage to use Dijkstra algorithm.

57

After generating this grid, assign the weight of the edge based on the fuel consumption estimation through the ship performance model according to weather conditions and traveling time. Search for a path to destination starting from each waypoint in the middle Isochrone using Dijkstra algorithm, with a cost function evaluating fuel consumption from departure to current position. These candidate paths obtained, however, would possess a different time of arrival since the binding between nodes is removed, and the length of edges between nodes, therefore, varies while traveling speed most of the time remains constant. In this approach, the optimal path then becomes the one with the closest time of arrival.

Figure 6. Case study voyages for comparison.

waypoints in steps, and nodes in each step are bound like a chain. The second half voyage optimization is not independent of the first half; therefore, they should be presented together as a complete route.

4 COMPARISON OF VARIOUS IMPROVED ISOCHRONE OPTIMIZATION METHODS 4.1

Case study of ship voyages

4.2

A chemical tanker with full-scale measurement of 2 voyages is used as a case study to compare the cap­ ability of voyage planning by the improved Isochrone optimization methods. The case study voyages are chosen for the ship with main particulars, as in Table 2, sailing in North Atlantic. A conventional weather routing system was installed on the ship to provide guidance on its sail planning. Combined with the ship master’s experience, the actual sailing routes are supposed to be more efficient than ordinary voyage planning systems. The case study voyages were measured in winter 2016, shown in Figure 6. For ship voyage optimization, the sailing time and fuel cost along candidate sub-paths should be estimated to select optimal waypoints. The estimation requires inputs of all encountered MetOcean environments (wind, wave, and current) around the searching area and the ship performance model that describes the ship’s speed and fuel consumption relationship in terms of encountered MetOcean conditions. In this study, all related MetOcean parameters used in the case study are extracted from ECMWF ERA-5 (2019) dataset, such as wind (wind speeds and directions) and wave information (wave height and period), and from http://marine.copernicus.eu/ (Copernicus 2019) server for current information. Furthermore, the ship energy performance model developed by Lang et al. (2020) is used for voyage optimization.

Table 2.

Results of optimization for a westbound voyage

A westbound case, Voyage 20161108, planned by a weather routing system from the shipping market, as well as original isochrone voyage optimization by Hagiwara (1989) are studied. Since the optimization result is sensitive to the parameter settings of iso­ chrone methods, for Voyage20161108 case study, the value of each parameter is listed in Table 3, and the details of each optimized voyage obtained by differ­ ent approaches are presented in Figure 7 and Figure 8. In Table 4, for each voyage, estimated time arrival, fuel consumption and travel distance in total are listed accordingly.

Principal particulars of the chemical tanker ship.

Length Loa Length Lpp Beam B Depth

178.4 m 174.8 m 32.2 m 17.0 m

Design draught Block coefficient Deadweight

10.98 m 0.8005 50752 t

Figure 7. Sailing speeds and encountered Hs along the routes.

It can be seen from Figure 7 that the weather con­ dition Voyage20161108 encountered is relatively calm, and Hs (significant wave height) is lower than

Results of case studies are based on the whole voyage, since Isochrone algorithm generates

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3 meters along the voyage. The optimal voyage gen­ erated by reversed sub-sector, reserving more nodes and half Dijkstra are similar in most of the part, only deviating from each other around the end stage; therefore, their traveling distance are close as well, from Table 4. It indicates that reserving more candi­ date nodes in each stage does not really influence the result since the node number is the only changing factor compared with reversed sub-sector. The reason why half Dijkstra is not improving could be that the first and second half planning is separated from each other as well as the cost evaluation func­ tion; generally, this method is initialized and restarted again in the middle stage isochrone. There­ fore, trying to combine these two parts could be a possible consideration for further improvement. The optimal power searching approach, however, shows the highest fuel consumption. The greedy algorithm it refers based on the assumption that the optimal solution obtained by all sub-tasks it divides into can lead to a global optimization in the end. However, in the isochrone algorithm, from the result, this assumption may be proven not to stand.

Table 3.

Table 4. Results of the proposed voyage optimization algorithm.

m

Δt½h�

ΔD

k

2

20

6

7

20

ETA[h]

Fuel [ton]

Distance [km]

Actual route Original Isochrone Reversed Sub-Sec More Node Opti Power A* Isochrone Half Dijkstra

164.25 164.30 164.77 164.39 164.66 164.54 164.74

117.722 118.641 122.715 120.307 144.130 116.110 118.570

3877.45 3812.11 3895.98 3894.16 4185.09 3909.47 3902.37

A* isochrone, on the other hand, in this case, shows the best performance. From the route, it is also the one close to the actual voyage. They all head south toward the destination instead of directly following the great circle route, and the reason would be the weather condition which can be proved by MetOcean data. As the assumption of Isochrone, the sailing speed remains constant unless an extreme condition is encountered; therefore, the speed of all methods is similar to straight lines. The main difference appears in the area at around -30°W longitude, the middle stage of the voyage, and is mainly caused by Hs. The weather condition at this time is presented in Figure 9, which gives the reason this approach stands out from others. The voyage A* isochrone generated meets the lowest Hs in the middle, and it is the only one sails the same direction as the actual route, which is operated manually. Besides this, during the whole voyage, it encounters a relatively steadier sea state. This shows that A* iso­ chrone has the capability to optimize voyage consid­ ering dynamic weather conditions, with the objective of energy efficiency.

Parameter of isochrone algorithm.

ΔC [degrees]

Category

4.3

Results of optimization for an eastbound voyage

The optimization results for an eastbound case, Voyage20160229, are presented in this section. In this case, values of each parameter keep the same as the westbound case, listed in Table 3, and details of optimized voyages obtained by different approaches are presented in Figure 10 and Figure 12. Weather conditions at the time when voyages behave most differently are presented in Figure 11. Estimated time arrival, fuel consumption, and travel distance are listed in Table 5. In this eastbound case, the encountered weather conditions were rougher. From Figure 12, Hs could even reach 10 meters during the actual voyage. Opti­ mal paths generated all head north to avoid extreme weather and then turn around. It is noticed that all generated voyage deviates from the actual route, and the reason could be the actual route is sailing much faster at the beginning, and the speed is not constant; thus, its weather condition is different all along, which consequently leads to different choices.

Figure 8. The optimized voyage from different approaches.

Figure 9. Optimized voyage with the weather condition in the middle stage of the voyage.

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Figure 10. The approaches.

optimized

voyage

from

nodes is similar to reversed subsector and slightly improves energy efficiency. A* isochrone gives the lowest fuel consumption voyage among the rest with a smoother turning. Meanwhile, the voyage from half Dijkstra deviates from others in the middle of the voyage and gives a lower fuel consumption, which might indicate this method has the capability to explore voyages differently than isochrone types. For this case, all strategies can improve Isochrone method to not only generate a smoother route but also meet the objective of energy efficiency and accurate ETA. A* isochrone has the best perform­ ance among them, saving approximately 14.4% fuel compared to the actual voyage case.

different

Table 5. Results of the proposed voyage optimization algorithm. Category

ETA[h]

Fuel [ton]

Distance [km]

Actual route Original Isochrone Reversed Sub-Sec More Node Opti Power A* Isochrone Half Dijkstra

159.00 159.84 160.44 158.90 159.66 158.02 157.17

173.719 175.522 161.509 155.883 166.157 147.546 148.024

3624.91 3757.49 3569.50 3684.01 3822.34 3583.96 3507.99

Figure 11. Optimized voyage with the weather condition in the middle stage of the voyage.

5 CONCLUSIONS Different strategies to improve the original Isochrone method for voyage optimization have been investi­ gated to design a ship’s sailing route in terms of vari­ ous objectives, such as minimum fuel consumption and accurate ETA. Five parameters significantly impact the voyage optimization results by the Iso­ chrone methods. Different strategies to improve the Isochrone algorithm are proposed, and their effi­ ciency is compared based on two case study voyages. The parameters within each improved algorithm are kept the same to make a comparison. It should be noted that those parameters should be adjusted for different improvement strategies, to achieve the best optimization results. It requires further study with possible quantified and formulated instructions. For the two case study voyages used for compari­ son, optimal power searching is not able to provide a globally optimized solution, and reserving more nodes is shown with slight improvement, together with using reversed sub-sector. Half Dijkstra behaves differently from other methods, and potential improve­ ment could be considered in better combining the Iso­ chrone and Dijkstra used in separate parts. A* isochrone optimization algorithm gives more reliable optimal voyage planning results. Moreover, to keep the efficiency of Isochrone algorithms, the computa­ tional cost of all purposed strategies does not increase significantly, and remains the same level as original

Figure 12. Sailing speeds and encountered Hs along the routes.

The optimal voyage from original Isochrone leads to a sharp turning. As results in Table 5, the optimal power strategy generates a route winding around the destination and therefore has a higher fuel consump­ tion. The performance of keeping more candidate

60

algorithm, which is within minutes. However, there is still potential to further improve it after the strategy with the best performance is determined. Also, it can be further investigated with more voyage cases in vari­ ous weather conditions, while as mentioned above, giving possible parameter adjustment instructions.

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ACKNOWLEDGEMENTS The authors would like to acknowledge the fund­ ing from the project AUTOBarge, European Union’s EU Framework Program for Research and Innovation Horizon 2020 under Grant Agreement No. 955768.

REFERENCES Bellman, R. (1952). On the theory of dynamic programming. Proceedings of the national Academy of Sciences, 38(8), 716–719. Calvert, S., Deakins, E., & Motte, R. (1991). A dynamic system for fuel optimization trans-ocean. The Journal of Navigation, 44(2), 233–265. Chen, H. (1978). A dynamic program for minimum cost ship routing under uncertainty (Doctoral dissertation, Massachusetts Institute of Technology). De Wit, C. (1990). Proposal for low cost ocean weather routing. The Journal of Navigation, 43(3), 428–439. Dijkstra, E.W. (1959). “A note on two problems in connex­ ion with graphs,” Numerische Mathematik, 1, 269–271. Hagiwara, H. (1989). “Weather routing of (sail-assisted) motor vessels,” Ph.D. dissertation, Delft University of Technology, Delft, the Netherlands.

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Estimation of nonlinear forces acting on floating bodies using machine learning C. Eskilsson, S. Pashami & A. Holst RISE - Research Institutes of Sweden, Borås, Sweden

J. Palm Sigma Energy & Marine AB, Gothenburg, Sweden

ABSTRACT: Numerical models used in the design of floating bodies routinely rely on linear hydrodynamics. Extensions for hydrodynamic nonlinearities can be approximated using e.g. Morison type drag and nonlinear Froude-Krylov forces. This paper aims to improve the approximation of nonlinear forces acting on floating bodies by using machine learning (ML). Many ML models are general function approximators and therefore suitable for representing such nonlinear correction terms. A hierarchical modelling approach is used to build mappings between higher-fidelity simulations and the linear method. The ML corrections are built up for FNPF, Euler and RANS simulations. Results for decay tests of a sphere in model scale using recurrent neural networks (RNN) are presented. The RNN algorithm is shown to satisfactory predict the correction terms if the most nonlinear case is used as training data. No difference in the performance of the RNN model is seen for the different hydrodynamic models.

1 INTRODUCTION

with a time-domain linear model, to be compared with more accurate CFD simulations that are in the order of 10000 to 100000 CPU hours per sea state (Eskilsson et al. 2014, Kim et al. 2016). As standards and regulations require Monte Carlo simulations of irregular sea states, linear hydrodynamic models will remain the tool of choice in practical engineering design of floating bodies for many years to come. Thus, there is much to gain by improving the linear hydrodynamic models to better and more reliably account for nonlinear effects.

Numerical models based on linear potential flow remain the tool-of-the-trade for ocean and marine engineering. The linear models are based on the underlying assumptions of: (i) inviscid and irrota­ tional flow and (ii) small amplitude waves and body motions as the acting forces are computed on the body at equilibrium position. Weak nonlinearities are routinely included through nonlinear Froude-Krylov approaches for the wave-body interaction (Giorgi & Ringwood 2017a) and Wheeler stretching for the wave kinematics (Wheeler 1970). Viscous effects are incorporated by the Morison approximation using drag coefficients (Morison et al. 1950). The drag coefficients are typically calibrated based on physical experiments or CFD simulations. The calibration of drag coefficients is however problematic, as it does not differentiate between different nonlinear effects. Therefore the calibrated coefficient becomes a fudge factor representing all nonlinear effects (Giorgi & Ringwood 2017b). Regardless of the inclusion of weak nonlinearities, linear models often struggle with simulations of: (i) survival conditions with large amplitude waves, and (ii) bodies working at reson­ ance such as wave energy converters (WECs). The key selling point of the linear models is the computational efficiency. A three hour irregular sea state is computed in the order of minutes to hours

1.1

Hierarchical numerical modelling

Four levels of hydrodynamic models will be used in this study: Reynolds-Averaged Navier Stokes (RANS) equations, Euler equations, fully nonlinear potential flow (FNPF) equations and linear potential flow (LPF) equations. The models are fundamentally different and were chosen because they capture different levels of physical phenomena. From the hierarchy of the models we expect the difference between the LPF and FNPF to provide an estimate for nonlinear potential flow effects (including radiation damping and Froude-Krylov). When comparing the LPF and Euler models we add rotational and, probably most importantly, pressure drag effects to the list of nonlinearities captured. Con­ tinuing to the RANS model we can add the viscous drag and surface friction effects. Finally, by comparing

DOI: 10.1201/9781003399759-7

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Figure 1. Overview of the hierarchical modelling approach.

between how the ML model compare to different levels of fidelity for the hydrodynamic modelling.

simulations of RANS and instances of overtopping using two-phase volume of fluid simulations (VOFRANS) we can find information about slamming and overtopping, see Figure 1. By comparing models of dif­ ferent fidelity we can define correction factors that can be added to improve the linear potential model later. The differentiation between nonlinear effects is import­ ant not only for an increased understanding of the hydrodynamic problem. Indeed, the differentiation is essential in order to apply suitable scaling factors to the nonlinear correction terms when going from model scale to full scale (Giorgi and Ringwood 2017b, Palm et al. 2018). 1.2

2 HYDRODYNAMIC MODELS 2.1

This paper uses computational fluid dynamics (CFD) as a fully nonlinear deterministic model of the wavestructure interaction. The CFD simulations are in the form of Reynolds Averaged Navier-Stokes simula­ tions with air-water interface capturing by the volume of fluid method (VOF-RANS). Using the single fluid approximation the two-phase incom­ pressible VOF-RANS equations read

Machine learning for sequential data

Many machine learning (ML) models are general function approximators and are therefore suitable for representing nonlinear correction terms. ML is widely used in all fields of engineering. In the fluid dynamics community ML has started to be used for turbulence modelling (Duraisamy et al. 2019, Wu et al. 2018) and for specific problems like lift and drag coeffi­ cients for vortex-induced vibrations (Raissi et al. 2019). Thus, ML is suitable for delivering the required nonlinear correction terms while keeping the computational overhead within acceptable limits. The success of machine learning, especially deep neural networks, is not limited to classical classifica­ tion tasks; it can, in many cases, model the informa­ tion available across a sequence of data. Recurrent neural networks (RNNs) are designed to process sequential data by learning their inter-dependencies in varying lengths. Long short-term memory (LSTM) (Hochreiter & Schmidhuber 1997) was introduced in 1997 to deal with some of the shortcomings of stand­ ard recurrent structures, such as vanishing gradient (Hochreiter 1998). LSTM has more control over which information should be kept and which should be forgotten through its internal memory nodes. 1.3

Navier-Stokes equations

where ~ u is the fluid velocity, and ~ ur ¼ ~ u ~ ug is the velocity relative to the grid velocity ~ ug. p denotes the g~ x, in which ρ is the total pressure and p0 ¼ p ρ~ fluid density and ~ g the acceleration of gravity. S is the viscous stress tensor and ~ fb is the body force. Finally, α 2 ½0; 1� denotes the phase fraction of the volume of fluid method. The phase fraction provides linear inter­ polation between the fluid density and kinematic vis­ cosity of water ( ρw , �w ) and air ( ρa ,�a ), where the indices w and a denotes water and air, respectively. The viscous term present in Equation (2) is evaluated using the the standard k ω-SST model, used together with continuous wall functions. Equations (1) - (3) are solved using the multi-phase interFoam model, which is part of the widely used open-source framework OpenFOAM (OpenFOAM 2020, Weller et al. 1998). OpenFOAM is based on a cell-centred 2nd order finite volume method on unstructured polyhedral cells. The framework supports parallel implementation and moving meshes. Open­ FOAM has been used extensively for floating bodies, such as ships (Wang et al. 2019), offshore platforms (Xia et al. 2017), floating offshore wind (Aliyar et al. 2022) and wave energy applications (Wang et al. 2018).

Scope of the paper

We outline the basis of the approach to improving LPF models with correction factors obtained from hierarchical numerical modelling with the aim to develop an hybrid LPF-ML model with better non­ linear performance than today’s approaches. The main focus of this paper is to investigate the perform­ ance of RNN for the simple test case of heave decay of a sphere, and more specifically see any differences

64

where D=Dt denote the material derivative. Equations (7) – (10) are solved using the commer­ cial tool SHIPFLOW-MOTIONS 7 (Flowtec 2022, Kjellberg 2013). MOTIONS solves the FNPF equa­ tions using an unsteady three-dimensional boundary element method (BEM) together with a mixed Euler­ ian–Lagrangian method for the free surface. The unsteady Bernoulli equation can be used to compute the pressure at any point in the domain

The forces acting on a floating body are in the CFD method estimated by direct summation of the total pressure p and the viscous shear stress τ on each cell surface P c of the body. For a body with total area A ¼ Ni¼1 Ai discretised into Nc cells, the equations of motion around the centre of gravity is described as

The equations of motion is obtained from Equa­ tions (4) and (5), with the difference Fv = 0 and Nc denotes the number of wetted panels, because only the water phase is considered. Please note that the number of wetted panels is not constant during the simulation.

~ is the velocity of where M is the mass matrix and U the floating body. Additionally ri is the position vector from the centre of gravity to the cell face-centre and ^ni is the unit outward-pointing normal of face i. Please note that separate definitions of the pressure force Fp and the viscous force Fv are presented in Equations (5) and (6). 2.2

2.4

The linear potential flow approach computes the floating body dynamics based on linear radiationdiffraction theory using Cummins equation (Cum­ mins 1962). The equations of motion for the six ~ may be written degrees of freedom X

Euler equations

The Euler equations are simply obtained by setting the viscous term to zero in Eq. (2), and to implement slip conditions on any wall boundaries. Hence, Fv = 0 in Equation (6). The same interFoam solver is used for the Euler and RANS simulations. 2.3

Linear potential flow

Fully nonlinear potential flow

The fully nonlinear potential flow model is obtained from the Euler equations by assuming the flow to be irrotational. We can then define the scalar velocity potential � as r� ¼ ~ u. Consequently the continuity equation (1) becomes the Laplace equation, which governs the fluid flow

where M∞ is the (constant) added mass matrix at infinite frequency, ~ Fe is the hydrodynamic excita­ tion force and ~ Fc is the stiffness force, with the linear stiffness matrix C. The radiation forces are denoted ~ Fr where K(t) are the impulse response functions (kernel functions) that enable the radi­ ation term to be computed from the convolution inte­ gral in Equation (15). To avoid truncation errors, the time window Tirf should be chosen large enough so that KðTirf Þ ≈ Kð∞Þ ¼ 0, see e.g. Armesto et al. (2015). The coefficients of excitation force, added mass and radiation damping were obtained from Capytain (Ancellin & Dias 2019). The infinite frequency added mass was obtained from Ogilivie’s relation (Ogil­ vie 1964). Finally, Equation (12) is solved using the in-house MoodyMarine model (Palm & Eskils­ son 2021).

In contrast to the RANS and Euler equations where the free surface is implicit in the model through the use of the VOF approach, the FNPF for­ mulation requires boundary conditions at the free surface. These are defined as

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Figure 3. Investigation of performance of ML based cor­ rection factors. Figure 2. Generating ML based correction factors to improve LPF simulations.

3 A HIERARCHICAL MODELLING APPROACH TO GENERATING ML BASED CORRECTION FACTORS where ~ δp corresponds to differences in pressure forces (the potential flow domain), and ~ δ� concerns differences due to viscous effects. Please note that only the shear-forces are captured as viscous effects in this method. Effects from induced drag (form drag and pressure drag) are considered within the pressure correction factor, as are pressure variations due to vortical structures in the fluid.

The hydrodynamic models used (RANS, Euler, FNPF and LPF) capture different levels of physics, and we use all four models to generate training data for the ML algorithm. When comparing time series from models of different fidelity in transient simulations, large differences are to be expected. To avoid propa­ gating these differences in the transient simulation results, the lower-fidelity method is to mimic the higher-fidelity models results by assigning the higherfidelity body response as a forced motion in the lowerfidelity model. This removes the transient error from the analysis but it also fundamentally changes what the lower-fidelity model predicts. The outline of the hierarchical approach to training data for ML based estimation of nonlinear forces is presented in Figure 2. However, in this initial study we focus on the per­ formance of the ML algorithm for a simple test case, see Figure 3. The higher-fidelity models will be all tested against the LPF model and we examine the per­ formance of the ML algorithms to capture the different physical phenomena. We once again stress that the training data obtained by the linear model predicts the linearised hydrodynamic forces acting on the body when it is following the fully nonlinear response from the higher fidelity simulations. 3.1

3.2

Machine learning algorithm

As we are to model time-series we use RNNs (Sher­ stinsky 2020, Lipton 2015). Time series forecasting is a standard application for RNNs and we use a LSTM network for sequence-to-one regression which means a prefix of a sequence up to a certain time is used to predict the target values for the upcoming time. The RNN is thus made up of a sequence input layer; an LSTM layer; a fully con­ nected layer, and finally a regression layer. Out-ofsample testing has been used for the evaluation of the trained models. The Deep Learning Toolbox in Matlab is used in this study (Mathworks 2022). 4 TEST CASE The methodology is tested using decay tests of a sphere in model scale. A reason for working with model scale rather than full scale is that model scale typically experience larger relative viscous forces. This is beneficial for the testing as we by extension get larger discrepancies between the hydrodynamic models in model scale compared with full (prototype) scale. The test case used in this paper originates from a highly accurate experimental campaign for a D = 0.3 m diameter sphere was presented by Kramer

Correction factors

From the RANS simulations we have two general­ ised forces (6x1): the pressure force ~ Fp , and the vis­ cous force ~ F� . From the Euler and FNPF simulations we only have ~ Fp . The linear hydrodynamic forces include added inertia force at infinite frequency ~ Fa, radiation damping force ~ Fr , hydrodynamic excitation force ~ Fe and stiffness force ~ Fc . In this work we focus on the following correction factors

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Figure 4. Decay tests of a sphere. Top: Photos of the sphere at the initial positions for the three drop cases. Bottom: Nor­ malised time series of recorded heave motion for the three drop height. From Kramer et al. (2021).

Figure 6. Computational mesh for body at equilibrium for the FNPF simulations.

Figure 5. Computational mesh for body at equilibrium for the RANS and Euler simulations.

oct-tree hexahedral dominated meshes from stl sur­ faces of the body. The mesh is a full 3D mesh made up from a 1.1M cells (Figure 5). There are refinement zones at the free surface as well as around the sphere and the boundary layer is resolved with five cells. Please note that we use the same mesh for all drop heights. This mesh is too coarse for the lar­ gest drop height – solution verification (Eça & Hoekstra 2014) gives 5% uncertainty in the decay coefficient for the 0.1D and 0.3D drops heights, but 20% uncertainty for the 0.5D case – but the mesh is nevertheless deemed sufficient for the purpose of testing ML performance. Additionally, we are using the same mesh for both the RANS and Euler simulations. The Euler simulation does not require the resolution of the boundary layer, but we use the same mesh anyway to avoid any differences arising from the spatial resolution.

et al. (2021). The uncertainties were found to be very low – on average only about 0.3% of the respective drop heights. The sphere has a mass of 7.0686 kg giving that it is half-submerged at equilibrium (the water has a density of ρw ¼ 1000 kg/m3). The sphere is dropped from three heights H0 ¼ ½0:1D; 0:3D; 0:5D� m. The sphere has an analytical resonance period of Te0 = 0.76 s. The initial positions and the measured time series of the heave motion is presented in Figure 4. 5 NUMERICAL SET-UP 5.1

RANS and Euler models

The computational mesh is created using the snappyHexMesh utility in OpenFOAM, which creates

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Figure 7. Comparison of hydrodynamic models to experimental data for the sphere decay test. Top panel shows the normal­ ized heave motion x3 =H0 , and bottom panel shows the normalized vertical force F3 =mg. (a) H0 = 0.1D, (b) H0 = 0.3D and (c) H0 = 0.5D. Experimental data from Kramer et al. (2021).

The numerical schemes used are second-order van Leer scheme for convection terms, second-order cen­ tral differences for diffusion terms, while the turbu­ lence equations are solved using the first-order upwind method. The time-stepping is carried out using the first-order backward Euler scheme with a CFL number of 0.9 and a maximum time step of 0.001 s. 5.2

For each hydrodynamic case, we use one drop height for training and two for testing, given that we have three set-ups per case. The predictors and tar­ gets are normalised to have zero mean and unit vari­ ance, but the results are later presented in dimensional form. 6 RESULTS AND DISCUSSION

FNPF model

We initially look into the nonlinear forces obtained with the deterministic models before looking into the prediction of the nonlinear forces using RNN.

The computational mesh for the FNPF employs sym­ metry around the xz plane and is made up by 1 900 quadrilaterals on the body. The free surface reso­ lution uses adaptive mesh refinement and changes from some 2400 quadrilaterals at t = 0 s to roughly 19000 quadrilaterals at the end of the computations. The mesh at equilibrium is presented in Figure 6. SHIPFLOW-MOTIONS employs a fourth-order accurate time-stepping scheme. In this study we use a fixed time step of 0.001 s. 5.3

6.1

The results from the hierarchical modelling are presented in Figures 7 and 8. As expected, Figure 7 shows that increasing drop height increases the differences between the model results. Most noticeable is that the 0.5D case has a somewhat longer response period than the smal­ ler amplitude motions. This is consistent through­ out the models, with the exception of the LPF model that predicts a response constant period for all drop height. The results are similar to the results presented in Kramer et al. (2021), and the higher-fidelity models matches well with the experimental results. The largest discrepancies between the models is seen for the 0.5D case, which is not surprising as the behavior becomes highly nonlinear when the sphere is fully out of the water in the 0.5D case. Figure 7 shows that the viscous force contribution is practically negligible for the 0.1D and 0.3D cases, while some viscous effects are present in the 0.5D case. This is deduced by noting how the CFD body motion is more damped than the Euler results for 0.5D in Figure 7(c), while being almost perfectly matching in 0.1D and 0.3D drop heights. Remember that the CFD and Euler simulations were made in the same numerical framework, with the same computa­ tional mesh. We therefore expect the numerical effects to be negligible when we compare these two

LPF model

As mentioned above, the hydrodynamic coefficients for the LPF solver are computed using the Capytaine software. The input geometry to Capytaine is a stl surface of the sphere made up of almost 20000 tri­ angles. The MoodyMarine software uses a 3rd order explicit Runga-Kutta scheme to advance in time with a fixed time step of 0.001 s. 5.4

Hierarchical modelling results

RNN model

As mentioned above, the RNN is made up of a sequence input, a LSTM layer, a fully connected layer and a regression layer. The sequence layer takes six fea­ tures as input after removing the columns containing only zeros from the higher-fidelity data. The LSTM layer is set to have 128 hidden units, which was found to work well, but we mention that the number has not been optimised. The model is trained using the adaptive moment estimation (adam) optimizer (Kingma & Ba 2014), with a learning rate of 0.001-0.01 and 100-500 epochs.

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Figure 8. LPF obtained normalized vertical forces for decay tests with prescribed motion given by higher fidelity models. From left column: RANS, middle column: Euler and right column: FNPF. The different drop heights are top row: H0 = 0.1D, middle row: H0 = 0.3D and bottom row: H0 = 0.5D.

model predictions with each other, regardless of their absolute error (Palm et al. 2018). The FNPF model struggles a bit with the 0.5D case as apparent from the recorded vertical force. The discontinuity in force at t =0.61 s is when the layer of water disappears from the sphere, and the secondary peak at t ≈ 0:8 s is caused when the sphere nearly leaves the free surface. It is possible to tune the free surface mesh discretisation to improve the performance of the FNPF model but, again, we keep the slightly sub-optimal mesh and focus on the RNN performance. The nonlinearity of the hydrostatic pressure at dif­ ferent submersion levels of a spherical shape is clearly manifested in the results. The stiffness matrix C was linearised at the equilibrium where the water plane area is maximised (C33 ¼ 0:25D2 πρg). Therefore Fc in the linear model force predictions should be con­ sidered an upper limit of the restoring force. This is also why Fc =mg41 in the 0.5D drop cases in Figure 8. This nonlinearity is considered in all three high-fidelity models.

6.2

Machine learning results

The simulations presented in the previous section are here used as training and testing data for the RNN algorithm. Small-amplitude high-frequency noise is observed in Figure 8, which is not surprising consid­ ering the VOF nature of the simulations. The train­ ing data is thus slightly smoothed using a Gaussianweighted moving average filter. Figure 9 first shows the viscous force correction prediction, when training and testing on the LPFRANS model results. Three different models were trained using one of the drop test data sets each. The viscous force correction is shown to be very well predicted if the 0.5D case is used as training data. The models trained on smaller amplitude motion fail to predict the 0.5D data set with good accuracy. This is in line with the deterministic results (Figure 7) where viscous effects were only visible in the 0.5D case. This may also explain why there is a slight over-prediction of the viscous part in the 0.1D case for the 0.5D-trained model, see top graph in Figure 9(c).

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Figure 9. Prediction (dashed line) vs testing data (green line) of ~ δ� using training data (red line) from RANS model with: (a) H0 = 0.1D, (b) H0 = 0.3D and (c) H0 = 0.5D.

Figure 10. Prediction (dashed line) vs testing data (green line) of ~ δp using training data (red line) from RANS model with: (a) H0 = 0.1D, (b) H0=0.3D and (c) H0 = 0.5D.

Figure 11. Prediction (dashed line) vs testing data (green line) of ~ δp using training data (red line) from Euler model with: (a) H0=0.1D, (b) H0 = 0.3D and (c) H0 = 0.5D.

Figure 12. Prediction (dashed line) vs testing data (green line) of ~ δp using training data (red line) from FNPF model with: (a) H0 = 0.1D, (b) H0 = 0.3D and (c) H0 = 0.5D.

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ACKNOWLEDGEMENTS

It turns out that, in general, the 0.5D case provides sufficient and diverse training data also for predicting the pressure force correction. Good estimates of the 0.3D pressure force correction are obtained for all models, as can be seen in Figures 10 to 12. The 0.1D performs worse, especially in the start-up. Even though ~ δp for the 0.1D case is over-predicted for the initial pressure peak, the overall prediction is satisfac­ tory as the magnitude of the correction factor is small for the 0.1D case. Training with 0.3D data provides only marginally better estimate on the 0.1D case, while missing some key parts of the 0.5D response. Training with 0.1D data grossly under-predicts the correction forces for the higher drop cases. Thus, the results imply that RNN-type ML methods are suitable for handling single time history data such as these decay tests, provided that the model is exposed to the largest nonlinearity. When trained on smaller amplitudes, it has problems with accurately increasing the nonlinear effects in the response. The RNN model had no problem to capture all nonlinear effects in one go, as evident from the results for the RANS correction factor being as good as for the FNPF correction factor. If these conclusions stand when more complex cases including wave exci­ tation and several degrees of freedom in the motion remain to be seen. This is ongoing work.

Support for this work was given by the Swedish Energy Agency through Grant Nos. 50196-1 and 44423-2. Computations were performed on resources at the National Supercomputer Centre provided by the Swedish National Infrastructure for Computing (SNIC), partially funded by the Swedish Research Council through Grant Agreement No. 2018-05973. The SHIPFLOW-MOTIONS software was made avail­ able to this study by FLOWTECH International AB.

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7 CONCLUDING REMARKS This paper has presented the first steps towards a hierarchical modelling approach with ML to differen­ tiate between nonlinear effects in marine hydrodynam­ ics. Decay tests of a spherical shape at three amplitudes (0.1D, 0.3D, 0.5D) were modelled numer­ ically using four levels of computational fidelity: Rey­ nolds Averaged Navier-Stokes simulations (RANS), Euler simulations (Euler), Fully nonlinear potential flow (FNPF), and linear potential flow (LPF). Trained machine learning algorithms using RNNs were for each high-fidelity model used to provide correction factors to the LPF method. The vertical forces from pressure and viscosity (RANS only) were predicted. The results showed that it is possible to get a relatively good correction model for ~ δp and ~ δ� in the smaller drop heights by training on the largest drop height results only. Training on smaller amplitudes gave less accurate results for the 0.5D case. This study is a subset of a larger work package which evaluates different classes of machine learning approaches to establish what can and should be used in marine applications. Work on combining different cor­ rections to approximate the magnitude of separate non­ linearities is also ongoing. Finally it should be mentioned that the trained machine learning algorithms are to be integrated into MoodyMarine, producing a much needed ML-corrected linear model for marine renewable energy applications.

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Comparative study of ship design load on various environmental conditions T. Firmandha & S. Anggara PT. Biro Klasifikasi Indonesia, Jakarta, Indonesia

ABSTRACT: The background formulation to predict ship design load provided by Classification Society rules is derived based on long-term sea-state data of the North Atlantic Ocean. This load may apply to ships sailing worldwide, but it may be conservative when used for designing the ships sailing in closed sea such as Indonesia. Therefore, a more proportional wave-induced load needs to be defined. To support this study, 35 years wave observation data has been collected to compose a scatter diagram of the Indo­ nesian waterway. By these data, the local wave load parameter has been determined based on the 10-8 probability exceedance level. The linear long-term analysis is performed on around 100 merchant ship models that are derived from 10 hull forms within the length of 50 m to 350 m. The results show that all ship model operated in Indonesia gives 70% lower wave induce bending moment compared to North Atlantic.

however, since the coverage is so wide, this study is only focused on obtaining wave parameter values (CW) in respect to Indonesia environmental condition.

1 INTRODUCTION The hull girder strength criteria of ship structures required by Class Society rules is developed based on wave-induced loads covering the worst antici­ pated wave environment. IACS Rec.34 (2001) explains that the minimum environmental conditions to be used in the class rules are the North Atlantic wave environment which is specifically aiming at ships with unrestricted service. Those assumption is clearly more conservative compared to restricted operation. Prasetyo et al. (2019) mentioned that North Atlantic wave heights are predominantly distributed in around 7,5 m with periods between 4,5 s - 12,5 s. While the Indonesian wave height is distributed half smaller, i.e., 3,5 m with period range of 2 s – 13,4 s. The different conditions between North Atlantic and the Indones­ ian environment will certainly affect the different needs of the strength of the ship’s structure. Thus, a new reference standard of wave load calculation is needed. Nowadays, wave loads prediction on ship struc­ tures has been widely carried out by a “direct cal­ culation” approach which is quite complicated and impractical. Meanwhile, classification society rules demand a prescriptive formulation approach so that it can be easily used by the user. The formula is embodied in the value of the wave parameter (CW), this parameter governs the value of ship global loading (wave bending). In addition to wave parameters, there are other parameters such as shape, non-dimensional parameters, and the influ­ ence of non-linear factors from wave bending,

2 METHODOLOGY 2.1

General approach

The wave parameter has been known in class rules set to represent the dynamic load for wave bending moment calculations. The technical background of its development has been released and many studies is established to polish the previous result. Mean­ while, the purpose of this study is particularly to obtain wave parameter (CW) values in accordance with the Indonesian waterway. CW has been derived from the direct calculation method by employing spectral analysis and combining

Figure 1. General workflow.

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the same manner methodology used in the preparation of the Global wave statistics. The compilation of the Indonesian map was derived from the History of ECMWF hindcast data located around 10°N to 10°S and 90°E to 140°E and the boundary of Indonesian waterways territory is defined based on the regulation of Indonesian Government (PM No.7: 2013), as can be seen in Figure 3. This data is used for further pro­ cessing in the current study

it with the Equivalent design wave (EDW). EDW is an equivalent wave or wave group representing the long-term response of the dominant load parameter under consideration. This approach has advantage for shortening the computation time through simpler cal­ culations. However, the accuracy of the methodology highly depends on how the design parameters have been chose Hauteclocque et al. (2012). A similar method is used by Hauteclocque et al. (2016) to create new wave load formulae for container ships. This overall methodology was also similar to the one used for the development of IACS common structural rules. The method may be separated into three main fol­ lowing steps:

2.3

One of the important aspects of long-term wave ana­ lysis is the ability to predict extreme wave intensities (M. Li et al. 2016). Some standards require different extreme wave periods for their designation. IACS determines to use an exceedance probability level (PEX) of 10-6 for normal operating conditions and 10-4 for fatigue assessment. When calculating design wave bending moments, it is recommended to use a return period of at least 20 years, corresponding to about 10-8 probability of exceedance per cycle. Furthermore, Weibull distribution is employed to see HW value as presented in Figure 4. As can be seen that the correlation between the natural logarithm of HW and PEX can be predicted by using linear interpol­ ation, Prasetyo et al. (2019), and it results wave height HW of 7.22 m for Indonesia and 19.9 m for North Atlantic.

– Development of local environment scatter diagram – Wave load analysis by direct calculation – Breakdown rules requirement and adjust wave parameter 2.2

Extreme wave prediction

Scatter diagram

This study uses 2 environmental conditions, i.e., North Atlantic as the basis for the development of IACS URS 11 and the Indonesia waterway as the basis for adjusting the new wave parameters. Wave scatter of North Atlantic is taken from IACS recommendation 34 at grid positions 8,9,15 and 16 of Global Wave Statistics (GWS) in Figure 2. Meanwhile, Kurniawan et al. (2014) proposed the Indonesia wave scatter by

Figure 2. IACS Rec. 34 maps of global wave statistic. Figure 4. Weibull plotting of North Atlantic and Indonesian scatter diagram.

2.4

Wave spectrum

The wave spectrum used in this study (as the same used in IACS URS11 technical background) is single PM (Pierson–Moskowitz) for a wave height of HS 7,22 m and a zero-crossing period TZ of 10,516 s. The cos2 spreading function has also been included. 2.5

Ship model

Several ships were modeled, considering the ship type composition under BKI ship register database.

Figure 3. Indonesian map of global wave statistics.

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This study used 10 models consisting of 3 Oil Tankers, 1 Bulk Carrier, 3 container ships, and 3 General Cargo. All ship, particularly for container ship, selected model has coefficient block (CB) ≥ 0.6, this aims to minimize the effect of non-linearity parameter which may occurs for slender hull form (Nitta et al: 1992). This aspect is not considered in this study since out of URS11 contexts. The model dimensions are then varied homotheti­ cally (ranging in length from 50 m to 350 m), bring­ ing a total of 100 ship models. The homothetic approach beneficially reduces modeling time and sufficiently represents all ships since the same type of ship has an identical hull form. A similar method has also been adopted by Hauteclocque & Derbanne (2017) in their study.

The MWV sagging (negative moment) is:

Equation (2) and (4) contains of: – – – –

3 RESULTS AND DISCUSSION 3.1

Wave bending moment

After carrying out direct calculations with Indones­ ian environmental conditions, wave vertical bending moment values were obtained for each ship model, the value varies based on the dimension, shape, and mass distribution of a typical ship. The longer ship experiences a greater bending moment. Figure 5 shows a comparison of the vertical wave bending moment values between North Atlantic and Indonesia. The value increases exponentially as the length increase. The difference between 2 environ­ mental conditions is more than 30% in general.

Scale factor L3 Wave Parameter CW Non-dimensional Formulation 0; 19 Non-linear factor: 1 for hogging and � � 11 CB þ0;7 for sagging 19 CB

B L

CB

�

Where L is ship length and B is breadth. The nonlinear factor was not explicit in IACS S11, because it was assumed that vertical bending moment is linear in hogging and that non-linear effects influence only the sagging moment. Nitta et al (1992) also men­ tioned that the value of hogging moments is 7% smaller than sagging. On the basis of the results of a number of non-linear calculations conducted. it has been noted that the differences between the max­ imum values of sagging and hogging moments increase as Cb becomes smaller. The non-linearity due to hull form also leads to change locations where maxima value occurs. 3.2.2 Derivation formula for wave parameter CW After taking out the non-linear factor in sagging con­ dition, the value of the wave parameter CW can be obtained as the following formula:

Figure 5. Sample of vertical wave bending moment for a homo-tethic model in the Indonesian environment.

3.3 3.2

Rules prescriptive formulae

Wave parameter

From the total of 10 ship models and their homo­ thetic that have been analyzed, more than 100 nondimensional wave parameters (CW) are obtained. The results show that the numerical wave parameter (CW) for Indonesian waterway is 30% lower than the value in unrestricted conditions. As can be seen in Figure 6, the average of CW for Indonesia (dot symbol) are distributed polynomial, increasing up to ship length 250 m then decrease, the maximum value of Indonesia’s CW is slightly shifted towards

After obtaining the vertical wave bending moment (MWV), one needs to break a non-dimensional wave parameter CW out by the following step-by-step procedure. 3.2.1 Break non-linear parameters out from the Mwv Rules formula The MWV hogging (positive moment) according to IACS URS 11 is:

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smaller ship lengths compared to NA. This may be due to different average wave period for both locations. Furthermore, if the maximum CW for the North Atlantic area is 10.75 (solid greed line) then CW for Indonesia is 7 (solid red line).

REFERENCES ANSYS-AQWA. 2022. Theory Manual v.20.0. UK: Cen­ tury Dynamics Ltd. Hauteclocque, G., Derbanne, Q., and El-Gharbaoui, A. 2012. Comparison of different equivalent design waves with spectral analysis. In ASME 2012 31st International Conference on Ocean, Offshore and Arctic Engineering (OMAE), number 83405. Pages 353–361 European Centre for Medium-Range Weather Forecasts (ECMWF). www.ecmwf.int Hauteclocque, G.D. & Derbanne Q. 2016. Generalized Wave Parameter for Rules Formulae. Copenhagen, Den­ mark: Proceedings of PRADS. Pages 787–794 Hauteclocque G.D., Monroy C., Bigot F. & Derbanne Q. 2016. New rules for container-ships Formulae for wave loads Copenhagen, Denmark: Proceedings of PRADS. Page 795–803 IACS Rec. 34. 2001. Standard Wave Data. Corrigenda. IACS URS11. 2020. Longitudinal Strength Standard. Rev 10. Kementerian Perhubungan Republik Indonesia: PM. No. 7 2013, Kewajiban klasifikasi bagi kapal berbendera Indonesia pada badan klasifikasi, Kementerian Perhu­ bungan Re-publik Indonesia, 2013 (in Indonesia). Kurniawan, M.A., Prasetyo, F.A. & Komariyah, S. 2014. Study on wave scatter mapping of Indonesia waterways based on hind-cast data, Proceedings of Asian-Pacific Technical Exchange and Advisory Meeting (TEAM 2014), Pages 256–262. M. Li, E. Boulougouris, I. Lazakis & G. Theotokatos. 2016. Analysis of the Wave-Induced Vertical Bending Moment and Comparison with the Class Imposed Design Loads for 4250 TEU Container ship. Inter­ national Conference on Maritime Safety and Operations, Glasgow, UK. Pages 137–143. Nitta, A., Arai, H., and Magaino, A., (1992) “Basis of IACS Unified Longitudinal Strength Standard”, Marine Strutures, Vol. 5, 1992, pp. 1–21, Vol.7, pp. 567-572, and Vol. 8, 1995, pp. 337-339. Prasetyo F.A., Firmandha T., Zakki A.F., Kurniawan M. A. & Komariyah S., Anggara S. 2019. The Effect of Local Environmental Condition on the Ship Construc­ tion Design Standard. Proceedings of PRADS. Pages 805–82. Prasetyo F.A., Osawa N., Kurniawan M.A. & Komariyah S. 2016. Comparative Study on Fatigue Damage Assessment Of A Structure Member In A Bulk Carrier Using Various Environmental Conditions. Pro­ ceedings of the ASME 2019 38th International (OMAE 2019). Paper No: OMAE2019-96760

Figure 6. Distribution of Indonesian wave parameters com­ pared to IACS URS11.

In the previous study, by Prasetyo et al. (2019), CW values were simply predicted for Indonesian waters by using a comparison between the average wave heights of HW and CW. For example, North Atlantic has a mean wave Hm of 3.4 m, this value is about one-third of CW 10.75. Using the same assumptions, if Indonesia’s Hm based on the scatter diagram is 2.29 m then the CW obtained is 3 times, that is 7.22. Close to the CW value from the direct calculation results of the semi-spectral method. This strengthens the statement expressed in the previous study 4 CONCLUSION The calculation of the vertical bending moment has been carried out using the semi-spectral method as the basis for deriving the value of the new wave par­ ameter based on Indonesian environmental condi­ tions. • The average response show that wave induce bending moment in Indonesia waterway is 30% lower than North Atlantic.

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On nonlinear wave loads of a mega-scale container ship using an elastic backbone model A. Hanaoka National Maritime Research Institute, Mitaka, Tokyo, Japan

ABSTRACT: This paper discusses nonlinearities in motions, vertical accelerations, and vertical bending moments of a mega-scale container ship in heading waves. Experiments have been conducted using a ship model with an elastic backbone, which represents the mega-scale container ship with more than 300 m length. The waves generated in the experiments represent those of 8 m and 12 m heights at the real scale. Experimen­ tal data for the regular waves shows that the effects of the ship speeds and the wave heights are not so signifi­ cant in the ship motions. On the other hand, the data of the vertical accelerations and the vertical bending moments shows obvious effects of the ship speeds and the wave heights. Exceedance probability calculated using the experimental data for an irregular wave shows similar trends, i.e. the exceedance probability of the vertical accelerations and the sagging bending moments is changed by the ship speeds.

1 INTRODUCTION

Firstly, the discussion is made about the effects of ship speeds and wave conditions on nonlinearities in the ship responses with regular waves. Linear calcu­ lations have been performed using a strip method with the same wave conditions as those for the experiments. The obtained results were compared with the experimental data. Secondly, investigations are performed for statistical values of the ship responses with an irregular wave. The statistical value to be investigated is exceedance probability calculated using the experimental data.

Since the estimation of wave loads is important for accurate assessment of structural strength of ship hulls, problems to achieve the accurate estimation of the wave loads have been addressed by many researchers for a long time (Senjanovic et al. 2008, Lee et al. 2012, Temarel et al. 2016, Guedes Soares & Duan 2019). Among the problems, nonlinearity in the wave loads is known to have to be considered in the estimation (Fonseca & Guedes Soares 2002, Wu & Hermundstad 2002, Fonseca & Guedes Soares (2005), Kukkanen 2012, Kukkanen & Matu­ siak 2014, Rajendran et al. 2015, Jiao et al. 2019). It is also known that the nonlinearity is significant especially in the case of slender ships such as con­ tainer ships moving in heading waves of large heights. This paper discusses the nonlinearities in responses of a mega-scale container ship in the heading waves. The ship responses to be discussed are ship motions, vertical accelerations, and vertical bending moments. Experiments have been conducted using a ship model with an elastic backbone, which represents the megascale container ship with more than 300 m length. The experimental data has been obtained about motions, vertical accelerations near the bow, and ver­ tical bending moments at several sections of the ship model in both regular waves and irregular waves. The waves generated in the experiments represent those of 8 m and 12 m heights at the real scale. Two different speeds of the ship have been considered in the experiment.

2 EXPERIMENTAL METHODS The ship model used in the experiments represents the mega-scale container ship with more than 300 m length. Principal characteristics of the ship model is shown in Table 1. The full loading condi­ tion is considered in the experiments. Weight distri­ bution of the ship model is shown in Figure 1. A schema of the model is shown in Figure 2. In this figure, A.P. and F.P. indicate the position of the aft and fore perpendiculars, respectively. The numbers between A.P. and F.P. indicate the longitudinal dis­ tances from the A.P. The neighboring numbers are 0.1L distant, where L represents the length of the ship model. Hereinafter, the longitudinal positions on the ship model are described by using a symbol “SS”, which stands for ship sections. For example, SS9.25 indicates the position which is 0.925L far away from A.P. in the longitudinal direction. The ship model is composed of seven segments rigidly fixed to an elastic

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Table 1.

The experiments were performed in the towing tank of National Maritime Research Institute in Japan. The Froude numbers corresponding to the towing speeds are 0.092 and 0.138. Table 2 shows conditions of regular waves in the tank tests. The wave heights shown in the table represent 8 and 12 m, respectively. The generated irregular wave corresponds to an ISSC spectrum shape of mean period of 11 s and sig­ nificant wave height of 12 m at the real scale. Sev­ eral runs in different wave time histories of the same spectrum for one speed were summed up to obtain statistical results in the irregular wave. All waves generated in the experiment are heading waves.

Principal characteristics of the ship model. Actual ship m

Length Breadth Depth Draft Displacement Scale ration

Ship model ton

320 50.0 30.0 14.1

m

kg

3.50 0.547 0.328 0.154 151516

198

91.4

3 DISCUSSION 3.1

Experimental data of the ship responses with regular waves has been recorded as time history. Fourier analysis has been performed to the time history to obtain linear amplitudes. In addition, linear calcula­ tions have been performed using the New Strip Method (Kashiwagi & Iwashita 2012) with the same conditions as those for the experiments. The calcu­ lated results were compared with the linear ampli­ tude obtained by the Fourier analysis. For the comparison, the linear amplitudes are nondimensio­ nalized. Hereinafter, Froude number is described as Fn in the text and the figures. In the figures of this section, the experimental data and the calculated results are denoted by dots and lines, respectively. Figures 3-4 show the results of heave and pitch amplitudes, respectively. In the figures, the symbols z and θ indicate the heave and pitch amplitude, respectively. The symbols ζ, λ, k and H are wave amp­ litudes, wavelengths, wave numbers and wave heights, respectively. For Fn = 0.092, there are little differences between experimental data for the wave heights 8 m and 12 m at the real scale. For Fn = 0.138, there are slight differences between the different wave heights in the range of wavelengths from 0.9L to 1.2L. All experimental data of the heave and pitch motions agree with the obtained results. These trends show that the effects of the ship speeds and the wave heights are not so significant in the ship motions. Figure 5 shows the results of vertical accelerations at SS9.25. In the figure, the symbol a and g indicate the vertical accelerations and the gravitational acceler­ ation, respectively. For Fn = 0.092, no significant dif­ ference can be observed between the experimental data for the wave heights 8 m and 12 m at the real scale. For Fn = 0.138, on the other hand, there are obvious differences between the different wave heights in the range of wave lengths from 0.8L to 1.0L. The differences show that the nondimensiona­ lized amplitudes of the vertical accelerations becomes larger for the smaller wave heights. This trend is simi­ lar to those observed in Miyake et al. (2004) and

Figure 1. Weight distribution of the ship model. The origin of x coordinate in this figure is at the aft perpendicular of the ship model. L represents the length of the ship model.

Figure 2. Schema of ship model.

backbone. The backbone has the same shape of the cross section. Thus, it has the same rigidity along the whole length. Strain gauges are attached on the elastic backbone to measure the strains at the sections between neighboring segments. Vertical bending moments are calculated from the measured strains. An accelerometer is put to measure vertical acceleration at SS9.25. The ship motions are measured by devices attached to the towing device. Table 2.

Conditions of regular waves in the tank tests.

Wave height

Wavelength

m 0.0875, 0.131

Nonlinearities in the ship responses with regular waves

0.4L*, 0.6L, 0.8L, 0.9L, 1.0L, 1.1L, 1.2L, 1.5L, 2.0L

* L: ship length

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Figure 5. Amplitudes of vertical accelerations at SS9.25. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figure 3. Heave amplitudes. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figure 6. Amplitudes of vertical bending moments at SS2.4. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figure 4. Pitch amplitudes. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

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Figure 7. Amplitudes of vertical bending moments at SS4.0. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figure 9. Amplitudes of vertical bending moments at SS7.2. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

especially in the range of wave lengths from 0.8L to 1.2L, in which discrepancies were also observed in the results of the ship motions. This trend in the experimental data and the calculated results is similar to that observed in Watanabe et al. (1989). Figures 6-9 show the results of vertical bending moments at respective sections. In the figure, the symbol M, ρ and B indicate the vertical bending moments, the water density and the breadth of the ship model, respect­ ively. Note that trends observed in the results at SS3.2 and 6.4 are similar to those at SS 4.0 and 5.6, respect­ ively. Differences can be observed in the vertical bending moments at all sections between the different wave heights in the range of wave lengths from 0.8L to 1.5L. This trend can be seen for both ship speeds. The differ­ ences are such that the vertical bending moments become larger as the wave height is higher. This is the nonlinearity of the vertical bending moments, which is caused by large changes of restoring forces due to the rapid change of the hull shapes under the free surface of the high waves (Miyake et al. 2004). In most cases, the obtained results are larger than the experimental data. The exceptions in which the calculated results are smal­ ler than the experimental data can be observed especially in the results at SS7.2 for Fn = 0.138 in the range of wave lengths from 0.8L to 1.2L. The similar trends were observed by Miyake et al. (2004).

Figure 8. Amplitudes of vertical bending moments at SS5.6. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

3.2 could be the nonlinearity due to the change of the wave heights. Discrepancies can be observed between the experimental data and the calculated results

Nonlinearities in exceedance probability of the ship responses with an irregular wave

Experimental data of the ship responses with irregular wave has been recorded as time history. Histograms have

80

been obtained from the time histories in order to calculate exceedance probability of the ship responses. In the figures of this section, the symbols P and ζ1/3 indicate exceedance probability and significant wave amplitude, respectively. Figures 10-11 show the exceedance probability of the ship motions. While the exceedance probability for Fn = 0.092 shows no significant difference between the downward and upward motions, the slight differences can be seen in the probability for Fn = 0.138. The dif­ ferences between the maxima of the downward and upward motions are not so large. The ship speed does not affect the exceedance probability significantly.

Figure 12. Exceedance probability of vertical accelerations at SS9.25. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figure 10. Exceedance probability of heave amplitudes. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figure 13. Exceedance probability of vertical bending moments at SS2.4. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figure 12 shows the exceedance probability of the vertical acceleration at SS9.25. The exceedance probability of the vertical accelerations is changed by the ship speeds. The change implies that larger accelerations occur more frequently for Fn = 0.138 in both downward and upward directions than Fn = 0.092. This may be the result of vibrations of the elastic backbone, which were caused by slamming around the bow of the ship model towed in the high waves.

Figure 11. Exceedance probability of pitch amplitudes. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

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Figure 16. Exceedance probability of vertical bending moments at SS7.2. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figure 14. Exceedance probability of vertical bending moments at SS4.0. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

4 CONCLUSIONS Experiments have been conducted using a ship model with an elastic backbone, which represents the mega-scale container ship with more than 300 m length. The experimental data has been obtained about the ship motions, the vertical accel­ erations near the bow, and the vertical bending moments at several sections of the ship model in both regular waves and irregular waves with the heights representing 8 m and 12 m at the real scale. The experimental data for the regular waves has been compared with the results of linear calculations by a strip method. In the comparison, the experimen­ tal data of the ship motions agrees with the calcu­ lated results. No significant effects of the ship speeds and the wave heights can be observed in the ship motions. On the other hand, the experimental data of the vertical accelerations and the vertical bending moments shows the differences from the obtained results. The data also shows effects of the ship speeds and the wave heights. The exceedance probability of the ship motions shows slight differences between the downward and upward motions and is not affected by the ship speed significantly. The exceedance probability of the ver­ tical accelerations and the sagging moments is changed by the ship speeds. The change implies that the larger magnitudes occur more frequently for Fn = 0.138 in both downward and upward directions than Fn = 0.092. This trend is not so obvious in the results of the hogging moments. Due to the slam­ ming, the differences between the sagging and hog­ ging moments are significant at all sections.

Figure 15. Exceedance probability of vertical bending moments at SS5.6. The upper and lower figures show the results of Fn = 0.092 and 0.138, respectively.

Figures 13-16 show the exceedance probability of vertical bending moments at respective sections. The differences between the sagging and hogging moments are significant at all sections. The differ­ ences are caused by the slamming. It is obvious that larger sagging moments occur more frequently for Fn = 0.138 than Fn = 0.092. This trend is not so obvious in the results of the hogging moments.

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REFERENCES

Lee, Y., Chan, H., Pu, Y., Incecik, A. & Dow, R. S., 2012. Global wave loads on a damaged ship. Ships and Off­ shore Structures 7(3): 237–268. Miyake, R., Mizokami, S., Ogawa, Y., Zhu, T. & Kumano, A. 2004. Studies on wave loads acting on a large container ship in large waves. Journal of The Society of Naval Architecture of Japan 195: 185–194. Rajendran, S., Fonseca, N. & Guedes Soares, C. 2015. Sim­ plified body nonlinear time domain calculation of verti­ cal ship motions and wave loads in large amplitude waves. Ocean Engineering 107(1): 157–177. Senjanovic, I., Malenica S. & Tomasevic, S. 2008. Investi­ gation of ship hydroelasticity. Ocean Engineering 35: 523–535. Temarel, P., Bai, W., Bruns, A., Derbanne, Q., Dessi, D., Dhavalikar, S., Fonseca, N., Fukasawa, T., Gu, X., Nestegard, A., Papanikolaou, A., Parunov, J., Song, K. H. & Wang, S. 2016. Prediction of wave-induced loads on ship: Progress and challenges. Ocean Engineering 119(1): 274–308. Watanabe, I., Ueno, M. & Sawada, H. 1989. Effects of bow flare shape to the wave loads of a container ship. Journal of The Society of Naval Architecture of Japan 166: 259–266. Wu, M. & Hermundstad, O. A. 2002. Time-domain simula­ tion of wave-induced nonlinear motions and loads and its applications in ship design. Marine Structures 15(6): 561–597.

Fonseca, N. & Guedes Soares, C., 2002. Comparison of numerical and experimental results of nonlinear wave-induced vertical ship motions and loads. Journal of Marine Science and Technology 6: 193–204. Fonseca, N. & Guedes Soares, C., 2005. Comparison between experimental and numerical results of the non­ linear vertical ship motions and loads on a containership in regular waves. International Shipbuilding Progress 52(1): 57–89. Guedes Soares, C. & Duan, W. 2019. Wave loads on ships and offshore structures. Journal of Marine Science and Application 17: 281–283. Jiao, J., Yu, H., Chen, C. & Ren, H., 2019. Time-domain numerical and segmented model experimental study on ship hydroelastic responses and whipping loads in harsh irregular seaways. Ocean Engineering 185(1): 59–81. Kashiwagi, M., & Iwashita, H., 2012. Ship Motions in Sea­ keeping, No.4 in the Series of Naval Architecture & Ocean Engineering. Seizandou-shoten. (in Japanese) Kukkanen, T., 2012. Numerical and experimental studies of nonlinear wave loads of ships. Espoo: VTT Technical Research Centre of Finland. Kukkanen, T. & Matusiak, J., 2014. Nonlinear hull girder loads of a RoPax ship. Ocean Engineering 75(1): 1–14.

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Encounter spectra computation of heave motion based on full-scale measurements using ANN M. Katalinić, P. Matić & N. Assani Faculty of Maritime studies, University of Split, Split, Croatia

J. Parunov Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia

ABSTRACT: Simultaneous measurements of waves and ship heave response in various headings were per­ formed previously in a full-scale measurement campaign with a research vessel in the Adriatic Sea. Spectral moments were measured and calculated and some discrepancies were found. The aim of the present study is to calculate the exact heave response spectra for three various headings, including following seas for which the pro­ cedure is cumbersome and often avoided. Remapping of measured wave spectrum from absolute to encounter frequencies is shown, as well as the remapping of RAOs when calculating the response spectra. Differences between the calculated and measured response spectra are modelled using Artificial Neural Network (ANN) to test if they can serve as a suitable tool either to correct the error or to model the response spectrum solely based on the input wave spectrum. The overall aim is to contribute to reduction of uncertainties in seakeeping calculations.

1 INTRODUCTION

advancing in the same direction as the ship). Specific to present work is that both waves and ship motion data were simultaneously measured during a previous conducted full-scale measurement campaign (Gledić et al. 2022). Numerically calculated RAOs were obtained by a 3D panel method. Spectral moments obtained by measurement and calculations were com­ pared and found to be in fair agreement except for stern quartering and following seas. The present study aims to explore reasons for these discrepancies. The calculated RAOs are given in the absolute frequency domain, but the measured (full-scale trials) ship response is noted in the encounter fre­ quency domain. In order to have the calculated and the measured response spectrum comparable, they both have to be stated in the encounter frequency domain in which the reference is the ship and the waves are such “as the ship sees them” while on arbitrary speed and heading. The problem is math­ ematically governed by a Doppler shift with a straightforward definition for bow seas, with no influence for beam seas, but with a more compli­ cated calculation for stern quartering seas. The complication arises due to the fact that the encoun­ ter response spectrum, at stern quartering seas, draws energy from three different absolute fre­ quencies. Additionally, apart from the physical transformation the spectral notation had to be resampled to facilitate ANN training.

The uncertainty of ship response prediction is currently a topic of research in the international research com­ munity (Parunov et al. 2022). The ability of Artificial Neural Network (ANN) as an alternative approach to evaluate ship response and to reduce uncertainty in seakeeping calculation is investigated in the present paper. The applied methodology relies on the traditional definition of the seakeeping problem in the spectral domain. The approach uses the assumption of linear dependency between input waves and ship response described by a transfer function, equivalently RAO (response amplitude operator). The concept is wellknow and theoretically rather simple, but there are practical concerns to be considered especially for fol­ lowing or stern quartering seas. The problem of dis­ crepancies between measured and calculated spectrum results was analysed by Nielsen et al. (2021) using an optimization method to minimize difference between the two. In this study, heave motion calculations and fullscale measurements are analysed and compared for three various ship-to-wave headings: bow seas (µ = 135°), beam seas (µ = 90°) and stern quartering seas (µ = 45°); with µ denoting relative ship to wave heading angle (180° - head seas, i.e. waves advancing against the ship; 0° - following seas, i.e. wave

DOI: 10.1201/9781003399759-10

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An ANN was used to model the difference between measured and calculated heave response spectra. In Katalinić et al. (2017) and Mudronja et al. (2017) it has been shown that ANNs can out­ perform regression based statistical models in mod­ elling significant wave height and wave period and in Matić et al. (2020) a feed-forward ANN was suc­ cessfully used to predict ships’ response in the time domain. In Kawai et al, (2021) a convolutional neural network was used for directional wave spec­ trum estimation from the hull responses and in Lim and Jo (2020) a neural network was used to predict the response amplitude operator (RAO) of a ship.

The measurement campaign yielded simultan­ eous time series of ship heave, pitch and roll and wave elevation and direction. Heave motion for the three specified courses is used for the present study. The time series for heave and wave eleva­ tion are transferred to the spectral domain by means of FFT with smoothening (Figures 3 and 4).

2 DATA The paper uses the following data: • Ship response in heave during full-scale measurements. • Wave buoy data during full-scale measurements. • Calculated RAOs - 3D panel method (Hydrostar) Figure 2. Freely floating wave buoy.

2.1

Ship and wave buoy data

The ship was a 36.3-meter-long and 340 tons of dis­ placement research vessel. She was assigned specifically to perform the seakeeping measurements. The vessel run nine different, uniformly spaced ship-to-wave head­ ings, 30 min each, while maintaining a steady speed of 7 (±0.2) knots. The trails were carefully planned and executed on February 10th, 2021, during “choppy seas”, in conditions of south-eastern swell seas in diminishing (sirocco/jugo) wind, that would cause “measurable” ship response and would not put the ship and crew in harm in the most unfavourable courses. The experiment is described in detail in Katalinić et al (2022). Figure 3. Wave spectrums S0 (ω0) from time series in the absolute frequency domain.

Figure 1. Left. Research vessel BIOS DVA - Institute of oceanography and fisheries, split, croatia. Right - the IMU sensor SBG Ellipse2N.

During the measurements on-board, a freefloating wave buoy was deployed in the vicinity measuring wave elevations.

Figure 4. Heave response Sr (ωe) from time series in the encounter frequency domain.

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It is important to note that:

on encounter heading and speed, but the total energy must remain the same.

• the response spectrum is noted in the wave encounter frequency domain SR (ωe) • wave spectrum is noted in the absolute wave fre­ quency domain S0 (ω0) 2.2

where ω0 and ωe are respectively the absolute and the encounter frequency, while S0 and Se are the spectra ordinates in the absolute and encounter domain (Gaglione et al. 2019). In deep water, relation between ω0 and ωe is given by

Calculated RAOs

The RAOs for heave are calculated as presented in detail in Gledić et al (2022) by a seakeeping analysis in HydroSTAR, which is a 3D diffraction/radiation potential theory software for diverse types of wavebody interaction problems (Bureau Veritas, 2021).

where v is the advance speed of the ship [m/s] and µ is the relative ship-to-wave heading (0° denotes sterns seas, and 180° denotes head seas). 3.1.1 Bow seas For an arbitrarily selected frequency step and range of ωe a reverse relation is available as follows

Figure 5. Model in HydroSTAR.

Equation (4) can be conveniently used to simul­ taneously resample during re-mapping in order to obtain the same resolution as of the measured response spectra as will be needed to provide later on to the ANN.

The calculation yielded RAOs shown in Figure 6.

3.1.2 Following and stern quartering seas In following or stern quartering seas Equations (4) and (5) are not valid. Up to ωe < 1/(4ψ) three ω0,i solutions exist for one ωe

Figure 6. Calculated RAOs.

3 METHODOLOGY 3.1

Mapping wave spectra between the absolute and the encounter domain

Mapping between the absolute and the encountered frequency domain has been known for decades (e.g. Beck et al. 1989). The spectral peak is shifted in fre­ quency and its shape flattened or narrow deepening

Figure 7. Absolute ω0 to encounter ωe frequency relation. Source: Gaglione et al (2019).

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This causes mathematical singularities and the numerical calculation can be cumbersome. Thus, a majority of authors (e.g. Beck et al. 1989) rec­ ommend avoiding the remapping and advise using the absolute frequencies wave spectra when calcu­ lating response spectra, which in turn yields a socalled pseudo spectra. The pseudo-spectra will have no physical meaning as to the distribution of energy over frequencies, but the spectral moments are correct as well as the dependent significant quantities. The procedure for actual remapping of wave spec­ tra from the absolute to the encounter frequency domain is explained in detail in Gaglione et al. (2019) and Nielsen (2021). For following or stern quartering seas, the problem is divided into two regions:

Figure 8. Relationship between absolute and encounter wave frequencies. Source: Gaglione et al (2019).

3.2

Calculation of response spectra

The calculated response spectra in the encounter fre­ quency domain is given by

• For ωe < 1/(4ψ):

where j denotes the appropriate heading. Such calculation is needed to directly compare the calculated response spectra Sr_calc(ωe) and the meas­ ured response spectra Sr_trial(ωe) recorded during full- scale measurements at sea. The response spectra calculation, as given in Equation (10), follows the same concept as the pro­ cedure presented in paragraph 3.1, but it is necessary to consider that appropriate values of RAO must also be numerically identified (interpolated) and extracted from input RAOs (Figure 6), as per ω0i, and multiplied with an appropriate S0i(ω0i) prior their assembly into the resulting Sr_calc(ωe). Specifically, in region ωe < 1/(4ψ) this means:

and

• For ωe > 1/(4ψ):

and

3.3

The numerical procedure implies several loops to distinguish the correct region. The programming complicates further if it is automated for arbitrary speed and direction or if directional spectra is handled. The process can be easily visualized in Figure 8. In a practical sense, if the data is handled numer­ ically, for identified ω01, ω02, ω03 the energy from the absolute frequency domain S01(ω01), S02(ω02), S02(ω03) has to be interpolated between the existing record of S0(ω0). A practical validation of procedure is always needed according to Equation (1).

Modelling correction between the calculated and measured response spectra with ANN

The difference between calculated response spectra and Sr_calc(ωe) and response spectra measured during full-scale Sr_trial(ωe) has been modelled by means of ANN to test if they can prove worthy as a tool to correct traditional calculations and obtain a more accurate result. ANN represent an artificial structure that mimics the process that occurs in a brain of the living beings (Haykin 1999). Briefly described, it is a structure that consists of certain number of interconnected artificial neurons that calculates the response based

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in the input information. The main property of ANN is the ability to learn from example, meaning that the ANN is data-dependent. It has to have a set of examples, i.e. input-output data pairs based on which the network can be trained to perform certain function. To train the network means to adjust the network parameters, i.e. the weights (w) with a goal of producing the least possible error, i.e. to minim­ ize the cost function which is given in a form of root mean square error (RMSE), mean absolute error (MAE), sum of square errors (SSE), or similar. An ANN subtype called Multi-Layer Perceptron (MLP), presented in Figure 9, represents the most commonly used network for modelling purposes and is used in this study. The basic algorithm devel­ oped for training MLP network is the error back­ propagation (EBP). Many improvements have been made to the original algorithm to ameliorate its slow convergence. Bayesian-regularization (BR) algorithm was used to train the network in Matlab. Bayesian regulariza­ tion (BR) can be interpreted as an expanded version of Levenberg-Marquardt (LM) algorithm. LM algo­ rithm is one of the versions of the EBP algorithm, proven to be the fastest and most appropriate algo­ rithm for training networks containing up to hun­ dreds of adjustable parameters, and is explained in detail in Foresee & Hagan (1997). Bayesian regularization is based on the modifica­ tion of the cost function. LM algorithm aims to reduce the sum of squared errors (SSE) and regular­ ization adds an additional term SSW, i.e. the sum of square adjustable network parameters (w). In this way, the algorithm minimizes both the error and the number of adjustable network parameters needed to achieve a minimum error.

should reduce the size of the network to the optimal value while optimizing the performance in terms of an error reduction. Furthermore, a feed-forward arti­ ficial neural network with single hidden layer can approximate a continuous function when using enough neurons as stated in Cybenko (1989). Thus, it is initially assumed that an MLP with 50 neurons in a single hidden layer should enable satisfactory model performance once the model is trained using the BR algorithm. During the training, it has not been noticed that the network would benefit from a larger number of neurons since the effective number of network parameters would end up around 50% of the number used in total. In this study, an MLP-ANN was trained to model the relationship between the real heave spectrum based on the calculated one, thus improving the accuracy of the calculations (ANN_1). Furthermore, an ANN was also used to learn how to output meas­ ured response spectra based on the absolute wave spectrum (ANN_2). The model (ANN_1) uses angular frequency (ωe) and calculated heave spectrum (Sr_calc) as inputs, and therefore two neurons of the input layer to deter­ mine the real/measured heave spectrum (Sr_trial) at the output, thus using one neuron at the output layer. The measured heave spectrum is regarded as the real spectrum, i.e. the ship’s response.

In a second set of experiments, the neural network was trained to model the relationship between the absolute wave frequency wave spectrum S0(ω0) to the measured ship heave response spectrum Sr_trial (ωe) defined by Equation (13) where S0 ðω0 Þ denotes that the absolute frequency domain wave spectrum was resampled to the same spacing in which the measured response spectrum was available, i.e. S0 (ω0) was resampled and interpolated to the ωe posi­ tions. This was the only preparation done for ANN_2.

The available data for each of the three headings were initially divided into two sets, i.e. a training and a testing set, with both subsets being fair repre­ sentatives of the initial set. The division of 1304 samples for each heading were performed randomly, in ratio 70:30, producing the training set containing 914, and test set containing 392 samples. Each sample contains three values, two input values of the input variables, and one output value of the output variable. The idea was to test the model performances on an independent set of examples, which were not introduced to the network during the training thus

Figure 9. Artificial neural network model.

One of the benefits of the BR algorithm is that it optimizes the number of network parameters while adjusting their values to reduce the error, as explained in Foresee & Hagan (1997). If an adequate number of hidden neurons (h) is used, the algorithm

89

testing the generalization ability of the model. The training data set was additionally divided into train­ ing, validation and test sets in accordance with the Levenberg-Marquardt training algorithm, in order to make use of early stopping method, as explained in Beale, Hagan & Demuth (2010). This method is used as prevention for overtraining, i.e. to improve generalization abilities of the model.

Although mathematically sound, this can be argu­ able since, as will be seen later, there is energy in the measured response spectrum Sr_trial(ωe) at ωe > 0.99 rad/s which would mean that if the seakeeping lin­ earity assumption in the spectral domain is valid that some wave energy had to exist in that region. For clarity, the absolute-to-encounter wave fre­ quency relation for the specific vessel at stern quar­ tering seas heading is shown in Figure 11.

4 RESULTS AND DISCUSSION 4.1

Wave spectrum remapped to encounter frequency domain

Based on the methodology presented in paragraph 3.1 the three wave spectra (Figure 3) measured by a stationary (although free-floating) wave buoy in the absolute frequency domain were remapped in the encounter frequency domain corresponding to v = 7 knots and relative ship to wave headings µ = 135° (bow seas), µ = 90° (beam seas), µ = 45° (stern quar­ tering seas). Encounter wave spectra are presented in Figure 10. For beam seas (µ = 90°) the spectra remain unchanged, for bow waves (µ = 135°) the spectral peak expectedly flattens and moves to higher fre­ quencies as the ship advances into the waves, for stern quartering seas (µ = 45°) the spectral peak expectedly narrows and shifts to lower frequencies (from ω0 = 1.08 to ωe = 0.78 rad/s), but a second, perhaps unexpected, peak happens at ωe =1/4ψ = 0.99 rad/s. At this frequency the energy corres­ ponding to ω01, ω02, ω03 concentrates. This is com­ pletely in line with the mathematical model presented in Section 3.1. For any energy to transfer from ωe > 0.99 rad/s, where only ω03 contributes, there had to be energy recorded in S0(ω0) at fre­ quencies ω0 > 4.78 rad/s and there is none.

Figure 11. Absolute ω0 to encounter ωe frequency relation for the case study vessel at v = 7 knots. Beam to stern quar­ tering and following seas.

It is notable to mention that all calculations are aligned on the same frequency axis (either directly resampled or resampled simultaneously during remapping) to 0 < ωe < 9.96 rad/s with step of about 0.004 rad/s. High resolution is necessary “to catch” the data around the peak on Figure 11 where small differences in input ωe yield larger differences in ω01 and ω02. This can be seen in Figure 12. 4.2

Response spectrum calculation

The calculated response spectra Sr_calc(ωe) is obtained from encounter wave spectra Se(ωe) pre­ sented in Figure 10 and based on RAOs presented in Figure 6 following the methodology in Section 3.2. The obtained calculated response spectra Sr_calc(ωe) are presented in Figure 13. These can be directly compared to measured heave response spectra Sr_trial (ωe) recorded during full-scale measurements, pre­ sented in Figure 14. 4.3

ANN correction modelling between calculated and measured heave response spectra

Different quality (error) measures are used to evalu­ ate different aspects of model accuracy, i.e. to valid­ ate the model. According to Legates and McCabe (1999) these should include at least one relative error measure, such as the coefficient of efficiency (CE) or correlation coefficient (R), and at least one absolute error measure, such as Root Mean Squared

Figure 10. Encounter wave spectra for the three analysed ship-to-wave headings.

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performance of the model ANN_1, RMSE, MAE, R and CE were calculated for the three different headings with values presented in Table 1, where values of RMSE and MAE closer to zero, and R and CE closer to 1, indicate better performance of the model. Table 1. Numerical evaluation of the model ANN_1 performance. Heading

RMSE

MAE

R

CE

45° 90° 135°

0.000212 0.000181 0.000287

0.000075 0.000106 0.000181

0.999874 0.999984 0.999993

0.999730 0.999968 0.999986

Figure 12. Absolute to S0(ω0) to encounter Se(ωe) wave spectrum for v = 7 knots and μ = 45 deg (Stern quartering seas).

Figure 15. Graphical evaluation of the model ANN_1 response for the heading angle of 90°.

Figure 13. Calculated heave response spectra for the three analysed headings.

Figure 16. Graphical evaluation of the model ANN_1 response for the heading angle of 45°.

Graphical evaluation of the ANN model response in comparison to the measured data is presented in Figures 15 to 17. These figures show how well the neural network (NN) models the measured response spectrum based on the input, calculated, response spectrum acting as its black-box correction. Once the network was trained on the extracted, output, training data points (70% of available data) it suc­ cessfully predicted (blue points-line in Figures 15– 17) the extracted validation points that were not ini­ tially provided into the network (red points in Fig­ ures 15–17).

Figure 14. Measured heave response spectra for the three analysed headings.

Error (RMSE), or Mean Absolute Error (MAE). Beside numerical, graphical evaluation, i.e. the comparison of the model response and measured values is also desirable. Therefore, to evaluate the

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5 CONCLUSIONS The paper is a part of a wider research efforts to model uncertainties related to ship response in waves. The present study provides the necessary steps to make the theoretical heave response, calcu­ lated based on RAOs from a 3D panel method, com­ parable to the measured response spectra which was recorded during full-scale measurements on a research vessel in choppy seas. The procedures involve a cumbersome calculation of remapping the wave spectrum from the absolute to encounter frequency domain and calculating the response spectrum, with remapping RAOs, in the encounter domain. Difficulties arise due to the fact that in following seas there are several absolute wave and RAO frequencies corresponding to a single encounter frequency. Observed discrepancies could serve as discussion on the limitation and applicability of the traditional linear seakeeping approach espe­ cially in stern quartering (and following) seas. An approach using artificial neural networks was tested to assess if they could be a useful tool for cor­ rection of error of traditionally obtained RAOs. The case study for heave, at the analysed speed and three analysed headings, showed that the ANN performed excellent. To train ANN, full-scale motion data should be available. With constant improvement of sensor and communication capabilities full-scale data becomes ever more so. In future longer cam­ paigns, other motions and headings at different sea states should be analysed.

Figure 17. Graphical evaluation of the model ANN_1 response for the heading angle of 135°.

For the calculated heave spectra (Sr_calc) and corresponding angular frequency (ωe) at the input, the ANN was successfully trained to produce the measured encounter response spectra (Sr_trial) for all of the analysed headings, i.e. angles of 45°, 90° and 135° as shown in Figures 15–17 and in Table 1. The results are only slightly worse for heading angle 135° as it can be noted based on the numerical evalu­ ation presented in Table 1. 4.4

Additional ANN test to model measured response spectrum based on absolute frequency domain wave spectra

To evaluate the performance of the model ANN_2, RMSE, MAE, R and CE were also cal­ culated for three different headings with values in Table 2 and agreement was graphically valid­ ated between model response and measured data.

ACKNOWLEDGEMENTS This work has been supported by the Croatian Sci­ ence Foundation project lP-2019-04-2085 and pro­ ject “Functional integration of the University of Split through development of scientific and research infra­ structure” KK.01.1.1.02.0018. The work was also supported and done in cooper­ ation with the Institute of Oceanography and Fisher­ ies, Split, Croatia who has enabled the research vessel “Bios dva” for full-scale measurements.

Table 2. Numerical evaluation of the model ANN_2 performance. Heading

RMSE

MAE

R

CE

45° 90° 135°

0.000184 0.000104 0.000137

0.000112 0.000070 0.000084

0.999013 0.999995 0.999998

0.999813 0.999989 0.999995

REFERENCES

For the absolute wave spectra S0(ω0) and the corres­ ponding angular frequency ω0 at the input, the ANN was successfully trained to produce the full scale meas­ ured spectra Sr_trial((ωe) for all of the analysed head­ ings, i.e. angles of 45°, 90° and 135°. Errors, in terms of statistics, as presented in Table 2, are again negli­ gible indicating accurate modelling. Graphical plots are not provided as they are almost identical to graphs in Figures 15–17 showing that ANN_2, based on the pro­ vided training data points, successfully predicted the validation data points that were not provided to her beforehand.

Beale, M. H., Hagan, M. T., & Demuth, H. B. 2010. Neural network toolbox. User’s Guide, MathWorks, 2, 77–81. Beck R, Cummins W, Dalzell J, Mandel P, Webster W. 1989. Vol. III: Motions in waves and controllability. In: Lewis E, editor. Principles of naval architecture, Second revision. Jersey City, NJ, USA: SNAME; p. 1–188. Bureau Veritas (BV). 2021. HydroSTAR Manual. Research Department, Bureau Veritas, Neuilly-SurSeine, France. Cybenko, G. 1989. Approximation by superpositions of a sigmoid-al function, Mathematics of Control, Signals, and Systems, vol. 2, no. 4, pp. 303–314.

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Foresee, F.D. & Hagan, M.T. 1997. Gauss-Newton approx-imation to Bayesian learning. ICNN’97, Volume: 3, 1930–1935. DOI: 10.1109/ICNN.1997.614194 Gaglione, S., Pennino, S., Piscopo, V., Scamardella, A., 2019. Absolute sea spectrum resampling from encounter wave time history. IMEKO TC-19, International Work­ shop on Metrology for the Sea, Genoa, Italy, pp. 86–91. Gledić, I., Petranović, T., Katalinić, M., Vujičić, S., Matić, P., Ćatipović, I. and Parunov, J., 2022. Compari­ son of full-scale measurements and seakeeping calcula­ tions for two research vessels in the Adriatic Sea. Ocean Engineering, 266, p.113135. Haykin, S. 1999. Neural Networks: A Comprehensive Foundation. Pearson Education Inc. Second Edition, 2004. Katalinić, M., Matić, P., Petranović, T. and Parunov, J., 2022. Full-scale measurements of ship motion in rough seas in the Adriatic Sea. Trends in Maritime Technology and Engineering Volume 1, pp.365–371. Katalinić, M., Mudronja, L. and Matić, P. 2017. Data Based Modelling of the Mean Wave Period in the Adriatic Sea. Book of Proceedings of the 7th International Maritime Science Confer-ence, Solin, Croatia, 2017, pp 308–318. Kawai, T., Kawamura, Y., Okada, T., Mitsuyuki, T. and Chen, X., 2021. Sea state estimation using monitor­ ing data by convolutional neural network (CNN).

Journal of Marine Science and Technology, 26(3), pp.947–962. Lim, J.H. and Jo, H.J., 2020. Prediction of barge ship roll response amplitude operator using machine learning techniques. Journal of Ocean Engineering and Technol­ ogy, 34(3),pp.167–179. D.R. Legates, G.J. McCabe Jr., 1999. Evaluating the use of “goodness-of-fit” measures in hydrologic and hydrocli­ matic model validation, Water Resources Research, Vol. 35, No. 1, pages: 233–241. Matić, P. and Katalinić, M. 2020. Artificial neural network boat seakeeping model based on full scale measurements, ICTS 2020 Maritime, transport and logistics science con­ ference proceedings, Portoroz, 226–230. Mudronja, L., Matić, P. and Katalinić, M. 2017. Data-Based Modelling of Significant Wave Height in the Adriatic Sea. Trans-actions on maritime science, 6(01), pp.5–13. Nielsen, U.D., Mounet, R.E. and Brodtkorb, A.H., 2021. Tuning of transfer functions for analysis of wave–ship interactions. Marine Structures, 79, p.103029. Parunov, J., Guedes Soares, C., Hirdaris, S. & Wang, X. (2022) Uncertainties in modelling the low-frequency wave-induced global loads in ships. Marine Structures, 86, 103307, 21 doi:10.1016/j. marstruc.2022.103307.

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Data-driven ship fatigue assessment based on pitch and heave motions X. Lang, J.W. Ringsberg & W. Mao Department of Mechanics and Maritime Sciences, Chalmers University of Technology, Gothenburg, Sweden

D. Wu & C. Zhang National Waterborne Transport Safety Center, Wuhan University of Technology, Wuhan, China

ABSTRACT: Ocean-crossing ship structures continuously suffer from wave-induced loads when sailing at sea. The encountered wave loads cause significant variations in ship structural stresses, leading to accumu­ lated fatigue damage. It is common today to use the spectral method for direct fatigue calculation when evalu­ ating ship fatigue, where large inherent uncertainties still exist. This paper investigates the machine learning technique to establish model for a 2800TEU container vessel fatigue assessment. The measurement data of three years cross-Atlantic sailing demonstrates and validates the machine learning model. In this investigation, the motions of the ship are used as inputs to build machine learning model. The fatigue damage amounts predicted using machine learning model were compared with those obtained from full-scale measurements and direct fatigue calculation. The pros and cons of the methods are compared in terms of capability, robust­ ness, and accuracy of the prediction.

1 INTRODUCTION

marine technology research community, uncertainties related to fatigue methods, wave load, and structural stress analysis have been extensively investigated during the last decades (Fricke, 2017; Gaidai et al., 2019). With the digitization of the current shipping indus­ try, large quantities of data are collected to monitor stress signals of a ship indicating fatigue accumula­ tion (Mao, 2014; Storhaug, 2014). The machine learn­ ing and data-driven approaches have been applied to fatigue analysis and life prediction of structures (Bao et al., 2021; Feng et al., 2023; He et al., 2021; Masoudi Nejad et al., 2022; Yan et al., 2019). How­ ever, almost no work has been conducted using machine learning techniques to investigate how ship fatigue damage accumulated in terms of ship motions. Especially most of the existing ships are now equipped with sensors for measuring 6 DOF motions. A reliable motions-based fatigue assessment method can have accurate fatigue damage monitoring during operations, without investing in sensors to measure structural stress in subsequent voyages. In this study, a data-driven model is constructed to estimate a ship’s fatigue damage under different sea states. The ship motion responses’ statistics are con­ sidered as the input features. The proposed model is developed and validated using the full-scale measure­ ments of a North Atlantic container vessel. The onboard monitoring system collected a database containing gen­ eral ship operation profiles, structural stresses and 6 DOF motion responses. While the encountered

Fatigue damage accumulated on ships when sailing at sea is an important challenge for ship safety. Different fatigue assessment methods were proposed to predict fatigue life of ship structures. If time series of detailed structural stresses are available, the corresponding fatigue damage can be straightforwardly estimated by rainflow couting method based on Palmgren–Miner’s rule. This approach is often called the time-domain fatigue analysis. The definition of the rainflow count method has been detailed in Rychlik (1987), and Rychlik (1993). For ship fatigue design and damage monitoring, the time series of stresses are not available. To simulate those time series for large amounts of sea states requires expensive computation efforts. Nowdays, ship fatigue design is commonly conducted in the frequency domain by narrow-band approximation (NBA) assum­ ing Gaussian random processes (Rychlik, 1987). How­ ever, significant uncertainties remain in current fatigue assessment methods (Fricke et al., 2002), wave load and structural stress calculations (Li et al., 2014), and input wave statistics (Mao et al., 2010). These uncer­ tainties in the fatigue design methods of a ship have contributed to large fatigue failures (cracks) in presentday ship structures (Mao, 2014). The investigation by Jordan & Cochran (1978), as well as that of Jordan & Knight (1978) for commercial ships in the United States, showed that there were, on average, 86 struc­ tural failures (cracks) per ship at any inspection. In the

DOI: 10.1201/9781003399759-11

95

metocean environments, such as significant wave height, wave period, wave are extracted from the public reanaly­ sis hindcast datasets. To make the complete of this paper, the following Section 2 presents basic approaches and models for ship spectral fatigue assessment. The full-scale measurement and data analysis are given in Section 3. The machine learning architecture for spectral fatigue assessment is presented in Section 4, followed by the results in Section 5.

The frequency domain RAOs of structural stress response can be easily employed to get a ship’s short-term stress response spectrum. For fatigue assessment during a ship’s sailing life, the long-term wave conditions are assumed to be composed of a series of short-term stationary sea/wave states. Each stationary sea state can last for between 30 minutes to 6 hours. The sea states are normally described its statistical property, e.g., the significant wave height Hs, the wave period Tz, the wave spec­ trum, and sometimes the spreading angles. Different wave spectra have been developed for such purpose. In this study, the ISSC wave spectrum (Tucker, 1991) is used and defined as:

2 SPECTRAL FATIGUE ASSEEMENT FOR SHIP STRUCTURE 2.1

Ship fatigue and S-N method

Ship fatigue problem is recognized as a high cycle fatigue accumulation process, where fatigue damage is normally estimated by the linear Palmgren-Miner law based on a specific S-N curve (DNV, 2010) as: Finally, the ship stress response spectrum can be estimated by multiplying the square of stress RAOs with the encountered wave spectrum. But the encountered wave spectra for certain wave fre­ quency can be infinite for following sea operations, leading to unreliable results of stress response spec­ tra (Mao et al. 2010). While the stress spectral moments λn , n = 0, 1, 2, …, can still be easily esti­ mated by:

where α and m are the S-N curve parameters and are dependant on structural materials, geometries, working environments and method of fabrication (welding details). In this study, the Ib S-N curve with α ¼ 1012:76 and m ¼ 3 (DNV, 2010) is assumed in the following analysis. The ni is the number of stress cycle ranges Si, which can be estimated by the rainflow counting method if structural stress signals are avail­ able. The non-recursive definition rainflow counting method, given by Rychlik (1987), is used in this study. 2.2

For a stationary sea state of time interval T, if the stress signals are assumed to be a narrow band Gaussian process, its stress ranges follow Rayleigh distribution. Then, fatigue damage accumulated during this period can be approximated by the socalled narrow band approximation as:

Direct fatigue estimation by the spectral method

In addition to the rainflow counting method, Figure 1 presents a schematic workflow for a direct fatigue estimation method, based on Response Amplitude Operators (RAOs) combined with encountered wave conditions.

where fz is the frequency of zero-crossing, i.e., the frequency of stress cycles, and Γ() is the gamma function. The only variables in the above narrow band approximation are the spectral moments, which can be easily estimated by Equation 3. 3 FULL-SCALE MEASUREMENT 3.1

Case study ship

In this study, the full-scale measurements from the hull monitoring system are employed to investigate the effect of horizontal bending and torsion to

Figure 1. Workflow of the conventional ship fatigue damage calculation method by spectral fatigue assessment.

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a container ship’s total fatigue life. The hull monitor­ ing system was launched in a North Atlantic sailing 2800TEU container vessel. The case study ship was built in 1998; the main characteristics are listed in Table 1.

of stress measured on port and starboard sides. A Motion Response Unit (MRU) was also installed at the central line and measured the container vessel’s 6 DOF motions. The MRU measurements include 16 variables, i.e., motions Z ¼ ½z2 ; z3 ; z4 ; z5 ; z6 �, motions velocity Z_ ¼ ½z_ 2 ; z_ 3 ; z_ 4 ; z_ 5 ; z_ 6 �, and motions acceler­ € ¼ ½€z1 ; €z2 ; €z3 ; €z4 ; €z5 ; €z6 �. Where z1 to z6 denote ation Z surge, heave, sway, roll, yaw and pitch, respectively. The case study ship operates the trade between Western Europe and Quebec in Canada. A total of 48 complete sailing voyages from September 2007 to February 2010 are selected from the full-scale measurement and applied in the fatigue modeling and analysis. The other voyages were not included in this study for missing more than 50% measurements. The measurement frequency of strain/stress and ship motions are 25 Hz, and the frequency for other oper­ ational parameters is 1 Hz. The North Atlantic is considered as one of the harshest sea environments in the world. Especially during winter seasons, more low-pressure systems are expected for every cross-Atlantic voyage, and 15% ~25% of the significant wave height encoun­ tered are larger than 5 meters (Mao, 2010). And all 48 case study voyages are presented in Figure 3 for winter sailing and summer sailing, respectively.

Table 1. Main characteristics of the case study North Atlantic sailing container vessel. Parameter

Symbol

Magnitude

Max. TEU Length between perpendicular Moulded breadth Moulded depth Design draft Block coefficient Deadweight Service speed Building year

Lpp B D T CB DWT Vservice -

2800 232 [m] 32.2 [m] 19.0 [m] 10.78 [m] 0.685 40900 [tonnes] 21.3 [knots] 1998

The hull monitoring system followed the DNV hull monitoring rules. It recorded real-time data of GPS position (longitude and latitude), ship motions, and operational profiles, such as ship speed over ground and ship heading. In order to separate the stresses caused due to torsion and bending moments, the strain sensors were arranged at different locations of the same cross-section. There are four sensors in the 2800TEU container ship, and they are located at the middle section (118.7 m from AP) and after section (50.3 m from AP) close to the engine room bulkhead, respectively. The sensors are placed on the stiffener web and measure the nominal longitudinal strain of both the port and starbo ard sides. The sensor on the starboard side amidships is shown in Figure 2.

Figure 3. The case study voyages from the 2800TEU con­ tainer ship, for (a) winter sailing and (b) summer sailing. The black line is the measured raw GPS position, and the red frame is the selected analysis legs by voyage spatial boundaries [55� W; 5� W] filtered.

It is visible that the container vessel has chosen alternative routes deviate notably from the shortest route, to avoid heave weather during the winter navi­ gations, as Figure 3a shows. The focus of the present study is in open sea area, therefore, the measure­ ments closed to coasts and shallow water area are excluded by two spatial boundaries, i.e., 55� W and 5� W longitude as indicated by red vertical line in Figure 3. The applied spatial boundaries can also disregard the abnormal stress samples in ports and coastal areas. Figure 4 shows one voyage example of

Figure 2. The 2800TEU container ship employed in the case study, and the layout of the strain sensor location at midship cross-section with measurement position in star­ board upper deck.

In this study, only the vertical bending induced stress at middle section are applied for the fatigue ana­ lysis, thus the considered stress values are the average

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the response stress (25 Hz) for voyage 2008-04-24. The blue frame is the selected span by the spatial boundaries, and the mean value of the stresses are set to 0. It is clearly shown that the applied span only consists of the induced stress in the normal navigation. And the low-fluctuation stress during anchor/drifting or calm water sailing near berth have been excluded.

Figure 4. Time series of stress measured midship by the sen­ sors relating to vertical bending induced strain with SCF = 2, along the example voyage 2008-04-24. The blue frame is the selected study stress span by voyage spatial boundary fil­ tered. The mean stress is set to 0.

3.2

Figure 5. (a) the first westbound (from EU to Canada) winter voyage 2008-01-06 stress measurement span for the fatigue analysis after geographic boundaries filtering, and this investi­ gation window include 277 stationary sea states, (b) total 277 normalized spectra estimated from 30-minute long stationary sea state stress measurement during this voyage.

Data analysis

In this study, a stationary sea state is assumed to be 30 minutes. After data synchronization and geo­ graphic boundaries filtering for deep-sea conditions, and the final applied sample size for fatigue machine learning modeling is 10377. For the i-th 30-minute stationary sea state, the fatigue damage accumulation di is calculated by the rainflow count method based on the stress measure­ ments. In addition to the ship stress response caused by wave-induced hydrodynamic loads, container vessels also need to consider two other waveinduced vibrations: springing and whipping. The springing response is a resonant hull girder response that depends on the structural dynamics and the wave frequencies. And the transient vibratory response, i.e., whipping of the vessel, is excited by the large and frequent slamming loads. Figure 5 pre­ sent the normalized stress spectra (λ0 ¼ 1), by using the FFT to transfer the response into the power spec­ tral density, based on the stress measurements of the 2800TEU container ship’s first westbound winter voyage in 2008. After geographic boundaries filter­ ing, the voyage 2008-01-06 includes 277 stationary sea states, and the blue frame in Figure 5a indicate the 150-th 30-minute sea state. In Figure 5b, the typical multi-peak character of the spectra can be observed from the investigated 277 spectrums. The first spectral peak is related to the wave frequency ship responses, while the second peak is caused by the ship whipping and springing. And some noise with very low energy causes the third peak in the figure. Here, the wave-induced hydrodynamic loads (main ship responses) are denoted as WF responses, while the wave-induced vibrations (whipping and springing together) are

denoted as the high frequency (HF) responses. In this study, not only the damage dWF caused by the WF responses, but also the damage d TF due to the total frequency (TF = WF+HF) are considered. The WF response is extracted from the total response by a Fourier analysis with a frequency range from 0 rad/s to 3 rad/s, and the frequency of TF is between 0 rad/s to 6 rad/s. For each stationary sea state, the statistics of the 16 motion variables are extracted based on the 30minute window measurement signal. The statistics consist of the mean value, standard deviation, skew­ € meas­ ness and kurtosis. The motion accelerations Z urement within the 150-th sea state window in Figure 5a are depicted in Figure 6. The solid line is the mean value, and the dashed line is the one-time € The mean values standard deviation boundaries of Z. € within this 30-minute of all motion accelerations Z interval are around 0, as Figure 6 shows. It indicates that the mean value may not determine the fatigue estimation, and the correlation analysis will be con­ ducted in Section 4 to select input features for fatigue machine learning modeling. The encountered metocean environments, i.e., the mean wave direction Dwave , mean wave period Tz , significant wave height Hs , are extracted from the ERA5 reanalysis dataset hourly with 0:5� � 0:5� spatial resolution (Copernicus 2019). The current velocity Ucurrent and Vcurrent are obtained from the

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Figure 7. Fatigue damage dTF and dWF along the 48 case study voyages from September 2007 to February 2010, esti­ mated by the rainflow count method based on the measured stress. The white frames represent for the westbound voy­ age’s sailing span, and the blue frames stand for the east­ bound voyages. Figure 6. The 6 DOF motion acceleration measurement within the 150-th stationary sea state for voyage 2008-0106, where €z1 to €z6 denote the acceleration of surge, heave, sway, roll, yaw and pitch, respectively.

loaded condition is considered in the hydrodynamic analysis. The global stresses on the hull girder are then estimated using the beam theory. The RAOs of the local stresses are computed by multiplying the girder stresses by a stress concentration factor (SCF), which is considered to be 2 for this container vessel. Figure 8 present the RAOs polar heatmap for one example speed, i.e., V ¼ 12 m/s. The angle of the polar diagram is the relative wave angle θ, and the radius is the encountered wave frequency ω ranged from 0.2895 rad/s to 1.63 rad/s. The color of the heatmap represents the value of the RAOs. The darker the color, the larger the value of Hσ ðωjV; θÞ, and the larger the stress response in the same Hs and Tz condition.

Copernicus Marine Service, with 0:083� � 0:083� geographical resolution, and the temporal resolution is 24 hours (CMEMS, 2021). The ship speed through water is then estimated by the measured speed over ground and current conditions, following the ISO (2015) guidelines. 3.3

Rainflow count fatigue damage and RAOs for narrow-band spectral method

The fatigue damage along the 30-minute stationary sea states for the case study 48 voyages are esti­ mated by the rainflow count method, based on the measured stresses. The total frequency induced damage d TF and wave loads induced damage dWF are presented in sailing sequency, i.e., from September 2007 to Feb­ ruary 2010, in Figure 7. The white frames represent for the westbound voyages, and the blue frames stand for the eastbound voyages. As shown in Figure 7, the larger fatigue damage mainly occurs in the winter sailing westbound voyages, because the weather encountered during winter is harsher and there are more large waves. And the storms in North Atlantic are always blowing from the west to east, thus the relative wave angle of westbound sail­ ing are rather stable close to head sea. The total fre­ quency induced fatigue damages dTF are always larger than the dWF , since the latter has not con­ sidered the high frequency response induced fatigue damage, caused by whipping and springing. And the differences d TF dWF are more obvious when the weather conditions are worse, because larger waves will excite more significant vibrations, caus­ ing high frequency responses to contribute more fatigue damage. The case study container vessel’s RAOs are are obtained by first evaluating wave bending moments using a hydrodynamic analysis based on a 2D poten­ tial theory and a ship model with 20 strips (Mao et. al., 2009). In order to determine the wave loads applied to the hull of this container ship, the fully-

Figure 8. The transfer function (RAOs) polar heatmap of the amidship deck longitudinal stiffener for the case study container vessel, at speed though water V ¼ 12 m/s. The ω ranged from 0.2895 to 1.63 stands for the angular frequency of regular waves, and 0° for head sea and 180° for follow­ ing sea.

4 MACHINE LEARNING MODEL ESTABLISHMENT 4.1

Input features

The machine learning model is established to describe a ship’s fatigue damage accumulated during

99

a sea state in terms of the input features (denoted as X). The prediction target is the observed ship fatigue damage in the assumed stationary 30-minutes sea states, i.e., D ¼ ½d1 ; d2 . . . dn �, where n is the sample size. The observed fatigue damage is esti­ mated by the rainflow count method for each 30minutes stress signal.

Figure 9. The Pearson correlation coefficient of different statistics of 16 MRU measured variables relative to fatigue damage dTF , different color frames stand for mean value, standard deviation, skewness, and kurtosis.

As discussion in Section 3, the mean value, stand­ ard deviation, skewness and kurtosis of the 16 MRU measured variables are extracted from each 30minute stationary sea state. Therefore, total 16�4 features can be chosen for the ship motion based fatigue machine learning modeling. But if the aver­ age values are around 0 as shown in Figure 6, then the correlation between the average value of motion and fatigue damage is not high. In order to reduce the dimension of the features, the Pearson correlation coefficients of different statistics relative to fatigue damage are calculated and compared in the histo­ gram of Figure 9. Here, the fatigue damage is d TF due to the total frequency (TF = WF+HF), and the Pearson correlation coefficients relative to dWF are similar. As shown in Figure 9, the standard deviation of Z, € have the highest correlation with the fatigue Z_ and Z damage d TF . Among them, as marked in the figure, the correlation coefficient of the pitch related meas­ urements (z6 ; z_ 6 ; €z6 ) standard deviation exceeds 0.8, and the standard deviation’s correlation coefficient of the heave motions (z2 ; z_ 2 ; €z2 ) are also relatively high, around 0.6. The motion’s mean value, skewness and kurtosis have quite small correlation coefficients, and they are not strongly correlated to the fatigue damage (even negative correlation). Therefore, for the ship motions based machine learning model, the applied input features are X ¼ ½varðz6 Þ; varð_z6 Þ; varð€z6 Þ; varðz2 Þ; varð_z2 Þ; varð€z2 Þ�; pitch and heave related variables. 4.2

performance compared to a single model. Boosting is a technique that generates weak evaluators one by one to reduce residual error after numerous iter­ ations. It is similar to the bagging method, which simultaneously establishes multiple independent weak evaluators. XGBoost is among the gradient tree-boosting methods (Friedman et al., 2000). Each weak evaluator in XGBoost is a tree. The applied XGBoost algorithm is to establish the relationship between the prediction target, i.e., the rainflow count fatigue damage di and the input fea­ tures. Assuming that the ensemble model has a total number of K decision trees, the prediction for this ith sample d^i is:

where n is the total number of the training samples, fk is the k-th decision tree. Standard objective func­ tions, such as error rate, mean square error, etc., can only evaluate the performance of a model and cannot assess its computing efficiency. The object­ ive function of XGBoost, which includes model complexity to quantify efficiency, is defined as follows:

where the first component is the conventional loss function, which quantifies the residual between the actual damage di , and the predicted values d^i , often the root mean square error (RMSE). The second component indicates the model complexity, which is measured based on the structure of the tree. More details of XGBoost algorithm and the hyperparameters, are elaborated in Chen and Guestrin (2016). 4.3

Model establishment

For the fatigue machine learning model establish­ ment, the workflow is presented in Figure 10. The test set is independent voyages in Figure 3, held out from the pre-processed dataset for unseen data

XGBoost algorithm

In this study, the eXtreme Gradient Boosting (XGBoost) algorithm is applied for the machine learning modeling. The XGboost algorithm is one of the ensemble machine learning techniques. It com­ bines the modeling results of numerous weak evalu­ ators to achieve superior regression or classification

Figure 10. Workflow to establsih the proposed faigue machine learning model.

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evaluation. And the rest of the dataset is split into a training set to establish the model. This study applies Bayesian optimization to find the optimal hyperparameters of the XGBoost machine learning modeling. It has recently been util­ ized in machine learning for hyperparameter adjust­ ment. Bayesian optimization is a method that uses the Bayesian theorem to create adaptive data for hyperparameters and uses surrogate models to find the best values for the hyperparameters. The 10-fold cross-validation procedure prevents model overfit­ ting and ensures that the selected hyperparameter combination values are close to the ideal values. 5 RESULTS AND DISCUSSION 5.1

Uncerstainties of narrow-band spectral method

The narrow-band fatigue assessment method intro­ duced in Section 2, is first employed to evaluate the fatigue damage. This study does not employ the cor­ rection methods to split the stress response spectrum for the spectral method; therefore, the narrow-band approach is only used to investigate wave frequency loads induced fatigue dWF .

Figure 11. Fatigue damage along the 48 case study voy­ ages from September 2007 to February 2010, estimated by the rainflow count method based on the measured stress (dWF ), and RAOs+wave spectrum method dnb .

The observed fatigue damage in time series of dWF , and dnb predicted by narrow-band approxima­ tion are presented in Figure 11. And the encountered significant wave height Hs along the sailing route is also shown in Figure 12.

over-estimation becomes more and more serious. This shows that when the wave height and relative wave angle are large, the traditional spectral method tend to get higher fatigue damage with a lot of uncer­ tainties. Only for the westbound sailing with calm sea condition, the spectral method can give relatively good estimation. 5.2

Fatigue prediction by proposed machine learning method

First, each of the 48 voyages is individually used as an unseen test for case study. After the best combin­ ation of hyperparameters are obtained by Bayesian optimization on the training set, the machine learn­ ing models have been established 48 times. Figures 13 and 14 show the prediction results of machine learning model on test set of different voy­ ages. Where Figure 13 is the prediction result of total frequency measured stress induced fatigue dTF , and Figure 14 presents the result of wave frequency induced fatigue dWF .

Figure 13. Fatigue damage along the 48 case study voyages from September 2007 to February 2010, estimated by the rainflow count method based on the total frequency measured stress dTF , and proposed machine learning model dMotion .

Obviously, compared to the results of spectral method in Figure 11, the machine learning model have very robust predictive ability. Even though each voyage is unseen data at the time of prediction, the machine learning prediction results have no obvi­ ous over-estimation.

Figure 14. Fatigue damage along the 48 case study voy­ ages from September 2007 to February 2010, estimated by the rainflow count method based on the wave frequency measured stress dWF , and proposed machine learning model dMotion .

Figure 12. Encountered significant wave height Hs along the 48 case study voyages from September 2007 to February 2010.

It is evident that the spectral method has signifi­ cantly over-estimated the fatigue damage compared to the observed value by rainflow count. Especially for some eastbound voyages in winter, when the wave conditions are harsher as Figure 12 shows, the

5.3

Fatigue prediction for future voyages

Except for each individual unseen voyage as a test set, the fatigue assessment model should have the

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ability to predict and monitor the fatigue for a period of time in the future or the entire life cycle of the ship. In this part, the 33 voyages before Febru­ ary 2009 are applied as the training set, and the rest of the around 30% dataset, i.e., 15 complete unseen voyages from February 2009 to February 2010 (oneyear), are used as the test set. The split is more in line with the needs of the shipping industry, which uses past data and nearly weather conditions forecast to optimize the ship operation. The final year 15 voyages selection is not solely based on the data split method that matches the 70% training and 30% test sets most commonly used by the machine learning community. Moreover, as seen in Figure 12, the sailing data before February 2009 covered from calm sea conditions to harsh condi­ tions with Hs of more than 9 meters. Even if many wave heights exceed 5 meters or even 8 meters in the subsequent test set voyages, machine learning model based on the previous training data will give robust predictions.

till the end of February 2010 by dTF and 0.0159 by dWF . While the fatigue accumulation by dnb for wave frequency induced fatigue damage reaches 0.037, around more than 100% over prediction than the observed fatigue dWF .

Figure 16. Accumulated fatigue damage along the 15 unseen voyages from February 2009 to February 2010, esti­ mated by the rainflow count method based on the total fre­ quency measured stress dTF , and machine learning model dMotion .

Figure 17. Accumulated fatigue damage along the 15 unseen voyages from February 2009 to February 2010, esti­ mated by the rainflow count method based on the wave fre­ quency measured stress dWF , RAOs+wave spectrum method dnb , and machine learning model dMotion .

Figure 15. The histogram of damage estimation errors by narrow-band method dnb , and motion-based machine learn­ ing model dMotion , in comparison with the reference rain­ flow damage dWF .

The prediction error between machine learning model and the wave frequency induced damage dWF are presented in Figure 15. The histogram of damage estimation errors by dnb are also plotted as compari­ son. As shown in the figure, the prediction errors of machine learning model are more uniformly distrib­ uted on the positive and negative sides of zero, and there is no significant over-estimation like spectral method dnb . This shows that the cumulative error of machine learning could not have obvious overestimation or under-estimation. The accumulated fatigue damage during this one year sailing by real measurement rainflow count dTF , dWF , spectral methods dnb , and machine learn­ ing model d Moment are presented in Figure 16 and Figure 17. During the 15 cross-Altlantic navigations, the fatigue damage has accumulated to around 0.019

The machine learning model prediction for these 15 unseen voyages, as shown in Figure 17, has much better predictions for the wave frequency induced damage d WF accumulation, compared to the narrowband method. The machine learning model dMoment has almost the same increase trend as observed fatigue. The prediction of dMotion has a slightly larger error for total frequency induced fatigue dTF , consid­ ering the high frequency stress caused by wave induced vibration. The cumulative error of the over prediction is about 16% for machine learning predic­ tion on d TF , but it is still follow the overall trend well. It can be seen that whipping and springing do add uncertainty and noise to the machine learning model. 6 CONCLUSIONS This study applied machine learning techniques to predict a North Atlantic sailing 2800TEU container vessel’s fatigue damage, based on the full-scale measurements data of the onboard hull monitoring system. The recorded ship’s heave and pitch motion responses (z2 ; z6 ), and the related velocity (_z2 , z_ 6 ) and acceleration (€z2 ; €z6 ) are deployed as input fea­ tures. Only these two motions are selected since they have the highest Pearson correlation coefficient rela­ tive to fatigue damage.

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The modeling targets are wave frequency induced fatigue damage dWF , and the total fre­ quency induced fatigue damage d TF considering the high frequency whipping and springing. Based on the comparison on the test set of individual unseen voyage, the proposed machine learning model have achieved much better predictive ability than the widely used spectral method for direct fatigue cal­ culation. It is also found that the high frequency whipping and springing has introduced uncertain­ ties and noise into the model establishment. That the machine learning models predictive ability on dWF are slightly better than dTF . The proposed models are also applied for longterm unseen voyages prediction. That the machine learning model still give quite good prediction for nearly one-year unseen sailing data. Especially for the wave induced fatigue damage, the prediction are almost consistent with the observation d WF . And the prediction has achieved at least 100% accuracy increase compared to the traditional spec­ tral method. It shows that the machine learning fatigue models are robust and suitable for ship fatigue monitoring. Based on the improvement, the ship can be operated wisely, leading to less main­ tenance, extended service life and enhanced onboard crew/cargo safety.

REFERENCES Bao. H., Wu. S., Wu, Z., Kang, G., Peng, X. & Withers, P. J. 2021. A machine-learning fatigue life prediction approach of additively manufactured metals. Engineer­ ing Fracture Mechanics. Vol. 242. Chen, T.Q. & Guestrin, C. 2016. XGBoost: A Scalable Tree Boosting System. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Dis­ covery and Data Mining, pp.785–794. CMEMS. 2021. Copernicus Marine Service: Global Ocean Physics Reanalysis. https://marine.copernicus.eu/. Copernicus. 2019. Copernicus Climate Change Service (C3S): ERA5 Fifth generation of ECMWF atmospheric reanalyses of the global climate. https://cds.climate. copernicus.eu/cdsapp#!/home. DNV. 2010. Classification Note No. 30.7: Fatigue Assess­ ment of Ship Structure. Det Norske Veritas. Feng, C., Su, M., Xu, L., Zhao, L., Han, Y. & Peng, C. 2023. A novel generalization ability-enhanced approach for corrosion fatigue life prediction of marine welded structures. International Journal of Fatigue. Vol. 166. Fricke, W., Cui, W., Kierkegaard, H., Kihl, D., Koval, M., Mikkola, T., Parmentier, G., Toyosada, M., & Yoon, J.H. 2002. Comparative fatigue strength assessment of a structural detail in a container ship using various approaches of classification societies. Marine Struc­ tures, 15 (1), 1–13. Fricke, W. 2017. Fatigue and fracture of ship structures. In Encyclopedia of maritime and offshore engineering, pp. 1–12. John Wiley & Sons, Ltd. Friedman, J., Hastie, T. & Tibshirani, R. 2000. Additive logistic regression: a statistical view of boosting (with discussion and a rejoinder by the authors). Ann. Statist. Vol. 28(2),pp.337–407.

Gaidai, O., Storhaug, G., Naess, A., Ye, R., Cheng, Y. & Xu, X. 2019. Efficient fatigue assessment of ship struc­ tural details. Ships and Offshore Structures, 1–8. He, L., Wang, Z.L., Akebono, H. & Sugeta, A. 2021. Machine learning-based predictions of fatigue life and fatigue limit for steels, Journal of Materials Science & Technology. Vol. 90, pp.9–19. ISO. 2015. Ships and marine technology - guidelines for the assessment of speed and power performance by ana­ lysis of speed trial data 15016. Jordan, C. R., & Cochran, C. S. 1978. In-service perform­ ance of structural details. Ship Structure Committee, Report SSC-272, US. Coast Guard, Washington, DC. Jordan, C. R., & Knight, L. T. 1978. Further survey of in-service performance of structural details. Ship Struc­ ture Committee, Report SSC-294, US. Coast Guard, Washington, DC. Li, Z., Mao, W., Ringsberg, J. W., Johnson, E., & Storhaug, G. 2014. A comparative study of fatigue assess­ ments of container ship structures using various direct cal­ culation approaches. Ocean Engineering, 82, 65–74. Mao, W., Ringsberg, J., Rychlik, I. & Storhaug, G. 2009. Comparison between a fatigue model for voyage planning and measurements of a container vessel. Proceedings of international conference on Ocean Offshore and Arctic Engineering (OMAE), Honolulu, Hawaii, May 31 June 5, 2009. Mao, W. 2010. Fatigue Assessment and Extreme Prediction of Ship Structures. Doctoral (PhD) Thesis, Department of Mathematical Sciences, Chalmers University of Tech­ nology, Gothenburg, Sweden. Mao, W., Ringsberg, J.W., Rychlik, I. & Storhaug, G. 2010. Development of a fatigue model useful in ship routing design. Journal of Ship Research, Vol. 54 (4), pp.281–293. Mao, W. 2014. Development of a spectral method and a statistical wave model for crack propagation prediction in ship structures, Journal of Ship Research, Vol. 58 (2), pp.106–116. Nejad, R.M., Sina, N., Moghadam, D.G., Branco, R., Macek, W. & Berto, F. 2022. Artificial neural network based fatigue life assessment of friction stir welding AA2024-T351 aluminum alloy and multi-objective opti­ mization of welding parameters. International Journal of Fatigue. Vol. 160. Pujol, J.C.F. & Pinto, J.M.A. 2011. A neural network approach to fatigue life prediction, International Journal of Fatigue. Vol. 33, pp.313–322. Rychlik, I. 1987. A new definition of the rainflow cycle counting method. International Journal of Fatigue. Vol. 9 (2), pp. 119–121. Rychlik, I. 1993. Note of cycle counts in irregular loads. Fatigue Fracture of Engineering Materials and Struc­ tures, 16 (4), 377–390. Storhaug, G., Moe, E. & Piedras Lopes, T.A. 2007. Whipping measurements onboard a midsize container vessel operat­ ing in the North Atlantic. Marintec China Proceedings (RINA, CMP and SNAME), Proceedings of international symposium on ship design and construction, pp.55–70. Yan, W., Deng, L., Zhang, F., Li, T. & Li, S. 2019. Prob­ abilistic machine learning approach to bridge fatigue failure analysis due to vehicular overloading, Engineer­ ing Structures. Vol. 193, pp.91–99. Zhang, M., Sun, C.N., Zhang, X., Goh, P.C., Wei, J., Hardacre, D. & Li, H. 2019. High cycle fatigue life pre­ diction of laser additive manufactured stainless steel: A machine learning approach, International Journal of Fatigue. Vol. 128.

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Numerical simulation of ship motion and non-linear sea loads of a modern frigate in regular waves Ziwen Zhang, Ning Ma & Qiqi Shi State Key Laboratory of Ocean Engineering, Shanghai, China School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China Institute of Marine Equipment, Shanghai Jiao Tong University, Shanghai, China

ABSTRACT: Ship design, especially design of warships, requires precise predictions of sea loads in extreme wave conditions. In recent years, with the enlargement tendency of ships, the stiffness of ship hull has become weaker, resulting in more significant non-linear springing and whipping loads, which should be paid more attention to. This paper presents a numerical study of non-linear sea loads of a modern Canadian frigate in regular head seas based on a time-domain non-linear hydro-elastic prediction program. Non-linear Froude-Krylov forces, slamming forces and restoring forces are taken into consideration. Transfer matrix method is used to solve natural vibration of dry mode as an input of the hydro-elastic program. The numerical results are provided for comparison with model tests conducted by McTaggart et al. This numerical study will also be a part of the benchmark study of MARSTRUCT, which will provide a reference for an appropriate uncertainty analysis procedure in the future.

1 INTRODUCTION Quantifying uncertainties in computing global wave loads on ships is a necessary prerequisite for efficient application of structural reliability methods. While benchmark studies of uncertainties in wave loads for rigid ships have been published in the past, the same is not true regarding hydro-elastic responses. The main challenges are related to the different aspects of prop­ erly modelling slamming load and whipping response. There are several methods of different complexity that may be employed for slamming computation, ranging from simplified added mass variation method to the CFD methods coupled with ship seakeeping analysis. Ship whipping response may be determined using either 1D Timoshenko beam method either 3D FEM. Different approaches are to be compared within this benchmark study to find mutual differences, quantify modelling uncertainties and eventually propose the most appropriate analysis procedure. As a part of this benchmark study, an Euler beam model is adopted with variable cross-section to simu­ late the natural modes of vibration. Transfer matrix method is used for solving the natural frequencies. A traditional 2-D model proposed by Bishop & Price (Bishop & Price 1979) is used for hydroelastic calcula­ tion. The ship motions are solved by 2-D STF strip method (Salvesen 1971), with the 2-D potential is simulated through Frank (Frank 1967) close-fit method. The radiation and diffraction force are

calculated on mean wet surface, while Froude-Krylov force and restoring force is integrated on the instantan­ eous wet surface, which is modeled through 3-order Non-Uniform Rational B-Splines (NURBS) surface. Impulse Response Function (IRF) method raised by Cummins (Cummins 1962) is applied for simulation in time domain. The hydroelastic model provides waveinduced vertical bending moment (VBM) with a sum of the components in first three order vibration modes. Whipping bending moment (WBM) has been simu­ lated based on slamming simulation with simplified theory (Ochi 1971, 1973). The simulation results of vertical ship motions, VBM of midship section, and WBM induced by 3 different wave heights are com­ pared with experimental data given by McTaggart (McTaggart 1997). 2 SHIP MODEL A modern Canadian Patrol Frigate (CPF) is tested in the benchmark study. In the numerical model, the information below is used. All the data are in full scale and all units below are in International System of Units (SI) if not mentioned. The main dimensions of the CPF are listed in Table 1. The whole ship hull is divided into 21 sec­ tions, which were named after Station No. 0 to 20 from fore perpendicular (FP) to after perpendicular (AP). As Figure 1 shows, the mass distribution of the hull is given as an important input.

DOI: 10.1201/9781003399759-12

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Another input for hydro-elastic ship model is the stiffness. Stiffness of specific sections have been listed in Table 2. The stiffness is assumed to be con­ tinuous and linear varied along the lengthwise. The geometry detail is given as an IGES file. The original Non-Uniform Rational B-Splines (NURBS)

surface of the CPF is divided into 15096 faces, as Figure 2 shows. Each face is a 3-order NURBS face container 9 subfaces. A python program is developed to segment the ship hull into different sections based on the IGES-format file.

Figure 1. Mass distribution of CPF.

Table 1.

Main Dimensions LPP LOA Beam Midships Draft Trim by bow

Table 2.

3 CALCULATION OF NATURAL VIBRATION MODES

Main dimensions of CPF (Full Scale). Value 124.5 m 134.7 m 14.8 m 4.97 m 0.04 m

Main Dimensions KG LCG GM Displacement, Δ Roll period (wet mode)

Euler beam model with variable cross-section is used to simulate the natural mode of vibration. Transfer matrix method is used for solving the nat­ ural frequencies. The variable cross-section model considered the hull beam us divided into 21 isometric Euler beams connecting to each other. The length, mass density and stiffness of the i-th section (i=0, 2…20) is defined as li, ðρAÞi and ðEI Þi , respectively, as shown in Figure 3. The definition is listed in Equation (1) below:

Value 6.26 m 59.45 m 1.08 m 4655 ton 12.3 s

Main dimensions of CPF (Full Scale).

Station No.

Vertical Bending EI (GN·m2)

2.5 5.0 7.5 10.0 13.7

1364 1799 2080 2521 2151

The balance equation of Euler-Bernoulli beam is Equation (2):

The Modes Function of the i-th section of a specific Yi(x) order is defined as Equation (3) below

Figure 2. NURBS faces of the CPF Hull.

Amongst which Xi ¼ ω2 � ðρAÞi =ðEI Þi , ω is the natural frequency of the whole variable cross-section

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Figure 3. Euler beam model with variable cross-section.

Euler beam. Ai ; Bi ; Ci ; Di are four parameters of the i-th section. Considering that the displacements, angles, moments and shear forces are continuous at the con­ necting points, these features of i-th and (i+1)-th section should be the same at the at the convergence of the two beam sections, thus we have:

Therefore,

Through boundary conditions of free beam Equation (8), the i-th order natural frequencies ωi can be solved.

According to Equation (4), set ½ A�i ¼ ½Ai ; Bi ; Ci ; Di �T , we can use a transfer matrix ½Z �i :

The parameters in the matrix are:

Replacing Xi in Equation (3) with the natural fre­ quencies ωi , the natural modes can be solved. The first 3-order natural frequencies of dry modes are 14.08 rad/s, 34.80 rad/s, 64.32 rad/s, respectively. And the natural frequencies of wet modes are 7.86 rad/s, 18.76 rad/s, and 34.57 rad/s, respectively. Comparing with the experimental results of natural frequencies, the bias of the numerical results above mainly come from the lack of stiffness data, since we only got data of 6 stations (ST.0, ST.2.5, ST.5.0, ST.7.5, ST.10.0 and ST.13.7) while the data of other stations are linear interpolated by the authors. Addition­ ally, the Euler beam model also credits to the bias since the shearing force are not considered. Figure 4 shows the simulation results of natural modes of the CPF. 4 PREDICTION OF SHIP MOTIONS AND WAVE-INDUCED VERTICAL BENDING MOMENTM 4.1

According to the continuous condition, Equation (6) can be derived:

Figure captions

Two coordinates are used in this study, as Figure 5 shows. One is the coordinate fixed to earth (Earth Coord­ inate, O-x0y0z0) and the other is fixed to the gravity center of the ship (Ship Coordinate, G-xyz). The discuss below is analyzed in the latter one, if not mentioned.

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Figure 4. Normalized natural modes of First 3 orders vertical vibration of CPF.

� � Yj i represents the j-th order natural mode of the i-th section, and pj represents the j-th order principal coordinate. Considering the hull as a linear system, based on basic equation of structural dynamics we have:

Figure 5. Two coordinates used in this study.

4.2

With Equation (11), it can be transformed into:

Hydro-elastic model

The two-dimensional traditional hydro-elastic model raised by Bishop and Price is used in this study. Only vertical motions and loads are taken into con­ sideration. The CPF hull is divided into 21 strips and assumed as an elastomer, which has 5 natural modes. Mode 0 represents heave motion, Mode 1 represents pitch motion, and Mode 2-4 represent first 3 orders of natural vibration modes. Based on the results in Chapter 3, we have:

A ¼ Y T MY is the generalized structural mass matrix, B ¼ Y T CY is the generalized structural damping matrix, C ¼ Y T KY is the generalized struc­ tural stiffness matrix. Y T F is the generalized force acting on the ship hull. The structural damping coef­ ficients are given by McTaggart (1997), which equal 2.7%, 2.5% and 3.1%, respectively for the first three modes of vertical bending. 4.3

For Mode 0 and Mode 1, from AP to FP the nat­ ural modes are:

For Mode 2-4, the natural modes are given in Chapter 3. The vertical motion displacement of i-th section is described by natural modes and principal coordinates as Equation (11):

Time-domain weakly non-linear strip method

According to the ITTC Recommended Procedure 7.5-02-07-02.5 Verification and Validation of Linear and Weakly Nonlinear Seakeeping Computer Codes, nonlinear methods can be classified into four main levels, as shown in Table 3. Level 2 is adopted in this study. All waves are assumed to be with minute ampli­ tude (Airy wave model) in the study. The STF strip method is applied for solving the generalized forces resulted from regular head wave, and the depth of water is supposed to be infinite. The wave-induced force is divided into Froude-Krylov (F-K) force FFK , radiation force Frad and diffraction force Fdiff . The gravity force and buoyance are inte­ grated as the restoring force Fres. Slamming force is not considered in this part. The F-K force and restor­ ing force are simulated on instantaneous wet surface in time-domain, which brings the weak nonlinearity,

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while the radiation force and diffraction force are cal­ culated on mean wet surface. Table 3. Categorization of nonlinear methods (ITTC 2021).

dr ðiωe Þ is the diffraction force in frequency domain. The restoring force and F-K force are simulated on the instantaneous wet surface, which can be described through the 15096 NURBS faces men­ tioned in Chapter 1:

Sw(t) represents the instantaneous wet surface at time t, ~ n is the normal vector of a specific NURBS face, dS is the area of the NURBS face. P is the com­ posed pressure caused by wave and still water (Zhang 2018), its value can be calculated through Equation (20), and P0 is P at initial time in still water, which is replaced for the gravity influence in the restoring force term.

Thus, Equation (13) can be transformed as below.

Based on the IRF method raised by Cummins, the time-domain radiation force acting on the r-th mode can be described with time delay function:

4.4

Ark ð∞Þ is the added mass at infinite encounter U0 0 frequency, BU rk and Crk are the added damping and added restoring coefficient caused by ship speed U0 . These two terms can be calculated through STF method, and will be demonstrated in detail in Chapter 4.4. Rrk ðtÞ is the time delay function of radiation force:

Potential calculation through frank close-fit method

The precision of STF strip method is based on the precision of 2-D potential calculation at each section. To calculate the 2-D potential, is to find out the solu­ tions to Equation (21) below:

As to diffraction force, the time-domain force acting on the r-th mode can be described with time delay function as well:

ζ ðtÞ ¼ Acosðkx þ ωe tÞ represents the elevation of a regular head wave, and Qr ðtÞ is the time delay function of diffraction force:

The close fit method raised by Frank (Frank 1967) is used. The contour of a 2-D section is seg­ mented into n lines between n+1 source points, as Figure 6 shows. For each source point, the Green function below is adopted:

109

2D The m2D 3 represents 2-D added mass and N3 rep­ resents 2-D added damping. The subtitle ‘3ʹ repre­ sents heave mode. 2D With m2D 3 and N3 , the 3-D hydrodynamics coef­ ficients can be obtained based on STF strip method and its expansion to hydro-elastic raised by Bishop & Price, as Equation (26) shows.

Figure 6. Diagram of a 2-D section.

ε ¼ ðy; zÞ represents the coordinate of a specific point P in the field, while ξ ¼ ðη; ζ Þ represents the coordinate of a source point Q,P:V: represents the Cauchy principal value, k means the wave number. For the potential at point P:

Aij and Bij are added mass and damping matrix, respectively.wi is the i-th natural mode function, i.e. the A00 is the added mass of heave motion (A33 in rigid body assumption). A Wigley III standard model is taken for verifi­ cation. The experimental data (Journee 1992) released by Journee in 1992 is used for compari­ son with the numerical simulation, as Figure 7 shows. The ‘irregular frequency’ problem in close-fit method, as Frank mentioned (Frank 1967), has also been paid attention to. Since the ‘irregular fre­ quency’ phenomenon is significant in the potential calculation, as shown in Figure 8, a moving average numerical method based on convolution Equation (27) is used for the elimination of irregular frequencies.

Then considering the surface boundary condi­ tion, we can have discrete equations for the central points Pi of each line li :

win represents time window vector, n=10 is discrete step,� represents the convolution. The results after irregular frequencies eliminations are shown in Figure 8 as well for comparison.

With the solution of Equation (24), the 2-D potential can be obtained, and the hydrodynam­ ics coefficient can be calculated through Equa­ tion (25):

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Figure 7. Numerical & experimental results of nondimensional added mass and damping.

4.5

Figure 8. Comparison with original added mass & damp­ ing results with results after irregular frequencies elimin­ ations (the CPF, Fr=0.12, Heave Mode).

Ship motion and linear wave-induced vertical bending moment results

Equation (14) can be reorganized as the form below:

The comparison between numerical results and experimental results (McTaggart 1997) of linear wave-induced vertical bending moment at midship section are shown in Figure 11. All the results are non-dimensional with the same way of McTaggart’s paper. Experimental data with three different wave heights. As Figures 9-11 show, the numerical results of ship motions fit well with experimental data, especially the results of pitch motion. Although the linear wave-induced vertical bending moment is 0-50% higher than the experi­ mental value, it fits well with the numerical results based on strip method conducted by McTaggart in 1997. Thus, the bias maybe a systematical error resulted from strip methods. Several other factors may also result in the bias:

The Newmark-β numerical method Equation (29) is applied for solving Equation 814).

Among Equation (29), the calculation parameters a0 to a5 are determined by α & β:

Two-order precision is promised by setting α ¼ 1=2 and β ¼ 1=6. Finally, the results of ship motions can be obtained. The comparison between numerical results and experimental results (McTaggart 1997) are shown in Figures 9 and 10. The linear wave-induced vertical bending moment at midship section can be obtained immedi­ ately through Equation (30):

1. The FK and restoring force are acting on instant­ aneous wet surface while the radiation force and diffraction force are still calculated on the mean wet surface. There may be an error without the non-linear correction of the latter when wave heights are large. 2. When solving the natural modes of the hull girder, an Euler beam model with variable section

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Figure 9. Non-dimensional numerical and experimental results of heave motion (the CPF, Fr=0.12, head wave).

Figure 10. Non-dimensional numerical and experimental results of pitch motion (the CPF, Fr=0.12, head wave).

Figure 11. Non-dimensional numerical and experimental results of linear wave-induced vertical bending moment (the CPF, Fr=0.12, head wave).

is used to calculate by the transfer matrix method, which ignores the effect of vertical shear force. 3. The time-domain slicing method based on Cum­ mins’ impulse response theory may lead to numerical bias when selecting the truncation of the delay function.

5 SLAMMING PRESSURE AND WHIPPING BENDING MOMENT In Chapter 4, the slamming force and the VBM resulted from slamming are excluded in Equation (14). However, when slamming happens, the transi­ ent pressure on the ship hull may lead to local and

112

pffiffiffiffiffiffiffiffiffi gLPP is the critical speed raised by Ochi, kb is determined as 4 (Lin 2021). B(x)is the mean beam length of the section, l(x) is length of the section, kf is determined from Stavovy’s theory (Stavovy & Chuang 1975), S(x) is the area of wet surface based on the NURBS faces near the instantaneous water­ line. The numerical results have been reorganized as non-linear data and compared with the experimental data, as Figure 12 shows. Since the first condition has quite a long wave length (λ ¼ 241:3 m), and the wave height of 3 dif­ ferent test conditions are proportional to wave length, the wave height reaches more than 8-16 meters high, which is far beyond the condition satis­ fying model’s linear assumption. Therefore, the result of the first condition (ω ¼ 0:505 rad/s) has been excluded in Figure 12. Compared with the results of ship motions and linear vertical bending moments, the results of whip­ ping bending moment seem not to be satisfactory. Errors may be resulted from:

global effects, which is so-called whipping. Whip­ ping may lead to a more severe 2-nodes vibration with high frequency and this part of VBM is defined as the whipping bending moment (WBM). Thus, slamming needs to be taken into consider­ ation, and the dynamic equation will be transformed to Equation (31):

In order to simulate whipping phenomenon, an approximate method (Gu, Shen & Moan 2003, Lin 2021, Stavovy & Chuang 1975) has been applied for calculating the slamming force Fslam. Firstly, slamming is classified into two main cat­ egories, one is bottom slamming and another is flare slamming. An assumption has been adopted that the bottom slamming only happens when the hull is just emerging into waves (the draught is less than 0.1*draft in still water) and the impact speed exceeds the thresh­ old speed set by Ochi’s experiments. Otherwise, when the hull motion approaching to wave elevation and the bottom is still emerged, the flare slamming happens. Since slamming happens in a very short period of time, the slamming impact is considered as a constant in a time step. Define zr ¼ zship Acos ðωe t þ kxÞ, the model of the transient slamming impact can be con­ cluded as Equation (32):

1. The required wave height condition is proportional to the wave length, resulting in a relatively low wave height in the middle-high frequency region. The relative speed may not be enough to reach the threshold (critical speed of Ochi), so that the bottom slamming is relatively weaker than real. 2. The semi-empirical formula and 2-D model is relatively rough. Several researches (Lin 2021) mentioned the results may have great deviations using the simplified model as well.

6 CONCLUSIONS As part of the benchmark study of MARSTRUCT, a traditional 2-D model proposed by Bishop & Price is used for hydroelastic calculation. An Euler beam model with variable cross-section is used to simulate the natural modes of vibration based on transfer matrix method. The ship motions are solved by 2-D time-domain method, based on STF strip method and IRF method. The 2-D potential is

dzship dt

Among them,dzdtr ¼ resents the vertical

þ Aωe sinðωe t þ kxÞ rep­ relative speed,Vcr=0.093.

Figure 12. Non-dimensional numerical and experimental results of whipping bending moment (the CPF, Fr=0.12, head wave).

113

simulated through Frank close-fit method. The radi­ ation and diffraction force are assumed to act on mean wet surface, while Froude-Krylov force and restoring force is integrated on the instantaneous wet surface, which is modeled through 3-order NURBS surface. The hydroelastic model provides VBM with a sum of the components in first three order vibration modes. Whipping bending moment has also been simulated based on slamming simulation with Ochi’s semi-empirical formula and Stavovy’s theory. The numerical results of ship motions fit well with experimental data, especially the results of pitch motion. The outcome of linear wave-induced vertical bending moment fits well with the numerical results based on strip method, and both are 0-50% higher than the experimental value. Compared with the results of ship motions and linear vertical bend­ ing moments, the results of whipping bending moment seem not to be satisfactory based on the existing simplified 2-D model. Several factors may lead to the bias have been analyzed.

Frank W. 1967. Oscillation of Cylinders in or below the Free Surface of Deep Fluids. Oscillation of Cylinders in Or Below the Free Surface of Deep Fluids. Gu X K, Shen J W, Moan T. 2003. Efficient and Simplified Time Domain Simulation of Nonlinear Responses of Ships in Waves. Journal of Ship Research, 47(3): p.62–73. ITTC. 2021: ITTC Recommended Procedures and Guidelines 7.5-02-07-02.5: Verification and Validation of Linear and Weakly Nonlinear Seakeeping Computer Codes. Journee I. 1992. Experiments and Calculations on Four Wigley Hull forms. SNAME Report. Lin Y. 2021. Numerical Prediction and Experimental Study on Nonlinear Hydroelastic Responses of Large Ships. Shanghai, China; Shanghai Jiao Tong University. Mctaggart K, Datta I, Stirling A, et al. 1997. Motions and Loads of a Hydroelastic Frigate Model in Severe Seas. Transactions SNAME. Ochi M K, Motter L E. 1971. A Method to Estimate the Slamming Characteristics for Ship Design. Marine Technology, 8(2): 219–32. Ochi M. 1973. Prediction of Slamming Characteristics and Hull Responses for Ship Design. Transactions SNAME, 81. Salvesen N, Tuck E O, Faltinsen O M. 1971. Ship Motions and Sea Loads. Transactions SNAME, 78. Standardization Administration of China. 2008. The Initial Graphics Exchange Specification (IGES). Beijing: China Standard Press: 484. Stavovy A B, Chuang S L. 1975. Analytical Determination of Slamming Pressures for High-Speed Vehicles in Waves. Zhang K. 2018. Research on Nonlinear Wave Loads’ Timedomain Hydroelastic Analysis Method and Application of Ships with Large Bow Flare. Harbin, China; Harbin Engineering University.

REFERENCES Bedel J W, Lee C M. 1971. Numerical Calculation of the Added Mass and Damping Coefficients of Cylinders Oscillating in or Below a Free Surface. NSRDC Report. Bishop R, Price W G. 1979. Hydroelasticity of Ships. Cam­ bridge University Press. Cummins W E. 1962. The Impulse Response Function and Ship Motions. Schiffstechnik, 9: 101–9.

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Slamming

Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Abnormal wave slamming impact of stiffened cylinders S. Bjørgo Fimreite, ZL. Yu & J. Amdahl Centre for Autonomous Marine Operations and Systems (AMOS, Norwegian University of Science and Technology (NTNU), Trondheim, Norway Department of Marine Technology, Norwegian University of Science and Technology (NTNU), Trondheim, Norway

ABSTRACT: The structural response of a stiffened cylinder due to slamming impact is studied using the ALE method in LS-Dyna. The numerical modelling approach is in the first place verified against experimental results from unstiffened plate drop tests. Afterwards, structural responses of a stiffened cylinder are investi­ gated by simulations with a range of impact speeds. The deformation mode, maximum deformation and slam­ ming pressure from each impact is studied. Pressure impulses from impact simulations of both rigid and deformable structures are reapplied on the stiffened cylinder to reproduce the structural damage. The obtained deformations are compared with those from the coupled simulation. Important characteristics of the slamming loads are identified and discussed. The structural response due to a relevant design load is also investigated and compared against the coupled simulation.

1 INTRODUCTION

to start deforming. The change in geometry however also leads to changes in the pressure distribution. Fal­ tinsen (1997) presented an analytical model for describing the structural response of plates due to slamming, coupling elastic vibrations of an Euler beam with the boundary value problem for an incom­ pressible fluid. As the fluid-structure interaction (FSI) of elastic vibrations and slamming loads have been better understood, recent research has also focused on describing the coupling between hydrodynamic loads and plastic deformations. Truong et al. (2022) presented a numerical study of slamming pressures on stiffened panels experiencing permanent deform­ ations. Abrahamsen et al. (2020) presented slamming experiments of an unstiffened plate experiencing large permanent deformations. An analytical approach for describing the coupled response using mode shapes from dry calculations was also intro­ duced. Yu et al. (2019a) also presented an analytical approach for describing the plastic deformations of plates, assuming a beam model for plate strips. The beam equation is solved coupled with the 2D bound­ ary value problem, describing the deformations of the strips in three separate stages. This approach is validated against numerical simulations and experi­ mental results in Yu et al. (2019b) and Yu et al. (2021), including the experiments used for validation of the numerical modeling in this paper. Recommendations for design of vertical offshore structures against slamming loads are presented in DNV GL OTG-14 (2019). The recommended prac­ tice gives ULS pressures with a maximum amplitude

Offshore wind parks are expected to be an important segment in the green energy production of the future. As the number of new developments increases, wind parks are also designed in more exposed seas. This exposes the structures to harsher conditions including large wave impacts. In extreme conditions, the waves can be expected to induce such large loads that per­ manent deformations of the structures are instigated. In order to ensure safety and reliability, methods for analyzing the hydrodynamic loads and the plastic structural responses are required. Due to the nature of wave slamming, these methods should solve for loads and deformations simultaneously. Theoretical studies of slamming impact go back to the 1930s, when von Karman (1929) and Wagner (1932) developed methods for estimating the slam­ ming pressures on rigid structures. Both methods use a flat plate approach to describe the instantaneous wet surface of a body entering water, but von Karman neglected the uprising of the free surface close to the body. In the following decades, the models have been developed further to be applicable for more complex geometries and to include previously neglected effects. Greenhow & Yanbao (1987) used both von Karman and Wagner approaches to calculate slam­ ming pressures and forces on a rigid cylinder. In the 1990s, the focus shifted from studying rigid to deformable impact, aiming to describe the coupling between the hydrodynamic loads and the structural deformations. When slamming occurs on a deformable structure, the hydrodynamic loads cause the structure

DOI: 10.1201/9781003399759-13

117

related to the position on the structure relative to the 100 year wave upwell. For design purposes, the pres­ sure loads should include typical safety factors for ULS design. The following study investigates structural response of a stiffened cylinder due to extreme slam­ ming impact. The cylinder is representative of a section of a cylindrical column in offshore struc­ tures like floating wind turbine foundations. Wave slamming events are idealized as the impact between a moving structure and still water. Numerical simula­ tions of the impacts are performed using the mechan­ ical solver in the finite element program LS-Dyna, using the available ALE method to model the fluidstructure interactions. In addition, structural response due to different types of slamming pressures is also investigated.

Figure 1. Numerical model of flat plate impact.

The fluid domains are modeled with 1 point ALE multi-material solid elements. Both the air and water domains have length, width and height 1500 × 1020 × 200 mm3 respectively. The material properties for water and air are presented in Table 1. Both mater­ ials are defined as null materials. Air is defined with a linear polynomial equation of state while water is defined with a Gruneisen equation of state. Mesh size 5 mm is used for both the Lagrangian structures and the Eulerian fluid domains based on a convergence study and recommendations of a global mesh size in ALE modeling.

2 VALIDATION OF NUMERICAL METHOD To ensure realistic results from the numerical simu­ lations, an initial model of experimental model tests is created to verify the modeling method. Model tests of slamming impact of unstiffened plates, causing large Updated with revised simu­ lations of flat impact. Presenting maximum deformations of the plates, were presented in Abrahamsen et al. (2020). In the tests, an unstif­ fened aluminum plate was clamped into a rigid test rig. The slamming impact was caused by dropping the impactor into still water. Tests were conducted with four different drop heights for flat impact. In addition, tests with a 4° deadrise angle between the plate and free surface were con­ ducted with two different drop heights. DIC measurements of the plate deformations were captured using two highspeed cameras. 2.1

Table 1. Material properties for the fluids, as given by Truong et al. (2022).

Numerical modelling approach

The numerical model of flat impact is illustrated in Figure 1. The test rig is idealized as a flat plate with an adjusted density to account for the total mass of the rig. The deformable plate is quadratic with side lengths 220 mm, while the bottom surface of the test rig is modeled with outer dimensions 340 × 500 mm2. Both plates are modeled with Belytschko-Tsay shell elements with a thickness of 0.6 mm. The plate representing the rigid test rig is mod­ eled with a rigid steel material with Young’s modu­ lus 210 GPa and Poisson’s ratio 0.3. To account for the total mass of the structure, the density is increased to 1.91 × 106 kg/m3. The simplified John­ son-Cook material model is fitted to the experimen­ tal material curve for the deformable aluminum plate, illustrated in Figure 2. Strain rate effects are neglected. In addition, the density, Young’s modu­ lus and Poisson’s ratio are taken as 2720 kg/m3, 72 GPa and 0.32, respectively.

Parameter

Air

Parameter

Water

Density ρ [kg/m3] C0 [MPa] C1 [MPa] C2 [MPa] C3 [MPa] C4 [MPa] C5 [MPa] C6 [MPa] E0* [MPa] V0** [-]

1.225 0.0 0.0 0.0 0.0 0.4 0.4 0.0 0.25 1.0

Density ρ [kg/m3] c*** [m/s] S1[-] S2 [-] S3 [-] γ0 [-] a**** [-] E0* [MPa] V0** [-]

1025 1480 1.921 -0.096 0.0 0.35 0.0 0.2895 1.0

* Initial internal energy ** Initial relative volume *** Sound of speed in fluid **** First-order volume correction

The fluid-structure interactions are defined using the Lagrange in solid constraint card, applying pen­ alty coupling between the Eulerian master part and the Lagrangian structures. A penalty factor of 0.1 for compressional coupling is applied. 3 coupling points are distributed over the Lagrangian surface elements, and coupling is applied when at least 30% of an element is made up of water. For more accurate

118

Figure 2. Curve fit of simplified Johnson-Cook model to experimental stress-strain curve.

Figure 5. Deformation of the point reaching maximum per­ manent deformation due to angled impact. Dashed lines represent experimental data.

modeling of the membrane actions of the thin struc­ ture, the control parameters used by Storheim (2016) have been included. In the experimental tests, the impactor was released from different heights, giving different velocities at the time of impact. As simulations of the plate falling through the air from a high drop height are computationally demanding with little reward, the plates are instead initially placed 5 mm above the free surface and impact speeds calculated assuming energy conservation of the different drop tests are applied as initial velocities of the plates. 2.2

Figure 3. Deformation of plate midpoint due to flat impact. Dashed lines represent experimental data.

Comparison of results

The development of maximum deformations during the impacts are illustrated in Figure 3 for flat impact and Figure 5 for angled impact. During flat impact, the maximum deformations occur at the center of the plate. The numerical simulations overestimate both maximum and permanent deformations of the plate. This is however explained by the deformation modes of the two approaches, as Figure 4 illustrates the deformations of the midline of the plate. The numer­ ical overestimations are caused by the more triangular shape of the deformations, overestimating the deform­ ations at the middle but giving good results for the non-central parts. The deformations of the point with largest per­ manent deformation due to the angled impact is illustrated in Figure 5. While the maximum deform­ ations are slightly underestimated, the permanent deformations of the plate are very well captured by the numerical simulations. Figure 6 also illustrates that there is good agreement in the deformation mode of the numerical simulations and the experi­ ments for angled impact.

Figure 4. Maximum deformation of the plate midline due to flat impact.

119

3.1

Figure 6. Maximum deformation of the plate midline due to angled impact.

For both flat and angled impact, the numerical simulations lack the elastic recovery of the plate after maximum deformations are reached which was observed in the experiments. The agreement between the deformations of the numerical simulations and the experimental results is nevertheless considered satis­ factory and the modelling approach acceptable for fur­ ther studies.

Slamming impact

Wave slamming events on the stiffened cylinder is studied by simulating a drop of the cylinder into still water, in a manner representative of the model tests described in the previous section. The wave impact velocity is idealized as an initial impact speed for the cylinder, applied as a constant speed for the global structure. The general approach described in the previous section for modeling the fluids and the fluid-structure coupling is also used for modeling cylinder impact. In the cylinder model, both the water and air domains have length, width and height 8 × 24 × 6 m3, and are meshed with solid elements with side length 100 mm. Wave impact on the middle 3 m long section of the cylinder is studied. FSI coupling is therefore only defined for the middle section of the cylinder accentuated in Figure 7, while the outer sections provide realistic boundary conditions for the impacted section. As the damage due to slamming is expected to be local, rigid bodies are created at the openings of the cylinder to represent connections to a longer undam­ aged column. The cylinder is initially located 10 mm above the free surface. To study a range of different structural responses, six impact speeds between 5 and 30 m/s are applied in the simulations, with steps of 5 m/s.

3 STIFFENED CYLINDER MODEL An 8 m long section of a cylindrical column is modeled, illustrated in the full impact model in Figure 7. All geometrical dimensions of the cylinder are given in Table 2. The cylinder is modeled using Belytschko-Tsay elements and a quadratic mesh with mesh size 100 mm determined from a convergence study. Material properties of typical offshore steels given in Storheim (2016) are used. The density, Young’s modulus and Poisson’s ratio are set to 7850 kg/m3, 204 GPa and 0.3, respectively. The power law hardening material model is applied with initial yield stress 338 MPa, strength coefficient K = 758 MPa and hardening exponent n = 0.19. Figure 7. Numerical model of slamming impact of the stiffened cylinder. Table 2.

Geometrical dimensions of stiffened cylinder. Value

Dimension Diameter D Length L Thickness t Stiffener spacing l Stiffener web height h Stiffener web thickness ts

mm 12,000 8000 25 1000 500 25

4 RESULTS FROM SLAMMING IMPACT SIMULATIONS Deformation histories of the nodes reaching max­ imum permanent deformation in each of the impact simulations are illustrated in Figure 8. The struc­ tures reach maximum deformation a while after impact, after which they recover to a slightly lower permanent deformation. The magnitude of the deformations increases non-linearly with increasing impact speed.

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Figure 8. Maximum cylinders.

deformations

of

the

stiffened

deformation also moves away from the centerline. This change in deformation mode gives more spa­ tial variations in the slamming pressures. Average slamming pressures on panels 2-1 and 2-2 in Figure 11 during 10 m/s impact are reported in Figure 10. The pressure histories are representative for panels along the centerline (panels 1-2, 2-2 and 3-2) and on the sides. The panel along the centerline experiences a high peak pressure due to the initial flat impact of the bottom. During the remainder of the impact, lower pressures in the range of 1 MPa are sustained. The time instance of maximum pres­ sure on the side panel is somewhat delayed compared to the middle panel due to the vertical positions on the panel. This delay is decreased for increasing impact speeds. The side panel is not greatly affected by the peak pressure of the middle panel but has similar sustained pressure for the dura-tion. The results illustrated in Figure 10 are representative of all impact speeds. However, the magnitude of both peak pressure on the middle panels and sustained pressures increase with increasing impact speed.

Figure 9. Effective plastic strains in the cylinder after 30 m/s impact.

The deformations due to the 30 m/s impact is dis­ tinct from the other simulations in that deformations develop in the later stages of the impact. This is however not caused by increasing slamming pres­ sures. Instead, the continued loading after the peak pressure causes the stiffeners to buckle, causing deformation growth, whereby two plastic hinges are formed, see Figure 9. The hinges act as boundary conditions of the deformed stiffened cylinder in the center, which on one hand prevent deformation development outward in the circumferential direc­ tion, but on the other hand cause deformation growth inward in the radial direction due to the postbuckling reduction of structural resistance. Impact with the four lowest impact speeds cause similar deformations to the cylinder: some deform­ ation of the area covered by the panels in Figure 11 and a local indentation along the centerline on both sides of the central stiffener. The maximum deform­ ations occur in the middle of these indentations. With impact speeds higher than 20 m/s, more damage is caused to a larger portion of the impact­ ing section and no clear indentations may be observed. The location of the absolute maximum

Figure 10. Slamming pressures on the middle and one of the central panels during 10 m/s impact.

The average pressure of over all nine panels is also illustrated in Figure 10. The spatial average pressure of the large panel has an initial pressure peak like the centerline panel, but with lower peak pressure. For the duration of the impact, the average pressure is similar to the side panel pressure. 5 REPRODUCTION OF SLAMMING DAMAGE WITH PRESSURE IMPULSES During slamming impact, the structure is subjected to complex pressure distributions with large spatial and temporal variations. While coupled water impact simulations give good information about the structural response due to slamming, these simulations are com­ putationally demanding. For design purposes, it is

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helpful to have simplified methods such as application of pressure loads. The structural response of the undeformed cylinder due to different slamming pres­ sures is therefore studied by applying pressure impulses as distributed loads. Only the structural part of the model is used for this. Four different ways of generating representative pressure histories are used.

5.1

Figure 11. 1 × 1 m2 panels on the bottom of the cylinder, used for calculation and application of pressure load.

The first two approaches are extraction of pressures directly from the coupled ALE simulations as follows. P1: The 3 × 3 m2 slamming exposed region is considered as a whole, where only one representative spatially average pressure is extracted. P2: The 3 × 3 m2 slamming exposed region is subdivided into 9 panels, each of size 1 × 1 m2, as illustrated in Figure 11. The time history of the aver­ age pressure on each panel is extracted. The industry practice in design against slamming is often to use pressure histories from design rules e.g. DNV GL OTG-14 (2019) or those measured from slamming experiments with rigid bodies. The fluid structure coupling effects are therefore not con­ sidered. In correspondence to that, two more average pressure curves P3 and P4 are generated. P3: ALE slamming simulations are also performed with a rigid cylinder, and the average pressure of the 3 × 3 m2 slamming exposed region is extracted. P4: The design pressure from DNV GL OTG-14 (2019) for a 3 × 3 m2 panel of an undeformed structure is calculated, including a material factor of 1.15, a load factor of 1.3 and a tensile factor of 1.3 according to DNV (2022). The design pressure impulse is compar­ able to a 20 m/s impact by use of Equation 2, where p is the peak pressure, ρ is the water density, Cp is a pressure coefficient taken as 5.15 and V is the impact velocity.

Structural response

The maximum deformations of the structure due to reapplication of the pressure loads on the nine panels, approach P2, are illustrated in Figure 12. Results from the coupled simulations are presented with full lines and pressure load responses as dashed lines. For the four lowest impact speeds, there is very good agreement between the two approaches. Both deformation magnitudes and variations are well cap­ tured. The dry pressure application models do how­ ever include some higher frequency vibrations not found in the wet impact. For the two highest impact speeds, there is less agreement between the two approaches. In addition to underestimating the deformation magnitudes, the development of the deformations is also in disagree­ ment. As mentioned in the previous section, these cylinders are damaged in a less regular mode than the lower impact speeds. While the deformation modes are not captured with the pressure loads. The disagreements between the coupled wet responses and the dry responses are expected to be caused by spatial pressure variations not captured with the nine panels. Better agreement is therefore expected if more panels of a smaller size for pressure calcula­ tions and reapplication are used.

Figure 12. Maximum deformations of the stiffened cylin­ der, due to coupled impact simulations (full lines) and application of P2 pressure loads (dashed lines).

Figure 13 illustrates the response of the undam­ aged cylinder due to a range of different pressure application approaches. The coupled simulation graph illustrates the deformations from the 20 m/s impact simulation. Pressure impulses P1 and P2 are extracted from the deformable impact simulation, spatially averaged over the 3 × 3 m2 panel and 1 × 1 m2 panels, respectively. The P3 pressure impulse repre­ sents an impact simulation with a rigid structure while P4 represents the OTG-14 design load includ­ ing safety factors.

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The response due to the large panel pressure impulse from the coupled simulation severely under­ estimates the response of the structure, with the max­ imum dry deformation being about 1/3 of the wet response. The spatial average pressure impulse does not capture any pressure variations across the bottom section of the cylinder. This causes similar underesti­ mations of the structural response as is observed with the nine smaller panels for the two highest impact speeds. Overall, this illustrates the importance of the spatial distributions of the pressures as a parameter in the slamming description.

spatial variations, will therefore allow for good calcula­ tions of the structural responses. Finally, the structural response due to a comparable design pressure is also included in Figure 13. The design load is applied with a total safety factor of 1.794. Despite this, the design load is found to underestimate the structural response of the coupled simulation, which is nonconservative. There are however some uncertainties related to the design loads. No direct reasoning for the creation of the design pressure impulses is available in the open literature, and an earlier version of the tech­ nical guidance uses almost twice as high maximum pressure magnitudes as the current version. Experi­ ments by Kvålsvold, Faltinsen & Aarsnes (1997) also illustrated the difficulty of accurate relations between impact velocity and maximum slamming pressure according to Equation 2. Further investiga­ tions of design loads and their applications are nevertheless necessary. 6 CONCLUSIONS

Figure 13. Structural response due to pressure loads on a 3 × 3 m2 panel, compared to 20 m/s impact damage.

Applying loads measured during a rigid impact to a deformable structure leads to an uncoupled approach for estimation of slamming response and has been a typical method for design against slam­ ming. The response due to the large panel rigid pres­ sure impulse overestimates the maximum deformation by about 49%, and even larger over­ estimations are found when using the nine smaller panels separately from the rigid impact as well. This approach gives conservative estimations of the slam­ ming response which may be used for design, but due to the magnitude of the overestimations, design based on this will be unnecessarily conservative. The difference in the response due to deformable and rigid pressure impulses is caused by the hydro­ plastic effects, i.e. the changes in the hydrodynamic pressures due to the structural deformations. In industry applications, application of rigid slam­ ming loads has been a popular approach. In order to correct the overestimations caused by this approach, the structures are often corrected with added masses. It is however not clear how large the applied added mass should be, which introduces significant uncertainty. By direct application of the coupled slamming pressures, the deformations were however captured without intro­ ducing any corrections. A sufficient description of the slamming pressures, including both the temporal and

Abnormal wave slamming events on stiffened cylinders has been studied, idealized numerically as a drop of the structure into still water. The impact has been modeled using the ALE approach in LS-Dyna with full coupling between the structural and the fluid analysis. The struc­ tural response due to pressure loadings representative of slamming impact has also been studied. The deformations of the stiffened cylinder due to slamming impact increase non-linearly with the impact speed. For impact speeds up to a limit, here 20 m/s, the deformations from every impact have similar modes with varying magnitude. At high impact speeds, the deformations become less regular and differ increasingly more from the initial deform­ ation mode at low speeds. The structural response due to the slamming impact is very well recreated by reapplying the pres­ sures from the impact to an undeformed structure. The accuracy of this approach is highly dependent on the spatial distribution of the pressures, requiring a sufficient representation of the slamming pressures. This is supported by disagreements between impact results and results from pressure reapplication when the spatial average pressures are calculated over too large panels of the structure. A typical slamming design load is found to underestimate the structural response of a coupled impact simulation. The deformations from an uncoupled approach, applying pressure loads from a rigid impact simulation to a deformable struc­ ture, is however found to overestimate the response from a corresponding coupled simula­ tion. Both approaches cause uncertainties in the design process of similar structures. Further investigations into design principles should there­ fore be conducted.

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ACKNOWLEDGEMENT The authors gratefully acknowledge the financial support by Research Council of Norway via the Centers of Excellence funding scheme, project number 223254 – NTNU AMOS. The authors would also like to thank the support from high per­ formance computation resources from the Norwe­ gian nation e-infrastructures, Project NN9585K – Accidental actions on strait crossings and offshore platforms.

REFERENCES Abrahamsen, B.C., Alsos, H.S., Aune, V., Fagerholt, E., Faltinsen, O.M. & Hellan, Ø. 2020. Hydroplastic response of a square plate due to impact on calm water. Physics of Fluids 32(8):82–103. DNV 2022. DNV-RP-C208 Determination of structural capacity by non-linear finite element analysis methods. DNV GL 2019. DNVGL-OTG-14 Horizontal wave impact loads for column stabilized units. Faltinsen, O.M. 1997. The effect of hydroelasticity on ship slamming. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Schiences 355(1724):575–591. Greenhow, M. & Yanbao, L. 1987. Added masses for circular cylinders near or penetrating fluid boundar­ ies – review, extension and application to waterentry, -exit and slamming. Ocean engineering 14 (4):325–348.

Kvålsvold, J., Faltinsen, O.M. & Aarsnes, J.V. 1997. Effect of structural elasticity on slamming against wetdecks of multihull vessels. Journal of Ship and Ocean Technol­ ogy 1(1):1–14. Storheim, M. 2016. Structural response in ship-platform and ship-ice collisions. PhD Thesis, Norwegian Univer­ sity of Science and Technology. Truong, D.D., Jang, B.S., Ju, H.B. & Han, S.W. 2022. Pre­ diction of slamming pressure considering fluid-structure interaction. Part 1: numerical simulations. Ships and Offshore Structures 17(1):7–28. von Karman, T. 1929. The impact on seaplane floats during landing. Washington, DC.: National Advisoty Commit­ tee for Aeronautics. Wagner, H. 1932. Über stoß- und gleitvorgänge an der oberfläche von flüssigkeiten. Journal of Applied Math­ ematics and Mechanics 12(4):103–215. Yu, Z., Amdahl, J., Greco, M & Xu, H. 2019a. Hydroplastic response of beams and stiffened panels sub­ jected to extreme water slamming at small impact angles, part i: An analytical solution. Marine Struc­ tures 65:53–74. Yu, Z., Amdahl, J., Greco, M & Xu, H. 2019b. Hydroplastic response of beams and stiffened panels subjected to extreme water slamming at small impact angles, part ii: Numerical verification and analysis. Marine Struc­ tures 65:114–133. Yu, Z., Cao, A. & Amdahl, J. 2021. Experimental and numerical validation of an analytical hydro-plastic model for the prediction of structural damage in extreme water slamming. In Jørgen Amdahl & C. Guedes Soares (ed.), Developments in the Analysis and Design of Marine Structure; Proc. intern. conf., Trondheim, 7-9 June 2021. Leiden: Balkema.

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Comparison between deforming meshes and overset meshes for water entry of a wedge M.F. Silveira, S. Wang & C. Guedes Soares Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

ABSTRACT: Water entry of a wedge into calm water is investigated by multiphase simulations using the open source CFD toolbox OpenFOAM. Comparisons are made between the dynamic and kinematic results using deforming meshes and overset meshes. Firstly, the simulation of a wedge with a deadrise of 30º is simu­ lated using the overset grids with different resolutions. Uncertainty analysis is made for the maximum acceleration and pressure coefficient of the wedge during water entry. The numerical predictions of vertical displacement, vel­ ocity, acceleration and slamming pressures are also validated against the published experiments. Then the results from the fine resolution with overset grids are compared to the ones from the morphing meshes. In general, the results are in accordance with each other, and differences are observed mostly at the peaks, however the overset mesh method requires more of the CPU than the morphing method.

1 INTRODUCTION When a ship is navigating in stormy seas, slamming is a typical occurrence, and it has complicated phys­ ical implications on ship structures, occurring when a ship’s bottom rapidly strikes the water. Similar slamming can also happen on the connecting struc­ ture between a catamaran’s two hulls or the under­ side of the deck of offshore rigs (Wang & Guedes Soares, 2017; Hallak et al, 2022). These loads have large amplitude and very short duration and might be responsible for causing structural damage and transient response (Wang, et al. 2021a; Wang, et al. 2021b). Assessing the water entry impact, however, is a challenge even nowadays. The most reliable method to study and evaluate this is by performing experiments, but it is unrealistic to measure the pressure this way since it would be necessary to cover all hull-sensitive areas with pressure sensors and it would limit the loca­ tions where these measurements can be taken. This is why several alternatives are in development, such as finite element analysis models together with CFD simulations, computational models, etc, but are also still improving and requiring validation from the exist­ ing experimental known results (Huang, et al., 2021). As an important application in naval architecture, the dynamic variation of pressure on surface ships and offshore structures during the water entry pro­ cess has been a long-lasting topic. The pioneering study is attributed to von Karman (1929), who aimed to develop a method capable of obtaining the

impact force on a seaplane landing on the water sur­ face by proposing the application of momentum variation to compute the hydrodynamic force acting on a bluff body penetrating the liquid surface. This depends on the speed rate and wetted area. Later on, Wagner (1932) established another groundwork by developing the theoretical models on the ideal­ ized problem of a two-dimensional wedge entering the water and assuming it to rise as jet flows and hit on the walls of the wedge, to have irrotational flow and applicable for linear boundary conditions neg­ lecting gravity (Huang, et al., 2021; Shen, et al., 2016). Wagner’s asymptotic solution has gone through adaptations, and it was applied to various research and practical conditions. It can be applied for obtaining the water entry loads and coupling them with structural solution, but it also neglects some hydrodynamic phenomena, causing the mod­ elled flow field to be unrealistic. One example of these phenomena is the flow separation that typic­ ally occurs for wedge bodies. Later on, Zhao & Faltinsen (1993) developed the Boundary Element Method (BEM), a numerical method for studying water entry of a two-dimensional body of arbitrary cross-section which relies on the panel method with potential flow theory to obtain the Froude-Krylov forces by integrating the pressure of each discretised panel. Although BEM is still widely used as a reference, nowadays, the solution neglects the effects of gravity, which limits the application to the water entry process where gravity can be not accounted for. Sun & Faltinsen (2007) extended the

DOI: 10.1201/9781003399759-14

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method to include gravity, while Wu et al (2010) included the nonlinear velocity potential flow theory. The applications of some simplified approaches to water impact problems of simple structures, including 2D wedge-shape sections, 2D bow-flared sections and 3D cones, were reviewed and compared by Wang & Guedes Soares (2014). Still, these methods do not account for viscous effects and are still focused on 2D cases or simple 3D geometries. Rather than solving the problem numerically using BEM, one method that has shown to be promising and it is widely studied recently is by using volumetric methods, also known as Computational Fluid Dynam­ ics (CFD). The first time this approach has been applied to a ship section’s slamming was in the studies of Arai et al. (1994), where the Euler solver, consider­ ing inviscid and incompressible fluid, and the frac­ tional volume of fluid method (VOF) was used to compute the transient deformation of the free surface, and compared with the present analytical and experi­ mental results. Moreover, the development of these CFD methodologies made it possible to be one of the few methods that account for viscosity with the option to couple a turbulence model to solve the NavierStokes equation. One branch of CFD is the mesh-based volumes, also known as the Finite Volume Method (FVM), it is a method that relies on the discretization of the fluid domain. It can account for complex structures by a closed curve (for a two-dimensional approach) or surface (for a three-dimensional approach) in con­ tact with numerous neighbouring cells (Huang, et al., 2021). This approach allows not only be applied to predict fluid behaviour, structural loads, and deformations (Pena, et al., 2019) with good accuracy and with viscous and turbulent flows being modelled (Khojasteh, et al., 2020), but can also account for and compare different effects on slam­ ming impacts. The studies of the effects of compressibility and air cavity were investigated for a free-falling wedge cylinder made using the software OpenFOAM in Wang & Guedes Soares (2020), for instance, and the effects of three-dimensionality were further researched using the same software in Wang et al. (2021b). These distinct aspects not only were studied but also validated with the recent experimental stud­ ies of the same wedge from Wang et al. (2015), Cointe & Armand (1987), and also, the analytical results from Korobkin (2005). Most of these studies only dealt with two-dimensional structures and the application for them is in the strip theory, widely used in ship motions research (Wang et al. 2021b). One challenge for the FVM in the context of slamming is the large displacement that the solid body can face, especially in the case of free-falling objects or structures. There is a limit on the body’s offset in the deformation mesh method because a large amplitude motion will distort the cells and crash the simulation. There are some ways to get

around this. One of them is by considering the whole domain to move while the body is fixed, which was done in the studies of Huang et al. (2020) for resist­ ance. The problem with this approach is applicable when the motion is prescribed, but for free-falling structures, it should be defined for each interaction. Another approach is the one used, for instance, in the studies of Wang et al. (2021b), where the struc­ ture is placed right above the waterline, avoiding large motion amplitudes. Although this can be fairly applied for small structures and when the velocity before hitting the water is well-known, the motion can start from a big dropping height and with a con­ siderable air resistance to be accounted for before water entry. Another approach to deal with this challenge is to incorporate an advanced mesh approach. One of them is the Immerse Boundary Method (IBM), which considers the body as a closed wall boundary moving along the domain and deactivates the internal fluid cells, while the external ones are com­ puted. The original IBM, however, does not have specialized boundary cells outside the moving wall (Huang, et al., 2021; Mittal & Laccarino, 2005), and this oversimplifies the boundary-layer effect of the geometry and generates uncertainties in the analysis. Zheng et al. (2020) used IBM to simulate the water entry process of a wedge body and applied a ghostcell approach to improving the original IBM’s inef­ fective boundary layer modelling, with results in good accordance with the benchmark experiments of Yettou et al. (2006). On the other hand, a widely applied method in CFD is the overset (chimera) mesh, where two domains are created: one for the background con­ taining information about the fluid and another, for the geometry (chimera). Water entry studies using this approach have been validated and shown to have proven to be adaptable and effective for movements of high amplitude (Shen, et al., 2016; Ma, et al., 2018). This makes it appear as though the overset mesh approach is better appropriate for the slamming problem. Considering the water entry problems, Wang et al. (2021a) estimated the numer­ ical uncertainty due to discretization on the Arbitrary Lagrangian-Eulerian (ALE). The study concluded that the uncertainty due to discretization in CFD was case-specific and parameter specific. A constant Courant-Friedrichs-Lewy (CFL) number-based dis­ cretization approach was recommended. The objective of this study is to simulate the water entry problems using the overset meshes solver, overInterDyMFoam in OpenFOAM v1912 and to compare the results with the ones from the morphing meshes solver and available experimental data. Comparisons are made on the vertical displace­ ment, velocity, acceleration, slamming pressures and CPU expenses. Numerical uncertainty due to discret­ ization is also performed to assure the accuracy of the solver.

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2 NUMERICAL METHOD Open Fields Operation and Manipulation, also known as OpenFOAM, is a C++ toolbox for the development of customized numerical solvers, and pre-/post-processing utilities for the solution of con­ tinuum mechanics problems. For this project, a multiphase simulation based on the VOF method is used. This method is relevant for engineering applica­ tions since it is applied when two immiscible fluids set the scenario studied (Manafpour & Hamzeh, 2017). Furthermore, this solver is applied over the chimera meshing technique so it can deal with large amplitude motion, which is the case of the majority of slamming cases. The tool from OpenFOAM that supports the VOF method and overset meshes is the overInterDyM­ Foam solver, which stands for overset and dynamic meshing version of interFoam. For two or more immiscible fluids, the VOF method is a surfacetracking method where the location of the fluid inter­ face is of interest. The fluids in this model share a single set of momentum equations, and the volume fraction of each fluid in each computational cell is monitored over the entire domain. The physical for­ mulation of this problem of two isothermal, incom­ pressible and immiscible fluids is based on the continuity, momentum and interface capturing advec­ tion equations below, respectively (Weller et al., 1998; Rusche 2002).

where ρ is the fluid density, V is the fluid velocity vector, τ the� viscous stress tensor h i� defined as τ ¼ 2μS ¼ 2μ 0:5 ðrV Þ þ ðrV ÞT , μ the fluid dynamic viscosity, P the scalar pressure, F σ the volu­ metric surface tension force, g the gravitational accel­ eration vector, r � ðV C αð1 αÞÞ an anti-diffusion heuristic term and α the interface capturing. Regarding the floating body, it is modelled as a free rigid body where the forces considered are gravity and the surface forces of pressure and shear stress. At each time step, the six DoF solver from OpenFOAM performs the integration of pressure and viscous stress component over the wetted surface SH to assess the resultant force and moment around the CG. The accelerations are obtained by dividing both resultants by their respective inertia term and it can be integrated into velocity and displacement by the Newmark integration using γ = 0.5 and β = 0.25.

A general implementation for the use of uncon­ nected (also known as Chimera) meshes is the overset framework in OpenFOAM. In this approach, two independent and disconnected meshes are created: the background and overset (chimera). This method is very helpful in situations involving mesh motion and interactions. It avoids the problems and instabilities associated with deforming meshes (Tisovska, 2019). The principle of this meshing approach is to give, at each time step, a label to each cell of both domains, where it describes if the cell is calculated, where the equations are solved for this cell; hole, where there is no computation for this cell; or interpolated, which is when the value is computed from the nearest elem­ ents of the other domain (background or overset). Between the cells, there are the donors, which are the ones that provide values, and the acceptors, whose value gets set from interpolation. The sim­ plest one and also the fastest that is considered in this project is the “cellVolumeWeight”, which uses weights proportional to the volume of the acceptor cell inside a given donor cell and normalized to the total volume of the acceptor. 3 WATER ENTRY OF A WEDGE SECTION 3.1

Simulation setting

The open-source CFD software OpenFOAM is used to perform simulations of the 2D wedge free-falling into calm water using overset (chimera) mesh approach. The numerical results are validated by comparing the slamming pressures and vertical motions with the experimental study of Wang et al. (2015). Figure 1 shows the dimensions, the geometry of the wedge cylinder and the arrangement of the pressure sensors applied in the experiments. The present study focuses on the comparisons between the overset meshes and morphing meshes by considering the 2D case instead of the 3D case which requires very high CPU expenses. It was also mentioned by Wang et al. (2021b) that the three-dimensional effects had negligible effects on the pressures for the section in the middle of the 3D body.

Figure 1. Arrangement of the prismatic wedge and sensors from Wang et al. (2015). Units in mm.

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Table 1.

Parameters of the wedge and simulation.

Parameter

Value

Deadrise angle Mass Initial velocity Gravity Water density Water kinematic viscosity Air density Air kinematic viscosity

30° 161.5 g 2.5 m/s 9.8065 m/s2 998.2 kg/m3 1 mm²/s 1 kg/m3 1.58 mm2/2

Figure 2. Background mesh with dimensions in mm and the cell distribution in the x-y plane.

Figure 3. Details on the medium overset meshing.

Three simulations are performed with different mesh sizes and following the division shown in Figure 2 with the background meshes and the simple grading ratios. The smallest cell size in the water entry region is 4 mm (coarse), 2 mm (medium) and 1 mm (fine) respectively. It means that a constant refinement ratio of 2 is used.

In the simulation, the wedge is placed 0.01 m above the calm water surface, and it enters the water with an initial velocity of 2.5 m/s. The motion of the wedge is restrained to the vertical axis. The mass is also defined based on the 32.3 kg/m from Wang et al. (2015). The simulation’s parameters are shown in Table 1. The details of the meshes around the wedge section for the medium model are shown in Figure 3. Even though the simulation is two-dimensional, OpenFOAM performs as a three-dimensional extru­ sion with a thickness of 0.01 m. The empty boundary condition is applied to the front and back faces. Wall boundary condition was given to the right face and the atmosphere standard boundary condition is applied to the top face, while the left face is a symmetry plane. Pressure sensors (patch Probes) are used on the locations of P4, P5 and P6 as illustrated in Figure 2. The Courant number is kept constant at 0.25 based on a timestep of 10-5 s. The overset mesh with the wedge is refined and extruded around the wedge, as shown in Figure 3. The simulations are performed on a regular desktop equipped with an Intel Core i7-

Figure 4. Comparison of pressure results from CFD with different validation data from Wang et al. (2015).

Figure 5. Comparison between kinematics results from CFD with validation data from Wang et al. (2015).

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4570 @ 3.2 GHz with 94.1 GB of RAM and using 4 parallel simulations. Further details of the meshing and CPU time can be seen in Table 1. It is notable that with the increase in the number of cells, the CPU is also more demanding by taking longer to process the whole simulation. 3.2

and also calculate a corrected uncertainty, Uic. With this, the maximum acceleration and maximum slam­ ming pressure coefficient, Cp = p(t)/[0.5ρV²(t)], where p(t) and V(t) are the instantaneous pressure and velocity, and ρ is the water density, are ana­ lysed by this method, and the summary is shown in Table 2.

Model validation

Figure 4 compared the numerical time history of the pressures P4, P5, and P6 from OpenFOAM simulations using overset meshes with the experimental results and the ones from the 2D Boundary Element Method (BEM) of Wang et al. (2015) studies. The results from the BEM method are denoted by “numerical”. In general, the results are in good agreement among the three approaches, although deviations were observed on peak values. The punctual differ­ ences that appear are the appear delay to the begin­ ning of the ascension and the peak, which can be explained by the initial shift given to not disturb the background mesh in the simulation. The peaks differ from each other due to the high uncertainties present in the experimental and numerical analysis, as con­ cluded by Wang et al. (2021b), which can also be noted in the large uncertainty bar presented in the results of Wang et al. (2015). The OpenFOAM results of vertical acceleration, velocity, and wedge draft are compared in Figure 5. In the draft time series, due to the high similarity of the experimental results and the numerical ones (BEM), just the experimental were considered. In all three cases, the 2D simulations are in better agree­ ment with the numerical solutions, even considering the three different meshes. As for the difference concerning the experimen­ tal result and the numerical ones, the acceleration peak is notably higher, resulting also in differences in behaviour in the velocity time series. This has to do with three-dimensional effects neglected in the numerical methods, which were tested by Wang et al. (2021b) performing the same simulation but considering it three-dimensional and applying the actual length in a 3D domain. It is expected that this model should agree better with the twodimensional BEM numerical approach, which has few deviations due to potential flow consideration. To perform the convergence analysis and quan­ tify the discretization errors, the constant CourantFriedrichs-Lewy (CFL) number is used, along with the approach of a correction factor based on Richardson extrapolations following ITTC (2017) standards, with the modification proposed in Wang et al. (2021a). The uncertainties associated can be assessed by a factor of safety approach (Roache, 2003), which can be used to define the uncertainty Ui with a factor of safety of FS = 1.25 for careful grid studies to bound simulation error. The factor of safety strategy, albeit not suggested by Roache (2003), can be employed in circumstances when the answer is corrected with an error estimate from RE

Table 2. Uncertainty calculation for maximum acceler­ ation and pressure coefficient. Uncertainty calculation Output values

Maximum accel­ Maximum eration [m/s2] Cp

Ø1 (fine) 43.77

6.801

Ø2 (mid) 42.66 Ø3 44.41 (coarse)

6.925

Refinement ratio r

2

5.948 2

-0.6345

-0.1275

Convergence ratio

ϵ21/ϵ32

Order of accuracy

p

0.6564

2.9709

Approximate relative error

e21a

0.0254

-0.0182

e32a

0.0410

-0.1411

Extrapolated relative error

e21ext

-0.0421

0.0027

e32ext

0.0766

-0.0201

GCI21fine

0.0550

-0.0033

GCI32fine

0.0889

-0.0256

U1

5.498%

0.331%

U2

8.893%

2.564%

U1c U2c

1.100% 1.779%

0.066% 0.513%

Grid conver­ gence index (GCI) Uncertainty

Corrected Uncertainty

3.3

Comparison with deforming mesh method

The same scenarios were investigated by Wang et al. (2021b), but instead of chimera meshes, the morphing mesh was applied using the same soft­ ware for reducing computational expense. The simulations from this investigation were recreated using parallelization to reduce the computational time. A comparison between the computational time of using chimera and deforming meshes using simi­ lar mesh size processors is shown in Table 3. It is seen that the overset mesh method requires more of the CPU than the morphing method, even by using 4 parallel processors to run the simulations. This happens mostly due to the drawback of additional computational load introduced by the interpolation process between domains (Berton, et al., 2017).

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Figure 6. Pressure comparison between fine deforming and overset meshes.

Figure 7. Comparison between kinematics of the wedge for deforming and overset meshes.

Figure 8. Contour plot of alpha.water over time for fine deforming (left) and overset (right) meshes.

Table 3.

Coarse Medium Fine

Comparison between CPU time, in hours. Deforming

Overset

Difference

0.035 0.462 4.896

0.11 1.07 37.34

214% 132% 663%

Comparisons are made between the dynamic and kinematic results of the wedge water entry problem. The first is the pressure from the sensors P4, P5 and P6, as shown in Figure 6, by considering the finest mesh of both methods. In general, the results agree with each other, and differences are observed mostly at the peaks. The kinematics results are compared in the plots of Figure 7. There are differences of 4.2% between the peaks of acceleration but mostly show the same results. This variation is even less perceptible in the velocity and penetration results shown in Figure 7. While for P4 and P5 this difference remains at 3.4% and 4.4%, respectively, for P6 this was more

discrepant, with about a 10% difference. These regions are known to variate to a great extent for how refined the mesh and time step are, and the difference between resolutions observed in the comparison between con­ tour plot in Figure 8 and also, in the results related to dynamics, such as acceleration – as a result from force integration – and pressures, can be an explanation for that together with the convergence velocity, which tends to be faster for overset mesh approach (Lopez Mejia, et al., 2021). Figure 8 shows the water surface elevations at different time moments during the water entry process. For each subplot, the left part is from the morphing meshes simulation and the right one is from the overset meshes simulation. It is seen that the evolu­ tion of the wave jet is quite similar for both models, although slight deviations are observed on the water spray in the later stage. 4 CONCLUSIONS The study performed the CFD simulations of the free falling of a wedge into calm water using the open-source CFD toolbox OpenFOAM with overset

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meshes. The vertical displacement, acceleration, velocity and slamming pressures are compared with available experimental data and BEM results, show­ ing good agreement except for the peak values. The differences in the peaks are mainly due to the uncer­ tainties in the model tests and the numerical predic­ tions, and also the 3D effects which were found in the previous study. The uncertainty analysis is performed by using three models with different resolutions The convergence and uncertainty quantification are estimated following the ITTC guidelines with a constant refinement ratio of 2 but with a constant CFL number. Both the maximum acceleration and pressure coefficient show oscillatory convergence, even though the corrected uncertainties are below 2 %. Comparisons between morphing and overset mesh are also done and mostly the results are the same. Differences can be found at peaks of pres­ sures. When the fine resolutions are used, the max­ imum difference is about 10% which is for the peak value of the pressure P6. The differences in vertical motions are all below 5%. However, the CPU expense for overset meshes is about 663% higher. The study shows that the overset meshes achieve the similar accuracy as the morphing meshes when the fine resolution is applied. Even though, the computa­ tional time for morphing mesh is fairly lower than chi­ mera, the simulation may crash due to large motion of the body. The overset mesh is recommended for study­ ing the water entry cases with a large displacement.

ACKNOWLEDGEMENTS This work contributes to the Strategic Research Plan of the Centre for Marine Technology and Ocean Engineering (CENTEC), which is financed by FCT under contract UIDB/UIDP/00134/2020. The numer­ ical simulations are also supported by INCD through the advanced computational resource project from FCT under contract CPCA/A1/407670/2021.

REFERENCES Arai, M., Cheng, L.-Y., & Inoue, Y. 1994. A Computing Method for the Analysis of Water Impact of Arbitrary Shaped Bodies. Journal of the Society of Naval Archi­ tects of Japan, 1994(176), 233–240. Berton, A., D’Orrico, F., & Sideri, M. 2017. Overset grids for fluid dynamics analysis of internal combustion engines. In Energy Procedia, 126, 979–986. Cointe, R., & Armand, J.-L. 1987. Hydrodynamic Impact Analysis of a Cylinder. Journal of Offshore Mechanics and Arctic Engineering, 109(3), 237. Greenshields, C. J. (2015). OpenFOAM, The open source CFD Toolbox. User Guide. Hallak, T. S.; Teixeira, A. P., and Guedes Soares, C. 2022. Epistemic uncertainties on the estimation of minimum air gap for semi-submersible platforms. Marine Struc­ tures. 85, 103244.

Huang, L., Tavakoli, S., Li, M., Dolatshah, A., Pena, B., Ding, B., & Dashtimanesh, A. 2021. CFD analyses on the water entry process of a freefall lifeboat. In Ocean Engineering, 232, 109115. Huang, L., Tuhkuri, J., Igrec, B., Li, M., Stagonas, D., Toffoli, A., Cardiff, P., & Thomas, G. 2020. Ship resist­ ance when operating in floating ice floes: A combined CFD& DEM approach. In Marine Structures, 74, 102817. Khojasteh, D., Tavakoli, S., Dashtimanesh, A., Dolatshah, A., Huang, L., Glamore, W., Sadat-Noori, M., & Iglesias, G. 2020. Numerical analysis of shipping water impacting a step structure. Ocean Engineering, 209, 107517. Korobin, A. 2004. Analytical models of water impact. European Journal of Applied Mathematics, 15(6), 821–838. Lopez Mejia OD, Mejia OE, Escorcia KM, Suarez F, Laín S. 2021. Comparison of Sliding and Overset Mesh Techniques in the Simulation of a Vertical Axis Turbine for Hydrokinetic Applications. Processes, 9(11), 1933. Ma, Z. H., Qian, L., Martínez-Ferrer, P. J., Causon, D. M., Mingham, C. G., & Bai, W. 2018. An overset mesh based multiphase flow solver for water entry problems. Computers & Fluids, 172, 689–705. Manafpour, M. & Ebrahimnezhadian, H. 2017. The Multi­ phase Capability of OpenFoam CFD Toolbox in Solving Flow Field in Hydraulic Structure. Verlag der Tech­ nischen Universität Graz. Mittal, R., & Iaccarino, G. 2005. Immersed Boundary Methods. Annual Review of Fluid Mechanics, 37(1), 239–261. Pena, B., Muk-Pavic, E., Thomas, G., & Fitzsimmons, P. 2019. Numerical analysis of a leading edge tubercle hydrofoil in turbulent regime. Journal of Fluid Mechan­ ics, 878, 292–305. Roache, P. J. 2003. Criticisms of the “Correction Factor” Verification Method. J. Fluids Eng, 125(4), 732–733. Rusche, H. 2002. Computational Fluid Dynamics of Dis­ persed Two-Phase Flows at High Phase Fractions. Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London, UK. Shen, Z., Hsieh, Y.-F., Ge, Z., Korpus, R., & Huan, J. 2016. Slamming Load Prediction Using Overset CFD Methods. In Offshore Technology Conference. OTC. Sun, H., & Faltinsen, O. M. 2006. The influence of gravity on the performance of planing vessels in calm water. Journal of Engineering Mathematics, 58, 91–107. Sun, H., & Faltinsen, O. M. 2006. The influence of gravity on the performance of planing vessels in calm water. Journal of Engineering Mathematics, 58(1-4), 91–107. Tisovska, P. 2019. Description of the overset mesh approach in ESI version of OpenFOAM. In Proceedings of CFD with OpenSource Software, Edited by Nilsson. H. von Kármán, Th. 1929 The impact on seaplane floats during landing. National Advisory Committee on Aero­ nautics, Washington, DC. Wagner, H. 1932. Über Stoß- und Gleitvorgänge an der Ober­ fläche von Flüssigkeiten. ZAMM - Zeitschrift Für Ange­ wandte Mathematik Und Mechanik, 12(4), 193–215. Wang, J., Lugni, C., & Faltinsen, O. M. 2015. Experimental and numerical investigation of a freefall wedge verti­ cally entering the water surface. Applied Ocean Research, 51, 181–203. Wang, S., & Guedes Soares, C. 2014. Comparison of simpli­ fied approaches and numerical tools to predict the loads on bottom slamming of ship structures. Guedes Soares, C.

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& Lopez Pena F., (Eds.). Developments in Maritime Transportation and Exploitation of Sea Resources, Taylor & Francis Group London, UK; 157–168. Wang, S., & Guedes Soares, C. 2017. Review of ship slam­ ming loads and responses. Journal of Marine Science and Application, 16(4), 427–445. Wang, S., & Guedes Soares, C. 2020. Effects of compressibil­ ity, three-dimensionality and air cavity on a free-falling wedge cylinder. Ocean Engineering, 217, 107589. Wang, S.; Islam, H., & Guedes Soares, C. 2021a Uncer­ tainty due to discretization on the ALE algorithm for predicting water slamming loads. Marine Structures. 80, 103086. Wang, S., Xiang, G., & Guedes Soares, C. 2021b. Assess­ ment of three-dimensional effects on slamming load pre­ dictions using OpenFoam. Applied Ocean Research, 112, 102646. Weller, H.G., Tabor, G., Jasak, H. & Fureby, C., 1998. A tensorial approach to computational continuum

mechanics using object-oriented techniques. Com­ puters in Physics, 12(6), pp.620–631. Wu, G.-X., Xu, G.-D., & Duan, W.-Y. 2010. A summary of water entry problem of a wedge based on the fully non­ linear velocity potential theory. Journal of Hydrodynam­ ics, 22(5), 859–864. Yettou, E.-M., Desrochers, A., & Champoux, Y. 2007. A new analytical model for pressure estimation of symmetrical water impact of a rigid wedge at variable velocities. Journal of Fluids and Structures, 23(3), 501–522. Zhao, R., & Faltinsen, O. 1993. Water entry of two-dimensional bodies. Journal of Fluid Mechanics, 246, 593–612. Zheng, K., Zhao, X., Yang, Z., Lv, C., Duan, S., Lin, W., & Fang, Z. 2019. Numerical simulation of water entry of a wedge using a modified ghost-cell immersed boundary method. Journal of Marine Science and Technology, 25 (2), 589–608.

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Experimental investigation on oblique entry of trimaran cross deck structure Shou-qi Tang China Ship Development and Design Center, Wuhan, China

Shi-li Sun, Hui-long Ren & Xue-qian Zhou College of Shipbuilding Engineering of Harbin Engineering University, Harbin, China International Joint Laboratory of Naval Architecture and Offshore Technology between Harbin Engineering University and the University of Lisbon, Harbin, China

ABSTRACT: The slamming problem is common when a trimaran sails in rough sea conditions, especially for the cross deck structure, and the slamming with non-zero angles between the cross deck structure and the water surface will occur. In the present study, an experimental study of oblique entry of trimaran cross deck structure is carried out, and the relationship between angle, velocity and the slamming load are explored. The trimaran section is released from different heights with various angles above the water and falls into water. Then experi­ ment is also repeated for various falling heights. The relationships between the distribution of the slamming pres­ sures with angle and the velocity are discussed, based on the results and analysis, some conclusions are drawn.

1 INTRODUCTION Compared with monohulls, the trimaran has unique and excellent advantages (Ali & Brack, 2022), such as faster speed, better stability, and easier maneuver­ ability (Andrews & Zhang, 1995), However, when a trimaran performs missions on high seas, the slam­ ming phenomena is inevitable and threat the struc­ tural safety of the hull seriously (Dias & Ghidaglia, 2018). The cross deck structure plays an important role in connecting the main hull and the side hull, at the same time, the shape of the cross deck structure is usually relatively straight, Therefore, the slamming problem of the cross deck is more worthy of study. The pioneering work was done by Wagner (1932). A plate-fitting method based on von Kármán’s research was proposed to solve the slamming problem, which laid the foundation for understanding fluid and struc­ tural interactions in slamming theory. This theory has been widely used in fluid-solid collision problems. Typical works include Armand & Cointe (1987), Howison et al. (1991), Scolan & Korobkin (2001, 2012), Korobkin & Scolan (2006), Oliver (2007) and so on. The analytical solution is valid for the kind of simple problems in the existing study mentioned above, However, for the slamming load on the con­ necting bridge of the trimaran with a complex profile, the experiment is still the most effective method at pre­ sent. On the one hand, the phenomenon can be observed directly and experimental data can be obtained. On the other hand, it will also inspire the

establishment of the analytical model. For this reason, a series of experiments have been conducted by researchers on the slamming load. An experimental study of three-dimensional water impact at constant speed was presented by Alaoui et al. (2012), and the results for several axisymmetric shapes are presented and discussed. Good agreements between the theoretical model, numerical results and available experimental measurements have been obtained. Huera-Huarte et al (2011) used a Slingshot Impact Testing System (SITS) to carry out a series of experiments in order to study the slamming force on flat plates during free surface impacts. A series of experiments on the water entry of rigid and elastic wedges have been carried out for deadrise angles ran­ ging from 0 degrees to 45 degrees and for drop heights ranging from 0.1 m to 1.0 m by Duan et al (2020). A 1:64 scaled segmented ship model with U-shape open cross-section backbone was newly designed by Lin et al (2020) to simultaneously satisfy the laws of elastic similarity of vertical, horizontal, and torsional stiffness to study the asymmetric impact effects on hydroelastic responses. Jia & Zong (2022) put forward a series of experiments to study the different outrigger positions in calm water and regular waves. An experimental study was performed by Shabani et al (2019) to quantify slamming pressures in the archways between the bow and main hulls and the Centre bow slamming force on a 112m wavepiercing catamaran. In the experiment, it was found that increasing the wet-deck height will

DOI: 10.1201/9781003399759-15

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increase the motions but reduce the maximum slamming load in moderate waves. Catamaran wet-deck slamming was also experimentally investigated by Swidan et al. (2016, 2017) using a servo hydraulic slam testing system and the relationships between the peak force magnitudes, relative impact angle and vertical velocity were observed. Tang et al. (2022) presented an experi­ mental procedure together with a novel model design to study the air cushion effect during ver­ tical water entry of the trimaran section. A set of experiments were taken to study the relation­ ship between the drop heights as well as heel angles and slamming pressure by Duan et al. (2022). It can be seen in the review of the existing studies that the research on the problem of water slamming with non-zero angles mainly focuses on the wedge or other symmetrical simple shape models. However, for the slamming problem of the trimaran cross deck, it is more about the vertical falling body slam­ ming, and the study of the slamming time history of the whole ship model in different wave environ­ ments. In this paper, the slamming problem of the trimaran cross deck with angle into the water is experimentally studied to explore the relationship between the numerical discreteness, distribution of the cross deck slamming and the angle as well as the height. 2 MODEL DESIGN AND EXPERIMENT PROCEDURE 2.1

In this paper, according to a certain scale ratio, the geometric dimensions of the experimental model are obtained based on the profile of a trimaran section with beam B=793mm, height D=273.5mm, and sec­ tion length T=200mm., the external profile of the model structure was ensured to be geometrically similar to the ship. The profile, dimensions and experimental model are as shown in Figure 1.

Experiment design

The vertical velocity between the real trimaran cross section and the incident wave is predicted by the three-dimensional wave load calculation method soft­ ware COMPASS-WALCS developed by the China Classification Society at first then through the dynamic similarity between the real ship and the model, that is to say, the vertical falling velocity of the model and the relative vertical velocitypofffiffiffiffithe ffi � real ship conform to a certain scale ratio 1 : 50 , then the water entry velocity of the model can be con­ verted, by the free fall formula, the height of the experiment can be obtained, the test conditions in this study are finally determined as shown in Table 1. Table 1.

Experiment cases.

Case no.

Oblique Angle

Falling height

Case ID

1 2 3

0°

25mm 115mm 204mm

0Dh1 0Dh2 0Dh3

4 5 6

3°

25mm 115mm 204mm

3Dh1 3Dh2 3Dh3

25mm 115mm 204mm

5Dh1 5Dh2 5Dh3

7 8 9

2.3

Model design

Figure 1. Experiment model.

2.2

5°

Experiment device and sensor measuring point layout

Figure 2 shows the positions of 10 probe-type pressure sensors with a range of ± 500 kPa to measure the slam­ ming pressure. Measurement points P1–P4 are on the same level. Measurement points P1 and P2 are sym­ metrical to P3 and P4 for the purposes of verifying the symmetricity of the experiments. However, P5, P6, and P7 are not symmetrical to P10, P9, and P8. These pres­ sure sensors are staggered on both sides of the model to provide more measurement points. The displacement sensor, S1, has a measurement range of 0 mm to 3000 mm. The falling velocity of the model is obtained by differentiating the displacement. The sampling rate of the data acquisition system is set to 1000 Hz. The test is completed using a falling experimental device, as shown in Figure 3.

Figure 2. Sensor measuring points layout.

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black and white balance was checked and the FPS (Frames Per Second) was set to 1000. (4) Model release and data acquisition. The model is released onto the still water surface, and the test data are collected by a data acquisition system. Once a test is complete, the model is hoisted again and released again after the water surface is calm. For each case, the test is repeated multiple times in order to reduce random errors.

3 EXPERIMENTAL RESULTS AND ANALYSIS 3.1

Figure 3. Experiment device (1, Water tank; 2, Braced frame; 3, The free-falling frame; 4, The catamaran section Model).

Repeatability verification

Each case was carried out more than once to ensure the accuracy of the results, T1~T4 in Figure 5 shows the measures of 4 tests for the same cases. It can be seen in the figures and Table 2 that the repeatability is excellent in the pressure at P9 for Case 0Dh2, the pressure at P6 for 3Dh3, P5 for 5Dh1 and the vel­ ocity for Case 0Dh1and Case 3Dh2.

The falling frame “3” and the model “4” are fixed by a pair of matching fixtures, and the typical match­ ing fixture is shown in Figure 4.

Figure 4. Typical matching fixtures.

2.4

Experimental operation process

(1) Calibrate the sensor. The sensor is calibrated before use to check whether its measurement accuracy meets the standard. Calibration is done to check the proportional relationship between the sensor output and the given accurate input. (2) Place the falling body test stand and adjust the height of the model. When placing the falling body test frame, it is ensured that there is no inclination, then through changing the matching fixtures to achieve the change of the angle. (3) Set the high-speed camera. during the experi­ ment, using the high-speed camera to capture the motion of the free surface during the falling test. In an attempt to improve the quality of the images to facilitate the analysis of results, the

Figure 5. Results of multiple tests for the pressure and vel­ ocity for the same cases.

3.2

Analysis and discussion

3.2.1 Analysis of the time history of slamming pressure During the experiment, the high-speed camera was used to capture typical instants of vertical and oblique water entries, and the slamming of the tri­ maran cross deck is analyzed based on the phenom­ enon and the experimental data.

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Through the pictures taken at typical moments, the free surface changes of the main hull, cross deck, side hull and free surface can be clearly observed, as shown in Figure 6(a)~(f). For 0° vertical water entry slamming, Model release and contact with the water surface, The liquid surface is pushed due to the fall­ ing motion of the model, forming an outward jet, In the process of entering the water, there is an energy accumulation and the liquid level climbs up along the main hull. When the cross deck entered the water, the slammed occurred, the gas liquid mixture is produced at the middle of the connecting bridge splashes outward. Table 2. case.

The deviation of velocity and pressure of one

/

P9 0Dh2 Pressure kPa

P6 3Dh3 Pressure kPa

P5 5Dh1 Pressure kPa

Symmetry Verification Pressure kPa

Velocity 0Dh1 mm/s

Value

The Maximum Deviation

T1

11.33

T2 T3 T4

11.41 4.20% 10.93 11.19

T1

18.14

T2 T3 T4

17.45 3.80% 17.57 17.73

T1 T2 T3 T4 P1 P4

10.42 10.40 2.30% 10.64 10.54 8.3 4.10% 7.96

P2 P3

6.33 0.80% 6.38

T1 1069.98

Figure 6. Free surface at typical instants during the entry.

no longer climbs along the surface of the main hull, but only along the side of the main hull, When the left and right cross deck respectively contact with the liquid level, the gas-liquid mixture will also be formed and then ejects outwards, while large bubbles will be formed under the cross deck. In addition, it can be found that most of the splashes produced by slamming are mainly located near the outer corner point, it is because that when the cross deck slam­ ming occurs the liquid level is pushed to the two sides, so it is not easy to run out, and a greater impact will be produced on the corner, this phenom­ enon is more obvious during the oblique entry.

2.90%

T2 1101.93 Velocity 3Dh2 mm/s

T1 1347.77 0.64% T2 1356.4

At the same time, there are large and uniform bubbles forming under the connecting bridge. As the model continues to fall, more gas-liquid mixture is ejected outward and gradually breaks up. Figure 7 and Figure 8 show the typical moments of slamming of the oblique connecting bridges with an inclination of 3° and 5° respectively. Through comparison, it can be seen that the phenomenon of slamming of falling bodies at two angles is similar. When the main hull enters the water, the liquid will also be discharged outward to form an outward jet, and the jet on the left side is slightly more obvious than that on the right side, As the model continues to fall, the liquid level will also climb which is obvi­ ously different from the vertical fall, The liquid level

Figure 7. Free surface at typical instants during the entry of the segment with an inclination of 3°.

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the shorter the time interval between “double peaks” is; what`s more, Whether it is inclined into the water at 3° or 5°, when the height increases from h2 (115mm) to h3 (204mm), Δt barely changes. This shows that for large falling heights, the change of angle has little influence on Δt.

Figure 10. Δt at different cases. Figure 8. Free surface at typical instants during the entry of the segment with an inclination of 5°.

The time histories of slamming pressure at P6 and P9 show that there is an obvious “double peak” phe­ nomenon in the process of slamming in the cases of oblique water entry, which is related to the height of the falling body. The time history curve of slamming pressure is shown in the figure below Figure 9.

3.2.2 Analysis of slamming pressure peak value The peak slamming pressure at P5, P6, P7, P8, P9 and P10 on the cross deck is shown in Figure 11. It can be seen that the larger the initial inclination angle is, the greater the peak value of slamming pressure is. This phenomenon becomes more signifi­ cant with the increase of the falling height, espe­ cially for a very small height (h1=25mm), the peak values of slamming pressure at P6, P7 and P8 are almost the same at different angles, and the peak values of slamming pressure at P5, P9 and P10 are also relatively close. When the falling angles are the same, the higher the falling height, the greater the peak value of slamming pressure

Figure 9. P6 and P9 slamming pressure.

Denote the instant when the slamming pressure of P6 reaches the peak value as t1, the time when the slamming pressure at P9 reaches the peak value as t2, and the time interval of the “double peak” phe­ nomenon as Δt, as is shown in the below figure, we can find that, The time interval of “double peak” phenomenon at 5° is significantly longer than that at the tilt angle of 3°, The higher the falling height is,

Figure 11. Slamming pressure peak value.

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P5~P7 on the right side of the connecting bridge contact the water surface first, and then the P8~P10 measuring points, in order to study the distribution of slamming pressure in the oblique water entry of the trimaran cross deck, the pressure versus falling height at these points are measured and plotted in Figure 12. The same patterns as above can be observed. When the falling height is consistent, the peak value of slamming pressure increases with the initial inclination angle, and this “increase” effect is weaker when the falling height is low.

3.2.3 Analysis of slamming pressure coefficient The slamming pressure coefficient, as a dimensionless value, is more useful for comparison. in theory, the slamming pressure coefficient k of the model and the full-scale vessel as a dimensionless value are the same, through the slamming pressure coefficient, the slamming pressure of the corresponding real scale ship can be calculated.

where, k, P, ρ, and V represent the pressure coeffi­ cient, slamming pressure, the density of water, and the maximum velocity of the model, respectively.

Figure 13. Slamming pressure coefficient.

It can be seen in Figure 13 that, regardless of the initial inclination angle, the increase in the falling height will reduce the slamming pressure coefficient. At the same height, the increase of angle will also reduce the coefficient. This “reducing” effect weak­ ens as the height increases. 4 CONCLUSIONS

Figure 12. The distribution of slamming pressure.

In addition, according to these curves, regardless of the initial inclination angle, the closer to the corner, the greater the peak value of slamming pres­ sure, that is, P7 on the right and P10 on the left have the greatest peak value, especially when the falling height is low, this phenomenon is more obvious.

In this paper, the falling slamming experiment is carried out using the model of the trimaran cross section, in order to study the relationship between height angle and slamming of the cross deck. At the same time, the experimental data and photos of typical moments are used to analyze from both numerical and phenomenological aspects.

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(1) Compared with the vertical water entry slam­ ming, the oblique water entry slamming has a “double peak” phenomenon of the slamming peak values of the cross deck on both sides, The increase of the initial tilt angle will increase the time interval between the two peaks. This “increase” effect decreases as the height increases, for oblique water entry, the maximum slamming value appears near the outer corner (P7, P10) on each side, which should be paid attention to during actual construction and verification. (2) For the same water entry angle, the peak value of slamming pressure increases with the increase of falling height. For the same falling height, the greater the inclination angle is, the greater the peak value of slamming pressure is, and this phe­ nomenon is more obvious when the falling height is higher. (3) For small falling heights, the angle has a great influence on the slamming pressure coefficient. The smaller the angle is, the greater the slam­ ming pressure coefficient is. With the increase of falling height, the influence of angle change on the slamming pressure coefficient gradually weakens.

REFERENCES Alaoui, A. E. M., et al. (2012). “Experimental study of coefficients during vertical water entry of axisymmetric rigid shapes at constant speeds.” Applied Ocean Research 37: 183–197. Ali, D., Burak, Y. (2022). “Numerical Prediction of Form Factor and Wave Interference of A Trimaran for Differ­ ent Outrigger Positions.” China Ocean Engineering, 36(02):279–288. Andrews, D. J., Zhang, J. W. (1995) Trimaran ships: the configuration for the frigate of the future. Naval Engin­ eers Journal, 107(3):77–93. Armand, J. L., Cointe, R (1987) Hydrodynamic impact ana­ lysis of acylinder. Journal of Offshore Mechanics and Arctic Engineering, 109(3):237–243. Dias, F. and J.-M. Ghidaglia (2018). Slamming: Recent Progress in the Evaluation of Impact Pressures. Annual Review of Fluid Mechanics, Vol 50. S. H. Davis and P. Moin. 50: 243–273.

Duan, L. L., et al. (2020). “Experimental study on the propagation characteristics of the slamming pressures.” Ocean Engineering 217: 16. Duan, W. Y., et al. (2022). “Experimental study of slam­ ming pressure for a trimaran section with different drop heights and heel angles.” Ocean Engineering, 263. Howison, S.D., Ockendon, J. R., Wilson, S. K (1991). Incompressible water-entry problems at small deadrise angles. Journal of Fluid Mechanics, 222:215–230. Huera-Huarte, F. J., et al. (2011). “Experimental investiga­ tion of water slamming loads on panels.” Ocean Engin­ eering 38(11-12): 1347–1355. Jia, J. B., Zong, Z. (2022). “Experimental Study on the Configuration Hydrodynamics of Trimaran Ship” Jour­ nal of Marine Science and Application, 21(03):46–55. Korobkin, A.A., Scolan, Y. M. (2006) Three-dimensional theory of water impact.Part 2. Linearized Wagner problem. Journal of Fluid Mechanics, 549:343–373. Lin, Y., et al. (2020). “Experimental study on the asymmet­ ric impact loads and hydroelastic responses of a very large container ship.” International Journal of Naval Architecture and Ocean Engineering 12: 226–240. Oliver, J. M. (2007) Second-order Wagner theory fortwo-dimensional water-entry problemsat small dead­ rise angles. Journal of Fluid Mechanics, 572, 59–85. Scolan, Y. M., Korobkin, A. A (2001) Three-dimensional theory of water impact, Part 1: Inverse Wagner problem. Journal of Fluid Mechanics, 440:293–326. Scolan, Y. M., Korobkin, A. A (2012) Hydrodynamic impact (Wagner) problem and Galin’s theorem. 27th International Workshop on Water Waves and Floating Bodies, Copenhagen, Denmark, 22–25. Shabani, B., et al. (2019). “Slam loads and pressures acting on high-speed wave-piercing catamarans in regular waves.” Marine Structures 66: 136–153. Shabani, B., et al. (2018). “The effect of centre bow and wet-deck geometry on wet-deck slamming loads and vertical bending moments of wave-piercing catamarans.” Ocean Engineering 169: 401–417. Swidan, A., et al. (2017). “Wetdeck slamming loads on a developed catamaran hullform - experimental investigation.” Ships and Offshore Structures 12(5): 653–661. Swidan, A., et al. (2016). “Experimental drop test investi­ gation into wetdeck slamming loads on a generic cata­ maran hullform.” Ocean Engineering 117: 143–153. Tang, S. Q., Zhang, Y. et al. (2022). “Experimental investi­ gation on the air-cushion effect during free fall of a trimaran section using an air escape control method” Ocean Engineering, 254(254). Wagner, H. (1932). The phenomena of impact and planning on water. Technical Report, 1366, NACA.

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Fluid-structure interaction analysis for bow-shaped structure subjected to slamming pressure K. Toh & D. Yanagihara Kyushu University, Fukuoka, Japan

K. Nagayama Namura Shipbuilding Co., Ltd., Saga, Japan

ABSTRACT: In this study, the fluid-structure interaction (FSI) analyses utilizing the Incompressible Computa­ tional Fluid Dynamics (ICFD) solver implemented in LS-DYNA are performed. In addition, the accuracy verifica­ tion not only of the fluid force calculation by water surface drop analyses using several wedge-shaped structures but also of the FSI analysis by a box-shaped structure is carried out. Assuming the bow flare slamming (BFS) behavior, the FSI analyses by the ICFD method of LS-DYNA on bow-shaped structures are also conducted. The differences of numerical results between the Arbitrary Lagrangian-Eulerian (ALE) method and the ICFD solver are investigated. The results show that the ICFD analysis gives the relatively reasonable values for time histories of fluid forces as well as pressure responses regardless of two-dimensional and three-dimensional structures. Through the comparison with theoretical values and empirical formulas, moreover, the discussions on the ICFD results are presented in detail.

1 INTRODUCTION In recent years, ships are becoming larger to optimize operating costs, and the bow flare angle of ship hulls; especially container ships, tends to increase. As a result, large impact loads are applied to the bow structural members, and the bow flare slamming (BFS) is one of the concerns. In the phenomena of structural response under such impact loads, there is the interin­ fluence between fluid and structure, such as changes in impact pressure due to the structural deformation, and the fluid-structure interaction (FSI) analysis is neces­ sary for the highly accurate strength investigation. Regarding the BFS phenomena, the problem of water surface impact of elastic wedges has been stud­ ied for some time (ex.: Faltinsen, 2002, Luo, et al. 2010). Recently the three-dimensional studies with considering the hydro-elastic phenomena have been carried out as the computational technology progress (ex.: Truong, et al. 2019, 2021, 2022, Wang & Guedes Soares 2016, Yamada et al. 2017, 2018, 2019a, 2019b, 2020). Wang & Guedes Soares, 2017) presented a comprehensive and detailed review of state of the art of knowledge on hull slamming. Our research group has been used the Arbitrary Lagrangian-Eulerian (ALE) method in LS-DYNA; a commercial structural analysis software, to perform two-dimensional FSI analyses assuming the BFS behavior, and proposed an FSI analysis method using

the ALE method through the evaluation procedure of the structural response by applying the obtained time histories of pressure response to a three-dimensional structure (Furuno, et al. 2014, 2015, Nakashima, et al. 2011, Yoshikawa, et al. 2013, 2017). In addition, the Incompressible Computational Fluid Dynamics (ICFD) solver (LSTC. 2014) has been implemented into LS-DYNA from 2012-2014, and the authors con­ ducted a fundamental study using the ICFD method (Nagayama, et al. 2019). As mentioned above, since the situation is being prepared for the FSI analysis to be performed more easily, in this study, the FSI analyses using the ICFD solver of LS-DYNA are performed. Also, the accuracy verification not only of the fluid force calculation by water surface drop analyses with some wedge-shaped structures (tow-dimensional analysis); see Section 3, but also of the FSI analysis by a water surface drop analysis with a box-shaped structure (threedimensional analysis); see Section 4, is carried out. Assuming the BFS behavior, furthermore, the FSI analyses by the ICFD solver of LS-DYNA on bowshaped structures (three-dimensional analysis); see Section 5, are conducted. Through these results by numerical simulations using the ICFD method in LS-DYNA and the comparison with the ALE results presented in the previous researches and/or theoret­ ical and empirical values, the validity of analysis results by the ICFD method is discussed.

DOI: 10.1201/9781003399759-16

141

Table 1.

Features of fluid analysis methods in LS-DYNA.

Solver

ALE

SPH

DEM

ICFD

Implementation year Governing equation

1993-1994 Mass conservation law Momentum conserva­ tion law Energy conservation law

2008-2011 — Momentum conser­ vation law —

2012-2014 Mass conservation law Momentum conser­ vation law —

Fluid characteristic

Compressible / Incompressible Finite volume method

2001-2002 Mass conservation law Momentum conser­ vation law Energy conservation law Compressible

Incompressible

Incompressible

Particle method

Particle method

Explicit method Eulerian and Lagrangian Hexa

Explicit method Meshfree method

Explicit method Lagrangian

Particle

Particle

Finite element method Implicit method Eulerian and Lagrangian Tetra

Penalty method

Penalty method Constraint method Weak — Particle method

(Automatic coupling)

Weak — ALE (VOF) method

Penalty method Constraint method Weak — Particle method

Possible Possible

Possible Possible

Possible Impossible

Possible (2D) Possible

Computational technique Time evolution method Space discretization procedure Recommended element type Coupling method with structure Coupling Turbulence model Analysis method of freesurface flow Multiphase flow Two-dimensional analysis

2 FSI METHODOLOGIES IN LS-DYNA LS-DYNA is a multi-physics solver, and various ana­ lysis methods other than the FEM (Finite Element Method), the ALE method, and the ICFD method are implemented. Among many analysis solvers in LSDYNA, the features of four methods, i.e., ALE, SPH (Smoothed Particle Hydrodynamics), DEM (Discrete Element Method), and ICFD, that are often used for the fluid analysis, are summarized in Table 1. As can be seen from Table 1, the ALE method has been implemented in LS-DYNA from the very beginning, and there are many researches applying its solver func­ tion to the BFS problem. On the other hand, as the ICFD method has been newly implemented to LSDYNA, studies using its solver function to the BFS problem are relatively limited. The ICFD method has an advantage over other methods in terms of the computational time, because it is the only analysis solver that adopts the implicit method as the time evolution method. The ICFD method solves the incompressible Navier-Stokes equa­ tions, and can be applied when the target fluid is regarded as an incompressible fluid. A fluid can be generally considered incompressible when the Mach number is lower than 0.3, and the phenomena such as slamming or sloshing are in this range. By introducing the incompressibility hypothesis, the equation of state (EOS) is not needed to define, as the incompressible fluid has a constant density. The ICFD method, which is based on the FEM as shown in Table 1, is suitable

Strong/Weak k-ε, LES Level set method

for the phenomena with relatively small deformation on the structural members, and the coupling between fluid and structure is automatically performed. In add­ ition, the remeshing of fluid part is automatically per­ formed too when a fluid element is greatly deformed. The ICFD solver uses a level set method, which is based on a fast and reliable technique, as the analysis method of free-surface flow so as to track and cor­ rectly represent moving interfaces, and considers the interplay such as surface-tension effects. In this study, based on the above points of view, the ICFD method is adopted to simulate the water impact behavior with considering the FSI effects. By the current ICFD solver of LS-DYNA, since it is dif­ ficult to stably perform the multiphase flow analysis with a large density difference such as water and air; particularly in three-dimensional analyses, the air part is treated as the void in this paper. 3 COMPARISON BETWEEN THEORETICAL/ EMPIRICAL VALUES AND ALE/ICFD RESULTS REGARDING WEDGE-SHAPED (TWO-DIMENSIONAL) STRUCTURE In this section, a series of drop analyses by the ICFD method is conducted using two-dimensional wedgeshaped structures, and the verification of the ICFD analysis is discussed through the comparison with a theoretical solution, an empirical formula, and simulation results by the ALE method.

142

3.1

Theoretical solution

Many methods have been proposed to calculate the slamming pressure on rigid bodies that penetrate the water surface with a certain velocity. As the earliest methods, von Karman (1929) and Wagner (1932) are well known. Regarding the impact load when the seaplane lands on the water, Karman focused on the change in momentum due to the change in the added mass when the two-dimensional wedge plunged into the water surface, and calculated the impact load. Based on Karman’s theory, on the other hand, Wagner introduced the calculation method of impact water pressure taking into account the piling up water when a two-dimensional wedge collides with the water surface. Comparing Wagner’s theory with Karman’s one, whether there is the piling water sur­ face is significant, and the impact load according to Wagner’s theory is approximately larger 2.5 times, i.e., (π/2) squared, than that of Karman’s one. In this paper, Wagner’s theory, which is closer to the actual phenomena of the BFS behavior, is considered as the theoretical solution. Wagner’s theory does not consider the effect of air, and is based on the following assumptions: ➢ The fluid is non-viscous and incompressible. ➢ The gravity can be neglected because the accel­ eration of fluid is much greater than that of gravity. ➢ The draft of wedge is much smaller than the wetted width. ➢ The impact velocity is constant. The pressure coefficient, Cp, based on Wagner’s theory is given by Equation 1:

3.3

3.4

where β is the deadrise angle, i.e., the wedge tip angle to the water surface. 3.2

Empirical formula

Chuang (Chuang, 1966, 1970) carried out a large number of impact experiments using rigid and elastic bodies, and proposed an empirical formula through these experimental results (Stavovy & Chuang 1976). The pressure coefficient, Cp, based on Stavovy & Chuang is given by Equation 2:

where k is a coefficient that varies with the deadrise angle, β, as follows:

Numerical simulation by ALE

Authors’ research group has used the ALE method in LS-DYNA to perform two-dimensional FSI ana­ lyses assuming the BFS phenomena. Yoshikawa et al. (Furuno, et al. 2014, 2015, Naka­ shima, et al. 2011, Yoshikawa, et al. 2013, 2017); former colleagues of authors, performed the FSI ana­ lyses under the BFS load to calculate the impact pressure and the structural response accurately. In these studies, the ALE method in LS-DYNA, which can simulate the fluid-structure coupling behavior, was applied for the impact problem of elastic wedges varying their deadrise angle. In these ALE analyses, two types of modeling for the air part were considered. One was modeling the air part with the fluid assuming air, and the other was modeling that with the void. As for the detailed information about these FSI analyses utilizing the ALE method in LS-DYNA, e.g., the calculation model and its mesh size for FEM, the calculation conditions, the material and physical properties of structure and fluid, and so on, please see the references (Furuno, et al. 2014, 2015, Nakashima, 2011, Yoshikawa, et al. 2013, 2017). Numerical simulation by ICFD

In order to investigate the simulation accuracy of the ICFD method in LS-DYNA, the drop analyses on the water surface using two-dimensional wedge-shaped structures varying their deadrise angle are conducted. The model for the ICFD analysis is shown in Figure 1. Each mesh size of fluid and structural parts is determined based on the results of convergence cal­ culations as well as previous studies with the ALE analysis, and that of fluid around structure is approxi­ mately 1 mm. Although the wedge-shaped structures are modeled by an elastic body assuming a material of steel, they can be almost regarded as a rigid body because two-dimensional solid elements are utilized in this simulation. As shown in Figure 1, the fluid part is only water, and the air part is treated as the void due to keeping the analysis stability. Table 2 shows the material and physical properties of structure (elas­ tic body) and fluid (water) in this ICFD analysis. As the boundary conditions, the left, right, top, and bottom edges of the analysis model are free-slip boundaries, and the boundary between fluid and struc­ ture is a non-slip boundary. The drop analyses varying

143

the deadrise angles, β, as 1, 2, 3, 5, 10, 15, and 30 degrees are performed under the initial velocity of 6.15 m/s, and the gravitational effects are not con­ sidered this time not only for comparison with Wagner’s theory but also for simplicity. An example of the slamming behavior to the water surface simulated by the ICFD analysis is shown in Figure 2. From Figure 2, it is found that the FSI ana­ lysis with the water splash can be performed. As shown in Equation 4, the dimensionless value of the maximum impact pressure, Pmax, which is obtained by the ICDF analysis, is considered as the pressure coefficient, Cp. Figure 1. Model for ICFD analysis of wedge-shaped structure. Table 2.

where ρf and v are the density of fluid (water) and the velocity, respectively. As the velocity, v, the ini­ tial velocity, i.e., 6.15 m/s, is adopted instead of the relative impact velocity between fluid and structure. Figure 3 shows the comparison between the pres­ sure coefficients, Cp, given by Equations 1-2 and 4. Based on the results obtained by the ALE analyses per­ formed in the previous researches (Furuno, et al. 2014, 2015, Yoshikawa, et al. 2017), the pressure coeffi­ cients, Cp, are calculated in the same way as Equa­ tion 4, and these results are also plotted in Figure 3. In Figure 3, solid and open square marks of the ALE results represent whether or not the air is considered. As can be seen from Figure 3, the ALE results without considering the air; that is, the air part is considered as the void, almost agree with Wagner’s theory. In contrast, the ALE results with considering the air effect almost agree with the empirical formula by Stavovy & Chuang. It can be also seen that the ICFD results and the empirical formula by Stavovy & Chuang are in good agreement except for the small range of the deadrise angle, β. When the deadrise angle, β, is 2 degrees or less, the value of pressure coefficient, Cp, decreases due to occurring the air entrainment in the empirical formula by Stavovy & Chuang. This phenomenon caused by the air cushion effects can be simulated by the ALE analysis with considering the air, but not by the ICFD analysis, because the air part in the ICFD analysis should be modeled by not air but void to proceed with the stable calculations in the current LS-DYNA. When the deadrise angle, β, is very small, the dif­ ference of the pressure values obtained with and without air is relatively large, so it is necessary to consider the air cushion effects so as to calculate the accurate impact pressure. Whereas, when the dead­ rise angle, β, is 2 degrees or more, the difference due to the air cushion effects is hardly observed between the ICFD results and the ALE results with consider­ ing the air in Figure 3, and the ICFD results show almost the same values as the experimental value by Stavovy & Chuang.

Material and physical properties.

(a) Structure (elastic body) Item

Symbol

Value

Unit

Young’s modulus Poisson’s ratio Density

E ν ρs

206000 0.3 7.85 × 10-9

MPa — ton/mm3

Item

Symbol

Value

Unit

Viscosity coef. Density

η ρf

1.5674 × 10-9 1.00 × 10-9

MPa·s ton/mm3

(b) Fluid (water)

Figure 2. Example of slamming behavior obtained by ICFD analysis at deadrise angle of 30 degrees.

Figure 3. Relationships between pressure coefficient, Cp, and deadrise angle, β.

144

From these results, except for the range where the deadrise angle, β, is very small, it can be confirmed that the ICFD analysis gives the relatively reasonable value about the fluid force of the two-dimensional structure. 4 ICFD ANALYSIS FOR BOX-SHAPED (THREE-DIMENSIONAL) STRUCTURE Truong (2019) used the ALE method of LS-DYNA to analyze the drop experiment of a box-shaped structure conducted by Mori (Mori, 1977). In this section, therefore, the similar analysis is carried out using the ICFD method, and the simulation accuracy of the three-dimensional analysis and the validity of the FSI analysis are investigated by comparing the ICDF results with not only the ALE results by Truong et al. but also the experimental results by Mori. 4.1

Table 3.

Material and physical properties.

(a) Structure (elastic body) Item

Symbol

Value

Unit

Young’s modulus Poisson’s ratio Density

E ν ρs

68700 0.3 2.7 × 10-9

MPa — ton/mm3

Symbol

Value

(b) Fluid (water)

Analysis model and conditions

Figure 4 shows the model for the ICFD analysis cre­ ated with reference to the analysis model utilizing Truong et al. based on Mori’s experimental works. The FE model including the internal members, such as longitudinal stiffeners, of the box-shaped structure is indicated in Figure 5, in which the upper plate is hidden so that the internal members can be confirmed. This box-shaped structure has two tee-bar stiffeners, and their dimensions are 94 × 6 + 56 × 6 mm. The structure is modeled by the elastic body, and its mesh size is about 10 mm × 10 mm. Table 3 shows the material and physical properties of structure (elastic body) and fluid (water) in this ICFD analysis. The boundary conditions are free-slip boundaries around the fluid part and a non-slip boundary between fluid and structure. Analysis model and con­ ditions other than the above are almost same as those described in Section 3.4. The structural responses are obtained from the ICFD analysis when the boxshaped structure freely falls from a height of 300 mm above the water surface.

Figure 4. Model for ICFD analysis of box-shaped structure.

Figure 5. FE model of box-shaped structure.

Item Viscosity coef. Density

4.2

η ρf

Unit -9

1.5674 × 10 1.00 × 10-9

MPa·s ton/mm3

Analysis results and discussion

The comparisons of impact pressure causing at two evaluation points, i.e., P1 and P2 shown in the right figure of Figure 5, between the experiment by Mori, the ALE results presented by Truong et al., and the ICFD results are indicated in Figure 6. It should be noted that the effect of air is considered in the ALE analysis by Truong et al. shown in Figure 6. In Figure 6, the ICFD results are larger than the ALE and the experimental results at both evaluation points. In this ICFD analysis, the air is not considered, while in the ALE analysis by Truong et al., the air was properly modeled, and the air cushion effects also existed in the experimental results by Mori. By con­ trast to the ALE and the experimental results, conse­ quently, the phenomenon of pressure reduction due to the entrainment of air does not occur in the ICFD ana­ lysis, so it is considered that the ICFD results show the relatively larger pressure values. Here, this ana­ lysis corresponds to the case where the deadrise angle, β, is zero. As can be seen from the results in Section 3, therefore, this is the most remarkable case where the ICFD method indicates the larger pressure values. However, in all ICFD, ALE, and experimental results, an almost similar tendency can be observed that the pressure fluctuation gradually decreases with the oscillation after reaching the peak pressure. Truong et al. also performed the ALE analysis without considering the air; namely with the air part as the void. Figure 7 shows the comparison of the impact pressure at P2 obtained by the ALE and the

145

ICFD simulations, which do not consider the air effect. From Figure 7, when the air is not taken into account in the ALE analysis, as the maximum pres­ sure value of the ICFD and the ALE results are almost the same, the analytical accuracy of the ICFD method may be the same level as the ALE method. From the above, it can be confirmed that the time histories of impact pressure by the ICFD method show the same trend as the ALE and the experimen­ tal results. In other words, although it is necessary to pay attention to the fact that the ICFD analysis gives higher pressure values in the range where the dead­ rise angle, β, is very small, the ICFD analysis gives the relatively reasonable results even under the three-dimensional and the FSI influences.

Figure 7. Comparison of pressure-time histories at evalu­ ation point P2 between ALE w/o air (void) by Truong and ICFD.

5.1

5 ICFD ANALYSIS FOR BOW-SHAPED (THREE-DIMENSIONAL) STRUCTURE Through the several investigations in the previous sections, it can be confirmed that the ICFD analysis enables the relatively valid FSI analyses about both two-dimensional and three-dimensional structures. In this section, thus, a series of drop analyses by the ICFD method is carried out using three-dimensional bow-shaped structures of an actual ship.

Figure 6. Comparisons of pressure-time histories between experiment by Mori, ALE w/ air by Truong, and ICFD. (This case corresponds to β = 0.).

Analysis model and conditions

Figure 8 shows the model for the ICFD analysis of the bow-shaped structure. The FE model including the internal members, such as transverse frames and longitudinal stiffeners, of the bow-shaped structure is indicated in Figure 9, in which the front plate is hidden so that the internal members can be con­ firmed. As for the modeling of the bow-shaped struc­ ture, since it is difficult to model and analyze a wide or whole area of an actual bow part, the ICFD ana­ lyses are performed for simplicity using bow-shaped structures whose shapes do not change in the longi­ tudinal direction this time. The yellow area sur­ rounded by white dotted lines shown in right figure of Figure 9 is the region for evaluating structural response, and the modeling range in the longitudinal direction is five times longer than this evaluating region. The bow shape of the FE model and its dimensions are determined based on an actual 14000 TEU container ship. The hull plates and other large plate members or the longitudinal stiffeners are mod­ eled by shell or beam elements, respectively. The structure is modeled by the elastic body, and its mesh division around the region for evaluating struc­ tural response is five between each longitudinal stiff­ ener space. Three FE models with flare angles of 25, 30, and 37 degrees are used for this ICFD simula­ tion. The flare width of 11.2 m is constant, so the height of bow-shaped structure, which is about 30 m, slightly changes when the flare angle changes. Table 4 shows the material and physical properties of structure (elastic body) and fluid (water) in this ICFD analysis. The boundary conditions or analysis model and conditions other than the above are completely or almost the same as those described in Section 4.1 or Section 3.4, respectively. Using the FE models with different flare angles, a series of drop analyses is per­ formed varying the forced velocities of 5, 7.5, 10, and 15 m/s.

146

5.2

Analysis results and discussion

Using the maximum value of the impact pressure in the evaluating region of flare plate seen in Figure 9, the pressure coefficients, Cp, obtained by the ICFD analyses are calculated from Equation 4. As the velocity, v, each forced velocity is adopted instead of the relative impact velocity between fluid and structure. The comparisons between these pressure coefficients, Cp, based on the ICFD results, Wagner’s theoretical solution, and Stavovy & Chuang’s empirical formula are shown in Figure 10. As can be seen from Figure 10, the pressure coefficients, C p , by the ICFD method are slightly smaller than the empirical formula by Stavovy & Chuang as the flare angle increases. The empir­ ical formula targets two-dimensional wedgeshaped structures, whereas this ICFD analysis targets three-dimensional structures. In the ICFD analysis, hence, the impact pressure decreases due to the escape of fluid in the longi­ tudinal direction. Although there are slight dif­ ferences in the pressure coefficients, C p , the ICFD results generally agree with Wagner’s the­ oretical solution and Stavovy & Chuang’s empir­ ical formula, so it is considered that the ICFD analysis provides the relatively appropriate pres­ sure values. The theoretical value of the bending stress caus­ ing in an element, which is here expressed by ‘ElmA’, located in the center of the evaluating region is obtained by the following procedure: (1) Find the time when von Mises stress in Elm-A reaches the maximum value. (2) At the time of (1), obtain the total pressure value causing in the panel, which includes ElmA, surrounded by both longitudinal and transverse structural members from the ICFD results. (3) For a rectangular panel with fixing its four sides under the uniformly distributed pressure, which is given as the average value of the total pres­ sure acquired in (2), derive the maximum bending stress in the X- and Y-directions. The X- and Y-directions use the element coordin­ ate system, and correspond to the ship length and depth directions, respectively. The bending stresses calculated by each stress in the X- and Y-directions on upper and lower surfaces of Elm-A are also obtained from the ICFD results. Table 5 shows the comparisons between the theoretical values according to the above procedure and the ICFD results at the forced velocity of 15 m/s. It can be confirmed that the errors of the ICFD results with respect to the theoretical values are almost less than 10% in Table 5, so the bending stress obtained from the ICFD analysis can be seen a reasonable value. From these results, if the

impact pressure acting on the hull plate of flare part in the BFS behavior can be accurately esti­ mated, it is possible to estimate the stress based on the deformation theory of the plate with fixing its circumference.

Figure 8. Model for ICFD analysis of bow-shaped structure.

Figure 9. FE model of bow-shaped structure.

Table 4.

Material and physical properties.

(a) Structure (elastic body) Item

Symbol

Value

Unit

Young’s modulus Poisson’s ratio Density

E ν ρs

206000 0.3 9.116 × 10-9

MPa — ton/mm3

Symbol

Value

Unit

(b) Fluid (water) Item Viscosity coef. Density

147

η ρf

-9

1.5674 × 10 1.00 × 10-9

MPa·s ton/mm3

Table 5. Comparisons of bending stresses between theor­ etical value and ICFD result at forced velocity of 15 m/s. (a) X-direction Flare angle [deg.]

Theoretical [MPa]

ICFD [MPa]

Error [%]

25 30 37

152.1 101.5 62.1

164.0 112.2 70.4

+7.8 +10.5 +6.7

Flare angle [deg.]

Theoretical [MPa]

ICFD [MPa]

Error [%]

25 30 37

506.9 338.3 207.1

488.3 351.9 221.0

-3.7 +4.0 +6.7

(b) Y-direction

ICFD analysis gives relatively reasonable results even under the three-dimensional and the FSI effects. • Comparing the impact pressure and the bend­ ing stress obtained from the drop analyses of bow-shaped structures with the theoretical and the empirical values, the ICFD analysis pro­ vides relatively appropriate structural responses. Based on the above results, the feasibility of the stress estimation in the BFS behavior by the ICFD analysis was demonstrated, but present findings are applied to specific conditions examined this time. It is pointed out that further studies are necessary to obtain more general conclusions. As future works, it is necessary to perform the ICFD analysis considering more realistic situations as follows: Figure 10. Relationships between pressure coefficient, Cp, and flare angle.

6 CONCLUSIONS In the present paper, using the ICFD solver in LSDYNA, the fluid analyses as well as the FSI analyses with wedge-, box-, and bow-shaped structures were performed. In this study, the following findings can be drawn: • Comparing the results by the drop analyses of wedge-shaped structures with the theoretical solu­ tion and the empirical formula, the fluid force cal­ culation by the ICFD analysis has the enough accuracy. • Comparing the results by the drop analysis of a box-shaped structure with the ALE results, the

■ ■ ■

Actual wave shape Shape change of bow part in longitudinal direction Elasto-plastic structure

In the future, it is desirable to develop the stress evaluation method in the BFS behavior that does not require the numerical simulations such as the ALE or the ICDF analysis.

ACKNOWLEDGEMENTS This work was supported by JSPS KAKENHI Grant Number JP21K04523. Furthermore, this research was partly performed with the supports of JSPS KAKENHI Grant Number JP20K14960 and Grantin-aid of the Fundamental Research Developing Association for Shipbuilding and Offshore (REDAS). The authors are grateful to JSPS and REDAS for their supports.

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interaction. Part II: Derivation of empirical formulations. Marine Structures 75: 102700. Truong, D.D., Jang, B-S., Ju, H-B. & Han, S.W. 2022. Pre­ diction of slamming pressure considering fluid-structure interaction. Part I: Numerical simulations. Ships and Offshore Structures 17(1): 7–28. von Karman, T. 1929. The impact on seaplane floats during landing. National Advisory Committee for Aeronautics. Technical Note No. 321. Wagner, H. 1932. Uber Stossund Gleitvergange an der Oberflache von Flussigkeiten. Zeitschrift fuer Ange­ wandte Mathematik und Mechanik 12: 193–215. Wang, S. & Guedes Soares, C. 2016. Experimental and numerical study of the slamming load on the bow of a chemical tanker in irregular waves. Ocean Engineer­ ing 111: 369–383. Yamada, Y. & Kameya K. 2017. A fundamental study on the dynamic response of hull girder of container ships due to slamming load. Proceedings of the ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering. OMAE2017-61068. Yamada, Y. & Kameya K. 2018. A study on the dynamic ultimate strength of global hull girder of container ships subjected to hogging moment. Proceedings of the ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering. OMAE2018-77402. Yamada, Y. 2019a. Approach to simulate dynamic elasto-plastic whipping response of global hull girder of a large container ship due to slamming load. Proceed­ ings of the Twenty-ninth (2019) International Ocean and Polar Engineering Conference 3: 2808–2816. Yamada, Y. 2019b. Dynamic Collapse Mechanism of Global Hull Girder of Container Ships Subjected to Hogging Moment. Journal of Offshore Mechanics and Arctic Engineering 141(5): 051605. Yamada, Y., Takamoto, K., Nakanishi, T., Chong, M. & Komoriyama, Y. 2020. Numerical study on the slam­ ming impact of stiffened flat panel using ICFD method: Effect of structural rigidity on the slamming impact. Proceedings of the ASME 2020 39th International Con­ ference on Ocean, Offshore and Arctic Engineering. OMAE2020-18242. Yoshikawa, T. & Maeda, M. 2013. Numerical simulation of structural response under bow flare slamming load. Pro­ ceedings of the 4th International Conference on Marine Structures: 25–33. Yoshikawa, T., Miyake, R., Yoshida, T. & Maeda, M. 2017. Numerical Simulation of Structural Response under Bow Flare Slamming Load. Journal of the Japan Soci­ ety of Naval Architects and Ocean Engineers 26: 267–276 (in Japanese).

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Dynamics and vibration

Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Motion analysis of a floating horizontal set of interconnected plates based on computer vision target tracking technique I.B.S. Bispo, P. Amouzadrad, S.C. Mohapatra & C. Guedes Soares Centre for Marine Technology and Ocean Engineering (CENTEC), Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

ABSTRACT: The interaction between a large freely floating articulated structure and regular waves is investigated by model tests based on the computer vision target tracking approach. The detailed description of the experimental setup and measurement techniques are presented. Tests are conducted on a wave flume to provide a benchmark of comparison and establish some of the limitations of the measuring method presented herein. Further, the vertical displacement of the freely floating interconnected structure for different wave periods is registered using the presented technique, showing the usefulness of the method within the hydroe­ lasticity framework.

1 INTRODUCTION The field of marine structure industry has demon­ strated a growing interest in integrating Very Large Floating Structures (VLFSs) for various purposes, such as airports, bridges, storage facilities, emer­ gency bases, terminals, and aquaculture bases. The VLFSs can be used as an effective option for the utilization of ocean space due to their costeffectiveness and the fact that they can be environ­ mentally friendly, not obstruct waves and currents, having natural flexibility of the structure, especially when considering an articulated VLFS (Yoon et al. 2014; Bispo et al. 2022). A key feature of these flexible structures is the coupling between their deflection and the fluid field. Thus, the analysis of the hydroelastic response of such structures based on the experi­ mental investigation can provide insights into the design and effective analysis of engineering structures applications in the offshore region (see Ding et al. 2021). In the present paper, the tests are carried out to analyse the motions of floating articulated plates that emulate a flexible structure, an important feature for structural analysis in engineering applications in the marine environment. The studies on different aspects of the hydroelas­ tic analysis of floating structures using different experimental and hybrid numerical methodologies were investigated. For instance, the predictions of the hydroelastic behaviour of a very large boxshaped flexible structure based on theoretical and experimental were presented by Kagemoto et al.

(1998) and in their analysis, the structure was div­ ided into several substructures, in which the con­ tinuous deformation was approximated by the succession of a discrete displacement of each sub­ structure, being the vertical displacement measured by an optical system composed of target LEDs attached on the model and a CCD camera to acquire their motion. A 3D experiment was implemented to investigate and assess the structural response (connectors’ internal forces and mooring lines’ tensions) of a Floating Breakwater (FB) along with its wave attenuation effectiveness under the action of perpen­ dicular and oblique regular and irregular waves in Loukogeorgaki et al. (2014). Cheng et al. (2015) studied the hydroelastic response of the VLFS with perforated, nonperforated or dual-submerged horizontal plates based on numerical and experimental results, by consider­ ing the variation of the water depth. Sun et al. (2022) investigated the mooring force and motion responses of kelp-box type floating breakwater with different arrangements by conducting model tests and com­ paring the results to a numerical set-up. The hydroelastic response, connector loads, and dynamic characteristics of an articulated floating system composed of an 8-modules VLFS near a typical island were studied by Ding et al. (2020a; 2021), employing a noncontact six-degree of free­ dom (DOF) motion capture system to acquire the motion of the structure. A chain-type floating system with a 5-module (chain-type floating system connected by hinges) and a three-modular floating platform were investigated by Ding et al.

DOI: 10.1201/9781003399759-17

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(2020b; 2022) based on coupling numerical models to reduced scale experiments, in which a 6 DOF motion system is used to capture the 3D displacements.) Riggs & Ertekin (1993) applied 2D and 3D hydroelastic theories to predict the dynamic response of 5- and 16-module VLFSs in regular waves where a rigid module and flexible con­ nector were used for a structural model for 3D based on the double composite source distribution technique. Xu et al. (2018) studied the dynamics of a multimodule floating airport with flexible connectors by applying the network theory to analyze the nonlinear dynamics of floating airports and the connector force along with the mooring system. Seithe and el Moctar (2019) simulated the motions responses of multibody offshore structures based on computational fluid dynamics (CFD), validating the results with experiments in reduced scale and evaluating the motion of the multiple bodies with the measured motion in 6 DOF obtained from a camera-target tracking system. Cui et al (2019) have performed an experimental investigation of the hydrodynamic performance of a box-floating type breakwater in different terrains in a 2D wave flume, capturing the motions of the model by employing a 6 DOF system, with reflective targets and a tracking camera. Diamantoulaki & Angelides (2010) investigated the vertical translations of an array of floating breakwaters connected by hinges based on hydro­ dynamic formulations. Jiang et al. (2021) studied the effect of head seas where waves travelled per­ pendicularly to the rotation axes of hinged joints, and the hinge forces were on an articulated modular floating structure utilizing the numerical model AQWA solver. A numerical procedure coupled between the BEM and FEM was proposed to analyze floating plate structures with multiple hinge connections by Yoon et al. (2014). Bispo et al. (2022a, 2022b) studied numerically based on the ANSYS AQWA solver of wave interaction with an articulated floating struc­ ture composed of 20 hinged plates and a wave energy converter type attachment of a moored hori­ zontal submerged set of articulated plates. The numerical model approaches in the hydroe­ lastic analysis of VLFSs were reviewed by Bispo et al. (2021). Moreover, the hydroelastic response to the effect of mooring lines was investigated based on BIEM in Mohapatra & Guedes Soares (2021) and analytically (see Mohapatra & Guedes Soares, 2019). Advancing further the previous numerical model of Bispo et al. (2022a), in the present paper, model tests are conducted to analyze the dynamic response by applying a method of displacement measurement, making it possible to investigate the

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hydroelastic behaviour of the articulated flexible structure in constant water depth and regular waves. The test results on the deflection of the structure are analyzed based on the computer vision target tracking technique, firstly verified with com­ parisons to 2D wave measurements in a wave flume. The input values and the number plates adopted by Bispo et al. (2022a) are used in the pre­ sent model tests. 2 EXPERIMENTAL SETUP 2.1

Experimental equipment

The model tests are conducted in a wave flume at the National Laboratory for Civil Engineering (LNEC), Portugal. As presented in the diagram of Figure 1, a piston-flap type wavemaker is installed at one end of the flume to generate regular waves whilst, at the other end of it, a rock slope “beach” area is provided to effectively absorb the incident wave energy, redu­ cing reflection. In this experimental study, the length, width, and height of the wave flume are 35 m, 0.62 m, and 1 m, respectively, keeping the water depth at constant 0.35m. The free surface elevation inside the wave flume is measured by 8 wave gauges (WGs) and the volt­ age signals are captured by a NI-9205 module, from National Instruments and sent to a computer at a sampling frequency of 40 Hz. The wave gauges are positioned along the flume according to the following: #8 is placed 3.2 m before the front edge of the rock slope absorption area, whilst wave gauge #4 is placed 4.65 m in front of wave gauge #5. The distances between wave gauges #1 and #2, #2 and #3, and #3 and #4 are 30 cm, 20 cm, and 38 cm, respectively. The distances between wave gauges #5 and #6, #6 and #7, #7 and #8 are 30 cm, 23 cm, and 35 cm, respectively. In between the two sets of wave gauges, as shown in Figure 1, there is a region for installing models, equipped with a glass window so that the behaviour of each experiment can be visually evaluated and, in the case of this study, be recorded by a digital camera at 60 fps (frames per second). The digital camera is positioned perpendicularly to the window glass, so that the frame has the model centred, at a distance that minimizes the distortion to a minimum and at a height that has the centre of the camera lens coincident with the water level in the flume. Two experiments are conducted at the described wave flume, aiming at extracting 2D positions of targets placed either floating over the free surface or the articulated structure model. These two apparatuses composed of the targets, supporting frame, and model are described shortly in the following.

Figure 1. Side view diagram of the wave flume depicting the floating articulated structure with the tracking targets, wave gauges arrangement and accessory equipment.

2.2

Benchmarking by wave measuring targets vs wave gauges measurement

To establish a basis of comparison to the measuring technique presented herein, a first series of tests is performed by placing two floating targets in the wave flume, with the centre of the supporting frame aligned with the centre of the flume, in the middle of the region between WGs #4 and #5. Perpendicular to the length of the tank, in front of the region where the targets were placed, a camera was positioned to record the motion of the floating targets, which are spheres made of a thin layer of ABS (Acrylonitrile Butadiene Styrene) plastic, with a plastic tube passing through its centre from top to bottom. Nylon strings are fixed vertically along the supporting stainless-steel frame from the top to the bottom of the flume, holding each target in place by running inside the tubes, avoiding any drift due to viscous effects caused by the interactions between the wave flow and the spheres. It can be seen in Figure 2 the arrangement of the floating targets during an experiment. Both pictures present the same video frame and the right one also shows the tracked box (blue) and the circle (magenta) resulting from the target identification, which ultimately results in the output of the com­ puter vision routine: the vertical and horizontal coordinates of the centre of the identified circles, with the water level as a reference.

Figure 2. Floating tracking targets move vertically along the nylon string as a wave passes.

For the floating targets tests, only regular waves are considered to be generated, being their character­ istics provided in Table 1.

Table 1. Adopted values of wave parameters for compari­ son of target tracking to wave gauges. Parameters

Cases

Wave height H (mm) Wave period T (s)

40 1.0

2.3

40 2.0

40 3.0

80 1.0

80 2.0

80 3.0

Details of the interconnected floating structure experiment

The floating horizontal interconnected structure is com­ posed of 20 wooden plates coated with silicone and joined by interweaved sheets of EVA (Ethylene Vinyl Acetate), acting as hinges with their rotation motions limited by the thickness of the plates. Each plate is 7mm x 59 mm x 526 mm (height x width x length) and the average mass of the plates is m = 0,119 kg. This model, which can be seen in Figure 3 (top picture: side view; bottom left: birds-eye view; bottom right: perspective), is conceived to have geo­ metrical, Froude and Cauchy similitudes in a scale of 1:57 to the prototype (see Chakrabarti (1994)). To achieve elastic similarity, flexible EVA is chosen as a suitable material by having an elastic modulus (approx. E = 3 × 107 N/m2) that would properly emulate the real scale articulations. Two stainless-steel frames with five stiffened steel wires each are used to prevent the model from drifting. These wires run from top to bottom of the frames, each passing through steel rings screwed to the aft and fore ends of the VLFS model, as can be seen in Figure 3 (bottom pictures). The supporting frames have steel plates on their tops so that a heavy weight can be placed over them, holding both in their positions on the top of the flume and preventing any horizontal motion (see Guo et al. 2022). This experimental test mainly focuses on the vertical motion of the floating horizontal

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interconnected plates for different wave periods. The fore of the model is placed 2.65 m apart from WG #4.

rectangles in Figure 4), a search for the target is per­ formed by filtering the colours of the image. This fil­ tering can extract only certain portions of the image within a range of shades of a determined colour, such as the orange chosen for the used targets. After this procedure, only the area with a range of colours, such as the target, is visible. The image is then converted to grayscale and the Hough Circle Transform can be applied to the region of interest of the monochromatic frame with specified parameters, such as a range of values for the radius of the circle for the algorithm to look for. This last step returns the horizontal and vertical coordinates of the centre of the found circle (magenta circles in Figure 4) and, since the target is fixed to the structure, the same dis­ placement experienced by a specific tracked target can be observed by the respective plate, ultimately leading to the vertical displacement of each segment of the VLFS model.

Figure 3. Floating articulated structure coated with silicone and interweaved with EVA.

In the context of the present model tests, the con­ sidered incident waves are regular and their charac­ teristics are provided in Table 2.

Figure 4. Camera view of the VLFS model with targets identified by the tracking algorithm.

4 BENCHMARKING AND MODEL TEST RESULTS Table 2. Wave characteristics adopted for the floating horizontal interconnected plates model. Parameters

Cases

Wave height H (mm) Wave period T (s)

40, 60 and 80 0.8 to 3.0, 0.2 steps

3 TARGET TRACKING TECHNIQUE The computational routines for image processing of the video frames are written using Python 3.9, employing OpenCV 4.5.0 library for the tracking and identification algorithms. The video recorded for each test is processed by reading each frame as an image, such as the one pre­ sented in Figure 4. In each of these frames, prespecified regions are selected to be tracked by a tracking algorithm and, in these regions (blue

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In Figure 5 the case of benchmark defined by H = 40 mm and T = 1 s is presented. The blue and red lines are the measurements performed by Wave Gauges #4 and #8, respectively. The yellow and purple lines are the measurements obtained by the target tracking technique presented herein. It can be seen that the measurements are fairly close to the ones per­ formed by the wave gauges and, although less noisy, they are somehow constricted by the resolution of the recordings, which is directly related to the accuracy of the measurements. Another case is presented in Figure 6, in which is shown that the peaks and valleys are well correlated, although some effect from the bottom seems to be present and is captured differently by each device. Since there is a larger discrepancy between the two wave gauges and the targets present an intermediate configuration of the propagated wave, it is reasonable to assume that this effect is increasing as it propagates along the wave flume, being more significant when

Table 3. Relative error of measurements of wave height (H) and wave period (T) obtained using the target tracking tech­ nique and the traditional wave gauge method. Parameter

% H Difference

% T Difference

Case

Target #1

Target #2

WG #4

Target #1

Target #2

WG #4

H=4; T=1 H=4; T=2 H=4; T=3 H=8; T=1 H=8; T=2 H=8; T=3

8.1% 12.1% 4.4% 13.7% 15.6% 12.7%

8.8% 14.4% 6.0% 13.7% 18.6% 6.8%

6.8% 0.4% 4.2% 10.2% 4.3% 3.9%

0.4% 0.4% 1.1% 0.4% 0.4% 1.1%

0.4% 0.4% 1.1% 0.4% 0.4% 1.1%

1.5% 1.5% 0.4% 1.5% 1.5% 0.4%

Figure 5. Time series of wave measurements for bench­ marking case with H=40 mm and T=1.0 s.

Figure 6. Time series of wave measurements for bench­ marking case with H=80 mm and T=3.0 s.

the wave is closer to the rock slope at the end of the flume. Since the bottom is at a constant depth, its effect on the wave is not investigated here. Parameters defined for the wave generation are recovered from the measurements and a comparison of results is presented in Table 3, where the relative errors of the parameters of the wave, H and T, are shown. It can be seen that the wave height measured by the target tracking technique is always larger than the same measurement performed by the wave gauge, which also presents some overestimation of values. On the other hand, good results were shown when consid­ ering the wave period, probably due to the smoothness of the measurements using the targets when compared to the wave gauge, which presents a noisier signal. The target tracking technique seems to exhibit a tendency of providing better estimates of the wave height for larger wave periods, at the cost of reducing the accuracy of the wave period estimate. This tradeoff between the estimations of wave height and period can be seen also with the decrease of the rela­ tive error of period measurements when the respective estimations of wave height present a larger difference. From these, one can conclude that a larger wave period will induce better estimations of the wave

height, which, in turn, implies that a more energetic wave leads to better measurements. Although the results display some of the limitations of the presented technique, they also reveal that they can describe the oscillatory nature of the waves with a good sampling frequency, reproducing the wave pro­ files, as required, with acceptable accuracy. These results support the choice of using it as a method to acquire the vertical motion of the VLFS using a simple setup with no special equipment required, making it a reduced-cost solution to motion analysis data acquirement. 4.1

Deflection of freely floating articulated structure

Figure 7 shows the case of a regular wave with H = 60mm and T = 2.2s incident to the structure, in which the deflection of the structure is measured by the target tracking technique, resulting in a time series for each plate (corresponding to each target – for visualization purposes, only odd-numbered plates are presented in this figure). It can be seen that the first half of the structure, corresponding to plates #1 to #9, has a higher motion

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amplitude than the other half of the VLFS model, and the oscillatory nature of the motion is main­ tained, having the same period of oscillation of the wave. This is because the wave is coming towards the structure from plate #1 side, and each plate after that will absorb less energy, so the motion amplitude is expected to be reduced along the structure.

It is noticeable that for most of the wave periods, there is a peak of motion in plates #2 to #9 and that the difference of the motion from the fore plates and the aft plates reduces as the wave period increases, with a clear exception to T=3.0s. In Figure 8 the left half of the model presents a larger variation than the right part and, although this trend is also present in Figure 9, the difference is reduced in terms of amplitude. Cases of a large discrepancy, such as T=3.0s, need further investigation in future works, consider­ ing also variations in wave height.

Figure 7. Time series of the vertical displacement of the odd-numbered plates of the VLFS model. Case with H=60 mm and T=2.2 s. Figure 9. Normalized vertical displacement along the VLFS model for H = 60 mm with wave periods from 2.0s to 3.0s.

The mentioned behaviour can be verified also in Figure 8 and Figure 9, in which the vertical displace­ ments of each plate (ξ3), normalized by the wave amplitude (Aw), of the VLFS model for H = 60 mm and all wave periods are presented along the normal­ ized length of the structure (markers positioned at the centre of mass of each plate). The wave periods are divided into two equal-size sets for a better view of each picture, organized in ascending order of wave periods.

Figure 8. Normalized vertical displacement along the VLFS model for H = 60 mm with wave periods from 0.8 s to 1.8 s.

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5 CONCLUSIONS In the present work, model tests are conducted to acquire motion data of the floating articulated structure in con­ stant water depth. These experiments are performed by measuring the model vertical displacement employing a target tracking technique based on computer vision algorithms. The dimensions, parameters and number plates adopted by Bispo et al. (2022) are used in the pre­ sent model tests. The simulations of wave elevation and vertical displacement for the freely floating structure are analysed and the results are summarized as follows: 1. The wave elevation results of wave measure­ ments between the wave gauges and target track­ ing technique are relatively close to each other for different wave heights and wave periods. Wave parameters can be successfully recovered from the measured oscillation of the targets. 2. Deflection results indicate that shorter waves induce similar vertical motion along the structure, while longer waves lead to a higher response in the fore plates of the model when compared to the other half of it. The longer the wave, the lesser the difference in motion amplitude between the two halves of the model. The exception is made to the longest wave considered, which needs further investigation in the future.

3. The presented technique of measurement of the ver­ tical displacement, via target tracking using com­ puter vision algorithms, can be applied for motion analysis of models, reducing the costs of performing experiments in wave flumes. It is also shown that it can be successfully applied to the measurement of 2D waves, even if non-linearities are present.

ACKNOWLEDGEMENTS The work was performed within the project Hydroelas­ tic behaviour of horizontal flexible floating structures for applications to Floating Breakwaters and Wave Energy Converters (HYDROELASTWEB), which is funded by the Portuguese Foundation for Science and Technology (Fundação para a Ciência e a Tecnologia FCT) under contract PTDC/ECI-EGC/31488/2017. The second author has been funded by the project Hydroelastic behaviour of horizontal flexible floating structures for applications to Floating Breakwaters and Wave Energy Converters (HYDROELASTWEB), which is the Portuguese Foundation for Science and Technology (Fundação para a Ciência e a Tecnologia FCT) under contract PTDC/ECI-EGC/31488/2017. The third author has been contracted as a Researcher by FCT, through Scientific Employment Stimulus, Individual support under Contract No. CEECIND/ 04879/2017. This work contributes to the Strategic Research Plan of the Centre for Marine Technology and Ocean Engineering (CENTEC), which is financed by FCT under contract UIDB/UIDP/00134/2020.

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Wind-induced vibration characteristics of typical guide rail frame structure in open area of large cruise ships X.L. Feng, J. Gan, Y. Zhu & Z.H. Chen School of Naval Architecture, Ocean and Energy Power Engineering, Wuhan University of Technology, Wuhan, China

W.G. Wu Green & Smart River-Sea-Going Ship, Cruise Ship and Yacht Research Center, Wuhan University of Technology, Wuhan, China

ABSTRACT: Cruise ships have high and plump superstructures, resulting in the vibration of open areas under complex wind fields, thereby significantly impacting the comfort of luxury ships. In this paper, the typical guide rail structure in the open area of a cruise ship was selected as the research object. The influence of the fluid-solid coupling method parameters was explored based on wind tunnel tests, and the method’s accuracy was verified. The results of the whole-ship wind field were applied as the wind field input of the open area by the subdomain method. Then, wind-induced vibration analysis of the guide rail structure in the open area was carried out under different wind conditions. The results revealed that employing the turbulence model SST k-w obtained minimal error between the simulation and test, and the vibration frequency of the guide rail structure was mainly concen­ trated at 0.8-10.1 Hz. The most unfavorable wind angles were 0° and 120°. This study could provide a reference for wind-induced vibration prediction and comfort design of typical structures in open areas of large cruise ships.

1 INTRODUCTION Large cruise ships, as high value-added ships, have high standards for the comfort of passengers, and the standards of vibration and noise are stringent. How­ ever, luxury cruise ships have high and plump super­ structures, which are wind-sensitive structures. There are rich and dense entertainment facilities in the open area. The bow and the buildings in front of the open area will affect the wind field of the rear building structure. Hence, there will be a complex wind field distribution, which will lead to complex wind-induced vibration of the structure in the open area, affecting the comfort of tourists. Therefore, it is critical to assess the effects of wind-induced vibration in the open area of cruise ships for comfort construction. Many researchers at home and abroad have studied the wind field characteristics of ships. Jasna et al. (2020) used Star CCM+ to study the impact of con­ tainer layout on the wind field on container ships. Jans­ sen et al. (2015) used Fluent to calculate and investigate the effect of a simplified container ship model on the wind field. Chen et al. (2015) took the inland river cruise ship as the research object, used CFD simulation to calculate the hull’s drag, lift, and yaw moment under wind angles of 0° to 180°, and compared the results with the empirical formula. Wang

et al. (2020) used Star CCM+ to calculate the threepart force coefficient and wind pressure of a large cruise ship in the wind field from 0° to 180°, and com­ pared them with wind tunnel tests. Zhang (2012) stud­ ied the influence of container spacing and arrangement on the wind field by CFD. Robertson et al. (2011) used CFD numerical simulation to divide a warship into unstructured tetrahedron meshes, solved the wind field characteristics at the corners of the ship’s superstructure steps, and compared them with wind tunnel tests. Briz­ zolara et al. (2006) used the CFD method to analyze the wind load of the superstructure of large merchant ships, focusing on the main deck’s suction zone caused by a negative pressure field. The above researchers used CFD to calculate the ship wind field and mainly studied the overall or local wind load distribution, streamline distribution, and three-component coeffi­ cient of the ship under different wind conditions. Regarding the research on wind-induced vibration, Yoshie et al. (1997) conducted a series of aeroelastic model tests to investigate the effect of building corner shape correction on aerodynamic stability. Taniike et al. (1988) studied the interference effect of windward build­ ings of different widths. The crosswind response of the downstream structure was affected by the alternating wake generated by the interference effect between build­ ings. Hu et al. (2020) investigated the cross-flow

DOI: 10.1201/9781003399759-18

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vibration of two cylinders arranged in series. Based on obtaining the natural frequency and damping ratio of the cylinders, the wind-induced vibration of the upstream and downstream cylinders was measured by a laser vib­ rometer to investigate the interference effect between the two cylinders at different spacings and wind speeds. Tang et al. (2004) conducted a comprehensive study on the interference mechanism of translational and tor­ sional responses using a three-dimensional aeroelastic high-rise building model. The wake of the upstream building will amplify translational and torsional vibra­ tions. In addition, the interference effect depends on the location of the downstream buildings. Zhao (2017) car­ ried out a rigid pressure measurement test and aeroelas­ tic model test on a special-shaped high-rise building structure with a swimming pool and analyzed the spa­ tial correlation of the wind load on the surface of the building structure. Yu et al. (2004) made an elastic model for the suspended glass curtain wall structure between the two floors of a high-rise building and measured the velocity and displacement of the meas­ uring points with a laser vibrometer. To explore the application of the two-way fluid-solid coupling method in the external flow field of the ship, Qi et al. (2004) carried out numerical simulation calculations on the single rearview mirror of the vessel. They ana­ lyzed the dynamic characteristics of the rearview mirror when the ship sailed in the shallow sea at dif­ ferent speeds. Based on the Ansys Workbench, Zhang (2018) carried out one-way and two-way fluidstructure coupling calculations for the elastic structure model of high-rise buildings under the effect of wind velocity at different recurrence periods. For the prob­ lem of wind-induced vibration of building structures, the above scholars adopted the wind-induced vibra­ tion test of the elastic model, the noncontact laser vib­ rometer for data acquisition, and the fluid-solid coupling method for numerical calculation. However, to the best of our knowledge, studies on the wind-induced vibration of large cruise ships have rarely been reported. The wind vibration calculation method for typical structure in the open area was dis­ cussed and verified based on wind tunnel tests. The wind field input of the open area was captured by CFD calculations of the cruise ship. The calculation process of wind-induced vibration of the structure in the open area of a large cruise ship was proposed, and the windinduced vibration characteristics of the typical guide rail structure in the open area were analyzed. The study could provide a reference method for wind vibra­ tion control of open area structures of cruise ships. 2 WIND TUNNEL TEST AND NUMERICAL METHOD OF TYPICAL STRUCTURES IN OPEN AREA 2.1

Wind tunnel test of typical structure in open area

Wind tunnel tests were carried out in the open wind tunnel at Wuhan University of Technology, China,

mainly including rigid force test and wind-induced vibration test. The plate structure was chosen, with the bottom fixed as the boundary condition and steel as the material, according to the design requirements of typical structural qualities, boundary conditions, and material properties of the open area. The Rey­ nolds number of the open area is set as Re54 � 105 . The structural size and measuring point arrange­ ment are shown in Figure 1. As shown in Figure 2, the turntable is designed to fix the bottom of the plate structure. Moreover, wind angles could be obtained by adjusting the rotation angle of the turntable.

Figure 1. Model size and measuring point layout.

Figure 2. Turntable and wind direction angle condition.

Wind tunnels were set with different wind speed conditions, including 15 m/s, 20 m/s, 25 m/s, and 30 m/s. Four wind directions of 0°, 30°, 60°, and 90° were set. A pitot tube and impeller anemometer were used to measure the incoming wind speed to ensure the accuracy of the input wind speed in the test sec­ tion. The wind load tests employed a six-component balance with a measurement range of 400 N/40 N m.

Figure 3. Plate structure vibration test.

As shown in Figure 3(a), vibration displacement was measured by a KathMatic Kv laser vibrometer to obtain the vibration displacement variation trend of the monitoring points under different wind condi­ tions (wind speed, wind direction angle, spatial pos­ ition). As shown in Figure 3(b), the vibration of the plate structure under various wind conditions was tested by the DIC system. The surface speckles were created on the test shooting surface and turned on the spotlight. After focusing, two high-speed cam­ eras mounted on the cloud platform were used for shooting. After software postprocessing, the vibra­ tion results were extracted.

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2.2

Study of wind-induced vibration simulation methods for typical structure in open area

The calculation domain of the corresponding size of the WHUT wind tunnel was established, as shown in Figure 4. The inlet of the computational domain was set as a velocity inlet, and the velocity was consistent with the wind velocity of the test. The wind tunnel outlet was set as the static pressure outlet. As the wind tunnel was open, the six sides of the square domain of the test section were set as static pressure outlets. The wall surfaces of the wind tunnel and test platform were non-slip wall surfaces. The surface of the plate struc­ ture was set as the fluid-solid coupling interface and non-slip wall. Adopt the Cartesian cut volume and polyhedral mixed mesh, and two local mesh refine­ ments around the plate structure and the wind tunnel wall were set. Ten layers of prism meshes with a growth rate of 1.1 were distributed on the surface of the plate, and the non-dimensional wall distanceyþ 55. Four sets of mesh refinement schemes were subse­ quently adopted for exploration.

information. The dynamic meshes were inserted around the structural surface to form uniform deformation. The structural simulation was set in the structural finite element software MSC Marc, and the setting parameters were consistent with the wind tunnel test. The structure and fluid contact surface were the fluid-structure coupling interface of co-simulation. The time step and iterations were set to be consistent with the fluid setting. Set the path of the fluid and structure calculation in the cosimulation interface Msc Cosim. The interface is used to exchange information between the fluid and the structure to achieve the interpolation calculation.

Figure 5. Fluid-solid coupling parameters analysis. Figure 4. Calculation domain of wind tunnel. Table 1.

Four mesh refinement schemes.

Mesh Scheme

y/L x/L (�10 2 ) (�10 4 )

z/L (�10 3 )

Number of mesh (�106 )

Scheme I Scheme II Scheme III Scheme IV

1.92 1.60 1.28 1.12

3.84(20%) 3.21(20%) 2.56(20%) 2.25(20%)

0.88 1.27 1.73 2.45

9.23(4.8%) 7.69(4.8%) 6.15(4.8%) 5.41(4.8%)

The mesh size is dimensionless, where x is the basic size of the mesh, y is the near-wall refinement size of the structure, z is the near-wall size of the wind tunnel, and L is the characteristic length of the computational domain, which is taken as the max­ imum value of the length, width, and height of the computational domain. Set the physical flow time to 8 s, and the time step was set to 0.01 s, 0.005 s, 0.001 s, and 0.0005 s. SST k ω, Realizable k ε and Des-SST k ω models were util­ ized for turbulence model analysis. The interface was set to exchange fluid pressure and structural displacement

Figure 6. Trend of vibration displacement with wind speed increasing.

As shown in Figure 5, with the mesh scheme’s con­ tinuous refinement, the drag coefficient and the vibra­ tion frequency calculation errors decrease first and then increase. The error of mesh scheme III is the minimum. The error of the vibration frequency decreases grad­ ually with mesh densification, and the difference between scheme IV and scheme III is insignificant. The

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dominant frequency measured by the test is 11 Hz, while the dominant frequency calculated by the simulation is 11.9 Hz, and the relative error of vibra­ tion frequency is small. The vibration cloud diagram of the simulation achieved a perfect match with the experimental result. 3 ANALYSIS OF LARGE CRUISE SHIP WIND FIELD AND SUBDOMAIN WIND FIELD MAPPING IN OPEN AREA 3.1

Velocity mapping based on subdomain method

The subdomain method can use the analysis results of the whole domain as the initial or boundary condi­ tions of another subdomain analysis. This method is suitable for the refined calculation of the flow field of the local target building in the building group with the group shielding effect. Through the CFD calcula­ tion of the building group, the wind environment dis­ tribution of the target building, that is, the small square building, in the building group can be obtained. The calculation results of the whole domain are used as the initial conditions to carry out the CFD calculation of the target building subdomain. Figure 7. Vibration cloud diagram and displacement fre­ quency response curve.

average error index of the above three evaluation indexes shows that mesh scheme III is the best refine­ ment scheme. The error of the drag coefficient is the smallest when the time step is 0.0005 s. The average vibration error decreases with decreasing time steps. The average error increases first and then decreases with the decrease of the time step. Considering the cal­ culation cost and accuracy, The following co-simula­ tion calculations used the time step of 0.001 s. As shown in Figure 5(c), the SST k ω model has a minimal error between the calculated drag coefficient and the average vibration value, and the vibration frequency calculation error of Realizable is the smallest. After averaging the three errors, it can be found that the SST k ω model has the minimal error. The following wind-induced vibration numer­ ical simulation adopted the SST k ω model. Each simulation was computed in parallel on the HPC using 56 processors. The co-simulation of each test condition using the above-selected parameters takes approximately 142 to 158 hours. Figure 6 compares the wind tunnel experimental data and computed results. The results of the numer­ ical calculations matched well with the experimental data at wind speeds of 10-20 m/s. As the increase of wind speed, the average displacement results obtained by the test are gradually larger than the simulation, and the results are in good agreement under different wind angles. The frequency domain curve of the vibration displacement of the structure was tested by DIC at a wind angle of 60°. The

Figure 8. Whole domain and Subdomain.

Figure 9. The velocity cloud diagram of the transverse sec­ tion at 0.015 m position.

As shown in Figure 9, the velocity cross-section cloud diagram is obtained at a distance of 0.015 m from the inlet, where the wind field distribu­ tions of the two cases are basically the same. Figure 10 shows the pressure curve of the target building struc­ ture distributed along the central ring of the building.

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right sides and top side were treated as symmetry planes. The dimensions of the calculation domain were as follows Figure 11(b), and the blocking rate calculated according to the longitudinal section area of the ship was 4.3 % less than 5 %. Due to the var­ iety of structural forms in the open area of cruise ships, a typical guide rail frame structure was selected for wind-induced vibration research. The guide rail structure is located in the open area in front of the chimney, as shown in Figure 11(d). Its structural fea­ ture is the track above the support of the circumferen­ tially arranged pillars. The track is used to arrange the sightseeing cable car, and its subdomain was estab­ lished, as shown in Figure 11(e), so that the wind field could be input to the subdomain 11(c) based on the wind field results calculated in the whole domain.

Figure 10. Wind pressure circumferential distribution curve.

The negative pressure amplitude of the subdomain at positions 1-7 in front of the structure was slightly smaller than that of the whole domain. The nega­ tive pressure amplitude of the subdomain at posi­ tions 11-17 in the rear was slightly larger than that of the whole area. The subdomain’s predicted pressure values exhibited consistent variation with the whole domain. 3.2

Wind field analysis of large cruise ship

The CFD calculation domain of the ship above the waterline was established. The inlet was set as the exponential velocity profile, as shown in Figure 11(a). The roughness coefficient was taken as 0.1, the refer­ ence elevation was 10 m above the ocean, and differ­ ent reference wind speeds were set. The bottom of the domain was set as a slip wall, and the hull and super­ structure were set as non-slip walls. The outlet was set as a static pressure outlet. The domain’s left and

Figure 12. Mesh division of whole domain.

Figure 12 shows the mesh division of the whole ship’s computational domain and local mesh refine­ ment. Because the cruise ship is a complex bluff body structure, the Cartesian cutting body, and the polyhedral

Figure 11. Whole ship domain and open area subdomain of large cruise ships.

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body combined mesh can be used to simulate the com­ plex boundary. Wang et al. (2020) conducted the wind tunnel test and numerical simulation for this cruise ship. They conducted a mesh dimensionless size con­ vergence analysis based on the aerodynamic coeffi­ cients measured in the test. According to this document’s mesh convergence analysis results, the basic size/LOA was selected as 0.035. The basic mesh size was 11.3 m, the outermost densified size was 5.65 m, the inner refinement area was 2.82 m, and the ship wall refinement size was 0.375 m. Moreover, eight layers of prism meshes with a growth rate of 1.05 were distributed on the ship’s surface, and a total of 4.51 million meshes were set at a wind angle of 0 °. The analysis was a transient analysis with a time step of 0.001 s and a set of 8000 iterations. The Realizable k ε turbulence model was utilized for the closure of the RANS equations. Under different wind angles, the wind field in the open area is not only directly affected by the wake gen­ erated by the flow separation in the front area, but also by the flow separation on the left and right sides of the whole ship. The wind field of the whole ship is differ­ ent at different wind angles. Under the influence of the flow separation of the whole ship and the front struc­ ture’s wake, the open area’s wind field has complex and varied wind field inputs. It can be seen from the subdomain method that the wind field in the open area is greatly affected by the wind angles under the whole ship wind environment. At wind angles of 30° and 60°, wind field input in the subdomain is affected by the open area ahead to form low-speed regions on the lower right, while at 150° and 180°, they are signifi­ cantly shielded by the chimney, forming low-speed regions on the left and in the middle. 4 WIND-INDUCED VIBRATION ANALYSIS OF TYPICAL GUIDE RAIL STRUCTURE IN OPEN AREA 4.1

Modal analysis of guide rail frame structure

The structure’s material properties are steel, elastic modulus E = 2.06×105MPa. Poisson’s ratio is 0.3, and the density is 7.85×10-9t/mm3.

Figure 13. Wind speed distribution of whole ship and open area under different wind angles. Figure 14. Guide rail frame structure in open area.

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structure’s natural frequencies are mainly concentrated in the low-frequency band, and the maximum vibration occurs at the front and rear sides of the track and the top of the pillar. 4.2

Figure 15. Typical vibration mode and natural frequency of guide rail.

The modal calculation was carried out using the finite element method. Three typical overall vibration modes are selected, as shown in Figure 15. The third order is the offset to the upper left corner and lower right corner. The seventh order is the inwards bending of the left and right side tracks, while the front and rear tracks are offset to the front side. The thirteenth order is mainly the inwards bending of the tracks. The

Parameter settings of the subdomain

The coordinate system of the subdomain was consist­ ent with the whole domain to ensure the accuracy of the wind field input of the subdomain. As shown in Figure 11(c), the inlet was set as the vector mapping of the whole domain, and other boundary conditions of the subdomain were consistent with the boundary of the whole domain. The guide rail was set as a non-slip wall surface that was a fluid-solid coupling interface. The two-way fluid-solid coupling calculations have been carried out according to the aforementioned fluid calculation settings, such as mesh division, step size, turbulence model, and structural finite element settings.

Figure 16. Frequency domain curves of acceleration at different positions.

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4.3

Wind-induced vibration analysis of guide rail structure

The vibration monitoring points are shown in Figure 14. Figure 16 shows the frequency domain curves of the vibration acceleration of the monitor­ ing points at different track positions. It can be seen that the dominant vibration frequencies of tracks F and B are consistent, both of which are 1.47 Hz, close to the third-order natural frequency of the structure of 1.52 Hz, and both have significant vibra­ tion peaks at 7.2 Hz. The dominant frequency of track L and track R is 0.8Hz, and both have a vibra­ tion peak at 9.26Hz. Compared with tracks F and B, track L and track R have peaks at 27.5Hz, and track B has the most abundant vibration frequency among the four cases, mainly because track B is affected by the complex incoming flow. The vibrations of the four tracks are concentrated in the low-frequency band within 10.1Hz. From the frequency domain curve of the vibra­ tion acceleration of the vertical monitoring points of the pillar, it can be seen that the distribution of the vibration peak frequency components of point 5 and point 6 is basically the same, with peaks at 1.48 Hz, 8.6 Hz, and 10.1 Hz. The difference is that the dominant frequency of point 5 is 10.1 Hz, while point 6 is 1.47 Hz. Point 7 at the vertex position is most affected by the nearby track vibration and the influence of the wind field ahead. The vibration fre­ quency components are different from that of the lower pillar. 4.4

Figure 17. Frequency domain curves of track under differ­ ent wind speeds.

Wind-induced vibration analysis of guide rail under different offshore wind conditions

Figure 17 shows that the vibration frequency is roughly the same under each wind speed condition. The fundamental frequency of the track is close to the third-order natural frequency of 1.52 Hz, with vibration peaks at 7.2. The dominant vibration fre­ quencies under 15 m/s, 20 m/s, and 25 m/s wind speeds are 8.6 Hz, which is close to the 26th-order natural frequency of 8.7 Hz, while under 30 m/s wind speed conditions, the dominant vibration fre­ quency is 1.47 Hz, which is close to the third-order natural frequency. As shown in Figure 18, the vibration frequency of the pillar under various wind speed conditions is basically the same. The fundamental frequency is close to the third-order natural frequency, with vibra­ tion frequency components at 8.6 Hz and 10.1 Hz. The dominant vibration frequency at 15 m/s is 10.1Hz. The dominant frequency of vibration is 8.64 Hz at 20 m/s, and the amplitude of 8.66 Hz at 25 m/s is roughly equivalent to that of 10.1 Hz, while the dominant frequency of vibration is 1.5 Hz at 30 m/s wind speed. The change rule here is consistent with that of the track and tends to the large amplitude of low frequency with the increase in wind speed.

Figure 18. Frequency domain curves of pillar under differ­ ent wind speeds.

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Figure 19. Frequency domain curves of track vibration displacement under Figure 20. Vibration displacement ampli­ tude of track under different wind angles. different wind angles.

Figure 21. Frequency domain curves of pillar vibration displacement under Figure 22. Vibration displacement ampli­ tude of track under different wind angles. different wind angles.

Figure 19 shows the track’s vibration displace­ ment curves of the frequency domain under different wind angles. The dominant vibration frequency and vibration displacement amplitude are the largest at a wind angle of 0°, and the dominant vibration fre­ quency is 3.6 Hz, which is close to the seventhorder natural frequency of 3.611 Hz. The dominant vibration frequency at a wind angle of 30° is con­ sistent with that at 180° wind angle, which is 1.39 Hz, close to the second-order natural frequency of 1.34 Hz. The vibration frequency at a wind angle of 60°is similar to that at a 120° wind angle. There are two main vibration frequency components, 1.49 Hz and 2.07 Hz, close to the third-order of 1.52 Hz and fourth-order natural frequencies of 2.095 Hz. The vibration frequencies are 1.69 Hz and 1.6 Hz at wind angles of 90° and 150°, respectively. Figure 21 shows the frequency domain curve of vibration displacement at the monitoring point of the pillar under different wind angles. The dominant fre­ quency of the pillar vibration under each wind angle is relatively close, between 1.2-1.39 Hz, close to the first-order natural frequency of 1.234 Hz and the second-order natural frequency of 1.34 Hz. The dominant frequency of vibration is the largest at a wind angle of 180°, and the vibration displacement of the dominant frequency is the smallest at this wind angle. The dominant frequency is the smallest at a wind angle of 120°, and the vibration amplitude of the dominant frequency is the largest at this wind angle, followed by the wind angle of 90°.

Figures 20 and 22 shows the distribution map of the vibration displacement amplitude of the track’s horizontal position and the pillar’s vertical position under different wind angles. The displacement amplitude of the track structure is the largest at wind angles of 0° and 120°. The displacement amp­ litude of left track L is the largest under the 0° wind direction, while the amplitude of displacement of rear track B is the largest at a wind angle of 120°, and the amplitude of displacement of track F under each wind direction is the smallest, followed by track R. The pillar’s vibration displacement amplitude increases with the pillar height in each wind direc­ tion and is consistent with the track structure. The vibration displacement amplitude is the largest at wind angles of 0° and 120°. 5 CONCLUSIONS This paper takes the typical structure of a cruise ship open area as the research object. First, the fluid-solid coupling numerical method was explored and verified based on wind tunnel tests. Then, the initial wind field was extracted by using the subdo­ main method based on the wind environment calcu­ lation of the whole ship. Finally, the wind-induced vibration of the guide rail frame structure in the open area was analyzed. The conclusions are as follows:

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(1) Based on the experimental data, it was found that the error between the mesh refinement scheme III and the test is minimal. The turbu­ lence model was discussed, and it found that the error of simulation calculation of the SST k ω model is minimal. (2) It was verified that the subdomain method has good applicability and accuracy. The wind field input and distribution in the open entertainment area will be directly affected by the wake gener­ ated by the flow separation in the front area and the diversion of the left and right sides of the whole ship. The wind field input varies signifi­ cantly under different wind directions. (3) From the wind-induced vibration characteristics of the guide rail frame, the dominant frequency of vibration acceleration of the front side track is con­ sistent with that of the rear side track, the left side is consistent with that of the right side, and the vibration frequency of the guide rail frame struc­ ture is mainly concentrated at 0.8-10.1Hz. The wind field in the open area significantly impacts the vibration of the guide rail’s rear side and the pillar’s top. (4) The frequency distribution of the vibration acceler­ ation of the track structure is consistent at each wind speed, except that the dominant frequency is not consistent and tends to approach the thirdorder natural frequency gradually. The dominant frequency of vibration displacement and the amp­ litude of vibration displacement of the track struc­ ture are the largest at a wind angle of 0°, close to the seventh natural frequency. The dominant fre­ quency of vibration displacement of the pillar is the largest at a wind angle of 180°, and the domin­ ant frequency of vibration is the smallest at a wind angle of 120°, but the amplitude is the largest. The wind angles of 0° and 120° are the most unfavor­ able wind direction angles.

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Advances in the Analysis and Design of Marine Structures – Ringsberg & Guedes Soares (Eds) © 2023 The Author(s), ISBN 978-1-032-50636-4

Model test of a dual-spar floating wind farm in regular waves Z. Jiang & G. Liang Department of Engineering Sciences, University of Agder, Grimstad, Norway

T. Lopez-Olocco & A. Medina-Manuel CEHINAV, DACSON, ETSIN, Universidad Politécnica de Madrid, Spain

L.A. Saavedra-Ynocente Laboratorio de Dinámica del Buque, Canal de Ensayos Hidrodinámicos el Pardo (INTA-CEHIPAR), Spain

A. Souto-Iglesias CEHINAV, DACSON, ETSIN, Universidad Politécnica de Madrid, Spain

ABSTRACT: A floating wind farm with shared moorings has the potential to reduce capital expenditure but may face structural dynamics issues. We selected a prototype wind farm that consists of two spar floating wind turbines with shared moorings and carried out model tests with a scale factor of 1:47. Rigid-body motions of one spar and mooring line tensions were measured. In this paper, the test setup is described, and results from the decay and regular-wave tests are discussed. In regular waves, the spar motions in surge, heave, and pitch are dominated by wave frequencies and the extreme motion ranges are acceptable. Compared with the baseline, the clump weight affects the mean position of platform motion; it also reduces the dynamic tension of the shared line but causes higher mean and maximum tension in the single lines. This paper contrib­ utes to an improved understanding of complex floating systems in offshore environments.

1 INTRODUCTION It has been more than a decade since the world’s first megawatt-scale floating wind turbine (FWT), Hywind Demo, was commissioned in 2008. Today, although the technology readiness levels of several FWT con­ cepts are high, many technical and economic chal­ lenges still exist for development of floating offshore wind farms (FOWFs); see (Jiang 2021). For example, to reduce the capital investment costs, solutions like shared anchors and concrete platforms are adopted in the Hywind Tampen farm which consists of eleven 8-MW spar FWTs. Shared mooring stands as an alternative mooring solution with the potential of reducing the total number of mooring lines and anchors in an FOWF. Previously, pilot-scale FOWFs with shared moorings have been studied by several researchers, e.g., (Con­ nolly & Hall 2019), (Liang et al. 2021). The foci of these references are on design and dynamics of the shared mooring systems and numerical methods are applied. To the authors’ knowledge, no experimental work has been carried out to address FOWFs with shared mooring. For such a multibody structural system, hydrodynamic model tests are useful means and provide additional insights into the physical

behavior of the system. To address this need, this paper documents interesting outcomes of a test cam­ paign carried out at CEHIPAR in June 2022. 2 THE FLOATING WIND FARM WITH SHARED MOORING The FOWF is a pilot-scale wind farm that consists of two spar FWTs. Each FWT has a draught of 120 m as specified for OC3-Hywind (Jonkman 2010). The two FWTs are placed along the xg-axis with an initial turbine spacing of 750 m, approximately six times the rotor diameter. Two shared mooring config­ urations, ‘baseline’ and ‘clump’ are tested for the FOWF. As shown in Figure 1, each FWT is moored to the seabed by two single lines. For the baseline configuration, the two FWTs are connected together through a shared line (Line 5). For the clump config­ uration, a clump weight is added to the shared line; see Figure 1. The projected angle between any two adjacent mooring lines is 120 deg in the xy-plan. For large-scale FOWFs with shared moorings, dif­ ferent mooring configurations can be adopted, and the present model can be regarded as the baseline. The baseline configuration was designed and studied

DOI: 10.1201/9781003399759-19

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parameters where the subscript p denotes ‘full scale’ and the subscript m denotes ‘model scale’. Two con­ figurations are considered, i.e., the first one with a pure shared line and the second one with a clump weight attached to the middle of the shared line. Table 2. Conversion of physical variables according to the Froude similarity laws.

Figure 1. Schematic of a dual spar floating wind farm.

Table 1. design.

Mooring properties of the selected single line

Parameter Material Length [m] Diameter [mm] Sheath thickness [mm] Mass per unit length [kg/m] Submerged weight [N/m] Extensional stiffness [N] Minimum breaking strength [N]

Wire segment

Chain segment

Sheathed steel wire 250 95 10 47.39 360.42 8.47E+08 9.34E+06

R3 Stud­ less chain 415 140 392.00 3535.94 1.53E+09 1.43E+07

previously (Liang et al. 2021, Liang et al. 2022). To accommodate the water depth of the ocean basin, a full-scale water depth of 235 m is considered. Therefore, the single lines of a spar FWT of OC3 Hywind were redesigned for the updated water depth. Details of the single line design can be found in (Lopez-Olocco et al. 2022). Mooring properties of the selected single line design are presented in Table 1. The unstretched length of the shared line is 739.6 m. The shared line is made of steel wire rope and its material properties is kept the same as the wire segment of the single lines. 3 MODEL TEST 3.1

General

Inertial and gravitational forces are important when it comes to testing of a floating structure in experi­ mental environments. Therefore, the Froude similar­ ity laws are chosen to scale down the mass and geometry of the OC3-Hywind spar platform (Jonk­ man 2010). Considering the depth limitation of the wave basin, a scale factor λ of 47 is selected. All structural elements of the dual-spar FOWF, including the spars and the mooring lines are scaled down based on this scale factor. Table 2 lists the the rela­ tion between the full- and model-scale physical

Parameter

Symbol

Scale factor

Length Linear velocity Linear acceleration Angle Angular velocity Period Density Displacement

Lp =Lm vp =vm ap =am θp =θm �p =�m Tp =Tm ρp =ρm Δp =Δm

λ λ1=2 1 1 λ 1=2 λ1=2 β βλ3

Figure 2. Image of the two FWT models during model tests; waves propagate from the right to the left.

Table 3.

Mass and geometrical properties of the FWT.

Parameter

Full-scale

Model-scale

Mass [kg] Ixx [kgm2 ] Iyy [kgm2 ] Izz [kgm2 ] CoG [m] D1 [m] D2 [m] Draught [m]

8.2372E+06 1.93E+10 1.93E+10 1.916E+8 -78 9.5 6.5 -120

77.721 84.39 84.38 0.69 -1.649 0.140 0.200 -2.54

3.2

Model test set-up and instrumentation

The full- and model-scale properties of the spar models are listed in Table 3. Here, CoG stands for center of gravity with the origin of the coordinate system at the mean water level. D1 and D2 refers to the lower and upper diameter of the spar, respectively. The

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Table 4.

List of wave conditions of the regular wave tests. Full-scale

Table 5. Natural periods of Spar 1 identified from decay tests [s].

Model-scale

Parameter

H [m]

T [s]

H [m]

T [s]

Operational Extreme

2.25 10.0

9.60 16.0

0.053 0.213

1.40 2.33

mass distribution of the model represents that of OC3 Hywind, but rotor blades and wind loads were not explicitly addressed in the model tests. A thermoplastic material, polyvinyl chloride (PVC), was used as the main construction material. Lead panels were used as ballast and put inside each spar floater model. Add­ itionally, to achieve the desired mass properties of the model, an aluminum structure with distributed weights was mounted on top of each PVC main structure. Prior to the test, calibrated springs were attached to the anchor of the single lines to reproduce equivalent stiffness of the single lines and the shared line. Special care was taken to place the anchors at the exact loca­ tions as shown in the layout (Figure 1). In the second configuration, a cylindrical clumped weight was placed at the middle of the shared line. The submerged weight of the clump was 15 tonnes on full scale. Details of the model scaling and mooring system prop­ erties can be found in (Liang et al. 2022) and (LopezOlocco et al. 2022). During the tests, the following measurements were obtained: six degree-of-freedom (DOF) motions of Spar 1; wave elevation at two different locations, i.e., one next to Spar 2 and the other at midway between the two spars; mooring tensions at two fairleads of the shared line and at three fairleads of the single lines of Spar 1. All the instrumentation was calibrated before the formal tests to ensure the quality and reliability of the acquired data. For the motions a camera based optical tracking system was used. The water surface elevation was measured by means of a capacitance and an ultra­ sonic wave probe. Finally, the fairlead tension was obtained with the use of four one-component HBM load cells with an strain gauge full bridge. More details regarding the instrumentation and its uncertain­ ties can be found in (Lopez-Olocco et al. 2022). 3.3

Test matrix

The test matrix of the load cases selected from the experimental campaign is detailed in Table 4. Two regular-wave conditions were tested for each moor­ ing configuration. The operational condition repre­ sents mild sea states and the extreme condition represents survival conditions. No additional envir­ onmental load, e.g., wind or current, was considered. In addition, decay tests for the 6 DOF in Spar 1 were performed to obtain the natural periods and damping level of the FOWF in different DOFs.

DOF

Baseline

Clump

Surge Sway Heave Roll Pitch Yaw

142.71 83.88 30.55 31.52 31.63 23.93

134.21 79.85 30.59 31.40 31.47 22.40

4 RESULTS In this section, we present results from the model tests in the order of decay test, and regular wave tests in both operational and extreme conditions. A focus is placed on the time series. 4.1

Natural periods of the dual-spar system

Natural periods, mode shapes, and damping of a moored marine structure are important properties that characterize the structural dynamics. For an inter­ connected dual-spar FOWF with a mooring design for a deepwater site (water depth=320 m), Liang et al. (2021) linearized the system stiffness and identified 12 natural periods and mode shapes. Here, the natural periods identified from the free decay excitations of Spar 1 are listed in Table 5. Due to test limitation, it was not possible to excite or measure both spars, and only six modes are obtained. The identified surge mode has a natural period of more than 130 seconds; the other surge mode exists with a much lower natural period. Compared with the baseline configuration, the shared line configuration with the clump weight causes an approximately 6.1% reduction in the surge natural period and 4.7% reduction in the sway natural period, whereas the influence on the other DOFs is relatively small. This observation is expected, as the clumped weight (15 tonnes on full scale) increases the mooring stiffness. 4.2

Regular wave test

In this section, the time histories of the model tests are presented on full scale and discussed. Among the vari­ ous response variables, the platform motion response in surge, heave and pitch DOFs and the mooring ten­ sion responses of both the single and shared lines at fairleads are selected. The platform motion response of Spar 1 is presented with respect to its static equilib­ rium positions in two configurations, respectively. 4.2.1 Operational wave condition The absolute elevations of the generated regular waves were obtained from a measurement point located next to Spar 2 and are shown in Figure 3. The presented sea state is called operational condition because it has a relatively high probability of occurrence and the FWTs

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are expected to be in operation. Although the wave ele­ vations are regular, small fluctuations are observed for the peak wave amplitudes for both configurations.

Figure 3. Comparison of the wave elevation, operational wave condition.

Figure 5. Comparison of the mooring tension, operational wave condition.

Figure 4. Comparison of the platform motion, Spar 1, operational wave condition.

Figure 4a presents the platform surge motion of Spar 1. As shown, the surge motion of the spar is dominated by wave frequency-induced first-order motions. In addition, slowly-varying motion of the spar can be clearly observed in the time history of both cases. This slowly-varying period corresponds to the longer surge mode of the dual-spar system. Compared to the baseline configuration, the clump configuration reduces the motion range and results in lower second-order responses due to the increase in the mooring stiffness in surge. Figure 4b and Figure 4c shows the platform heave motion and pitch motion, respectively. Although both the heave and pitch DOFs are dominated by wavefrequency responses, the heave motion is more affected by the clump weight than the pitch motion. As shown in Figure 4b, the response maxima of the platform heave under the clump case is slightly higher than those under the baseline case. This obser­ vation is because the presence of the clump weight draws the two spar FWTs closer to each other and reduces the vertical mooring stiffness provided by the two single lines. Compared to the shared line, the single lines play a more important role in providing restoring stiffness in the heave DOF in addition to the hydrostatic stiffness. As shown in Figure 4c, the dif­ ference between the baseline case and the clump case in the mean and maximum values of the platformpitch motion is limited. This observation is expected as the present shared mooring system, with or without additional clump weight on the shared line, provide

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limited pitch restoring stiffness for the spar platform motion in waves. The restoring moment created by the buoyancy and gravity forces govern. Time histories of the top tension of the mooring Line 1 and the shared line (at the attachment point of Spar 1) is presented in Figure 5a and Figure 5b, respectively. As the vertical positions of the fairleads lie between the buoyancy and gravity centers of Spar 1, the fairlead motion is strongly affected by the rigid body motions of the spar buoy in surge, heave, and pitch DOFs. As these motions are dominated by the wave period of the regular waves, the oscillation periods of Tension T1 and Tension T5f1 are also close to this period. In addition, two interesting observa­ tions can be found from the figures by comparing the tension responses of the baseline and clump weight configurations. First, the mean tension in both the single line and the shared line increases after the clump weight is used. The mean tension is influenced by the pretension of the moorings in the static position and the mean drift forces of the waves. As the clump weight is placed in the middle of the shared line, this additional weight has a more appreciable influence on the shared line than on the single lines. The pretension and the mean tension are increased by the submerged weight of the clump (15 tonnes). Second, for this operational sea state, the dynamic tension of the shared line is significantly reduced (> 70%) after the clump weight is used, whereas dynamic tension of the single line is on a similar level. Although the effect of the clump will depend on the weight, number, and location, this observation indicates the potential of clump in reducing the dynamic tension and hence fatigue damages of the shared line. 4.2.2 Extreme wave condition In the following, time histories from the extreme sea state (H=10 m, T=16 s) are analyzed. As shown in Figure 6, although the measured wave signal is quite regular and has better quality than that of the oper­ ational sea state, the measured wave height slightly exceeds 10 m on full scale. As these waves reflect the actual measurements and are not the calibrated waves, the small difference (