Flexible Engineering Toward Green Aircraft: CAE Tools for Sustainable Mobility [1 ed.] 3030365131, 9783030365134

Table of contents :
Preface
References
Contents
Designing the Next Generation of Aircraft with Simulation
1 An Unexpressed Potential
2 A New Era for Aerospace
3 A Concrete Impact
4 The Importance of Universities and Startups
5 The Future
Aeroelastic Wind Tunnel Tests of the RIBES Wing Model
1 Introduction
2 The Wing Model and the Wind Tunnel Facility
3 Experimental Setup and Flow Visualization
4 Force and Pressure Measurements
5 Stress and Displacement Measurements
6 Conclusion
References
Validation of High Fidelity Computational Methods for Aeronautical FSI Analyses
1 Introduction
2 Theoretical Background
2.1 Radial Basis Functions
3 Strategies of the FSI Numerical Analyses
3.1 Structural Modes Embedding
3.2 2-Way FSI Coupling
4 Test Case Description and Numerical Models
4.1 Experimental Reference Data
4.2 FEM Model and Solution
4.3 RBF Setup
4.4 CFD Configuration
5 Results
5.1 Aerodynamic Coefficients
6 Conclusions
References
High-Fidelity Static Aeroelastic Simulations of the Common Research Model
1 Introduction
2 Computational Aeroelasticity Tool
2.1 Flow Solver
2.2 Structural Solver
2.3 Fluid-Structure Interface
2.4 Mesh Deformation
2.5 Convergence of Coupled Solution
3 Test Case
3.1 Geometry
3.2 Structural Models
3.3 Aerodynamic Models
3.4 Flow Conditions
4 Results
4.1 Convergence Study of Static Aeroelastic Solution
4.2 Solution of Wing-Fuselage Geometry with Wing-Box Structural Model at M = 0.85
4.3 Solution of Wing-Only Geometry with Beam Stick and Wing-Box Structural Models
5 Conclusion
References
Aero-elastic Simulations Using the NSMB CFD Solver Including results for a Strut Braced Wing Aircraft
1 Introduction
2 The NSMB CFD Solver
3 The B2000++ Solver
4 Geometric Multi-region Coupling
4.1 Single-Region Coupling Procedure
4.2 Multi-region Coupling Procedure
4.3 Definition of Coupling Regions
5 Remeshing Algorithm
6 Validation Calculations
6.1 MDO Aircraft
6.2 AGARD 445.6 Wing
7 Strut Braced Wing Aircraft
7.1 PROTEUS
7.2 Aero-elastic Coupling
7.3 AMLoad
7.4 Results
8 Conclusions
References
Semi-Analytical Modeling of Non-stationary Fluid-Structure Interaction
1 Introduction
2 Theoretical Framework
2.1 Assumptions
2.2 Mathematical Formulation
2.3 Solution
3 Illustration of the Application of the Methodology: Analysis of the Shock Response of a Submerged Cylindrical Shell Subjected to a Double-Front Shock Wave
4 Conclusions
References
Fluid Structure Modelling of Ground Excited Vibrations by Mesh Morphing and Modal Superposition
1 Introduction
2 Theoretical Background
2.1 Formulation of Modal Theory
2.2 Radial Basis Functions
3 Problem Definition
3.1 Formulation of the Problem
3.2 Coupling Parametric Space
4 Method
5 Validation
6 Results and Discussion
7 Conclusions
References
Unsteady FSI Analysis of a Square Array of Tubes in Water Crossflow
1 Introduction
2 Structure of the Document
3 Experimental Set-up
4 Background on RBF Mesh Morphing and Modal Superposition FSI Method
5 Numerical Strategy and Used Numerical Means
6 Isolated Tube Case
6.1 CAD and FEM Case Set-up
7 RBF Solutions Set-up
7.1 CAD and CFD Case Set-up
7.2 Results of the Single Tube Case
8 Tube Array Case
8.1 CAD Case Set-up
8.2 RBF Solutions Set-up
8.3 CFD Case Set-up
9 Results of the Tube Array Configuration
10 Conclusions
References
Risk Measures Applied to Robust Aerodynamic Shape Design Optimization
1 Introduction
2 Risk Measures for Robust Design Optimization
3 Optimization Algorithm
4 Flow Field Analysis Method
5 Airfoil Shape Handling in the Presence of Uncertainties
6 Error Handling
7 Design Optimization Example
7.1 Aerodynamic Design Problem Description
7.2 Deterministic Problem Solution
7.3 Robust Design Optimization
8 Conclusions
References
Aero-structural Optimization of a MALE Configuration in the AGILE MDO Framework
1 Introduction
2 Requirements Definition
3 Preliminary Design Workflow
3.1 Engine Deck Definition
3.2 System Design
3.3 Handling Quality Evaluation
4 Aero-elastic Shape Optimization
4.1 Analysis and Tool-Chain Preparation
4.2 Optimization Work-Flow Application
4.3 Analysis Model Update
4.4 Automated Workflow Execution
5 Workflow Results
6 Conclusion and Future Prospects
References

Citation preview

Lecture Notes in Applied and Computational Mechanics 92

Marco Evangelos Biancolini Ubaldo Cella   Editors

Flexible Engineering Toward Green Aircraft CAE Tools for Sustainable Mobility

Lecture Notes in Applied and Computational Mechanics Volume 92

Series Editors Peter Wriggers, Institut für Baumechanik und Numerische Mechanik, Leibniz Universität Hannover, Hannover, Niedersachsen, Germany Peter Eberhard, Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany

This series aims to report new developments in applied and computational mechanics - quickly, informally and at a high level. This includes the fields of fluid, solid and structural mechanics, dynamics and control, and related disciplines. The applied methods can be of analytical, numerical and computational nature. The series scope includes monographs, professional books, selected contributions from specialized conferences or workshops, edited volumes, as well as outstanding advanced textbooks. Indexed by EI-Compendex, SCOPUS, Zentralblatt Math, Ulrich’s, Current Mathematical Publications, Mathematical Reviews and MetaPress.

More information about this series at http://www.springer.com/series/4623

Marco Evangelos Biancolini Ubaldo Cella •

Editors

Flexible Engineering Toward Green Aircraft CAE Tools for Sustainable Mobility

123

Editors Marco Evangelos Biancolini University of Rome “Tor Vergata” Rome, Italy

Ubaldo Cella University of Rome “Tor Vergata” Rome, Italy

ISSN 1613-7736 ISSN 1860-0816 (electronic) Lecture Notes in Applied and Computational Mechanics ISBN 978-3-030-36513-4 ISBN 978-3-030-36514-1 (eBook) https://doi.org/10.1007/978-3-030-36514-1 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Modern engineering design and analysis tools very often involve shape modifications for which an opportune geometric parametrization strategy is required. The efficiency of the technique selected to perform this task acquires particular importance when the shape updating process has to be integrated in automatic procedures. In aerospace engineering, there are many disciplines in which this aspect plays a crucial role. Some of the most sensitive to the reliability of shape parametrization methods are the ones involving numerical shape optimization procedures and fluid–structure interaction (FSI) analyses. FSI analysis capabilities, in particular, are fundamental in estimating the real operative behavior of aircraft. The flexibility of structures, in fact, affects the flight performances in all conditions. The capability to evaluate the interaction between aerodynamic, inertia and elastic forces is important to avoid drag penalties, control system efficiency loss and generation of potentially dangerous phenomena as divergence, control reversal and flutter. When this multi-disciplinary analysis capability, historically demanded to a verification stage, is adopted in the design process, it can be decisive in improving the performances and in reducing the time to market of the product. Such capability, nevertheless, is related to the performances of the FSI analysis methods available and to the numerical tools adopted to implement the coupling procedures of the solvers involved. Several approaches can be used in the implementation of geometric parameterizations. A strategy that offers several advantages consists in applying the shape modification directly to the numerical domain by mesh morphing techniques. The quality of the morphing process depends on the algorithm adopted to implement the smoothing action. Radial basis functions (RBF) are recognized to offer one of the most efficient mathematical frameworks to face this task. The Department of Enterprise Engineering “Mario Lucertini” of the University of Rome “Tor Vergata” has long experience in developing numerical methods based on RBFs [1]. From 2009, furthermore, it is active in the field of mesh morphing in cooperation with the company RBF Morph that launched to the market the first commercial mesh morphing software based on radial basis functions. Today, the software is very mature. The know-how at its base is the result of a long development process v

vi

Preface

within several research programs. In the aerospace field, it was the core technology of two EU-funded research projects that inspired the publication of the present book: RBF4AERO (www.rbf4aero.eu) and RIBES (www.ribes-project.eu) (Fig. 1). Object of RBF4AERO was the development of a benchmark technology numerical platform for aircraft design setting up methodologies based on RBF mesh morphing techniques [2] (Fig. 2). The platform can face some of the most relevant aircraft design problems as FSI analysis [3], icing growth [4] and shape optimization combining adjoint and evolutionary-based algorithms [5, 6]. The RBF4AERO platform was implemented on HPC Cloud to set up the experiment titled “Cross-Solver Cloud-based Tool for Aeronautical FSI Applications” within the EU-funded FORTISSIMO project (https://www.fortissimoproject.eu/experiments/906).

Fig. 1 RBF4AERO and RIBES projects logos

Fig. 2 The RBF4AERO platform allows to perform advanced shape optimization of multi-physics and multi-objective coupled analyses

Preface

vii

The main topic of the RIBES project was the reduction in the uncertainness in aeroelastic analyses based on coupled CFD–CSM numerical methodologies [7]. In addition, the project was addressed to the implementation of an RBF-based workflow for shape structural optimization (Fig. 3) and to the experimental validation of FSI methodologies [8]. A significant part of the budget was allocated to the setup of the wind tunnel experimental campaign. In this task, an extensive set of aeroelastic measurements, using an opportunely designed wing model with a typical metallic aeronautical wing box structure, was performed. The numerical experimental activity based on the RIBES wing continued after the conclusion of the project by a collaboration with the University of Rome “La Sapienza” within which the modal properties of the model were investigated and used to validate numerical configurations [9]. At the conclusion of the projects, the University of Rome “Tor Vergata” decided to organize a one-day workshop aimed to present the results of the two programs and to highlight the status of researches in the aerospace field focusing on the recent achievements in analysis and design methodologies adopted in the aircraft design process. December 14, 2017, the event Flexible Engineering Toward Green Aircraft organized by the editors of the present book was hosted by the university in Rome. Professors and engineers active in EU research programs, from several academic, industry and research institutions, joined the workshop to provide an overview on topics as Aeroelasticity, Aerodynamic Robust Design and Shape Optimization. A significant part of the agenda was dedicated to the progress of “high-fidelity” CFD–CSM coupling and to the description of the recent experimental activities performed to support the validation of numerical fluid–structure interaction analysis methodologies. An online service was setup to provide the possibility to follow the event remotely. The presentations of the works were published on the RIBES Web site (http://ribes-project.eu/flexible-engineering/).

DOE elements definition

FEM model setup

Mesh parameterization

Input files setup

Bulk data file modification

Yes DOE table completed?

FEM analysis Mesh morphing Design variables update

No

Optimized solution Optimization procedure

Fig. 3 The RIBES optimization platform automates FEA calculation

Response Surface

Optimization algorithm

viii

Preface

The success of the workshop suggested to collect relevant contributions to the topics covered by the event and to propose to Springer the publication of a book. The call for contributions, that ended in November 2018, was open to the scientific community in the field of aerospace research and was not limited to the participants of the workshop. It allowed to collect ten high-quality papers that were selected after a peer-reviewed process that involved members of academic institutions and researchers active in EU programs. The book is opened by Paolo Colombo, Global Industry Director of Aerospace and Defense at ANSYS Inc., who provides an overview about “Designing the Next Generation of Aircraft with Simulation” focusing on its unexpressed potential and outlining the future from the perspective of a leading engineering software provider. The static aeroelastic problem is faced exploring the experimental campaign on the RIBES test article, given in the chapter “Aeroelastic Wind Tunnel Tests of the RIBES Wing Model,” and then by methodological and practical applications in chapters “Validation of High Fidelity Computational Methods for Aeronautical FSI Analyses,” “High-Fidelity Static Aeroelastic Simulations of the Common Research Model” and “Aero-elastic Simulations Using the NSMB CFD Solver Including results for a Strut Braced Wing Aircraft,” where the solutions of different benchmarks (HIRENASD, AGARD, CRM, …) are proposed together with relevant industrial applications. Transient aeroelastic problem solutions follow starting from chapter “Semi-Analytical Modeling of Non-stationary Fluid-Structure Interaction,” where a semi-analytical approach is proposed. Chapter “Fluid Structure Modelling of Ground Excited Vibrations by Mesh Morphing and Modal Superposition” covers flow-induced vibrations, and chapter “Unsteady FSI Analysis of a Square Array of Tubes in Water Crossflow” is about a benchmark of water crossflow interaction. The book ends with two chapters about optimization: The robust aerodynamic shape optimization is presented in chapter “Risk Measures Applied to Robust Aerodynamic Shape Design Optimization,” and a complex aerostructural optimization of a MALE configuration is given in chapter “Aero-structural Optimization of a MALE Configuration in the AGILE MDO Framework”. Rome, Italy

Marco Evangelos Biancolini Ubaldo Cella

References 1. Biancolini, M.E. (2018). Fast radial basis functions for engineering applications. Springer International Publishing. 2. Bernaschi, M., Sabellico, A., Urso, G., Costa, E., Porziani, S., Lagasco, F., Groth, C., et al. (2016). The RBF4AERO benchmark technology platform. In VII ECCOMAS congress, Crete Island, Greece, June 5–10, 2016.

Preface

ix

3. Andrejašič, M., Eržen, A., Costa, E., Porziani, S., Biancolini, M. E., & Groth, C. (2016). A mesh morphing based FSI method used in aeronautical optimization applications. In VII ECCOMAS congress, Crete Island, Greece, June 5–10, 2016. 4. Costa, E., Biancolini, M. E., Groth, C., Travostino, G., & D’Agostini, G. (2014). Reliable mesh morphing approach to handle icing simulations on complex models. In 4th EASN association international workshop on flight physics and aircraft design, Aachen, Germany, October 27–29, 2014. 5. Kapsoulis, D. H., Asouti, V. G., Giannakoglou, K. C., Porziani, S., Costa, E., Groth, C., Cella, U., et al. (2016). Evolutionary aerodynamic shape optimization through the RBF4AERO platform. In VII ECCOMAS congress, Crete Island, Greece, June 5–10, 2016. 6. Andrejašič, M., Eržen, A., Costa, E., Porziani, S., Papoutsis-Kiachagias, E. M., Kapsoulis, D. H. et al. (2016) Adjoint and EAs based aerodynamic shape optimization on industrial test cases using RBF4AERO platform. In 4th OpenFOAM User conference, Cologne, Germany, October 11–13, 2016. 7. Biancolini, M. E., Chiappa, A., Giorgetti, F., Groth, C., Cella, U., & Salvini, P. (2018). A balanced load mapping method based on radial basis functions and fuzzy sets. International Journal for Numerical Methods in Engineering, 115(12), 1411–1429. 8. Cella, U., Biancolini, M. E., Groth, C., Chiappa, A., & Beltramme, D. (2015). Development and validation of numerical tools for FSI analysis and structural optimization: the EU RIBES project status. In 44th AIAS national congress, number AIAS 2015-562, Messina, Italy, September 2–5, 2015. 9. Groth, C., Porziani, S., Chiappa, A., Giorgetti, F., Cella, U., Nicolosi, F., et al. (2018) Structural validation of a realistic wing structure: the RIBES test article. Procedia Structural Integrity, 12, 448–456. In 47th AIAS national congress.

Contents

Designing the Next Generation of Aircraft with Simulation . . . . . . . . . . Paolo Colombo

1

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model . . . . . . . . . . F. Nicolosi, V. Cusati, D. Ciliberti, Pierluigi Della Vecchia and S. Corcione

9

Validation of High Fidelity Computational Methods for Aeronautical FSI Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corrado Groth, Marco Evangelos Biancolini, Emiliano Costa and Ubaldo Cella High-Fidelity Static Aeroelastic Simulations of the Common Research Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan Navrátil Aero-elastic Simulations Using the NSMB CFD Solver Including results for a Strut Braced Wing Aircraft . . . . . . . . . . . . . . . . . . . . . . . . J. B. Vos, D. Charbonnier, T. Ludwig, S. Merazzi, H. Timmermans, D. Rajpal and A. Gehri Semi-Analytical Modeling of Non-stationary Fluid-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Serguei Iakovlev

29

49

71

95

Fluid Structure Modelling of Ground Excited Vibrations by Mesh Morphing and Modal Superposition . . . . . . . . . . . . . . . . . . . . 111 A. Martinez-Pascual, Marco Evangelos Biancolini and J. Ortega-Casanova Unsteady FSI Analysis of a Square Array of Tubes in Water Crossflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Emiliano Costa, Corrado Groth, Jacques Lavedrine, Domenico Caridi, Gaëtan Dupain and Marco Evangelos Biancolini

xi

xii

Contents

Risk Measures Applied to Robust Aerodynamic Shape Design Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Domenico Quagliarella, Elisa Morales Tirado and Andrea Bornaccioni Aero-structural Optimization of a MALE Configuration in the AGILE MDO Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Reinhold Maierl, Alessandro Gastaldi, Jan-Niclas Walther and Aidan Jungo

Designing the Next Generation of Aircraft with Simulation Paolo Colombo

Abstract The aerospace industry is the one who is building the most complex machines on Earth. The Boeing 787 Dreamliner, for example, is made of more than 2 million parts that must be designed, tested, assembled and maintained to match the very strong safety and reliability aviation standard. An engine is made of 40,000 parts that are overhauled and replaced every few thousand hours of flight. It is not a surprise that, having to deal with such a complexity, this industry was, together with the automotive one, among the first using the power of numerical simulation. This article is a summary of the opening I’ve made for the workshop “Flexible Engineering Toward Green Aircraft” (December 14, 2017—University of Rome “Tor Vergata”) and is based on the observations I’ve made on how simulation is used while playing my role of aerospace industry director for ANSYS, visiting universities, research centres and customers all over the world.

1 An Unexpressed Potential Despite of this very early adoption, the way simulation is used today by the aerospace industry is far from being optimal. A partial justification is related to the very high accuracy expected from numerical tools, to fulfil the necessities of design and certification processes of such complex machines, that require deep and accurate validation processes by trained and experienced engineers capable to manage, with adequate confidence, experimental measurements and numerical solutions of complex phenomena. The large long-term investment involved in aircraft programs induces, furthermore, a conservative approach in the integration of new simulation technologies in the industry design process. As a consequence, there is a strong legacy from the past in terms of tools and workflows, deeply embedded into well known and certified processes. This makes it very difficult to change them, and there is always a strong resistance when someone try to do it. The typical outcomes are teams working in P. Colombo (B) Aerospace & Defense, ANSYS Inc, c/o Ansys via G. B. Pergolesi 25, 20142 Milano, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_1

1

2

P. Colombo

silos, claiming that they perform multi-physic simulation while the reality is that they manage multiple physics simulation in a sequential way. This is making more difficult to explore a phenomenon in its full complexity and above all to optimize an asset or a system’s performances. We see a better situation when talking about CFD, FEA and Fluid-structure interaction. In the aerospace field, disciplines related to aerodynamics and structures are very sensitive to innovation and historically represented the technology drivers for all others engineering fields. In these cases numerical simulations are often adopted also in the preliminary design phase. In other disciplines, as electrical, electronics, systems and software, engineers are more focusing on physical testing rather than on simulations. When they use simulation, it’s often strictly addressed to power and signal integrity or EMI, and seldom considering also the fluid and mechanical implications. Simulation is rarely used in the pre-design phase and is still the domain of a small group of high skilled experts who are overwhelmed by requests and unable to manage all of them. But is in the pre-design phase that some key decisions are taken, the ones that will strongly influence how the project will evolve, locking up to 80% of the total development costs (Fig. 1). Process automation is often seen as a loss of control and not as a way to improve speed and efficiency; In-house made codes, created when they were indispensable to solve specific problems, are kept in place even when the task could be performed much better from a commercial one, despite of how slow, limited and difficult to maintain they are today. Low fidelity models are used because of their speed, instead of using high fidelity models combined with high performance computing (HPC).

Fig. 1 CFD simulation—cooling for an electronic rack done with ANSYS discovery live. This tool is used during pre-design as it has a very simple interface and visualizes CFD results in seconds, helping to take design decisions quickly

Designing the Next Generation of Aircraft …

3

In general, the risk adverse attitude and the high costs in money and time of certification have created a dualism: aerospace companies are squeezed between their willing to innovate, their fear of failure and its consequences. This has led to an extremely conservative approach, strongly based on incremental innovation, physical demonstrators and fly tests, all certification oriented. This is going to change.

2 A New Era for Aerospace We have already seen it happening in the space sector, where market leaders have been challenged by private funded small startups with no history and no experience on paper. They came with a fresh attitude, based on the rapid insertion of new technologies and the full exploitation of the benefits brought by multi-physics and multi-domain simulation platform. Their motto is to fail quickly but virtually, learn fast and optimize everything to succeed in the minimum amount of time. They have set a new innovation pace in the industry, forcing everybody to leave their safe positions and adapt or take the risk of dying (Fig. 2). Something similar is coming in aviation, led by two different reasons, thus strongly connected. The first one is about global trends. The unpredictability of fuel cost and the economic impact it can have on airliners, together with the regulators’ push for more efficient, quieter and environmentally friendly aircraft, is among the most important one. On the same time, the growing demand for new aircraft is pushing OEMs to look beyond aircraft design and optimize production and maintenance, in order to stay competitive on the market and be able to deliver in a reasonable timeframe cheaper to buy and operate products. The second one is about technology. In the last decade we have seen emerging technologies like additive manufacturing, multifunctional composite materials, high speed connectivity and big data analytics,

Fig. 2 Benefit of simulation on new initiatives like electric vehicles—SAE 2018

4

P. Colombo

high power electrical systems to mature and become ready for aerospace applications. They are bringing new opportunities, as Urban Air Mobility and electric propulsion, but also huge challenges as they imply taking the risk of using technologies and solutions where we don’t have decades of experience and flight data to rely on. The Flightpath 2050—Europe’s vision for aviation document states clearly that “Breakthrough technology will be required to secure future competitive advantage, most notably in terms of energy, management of complexity and environmental performance”. In this fast pacing scenario, simulation appears to be the way to build the missing knowledge and understanding of these new technologies, in a way that is both quick enough and economically doable. It must be, however, highlighted that the aviation industry, unlike other high technology environments, must settle both market and regulation requirements in a peculiar manner. Comparing, for instance, with other competitive markets as automotive or comparable high technological environments as the space one (with which the aviation industry has major similitudes), the requirements for the tools adopted in the aircraft development process follow very different demanding requests from regulations, metrics of damage and life cycle of final products. Aircrafts design is based on reliability of numerical simulation and analytical methods, often linked to airworthiness certification regulations, deeply validated by experiments. Changing the numerical tools within the development phase or during the certification process is a risky procedure. Innovations in both design methodologies and technologies are demanded, in the aviation industry, to research activities following different paths from the development one. From this point of view the aerospace field has historically represented, and still represents, the technology driver for most of other fields of engineering but it is also the most conservative in their integration in the development phase. Referring to technologies in general, a typical example, emblematic of such a resilience, is the adoption of composite materials in the production of aircrafts. The high potential strength of carbon fibre was realized in a process developed in the aerospace field several decades ago but the deep adoption of composite materials in aircraft primary structures belongs to the new millennium. The behaviour described relate also to numerical simulations which follow similar criteria. This scenario contributes to justify the conservative approach observed in their integration at development stages but the competitiveness of modern markets is forcing aviation industry to speedup this process.

3 A Concrete Impact Consolidated simulation platforms support the realization of significant benefits, contributing in reducing both development time and overall product cost. In order to achieve such benefits, companies have to evolve the way they are implementing simulation and start looking to a real consolidated platform, where workflows can be optimized and automated, real multi-physics and multi-domain analysis can be performed together with hardware and software in the loop; modern IT architectures can be exploited to speed up calculations, including HPC and GPU. This approach

Designing the Next Generation of Aircraft …

5

has also the benefit to significantly reduce the total software cost of ownership. Another interesting trend is that simulation use is expanding from the small set of experts in the advanced design phase to both the extreme of the process: pre-design, with real time solving codes like we demonstrated with ANSYS Discovery live, and maintenance through high fidelity physics based digital twin. ANSYS Twin Builder, as an example, has been designed to improve predictive maintenance outcomes, to save on warranty and insurance costs and optimize product’s operations. The tool combines a multidomain systems modeler with extensive 0D application-specific libraries, 3D physics solvers and reduced-order model (ROM) capabilities. When combined with embedded software development tools, Twin Builder allow to reuse existing components and quickly create a systems model of a new product. Including rapid human-machine interface (HMI) prototyping, systems optimization and XiL validation tools is then easier to check the performances of the entire system (Fig. 3).

Fig. 3 ANSYS twin builder is a tool to build, validate and deploy complete systems simulations and digital twins for predictive maintenance

6

P. Colombo

This must be then integrated with industrial internet of things (IIoT) platforms that contains runtime deployment options, allowing to perform predictive maintenance on the physical product and provide feedback to the development about how each asset is used in the reality, to design a better product for tomorrow.

4 The Importance of Universities and Startups In this fast-paced evolving scenario, the importance of universities and startups is growing. Every time we want to go for radical innovation, we need to produce new ideas and knowledge, and to be bold enough to try. Universities can play a central role, above all if they focus their effort on a specific theme, strongly partnering with the industry to orient their effort toward an industry gap, and among themselves to increase the capacity and the scope of their research. The goals should not be only to develop new technologies, but also to find ways to better use the existing one, while working on the gaps. They have the responsibility to train our future engineers and inspire them to challenge the status quo. These are the new generations that are creating startups, often together with seasoned and very experienced people coming from the industry and tired of the limitation they have found. For this reason, universities should facilitate a smoot introduction of the next engineers generation inside companies completing the educational programs with the use of the “state of the art” of design tools allowing industries to benefit from a reduction of a long training requirements (Fig. 4). Startups are for this market (and sometimes for its key players who are supporting them) what Skunkworks was for Lockheed under Kelly Johnson: an environment free of conditioning where revolutionary ideas can be tested, risks are taken and failure is tolerated. In order to minimize the cost of such an approach and develop the technology as fast as possible, as they need to be able to sell quickly, simulation is widely used and a real multi-physic approach is the one preferred since the beginning.

5 The Future Despite of the early start, we have still a long journey to see simulation’s potential fully exploited by the industry in many fields of engineering even throw simulation has become reliable, accurate and able to predict with confidence complex phenomena and the behaviour of products in many operational conditions. Automotive is a bright example: F1 teams are able to explore hundreds of aerodynamic configurations in a week, looking for perfection in races where a fraction of a second per lap can make the difference between winning and losing; Electric powertrains are extensively simulated to look for performances optimization and safety. The benefit of deepen the adoption of simulations in aerospace is also unquestionable but its progress is related to the needs of certification authorities and to the strong investment required

Designing the Next Generation of Aircraft …

7

Fig. 4 The importance of startups when facing disruptive initiatives

by numerical procedures validation. Numerical analysis and analysts, in fact, enter in the certification process but their importance as key parts is an ongoing progress linked to safety and regulations evolutions. Aircraft programs, as already introduced, significantly differ from the design, manufacturing and certification process of any other products. Aircrafts usually need several years from their preliminary design to the enter in operation. No comparison with any other fields of engineering is then possible. Nevertheless, also in aerospace is felt the sense of urgency to take advantage of emerging technologies and business models, cope with more stringent rules and look for the entire assets’ life optimizations to keep the whole project’ costs as low as possible. Even though the time to market of an aircraft program might be marginally affected, digitalization is the way to achieve such results and certainly represents the key strategy to design better machines.

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model F. Nicolosi, V. Cusati, D. Ciliberti, Pierluigi Della Vecchia and S. Corcione

Abstract Aeroelastic wind tunnel tests on a half-wing model have been performed at the University of Naples “Federico II” to acquire data about aerodynamic forces, section pressure coefficient, stress, strain, and model displacement, to validate high fidelity Fluid-Structure Interaction approaches based on Reynolds-Averaged NavierStokes and Finite Element Method solutions investigated at the University of Rome “Tor Vergata”. Most of the available experimental databases of aeroelastic measurements, performed on aircraft wings, model full scale systems, focusing primarily on aerodynamic aspects rather than on structural similitudes. To investigate flow regimes that replicate realistic operating conditions, wind tunnel test campaigns involve the generation of relative high loads on models whose safe dimensioning force the adoption of structural configurations that lose any similitude with typical wing box topologies. The objective of this work is to generate a database of loads, pressure, stress, and deformation measurements that is significant for a realistic aeronautical design problem. At this aim, a wind tunnel model of a half-wing that replicates a typical metallic wing box structure and instrumented with pressure taps and strain gages has been investigated. All experimental data and numerical models are freely available to the scientific community at the website www.ribes-project.eu.

List of symbols (•)∞ c p q y η

Free-stream condition Wing local chord Static pressure Dynamic pressure Wing span station Non-dimensional span station

F. Nicolosi · V. Cusati · D. Ciliberti (B) · P. Della Vecchia · S. Corcione DAF Research Group, Department of Industrial Engineering, University of Naples “Federico II”, Via Claudio 21, 80125 Naples, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_2

9

10

C (•) Cp D L M N Re V

F. Nicolosi et al.

Aerodynamic coefficient Pressure coefficient Aerodynamic drag Aerodynamic lift Pitching moment Normal force Reynolds number Flow speed

Abbreviations AOA CFD CSM FEM FSI RANS

Angle Of Attack Computational Fluid Dynamics Computational Structural Mechanics Finite Element Method Fluid-Structure Interaction Reynolds-Averaged Navier-Stokes (equations)

1 Introduction The “RIBES” (Radial basis functions at fluid Interface Boundaries to Envelope flow results for advanced Structural analysis) project was led by the University of Rome “Tor Vergata” and was funded within the 7th European Union’s Research and Innovation funding program. The project started in December 2014 and had a duration of two years. Its scope was the development and validation of software tools for the improvement of accuracy in coupled CFD-CSM (Computational Fluid Dynamics— Computational Structural Mechanics) FSI analysis tools. Furthermore, it focused on the development of a structural shape optimization tool. The project was divided into three main topics: 1. Development of a load mapping procedure between CFD and FEM domains; 2. Setup of an experimental aeroelastic measurements campaign; 3. Development of a structural shape optimization procedure. This chapter deals with the second topic. The experimental campaign of the RIBES project had the objective of creating a database of measurements for the validation of FSI numerical analyses. A synthesis of the main achievements of the project on the three topics is reported in [1]. Further numerical/experimental activities performed after the conclusion of the project are described in [2] and [3]. Although several experimental static and dynamic public aeroelastic test cases are available (e.g. Agard 445.6 [4], HiReNASD [5], EuRAM [6], AePW [7]), at

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model

11

the date of the project start the literature concerning wind tunnel tests with realistic aeronautical wing structures was relatively poor. The RIBES experimental campaign was setup with the objective to contribute in this direction. The following data have been provided: • • • • •

Aerodynamic forces and coefficients evaluation Pressure measurements Strain and stress measurements Displacement measurements Flow visualization.

Section 2 describes the wing model, the wind tunnel facility and the instrumentation. Section 3 describes the setup of the experimental wind tunnel tests: boundary effects, data correction, and flow visualization (transition, stall). Section 4 deals with the measurements of the aerodynamic forces and moments, with an insight on pressure distributions. The strain and displacement measurement techniques are discussed in Sect. 5. Conclusions are drawn in Sect. 6. The complete set of experimental data (and numerical models developed by partners) are freely available to the scientific community at the website www.ribes-project.eu.

2 The Wing Model and the Wind Tunnel Facility The model is a tapered, unswept wing, characterized by a span length of 1600 mm, a root chord length of 600 mm, and tip chord length of 420 mm (0.7 taper ratio). To simplify manufacturing, no twist was adopted. The skin is obtained by lofting a single curvature surface between the root and the tip airfoils. The airfoil is the same along the wing span (Fig. 1). It was designed starting from the Göttingen 398, scaling the original shape to a relative thickness of 11%, and modifying the leading edge to improve the stall performance. The lower surface of the airfoil is quite flat, to help the assembly procedure keeping the nominal airfoil shape. The model has been sized to the target operative condition: flow speed equal to 40 m/s and about 60 Kgf of normal force. The wing box is a typical aeronautical structure with two C-shaped spars and ten ribs (Fig. 2). The front spar is located at 20% of the chord and is maintained orthogonal to the symmetry plane. The rear spar is located at 65% of the chord. The reference surface is 0.816 m2 , the mean aerodynamic chord is 0.515 m, the wing span is 1.60 m (Fig. 3). Model thickness to test section height ratio is below 4.1% at mean aerodynamic chord. The external skin is divided in four parts: an upper part, a lower part, a leading edge, and a V-shaped

Fig. 1 Wing airfoil

12

F. Nicolosi et al.

Fig. 2 Wing box

Fig. 3 Wing planform

trailing edge panel. They are joined to the structure by flush head CherryMAX rivets. The wing components were treated with Alodine to prevent corrosion and covered with a primer before painting. The model is connected to the wind tunnel balance by a flange and a tubular rod. The characteristics of the wind tunnel are reported in Table 1. A schematic of the wind tunnel is shown in Fig. 4. Aerodynamic forces and moments have been measured with an external balance, on which the model is held (Fig. 5), located outside the lateral wall. Full scale and accuracy data are reported in Table 2. The balance can measure normal force (lift), horizontal force (drag), pitching moment (about the spanwise direction, yaxis), bending moment (about the longitudinal direction, x-axis, with reference point in the balance center), yawing moment (about the vertical direction, z-axis). The reference system is reported in Fig. 6. From the measurement of force and moments, aerodynamic coefficients are derived and corrected for wind tunnel wall effects. See Sect. 3 for details. Above the balance system, an electric motor to control the model angle of attack is installed (Fig. 7). The angle of attack has been measured with an X-Bow electronic inclinometer with 0.01° accuracy.

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model Table 1 Wind tunnel characteristics

13

Tunnel type

closed circuit—closed test section

Test section dimensions

2.0 m × 1.4 m

Maximum speed

50 m/s

Turbulence level

0.1%

Temperature range

10–50 °C

Reynolds number

0.5–2.0 million (according to model size)

Dynamic pressure range

15–1200 Pa

Max stagnation pressure

104,700 Pa (ambient + dynamic)

Fig. 4 Wind tunnel schematics. Test section from A–A to B–B

Fig. 5 The wind tunnel balance used for the test of the RIBES wing model

14

F. Nicolosi et al.

Table 2 Wind tunnel balance characteristics Component

Range Min

Accuracy Max

Normal force (lift) L (Kgf)

−80

100

0.030

Horizontal force (Drag) D (Kgf)

−12

12

0.005

Pitching moment M (Kgf m)

−15

15

0.010

Bending moment M b (Kgf m)

−40

60

0.030

Yawing moment M y (Kgf m)

−8

8

0.006

Fig. 6 Reference system

Fig. 7 The electric motor, which rotates the model, above the wind tunnel balance

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model

15

3 Experimental Setup and Flow Visualization Test were performed in various conditions, with a Reynolds number, based on mean aerodynamic chord, from 1.0 to 1.4 million, and with a flow speed from 30 to 40 m/s. At high speed the angle of attack was limited to avoid excessive magnitude of the bending moment. The test matrix is reported in Table 3. Aerodynamic data has been corrected to account for wind tunnel effects, the most important being solid and wake blockages, streamlines curvature, and downwash change. Solid blockage is linked to the volume occupied by the model in the test section. Wake blockage is related to the wake thickness of the model. Streamline curvature and downwash variation are referred to the alteration of flow direction about the body. These effects are due to the presence of the walls around the model in a closed tunnel with a closed test section. Corrections are applied to get results as the model were in “free-air”. Because of solid and wake blockages, the dynamic pressure in the test section is increased by a factor estimated as 1.013. Because of the tunnel constraints, the angle of attack should be increased by an amount proportional to the lift coefficient. For instance, at AOA = 6° and C L = 0.80, the correction is 1° positive, meaning that the effective angle of attack is 7°. The corrections here applied are taken from Ref. [8]. They have been widely investigated and validated in previous experimental works [9–13]. Figure 8 shows the lift curve, drag polar, and pitching moment curves for TEST L30 with both corrected and uncorrected data. The only curve that is almost unchanged is the pitching moment curve, which is corrected only by the solid and wake blockage (dynamic pressure ratio) effect. The corrected lift curve has a decreased slope, whereas the drag polar has a significant correction due to the combined effects of blockage and downwash change. The model has no standoff, but it is distant about 10 mm from the lateral wall. This gap has been adequately sealed. Flow transition was investigated with the application of fluorescent oil at several span sections (Fig. 9). Since it is not possible to achieve a Reynolds number close Table 3 Test matrix Name

Flow speed

Reynolds no.

Measurement and conditions

Oil

30 m/s

1.06 million

Fluorescent oil visualization

TEST L30

30 m/s

1.06 million

Full polar, free transition. L, D, M, C p

TEST L35

35 m/s

1.25 million

AOA max 10°, free trans. L, D, M, C p

TEST L40

40 m/s

1.43 million

AOA max 8°, free trans. L, D, M, C p

TEST 7/8/13

35 m/s

1.25 million

Fixed transition, repeatability check (L, D, M)

TEST T30

30 m/s

1.06 million

Full polar, fixed trans. L, D, M, C p , strain

TEST T35

35 m/s

1.25 million

AOA max 10°, fixed trans. L, D, M, C p , strain

TEST T40

40 m/s

1.43 million

AOA max 8°, fixed trans. L, D, M, C p , strain

TEST F28

Variable

Variable

AOA = 4–6°, fixed trans. L, D, M, C p , strain

TEST Da6

40 m/s

1.43 million

AOA ≈ 7°, L ≈ 60 Kgf, fixed trans. L, D, M, strain, displacement (by laser)

16

F. Nicolosi et al.

(a) Lift curve.

(b) Drag polar.

(c) Pitching moment curve.

Fig. 8 Effects of wind tunnel correction on aerodynamic curves. Re = 1.06 million

to a typical flight condition of a general aviation or commercial aircraft, turbulence must be artificially added to the flow. Aluminum tape strips (3 layers, 0.6 mm total thickness) were applied to force transition within 2% chord length to prevent the generation of a laminar bubble, a typical low Reynolds phenomenon. Figure 10a shows the effect of transition on the lift curve at Re = 1.06 million (30 m/s flow speed). The curve linearity is lost at moderate angle of attack, where the slope is slightly increased for the effect of a separation and reattachment of the flow near the leading edge, commonly known as laminar bubble (see again Fig. 9), then it is reduced because of the trailing edge flow separation (mild stall). The addition of the strip tape inhibits the generation of the laminar bubble by forcing the flow transition at the leading edge. Moreover, it slightly increases the maximum lift coefficient because turbulence delays flow separation. The drawback is the additional aerodynamic drag, evaluated with the shift in the drag polar of Fig. 10b, due to

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model

17

(a) Free transition. Laminar separation and turbulent reattachment.

(b) Fixed transition. Flow is forced to be turbulent. Fig. 9 Fluorescent oil on wing sections C–D at AOA = 12°, 30 m/s flow speed

(a) Lift curve.

(b) Drag polar.

Fig. 10 Aerodynamic curves for free and fixed transition. Re = 1.06 million

18

F. Nicolosi et al.

the turbulent boundary layer, more stable than the laminar boundary layer, but also thicker and more dissipative. Flow visualization was performed by means of tufts to show separated zones at 30 m/s flow speed. Figure 11 shows an initial flow separation on the trailing edge of the inner sections at AOA = 10°, then the stall path proceeds towards the leading edge and extends outboard.

(a) AOA = 4°.

(b) AOA = 8°.

(c) AOA = 10°.

(d) AOA = 12°.

(e) AOA = 14°.

(f) AOA = 16°.

Fig. 11 Stall path visualization by means of tuft. Flow speed 30 m/s

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model

19

4 Force and Pressure Measurements Aerodynamic normal force (lift L) and moment (pitch M) data are reported against the corrected angle of attack in Fig. 12 at flow speeds 35 m/s and 40 m/s respectively. The design condition of the wing box is circled in the chart: the 60 Kgf load is achieved at about 7° corrected AOA at 40 m/s, corresponding to about 55 Kgf m of pitching moment. These aerodynamic forces are the resultant of pressure distribution that are described in the following. As regard the pressure field around the body, about 80 pressure taps of 1.0 mm diameter have been installed at several spanwise sections on the model. Tables 4, 5, and Fig. 13 show their location on the wing planform. Wing sections A, B, D, and F were instrumented with only 4 pressure taps each on the upper surface to check pressure distribution. Wing sections C and Section E were instrumented with 39 and 26 pressure taps respectively to evaluate chordwise pressure distribution and wing span loading. All pressure taps were connected with flexible tubes to a Scanivalve electronic pressure measurement system. The pressure signal from the Venturi system on the lateral wall, together with the pitot probe installed on the floor of the test section ahead of the model, provided evaluation of the dynamic pressure and flow speed in

Fig. 12 Lift force L and pitching moment M at two flow speeds

Table 4 Pressure taps location η

Chord (mm)

Distribution of pressure taps

160

0.100

582

4 (A1 to A4) TE → LE, lower → upper

450

0.281

549

4 (B1 to B4) TE → LE, lower → upper

3

600

0.375

533

39 (C1 to C39) TE → LE, lower → upper

4

990

0.619

488

4 (D1 to D4) TE → LE, lower → upper

E

5

1200

0.750

465

26 (E1 to E26) TE → LE, lower → upper

F

6

1500

0.938

431

4 (F1 to F4) TE → LE, lower → upper

ID

Sec.

A

1

B

2

C D

y (mm)

0.049

x/c

0.834

0.963

s/c

A2

A1

0.048

x/c

0.835

0.965

s/c

B2

B1

0.396

0.640

0.831

F2

s/c

F3

x/c

0.968

D1

F4

0.046

0.837

D2

0.177

0.368

0.610

0.171

0.361

0.647

D3

0.607

0.400

D4

0.172

0.360

0.647

B3

0.608

0.398

B4

0.173

0.358

0.648

A3

x/c

0.607

0.398

A4

s/c

0.302

0.306

0.282

y/c

0.042

0.077

0.085

0.062

y/c

0.050

0.083

0.089

0.066

y/c

0.046

0.081

0.087

0.065

y/c

s/c

C21

C20

C19 1.050

1.038

1.022

1.012

1.001

C18

0.987

C17

0.969

0.940

0.893

C16

C15

C14

C13

0.828

0.758

C12

0.684

C11

0.611

0.537

0.462

C10

C9

C8

C7

0.410

0.315

C6

0.243

C5

0.170

0.118

0.074

C4

C3

C2

C1

0.006

0.002

0.003

0.009

0.016

0.029

0.044

0.071

0.116

0.179

0.250

0.324

0.396

0.471

0.545

0.597

0.690

0.762

0.833

0.884

0.928

x/c

E19 E20 E21

−0.006 −0.017

E18

E17

E16

E15

E14

E13

E12

E11

E10

E9

E8

E7

E6

E5

E4

E3

E2

E1

0.010

0.018

0.025

0.032

0.041

0.053

0.066

0.077

0.082

0.085

0.082

0.077

0.068

0.062

0.049

0.037

0.024

0.014

0.006

y/c

1.204

1.114

1.034

1.010

0.957

0.912

0.874

0.855

0.828

0.814

0.803

0.794

0.782

0.764

0.733

0.698

0.620

0.467

0.319

0.196

0.025

s/c

0.141

0.052

0.001

0.015

0.063

0.105

0.142

0.161

0.188

0.202

0.213

0.221

0.233

0.252

0.282

0.317

0.395

0.548

0.694

0.816

0.983

x/c

(continued)

−0.031

−0.031

0.014

0.032

0.056

0.072

0.080

0.083

0.087

0.088

0.088

0.089

0.089

0.090

0.091

0.091

0.090

0.075

0.055

0.035

0.002

y/c

Table 5 Detailed pressure taps location, given in curvilinear abscissa and cartesian coordinates, unit chord. Curvilinear abscissa starts from upper trailing edge

20 F. Nicolosi et al.

F1

0.963

s/c

Table 5 (continued)

x/c

0.050

y/c

0.267 1.090 1.125 1.168 1.222 1.299 1.348 1.412 1.478 1.546 1.613 1.662 1.752 1.822 1.894 1.955 2.003 2.046

C23 C24 C25 C26 C27 C28 C29 C30 C31 C32 C33 C34 C35 C36 C37 C38 C39

s/c 1.068

C22

x/c

0.994

0.952

0.904

0.843

0.771

0.701

0.610

0.562

0.495

0.427

0.361

0.297

0.248

0.171

0.117

0.074

0.039

0.018 E24 E25 E26

−0.036 −0.036 −0.035

−0.010

−0.012

−0.013

−0.014

−0.016

−0.017

−0.021

−0.022

−0.023

−0.026

−0.028

−0.031

−0.033

E23

−0.036

1.781

1.642

1.555

1.407

s/c 1.307

E22

y/c −0.030

x/c

0.718

0.579

0.493

0.344

0.244

y/c

−0.013

−0.018

−0.021

−0.025

−0.027

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model 21

22

F. Nicolosi et al.

Fig. 13 Pressure taps location

the test section. Pressure coefficient is defined as C p = (p–p∞ )/q∞ . Figure 14 shows pressure coefficient for TEST T40 on wing sections C and E at several angles of attack. The unsmooth shape of the curves, especially near the leading edge, is due to errors in the assembly between the skin and the wing box. Some pressure taps close to the leading edge were lost during the application of the skin and the step caused by the transition strip may have altered the pressure readings of the C p peak. Also, the variations of the C p curve slope around 30% of the chord on each section seem an effect of surface waviness.

5 Stress and Displacement Measurements The model stress and deformation have been measured at 40 m/s airspeed, with a 6° geometrical angle of attack and a normal force of 60 Kgf, through 25 strain gages (16 uniaxial plus 3 rosettes with 3 signals) applied on the wing box and skin (Fig. 15 and Table 6). Some results are reported in Fig. 16, showing the local stresses, evaluated from the strain gages data, against the normal force N (that coincides with the lift force L) at several locations. The reference wind tunnel run is TEST T40. As expected from the pressure distribution data of Fig. 14, the front spar (marker 2) is the most stressed, while the rear spar (markers 3 and 4) is almost unloaded. Front spar thickening (markers 13 and 14) is almost equally stressed in tension (lower) and compression (upper). The highest stress is measured by sensor 15, located between the first and second stringer on the upper surface. It presents a double solution at about 20 Kgf of lift force, indicating structural instability. The rosettes (markers R16

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model

(a) Section C.

23

(b) Section E.

Fig. 14 Pressure coefficient distribution on two wing sections. Re = 1.43 million

24

F. Nicolosi et al.

Fig. 15 Location of some of the strain gages

and R19) measure shear flow. The high stress on rosette R16 is due to panel shear buckling. Other wing bays are loaded in a similar way. The model displacement has been measured at 11 span-wise stations and several chord-wise points by aiming a measurement laser (accuracy 0.05 mm) at specific markers. The laser was mounted on a track and manually moved from point to point. Thus, each acquired data point is obtained through a wind tunnel run. The geometric angle of attack was held constant at 6°, the flow speed at 39 m/s, the normal force 60.3 Kgf, the bending moment about 55 Kgf m. The assembly between the wing and the tubular sting caused a vertical translation and a rotation around the longitudinal axis at the application of the aerodynamic force. In test Da6, the values of these displacements are 3.6 mm and 1.45° respectively. They have been measured with additional high accuracy instruments, with a resolution of 0.01 mm for the translation and 0.01° for the rotation. Therefore, deformation measurements read by the laser have been corrected (Fig. 17). Since such errors involved rigid displacements, they did not affect strain gages measurements.

6 Conclusion A database to validate high-fidelity Fluid-Structure Interaction approaches, based on RANS and FEM analyses on behalf of the RIBES project, has been generated from the results of aeroelastic wind tunnel tests on a half-wing model. More in detail, the tests have provided a database of loads, pressure, stress, and displacement

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model

25

Table 6 Strain gages location ID

Bay

Position

Installation

Type

y (mm)

η

1

1

Between rib1-rib2

Front spar

Unidirectional

35.5

0.025

2

1

Between rib1-rib2

Front spar

Unidirectional

35.5

0.025

3

1

Between rib1-rib2

Rear spar

Unidirectional

35.5

0.025

4

1

Between rib1-rib2

Rear spar

Unidirectional

35.5

0.025

5

3

Between rib3-rib4

Front spar

Unidirectional

310

0.194

6

3

Between rib3-rib4

Front spar

Unidirectional

310

0.194

7

3

between rib3-rib4

Rear spar

Unidirectional

297

0.194

8

3

between rib3-rib4

Rear spar

Unidirectional

297

0.194

9

5

Between rib5-rib6

Front spar

Unidirectional

600

0.391

10

5

Between rib5-rib6

Front spar

Unidirectional

600

0.391

11

5

Between rib5-rib6

Rear spar

Unidirectional

598

0.391

12

5

Between rib5-rib6

Rear spar

Unidirectional

598

0.391

13

1

Between rib1-rib2

Front spar thickening

Unidirectional

35.5

0.025

14

1

Between rib1-rib2

Front spar thickening

Unidirectional

35.5

0.025

15

1

1st bay, bet. 1st and 2nd stringer

Upper skin

Unidirectional

35.5

0.025

16

1

1st bay, corr. to unidir. N.15

Lower skin

Rosette

35.5

0.025

17

2

2nd bay, bet. 1st and 2nd stringer

Upper skin

Unidirectional

169

0.106

18

2

2nd bay, bet. 2nd and 3rd stringer

Upper skin

Rosette

169

0.106

19

1

Between rib1-rib2

Front spar

Rosette

35.5

0.025

measurements that is significant for a realistic aeronautical design problem, since most of the data available in literature focus on aerodynamic aspects rather than on structural similitudes. It is the authors’ opinion that the experimental database, also presented on the project’s website, will be useful to other specialists involved in similar problems. Wing planform geometry and internal structure were fixed by the topic leader, the University of Rome “Tor Vergata”. Test conditions were set to comply with the objective of the experimental part of the research project, i.e. a realistic wing load with appreciable structural deformation. As regard the aerodynamic data, the necessary corrections for wind tunnel boundaries and low Reynolds number have been applied, ensuring a fully-turbulent flow over the wing model that simulate free-air conditions. Pressure measurements were taken at 6 spanwise stations. Some pressure taps at the leading edge were lost during the installation of the skin on the wing structure.

26

F. Nicolosi et al.

Fig. 16 Local stress measurements from strain gage data. TEST T40

Fig. 17 Wing displacement measured by laser. AOA = 6°, V = 40 m/s, L = 60.3 Kgf

Also, for the same reason, the test article’s airfoil shape is slightly different from the nominal contour. The altered leading edge curvature, surface waviness, and probably the presence of the transition strips, caused some “dirty” data on the chord-wise pressure coefficient distributions presented in the text.

Aeroelastic Wind Tunnel Tests of the RIBES Wing Model

27

As concern stress measurements, 19 strain gages were applied on wing spars, ribs, and skin. Results have been presented for a test at constant speed and variable angle of attack, and stress has been reported against the measured lift force. The highest stress gradients were measured by the sensors close to the wing root. Structural instability has been identified from a double solution at the measured lift of 20 Kgf. In case of numerical comparison, it is important to underline that it is difficult to achieve an accurate reproduction of shape (especially at leading edge), ensure a reasonable deformation (especially torsional), and model the constraint at root with connections through bolts. The installation of a high accuracy measurement laser on a track, mounted on the floor of the wind tunnel test section, enabled the possibility to measure wing displacement, by repeating test runs in the same flow conditions and pointing the laser at different spanwise stations. A rigid displacement, probably due to the coupling between the wing and the balance sting, has been registered and cleared from laser readings.

References 1. Biancolini, M. E., Cella, U., Groth, C., Chiappa, A., Giorgetti, F., & Nicolosi, F. (2018). Progresses in fluid structure interaction numerical analysis tools within the EU CS RIBES project. In Evolutionary and deterministic methods for design optimization and control with applications to industrial and societal problems, computational methods in applied sciences series. Springer International Publishing. https://doi.org/10.1007/978-3-319-89890-2_34. 2. Coppotelli, G., Di Giandomenico, F., Groth, C., Porziani, S., Chiappa, A., & Biancolini, M. E. (2019). On the structural updating using operational responses of a realistic wing model: the RIBES test article. In International Operational Modal Analysis Conference. Copenhagen, Denmark. 3. Groth, C., Porziani, S., Chiappa, A., Giorgetti, F., Cella U., Nicolosi F., et al. (2018). Structural validation of a realistic wing structure: The RIBES test article. In Procedia structural integrity (Vol. 12, pp 448–456). Elsevier. https://doi.org/10.1016/j.prostr.2018.11.073. 4. Yates, Jr. C. AGARD standard aeroelastic configurations for dynamic response I-Wing 445.6 (AGARD Report No. 765). Hampton, VA 23665-5225, USA: NASA Langley Research Center. 5. Chwalowski, P., Florance, J. P., Heeg, J., Wieseman, C.D., & Perry, B. (2011). Preliminary computational analysis of the HIRENASD configuration in preparation for the aeroelastic prediction workshop. In IFASD-2011-108. 6. Kuzmina, S. I., Ishmuratov, F., Zichenkov, M., & Chedrik, V. (2011). Integrated numerical and experimental investigations of the active/passive aeroelastic concepts on the european research aeroelastic model (EuRAM). Journal of Aeroelasticity and Structural Dynamics, 2(2). 7. Schuster, D. M., Heeg, J., Wieseman, C. D., & Chwalowski, P. (2013). analysis of test case computations and experiments for the first aeroelastic prediction workshop. In 51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition (p. 788). 8. Barlow, J. B., Rae, W. H., & Pope, A. (1999). Low-speed wind tunnel testing. (3rd ed.). New York: Wiley. 0471557749. 9. Nicolosi, F., Corcione, S., & Della Vecchia, P. (2016). Commuter aircraft aerodynamic characteristics through wind tunnel tests. Aircraft Engineering and Aerospace Technology, 88(4). https://doi.org/10.1108/aeat-01-2015-0008.

28

F. Nicolosi et al.

10. Nicolosi, F., Ciliberti, D., & Della Vecchia, P. (2016). Aerodynamic design guidelines of aircraft dorsal fin. In 34th AIAA Applied Aerodynamics Conference (pp. 1–13). https://doi.org/10.2514/ 6.2016-4330. 11. Nicolosi, F., Ciliberti, D., Della Vecchia, P., Corcione, S., & Cusati, V. (2017). A comprehensive review of vertical tail design. Aircraft Engineering and Aerospace Technology, 89(4). https:// doi.org/10.1108/aeat-11-2016-0213. 12. Ciliberti, D., Della Vecchia, P., Nicolosi, F., & De Marco, A. (2017). Aircraft directional stability and vertical tail design: A review of semi-empirical methods. Progress in Aerospace Sciences, 95. https://doi.org/10.1016/j.paerosci.2017.11.001. 13. Nicolosi, F., Ciliberti, D., Della Vecchia, P., & Corcione, S. (2018). Wind tunnel testing of a generic regional turboprop aircraft modular model and development of improved design guidelines. In 2018 Applied Aerodynamics Conference, AIAA AVIATION Forum, (AIAA 20182855). https://doi.org/10.2514/6.2018-2855.

Validation of High Fidelity Computational Methods for Aeronautical FSI Analyses Corrado Groth, Marco Evangelos Biancolini, Emiliano Costa and Ubaldo Cella

Abstract The aim of this paper is to compare and validate two computational methods to study typical aircrafts aeroelastic problems. The first one is the very wellestablished 2-way coupling approach, which envisages the use of a mesh morphing tool to update the CFD mesh according to the computed displacements in combination with a mapping algorithm to transfer the loads onto the FEM mesh. The second one is based on the embedding of structural modes, computed in advance by a FEM solver, directly into the CFD solver. It requires a morphing tool to deform the CFD mesh according to FEA computed modal shapes and a surface integration algorithm that allows to evaluate the modal forces acting on the CFD mesh. Modes embedding makes the CFD model intrinsically aeroelastic and thus capable to self-adapt its shape in the respect of the actual deformation, removing all the complexities related to the data exchange between solvers. Both methods were validated against a literature benchmark test case consisting in the prediction of the static aeroelastic equilibrium of the HIRENASD model using two of the meshes available in the “NASA First Aeroelasticity Workshop” database. The fidelity of both methods has been successfully validated, achieving a satisfactory agreement with experimental data.

C. Groth · M. E. Biancolini · U. Cella (B) University of Rome “Tor Vergata”, Rome, Italy e-mail: [email protected] C. Groth e-mail: [email protected] M. E. Biancolini e-mail: [email protected] E. Costa RINA Consulting, Rome, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_3

29

30

C. Groth et al.

List of Symbols β γ η η¨ ϕ ω CD CL FC F D h K M N n nodes ns Q s u C F Dm um x xC F D

coefficients of the polynomial correction vector of coefficients of the RBF vector of modal coordinates vector of modal accelerations radial basis function circular natural frequency drag coefficient lift coefficient vector of nodal loads computed on the CFD mesh multi-variate polynomial stiffness matrix of the system mass matrix of the system vector of modal forces number on nodes of the deformed surface mesh total number of source points (RBF centres) external load field over the entire structure interpolant function modal displacements for the m-th mode on the CFD mesh mode, eigenvector position of a point position of CFD surface mesh nodes

1 Introduction The demand for developing multi-disciplinary approach using high fidelity Computer-Aided Engineering (CAE) methods is today strongly rising in a widespread range of technical fields including aerospace, automotive, marine, product manufacturing and healthcare. This is even more true with the vision of modern design methods which are strongly oriented to work embedded in reliable numerical optimization procedures. The core of a multi-physics numerical investigation is the coupled-field analysis, which lets users to determine the combined effects of multiple physical phenomena as in case of fluid-structure, thermal-mechanical and electric-thermal interaction. Fluid-Structure Interaction (FSI) is the mechanism that evaluates the interaction of movable or deformable structures with an internal or a surrounding fluid flow [1] occurring at different length scales. Such an interaction can be the working principle of the component itself (reed valves action, parachute canopy unfolding, movement of a sheet of paper within a printing device) or can be exploited to finely tune components manufacturing in view of lightening a structure as in case of aircraft design. It is

Validation of High Fidelity Computational Methods …

31

a typical multi-physics phenomenon whose physical modelling implies the computation of both structural and fluid-dynamic solution and, when temperature effects are relevant, the thermal one as well. In general, the strategies to face the problem can be roughly grouped depending on the approach adopted to solve the governing equations (monolithic and partitioned methods) and upon the treatment of meshes (matching and non-matching mesh methods) [2]. FSI perspectives may vary depending on types of flow fields covered (compressible, incompressible, laminar, turbulent), types of applications, structural fields (thin-walled, rigid bodies, non-linear material), discretization schemes (finite volume, spectral methods, multi-body dynamics), flow modelling assumptions (continuum, statistical Lattice Boltzmann distribution) and calculation grid treatment (moving grid, fixed grid, immersed boundary) [3]. An important task in FSI analysis methods is the deformation of the Computational Fluid Dynamics (CFD) mesh that needs to be updated to replicate the elastic deformations of the structure. As demonstrated by Keyes’s work [4], a robust algorithm suitable to efficiently perform mesh morphing is based on Radial Basis Functions (RBF). In the proposed approach, RBF are used to update the CFD mesh according to the deformed shape calculated by a Finite Element Method (FEM) solver. Although several works demonstrated that RBF can be successfully adopted for the deformation of CFD meshes [5–8], its numerical cost has limited their actual application to tackle industrial relevant cases in the past (direct solution grows with the third power of n s , where n s is the number of RBF centres). To this end, many attempts have been devoted to the acceleration of such a method to deal with large RBF datasets [9, 10]. The numerical FSI approach proposed in the present paper makes use of the RBF technique by linking the CFD ANSYS ® Fluent ® solver to the RBF Morph™ tool [11]. The coupling between these codes has recently proven its powerful capabilities and effectiveness by solving with success challenging engineering applications such as surface vehicle and aircraft shape optimisation [12–17], sails trim optimization in conjunction with VPP [18, 19] and ice accretion on aircraft wings [20]. The proposed FSI modal approach fruitfully exploits the RBF Morph™ tool by enabling the adaptation of the shape of deformable parts according to modes superposition by means of the smoothing of the mesh directly during the progress of the computation. An intensive application of this FSI method has been tested within the RBF4AERO EU FP7 project [21, 22]. An example of shape optimization in which each explored design point is fully computed using the FSI approach is given in [23]. An important aspect related to the proper implementation of a modes embedding based FSI analysis is the verification of the modal base. Such information responds to one of the point left open at the end of the Aeroelastic Prediction Workshop (AePW) concerning the optimal number of modes to be used for static analysis. Among the works presented in that workshop, for instance, DLR used 20 modes to perform its aeroelastic analysis [24]. An overview of the performances of high fidelity FSI methods has been firstly published at IFASD 2011 Workshop [25] and two years later at IFASD 2013 Workshop [26] where all numerical tests were conducted using 30 modes. A modal base qualification procedure suitable for the identification of the minimal number of modes to be accounted to gain accurate values for wing deflection and aerodynamic coefficients has been recently presented in [16].

32

C. Groth et al.

The modal-based structural analysis is very efficient and can be adopted for transient problems which require an FSI evaluation at each time step, but it cannot handle nonlinear material parameters, contact problems and pre-stressed components. Such kind of situations can be properly managed using the 2-way FSI. Data exchanging can represent in that case a critical point, especially if a transient analysis has to be addressed. The improvement in performances for transient analyses using modal superposition can speed up the whole process. A full demonstration of this concept is given in [27], where the transient evolution of wing vibration after store separation has been investigated, and in [28] where the vibrational modes of a hydrofoil where evaluated underwater. The present work compares results achieved by comparing both the 2-way coupling and the modal approaches. The aim is to evaluate the accuracy of the two analysis configurations and to provide information about the limits in adopting the modal approximation with respect to face an aeroelastic problem by coupling fluid and structural solvers. The estimation of the opportune dimension of the modal base to be adopted, which as stated represents one of the main source of uncertainness in the setup of modal FSI approaches, is, furthermore, one of the targets of the work. This paper is structured as follows: in the next section a brief overview on the theoretical background is given, basic modal theory and RBF are introduced; at this point the FSI strategies employed in this paper are shown, demonstrating the workflow for both the 2-way and the modal superposition method; the HIRENASD testcase and its numerical setups are subsequently introduced and, finally, results of both methods are compared with regard to experimental data.

2 Theoretical Background A brief theoretical introduction on modal analysis is here provided. A complete description is given in [29]. For the modal analysis of a mechanical system, the following eigenvalue problem has to be solved: K u m = ω2 Mu m ,

(1)

where K is the stiffness matrix, ω2 is an eigenvalue, ω is a circular natural frequency, u m is the related eigenvector and M is the mass matrix of the system, stating that a vibration mode is a configuration in which a balance between elastic resistance and inertial loads occurs. Since mechanical systems are characteristically low-pass, the lowest frequency modes have the highest energy levels and, then, are physically prominent. Moreover, since the solution of the eigenvalue problem is a subspace of eigenvectors problem, the sign and the entity of each eigenvector may change depending on the algorithm adopted for the solution achievement [30]. Given that, for solution purposes, a convenient normalization is performed by imposing for each m-th mode u m a unit modal

Validation of High Fidelity Computational Methods …

33

mass to obtain u mT Mu m = 1

(2)

u mT K u m = ωm2 .

(3)

and then

One of the most important aspects of modal analysis is the spectral decomposition, which means that modes are orthogonal and form a basis in the modal coordinates (or displacements) η. The dynamic response of a mechanical system can be represented by the summation of the response of each mode where each mode acts as a single DoF dynamic system (i.e. mass and stiffness matrixes become diagonal) and then the following system relationship is valid η¨ + ω2 η = N

(4)

η¨ m + ωm2 ηm = Nm m = 1, 2, . . . , n

(5)

or alternatively

being η¨ m and ηm respectively the acceleration and displacement of the modal coordinate and Nm the force acting on the m-th mode. A linear static solution can be approximated as follows ωm2 ηm = Nm .

(6)

Modal forces N are obtained projecting external loads on each mode, i.e. performing the integral of the external load field Q over the entire structure weighted by the mode eigenvector u m Nm = u mT Q

(7)

where Q are external loads acting on the discrete system.

2.1 Radial Basis Functions Since their inception [31], RBF have been used as an interpolation tool of scattered data in a n-dimensional space. RBF can interpolate everywhere a scalar function defined at discrete points ensuring, at the same time, its exact value at original points. Their main purpose is to overcome the too severe constraints in data treatment by numerical methods existing in the 1970s, such as the minimalism of their framework and the simplicity of the shape of their containing region. Afterwards, RBF

34

C. Groth et al.

experienced a very rapid development consequent to their successful application in various scientific fields such as climate modelling, facial recognition, topographical map production, ocean floor mapping and medical imaging. RBF approach solved many cases where polynomial interpolation failed [32]. According to a first categorization, the several existing RBF can be nowadays classified on the basis of the type of support (global or compact) they have, namely the set of points where the chosen RBF is non-zero valued. In general, the solution of the RBF mathematical problem consists in the calculation of the scalar parameters (sought coefficients) of the linear system of order equal to the number of considered centres [33] (source points). According to the strategy adopted by RBF Morph™, the RBF system solution, determined after defining a set of source points with their displacement, is employed to operate the mesh morphing of a discretized domain. Operatively, once the RBF system coefficients have been calculated, the displacement of an arbitrary node of the mesh, either inside (interpolation) or outside (extrapolation) the influence domain of source points, can be expressed as the summation of the radial contribution of each. In such a way, a desired modification of the mesh nodes position (smoothing) can be rapidly applied preserving the mesh topology in terms of total number and type of the constituting elements. Figure 1 shows the localization of source nodes of the test case studied in this paper. To afford a three-dimensional study in x, y and z coordinates, the RBF Morph™ tool uses an RBF interpolant composed by a trial function containing the RBF ϕ and a multivariate polynomial corrector vector h of order m − 1 where m is said to be the order of ϕ, introduced with the aim to guarantee the existence of the solution when rigid movements need to be managed. If n s is the total number of introduced source points, the formulation of the interpolant is

Fig. 1 Source nodes for the RBF problem

Validation of High Fidelity Computational Methods …

s(x) =

ns 

  γi ϕ x − xki  + h(x)

35

(8)

i=1

where x is the vector identifying the position of a generic node belonging surface  to the  and/or volume mesh, xki is the i-th source node position vector and x − xki  is the Euclidean normalized distance between two points. The RBF fitting solution consists in the evaluation of the coefficients vector γ and of the weights of the polynomial corrector β so that, at source points, the interpolant function possesses the specified (known) values of displacement. The RBF approach for mesh morphing in three-dimensional space consists in the solution of the three problems, one for each component of the displacement field. ⎧ ns  

⎪ ⎪ γix ϕ x − xki  + β1x + β2x x + β3x y + β4x z ⎪ sx (x) = ⎪ ⎪ i=1 ⎪ ⎨ ns 

y  y y y y γi ϕ x − xki  + β1 + β2 x + β3 y + β4 z s y (x) = ⎪ i=1 ⎪ ⎪ ns  ⎪ 

⎪ ⎪ (x) = γiz ϕ x − xki  + β1z + β2z x + β3z y + β4z z. s ⎩ z

(9)

i=1

RBF methods offer several advantages that make them very attractive in the area of mesh smoothing. One of the most important is that, due to their meshless nature, only the grid points are moved regardless of the volume cells they belong to, as well as regardless of specific key features that can be involved in smoothing such as, for instance, non-matching interfaces. They also are particularly suitable for parallel implementation providing the capability to manage huge cases. Once the RBF solutions are available and shared in memory, each process (node of the cluster) can, in fact, smooth its nodes without taking care of what happens outside.

3 Strategies of the FSI Numerical Analyses Two approaches for FSI analyses, based on the implementation of RBF mesh morphing to transfer the FEM displacement fields from the structural grid to the CFD mesh, are proposed: the modal superposition method based on the embedding of the structural modal shapes within the CFD environment and the 2-way in which the structural and fluid dynamic solvers are coupled.

3.1 Structural Modes Embedding The workflow to handle FSI numerical studies according to structural modes embedding is based on three stages (Fig. 2):

36

C. Groth et al.

Fig. 2 Workflow of the modes embedding FSI approach

• structural modes of deformable parts normalised with respect to mass are calculated and extracted by means of a FEM analysis; • an RBF solution is calculated for each mode by applying the displacements of the corresponding mode to the deformable surfaces; • the stored RBF solutions are loaded at the beginning of the analysis to provide a parametric mesh formulation that makes the fluid dynamic configuration intrinsically aeroelastic. The structural response is evaluated directly in the modal space during the progress of the CFD computation. The mesh is made parametric according to the following relationship xC F D = xC F D0 +

n modes

ηm u C F D m

(10)

m=1

where xC F D0 are the positions of the nodes of the undeformed CFD mesh (baseline configuration), ηm are the (unknown) values of the modal coordinates and u C F Dm are the modal displacements for the generic retained m-th mode. The integration of the modal forces Nm is obtained from the simple summation over the nodes of nodal forces FC F D computed over all the faces

Validation of High Fidelity Computational Methods …

Nm =

n nodes

37

u C F Dmi · FC F Di

(11)

i=1

where the m-th modal load is a scalar obtained summing the dot product between the nodal mode displacement and the nodal load of each i-th node of n nodes nodes of the surface. Considering that a mass normalization criterion was defined for modes extraction, the modal coordinates are expressed, according to Eq. (6), as follows ηm =

Nm ωm2

(12)

The parametric CFD mesh can adapt its shape on the basis of actual loads according to the formula xC F D = xC F D0 +

n modes m=1

Nm u C F Dm ωm2

(13)

3.2 2-Way FSI Coupling The three stages of the workflow to handle FSI numerical studies employing the 2-way approach are sketched in Fig. 3. The loop begins with the CFD analysis of the rigid geometry up to a fully converged solution, it progresses computing the structural deformation adopting the pressure distribution of the fluid dynamic

Fig. 3 Workflow of the 2-way FSI approach

38

C. Groth et al.

solution as loading condition and it finishes adapting the CFD mesh to the estimated deformed shape using RBF Morph™. It is worth to mention that the iteration progresses up to the convergence characterising the steady state solution (usually 4–5 loops are enough).

4 Test Case Description and Numerical Models To demonstrate the effectiveness and efficiency of the proposed techniques in solving real world aircraft FSI problems, one of the configurations of the NASA Aeroelastic Prediction Workshop (AePW) was simulated. The public available experimental reference data, Finite Element Analysis (FEA) model and CFD grids were used to set-up the following descripted numerical activities that include a new run of the FEA model, the RBF set-up to transfer the structural solutions to the CFD mesh and the set-up of the CFD analysis.

4.1 Experimental Reference Data The test case is called HIgh REynolds Number Aero-Structural Dynamics (HIRENASD [34]). It consists of a tapered 34° swept back wing specifically designed to achieve high structural stiffness and distant modes in frequency domain. The airfoil adopted is the BAC3-11/RES/30/21 supercritical airfoil. The model, shown in Fig. 4, was tested in the European transonic cryogenic wind tunnel (ETW) at Mach numbers ranging from 0.7 to 0.88. The Reynolds number, based on a mean aerodynamic chord of 0.3445 m, ranged from 7 to 73 million.

4.2 FEM Model and Solution The structural model of the wing was generated adopting the FEM mesh available from the NASA database, which is composed of 357,545 nodes and 225,112 elements (Fig. 5). Materials, properties and constraints were left unchanged according to the original configuration provided by AePW committee. The first six mass normalized modes were extracted using the Nastran FEM solver. Wing modal shapes and frequencies are shown in Fig. 6, where B and FA respectively stand for out-of-plane bending and inplane fore-and-aft bending according to AePW mode classification. A perfect matching with AePW results is observed. The native mapping procedure implemented in the ANSYS ® Fluent ® solver was adopted to transfer the computed pressure as loads for the structural model. Only the mesh cells belonging to the wing wetted surfaces were involved in the load transfer process.

Validation of High Fidelity Computational Methods …

Fig. 4 HIRENASD wind tunnel model

Fig. 5 Structural mesh of the HIRENASD WT model

39

40

C. Groth et al.

(a) Mode 1 – 1B - 25.5 Hz

(b) Mode 2 – 2B - 80.2 Hz

(c) Mode 3 – 1FA - 106.1Hz

(d) Mode 4 – 3B - 160.3 Hz

(e) Mode 5 – 4B - 241.9 Hz

(f) Mode 6 – 2FA - 252.2 Hz

Fig. 6 Modal shapes and frequencies of the HIRENASD model

4.3 RBF Setup The FEM grid nodal displacements for the modal superposition FSI method and for the 2-way analysis were applied to the wing CFD surface mesh by means of RBF Morph™ using the data exported in Nastran format. In both wind tunnel and FEM model the fuselage fairing is mechanically uncoupled from the wing and slight motion of the root is allowed. A small portion of the fuselage around the wing of the CFD model was then left free to deform using a buffer region to absorb the required motion. Figure 7 illustrates the CFD model morphed according to the mode 2 superposed to the baseline configuration.

Fig. 7 Morphed configuration corresponding to second structural modal shape

Validation of High Fidelity Computational Methods …

41

The image evidences the buffer region around the wing root as well as the boxshaped encapsulation domain generated to delimit the volume of the morphing action. The total number of source nodes of the RBF set-up was about 4500.

4.4 CFD Configuration Two unstructured CFD meshes, available for the HIRENASD test case and having respectively 1.5 and 13.2 million of cells (Fig. 8), were used to setup the FSI analyses with the two descripted methods (2-way and modal superposition). A preliminary fully converged solution on the rigid model was first obtained. The two FSI configurations were then implemented by embedding the RBF solutions in the process for the modal superposition method and by exchanging the data with the structural solver for the 2-way approach. The integration of modal forces and the CFD mesh updating process were invoked, in the modal superposition setup, every 25 iterations. The flow conditions of the analyses referred to the ETW test number 132, to which corresponds Mach = 0.8 and Reynolds number = 7 million. Steady RANS computations, adopting the density based implicit solver, were run using the roeFDS flux scheme, the least squares cell based spatial discretization and second order upwind schemes for flow and modified turbulent viscosity. The turbulence model adopted was the Spalart-Allmaras single equation. A pressure far-field boundary condition was imposed to the hemispheric far field and an adiabatic no-slip wall condition on wing and fuselage. The angle of attack (AoA) was 1.5°.

5 Results The first evaluation concerned the capability to estimate the deformation of the wing and the number of modes required by the FSI modal approach to produce solutions comparable to the 2-way analysis. Such evaluation is quantified monitoring the displacement of a point located at the wing at tip. Table 1 reports the comparison between the tip displacement estimated by the 2-way model and the one estimated by the modal superposition approach populating the modal base up to six structural modal shapes. The lowest achievable difference was obtained once having included

Fig. 8 Detail of the two CFD grids

42

C. Groth et al.

Table 1 Basis verification of the first 6 modes with respect to 2-way

Tip displacement (mm)

Relative discrepancy (%)

2-way

14.44

–

1 mode

15.26

−5.64

2 modes

14.18

1.81

3 modes

14.18

1.80

4 modes

14.26

1.29

5 modes

14.26

1.29

6 modes

14.26

1.29

the first four modes. The slight difference observed between the solutions of the two methods might be also due to the uncertainness introduced in the pressure mapping interpolation process. Figure 9 reports the convergence history of the tip displacement using the 2-way approach. Four iterations between the fluid dynamic and the structural solver were sufficient to reach the static equilibrium. The maximum displacement obtained at wing tip is in good agreement with the displacement obtained during the AePW activities. In Fig. 10 the workshop numerical and experimental data are plotted together with the result obtained in our study. Figure 11 compares the estimated computed spanwise variation of the wing twist due to deformation (adopting both coarse and fine CFD meshes) with measurements. The total twist increase under load is close to half degree.

Fig. 9 Tip displacement convergence for 2-way coupling

Validation of High Fidelity Computational Methods …

43

Fig. 10 Validation of wing tip displacement

Fig. 11 Numerical and experimental wing twist distribution along the span

5.1 Aerodynamic Coefficients Table 2 compares the aerodynamic coefficients computed by the modal superposition FSI method populating the modal base with six modal shapes and with the solutions obtained on the rigid configuration. The values are in good agreement with the numerical solutions obtained with other solvers [26]. The comparison gives the

44 Table 2 Rigid and elastic (modal solution with 6 modes) aerodynamic coefficients (Mach = 0.8, Re = 7 million, AoA = 1.5°)

C. Groth et al. Rigid

Elastic

Difference (%)

CL

0.3568

0.3395

−4.85

CD

0.0137

0.0144

+5.11

Fig. 12 Spanwise location of pressure taps lines

sensitivity on how the wing flexibility, despite its high rigidity, impacts the aerodynamic solution. The rigid model over predicts the lift and under predicts the drag of about 5%. The effect is induced by the positive sweep angle of the wing that, when loaded, it generates an increase of the absolute value of the wing twist (quantified in Fig. 11) that unloads, at fixed incidence, the outer region of the wing. The pressure distribution was measured along several sections of the model. In this work we extracted, for comparison, the pressure at the stations located at 14.5% (Section 1), 32.3% (Section 2), 65.5% (Section 5) and 95.3% (Section 7) of the wing span (see Fig. 12). Figure 13 reports the comparison between the elastic solution of the modal superposition FSI analysis, using both the coarse and the fine mesh, and the measured pressure at the four stations indicated in Fig. 12. The numerical solutions are very similar with just very small differences in the shock regions. Both solutions slightly overpredicted the strength of the recompressions although the locations are reasonably correct. The effect of the wing elasticity is evidenced in Fig. 14 where the comparison between the rigid and the elastic solutions is reported. The differences are, as expected, more evident comparing the pressures in the outer region of the wing. The two numerical FSI approaches provided the same solution.

6 Conclusions Two aeroelastic numerical procedures based on RBF mesh morphing conceived to model the Fluid-Structure Interaction mechanism, were described and validated

Validation of High Fidelity Computational Methods …

45

(a) Section 1

(b) Section 2

(c) Section 5

(d) Section 7

Fig. 13 Validation of modal FSI pressure solution

against a well-known aeronautical test case: the wind tunnel aeroelastic measurements of the HIRENASD wing. The methods proposed were a 2-way direct coupling between fluid dynamic and structural solvers and a modal superposition-based approach. The latter method, which is suitable for linear problems, is implemented generating a parametric formulation of the computational domain combining a set of RBF solutions, each representing one natural modal shape of the structure. The stored solutions constitute the modal base of the FSI problem and are amplified according to modal coordinates that are computed during the progress of the CFD computation from the modal forces extracted integrating the pressure and friction forces on the wall boundaries. The morphing action is performed on the grid every prescribed number of iterations making the fluid dynamic numerical model intrinsically aeroelastic. The main advantage of this approach with respect to the 2-way coupling is the creation of a simpler and more robust configuration. The iteration with external solvers is no further required and the complexities associated to the codes coupling (i.e. inputs/outputs format conversion, mapping interpolation, runs managements) bypassed. In the modal FSI analysis the “interface” between the fluid

46

C. Groth et al.

(a) Section 1

(b) Section 2

(c) Section 5

(d) Section 7

Fig. 14 Comparison between rigid and elastic solutions

dynamic and the structural solution consists in just a single small vector representing the modal coordinates. The modal approach is widely adopted to face unsteady aeroelastic problems. The comparison between the two numerical solutions showed that the first four modal shapes, with which the modal base of the modal approach can be populated, were sufficient to get almost the same solution (in terms of wing deformation and aerodynamic solution) of the 2-way coupling. The flexibility of the wing showed to affect the aerodynamic coefficients in the order of 5%. The numerical solutions obtained are in a very good agreement with the solutions generated by other solvers and reported in literature. Both methods accurately captured the model deformation and provided a pressure distribution reasonably in agreement with measurements. Slight differences were observed in the region of the shocks and in the outer part of the wing.

Validation of High Fidelity Computational Methods …

47

References 1. Bungartz, H. J., & Schäfer, M. (Eds.) (2006). Lecture Notes in Computational Science and Engineering Vol. 53. Fluid-structure interaction: Modelling, simulation, optimization. Berlin: Springer. 2. Gene, H., Jin, W., & Anita, L. (2012). Numerical methods for fluid-structure interaction—A review. Computer Physics Communications, 12(2), 337–377. 3. Costa, E. (2012). Advanced FSI analysis within ANSYS Fluent by means of a UDF implemented explicit large displacements FEM solver. Ph.D. Dissertation, Program in Environment and Energy—Cycle XXIV, Industrial Engineering Dept., University of Rome Tor Vergata. 4. Keye, S. (2009). Fluid-structure coupled analysis of a transport aircraft and comparison to flight data. In 39th AIAA Fluid Dynamics Conference AIAA. https://doi.org/10.2514/6.2009-4198. 5. Jakobsson, S., & Amoignon, O. (2007). Mesh deformation using radial basis functions for gradient based aerodynamic shape optimization. Computers & Fluids, 36(6), 1119–1136. https:// doi.org/10.1016/j.compfluid.2006.11.002. 6. De Boer, A., Van der Schoot, M. S., & Bijl, H. (2007). Mesh deformation based on radial basis function interpolation. Computers & Structures, 85(11–14), 784–795. https://doi.org/10.1016/ j.compstruc.2007.01.013. 7. Van Zuijlen, A. H., De Boer, A., & Bijl, H. (2007). Higher-order time integration through smooth mesh deformation for 3D fluid–structure interaction simulations. Journal of Computational Physics, 224(1), 414–430. https://doi.org/10.1016/j.jcp.2007.03.024. 8. Cella U., Groth C., & Biancolini, M. E. (2016, September). Geometric parameterization strategies for shape optimization using RBF mesh morphing. In Lecture Notes series in Mechanical Engineering. Advances on mechanics, design engineering and manufacturing (pp. 537–545). https://doi.org/10.1007/978-3-319-45781-9_54. 9. Rendall, T. C. S., & Allen, C. B. (2009). Efficient mesh motion using radial basis functions with data reduction algorithms. Journal of Computational Physics, 228(17), 6231–6249. https://doi. org/10.1016/j.jcp.2009.05.013. 10. Rendall, T. C. S., & Allen, C. B. (2010). Reduced surface point selection options for efficient mesh deformation using radial basis functions. Journal of Computational Physics, 229(8), 2810–2820. https://doi.org/10.1016/j.jcp.2009.12.006. 11. RBF Morph, RBF Morph—Users’ Guide. (2016). 12. Khondge, A., & Sovani, S. (2012). An accurate, extensive, and rapid method for aerodynamics optimization: the 50:50:50 method. In SAE 2012 World Congress & Exhibition, Detroit, Michigan, USA, 24–26 April 2012. 13. Biancolini, M. E., Costa, E., Lagasco, F., Dariva, J., Cella, U., Mancini, M., & Travostino, G. (2013). Tecniche di Mesh Morphing per l’ottimizzazione in campo aeronautico. Analisi e Calcolo, No. 56. 14. Papoutsis-Kiachagias, E. M., Andrejasic, M., Porziani, S., Groth, C., Erzen, D., Biancolini, M. E., Costa, E., & Giannakoglou, K. C. (2016). Combining an RBF-based morpher with continuous adjoint for low-speed aeronautical optimization applications. In ECCOMAS Congress 2016, Crete Island, Greece, 5–10 June 2016, ISBN: 9786188284401. 15. Cella, U., & Biancolini, M. E. (2012). Aeroelastic analysis of aircraft wind-tunnel model coupling structural and fluid dynamic codes. AIAA Journal of Aircraft, 49(2), 407–414. 16. Biancolini, M. E., Cella, U., Groth, C., & Genta, M., Static aeroelastic analysis of an aircraft wind-tunnel model by means of modal RBF mesh updating. Journal of Aerospace Engineering, 29. https://doi.org/10.1061/(asce)as.1943-5525.0000627. 17. Groth, C., Cella, U., Costa, E., & Biancolini, M. E. (2019). Fast high fidelity CFD/CSM fluid structure interaction using RBF mesh morphing and modal superposition method. Aircraft Engineering and Aerospace Technology Journal. https://doi.org/10.1108/AEAT-09-20180246. 18. Biancolini, M. E., Viola, I. M., & Riotte, M. (2014). Sails trim optimisation using CFD and RBF mesh morphing. Computers & Fluids, 93C, 46–60.

48

C. Groth et al.

19. Viola, I. M., Biancolini, M. E., Sacher, M., Cella, U. (2015, March). A CFD-based wing sail optimisation method coupled to a VPP. In High Performance Yacht Design Conference, Auckland (NZ). 20. Groth, C., Costa, E., & Biancolini, M. (2019). RBF-based mesh morphing approach to perform icing simulations in the aviation sector. Aircraft Engineering and Aerospace Technology, 91(4), 620–633. https://doi.org/10.1108/AEAT-07-2018-0178. 21. RBF4AERO, http://www.rbf4aero.eu/, RBF4AERO website, retrieved 2018. 22. Kapsoulis, D. H, Asouti, V. G., Giannakoglou, K. C., Porziani, S., Costa, E., Groth, C., Cella, U., & Biancolini M. E. (2016). Evolutionary aerodynamic shape optimization through the RBF4AERO platform. In 7th ECCOMAS Congress 2016, 5–10 June 2016, Crete Island, Greece. 23. Andrejašiˇc, M., Eržen, D., Costa, E., Porziani, S., Biancolini, M. E., & Groth, C. (2016). A mesh morphing based FSI method used in aeronautical optimization applications. In ECCOMAS Congress 2016, Crete Island, Greece, 5–10 June 2016. ISBN: 9786188284401. 24. Markus, R., Static and forced motion simulations of the HIRENASD test case-approaches and results. In 1st Aeroelastic Prediction Workshop. https://c3.nasa.gov/dashlink/static/media/ dataset/HIRENASD_Ritter.pdf. 25. Chwalowski, P., Florance, J. P., Heeg, J., Wieseman, C. D., & Perry, B. (2011). Preliminary Computational Analysis of the HIRENASD Configuration in Preparation for the Aeroelastic Prediction Workshop, IFASD-2011-108. 26. Chwalowski, P., Heeg, J., Dalenbring, M., Jirasek, A., Ritter, M., & Hansen, T. (2013). In Collaborative HIRENASD Analyses to Eliminate Variations in Computational Results, IFASD2013–1D. 27. Reina, G., Della Sala, A., Biancolini, M. E., Groth, C., & Caridi, D., Store separation: theoretical investigation of wing aeroelastic response. Paper presented at Aircraft Structural Design Conference, Belfast. 28. Di Domenico, N., Groth, C., Wade, A., Berg, T., & Biancolini, M. E. (2018). Fluid structure interaction analysis: vortex shedding induced vibrations. Procedia Structural Integrity, 8, 422– 432, https://doi.org/10.1016/j.prostr.2017.12.042. 29. Meirovitch, L. (2010). Fundamentals of vibrations (Reissue edition). Waveland Press Inc. 30. Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002). Concepts and applications of finite element analysis (4th ed.). New York: Wiley. 31. Hardy, R. L. (1971). Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research, 76(8), 1905–1915. https://doi.org/10.1029/JB076i008p01905. 32. Sarra, S. A. (2011). Radial basis function approximation methods with extended precision floating point arithmetic. Engineering Analysis with Boundary Elements, 35(1). https://doi. org/10.1016/j.enganabound.2010.05.011. 33. Buhmann, M. D. (2003). Radial basis functions. New York: Cambridge University Press. 34. http://heinrich.lufmech.rwth-aachen.de/en.

High-Fidelity Static Aeroelastic Simulations of the Common Research Model Jan Navrátil

Abstract Current aircraft design leads to increased flexibility of the airframe as a result of modern materials application or aerodynamically efficient slender wings. The airframe flexibility influences the aerodynamic performance and it might significantly impact the aeroelastic effects, which can be more easily excited by rigid body motions than in case of stiffer structures. The potential aeroelastic phenomena can occur in large range of speeds involving transonic regime, where the non-linear flow effects significantly influence the flutter speed. Common aeroelastic analysis tools are mostly based on the linear theories for aerodynamic predictions, thus they fail to predict mentioned non-linear effect. This paper presents the first step in the design of high-fidelity aeroelastic simulation tool. Currently, it allows to perform static aeroelastic simulations by coupling Computational Fluid Dynamics solver with Matlab based Finite Element solver. The structural solver is a linear elasticity solver which is able to solve either models consisting of beam elements or arbitrary models using stiffness and mass matrices exported from Nastran solver. The aeroelastic interface is based on the Radial Basic Functions. The test case studied in this work is a static aeroelastic simulation of the Common Research Model in the transonic conditions. The structural models tested are a wing-box finite element model and a beam stick model which is statically equivalent to the wing-box model. The comparison of results using respective structural models shows good agreement in aerodynamic properties of the model wing at static equilibrium state.

1 Introduction In the early years of the aviation the solution of aeroelastic problems was sought by trial and error. Thus, problems were solved in late stage of the design process and often led to accidents during flight tests. In the course of the aeronautical engineering development, the theoretical investigations and research were conducted to J. Navrátil (B) Brno University of Technology, 616 69 Brno, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_4

49

50

J. Navrátil

understand the aeroelastic phenomena. The theories solving unsteady aerodynamics were established by Von Wagner [38] and Küssner [25]. The developments in the unsteady aerodynamics allowed to create theories predicting the wing flutter in subsonic speeds by Küssner [24] and later by Theodorsen [34]. The Theodorsen’s work created a basis for the strip theory, which has been further developed [40]. Nowadays, commonly applied methods for aerodynamic prediction in aeroelastic computations are based on the doublet-lattice method [33, 41]. This essentially linear method is capable of relatively accurate prediction in the subsonic conditions with no flow separation but fails in transonic flow or flow with extensive separation. Therefore, it has been common practice to correct results of the linear aerodynamic methods by the wind tunnel measurements. The advances in the development of Computational Fluid Dynamics (CFD), creating the standard tool for non-linear aerodynamic predictions, commenced an extensive research of the aeroelastic solvers which are based on coupling high-fidelity flow solvers with the already matured Finite Element Method. The potential of those solvers is reduction of the number of tunnel or flight tests during an aircraft design. The research in this field was probably started by Bendiksen [6], who has been followed by Lee-Rausch and Baitina [26], Alonso et al. [1], Thomson et al. [35], Feng et al. [14] and many others, focusing on both static and dynamic aeroelastic simulations. The recently established activity in NASA [17, 18] focusing on validation of tools for high-fidelity flutter predictions in transonic speeds highlights the effort given to the research in this field. Nowadays, the research focuses on wide range of computational methods involved in the aeroelastic simulations, ranging from mesh deformation methods [7, 19, 28, 42] through effective spatial and temporal coupling of the essentially different domains [10, 13, 32, 36, 39]. The paper presents application of a high-fidelity tool, designed and tested in frame of author’s dissertation thesis [30], for static aeroelastic simulations. The tool is based on the coupling of CFD solver with external finite element solver. The test case, used for evaluation of the tool properties and capabilities, is the Common Research Model in the transonic flow conditions employing two different structural models. In the first part of the paper a study of coupled solution convergence, considering several settings of the flow solver, is presented. Next, the application of two different structural models, wing-box and beam stick finite element models, coupled to wingfuselage and wing-only aerodynamic models, is studied. Transonic flow conditions at M = 0.85 are considered in the study.

2 Computational Aeroelasticity Tool The implementation of the computational aeroelasticity tool is based on the partitioned (coupled-field) formulation of the fluid-structure interaction. Therefore, it is possible to couple arbitrary separate flow and structure solvers independently of each other. Thus, the best suiting solver for particular domain and application can be employed.

High-Fidelity Static Aeroelastic Simulations …

51

The basic task of the tool, in this work programmed in Matlab environment, is sharing appropriate information between fluid and structure solvers on the defined interface. The formulation of the interface employing Radial Basis Functions (RBF) ensures capability to couple independently discretized domains differing in size by several orders of magnitude. At the same time, the formulation satisfies the conservation of energy and loads and it is accurate and efficient [8].

2.1 Flow Solver The flow solver used in the work is CFD code Edge [12]. It is a finite volume solver for unstructured grids which can solve 2D and 3D Euler and RANS equations, as well as the adjoint of the Euler and Navier-Stokes (frozen viscosity) equations [2]. The time integration uses the fourth order Runge-Kutta scheme. It employs localtime-stepping, local low-speed preconditioning, multi-grid and dual-time-stepping for steady-state and time-dependent problems. For the unsteady cases, the employed numerical scheme is a dual-time-stepping scheme [20]. The data structure of the code is edge-based. The solver can be run in parallel on a number of processors to efficiently solve large flow cases.

2.2 Structural Solver The structural solver, programmed in Matlab environment, is capable to solve linear elasticity problems. It works in two modes: beam finite element mode and prescribed stiffness and mass matrices mode. The purpose of this solver is to overcome cumbersome communication with commercial FEM packages via input files and to allow future development focused on direct communication between solvers. The first mode can solve static deformation and modal analysis of the model consisting of beam elements (either Euler-Bernoulli or Timoshenko). Its advantage is a possibility of easily changing model parameters, which is convenient in potential parameterization purposes, e.g. during aero-structural optimization. The second mode is capable of solving model consisting of arbitrary structural finite elements. The stiffness and mass matrices of the structural model must be provided. The dynamics of a mechanical system can be, for small displacements, described by a system of linear differential equations: M¨u + C˙u + Ku = F(t)

(1)

where M, C, K are the mass, damping and stiffness matrices, respectively, u is the vector of nodal displacements and rotations and F(t) is the vector of corresponding

52

J. Navrátil

nodal forces and moments. Solving a linear static elastic problem the system of equations reduces to the form: Ku = F, (2) In both solver modes, the loading by inertial forces is implemented. Inertial forces are calculated according to Newton’s second law of motion with an assumption that only inertial forces act on a structure. Thus, the inertial forces can be calculated from the equation: ¨ (3) Fi = Mu, where M, Fi and u¨ are mass matrix, vector of nodal inertial forces and vector of nodal accelerations, respectively. Used linear beam theory applies for small (or even infinitesimal) deflections and, as a result of a bending load, only transverse displacements are obtained. Therefore, large deflection due to bending causes a change of the beam length. This property is fixed in the beam structural solver by introduction of fictitious displacement uf in direction along the beam element. For reasonable values of element rotation angles θ due to transverse displacement, v the fictitious displacement is [31]: uf ≈ −

1 (v1 − v2 )2 , 2l

(4)

where v1 and v2 are transversal displacements of beam element nodes due to bending and l is the beam element length.

2.3 Fluid-Structure Interface The fluid-structure interface is defined using the partitioned formulation. It allows to combine the complex domains which are described by different approaches and their sizes may differ by several orders of magnitude. The difference in size of the systems is common in the real applications when usually a flow domain is much larger than a structural domain. The necessary information obtained by arbitrary flow and structural solvers is exchanged on the defined interface according to the chosen coupling scheme which should satisfy several criteria—conservation of energy and loads, accuracy and efficiency. The principle of the fluid-structure coupling is based on the conservation of the virtual work satisfying the conservation of energy. The virtual work performed by the aerodynamic load must be equal to the virtual work of the structural forces: δW = FsT δus = FfT δuf

(5)

where Fs is vector of forces acting on structural nodes, Ff is vector of forces at fluid nodes, us and uf are structural and fluid nodes displacement vectors, respectively.

High-Fidelity Static Aeroelastic Simulations …

53

The fluid-structure interface is expressed introducing a coupling matrix H, giving a relation between the displacement vectors of the fluid and the structure meshes: uf = Hus

(6)

The combination of Eqs. 5 and 6 gives a relation: Fs = HT Ff

(7)

Thus, if matrix H satisfies the mentioned conservation criteria, it can be used for the transformation of the structural displacements to the fluid mesh and, once it is transposed, for the transformation of the aerodynamic load to the forces acting on the structural nodes. In this work, the interface is defined using Radial Basis Functions (RBF) [5, 19] which are flexible and well-established tool for multivariate interpolation. The displacements at the fluid and the structure are approximated by an interpolant which has the form: N  γj φ(x − xj ) + h(x) (8) s(x) = j=1

where φ is a given basis function, coefficients γj ∈ R, the xj are centers with known values (structural points), h(x) are first degree polynomials and  ·  denotes Euclidean norm. Often it might be convenient to scale the basis function with a shape parameter ε, then the basis function is replaced by φ(r) = φ(εr). Common radial basis functions are given in Table 1. In [5], it is shown that the coupling matrix H is: H = Afs C−1 ss

(9)

Both matrices at the right-hand side come from RBF approach. The square interpolation matrix Css of size Ns × Ns (Ns is a number of structural nodes) consists,

Table 1 Common radial basis functions Radial basis function

φ(r)

Spline type Thin plate spline Multiquadric Inverse multiquadric Inverse quadric

|r|n , n odd |r|n log |r|, n even √ 1 + r2 √ ( 1 + r 2 )−1 (1 + r 2 )−1

Gaussian Multi-quadric biharmonic splines

e−r √ r 2 + a2 2

54

J. Navrátil

among others, of values φ(xsi − xsj ), while the radial basis function is evaluated only on structural nodes. The matrix Afs (of size Nf × Ns ) depends on both fluid and structural nodes. The interpolation by radial basis function is dependent on the radius of the support r, which can be varied by the shape parameter ε. A large support radius gives good approximation, but if the radius is too large it leads to singular matrices. On the other hand, a small support radius yields less accurate interpolation. In many cases, it would be useful to vary the radius from structure node to structure node but according to the theory it is not possible. In practice, the support radius should guarantee that enough points are covered, on the other hand, points far away should have no influence. Therefore, the support radius for fluid-structure interaction should be chosen as large as the maximum distance of all centers from their nearest neighbors in both meshes. Other important choice is a choice of the radial basis function itself. In this work, the Thin-plate spline is used. It belongs, according to [8], among the most robust, cost effective and accurate RBFs for this application. The unique solution of the interpolation can be guaranteed only if at least four structural points are not in plane [39]. This is not often fulfilled, e.g. if the simple beam structural model is used. In this case, it is possible to solve the linear system but the solution is not unique.

2.4 Mesh Deformation The motion of deformable surfaces of an aeroelastic model must be captured by a fluid computational grid prior to calculation of new flow solution. Two methods of moving geometry treatment exist, remeshing and grid deformation. The first mentioned allows to capture arbitrarily large geometry deformation, but at high computational cost connected with the recalculation of the entire volume mesh. Moreover, risk of physical conservation loss exists due to possible large changes in the grid which may lead to reduced local computational accuracy. Therefore, development of mesh deformation techniques, such as spring analogy [4], Laplace smoothing [15, 16, 27] and radial basis functions interpolation [9, 19], has began in recent years. Their advantage is conservation of mesh topology, nodal connectivity and generally lower computational cost compared to the remeshing. Commonly, mesh deformation methods suffer by high risk of inverted elements occurrence as result of large geometry deformation. The Radial Basis Function mesh deformation method was applied in this work. It applies the similar principle to the one used for defining the fluid-structure interface. In application for the fluid mesh deformation, control points are defined on the movable boundary of the fluid mesh. The boundary movement is interpolated into the fluid volume mesh. Compared to the spring analogy and Laplace methods, the computational cost is low, once the interpolation matrix is created. This method can handle large deformations and is applicable for structured, unstructured and hybrid meshes, because it is independent of the mesh connectivity. The precision of the

High-Fidelity Static Aeroelastic Simulations …

55

surface shape, in the deformed volume mesh, depends on the number of the surface nodes acting as RBF control points. The method performance depends on the RBF type and on the choice of the support radius.

2.5 Convergence of Coupled Solution The convergence criterion of the coupled solution is satisfied once the change between two consecutive normalized root mean square (RMS) values of fluid surface mesh nodes displacements is less than a prescribed value. The criterion must be satisfied for displacements in all directions: ufx , ufy and ufz and for one dimension is defined as (notation is without subscript defining direction): uˆ fj − uˆ fj−1 < ts

(10)

where ts is a convergence tolerance and uˆ fj in current jth iteration is defined as: uˆ fj =

ufRMS,j ufRMS,1

(11)

The ufRMS,j and ufRMS,1 are RMS of fluid surface mesh displacements in the given direction in j-th and the first coupling iteration, respectively, defined as follows:   N 1  u2 (12) ufRMS,j =  n i=1 fi ,j

3 Test Case The test case is transonic flow around the NASA Common Research Model (CRM) [37] considering the wing-fuselage and wing-only geometries. In the tests, two different structural models were applied—wing-box and beam stick models. The beam model is equivalent to the wing-box model in the sense of static deformation.

3.1 Geometry The CRM was originally intended for CFD validation studies [37], but it became a standard model for other applications including aerodynamic shape optimization [3, 29], aero-structural optimization [11, 22, 23] and aeroelastic tailoring [21].

56 Table 2 CRM specification Parameter Cruise Mach number Cruise lift coefficient Cruise altitude Wing span Reference wing area Reference wing area—wing-only geometry

J. Navrátil

Value 0.85 0.5 11,000 m 59.1 m 383.7 m2 332 m2

Fig. 1 CRM wing-fuselage geometry

The model corresponds to the Boeing 777 airliner, the relevant specification of the CRM are listed in Table 2. Since only symmetric flow conditions without a sideslip were considered, only half of airplane has been used in the test cases as it is shown in the model schematic drawing in Fig. 1. The wing-only configuration was created by cut in the position about 3 m from symmetry plane removing the fuselage. New symmetry plane was created in the wing root location.

3.2 Structural Models The finite element models of CRM wing structure are provided on the website.1 The available structural models differ in the mesh density and element topology. 1 http://commonresearchmodel.larc.nasa.gov/.

High-Fidelity Static Aeroelastic Simulations …

57

Fig. 2 Wing-box finite element model of CRM wing structure applied in test cases, overview (left) and detail with partly hidden skin (right)

Fig. 3 Beam stick finite element model of CRM wing structure applied in test cases

The coarse model v14 was chosen for the application in test cases. It consists of 4622 nodes connected to 8502 quad or tri shell elements (see Fig. 2). The wing main structure mass distribution was calculated from the finite element model. The masses of the engines, nacelles, control surfaces, flaps and fuel were not taken into account. A beam stick model of the wing structure was designed according to the inverse design method presented in [30]. The model is equivalent to the wing-box model meaning the static aeroelastic deformation of the wing is comparable in given operating conditions. The wing structure weight is neglected. The beam stick model (Fig. 3) consists of 22 nodes connected together by beam elements. Each node of the beam element is connected to two additional nodes by rigid elements. Those nodes are beneficial for coupling with aerodynamic surface, meaning the nodes allow reconstruction of rotations only by translational degrees of freedom. In the wing root, more additional rigid elements are required due to applied fluid-structure coupling interface, which is defined by radial basis functions method.

3.3 Aerodynamic Models Fluid computational grid was created from a geometry provided on CRM website which is a wing-fuselage configuration of a common transport aircraft. The model

58

J. Navrátil

Fig. 4 Aerodynamic grids of wing-only and wing-fuselage geometries

was originally designed for evaluating the ability of computational tools to predict drag, thus the shape of the wing is twisted and bended as it would be during the design cruise conditions. It was assumed the deformed shape of the wing is not in conflict with intended use of the model for verification of the aeroelastic tool. The wing-only hybrid unstructured grid consist of 2,644,786 nodes and about 7.6 millions tetrahedral and prismatic elements. In case of the wing-fuselage geometry, the mesh consist of 2,831,524 nodes and 8.5 millions elements. Both grids are shown in Fig. 4.

3.4 Flow Conditions The transonic viscous flow conditions listed in Table 3 were considered in the test case.

High-Fidelity Static Aeroelastic Simulations … Table 3 Flow conditions

59

Parameter

Value

Cruise Mach number Cruise altitude Static pressure Temperature Dynamic pressure Air density Reynolds number per unit length

0.85 11,000 m 22,632 Pa 216.65 K 11,450 Pa 0.364 kg m−3 5,184,000 m−1

4 Results In the first part of this section, the convergence of the static aeroelastic solution is studied. It is followed by application of the computational aeroelasticity tool using different aerodynamic and structural models.

4.1 Convergence Study of Static Aeroelastic Solution The influence of fluid solver settings on the convergence of the static aeroelastic solution was studied. In Table 4 the settings of the flow solver are listed. In general, well-converged flow solution was required at the end of the coupled solution, thus the target residual reduction of the flow variables about 7 orders of magnitude was prescribed. Then, the influence of the flow residuals reduction in each coupling iteration on the coupled solution was observed in terms of computational cost and the resultant aerodynamic forces. The residual reduction was limited by prescribed total number of flow solver iterations (inner iterations) in each coupling iteration. The set-up designated as “01” requires complete converged flow solution before the data

Table 4 Flow solver settings Set-up Number of grids in multi-grid solutiona Max. number of full multi-grid cycles Max. number of CFD iterations at each coupling iteration Order of residual decrease for converged solution a If

01

02

03

04

05

06

3

3

1

1

1

1

500

500

–

–

–

–

5000

500

500

250

125

65

−7

−7

−7

−7

−7

−7

>1 then full multi-grid solution was performed

60

J. Navrátil

exchange on the aeroelastic interface. In other cases the flow solution is limited by prescribed number of inner iterations. Therefore, partially converged flow solution can be expected mainly during first coupling iteration. After the coupled solution convergence criteria were satisfied, the reduction of the flow residuals was checked whether the prescribed value was reached. If needed, the flow solution was restarted to obtain the well-converged one as it was required. In this study, inviscid Euler flow was considered. The convergence criterion of the coupled solution was residual reduction about 4 orders of magnitude. The reduction of the inner iterations number has consequence of the computational cost saving in terms of the total number of CFD iterations. (It should be noted that the really comparable are only cases from “03” to “06”. The first and the second case employed full multi-grid solution using two coarser CFD grids, thus one iteration is cheaper on those grids than on the finest grid). The number of coupling iterations was reduced as well, as it is shown in Table 5. This is beneficial from the perspective of time saving for the communication and the mesh deformation. In this case, the lowest cost is suggested by the scheme “05” and is about three time higher than the cost of pure flow solution. The convergence was tested for other two flow conditions—at M = 0.6, α = 5◦ and at M = 0.88, α = 0◦ . The results given in Table 6 suggest that in this case the most efficient is scheme “06” from perspective of the needed CFD time. But the lowest number of the coupling iterations was achieved by scheme “03” (see Table 6). For the flow conditions at M = 0.88, α = 0◦ , the scheme “03” was the most efficient from both perspectives, flow solution computational cost and costs for communication and mesh deformation (Table 7). Table 5 Results of convergence study at M = 0.85 and α = 0◦ Set-up 01 02 03 04 CL CD Cm Total CFD iterationsa Total coupling iterations a In

05

06

0.2733 0.00496 −0.0927 40,702

0.2733 0.00496 −0.0927 7338

0.2733 0.00496 −0.0927 7415

0.2733 0.00496 −0.0927 4196

0.2733 0.00496 −0.0927 2114

0.2733 0.00496 −0.0927 2124

19

19

19

16

16

16

cases “01” and “02” the less expensive iteration on coarser CFD grids are included. The rigid CFD solution converged after 1249 iterations (including full multi-grid solution on coarser grids)

High-Fidelity Static Aeroelastic Simulations …

61

Table 6 Results of convergence study at M = 0.6 and α = 5◦ Set-up 03 04 05 CL CD Cm Total CFD iterationsa Total coupling iterations

06

0.6899 0.01979 −0.1195 2653

0.6899 0.01979 −0.1195 1989

0.6899 0.01979 −0.1194 1813

0.6899 0.01979 −0.1195 1737

7

8

9

19

a The

rigid CFD solution converged after 1512 iterations (including full multi-grid solution on coarser grids) Table 7 Results of convergence study at M = 0.88 and α = 0◦ Set-up 03 04 05 CL CD Cm Total CFD iterationsa Total coupling iterations

06

0.2770 0.00696 −0.0993 5411

0.2770 0.00696 −0.0993 7280

0.2770 0.00696 −0.0993 6227

0.2768 0.00698 −0.0993 5567

11

26

37

45

a The

rigid CFD solution converged after 1236 iterations (including full multi-grid solution on coarser grids)

4.2 Solution of Wing-Fuselage Geometry with Wing-Box Structural Model at M = 0.85 This test case represents the case when the aerodynamic model consists of two different types of surfaces—rigid (the fuselage) and elastic (the wing)—and the structural model is created by modelling the wing structure, only. Convergence of the coupled solution is presented in Fig. 5 and the aerodynamic forces convergence during the aeroelastic solution is shown in Fig. 6. The simulation has converged after 9 coupling iterations. Table 8 presents the computational cost and the resultant forces coefficients. The cost of the aeroelastic simulation is about 2 times higher compared to the aerodynamic solution. The drag coefficient in the aeroelastic simulation, in the cruise operating conditions, is about 6% higher compared to the aerodynamic solution of the rigid wing. The negative pitching moment, thus the balancing force of the horizontal tail unit, is about 43% higher comparing the same cases. Figure 7 presents contours of the pressure coefficient on upper surface of the wingfuselage configuration. Clearer insight on results is given in Fig. 8, where the pressure distribution is plotted over the chosen wing sections. The position of the sections is defined in the percent of the half-span counting from the airplane symmetry plane.

62

J. Navrátil

Order of residuals decrease

0 rms x−displ. rms y−displ. rms z−displ.

−1

−2

−3

−4

−5 1

3

5

7

9

11

13

15

17

19

21

Coupling iterations Fig. 5 Convergence of aeroelastic solution for given CL = 0.5, wing-fuselage geometry, wing-box structural model 0.65

0.035 RANS

RANS

0.6 0.03

CL

CD

0.55

0.5 0.025 0.45

0.4

0.02 1

2

3

4

5

6

7

Coupling iterations

8

9

10

1

2

3

4

5

6

7

8

9

10

Coupling iterations

Fig. 6 Convergence of aeroelastic solution—aerodynamic forces coefficients, the wing-fuselage geometry with the wing-box structural model

The result examination suggests that in aeroelastic simulations the wing torsional deformation together with higher required angle of incidence resulted in redistribution of pressure over wing surfaces. The effect of the wing torsional deformation is more obvious in the sections near the wing tip, where the suction was decreased as a result of the sectional angles of attack decrease (see left part of Fig. 9) compared to rigid cases. Plot of the loading distribution over the wing span in the right part of Fig. 9 shows that the wing deformation led to increased loading of the wing inboard part, while the outboard part was alleviated.

High-Fidelity Static Aeroelastic Simulations …

63

Table 8 Computational cost and resultant aerodynamic forces coefficients—wing-fuselage geometry, required CL = 0.5 Rigid Elastic Difference (%) No. of coupling iterations Total no. of CFD iterations α (◦ ) CL CD Cma

–

9

–

1495

3358

124

1.58 0.4998 0.0232 −0.0511

3.12 0.5001 0.0247 −0.0729

97 0 6 43

Fig. 7 Comparison of rigid (upper half) and aeroelastic (bottom half) solution—contours of the surface pressure coefficient distribution at static aeroelastic equilibrium state

4.3 Solution of Wing-Only Geometry with Beam Stick and Wing-Box Structural Models The intention of the test cases was to validate an ability of the computational aeroelastic tool to handle a simplified structural model. Evaluation was done at two free stream conditions: M = 0.85, CL = 0.5 and M = 0.6, α = 5◦ . The results of the static aeroelastic calculation were compared with the case using the wing-box model.

64

J. Navrátil Section at 10.4%

-1

Section at 35.2%

-1.5 -1

-0.5

-0.5

C

Cp

p

0 0

0.5 0.5 1

1

Rigid Elastic

Rigid Elastic

1.5

1.5 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x/c Section at 49.4%

-1

0.8

1

Section at 67.1%

-1

-0.5

-0.5

0

0

Cp

C

p

0.6

x/c

0.5

0.5 1

1

Rigid Elastic

Rigid Elastic

1.5

1.5 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

Section at 81.3%

-1

0.8

1

Section at 95.5%

-1

-0.5

-0.5

0

0

Cp

C

p

0.6

x/c

x/c

0.5

0.5

1

1

Rigid Elastic

1.5

Rigid Elastic

1.5 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x/c

0.6

0.8

1

x/c

5

6

0

4

Rigid Elastic

L

c .c

Twist angle [deg]

Fig. 8 Comparison of surface pressure coefficient distribution at chosen sections of the wing— CRM wing-fuselage geometry

-5

2

Rigid Elastic

0

-10 0

0.2

0.4

0.6

y/b

0.8

1

0

0.2

0.4

0.6

0.8

1

y/b

Fig. 9 Comparison of rigid and aeroelastic solutions—the wing twist angle and the wing loading distribution

High-Fidelity Static Aeroelastic Simulations …

65

Table 9 Comparison of aeroelastic simulation results using wing-box and beam stick structural models—flow at M = 0.85 and required CL = 0.5 Wing-box Beam stick Difference (%) No. of coupling iter. No. of CFD iter. CL CD Cm a a Pitch

15 4931 0.5008 0.0179 −0.0614

16 4830 0.5 0.0175 −0.0615

– – – −2.2 0.2

moment is related to the quarter point of the wing mean aerodynamic chord

Table 10 Comparison of aeroelastic simulation results using wing-box and beam stick structural models—flow at M = 0.6 and α = 5◦ Wing-box Beam stick Difference (%) No. of coupling iter. No. of CFD iter. CL CD Cm a a Pitch

7 1631 0.5972 0.0214 −0.0722

7 1220 0.5936 0.0214 −0.0725

– – 0.6 0.0 0.4

moment is related to the quarter point of the wing mean aerodynamic chord

The computational cost and the resultant values of aerodynamic forces are given in Tables 9 and 10. The results, in terms of the aerodynamic forces, of the static aeroelastic computation using the simplified structural model are in good agreement with results given by the simulation applying the wing-box model. The pressure coefficient contours on the upper wing surface for the case at M = 0.85 are presented in Fig. 10. The figure does not show any significant dissimilarity in the pressure contours comparing the deformed wings. Figure 11 shows plots of the chordwise pressure distribution at selected wing sections. The pressure distribution given by the aeroelastic simulation using the beam structural model agrees well with the reference results in all cases. The plot of maximum spanwise wing thickness distribution in Fig. 12 illustrates that application of essentially two dimensional structural model does not produce any unrealistic geometrical changes of the deformed wing. However, this kind of structural model in combination with RBF transformation method requires use of additional rigid elements in the location of the wing root, as it is shown in Fig. 3. If additional elements are not applied, significant change in wing thickness will occur as the result of the non-unique solution of the aero-structural coupling matrix. Application of the beam stick model produced maximum differences in wing deformation in the order of centimetres compared to solution with the wing box model. Comparison is presented in Fig. 13.

66

J. Navrátil

Fig. 10 Comparison of aeroelastic simulation results using wing-box (left half) and beam stick (right half) structural model—surface pressure distribution, flow at M = 0.85, CL = 0.5 (condition 1)

5 Conclusion The applicability of the computational aeroelasticity tool in the aerodynamic analysis of elastic airplane was tested on the cases employing different types of the aerodynamic and structural models. The test cases have shown that the tool is able to handle complex geometries, such as a wing-fuselage model of transport aircraft with the swept wing. Different aerodynamic and structural models were used in order to test the tool at various conditions. The considered aerodynamic models were wing-only and wing-fuselage configurations. The applied structural models were the wing-box and beam stick models representing the wing structure, the fuselage was assumed as rigid. The performed analyses have shown the effect of the wing flexibility on the aerodynamic characteristics and load distribution over the wing. The primary cause of the load redistribution is the wing twist due to aerodynamic forces. The beam stick structural model was employed in order to evaluate the ability of the computational aeroelasticity tool to handle simplified model of a wing structure. The results have shown that the simplified model can be employed. The aeroelastic solution using the reference wing-box model was compared with the one employing the beam model. In the test several flow conditions and flow models were considered. The results are comparable in respective cases.

High-Fidelity Static Aeroelastic Simulations …

67 Section at 27.7%

Section at 0%

-2

-2

-1.5

-1

Cp

-0.5

C

p

-1

0

0 Wingbox, conditions 1 Beam, conditions 1 Wingbox, conditions 2 Beam, conditions 2

0.5 1

1

1.5 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

0.8

1

0.8

1

x/c

x/c Section at 43.5%

Section at 63.3%

-2

-2

-1.5 -1 -0.5

C

Cp

p

-1

0

0

0.5 1

1

1.5 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

Section at 79.2%

-2

Section at 95%

-2

-1.5

-1.5 -1 -0.5

p

-1 -0.5

C

C

p

0.6

x/c

x/c

0

0

0.5

0.5

1

1

1.5

1.5 0

0.2

0.4

0.6

x/c

0.8

1

0

0.2

0.4

0.6

x/c

Fig. 11 Comparison of aeroelastic simulation results using wing-box and beam stick structural models—pressure distribution at different sections of CRM wing, flow at M = 0.85, CL = 0.5 (condition 1) and M = 0.6, α = 5◦ (condition 2)

The obvious continuation of the presented work is the application of the computational aeroelasticity tool for solving time-dependent dynamic aeroelastic problems. Thus, the tool must be implemented in the way that the communication between solvers is based on the direct approach via random-access memory. Therefore, the tool as well as the linear elasticity solver must be programmed in the language such as Fortran or C and implemented into the applied flow solver.

68

J. Navrátil 6 Wingbox, conditions 1 Beam, conditions 1 Wingbox, conditions 2 Beam, conditions 2

1.2

4

Twist angle [deg]

1.4

t [m]

1 0.8 0.6 0.4

2 0 -2 Wingbox, conditions 1 Beam, conditions 1 Wingbox, conditions 2 Beam, conditions 2

-4 -6

0.2

-8 0

0.2

0.4

0.6

y/b

0.8

1

0

0.2

0.4

0.6

0.8

1

y/b

Fig. 12 Comparison of aeroelastic simulation results using wing-box and beam stick structural models—maximum wing thickness (left) and wing twist angle (right), flow at M = 0.85, CL = 0.5 (condition 1) and M = 0.6, α = 5◦ (condition 2)

Fig. 13 Comparison of wing shape at static aeroelastic equilibrium—difference in shape of wing using beam stick structural model compared to wing using the wing box model, flow at M = 0.85, α = 2.35◦ (condition 1) and M = 0.6, α = 5◦ (condition 2)

The potential application of the presented tool is, besides aerodynamic analysis, in aerodynamic and aero-structural optimization of a flexible aircraft. Acknowledgements This work was funded and supported by The Ministry of Education, Youth and Sports, Czech Republic under the National Sustainability Programme I—Project LO1202.

High-Fidelity Static Aeroelastic Simulations …

69

References 1. Alonso, J. J., Martinelli, L., & Jameson, A. (1995). Multigrid unsteady Navier-Stokes calculations with aeroelastic applications. In AIAA paper. Citeseer. 2. Amoignon, O., & Berggren, M. (2003). Discrete adjoint-based shape optimization for an edgebased finite-volume solver. In Computational fluid and solid mechanics (pp. 2190–2193). Elsevier. 3. Amoignon, O., Navrátil, J., & Hradil, J. (2014). Study of parameterizations in the project CEDESA. AIAA Paper, 570. 4. Batina, J. T. (1990). Unsteady Euler airfoil solutions using unstructured dynamic meshes. AIAA Journal, 28(8). 5. Beckert, A., & Wenland, H. (2001). Multivariate interpolation for fluid-structure-interaction problems using radial basis functions. Aerospace Science and Technology, 5, 125–134. 6. Bendiksen, O. O. (1987). Transonic flutter analysis using the Euler equations. In 28th Structures, Structural Dynamics and Materials Conference (p. 911). 7. Blom, F. J. (2000). Considerations on the spring analogy. International Journal for Numerical Methods in Fluids, 32(6), 647–668. 8. Boer, A. d., van Zuijlen, A. H., & Bijl, H. (2007). Review of coupling methods for nonmatching meshes. Computer Methods in Applied Mechanics and Engineering, 196(8), 1515– 1525. https://doi.org/10.1016/j.cma.2006.03.017 9. De Boer, A., Van der Schoot, M., & Bijl, H. (2007). Mesh deformation based on radial basis function interpolation. Computers & Structures, 85(11), 784–795. 10. Dettmer, W. G., & Peri´c, D. (2013). A new staggered scheme for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 93(1), 1–22. 11. Dunning, P. D., Stanford, B. K., & Kim, H. A. (2014). Aerostructural level set topology optimization for a common research model wing. In 10th AIAA Multidisciplinary Design Optimization Conference, National Harbor, MD, USA, 13–17 January 2014 (pp. 1–21). 12. Eliasson, P. (2002). EDGE, a Navier-Stokes solver for unstructured grids. In Proceedings to Finite Volumes for Complex Applications III (pp. 527–534). ISBN 1 9039 9634 1. 13. Farhat, C., Van der Zee, K. G., & Geuzaine, P. (2006). Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity. Computer Methods in Applied Mechanics and Engineering, 195(17), 1973–2001. 14. Feng, Z., Soulaı, A., Saad, Y., et al. (2009). Nonlinear Krylov acceleration for CFD-based aeroelasticity. Journal of Fluids and Structures, 25(1), 26–41. 15. Field, D. A. (1988). Laplacian smoothing and Delaunay triangulations. Communications in Applied Numerical Methods, 4(6), 709–712. 16. Freitag, L. A. (1997). On combining Laplacian and optimization-based mesh smoothing techniques. In Trends in unstructured mesh generation (pp. 37–43). 17. Heeg, J., Chwalowski, P., Florance, J. P., Wieseman, C. D., & Schuster, D. M. (2013). Perry Boyd: Overview of the aeroelastic prediction workshop. In 51st AIAA Aerospace Sciences Meeting (p. 2002). Reston, Virginia: American Institute of Aeronautics and Astronautics. https:// doi.org/10.2514/6.2013-783. 18. Heeg, J., Chwalowski, P., Raveh, D. E., Dalenbring, M. J., & Jirasek, A. (2015). Plans and example results for the 2nd AIAA aeroelastic prediction workshop. In 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (p. 2013). Reston, Virginia: American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.2015-0437. 19. Jakobsson, S., & Amoignon, O. (2007). Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization. Computers & Fluids, 36(6), 1119–1136. https://doi.org/10.1016/j.compfluid.2006.11.002. 20. Jameson, A. (1991). Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. In 10th Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences. American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.1991-1596.

70

J. Navrátil

21. Jutte, C. V., Stanford, B. K., Wieseman, C. D., & Moore, J. B. (2014). Aeroelastic tailoring of the NASA common research model via novel material and structural configurations. In AIAA SciTech Conference (pp. 13–17). 22. Kenway, G. K., & Martins, J. R. (2014). Multipoint high-fidelity aerostructural optimization of a transport aircraft configuration. Journal of Aircraft, 51(1), 144–160. 23. Kenway, G. K., Martins, J. R., & Kennedy, G. J. (2014). Aerostructural optimization of the common research model configuration. Group (ADODG), 6(7), 8–9. 24. Küssner, H. G. (1929). Schwingungen von flugzeugflugeln. Jahrbuch der deutscher Versuchsanstalt für Luftfahrt (especially Section E3 Einfluss der Baustoff-Dämpfung) (pp. 319–320). 25. Küssner, H. G. (1936). Zusammenfassender bericht über den instationären auftrieb von flügeln. Luftfahrtforschung, 13(12), 410–424. 26. Lee-Rausch, E., & Baitina, J. T. (1996). Wing flutter computations using an aerodynamic model based on the Navier-Stokes equations. Journal of Aircraft, 33(6), 1139–1147. 27. Liakopoulos, P. I., & Giannakoglou, K. C. (2006). Unstructured remeshing using an efficient smoothing scheme approach. In ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, 5–8 September 2006. Delft University of Technology; European Community on Computational Methods in Applied Sciences (ECCOMAS). 28. Luke, E., Collins, E., & Blades, E. (2012). A fast mesh deformation method using explicit interpolation. Journal of Computational Physics, 231(2), 586–601. 29. Lyu, Z., Kenway, G. K., & Martins, J. R. (2014). Aerodynamic shape optimization investigations of the common research model wing benchmark. AIAA Journal, 53(4), 968–985. 30. Navrátil, J. (2016). New approaches in numerical aeroelasticity applied in aerodynamic optimization of elastic wing (Ph.D. thesis). Brno University of Technology. 31. Przemieniecki, J. (1985). Theory of matrix structural analysis. Dover Civil and Mechanical Engineering, Dover. https://books.google.cz/books?id=Pb8qTzqOKbAC. 32. Rendall, T., & Allen, C. (2008). Unified fluid-structure interpolation and mesh motion using radial basis functions. International Journal for Numerical Methods in Engineering, 74(10), 1519–1559. 33. Rodden, W. P. (1997). The development of the doublet-lattice method. Proceedings of the CEAS/International Forum on Aeroelasticity and Structural, 2, 1–7. 34. Theodorsen, T. (1935). General theory of aerodynamic instability and the mechanism of flutter (NACA Report 496). 35. Thompson, E., Kolonay, R., Eastep, F., & Camberos, J. (2007). Aeroelastic analysis with transpiration enabled Euler flow solver. In 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (p. 2331). 36. van Zuijlen, A. H., & Bijl, H. (2005). Implicit and explicit higher order time integration schemes for structural dynamics and fluid-structure interaction computations. Computers & Structures, 83(2), 93–105. 37. Vassberg, J., Dehaan, M., Rivers, M., & Wahls, R. (2008). Development of a common research model for applied CFD validation studies. In 26th AIAA Applied Aerodynamics Conference, Guidance, Navigation, and Control and Co-located Conferences. American Institute of Aeronautics and Astronautics. https://doi.org/10.2514/6.2008-6919. 38. Von Wagner, H. (1925). Dynamischer auftrieb von tragflugeln. Zeitchrift fuer Angewandte Mathematik und Mecdhanik, 5, 17. 39. Wendland, H. (2008). Hybrid methods for fluid-structure-interaction problems in aeroelasticity. In Meshfree methods for partial differential equations IV (pp. 335–358). Springer. 40. Yates, E. C., Jr. (1966). Modified-strip-analysis method for predicting wing flutter at subsonic to hypersonic speeds. Journal of Aircraft, 3(1), 25–29. 41. Yurkovich, R. (2003). Status of unsteady aerodynamic prediction for flutter of highperformance aircraft. Journal of Aircraft, 40(5), 832–842. 42. Zeng, D., & Ethier, C. R. (2005). A semi-torsional spring analogy model for updating unstructured meshes in 3d moving domains. Finite Elements in Analysis and Design, 41(11), 1118–1139.

Aero-elastic Simulations Using the NSMB CFD Solver Including results for a Strut Braced Wing Aircraft J. B. Vos, D. Charbonnier, T. Ludwig, S. Merazzi, H. Timmermans, D. Rajpal and A. Gehri

Abstract More then 10 years ago a large investment was made in extending the NSMB Navier Stokes Multi Block (NSMB) Computational Fluid Dynamics (CFD) towards Fluid Structure Interaction (FSI) simulations (Guillaume et al. in Fluid structure interaction simulation on the F/A-18 vertical tail, 2010 [1], Guillaume et al. in Aeronaut J 115:285–294, 2011 [2]). At that time a segregated approach was adopted using a loosely coupled approach. More recently NSMB was coupled to the opensource Finite Element Analysis environment B2000++ (http://www.smr.ch/products/ b2000/ [3]) in a strongly coupled approach. This has led to the possibility to perform both static and dynamic FSI simulations using either a modal or a FEM approach without the need to interrupt the simulation. Results of aero-elastic simulations for the MDO-aircraft, the AGARD445.6 wing and for a Strut Braced Wing configuration will be presented. J. B. Vos (B) · D. Charbonnier CFS Engineering, EPFL Innovation Park, Batiment A, 1015 Lausanne, Switzerland e-mail: [email protected] D. Charbonnier e-mail: [email protected] T. Ludwig · S. Merazzi SMR Engineering and Development, Blumenstrasse 14-16, 2502 Bienne, Switzerland e-mail: [email protected] S. Merazzi e-mail: [email protected] H. Timmermans NLR Netherlands Aerospace Centre, Postbus 90502, 1006 BM Amsterdam, The Netherlands e-mail: [email protected] D. Rajpal Delft University of Technology Faculty of Aerospace Engineering, Kluyverweg 1, 2629 Delft, The Netherlands e-mail: [email protected] A. Gehri Aerodynamics Department, RUAG Aviation, 6032 Emmen, Switzerland e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_5

71

72

J. B. Vos et al.

1 Introduction The Swiss Airforce operates since 1997 the F/A-18 C/D fighter. The usage of this aircraft is about three times more severe than the original design, requiring changes in the aircraft structure. To better understand the aerodynamic loads on the aircraft RUAG Aviation made a large investment in extending the Navier Stokes Multi Block (NSMB) Computational Fluid Dynamics (CFD) code to allow for Fluid Structure Interaction (FSI) simulations [1, 2]. Procedures were developed for both static and dynamic Fluid Structure Interaction. Static Fluid Structure Interaction was put in place using a segregated two solver approach. A CFD calculation was made and run for a certain number of steps, the calculation was stopped, the aerodynamic loads were extracted and used as input for Nastran. Nastran computed the displacements of the aircraft structure that were transformed to surface grid point displacements. The CFD solver was then re-started and used these displacements to re-compute the grid, and then continued the calculation on this new grid, typically for about 300–500 steps. Then the CFD solver was stopped and the process was repeated. In general 4 to 5 of these iterations were made to obtain a converged deformed geometry. Dynamic FSI simulations were made for the F/A-18 C/D fighter to study vertical tail buffeting, and employed a linear structural model based on a modal formulation. In this case the CFD solver and the Computational Structural Mechanics (CSM) solver are tightly coupled [1]. Over the past years several developments were made that are presented in this paper. The NSMB CFD solver was extended with the chimera method that permits easy changes of control surface deflections and that facilitates the mesh generation for complex geometries. The need to provide quickly aerodynamic loads for statically deformed wings led to the decision to couple the NSMB CFD solver with the open-source Finite Element Analysis environment B2000++ [3]. This has led to the possibility to perform both static and dynamic FSI simulations using either a modal or a FEM approach without the need to interrupt the simulation. This paper is organized as follows: first the CFD solver NSMB and the FEM Environment B2000++ are briefly presented. This is followed by a more detailed discussion on the different elements of the aero-elastic simulation environment. This environment has been validated using two known test cases, the MDO Aircraft [4] and the GARD445.6 wing [5]. Finally, in the frame of the EU funded H2020 project AGILE [6], the aero-elastic simulation environment was used to compute a Strut Braced Wing aircraft. Other approaches and methods for fluid structure interaction simulations can be found in the literature [7–9].

2 The NSMB CFD Solver The Navier Stokes Multi Block solver NSMB was initially developed in 1992 at the Swiss Federal Institute of Technology (EPFL) in Lausanne, and from 1993 onwards in the NSMB consortium composed of different universities, research establishments

Aero-elastic Simulations Using the NSMB CFD Solver Including …

73

and industries. Today NSMB is developed by IMF-Toulouse (IMFT, France), ICUBE (Strasbourg, France), University of Munchen (TUM, Germany), University of the Army in Munchen (Germany), Ariane Group (France), RUAG Aviation and CFS Engineering. A variety of papers have been published on NSMB, examples are in [1, 10–13]. NSMB is a parallelized CFD solver employing the cell-centered finite volume method using multi block structured grids to discretize the Navier Stokes equations. To simplify the mesh generation for complex geometries NSMB uses the patch grid (also known as the sliding mesh) approach and the chimera method. The chimera method is also used for simulations involving moving bodies. Space discretization schemes implemented in NSMB are the 2nd and 4th order central schemes with artificial dissipation and Roe and AUSM upwind schemes from 1st to 5th order. Time integration can be made using explicit Runge-Kutta schemes, or the semi-implicit LU-SGS scheme. Different methods are available to accelerate the convergence to steady state, as for example local time stepping, multigrid and full multigrid, and low Mach number preconditioning. Unsteady simulations are made using the dual time stepping approach or using the 3rd order Runge Kutta scheme. Turbulence is modelled using standard approaches as for example the algebraic Baldwin-Lomax model, the 1-equation Spalart model [14] (and several of it’s variants) and the k − ω family of models (including the Menter Shear Stress model [15]). Explicit Algebraic Reynolds Stress models and Reynolds Stress models have also been implemented, but are not used on a routine base. Transition to turbulence can be modelled by specifying transition lines or planes, or by solving the γ − Rθ transport equations [16]. For unsteady CFD simulations different Hybrid RANS-LES models are available. The Arbitrary Langerian Eulerian (ALE) approach is employed for simulations using moving or deforming grids. When using the ALE approach it is not necessary that all blocks are moving or deforming, it is possible to define different groups of blocks each having their own movement. This is in particular useful for multi-body simulations. To permit CFD simulations on deforming grids it is necessary to re-generate the grid. NSMB includes a remeshing algorithm [1] and recent developments are discussed in more detail in Sect. 5.

3 The B2000++ Solver B2000++ is a Finite Element Method (FEM) solver which is being developed by SMR Engineering & Development. A wide range of problems in aerospace engineering can be studied, such as global aircraft as well as components or sub-components such as stiffened panels. The element library comprises shell elements, beam elements, pointmass elements, rigid-body elements, as well as 2D and 3D elements. Linear static analysis, linear dynamic analysis, free-vibration analysis and buckling analysis can

74

J. B. Vos et al.

be selected, as well as nonlinear static and dynamic analysis. For strength analysis, several failure criteria for isotropic materials and laminated composites are available. A high computational effectiveness is one of B2000++’s strengths. Symmetric multi-processing (SMP) accelerates the element procedures, taking advantage of today’s multi-core CPU’s. The open-source matrix solver MUMPS provides distributed parallelism via MPI. Eigen-analysis is carried out with the implicitly restarted Lanczos solver that is implemented in the open-source package ARPACK. The modular architecture facilitates the implementation of numerical methods. New material and element formulations, essential and natural boundary conditions, and solution methods can be added, requiring no or only a few modifications to the existing code. This flexibility enables the adaptation of B2000++ to specific problems like coupled fully nonlinear FSI.

4 Geometric Multi-region Coupling To transfer the aerodynamic forces from the CFD wetted surface to the structural FEM model, and to transfer the displacements from the FEM model to the CFD wetted surface, it is necessary to employ a spatial coupling procedure as the CFD and FEM meshes are in general non-matching. The spatial coupling is implemented within the B2000++ FEM code and is capable of multi-region coupling. As example the F/A-18 wing consists of different control surfaces – inner leading edge flap, outer leading edge flap, trailing edge flap, and aileron – that can move independently. Thus, a multi-region spatial interpolation procedure is utilized where the wing surface and the control surfaces constitute different coupling regions, and C0 continuity constraints at the coupling region intersections are enforced (Fig. 1).

Fig. 1 Undeformed CFD wetted surface (yellow) and deformed CFD surface (silver) at the trailing edge flap and aileron (left) and at the wing tip (right)

Aero-elastic Simulations Using the NSMB CFD Solver Including …

75

4.1 Single-Region Coupling Procedure The following sets of points are distinguished: The set of structural grid points S , the set of fluid surface points F where the displacements shall be interpolated, and the set of fluid surface points P where the displacements are prescribed. P may be empty. The set of support nodes is H = S ∪ P, and without loss of generality it is assumed that the coordinates X H , displacements u H , and concentrated forces f H are arranged as follows:  XH =

XS XP



 ; uH =

uS uP



 ; fH =

fS fP

 (1)

Let n = |H | and m = |F |. The volume-spline coupling operator [17] G is a m × n matrix G = AB

(2)

⎤ 1 ||X1H − X1F || ||X2H − X1F || · · · ||XnH − X1F || ⎢1 ||X H − X F || ||X H − X F || · · · ||X H − X F ||⎥ n 1 2 2 2 2 ⎥ A=⎢ ⎣1 ⎦ ··· ··· ··· ··· H H F F H F 1 ||X1 − Xm || ||X2 − Xm || · · · ||Xn − Xm ||

(3)

where the m × (n + 1) matrix A is ⎡

and the (n + 1) × n matrix B B = (C−1 ){}{1}

(4)

is a submatrix where the first column is removed from the inverse of the (n + 1) × (n + 1) matrix C. ⎤ 0 1 1 1 ··· 1 ⎢1 0 ||X2H − X1H || ||X3H − X1H || · · · ||XnH − X1H ||⎥ ⎥ ⎢ ⎢1 ||X H − X H || 0 ||X3H − X2H || · · · ||XnH − X2H ||⎥ 1 2 ⎥ ⎢ C=⎢ H H H H 0 · · · ||XnH − X3H ||⎥ ⎥ ⎢1 ||X1 − X3 || ||X2 − X3 || ⎦ ⎣1 ··· ··· ··· ··· ··· 0 1 ||X1H − XnH || ||X2H − XnH || ||X3H − XnH || · · · ⎡

(5)

The displacements u F are interpolated as follows: u F = Gu H

(6)

76

J. B. Vos et al.

The forces acting on the structural grid points and on the constrained fluid surface nodes are obtained by applying the transpose of G to the forces on the unconstrained fluid surface mesh nodes: f H = GT f F

(7)

4.2 Multi-region Coupling Procedure Two coupling regions A and B are considered where some of the fluid surface mesh nodes belong to both regions (Fig. 2). To maintain inter-region continuity of the displacements, it is necessary to proceed in three steps. Note that steps 2 and step 3 are interchangeable. 1. F contains the fluid interface nodes, S contains the structural nodes of region A and B, and P = ∅. 2. F contains the fluid nodes of region A without the interface nodes, S contains the structural nodes of region A, and P contains the fluid interface nodes. 3. F contains the fluid nodes of region B without the interface nodes, S contains the structural nodes of region B, and P contains the fluid interface nodes. The transfer of the concentrated nodal forces from the fluid wetted surface to the structural grid points is carried out in the opposite order (steps 1 and 2 are interchangeable).

Fig. 2 Two coupling regions with structural points and fluid surface mesh nodes

Aero-elastic Simulations Using the NSMB CFD Solver Including …

77

1. F contains the fluid nodes of region A without the interface nodes, S contains the structural nodes of region A. P contains the fluid interface nodes, for which the calculated forces are added to the fluid forces. 2. F contains the fluid nodes of region B without the interface nodes, S contains the structural nodes of region B, and P contains the fluid interface nodes, for which the calculated forces are added to the fluid forces. 3. F contains the fluid interface nodes, S contains the structural nodes of region A and B, and P = ∅. The forces that are mapped from F consist of the fluid forces at these points plus the forces that were mapped in steps 1 and 2 to these points. Because the operator that maps the forces preserves the sum of forces and is the transpose of the operator that maps the displacements, this spatial coupling procedure is rigid and, therefore, energy-conservative [18], which is important in flutter analysis (in addition, the energy error that is due to the time integration is minimized with the strong coupling approach). The deformed CFD surface is smooth except at the coupling region intersections. It is also possible to enforce zero displacements at coupling region boundaries. For example, the structural FEM model may consist only of a wingbox which is clamped at the root, while the CFD model is a wingbody configuration. In this case, the set P contains all fluid surface points of the body, and u P = 0.

4.3 Definition of Coupling Regions To facilitate the definition of the different coupling regions, the interactive and graphical tool FSCON (Fluid-Structure CONnector) is being used. It allows to select and visualize individual parts of the CFD wetted surface and of the FEM model. For the CFD model, boundary codes are used to select the coupling regions, whereas nodesets, element sets, group codes, etc. are used for the FEM model. Thus, the coupling region definitions are independent of the mesh size and aircraft configuration. Figure 3 shows an example for the Strut Braced Wing configuration. Two different coupling regions are defined, one for the wing and the second one for the strut. To improve the quality of the spatial coupling for the torsional modes, the FEM model is augmented with nodes at the leading and trailing edges that are connected by rigid body elements to the wing box. A subset of the body fluid surface nodes is included in the coupling (colored in orange, the grey part is constrained to zero), since the FEM model is fixed at the symmetry plane and has nonzero displacements at the wing root. In this way possible problems with the re-meshing algorithm are avoided.

78

J. B. Vos et al.

Fig. 3 The FSCON graphical user-interface for the Strut Braced Wing aircraft

5 Remeshing Algorithm The remeshing algorithm implemented in NSMB is discussed in detail in [2], and is a combination of Volume Spline Interpolation (VSI) [19] and Transfinite Interpolation (TFI). It consists of the following steps: 1. collect information on coordinates and their displacements and create a list of so called prescribed points. These prescribed points are then used to compute the Volume Spline Coefficients by solving a linear matrix systems of equations. 2. compute the displacements of block edges using the VSI method. The advantage of working with the displacements instead of the coordinates is that if the displacement of the edges is close to zero, the displacement in the volume will be close to zero too, and the original coordinates are not changed. 3. use 2D TFI to generate the displacement of the coordinates on the block faces. 4. use 3D TFI to generate the displacement. of the coordinates in the volume. 5. sum the coordinates and displacements to obtain the new mesh. Several problems were encountered (and solved) for complex grids. For example it might occur that one block edge is shared between 3 blocks or more and not all blocks know about the mesh movement. At the start of a calculation all the information on these shared edges is collected and the displacements between these edges are exchanged during the remeshing process. The most time consuming part in the process summarized above is the solution of the linear matrix system of equations to compute the Volume Spline Coefficients, and the costs are proportional to 21 n 2 with n the number of prescribed points. A large effortwas made to limit the number of prescribed points through a suitable

Aero-elastic Simulations Using the NSMB CFD Solver Including …

79

combination of use of VSI and TFI. Today the remeshing of a F/A-18 third generation mesh having around 50 million points takes less than a minute elapse time on present day HPC clusters. A recent development in the remeshing procedure concerns the use of the chimera method. The chimera method implemented in NSMB employs blocks inside the structure (that are not computed) to determine intersections between chimera grids. If the structure is moving it means that not only the fluid mesh is moved but one needs also to move the mesh inside the structure. This has been implemented through an exchange of surface displacements between fluid and solid meshes. The second problem concerned the remeshing itself. When moving two aircraft components that have different overlapping grids it might happen that these movements influence each other, leading to negative volumes. The solution to this problem is to perform the remeshing procedure on each chimera grid independent of the other chimera grids.

6 Validation Calculations 6.1 MDO Aircraft The MDO aircraft is a large transport aircraft with a flexible wing resembling the Airbus 380, and was the result of an EU funded project in the 1990s. The geometry consists of a fuselage and a wing, and the wing has a jig-shape needed for static aeroelastic computations [4]. This case was also used in the EC funded project UNSI (Unsteady Viscous Flow in the Context of Fluid-Structure Interaction) that finished in the year 2000. The results of this project are published in a book [4]. Calculations were made for the so called Case A conditions that are summarized in Table 1. Both the modal approach (considering 10 elastic modes) and the FEM approach using B2000++ were used in the calculations. The grid employed in the calculations had 123 blocks and about 1.15 million cells, and the surface grid is shown in Fig. 4. All calculations were made using central space discretization scheme with artificial dissipation, the LU-SGS scheme for the time integration and the k − ω Menter

Table 1 MDO fuselage-wing Case A conditions Altitude 37,000 Mach 0.85 p∞ 21,662.3 ρ∞ 0.34832 T∞ 216 Re/meter 6.158 × 106 CL 0.4581

ft Pa kg/m3 K 1/m

80

J. B. Vos et al.

Fig. 4 Grid used for the MDO fuselage-wing calculations

Shear Stress turbulence model [15]. The calculations ran for 7000 steps, and the static wing deformation was executed 5 times, after respectively 4000, 4500, 5000, 5500 and 6000 steps. Figure 5 shows the wing position (including a zoom of the wing tip) at different iterations. The zoom of the wing tip shows clearly the different iteration steps: the first deformation (orange color) shows a large movement upwards followed by a downward movement for the 2nd deformation. The last two deformations (light blue and blue) are close to each other. Table 2 summarizes the computed angle of attack to reach C L = 0.4581 for this case, and the results are compared with results reported by Saab in [4]. Note that the results obtained by Saab are for the wing only. Figure 6 shows the deformation of the leading and trailing edge of the wing, together with the results obtained by Saab. η

Fig. 5 Static wing deformation of the MDO test case. Colors indicate the iteration (from red (undeformed) via green to blue (last iteration) Table 2 Computed angle of attack MDO test case A CFD approach CSM model Saab Euler NSMB turbulence NSMB turbulence

Modal Modal B2000++

α 0.7450 0.7145 0.5330

Aero-elastic Simulations Using the NSMB CFD Solver Including … 3

81

SAAB TE NSMB Modal TE NSMB B2000++ TE SAAB LE NSMB Modal LE NSMB B2000++ LE

2.5

delta z(m)

2

1.5

1

0.5

0

0

0.2

0.6

0.4

0.8

1

eta

Fig. 6 MDO aircraft test case A: deformation of the leading and trailing edge of the wing

is the normalized y-coordinate (y/ymax ), with y the coordinate in span direction. As can be seen the NSMB results obtained by using B2000++ are in good agreement with the results by Saab taking into account the different modeling assumptions. One can also observe that the maximum deflection of the leading edge of the wing near the tip is slightly less than 3 m. Figure 7 shows the computed C p at 4 stations along the wing, together with the results obtained by Saab. The agreement is good, keeping in mind that the Saab results are obtained for a wing only assuming inviscid flow. Differences between the Saab and NSMB results concern the shock position and the C p near the trailing edge. Both differences can be explained by the difference in modeling approach.

6.2 AGARD 445.6 Wing An important problem in the development and validation of dynamic aero-elastic simulation tools is the lack of available experimental data for assessment and validation. Reasons for this are that the experiments by definition are destructive and that they require special models for the correct scaling of the frequencies. One of the most cited experimental test case is the AGARD 445.6 wing [5]. The AGARD445.6 wing, made of mahogany, has a 45◦ quarter chord sweep, a half span of 2.5 ft, a root chord of 1.833 ft and a constant NACA64A004 symmetric profile. Flutter tests were carried out at the NASA Langley Transonic Dynamics Tunnel, were published in 1963 and re-published in 1987. Various wing models were tested (and broken) in air and Freon-12 for Mach numbers between 0.338 and 1.141.The case most often used

82

J. B. Vos et al. -1

-0.5

-0.5

0

Cp

Cp

-1

0.5 1

0.5 NSMB Modal NSMB B2000++ SAAB

0

0.2

0.4

0.6

0.8

1 0

1

x/c (a)  = 0.2

-1

0.2

0.4

0.6

0.8

1

0.8

1

x/c (b)  = 0.4

-0.5

0

Cp

Cp

NSMB Modal NSMB B2000++ SAAB

-1

-0.5

0.5 1

0

0 0.5

NSMB Modal NSMB B2000++ SAAB

0

0.2

0.4

x/c

0.6

0.8

1

(c)  = 0.6

1 0

NSMB Modal NSMB B2000++ SAAB

0.2

0.4

x/c

0.6

(d)  = 0.8

Fig. 7 MDO aircraft test case A: C p distributions at 4 stations on the wing Table 3 Material properties AGARD445.6 wing E1 3.15106 × 106 E2 4.16218 × 108 G 4.39218 × 108 ρ 381.98 ν 0.31

Pa Pa Pa kg/m3

in the literature is the so called weakened model 3 at zero angle of attack in air. The model was weakened by holes drilled through the surface of the original model to reduce its stiffness. The linear structural model for the AGARD 445.6 was build by SMR. The material properties for the structural model are summarized in Table 3, and were taken from [20]. In this Table E 1 and E 2 are the moduli of elasticity in the longitudinal and lateral directions, G the shear modulus, ν the Poisson ratio and ρ the wing density. Only the first four mode shapes are considered. Inviscid calculations were made using a 31 block grid having about 170,000 cells. Grid refinement studies were made indicating that this mesh was sufficient. Table 4

Aero-elastic Simulations Using the NSMB CFD Solver Including … Table 4 AGARD445.6 wing free stream conditions Mach 0.95 ρ∞ 0.061 α 0.0 p∞ 7000, 4600, 3500 U∞ 381, 309, 269

83

kg/m3 Pa m/s

lists the free stream values used in the calculations. Three different values of the free stream pressure were used, and due to the change in speed of sounds this results in three different free stream velocities and thus to three different values of the flutter speed index V f (respectively 0.383, 0.311 and 0.271). The experimental data [5] shows that at Mach = 0.95 the flutter boundary is around V f = 0.32, hence the highest pressure condition should yield flutter while for the other 2 conditions no flutter should be found. All calculations were made using the central scheme with artificial dissipation. The time integration was carried out using dual time stepping [21] employing the 2nd order implicit backward difference scheme. The LU-SGS scheme was used to converge the equations in the inner loop. The unsteady calculations were started from a steady state calculation. A 2% deflection of the first bending moment was given to the structure, and a ramping procedure was used during the first 25 outer time steps to impose this deflection smoothly. Different outer time steps were studied for the case with a free stream pressure of 7000 Pa, indicating that an outer time step of 10−3 is a good compromise. Using smaller time steps does not change the results significantly, while the results for larger time steps show differences, in particular in the amplitude of the oscillations. Figure 8 show the computed C L as function of time for the different calculations carried out. The left figure shows that flutter is obtained for the highest pressure, as was to be expected. The right figure shows the influence of the CFD-CSM coupling approach for the case with a free stream pressure of 4600 Pa. The weak coupling approach (couple the fluid and structure only each outer time step) leads to undamped oscillations while with the strong coupling approach (couple the fluid and structure each step in the inner time stepping loop) the oscillations are not amplified. With the strong coupling approach it is possible to switch off the CFD-CSM coupling after a certain number of inner loop steps. Performing the coupling only the first 25 steps of the inner time stepping loops yields results close to the results obtained by making the coupling each step.

7 Strut Braced Wing Aircraft In the frame of the European Funded H2020 project AGILE (Aircraft 3rd Generation MDO for Innovative Collaboration of Heterogeneous Teams of Experts) [6] aero-

84

J. B. Vos et al. 0.15 0.1

0.02

P 3500 Pa P 4600 Pa P 7000 Pa

0.01 0.005

0

CL

CL

0.05

0 -0.005

-0.05

-0.01

-0.1 -0.15 0

strong 25 steps weak

0.015

-0.015 0.1

0.2

0.3

time (s)

0.4

0.5

-0.02

0

0.1

0.2

0.3

0.4

0.5

time (s)

Fig. 8 Results AGARD445.6 case, left lift coefficient as function of time for different free stream pressures, right influence of coupling approach on the lift coefficient for a free stream pressure of 4600 Pa

elastic simulations were made for a Strut Braced Wing aircraft. The Strut Braced Wing (SBW) aircraft has received much attention over the last 10 years due to its potential to reduce emissions. However the SBW aircraft also poses several challenges, in particular the strong coupling between aerodynamics and structures [22]. The SBW aircraft designed in the AGILE project concerns a 90-seat passenger aircraft. The design choice’s and the top level aircraft requirements are discussed in the paper by Torrigiani [22]. Navier Stokes simulations were carried out for different wing configurations, showing that a wing with a super-critical airfoil and having a 16◦ sweep angle is the most efficient (compared to the initial configuration having a 0◦ sweep angle and a NACA009 airfoil). However these high-fidelity simulations did not take into account the deformation of the strut and the wing due to the aerodynamic loads. An aero-elastic tool chain has been set-up composed of the PROTEUS tool developed at the Delft University of Technology, the AMLoad tool developed by the Netherlands Aerospace Centre (NLR), the FSCON tool discussed in this paper and the NSMB CFD solver.

7.1 PROTEUS PROTEUS is an aeroelastic tool, developed at the Delft University of Technology. Figure 9 depicts the schematic representation of the framework of PROTEUS. To start with, the wing is discretized into multiple sections in the spanwise direction. For each spanwise section, one or more laminates can be defined in the chord wise direction. Using the laminate properties and the cross sectional geometry, a cross sectional modeler especially developed to deal with anisotropic shell cross-sections, converts the three-dimensional wing into a Timoshenko beam element. A geometrically nonlinear aeroelastic simulation is performed by coupling the nonlinear beam model to a vortex lattice aerodynamic model. A linearized dynamic aeroelastic anal-

Aero-elastic Simulations Using the NSMB CFD Solver Including … Analysis

Input Wing geometry and loadcases

Material properties

Laminate properties

85

Optimisation Lamination parameters and thickness

Post-processing

Optimiser

ABD matrices Cross-sectional geometry

Cross-sectional modeller

Static and dynamic objective and constraints

Cross-sectional strains

Timoshenko cross-sectional stiffness matrix

Wing geometry and loadcases

Nonlinear static aeroelastic analysis Wing deformations and nonlinear stiffness matrix

Wing geometry and loadcases

Static beam strains

Cross-sectional modeller

Dynamic beam strains

Linear dynamic aeroelastic analysis

Fig. 9 Framework of PROTEUS [23]

ysis is carried out around the nonlinear static equilibrium solution. The strains in the three-dimensional wing structure are obtained through the cross sectional modeler. Based on the applied strains, strength, buckling and the fatigue properties of the wing are calculated and fed to the optimizer as constraints. Since analytical derivatives of the objective and constraints with respect to the design variable are calculated with PROTEUS, an efficient gradient based optimizer can be used for optimization. A detailed description of PROTEUS is given the papers by Werter and De Breuker [23, 24].

7.2 Aero-elastic Coupling The aeroelastic chain used for the Strut Braced Wing aircraft in the AGILE project starts with PROTEUS, in which the aeroelastic stiffness and thickness optimization of the wing structure is carried out. The material properties used for optimization are given in Table 5. The optimization problem is shown in Table 6. The objective of the study is to minimize the structural weight of the wing. The wing is divided into 8 sections; 7 sections along the spanwise direction of the main wing and 1 section representing the strut. The sections each have 4 laminates: one laminate each for top and bottom skin, and for front and rear spar. This results in 32 unique laminates. Laminates can be unsymmetric and balanced but are chosen to be symmetric and

86

J. B. Vos et al.

Table 5 Material properties of AS4/3506 carbon/epoxy Property Value (GPa) E 11 E 22 G 12 ν12 Xt Xc Yt Yc S

Table 6 Optimization setup Type Objective Design variables Constraints

147 10.3 7 0.27 2.28 1.72 0.057 0.23 0.076

Parameter

# responses

Minimize wing mass Lamination parameter Laminate thickness Laminate feasibility Static strength Buckling Aeroelastic stability Local angle of attack

1 279 140 384/loadcase 1792/loadcase 10/loadcase 22/loadcase

unbalanced in this study. Every laminate is described by eight lamination parameters and one thickness variable, resulting in a total of 288 design variables. To ensure that lamination parameters represent a realistic ply distribution, feasibility equations formulated by Hammer et al. [25], Raju et al. [26] and Wu et al. [27] are applied. The modified Tsai Wu failure envelope [28] suitable for lamination parameter domain is used to assess the static strength of the laminate. The stability of the panel in buckling is based on idealized buckling model formulated by Dillinger et al. [29]. To guarantee the static and dynamic aeroelastic stability of the wing, the real part of the eigenvalues of the state matrix should be less than zero. The local angle of attack is constrained to a maximum of 12◦ and a minimum of −12◦ . Table 7 gives the information on the loadcases which are used for the current study. These loadcases, represent the flutter boundary, 2.5 g symmetric pull up maneuver and −1 g symmetric push down maneuver. The optimized stiffness and mass matrices are then fed to AMLoad in which a full aircraft MSC Nastran structural model is made in which the wing and strut are represented by matrices.

Aero-elastic Simulations Using the NSMB CFD Solver Including … Table 7 List of loadcases Loadcase ID V (m/s) 1 2 3

264 230 230

87

Altitude (m)

Load factor

11,000 11,000 11,000

1 2.5 −1

7.3 AMLoad Within the framework of the strut-braced wing, AMLoad, in-house developed by the Netherlands Aerospace Centre (NLR), was used both to convert stiffness and mass matrices obtained by PROTEUS to the B2000++ solver and to provide low/midfidelity static aeroelastic solutions. AMLoad is based on the vortex lattice panel aerodynamics, and includes correction factors for airfoil camber and for additional skin-friction drag components. As well known, panel aerodynamics do not take into account viscous effects meaning it cannot predict flow separation due to shocks at higher Mach numbers or high incidence angles. In order to increase the accuracy of the design by including viscous effects, high fidelity FSI simulations have been made using the NSMB CFD solver. To establish the connection within the framework, the output from the modal analysis using AMLoad’s structural model is used including the optimized stiffness and mass matrices from PROTEUS. Using the matrices inherently means that the detailed finite element properties are non-existing anymore. However, in order to perform high fidelity aeroelastic simulations, a 3D structural model is required in order to spline the model to the CFD mesh. Since it is not possible and common to reverse engineer the full detailed finite element model based on only the matrices another solution is proposed. In this solution, the simplified structural MSC Nastran model (existing of nodes in combination with the Direct Matrix Input (DMIG) cards) is extended using Rigid Body Elements (RBE’s). The RBE2 element is a rigid body connected to an arbitrary number of grid points. In this case, the structural nodes which include the structural dynamic matrices are connected to surrounding grid points representing the box structure of the wing (see Fig. 10). The independent degrees of freedom of the surrounding nodes are the six components of motion at a single grid point.

Fig. 10 Modeshapes of the 3D structural dynamic MSC Nastran model including RBE2 elements for splining, (left). First wing bending mode (clamped BC) and (right) first strut bending (clamped BC)

88

J. B. Vos et al.

Fig. 11 Structural bending mode splined to CFD model (left) and structural model (right) Fig. 12 Static aeroelastic solution under conditions; M = 0.78, angle of attack = 10◦ , altitude 11,000 m

These methods are applied for the full wing and strut with an increasing density of structural nodes along the span. This density increase along the span increases the accuracy of the mode shapes used in the high fidelity aeroelastic simulations, see Fig. 11. A restriction of using the rigid elements is the fact that local modes, e.g. local buckling modes or wing torsion at a specified spanwise location, cannot be captured accurately. However, these kind of local modes are not of significant influence for the aeroelastic simulation and therefore do not compromise the results. In addition to the conversion of the structural model, static Aeroelastic solutions have been computed by AMLoad as input to a mission simulation tool (the so called flexible polar). Figure 12 shows the static aeroelastic deformation obtained using the optimized structural model from Proteus. The deformation at the tip is about 95 cm using vortex lattice panel methods.

7.4 Results The geometric coupling tool described in 4 was used to couple the CFD to the FEM model, and the result is shown in Fig. 3. CFD simulations using NSMB were then made for different Mach numbers and different angles of attack. The freestream conditions are summarized in Table 8 and angles of attack considered were −5◦ , −2.5◦ , 0◦ , 2.5◦ , 5◦ and 10◦ . All NSMB calculations used the same setup: the 2nd order Roe scheme for the space discretization, the LU-SGS semi-implicit scheme for the time integration, and the k − ω Menter Shear Stress model to model

Aero-elastic Simulations Using the NSMB CFD Solver Including … Table 8 Strut Braced Wing calculation conditions Altitude 11,000 Mach 0.70, 0.78, 0.88 α −5.0, −2.5, 0.0, 2.5, 5.0, 10.0 p∞ 22,700 T∞ 217

89

m ◦

Pa K

the turbulence. All calculations were made on a grid having 327 blocks and about 4.9 million grid cells. The calculations without fluid structure interaction were run for 8000 steps. The aero-elastic calculations were first run for 3000 steps, and then the aero-elastic coupling was taken into account 10 times every 500 steps. Since the first deformations are in general overestimated under-relaxation was used in the first three coupling steps (the computed displacements were multiplied with factors of 0.5, 0.75 and 0.90 respectively). Figure 13 shows for the Mach = 0.78, α = 10◦ calculation the wing and strut positions at the start, after 5 deformations and at the end of the simulation (10 deformations). One can observe that the difference in wing and strut positions after 5 and 10 deformations is small. Table 9 shows that the wing tip deformation after 10 deformations was 0.578 m, after 5 deformations this was equal to 0.571 m. Figure 14 shows a front view of same case, with the undeformed wing on the left and the deformed wing on the right. One can clearly observe the upward movement of the wing tip for the deformed configuration. Figure 15 shows the wing tip profile, whereby the deformed profile has been translated 0.59 m in z-direction to match the undeformed wing. One can observe a small wing twist and a slightly forward movement of the wing. For several Mach numbers the same calculations were also made using AMLoad to allow a comparison of Low Fidelity (LowFi) and High Fidelity (HighFi) methods. Table 9 summarizes the computed vertical displacement as function of the Mach

Fig. 13 Wing and Strut position, case Mach = 0.78, α = 10◦ : green: undeformed position, blue: after 5 deformations, red: final position

90

J. B. Vos et al.

Table 9 Wing tip maximum vertical displacement in meters, Strut Braced Wing configuration α LowFi HighFi LowFi HighFi LowFi HighFi HighFi M = 0.20 M = 0.20 M = 0.60 M = 0.60 M = 0.78 M = 0.78 M = 0.88 −5.0 −2.5 0.0 2.5 5.0 10.0

−0.036 0.019 0.073 0.128 0.182 0.291

−0.026 0.026 0.079 0.133 0.186 0.291

−0.259 −0.019 0.228 0.472 0.715 1.203

−0.160 0.050 0.267 0.490 0.715 1.038

−0.189 0.026 0.241 0.457 0.672 1.103

−0.153 0.048 0.253 0.457 0.602 0.578

0.042 0.069 0.166 0.346

Z (m)

Fig. 14 Front view of the Strut Braced Wing aircraft, right: undeformed configuration, left: deformed configuration, case Mach = 0.78, α = 10◦ 0.95 0.9 0.85 0.8 0.75 0.7 17.2

Deformed Undeformed 17.4

17.6

17.8

18

18.2

18.4

18.6

18.8

19

X (m)

Fig. 15 Wing tip profile of the undeformed and deformed configuration, case Mach = 0.78, α = 10◦

number and angle of attack. One can observe that at Mach = 0.20 the computed displacements using the low and high fidelity methods are very close, except at α = −5◦ . At Mach = 0.60 and in particular at Mach = 0.78 the displacements predicted using the high fidelity method at an angle of attack of 10◦ are lower than the low fidelity results. It was observed that at this angle of attack the flow on the wing is largely separated, see Fig. 16, and this is not taken into account in the AMLoad simulations. For the Mach = 0.88 case there are shock waves on the wing, followed by a shock induced separation even at low angles of attack and for this reason displacements are smaller than the computed displacements at Mach = 0.78. Figure 17 shows the computed aerodynamic coefficients for the different Mach numbers, using both AMLoad and NSMB with and without aero-elastic deformation of the wing and strut. Accounting for wing deformation reduces slightly the lift coefficient but also leads to an important reduction in the drag coefficient. As a result the aerodynamic efficiency is slightly higher, in particular for the higher incidence angles.

Aero-elastic Simulations Using the NSMB CFD Solver Including …

91

Fig. 16 C p and skin friction lines, case Mach = 0.78, α = 10◦ 2

0.24

M 0.20 M 0.60 M 0.78 M 0.78 M 0.20 - FSI M 0.60 - FSI M 0.78 - FSI M 0.88 - FSI

0.22 0.2

1.5

0.18 0.16

CL

CD

1

0.5

M 0.20 M 0.60 M 0.78 M 0.88 M 0.20 - FSI M 0.60 - FSI M 0.78 - FSI M 0.88 - FSI M 0.20 LowFi M 0.60 LowFi M 0.78 LowFi

0

-0.5

-6

-4

-2

0

2

4

6

8

0.14 0.12 0.1 0.08 0.06 0.04

10

0.02

-6

-4

-2

0

alpha

(a) CL 20 15

CL/CD

10

2

4

alpha

(b) CD M 0.20 M 0.60 M 0.78 M 0.88 M 0.20 - FSI M 0.60 - FSI M 0.78 - FSI M 0.88 - FSI

5 0 -5 -10 -6

-4

-2

0

2 alpha

4

6

8

10

(c) CL/CD

Fig. 17 Computed aerodynamic coefficients with and without wing deformation

6

8

10

92

J. B. Vos et al.

When looking to the drag coefficient of the strut it can be observed that it is comparable to the drag coefficient without deformation at angles of attack below 0◦ , but it is 5–10% lower at higher angles of attack.

8 Conclusions Developments made to the NSMB CFD solver to improve the aero-elastic simulation capability were presented. NSMB was successfully coupled to the B2000++ open-source Finite Element Analysis Environment. A discussion on the geometrical coupling tool to transfer the aerodynamic loads from the CFD mesh to the structure FEM model, and to transfer the displacements from the FEM model back to the CFD grid was presented. The aero-elastic simulation environment was used to simulate the MDO aircraft and the AGARD445.6 wing test cases, and results are in good agreement with results found in literature. Finally the aero-elastic simulation environment was used to compute a Strut Braced Wing configuration. For this case the structural model was build using the PROTEUS software developed at the Delft University of Technology. This structural model was then translated into a Nastran model using the AMLoad software developed at the Netherlands Aerospace Centre NLR. The geometrical coupling tool discussed in this paper was then used to couple the FEM and CFD model. Calculations were made for various Mach numbers. Accounting for aero-elasticity reduced the lift and drag coefficient, but led to an increase in aerodynamic efficiency for higher Mach numbers. It was also observed that taking into account aero-elasticity led to a reduction of the drag of the strut. Comparing low and high fidelity simulation methods shows that low fidelity methods predict C L values close to the high fidelity methods for low Mach numbers; however large differences appear when shock wave and flow separations occur, in particular at high Mach numbers and high incidence angles. Acknowledgements The research on the Strut Braced Wing configuration presented in this paper has been performed in the framework of the AGILE project (Aircraft 3rd Generation MDO for Innovative Collaboration of Heterogeneous Teams of Experts) and has received funding from the European Union Horizon 2020 Programme under grant agreement no. 636202. The Swiss participation in the AGILE project was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number 15.0162. The authors are grateful to the partners of the AGILE consortium for their contribution and feedback.

Aero-elastic Simulations Using the NSMB CFD Solver Including …

93

References 1. Guillaume M., Gehri A., Stephani P., Vos J., & Manadanis G. (2010). Fluid structure interaction simulation on the F/A-18 vertical tail. AIAA-2010-4613, Chicago. 2. Guillaume, M., Gehri, A., Stephani, P., Vos, J., & Mandanis, G. (2011). F/A-18 vertical buffeting calculations using unsteady fluid structure interaction. The Aeronautical Journal, 115(1166), 285–294. 3. http://www.smr.ch/products/b2000/. 4. Haase, W., Selmin, V. and Winzell, B. (2003). Progress in computational fluid-structure interaction. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 81. Springer. 5. Yates, E. C., Jr. (1987). AGARD Standard aero-elastic configurations for dynamic response. I: Wing 445.6. AGARD R-765, 1988. Also published as NASA TM-100492. 6. Ciampa, P. D., & Nagel, B. (2018). AGILE the next generation of collaborative MDO: Achievements and open challenges. AIAA paper, 2018–3249. 7. de C. Henshaw, M. J., Badcock, K. J., Vio, G. A., Allen C. B, Chamberlain, J., Kaynes, I., et al. (2007). Non-linear aeroelastic prediction for aircraft applications. Progress in Aerospace Sciences, 43. 8. Afonso, F., Vale, J., Oliveira, E., Lau, F., & Suleman, A. (2017). A review on non-linear aeroelasticity of high aspect-ratio wings. Progress in Aerospace Sciences, 89, 40–57. 9. Mian, H. H., Wang, G., & Ye, Z.-Y. (2014). Numerical investigation of structural geometric nonlinearity effect in high-aspect-ratio wing using CFD/CSD coupling approach. Journal Fluids Structures, 49, 186–201. 10. Vos, J. B., Rizzi, A. W., Corjon, A., Chaput, E., & Soinne, E. (1998). Recent advances in aerodynamics inside the NSMB (Navier-Stokes Multiblock) consortium. AIAA paper, 98– 0225. 11. Vos, J. B., Sanchi, S., & Gehri, A. (2013). Drag prediction workshop 4 results using different grids including near-field/far-field drag analysis. Journal of Aircraft, 50(5), 1616–1627. 12. Vos, J. B., Bourgoing, A., Soler, J., & Rey, B. (2015). Earth re-entry capsule CFD simulations taking into account surface roughness and mass injection at the wall. International Journal of Aerodynamics, 5(1), 1–33. 13. Hoarau, Y., Pena, D., Vos, J. B., Charbonnier, D., Gehri, A., Braza, M., et al. (2016). Recent developments of the Navier Stokes Multi Block (NSMB) CFD solver. AIAA Paper, 2016–2056. 14. Spalart, P. R., & Allmaras, S. R. (1992). A one-equation turbulence model for aerodynamic flows. AIAA Paper, 92–0439. 15. Menter, F. R. (1993). Zonal two equation k − ω turbulence models for aerodynamic flows. AIAA paper, 93–2906. 16. Langtry, R., & Menter, F. (2009). Correlation-based transition modeling for unstructured parallized computational fluid dynamic codes. AIAA Journal, 47, 2894–2907. 17. Hounjet, M. H. L., & Meijer, J. J. (1995). Evaluation of elastomechanical and aerodynamic data transfer methods for non-planar configurations in computational aeroelastic analysis (pp. 18–19). TP 95690U, National Aerospace Laboratory NLR, Amsterdam, The Netherlands. 18. Beckert, A. (1997). Ein Beitrag zur Strömungs-Struktur-Kopplung für die Berechnung des aeroelastischen Gleichgewichtszustandes, Forschungsbericht-Deutsches Zentrum für Luft und Raumfahrt. 19. Spekreijse, S. P., Prananta, B. B., & Kok, J. C. (2002). A simple , robust and fast algorithm to compute deformations of multi-block structured grids. NLR-TP-2002-105. 20. Goura, G. S. L. (2001). Time marching analysis of flutter using computational fluid dynamics. Ph.D. thesis, University of Glasgow. 21. Jameson, A. (1991, June). Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. AIAA Paper, 91–1596. 22. Torrigiani, F., Bussemaker, J., Ciampa, P. D., Fiorite, M., Tomasella, F., Aigner, B., et al. (2018). Design of the Strut Braced Wing Aircraft in the AGILE collaborative MDO framework. ICAS. 23. Werter, N. P. M., & De Breuker, R. (2016). A novel dynamic aeroelastic framework for aeroelastic tailoring and structural optimisation. Composite Structures, 158, 369–386.

94

J. B. Vos et al.

24. Werter, N. P. M., & De Breuker, R. (2017). Continuous-time state-space unsteady aerodynamic modeling for efficient loads analysis. AIAA Journal, 56(3), 905–916. 25. Hammer, V. B., Bendsøe, M. P., & Pedersen, P. (1997). Parametrization in laminate design for optimal compliance. International Journal of Solids and Structures, 34(4), 415–434. 26. Gangadharan, R., Wu, Z., & Weaver, P. (2014). On further developments of feasible region of lamination parameters for symmetric composite laminates. In 55th AIAA/ASMe/ASCE/AHS/SC structures, structural dynamics, and materials conference. 27. Wu, Z., Gangadharan, R., & Weaver, P. (2015). Framework for the buckling optimization of variable-angle tow composite plates. AIAA Journal, 53(12), 3788–3804. 28. Khani, A., IJsselmuiden, S. T., Abdalla, M. M., & Gürdal, Z. (2011). Design of variable stiffness panels for maximum strength using lamination parameters. Composites Part B: Engineering, 42(3), 546–552. 29. Dillinger, J. K. S., Klimmek, T., Abdalla, M. M., & Gürdal, Z. (2013). Stiffness optimization of composite wings with aeroelastic constraints. Journal of Aircraft, 50(4), 1159–1168.

Semi-Analytical Modeling of Non-stationary Fluid-Structure Interaction Serguei Iakovlev

Abstract This chapter outlines the semi-analytical methodology that was developed over the past decade and a half to model transient fluid-structure interaction phenomena for thin-walled structures submerged in and/or filled with fluid. The theoretical framework of the methodology based on the use of the classical apparatus of mathematical physics is exposed first. Then, a demonstration of some of the capabilities of the methodology is presented as it is applied to an industrially relevant fluid-structure interaction problem. Specifically, the response of a submerged cylindrical shell to a double-front shock wave is considered, with the emphasis on the existence of certain resonance-like phenomena which result in a considerable increase of the maximum stress induced in the structure by such a loading. The outcomes of the modeling using both the 2D and 3D versions of the methodology are presented, and the differences between the results produced by these two approaches, a lower-fidelity one and a higher-fidelity one, are highlighted.

1 Introduction The idea of the methodology that is being discussed here originated in the late 1990s in response to the increasing need of the industry in a fast and reliable tool for the pre-design analysis of thin-walled structures that are in contact with fluids and are subjected to a transient loading. The ability of the tool to handle in an efficient manner extensive parametric studies with many hundreds or even thousands of scenarios was among the most critical requirements. The methodology was partially functional by 2005 [10–12], although some of its important capabilities were not fully developed until 2007 [14, 15]. Since then, the methodology has been continually developed in order to enable modeling of increasingly complex systems [16–27]. Today, it is a fully mature tool for the pre-design S. Iakovlev (B) Department of Engineering Mathematics and Internetworking, Dalhousie University, Halifax, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_6

95

96

S. Iakovlev

analysis of fluid-interacting structural systems with a proven record of successful application to a rather wide variety of industrial systems. As far as the place of the methodology within the contemporary shock-structure interaction research effort is concerned, it continues the well-established tradition [32] of developing reliable analytical frameworks that can be used either as a dedicated analysis tool or as a benchmark for validating fully numerical solutions, e.g. [4, 6–8, 37]. In both of these roles, the methodology provides a very useful complement to the more complex shock-structure interaction studies such as modeling the shock response of structures with advanced material characteristics, e.g. [1, 3, 29, 31, 34– 36, 39, 41, 43], analyzing the response of structures undergoing various complex shock loading scenarios, e.g. [2, 9, 30, 33, 40, 42], and high-fidelity modeling of fluid-interacting structures of a very specific geometry, e.g. [5, 28, 44] - no matter how diverse these and other applied studies that the current research and development needs of the industry motivate are, they can all benefit from using a tool of simplified analysis based on a reliable analytical or semi-analytical solution.

2 Theoretical Framework 2.1 Assumptions The efficiency of the methodology and, in fact, the very possibility of obtaining semianalytical solutions of the respective boundary-value problems come at the expense of introducing a number of simplifying assumptions. They are: • The structure has a classical geometry such as spherical or cylindrical; a wide variety of structures encountered in the industry either exactly or approximately satisfy this requirement; • The behaviour of the structure is linear; • The behaviour of the fluids is linear – they are assumed to be irrotational, inviscid and linearly compressible; this assumption limits the incident loading to acoustic and other pressure pulses and weak shock waves; • The structure has thin or moderately thick walls; this is the least limiting assumption of the model – the vast majority of the structures that are of interest in the present context satisfy this requirement.

2.2 Mathematical Formulation In our studies, we have developed two frameworks, a 2D one and a 3D one. Below, we summarize the 2D version, and the 3D one has all the same features but produces more complex resulting equations, e.g. [27].

Semi-Analytical Modeling of Non-stationary Fluid-Structure …

97

Fig. 1 Geometry of the problem

A circular cylindrical shell of radius r0 and thickness h0 is considered. It is assumed that h0  r0 and that the shell’s displacements are small compared to its thickness. The density of the shell material is ρs , and the sound speed in it is cs . The transverse and normal displacements of the middle surface of the shell are v∗ and w∗ , respectively. The shell is filled with and submerged into linearly compressible, irrotational and inviscid fluid, and it is assumed that the internal and external fluids have different properties, with the ρi and ci being the density of and the sound speed in the internal fluid, and ρe and ce being the respective parameters of the external fluid. The cylindrical coordinate system (ρ, θ ) based on the axis of the shell is employed, Fig. 1. The fluids are governed by the wave equations for the internal and external fluid velocity potentials, φi and φe , respectively, ∇ 2 φi =

1 ∂ 2 φi ci2 ∂τ 2

(1)

∇ 2 φe =

1 ∂ 2 φe . ce2 ∂τ 2

(2)

and

The internal potential has only one component, while the external one is comprised of three components, (3) φe = φ0 + φd + φr , where φ0 is the potential of the incident shock wave, φd is the diffraction potential, and φr is the radiation potential. The corresponding pressure components are pi , p0 , pd , and pr , and the total pressure on the shell surface, ps , is ps = (p0 + pd + pr − pi )|ρ=r0 .

(4)

98

S. Iakovlev

For the structure, any of the linear shell theories can be employed. Here we discuss the most traditional choice, i.e. the equations of the Kirchhoff-Love linear shell theory (e.g. [38]),   1 ∂ 3 w∗ 1 ∂w∗ 1 ∂ 2 v∗ 1 ∂ 2 v∗ 2 = + k0 − 2 + 2 r02 ∂θ 2 r0 ∂θ r02 ∂θ 3 r0 ∂θ 2   1 ∂ 4 w∗ 1 ∗ 1 ∂v∗ 1 ∂ 3 v∗ 2 = χ ps − w − + + k 0 r02 r02 ∂θ r02 ∂θ 4 r02 ∂θ 3

1 ∂ 2 v∗ , cs2 ∂τ 2

(5)

1 ∂ 2 w∗ , cs2 ∂τ 2

(6)

where k02 = h20 /(12r02 ) and χ = (r0 ρe ce2 )/(h0 ρs cs2 ). Although these equations are not always suitable for accurate modeling of all aspects of the interaction (in acoustical applications, for example, the Reissner-Mindlin shell theory is far more suitable, [19, 21, 25, 26]), in great many cases they still answer perfectly. The potential components must satisfy the boundary conditions on the interface,  ∂w∗ ∂φr  , =−  ∂ρ ρ=r0 ∂τ

 ∂φi  ∂w∗ =−  ∂ρ ρ=r0 ∂τ

and

  ∂φd  ∂φ0  = − , (7) ∂ρ ρ=r0 ∂ρ ρ=r0

the decay conditions at the infinity, φd −→ 0 and φr −→ 0 when ρ → ∞,

(8)

the condition at the center of the fluid domain, φi |ρ=0 < ∞, and the zero initial conditions. The shell can be subjected to an arbitrary transient loading, but one of the most typical loadings of practical interest is a shock wave originated at a point located at the distance R0 from the axis of the shell, with an exponential decay of the pressure behind the front. The pressure in such a wave is given by p0 =

pα SR −(τ −ce−1 (R∗ −SR ))λ−1 e H(τ − ce−1 (R∗ − SR )), R∗

(9)

 where R∗ = R20 + ρ 2 − 2R0 ρ cos θ , pα is the pressure in the front of the wave at the instant of its initial contact with the shell, λ is the exponential decay constant, SR = R0 − r0 is the stand-off of the source of the wave, and H is the Heaviside unit step function. We consider a dimensionless formulation of the problem normalizing all variables to r0 , ce , and ρe . A circumflex normally distinguishes a dimensionless variable from its dimensional counterpart, with the exception of the time t = τ ce /r0 , the radial coordinate r = ρ/r0 , and the shell displacements, w = w∗ /r0 and v = v∗ /r0 .

Semi-Analytical Modeling of Non-stationary Fluid-Structure …

99

2.3 Solution The Laplace transform with respect to the time was first applied to the dimensionless wave equations (1) and (2), and then the spatial variables were separated. This, after the relevant boundary conditions were imposed, yielded the general solutions for the potential components as Φˆ nd (r, θ, s) = Bn (s) en (r, s) cos nθ,

and

n = 0, 1, . . . ,

(10)

Φˆ nr (r, θ, s) = sWn (s) en (r, s) cos nθ,

(11)

Φˆ ni (r, θ, s) = −sWn (s) in (r, αs) cos nθ,

(12)

where α = ce /ci , In is the modified Bessel function of the first kind of order n, Kn is the modified Bessel function of the second kind of order n, the expansions  ∂ φˆ 0   ∂r 

= r=1

∞ 

bn (t) cos nθ

and

w=

n=0

∞ 

wn (t) cos nθ

(13)

n=0

are assumed, Φˆ nd , Φˆ nr , Φˆ ni , Wn and Bn are the Laplace transforms of φˆ nd , φˆ nr , φˆ ni , wn and bn , respectively, and en and in are the Laplace transforms of the response functions of the problem, ξne and ξni , given by Kn (rs) sKn (s)

(14)

In (rs) , sIn (s)

(15)

en (r, s) = − and in (r, s) =

where the prime denotes the derivative of the function with respect to its argument. The response functions are at the heart of the present methodology [11, 13, 15]. They do not depend on any of the parameters of the system, only on its geometry (and, therefore, they only need to be computed once). Physically, they represent all the fundamental features of the response of the fluid domains to the motion of and scattering by the structural surface. Their computation is anything but trivial, and it was discussed in much detail in [13] and [15]; several representative response functions are shown in Figs. 2 and 3. The striking difference in their behaviour is due to the very different physics of the phenomena they represent – the highly irregular nature of ξni reflects the multiple reflections and focusings that the internal pressure waves undergo when propagating in the inner fluid domain, in contrast with the very regular nature of ξne which represents the propagation of the scattered and radiated waves away from the shell in the outer, unconfined fluid domain.

100

S. Iakovlev

Fig. 2 Function ξ1i (r, t) for various r

Fig. 3 Function ξ1e (r, t) for various r

Inverting (10)–(12), the pressure components are obtained as

pˆ d =

∞  n=0

where

pˆ nd cos nθ, pˆ r =

∞ 

pˆ nr cos nθ, and pˆ i =

n=0

1 pˆ nd = − √ bn (t) − r

∞ 

pˆ ni cos nθ,

(16)

n=0

 0

t

bn (η)

dξne (r, t − η) dη, dη

(17)

Semi-Analytical Modeling of Non-stationary Fluid-Structure …

 pˆ nr and pˆ ni

t

=− 0

ρi ci = ρe ce

 0

t

d2 wn (η) e ξn (r, t − η) dη, dη2

d2 wn (η) i ξn (r, ci /ce (t − η)) dη. dη2

101

(18)

(19)

For the structural part of the solution, it is assumed that the shell displacements are expressed as v=

∞ 

vn sin nθ and w =

n=0

∞ 

wn cos nθ,

(20)

n=0

which yields, for each n, a system of two ordinary integro-differential equations, γ2

d2 vn + cn11 vn + cn12 wn = 0, dt 2

d2 wn + cn21 vn + cn22 wn dt 2   t 2 d wn (η) e 0 d = χˆ pˆ n + pˆ n − ξn (r, t − η) dη dη2 0   t 2  d wn (η) i ρi ci ξn (r, ci /ce (t − η)) dη  , + 2 ρe ce 0 dη r=1

(21)

γ2

(22)

where cn11 = n2 + k02 n2 , cn12 = cn21 = −n − k02 n3 ,

22 cmn = 1 + k02 n4 , and γ = ce /cs .

Solving these systems accomplishes the coupling between the hydrodynamic and structural parts. Although it is possible to do so in a closed form, it is incomparably more computationally efficient to handle the systems numerically (a simple explicit finite-difference scheme was used, and it performed very satisfactorily, e.g. [16]).

3 Illustration of the Application of the Methodology: Analysis of the Shock Response of a Submerged Cylindrical Shell Subjected to a Double-Front Shock Wave The motivation for this study was two-fold. First, it was prompted by the desire to advance the methodology in question to make it applicable to simulating shockstructure interaction in realistic environments, i.e. when reflective surfaces of various types are present in the proximity of the primary shock-responding structure. The

102

S. Iakovlev

first step in that direction was to understand how the interaction of the classical case of a single, stand-alone structure is altered when the incident shock wave has two or more fronts. Second, since the wave phenomena have been shown to play a dominant role in the dynamics of the stress state of the system in question (e.g. [10, 12, 16, 17]), it was only natural to expect that if the wave nature of the loading itself is more complex than that of the simple single-front loading, the stress state should also exhibit higher complexity, possibly with a higher than usual peak stress. The analysis that was carried out to address these points is summarized below (the details can be found in [20, 23, 27]). As far as the industrial systems that motivated the analysis in question are concerned, we mention such common scenarios as an underwater pipeline or storage tank (oil or otherwise), as well as certain types of heat exchangers. A thin steel shell was considered with h0 = 0.01 m, r0 = 1 m, cs = 5000 m/s, ρs = 7800 kg/m3 , and ν = 0.3, evacuated and submerged in water, ce = 1400 m/s and ρe = 1000 kg/m3 . The incident load was chosen as a sequence of two identical shock waves originated with the delay of Δt at the same source located at the distance of four radii of the shell from the shell’s surface (SR = 4r0 ), with the rate of exponential decay λ and the pressure in the front at the moment of the initial contact with the shell pα taken as 0.1314 ms and 250 kPa, respectively. In order to understand the basic physics of the interaction with such a loading, a 2D version of the problem was considered first. When there is only one incident wave, the stress state exhibits the well-known, regular wave patten, Fig. 4 (the compressive stress is shown as external to the shell, and the tensile as internal). When the second wave was present, however, the structural dynamics of the shell very much depended on when the wave arrived at the shell, and for certain values of Δt , the stress waves circumnavigating the shell superposed constructively, causing a much higher peak stress than in the single-front scenario, Fig. 5 (the stress was scaled in the same way as

Fig. 4 Dynamics of the stress state of the shell (σ22 ) for the single-front scenario

Semi-Analytical Modeling of Non-stationary Fluid-Structure …

103

Fig. 5 Dynamics of the stress state of the shell (σ22 ) for the double-front scenario with Δt = 1.70

for the single-front scenario in order to ensure the possibility of a direct comparison of the two image sequences). In order to understand the full implications of this effect, a parametric study for Δt was required. Such a study was carried out, and its outcomes are presented in Fig. 6 (the stress values corresponding to the 2D case were scaled down by the factor of 1.55 in order to make the 2D and 3D curves directly comparable). We note that the small values of Δt are of no interest in the present context because they simply represent scenarios that are more or less equivalent to the interaction with a single-front wave but with doubled pα . The existence of resonance-like effects in the double-front case is apparent – for certain values of Δt the peak stress in the system can be as high as 150% of the peak stress for the single-front scenario. This means that by simply changing the delay between the two wavefronts, it is possible to very considerably reduce the peak stress experienced by the structure. In practical terms, this means that, when the effects of shock waves are a concern, a very careful study needs to be carried out at the pre-design stage in order to determine the locations with respect to the structure’s immediate surroundings where it should not be placed. As interesting as these findings were, they were for the simplified two-dimensional case. Although they are fully reliable when the shock waves have a very distant source (SR /r0 1, the plane-front wave being the extreme case), before advocating the use of the methodology as a pre-design analysis tool for all relatively distant source locations, it was necessary to determine how the 2D outcomes change when the three-dimensionality of the loading is more pronounced than that of a shock wave with a near-plane front. To that end, a complete 3D modeling was carried out, and Figs. 7 and 8 illustrate the mechanism at work here. Much like in the 2D case, the stress waves originated at the instants of the initial contact of the wavefronts with the shell propagate in both axial and transverse directions, and often the latter waves

104

S. Iakovlev

Fig. 6 Peak stress in the shell subjected to two consecutive shock waves: results of the threedimensional analysis, black triangles, and the scaled results of the two-dimensional analysis, gray triangles; the dashed line marks the peak stress value for the corresponding single-front scenario

superpose constructively, an effect that can lead to the peak stress that exceeds, sometimes considerably, the peak stress seen for the single-front scenario, Fig. 6. Although there are some obvious similarities between the results of the parametric analysis for the 2D and 3D cases, there also are some rather significant differences. The most practically important one is that, unlike in the 2D case, there exists a rather wide range of Δt , 0.6 ≤ Δt ≤ 1.9, where the double-front peak stress exceeds the single-front one by at least 40% (and up to 60%), and the conditions that result in such a loading certainly need to be avoided. The outlined application of the methodology is just one of the many equally interesting problems that we have addressed over the years. However, it was chosen as an illustration of the capabilities of the methodology because it shows how the analysis of a system of industrial interest progressed from a (relatively) simple 2D modeling to higher-fidelity 3D studies. It is also a very suitable case to present here for another reason, namely because at the end, despite all the mathematical complexities of the modeling, the analysis that was carried out resulted in a very specific practical recommendation regarding the undesirable values of one of the parameters of the system; such an outcome is, perhaps, one of the happiest possible conclusions of any relationship between mathematics and engineering.

Semi-Analytical Modeling of Non-stationary Fluid-Structure …

105

(a)

(b)

(c) 5.42

MPa

0

0.68

Fig. 7 Stress state of the shell (σ22 ) for the double-front loading scenario with Δt = 0.50 at t = 0.70 (a), t = 1.10 (b), and t = 1.50 (c); the left column shows the front view of the shell, and the right column shows the back view

106

S. Iakovlev

(a)

(b)

(c)

3.03

MPa

0

1.54

Fig. 8 Stress state of the shell (σ11 ) for the double-front loading scenario with Δt = 0.50 at t = 0.70 (a), t = 1.10 (b), and t = 1.50 (c); the left column shows the front view of the shell, and the right column shows the back view

Semi-Analytical Modeling of Non-stationary Fluid-Structure …

107

4 Conclusions The semi-analytical methodology that was outlined here provides the practitioner with a tool that, due to its semi-analytical nature, is highly computationally efficient and very reliable. Although these benefits come at the expense of certain rather serious limitations, the methodology is very suitable for the use at the pre-design stage where the extent of the required parametric studies often makes the use of the high-fidelity numerical modeling impractical or plainly impossible. The developed methodology has been used extensively for almost 15 years to address many transient fluid-structure interaction problems encountered in industry, and the results have been invariably satisfying. Our aspiration is to continue developing the present approach to enable efficient and reliable simulations of more and more complex systems and/or interaction regimes. Acknowledgements The research program summarized here has been continually supported financially by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the Killam Trusts at Dalhousie University, Canada.

References 1. Batra, R. C., & Hassan, N. M. (2007). Response of fiber reinforced composites to underwater explosive loads. Composites Part B, 38, 448–468. 2. Brett, J. M., & Yiannakopolous, G. (2008). A study of explosive effects in close proximity to a submerged cylinder. International Journal of Impact Engineering, 35, 206–225. 3. Fan, Z., Liu, Y., & Xu, P. (2016). Blast resistance of metallic sandwich panels subjected to proximity underwater explosion. International Journal of Impact Engineering, 93, 128–135. 4. Geers, T. L. (1969). Excitation of an elastic cylindrical shell by a transient acoustic wave. Journal of Applied Mechanics, 36, 459–469. 5. Guo, G., Ji, X., Wen, Y., & Cui, X. (2017). A new shock factor of SWATH catamaran subjected to underwater explosion. Ocean Engineering, 130, 620–628. 6. Huang, H., & Wang, Y. F. (1970). Transient interaction of spherical acoustic waves and a cylindrical elastic shell. The Journal of the Acoustical Society of America, 48, 228–235. 7. Huang, H. (1979). Transient response of two fluid-coupled cylindrical elastic shells to an incident pressure pulse. Journal of Applied Mechanics, 46, 513–518. 8. Huang, H., & Mair, H. (1996). Neoclassical solution of transient interaction of plane acoustic waves with a spherical elastic shell. Shock and Vibration, 3, 85–98. 9. Hsu, C. Y., Liang, C. C., Nguyen, A. T., & Teng, T. L. (2014). A numerical study on the underwater explosion bubble pulsation and the collapse process. Ocean Engineering, 81, 29– 38. 10. Iakovlev, S. (2004). Influence of a rigid co-axial core on the stress-strain state of a submerged fluid-filled circular cylindrical shell subjected to a shock wave. Journal of Fluids and Structures, 19, 957–984. 11. Iakovlev, S. (2006). External shock loading on a submerged fluid-filled cylindrical shell. Journal of Fluids and Structures, 22, 997–1028. 12. Iakovlev, S. (2007). Submerged fluid-filled cylindrical shell subjected to a shock wave: Fluidstructure interaction effects. Journal of Fluids and Structures, 23, 117–142. 13. Iakovlev, S. (2007). Inverse Laplace transforms encountered in hyperbolic problems of nonstationary fluid-structure interaction. Canadian Mathematical Bulletin, 50, 547–566.

108

S. Iakovlev

14. Iakovlev, S. (2008). Interaction between a submerged evacuated cylindrical shell and a shock wave. Part I: Diffraction-radiation analysis. Journal of Fluids and Structures, 24, 1077–1097. 15. Iakovlev, S. (2008). Interaction between a submerged evacuated cylindrical shell and a shock wave. Part II: Numerical aspects of the solution. Journal of Fluids and Structures, 24, 1098– 1119. 16. Iakovlev, S. (2009). Interaction between an external shock wave and a cylindrical shell filled with and submerged into different fluids. Journal of Sound and Vibration, 322, 401–437. 17. Iakovlev, S., Gaudet, J., & MacDonald, B. (2010). Hydrodynamic fields induced by the shock response of a fluid-filled submerged cylindrical shell containing a rigid co-axial core. Journal of Sound and Vibration, 329, 3359–3381. 18. Iakovlev, S., Dooley, G., Williston, K., & Gaudet, J. (2011). Evolution of the reflection and focusing patterns and stress states in two-fluid cylindrical shell systems subjected to an external shock wave. Journal of Sound and Vibration, 330, 6254–6276. 19. Iakovlev, S., Santos, H. A. F. A., Willison, K., Murray, R., & Mitchell, M. (2013). Nonstationary radiation by a cylindrical shell: Numerical modeling using the Reissner-Mindlin theory. Journal of Fluids and Structures, 36, 50–69. 20. Iakovlev, S., Seaton, C. T., & Sigrist, J. F. (2013). Submerged circular cylindrical shell subjected to two consecutive shock waves: Resonance-like phenomena. Journal of Fluids and Structures, 42, 70–87. 21. Iakovlev, S., Santos, H. A. F. A., Lefieux, A., Schulman, B., & Williston, K. (2014). Transient radiation by a submerged fluid-filled cylindrical shell. Journal of Fluids and Structures, 50, 79–104. 22. Iakovlev, S. (2014). On the possibility of shock-induced cavitation in submerged cylindrical shell systems. Journal of Fluids and Structures, 50, 437–460. 23. Iakovlev, S. (2014). Resonance-like phenomena in a submerged cylindrical shell subjected to two consecutive shock waves: The effect of the inner fluid. Journal of Fluids and Structures, 50, 153–170. 24. Iakovlev, S., Mitchell, M., Lefieux, A., & Murray, R. (2014). Shock response of a two-fluid cylindrical shell system containing a rigid core. Journal of Computational Fluid, 96, 215–225. 25. Iakovlev, S., Pyke, D., & Santos, H. A. F. A. (2016). Semi-analytical technique for isolating the pseudo-Rayleigh component of the field induced by a transiently responding submerged cylindrical shell. Journal of Fluids and Structures, 65, 21–29. 26. Iakovlev, S., Lefieux, A., Huntemann, B., Santos, H. A. F. A., & Pyke, D. (2017). On the observability of the pseudo-Rayleigh waves on submerged cylindrical shells. Journal of Fluids and Structures, 69, 108–120. 27. Iakovlev, S. (2018). Structural analysis of a submerged cylindrical shell subjected to two consecutive spherical shock waves. Journal of Fluids and Structures, 76, 506–526. 28. Kim, J. H., & Shin, H. C. (2008). Application of the ALE technique for underwater explosion analysis of a submarine liquefied oxygen tank. Ocean Engineering, 35, 812–822. 29. Kim, C. H., & Shin, Y. (2013). Numerical simulation of surface shield effects to waterblast wave. Ocean Engineering, 60, 99–114. 30. Li, J., & Rong, J. L. (2012). Experimental and numerical investigation of the dynamic response of structures subjected to underwater explosion. European Journal of Mechanics-B/Fluids, 32, 59–69. 31. Liu, G., Liu, J., Wang, J., Pan, G., & Mao, H. (2017). A numerical method for double-plated structure completely filled with liquid subjected to underwater explosion. Marine Structures, 53, 164–180. 32. Mair, H. U. (1999). Benchmarks for submerged structure response to underwater explosion. Shock and Vibration, 6, 169–181. 33. Ming, F. R., Zhang, A. M., Xue, Y. Z., & Wang, S. P. (2016). Damage characteristics of ship structures subjected to shockwaves of underwater contact explosions. Ocean Engineering, 117, 359–382. 34. Nian, W., Subramaniam, K. V. L., & Andreopoulos, Y. (2012). Dynamic compaction of foam under blast loading considering fluid-structure interaction effects. International Journal of Impact Engineering, 50, 29–39.

Semi-Analytical Modeling of Non-stationary Fluid-Structure …

109

35. Panahi, B., Ghavanloo, E., & Daneshmand, F. (2011). Transient response of a submerged cylindrical foam core sandwich panel subjected to shock response. Materials & Design, 32, 2611–2620. 36. Schiffer, A., & Tagarielli, V. L. (2014). The one-dimensional response of a water-filled double hull to underwater blast: Experiments and Simulations. International Journal of Impact Engineering, 63, 177–187. 37. Sprague, M., & Geers, T. L. (1999). Response of empty and fluid-filled, submerged spherical shells to plane and spherical, step-exponential acoustic waves. Shock and Vibration, 6, 147–157. 38. Ugural, A. C. (1981). Stresses in plates and shells. New York: McGraw-Hill Book Company. 39. Wang, Z., Liang, X., Fallah, A. S., Liu, G., Louca, L. A., & Wang, L. (2013). A novel efficient method to evaluate the dynamic response of laminated plates subjected to underwater shock. Journal of Sound and Vibration, 332, 5618–5634. 40. Wang, H., Zhu, X., Cheng, Y. S., & Liu, J. (2014). Experimental and numerical investigation of ship structure subjected to close-in underwater shock wave and following gas bubble pulse. Marine Structures, 39, 90–117. 41. Xiao, F., Chen, Y., Wang, Y., Hua, H., & Zhu, D. (2014). Experimental research on the dynamic response of floating structures with coatings subjected to underwater explosion. Shock and Vibration, 2014, 705256. 42. Xie, W. F., Young, Y. L., & Liu, T. G. (2008). Multiphase modeling of dynamic fluid-structure interaction during close-in explosion. International Journal for Numerical Methods in Engineering, 74, 1019–1043. 43. Yin, C., Jin, Z., Chen, Y., & Hua, H. (2016). Shock mitigation effects of cellular cladding on submersible hull subjected to deep underwater explosion. Ocean Engineering, 117, 221–237. 44. Zhang, N., Zong, Z., & Zhang, W. (2014). Dynamic response of a surface ship structure subjected to an underwater explosion bubble. Marine Structures, 35, 26–44.

Fluid Structure Modelling of Ground Excited Vibrations by Mesh Morphing and Modal Superposition A. Martinez-Pascual, Marco Evangelos Biancolini and J. Ortega-Casanova

Abstract This paper presents a numerical approach for high fidelity modelling of ground excited vibrations of a structure interacting with surrounding fluid flow. The motion of the structure is represented directly on the CFD model mesh by embedding the structural modes using radial basis functions mesh morphing. Modal forces integrals are computed on the CFD mesh enabling a time marching FSI solution based on the weak approach. Ground vibration is represented by adding a rigid movement and related inertial loads using modal participation factors. The approach is validated by studying a cantilever beam vibrating in air excited by a transversal sine motion applied to the clamped end that is relevant for the design of flapping devices. Numerical results are successfully validated by comparing the coupled and uncoupled response computed according to the proposed approach with the analytic one and to a standard FEA solver.

1 Introduction The development of multiphysics computational techniques has broadened our comprehension of phenomena when interactions between different physical models occur. Fluid-Structure Interaction (FSI) is one technique which combines Computational Fluid Dynamics (CFD) and structural analysis and has a wide range of application in fields such as aerospace and biomedical engineering. Modelling flow past a heart valve or a boat sail and wing flutter are examples of this. FSI studies allow for the optimization of mechanical properties of solids/structures in order to exhibit A. Martinez-Pascual · J. Ortega-Casanova (B) Escuela de Ingenierías Industriales, Universidad de Málaga, Malaga, Spain e-mail: [email protected] A. Martinez-Pascual e-mail: [email protected] M. E. Biancolini University of Rome “Tor Vergata”, Rome, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_7

111

112

A. Martinez-Pascual et al.

a desired behaviour under certain flow conditions. Numerous approaches have been developed to compute FSI and they are classified according to the level of coupling between the governing equations of the structure and fluid systems. The monolithic approach combines the governing equations for fluid and structure in a single system simultaneously solving both component systems. The partitioned approach uses separate solvers for fluid and solid with an additional coupling scheme. Coupling is achieved by imposing dynamic conditions on the fluid-solid interface. When using a monolithic approach, the coupling is automatically satisfied at the expense of increasing the size of the system. The increase in computational effort required to solve this larger problem usually discards this approach for many industrial applications. Using a partitioned approach allows for different solution algorithms to be used in each solver, including reduced-order models (ROMs) that reduce the complexity of the problem and require less computational resources. Using weak coupling reduces the temporal accuracy at least one order with respect to the fluid and structure time integrators, and further restricts the stability limit for low-density-ratio problems [1]. In the present paper, the modal approach is used to obtain the deformation of a structure with ground induced motion. Structural dynamics is computed integrating the motion of modal coordinates of retained modes over the time; modal forces are computed integrating fluid forces, including both pressure and shear loads, over the wetted surfaces. Using modal superposition we are able to model the plate’s deformation, which is represented in the CFD mesh using Radial Basis Functions (RBFs) mesh morphing. The algorithm for mesh morphing based on RBFs has shown to be robust and efficient [2]. This FSI approach was implemented using the CFD ANSYS Fluent solver linked with the RBF Morph tool. Complex engineering problems such as vehicle shape optimization [3–5], sails trim optimization [6], pressure mapping [7], icing [8] and FEM results improvement [9] have been successfully and efficiently tackled with this approach. The applications of this FSI method has been intensively tested in the RBF4AERO (www.rbf4aero.eu) and RIBES (www.ribesproject.eu) projects. Calculating structural deformations via modal superposition is computationally efficient and provides an appropriate method for transient FSI problems which require mesh update at each time step. Using this approach, shorter computation times are required to solve challenging engineering problems, as demonstrated for the transient evolution of wing vibration after store separation [10] and in [11] where the vibrational modes of a hydrofoil where evaluated underwater. However, it is not valid for problems involving materials with nonlinear parameters, contact, or pre-stressed components. Strong coupling FSI should be used in this case. Fluid forces over the structure surface together with inertial loads are computed as modal forces and determine the amplitude for each modal shape [12]. The latter are superposed to obtain the deformation at each instant. This allows for a transient FSI solution to be obtained based on the weak approach. This approach allows for a great reduction in computational resources when linear theory can be applied, or assumed within a reasonable error. It has previously been used by Hoover et al. [13] to study swimming performance of 3-D heaving flexible plates.

Fluid Structure Modelling of Ground Excited Vibrations …

113

Many approaches have been used to comprehend the complex aerodynamic mechanisms involved in lift production during flapping flight. Rigid 2-D models have been used extensively [14–16] to understand the basic flow pattern around a heaving plate. These models have shown to be capable of capturing the basic vortex structure and three lift enhancing mechanisms: delayed stall [17], rotational circulation during stroke reversal [18, 19] and wake capture [20]. Although these models shed light on the main mechanisms, they are not complex enough to grasp all the phenomena involved in flapping flight. Therefore, understanding the role of flexibility in lift production seems vital in order to understand the underlying mechanisms of efficient flapping flight. Optimal rigidity leads to a higher efficiency at the expense of reduced lift. Insect wings show varied morphologies and structural properties according to their size and flight adaptations. They present anisotropic rigidity as they are composed of veins, fibres and membranes that stiffen the wing along the spanwise direction and allow deformation on the chord-wise direction [21–23]. Different approaches have been followed to model wing’s deformation and assess flexible flapping flight in terms of the aerodynamic forces and efficiency. Wing kinematics and deformation for dragonflies [24] and hoverflies [25] have been captured using high speed photogrammetry and latter imposed on the CFD mesh. Others authors followed the partitioned approach and studied anisotropic wings using loose coupling [26]. Hua et al. [27] studied how the locomotion of a flexible plate is influenced by the heaving amplitude and bending rigidity by means of the immersed-boundary method. Baudille and Biancolini [28, 29] proven how the numerical modelling of flexible beams interacting with a surrounding fluid can be achieved by embedding the structural finite element analysis (FEA) modes into the CFD solver Fluent. The Newmark scheme is in this case adopted and information are exchanged in a weak scheme at the end of each time step. In the present study, we analyse how a plate subject to a cyclic displacement of its support is deformed due to the variable inertial and fluid forces and how this affects its aerodynamic performance. For this specific flow condition we noticed that the movement of the plate is influenced more by the inertial loads than the fluid ones. This makes it possible to capture the structural dynamics by replacing fluid forces with an equivalent damping load. In Sect. 2 an overview of the mathematical formulation used is presented. We describe the formulation required for a modal analysis and give an introduction to the basic features of RBFs. In Sect. 3 the FSI problem is described, defining governing equations of the fluid region and establishing dimensionless parameters used to characterize structural features. In Sect. 4 we describe the procedure followed to perform the FSI analysis by embedding the modes on the CFD solver. In Sect. 5 we validate the capability of RBF Morph to model ground induced motion in order to use it to model the desired problem. In Sect. 6 we show the comparison between a coupled FSI response and an uncoupled but structurally damped response in order to determine the damping coefficient that causes the latter response mimic the first response with minimum error. In Sect. 7 the conclusion are summarised.

114

A. Martinez-Pascual et al.

2 Theoretical Background In this section we describe the equations that determine the dynamic response of a structure modelled by modal superposition. The mesh is morphed after each time step to account for the deformation and the corresponding node displacements are calculated by means of RBFs.

2.1 Formulation of Modal Theory The governing equations that determine the response of a linear system to an imposed acceleration α(t) in a direction D [30], with external loads FFSI caused by the FSI, in finite element form, is defined as follows: ¨ MX(t) + KX(t) = − MDα(t) + FFSI ,

(1)

where X(t) represents the evolution of the degrees of freedom of the system and M and K denote, respectively, the system mass, and stiffness matrices. Solving (1) directly, as in the approach by Baudille and Biancolini [28, 29], can be computationally inefficient, thus we transform the vector basis in order to obtain a set of ordinary differential equations. Instead, the following eigenvalue problem is solved: (2) (−ωn2 M + K)un = 0, where un is the eigenvector of the eigenvalue ωn2 . The subspace of eigenvectors form a basis U such that X(t) = Uq(t), (3) being q(t) the modal coordinates of vector X(t). The matrices M and K are symmetric and positive definite. After a convenient normalization that imposes a unit modal mass for each n-th mode un we obtain un T Mun = m n = 1,

un T Kun = ωn2 ,

(4)

The system response can be interpreted as a superposition of n single degree of freedom systems of unitary mass and stiffness ωn2 , oscillating with frequency f n :  1 kn ωn = fn = . 2π 2π m n

(5)

Substituting the vector decomposition of (3) into (1) and multiplying by UT we obtain the following set of single degree of freedom systems:

Fluid Structure Modelling of Ground Excited Vibrations …

q¨n (t) + ωn2 qn (t) = Fn ,

115

(6)

where Fn is the modal force acting on the n-th mode. To obtain this force we integrate the external loads over the structure and evaluate it in the new basis using un such that (7) Fn = un T (− MDα(t) + FFSI ). The linear static solution of the problem is approximated as ωn2 qn = Fn .

(8)

When the structural problem cannot be considered static, the modal coordinates are obtained from the following expression [31]:   q˙0 + ζ ωn q0 sin(ωd t) q(t) = e−ζ ωn t q0 cos(ωd t) + ωd    t 1 eb(t−τ )/2m n f (τ )sin[ωd (t − τ )]dτ , + e−ζ ωn t m n ωd 0

(9)

where q0 is the modal coordinate at the beginning of the time step, ζ is the damping ratio, m n is the modal mass of mode n (that is equal to 1.0 if mass normalization is enforced) and ωd is the corresponding damped frequency of ωn defined as  ωd = ωn 1 − ζ 2 .

(10)

The integral on (9) is known as Duhamel integral and states that the response of a linear system which is subject to an input force f (τ ) can be obtained by summing the differential response accumulated during the corresponding time interval. This formulation can only be applied when the force is assumed constant at each individual time step. In this case the values of q0 and q˙0 from the previous iteration are used in the following time step to obtain the updated values. The formulation used to obtain the modal coordinate can therefore be expressed as   q˙0 + ζ ωn q0 q0 cos(ωd Δt) + q(t + Δt) = e sin(ωd Δt) ωd 

 4ωd Fn (t) −ζ ωn Δt 2ζ ωn sin(ωd Δt) + 4ωd cos(ωd Δt) − e + . ωd ζ 2 ωn2 + 4ωd2 ζ 2 ωn2 + 4ωd2 (11) The modal force Fn (t) is comprised of two contributions: the fluid force FFSIn , that is computed on the CFD mesh by projecting the pressure and shear loads on the modal shape (detailed in Sect. 4), and the inertial force, that is obtained from the acceleration of the given motion and the modal participation factor Πn as −ζ ωn Δt

Fn (t) = −Πn α(t) + FFSIn .

(12)

116

A. Martinez-Pascual et al.

2.2 Radial Basis Functions RBF Morph is a mesh morpher, used to modify the mesh while conserving the topology (number and type of element), allowing small shape modifications and avoiding remeshing. This tool is based on RBFs, a method that consists on using a system of radial functions to produce a solution for a mesh modification from a set of source points and their displacements [32, 33]. RBFs are a powerful mathematical interpolation tool for n-dimensional space scattered data which allow to interpolate at any arbitrary point a scalar function defined at discrete points while ensuring its exact value at the original points. The interpolation function used consist of a RBF Φ, globally or compactly supported, and a multivariate polynomial corrector vector h and is defined as: s(x) =

js 

γ i Φ ( ||x − xki || ) + h(x),

(13)

i=1

where x is the position vector of a node, xki is the ith source node position vector, js is the total number of source points and || • || is the Euclidean norm. The polynomial h and the coefficient vector γ i are chosen according to the radial function used in order for the interpolant function to coincide with the specified values at the source points. A detailed description on RBFs and its applications is given in Biancolini [34]. As a conclusion, once the solution is known and shared in the memory of each calculation node of the cluster, each partition has the ability to smooth its nodes without taking care of what happens outside the partition because the smoother is a global point function, and the continuity at interfaces is implicitly guaranteed.

3 Problem Definition In this section we describe the dimensionless parameters that define the fluid problem and use them to define the FSI conditions. The rigidity of the structure, density ratios and heaving motion parameters are used to estimate the size of the deformation.

3.1 Formulation of the Problem A flat, 2-D, rectangular plate with chordwise length c, thickness e and uniform mechanical properties is immersed in the unsteady and incompressible flow of an incoming uniform current with constant speed U . The plate’s material is defined

Fluid Structure Modelling of Ground Excited Vibrations …

117

by its density ρs , Young’s modulus E and Poisson ratio ν. During the motion, the plate will deform due to inertial and aerodynamic forces. On the leading edge of the plate a sinusoidal heaving motion with amplitude h 0 and frequency f is imposed perpendicular to the current and is defined as: h = h 0 sin( 2 π f t),

(14)

where t represents time. Using the chord c and the free stream speed U as the reference length and velocity, respectively, the non-dimensional Navier-Stokes equations governing the incompressible flow can be written as ∇ · v = 0,

(15)

1 2 ∂v + v · ∇v = −∇ p + ∇ v, ∂t Re

(16)

while the boundary conditions to solve are: |x| → ∞, v → e x ,

p → 0,

S(x, t) = 0, v = 2 π St a cos(2 π St c t) e y ,

(17) (18)

where v is the non-dimensional velocity, p the non-dimensional relative pressure (scaled with ρ f U 2 , where ρ f is the fluid density), e x and e y are unit vectors which are parallel and perpendicular to free stream velocity, respectively, and S(x, t) = 0 define the non-dimensional position of the leading edge of the plate. To define the previous equations, the following dimensionless parameters have been used: Re =

ρf U c h0 f c f , Sta = , Stc = . μ U U

(19)

The Reynolds number is based on chord length c (μ is the fluid viscosity), while sinusoidal motion of the leading edge of the plate is characterized by two Strouhal numbers, one based on the amplitude h 0 and the other based on chord length c (Fig. 1). The fluid exerts pressure and viscous forces along the solid surface, which in non-dimensional form (scaled with 21 ρ f U 2 c) can be expressed as  F = −2 S

2 p n ds + Re





∧ n ds,

(20)

S

where n is the outward unit vector normal to the surface and = ∇ ∧ v is the non-dimensional vorticity.

118

A. Martinez-Pascual et al.

Fig. 1 Schematic of the problem. Original (dashed) and deformed (solid) shapes

3.2 Coupling Parametric Space In order to understand which phenomena is dominating the deformation, an estimation of the contribution of each fluid and inertial forces is required. Olivier and Dumas [35] estimated pressure (PLoads ) and inertial loads (ILoads ) during heaving motion for a cantilever beam in order to obtain dimensionless indicators of tip displacement: PLoads ∼ ρ f (2 π f h 0 )2 ,

(21)

ILoads ∼ (2 π f )2 h 0 .

(22)

From these approximations, dimensionless flexibilities with respect to pressure loads δ P and inertial loads δ I can be obtained and their numerical value is representative of the order of magnitude of the trailing edge deflection: δP =

ρ f (h 0 f )2 c3 , E Ic

(23)

δI =

ρs e h 0 f 2 c3 , E Ic

(24)

where Ic = I /b represents the area moment of inertia I over the span b. For a rectangular plate, Ic = e3 /12. Note that δ I is purely a structural parameter while δ P is representative of the fluid-solid interaction. To estimate the strength of the FSI, we define the pressure-to-inertia ratio Σ as the ratio between (23) and (24): Σ=

ρ f h0 δP = . δI ρs e

(25)

Fluid Structure Modelling of Ground Excited Vibrations …

119

4 Method Structural modes are embedded in the CFD model by creating a parametric mesh that models the deforming structure. The mesh update is performed directly on the CFD solver, saving considerable computation time. The workflow followed to perform FSI simulations by embedding structural modes has three steps: • Obtain natural shapes, frequencies and mass participation factors of the deformable structure from a FEA, see Fig. 2. • Calculate a RBF solution for each mode by displacing the nodes according to the modal shapes. • Load the RBF solutions at the beginning of the CFD analysis to generate a parametric mesh formulation that models flexible components. With this procedure we create as many RBF solutions as modes considered.For each solution, the nodal displacements of the mesh are saved and available for the following CFD computation. Using k modes, the parametric mesh is obtained defining the position of morphed nodes (XCFD ) as the sum of the original position in the undeformed mesh (XCFD0 ) plus a certain displacement according to XCFD = XCFD0 +

k 

qn uCFDn ,

(26)

n=1

where qn are the values of the modal coordinates and uCFDn are the previously computed modal displacements of the nth mode. The modal forces Fn determine the amplitude which each mode contribute to the total deformation. Modal forces corresponding to fluid and inertial loads are calculated at the beginning of each time step. Fluid forces are obtained as the summation of the dot product between the nodal mode displacement and nodal forces FCFD of each ith node of the j nodes of the surface or volume:

Fig. 2 First four modes of vibration for a rectangular plate

120

A. Martinez-Pascual et al.

FFSIn =

j 

u CFDni · FCFDi .

(27)

i=1

When a problem can be considered static, using (8) we can express the parametric mesh formulation as a function of loads according to the expression: XCFD = XCFD0 +

k  FFSI n=1

ωn2

n

uCFDn .

(28)

On the other hand, in transient simulations we should account for inertial terms. At each time step the mesh is updated according to (26). To update the modal coordinates we use (11), where modal loads are considered constant during each time step. The coupling for solid and fluid is computed sequentially only once per time step. The effect of ground vibration is modelled by applying a transient rigid movement to the whole structure and accounting for the inertial effect as a contribution to the modal forces, as detailed in (12). Fluent was configured to solve the transient laminar flow using a fixed time step with a pressure-based solver, coupled pressure-velocity scheme and second order discretization for pressure and momentum terms. A square domain of 20 × 20 chord lengths with 21,816 elements was used. Symmetry conditions were applied on top and bottom boundaries, left boundary set as velocity inlet and right boundary set as pressure outlet. The plate was initially positioned as shown in Fig. 3. Mesh depiction is shown in Fig. 4.

Fig. 3 Schematic of the non-dimensional computational domain

Fluid Structure Modelling of Ground Excited Vibrations …

121

Fig. 4 Mesh with different zoom levels

5 Validation In order to justify the procedure used in the present paper, it is worth to validate the capability of RBF Morph to model ground induced vibration. This vibration is added in the simulation with an extra term in each modal force (12), which is the product of the modal participation factor and the ground acceleration. To do so, we computed the modal forces on a heaving plate with no fluid force exchanged1 using RBF Morph and compared with the analytical and FEA results. RBF Morph was run on Ansys Fluent, the analytic solution was obtained with Matlab and the reference numerical solution was obtained with the NX Nastran solver. The plate’s length-to-thickness ratio is c/e = 50 and the motion is defined by the dimensionless parameters shown in Table 1, where Stc and Sta values are close to the optimal region for flight efficiency [36]. Clamped support was imposed on the leading edge to prevent the plate from pitching, while the top, bottom and trailing edge surfaces were free. The results for mode 1 are shown in Fig. 5, where it is depicted the temporal evolution of the normalized modal coordinate q1∗ obtained analytically, with FEA 1 The software allows to select wetted surfaces where surface integral are performed, in this case an

empty set is assigned.

122

A. Martinez-Pascual et al.

Table 1 Governing dimensionless parameters Parameter Re Sta Stc

Value 500 0.14 0.68

Fig. 5 Modal coordinate obtained from the analytical, FEM and RBF Morph solutions. The three methods produced identical results

and with RBF Morph. The period T = 1/ f is used to define dimensionless time t ∗ = t/T . As can be seen, it is clear that the three procedures coincide. To obtain qn∗ for mode n, the modal coordinate qn is scaled with the static modal response qn S : qn , qn S

(29)

(2 π f )2 h 0 Πn , ωn2

(30)

qn∗ = qn S =

where Πn is the participation factor of mode n which represents the contribution of this vibration mode. With the present sinusoidal motion, the amplitude A of the acceleration signal is (31) A = (2π f )2 h 0 . Deformations are due to inertial loads since fluid forces are neglected for the validation. RBF Morph has proven to be capable of modelling structural deformations caused by ground induced motion and previously proved to be capable of modelling

Fluid Structure Modelling of Ground Excited Vibrations …

123

FSI problems [3–6, 32]. As both inertial and fluid forces are computed as modal loads, the FSI study of a structure with ground induced motion can be assessed by means of RBF Morph.

6 Results and Discussion In this section we show how the proposed coupled approach allows to model the physical behaviour of an heaving flexible plate excited by a ground vibration. Work is done by the plate during the motion due to the pressure forces on the surface. Energy is hence transferred to the fluid and later dissipated in vortexes. This results in reduced oscillations since the pressure forces oppose the plate’s displacement and a modified temporal behaviour due to the unsteady phenomena. For the specific flow conditions herein investigated the effect of inertial terms dominates the problem. The interaction with fluid is small and can be represented by an equivalent viscous modal force. The unsteady phenomena dealt with in this problem are related with the vortex separation. The plate’s tip oscillations modifies the vortex pattern (see Fig. 6) and, consequently, alters the temporal pressure distribution along the plate’s surface and the instantaneous modal forces (see Fig. 7). It is worth to notice that with an appropriate damping coefficient ζ we are able to shift the uncoupled response, which does not account for fluid forces, towards

Fig. 6 Vorticity contours computed by the coupled approach

Fig. 7 Pressure contours computed by the coupled approach

124

A. Martinez-Pascual et al.

Fig. 8 Modal coordinates obtained for the uncoupled-undamped (—), uncoupled-damped (- -) and coupled-undamped ( ) cases

the coupled response. A comparison between the damped and the coupled response is shown in Fig. 8. For the present case, the conditions of the problem are defined by Σ = 5.52 × 10−3 , δ P = 1.92 × 10−4 and the dimensionless parameters given in Table 1. In order to obtain the most similar response, the mean error ε of the modal coordinates was obtained for each damping ratio as ε=

1 ti − t0

 t0

ti

|qd∗ (t) − qc∗ (t)| dt,

(32)

where qd∗ (t) and qc∗ (t) represent the normalized modal coordinate for the damped cases (dashed lines in Fig. 8) and the coupled case (red line in Fig. 8). This approach allows to decouple the FSI problem, since it is possible to compute the deformations without accounting for the fluid forces. An appropriate value of ζ should be chosen according to δ F , since this parameter is an indicator of the contribution of fluid forces to the deformation. Under the present conditions, the minimum error is obtained for ζ = 1.8%, as shown in Fig. 9.

7 Conclusions The approach to handle FSI simulations continues to be a matter of debate. The balance between the accuracy achieved by the coupling method and its computational cost will determine which method is more appropriated for each FSI problem. We have set-up a fast FSI method based on modes embedding capable to account also for

Fluid Structure Modelling of Ground Excited Vibrations … Fig. 9 Modal coordinate mean error against damping ratio

125

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15

0

0.01

0.02

0.03

0.04

ground motion. After validating the ability to represent such ground motion within the RBF Morph add-on of Fluent for the uncoupled vibration case (i.e. neglecting fluid forces), we explored how such method performs on the coupled case. The case of the heaving plate was successfully faced and it was possible to compute the fluid pattern released in the wake in a reasonable way. The validation of the full coupled study will be the subject of future studies. At the end of this study we have also verified that for the specific set of parameters herein investigated we noticed a small coupling. The motion of the flapping device is mainly influenced by the inertial loads, with a small contribution of fluid forces which can be estimated adopting an equivalent damping. Acknowledgements This research has been partially supported by the Ministerio de Economía y Competitividad of Spain Grants No. DPI2016-76151-C2-1-R and by the European Union within the RIBES project of the 7th Framework aeronautics programme JTI-CS-GRA (Joint Technology Initiatives-Clean Sky-Green Regional Aircraft) under Grant 632556.

References 1. Piperno, S., & Farhat, C. (2001). Partitioned procedures for the transient solution of coupled aeroelastic problems—Part II: Energy transfer analysis and three-dimensional applications. Computer Methods in Applied Mechanics and Engineering, 190(24), 3147–3170. https://doi. org/10.1016/S0045-7825(00)00386-8. 2. Keye, S. (2009). Fluid-structure-coupled analysis of a transport aircraft and comparison to flight data. In: 39th AIAA Fluid Dynamics Conference, , San Antonio, TX, USA 22–25 June 2009. https://doi.org/10.2514/6.2009-4198. 3. Biancolini, M. E., Cella, U., Groth, C., & Genta, M. (2016). Static aeroelastic analysis of an aircraft wind-tunnel model by means of modal RBF mesh updating. Journal of Aerospace Engineering, 29(6), 04016061. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000627. 4. Cella, U., & Biancolini, M. E. (2012). Aeroelastic analysis of aircraft wind-tunnel model coupling structural and fluid dynamic codes. AIAA Journal of Aircraft, 49(2), 407–414. https:// doi.org/10.2514/1.C031293.

126

A. Martinez-Pascual et al.

5. Papoutsis-Kiachagias, E. M., Andreja˘si˘c, M., Porziani, S., Groth, C., Er˘zen, D., Biancolini, M. E., Costa, E., & Giannakoglou, K. C. (2016). Combining an RBF-based morpher with continuous adjoint for low-speed aeronautical optimization applications. In: ECCOMAS Congress, Crete Island, Greece, 5–10 June 2016. https://doi.org/10.7712/100016.2270.15521. 6. Biancolini, M. E., Viola, I. M., & Riotte, M. (2014). Sails trim optimisation using CFD and RBF mesh morphing. Computers & Fluids, 93, 46–60. https://doi.org/10.1016/j.compfluid. 2014.01.007. 7. Biancolini, M. E., Chiappa, A., Giorgetti, F., Groth, C., Cella, U., & Salvini, P. (2018). A balanced load mapping method based on radial basis functions and fuzzy sets. International Journal for Numerical Methods in Engineering, 115(12), 1411–1429. https://doi.org/10.1002/ nme.5850. 8. Groth, C., Costa, E., & Biancolini, M. E. (2019). RBF-based mesh morphing approach to perform icing simulations in the aviation sector. Aircraft Engineering and Aerospace Technology, 91(4), 620–633. https://doi.org/10.1108/AEAT-07-2018-0178. 9. Chiappa, A., Groth, C., & Biancolini, M. E. (2019). Improvement of 2D finite element analysis stress results by radial basis functions and balance equations. International Journal of Mechanics, 13, 90–99. 10. Reina, G., Della-Sala, A., Biancolini, M. E., Groth, C., & Caridi, D.: Store separation: Theoretical investigation of wing aeroelastic response. In Aircraft Structural Design Conference, Belfast, Northern Ireland, 7–9 October 2014. 11. Di Domenico, N., Groth, C., Wade, A., Berg, T., & Biancolini, M. E. (2018). Fluid structure interaction analysis: Vortex shedding induced vibrations. Procedia Structural Integrity, 8, 422–432. https://doi.org/10.1016/j.prostr.2017.12.042. 12. Groth, C., Cella, U., Costa, E., & Biancolini, M. E. (2019). Fast high fidelity CFD/CSM fluid structure interaction using RBF mesh morphing and modal superposition method. Aircraft Engineering and Aerospace Technology journal, 91(6), 893–904. https://doi.org/10.1108/AEAT09-2018-0246. 13. Hoover, A. P., Cortez, R., Tytell, E. D., & Fauci, L. J. (2018). Swimming performance, resonance and shape evolution in heaving flexible panels. Journal of Fluid Mechanics, 847, 386–416. https://doi.org/10.1017/jfm.2018.305. 14. Martín-Alcántara, A., Fernández-Feria, R., & Sanmiguel-Rojas, E. (2015). Vortex flow structures and interactions for the optimum thrust efficiency of a heaving airfoil at different mean angles of attack. Physics of Fluids, 27(7), 073602. https://doi.org/10.1063/1.4926622. 15. Wang, Z. J. (2000). Vortex shedding and frequency selection in flapping flight. Journal of Fluid Mechanics, 410, 323–341. https://doi.org/10.1017/S0022112099008071. 16. Wang, Z. J., Birch, J. M., & Dickinson, M. H. (2004). Unsteady forces and flows in low Reynolds number hovering flight: Two-dimensional computations vs robotic wing experiments. Journal of Experimental Biology, 207(3), 449–460. https://doi.org/10.1242/jeb.00739. 17. van den Berg, C., & Ellington, C. P. (1997). The three-dimensional leading-edge vortex of a hovering model hawkmoth. Philosophical Transactions of the Royal Society B. Biological Sciences, 352, 329–340. https://doi.org/10.1098/rstb.1997.0024. 18. Harbig, R. R., Sheridan, J., & Thompson, M. C. (2013). Relationship between aerodynamic forces, flow structures and wing camber for rotating insect wing planforms. Journal of Fluid Mechanics, 730, 52–75. https://doi.org/10.1017/jfm.2013.335. 19. Sane, S. P., & Dickinson, M. H. (2002). The aerodynamic effects of wing rotation and a revised quasi-steady model of flapping flight. Journal of Experimental Biology, 205(8), 1087–1096. 20. Birch, J. M., & Dickinson, M. H. (2003). The influence of wing–wake interactions on the production of aerodynamic forces in flapping flight. Journal of Experimental Biology, 206(13), 2257–2272. https://doi.org/10.1242/jeb.00381. 21. Combes, S. A. (2010). Materials, structure, and dynamics of insect wings as bioinspiration for MAVs. In Encyclopedia of Aerospace Engineering, Chap. 3. American Cancer Society. https://doi.org/10.1002/9780470686652.eae404 22. Combes, S. A., & Daniel, T. L. (2003). Flexural stiffness in insect wings I. Scaling and the influence of wing venation. Journal of Experimental Biology, 206(17), 2979–2987. https:// doi.org/10.1242/jeb.00523.

Fluid Structure Modelling of Ground Excited Vibrations …

127

23. Combes, S. A., & Daniel, T. L. (2003). Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending. Journal of Experimental Biology, 206(17), 2989–2997. https:// doi.org/10.1242/jeb.00524. 24. Koehler, C., Wischgoll, T., Dong, H., & Gaston, Z. (2011). Vortex visualization in ultra low Reynolds number insect flight. IEEE Transactions on Visualization and Computer Graphics, 17(12), 2071–2079. https://doi.org/10.1109/TVCG.2011.260. 25. Du, G., & Sun, M. (2010). Effects of wing deformation on aerodynamic forces in hovering hoverflies. Journal of Experimental Biology, 213(13), 2273–2283. https://doi.org/10.1242/jeb. 040295. 26. Nakata, T., & Liu, H. (2012). A fluid-structure interaction model of insect flight with flexible wings. Journal of Computational Physics, 231(4), 1822–1847. https://doi.org/10.1016/j.jcp. 2011.11.005. 27. Hua, R. N., Zhu, L., & Lu, X. Y. (2013). Locomotion of a flapping flexible plate. Physics of Fluids, 25(12), 121901. https://doi.org/10.1063/1.4832857. 28. Baudille, R., & Biancolini, M. E. (2015, January). FSI makes FLUENT more flexible. Fluent News, 14. 29. Baudille, R., & Biancolini, M. E. (2008). A general approach for studying the motion of a cantilever beam interacting with a 2D fluid flow. Interaction and Multiscale Mechanics, 1(4), 449–465. https://doi.org/10.12989/imm.2008.1.4.449. 30. Sigrist, J. F., & Garreau, S. (2007). Dynamic analysis of fluid-structure interaction problems with modal methods using pressure-based fluid finite elements. Finite Element in Design and Analysis, 43(4), 287–300. https://doi.org/10.1016/j.finel.2006.10.002. 31. Meirovitch, L. (2001). Fundamentals of vibration. New York: McGraw-Hill. 32. de Boer, A., van der Schoot, M. S., & Bijl, H. (2007). Mesh deformation based on radial basis function interpolation. Computers and Structures, 85(11–14), 794–795. https://doi.org/ 10.1016/j.compstruc.2007.01.013. 33. Lodha, S. K., & Franke, R. (2002). Scattered data interpolation: Radial basis and other methods. In G. Farin, J. Hoschek, & M.-S. Kim (Eds.), Handbook of computer aided geometric design (pp. 389–401). North Holland: Elsevier. 34. Biancolini, M. E. (2017). Fast radial basis functions for engineering applications. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-75011-8. 35. Olivier, M., & Dumas, G. (2016). A parametric investigation of the propulsion of 2D chordwiseflexible flapping wings at low Reynolds number using numerical simulations. Journal of Fluids and Structures, 63, 210–237. https://doi.org/10.1016/j.jfluidstructs.2016.03.010. 36. Ortega-Casanova, J., & Fernandez-Feria, R. (2016). Analysis of the aerodynamic interaction between two plunging plates in tandem at low Reynolds number for maximum propulsive efficiency. Journal of Fluids and Structures, 63, 351–373. https://doi.org/10.1016/j.jfluidstructs. 2016.03.011.

Unsteady FSI Analysis of a Square Array of Tubes in Water Crossflow Emiliano Costa, Corrado Groth, Jacques Lavedrine, Domenico Caridi, Gaëtan Dupain and Marco Evangelos Biancolini

Abstract The present paper is addressed to the numerical analysis of fluid-structure instabilities in a flexible tubes bundle subjected to the loads induced by a water turbulent crossflow, using the arrangement presented in Weaver and Abd-Rabbo (J Fluids Eng, 1985 [1]) as benchmark. The physical phenomena involved by the water turbulent crossflow raise strong interest from the scientific community. The nuclear industry is particularly concerned as the design of reliable large-scale exchangers is of primary importance to ensure good performance of nuclear plants. As a matter of fact, their detailed simulation is characterised by challenging traits such as the large amplitude of the tubes vibrations, the strong coupling between water and tubes, the need for an accurate evaluation of the fluid damping and critical flow velocity which vibration instabilities arise at, as well as the complex transition of the fluid-structure behaviour. To tackle these challenges in an effective way, unsteady Fluid-Structure Interaction (FSI) studies were performed applying the mode-superposition approach by means of a mesh morphing technique founded on the mathematical framework of Radial Basis Functions (RBF). In particular, the computational outputs were gained by E. Costa (B) RINA Consulting, Rome, Italy e-mail: [email protected] C. Groth · M. E. Biancolini University of Rome “Tor Vergata”, Rome, Italy e-mail: [email protected] M. E. Biancolini e-mail: [email protected] J. Lavedrine ANSYS France, Montigny-le-Bretonneux, France e-mail: [email protected] D. Caridi ANSYS Italia Srl, Milan, Italy e-mail: [email protected] G. Dupain INSA Rouen Intern at ANSYS, Rouen, France e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_8

129

130

E. Costa et al.

employing a combined use of ANSYS® Fluent® , ANSYS® Mechanical™ and RBF Morph™ software. The two-equation realizable κ-ε turbulence model was adopted to run the U-RANS simulations on high-fidelity structured hexahedral meshes. The achieved numerical results were compared with well-documented experimental data, and a satisfying agreement was finally attained. Furthermore, the operative crossflow velocity guaranteeing the stable functioning of the tubes array was also identified. We demonstrated that the proposed modal approach, in combination with mesh morphing, allows designers to set-up an effective workflow to predict unsteady FSI problems that can be widely adopted for industrial applications under the hypothesis of linear structural behaviour.

List of symbols β γ ϕ ω C d, D ds F(t) g h K M Mint Q ns P q s Vu Vc xk xp {x} {x} ˙ {x} ¨ y+

Coefficients of the polynomial correction Vector of coefficient of the RBF Radial basis function Circular natural frequency (rad/s) Damping matrix Diameter of tubes Diameter of the steel rods Vector of externally applied forces Vector of displacement at source points Multi-variate polynomial Stiffness matrix of the system Mass matrix of the system Interpolation matrix Vector of modal forces Total number of source points (RBF centres) Constraint matrix Vector of modal coordinates Interpolant function Upstream inlet flow velocity (m/s) Critical or threshold flow velocity based on Vu (m/s) Position of RBF source points Position of a generic node Displacement vector Velocity vector Acceleration vector Dimensionless wall distance

Unsteady FSI Analysis of a Square Array of Tubes in Water …

131

Abbreviations CAE CAD CFD FEA FEM FSI GS IMQ IQ LSCB MQ RBF RMS Rn SC SST SAS TPSn U-RANS

Computer-Aided Engineering Computer-Aided Design Computational Fluid Dynamics Finite Element Analysis Finite Element Method Fluid-Structure Interaction Gaussian Inverse multiquadric Inverse quadratic Least Squares Cell Based Multi-Quadric Radial Basis Functions Root Mean Square Spline type System Coupling Shear-Stress Transport Scale-Adaptive Simulation Thin plate spline Unsteady Reynolds Averaged Navier Stokes

1 Introduction Nowadays, heat exchangers have become almost indispensable in several industries. In the nuclear and petrochemical sectors, for instance, many exchangers are arranged in the form of tubes networks. The possible configurations they can be grouped are two and are typically referred to as in-line, namely with a placement of the tubes in rows, and staggered, namely in the respect of a staggered placement. Such arrangements are illustrated in Fig. 1 on the left and right side respectively, through the temperature [T(°C)] contours plotted over a transversal (horizontal) plane cutting the simulation volume [2]. In the images of this figure x and y respectively are the streamwise direction, namely the one associated with the direction of flow, and the transverse direction, that is the one associated with the direction normal to the flow. The exchangers designed according to in-line arrangement enjoy the advantage to guarantee an improved heat transfer with respect to staggered ones but, to be effective, they need the flow to arrive transversally and to have a sufficient high rate [3]. In fact, for identical operating conditions the in-line arrangement of tube-fin head exchangers has heat transfer and pressure drop lower, of the order of 70%, than the values of staggered arrangement [4]. Though, they have the disadvantage of degrading the tubes over time due to vibrations induced by the crossflow that may cause not only erosion or fatigue, but also collisions between adjacent tubes.

132

E. Costa et al.

Fig. 1 In-line (left) and staggered (right) arrangement (from Herchang et al. [2])

A large number of works studying the interaction between cylinders and fluid flow are available in literature, and lots of models and theories were proposed during time to estimate the critical velocity at which the fluid-elastic instability occurs for tube heat exchangers. Interesting scientific reviews were written on such topics by Chen [5, 6], Gelbe et al. [7], Paidoussis [8], Weaver and Fitzpatrick [9] and Yeung [10] between others. Notables works based on experimental investigations include the ones of Chen [11], Connors [12], Andjelic [13], Pettigrew and Taylor [14], Lever and Weaver [15], while numerical investigations relying on fluid-structure simulations were carriedout by Ji et al. [16] and Duan et al. [17]. With particular reference to the nuclear sector, one of the main concerns is the mid-span leaks due tube-to-tube clashing which is typically caused by the excessive vibrations in tube banks which are induced by cross flow. Given such damage, the present work aims at simulating the hydro-elastic behaviour of an in-line configuration in order to determine under what conditions the shocks between the tubes are likely to occur and, so, to understand whether the exchanger is likely to deteriorate rapidly. To confirm the effectiveness and reliability of the proposed numerical FSI approach, whose workflow already demonstrated very good performances in handling steady analyses [18–20] and unsteady analyses [21], the experimental research described by Weaver and Abd-Rabbo [1] concerning a transient study was employed as reference.

2 Structure of the Document In the successive paragraphs the experimental set-up, the background on RBF mesh morphing, as well as the strategy and the workflow of the modal superposition method to handle FSI studies are respectively described. Then the two stages that were

Unsteady FSI Analysis of a Square Array of Tubes in Water …

133

performed in sequence to accomplish the FSI studies are described. In particular, in the first stage a single (isolated) tube was considered with a two-fold purpose: firstly to assess and properly improve the RBF solution set-up and, secondly, to compare the gained FSI outputs with those calculated using the standard two-way approach implemented in ANSYS® Workbench™ R15.0. In the second stage of the study twelve tubes were finally taken into account exploiting the best practices and the experience established in the first stage.

3 Experimental Set-up The experimental bench taken as reference is shown in Fig. 2. In specific, the left image of Fig. 2 allows to identify the tubes considered to behave as perfectly rigid, visualised through filled circles, and as flexible (deformable), depicted by means of empty circles. The right image of Fig. 2 depicts, from a side view, the main dimensions of flexible tubes, namely those on which FSI is applied and simulated, and how they were fixed to the experimental bench basement. With regards to experimental set-up [22], the tubes are made of acrylic and are supported by steel rods, whilst the lower extremity of steel tubes is fixed through a rigid support. Specifically, the length of acrylic tubes is equal to 295 mm while their diameter is d = 25.4 mm, the length of steel rods is equal to 105 mm, the diameter of steel (tubes) rods is ds = 6 mm, the vertical space between the top tubes surfaces and the upper surface of the experimental bench is equal to 5 mm and the gap between the centres of two adjacent tubes is 38.1 mm, both horizontally and vertically. Relating to the physical assumptions and properties for simulation, the fluid used was water regarded as Newtonian and incompressible, and flowing in an unsteady

Fig. 2 Experimental bench (from Weaver and Abd-Rabbo [1])

134 Table 1 Physical properties of materials and fluids used in the present study [23]

E. Costa et al. Physical property

Value

Water Density (kg m−3 ) Dynamic viscosity (kg

998.2 m−1

s−1 )

0.001003

Acrylic Density (kg m−3 )

1190

Young’s modulus (MPa)

2389

Poisson’s ratio

0.4

Steel Density (kg m−3 )

7850

Young’s modulus (MPa)

200

Poisson’s ratio

0.3

fashion in the x direction (see Fig. 2). The adopted values for the physical properties of all the materials included in the study are summarized in Table 1. It is worth to specify that for steel and water the values present by default in ANSYS® Workbench™ and ANSYS® Fluent® were adopted because they correspond to the values found in the literature [23, 24], whereas for acrylic properties, not listed in the ANSYS database, the used values were consistent with the data recovered in the literature [25]. With regard to the fluid flow since the fluid inlet velocity ranges from 0.32 to 0.9 m/s, imposing the tube diameter as reference length, the Reynolds number was estimated to range about from 8000 to 23,000. Given that, the flow behaviour experienced by the first column of tubes was expected to be “subcritical”, namely characterized by the wake completely turbulent and a laminar boundary separation [26]. As laminar-to-turbulent transition was likely to occur in the separated shear layer generated by the first column of cylinders, the downstream flow was likely to be mostly turbulent. Therefore, a turbulence model, hereinafter detailed, was used.

4 Background on RBF Mesh Morphing and Modal Superposition FSI Method A system of RBF is used to produce a solution for mesh morphing, once a set of source points and their displacements is defined. This approach is valid both for surfaces shape changes and volume mesh smoothing. RBF are a very powerful tool created for the interpolation of scattered data [27]. As a matter of fact, they are able to interpolate everywhere within the space a function that is defined at discrete points giving the exact value at original points. The interpolation quality, as well as its behaviour between points, depends on the kind of adopted basis.

Unsteady FSI Analysis of a Square Array of Tubes in Water … Table 2 Typical radial basis functions

135

RBF with global support

ϕ(r )

Rn

r n , n odd

TPSn

r n log(r ), n even √ 1 + r2

MQ

√ 1 1+r 2 1 1+r 2 2 −r e

IMQ IQ GS RBF with compact support Wendland

C0

(C0)

Wendland C2 (C2) Wendland C4 (C4)

ϕ(r ) = f (ξ ), ξ ≤ 1, ξ = (1 − ξ )

r R

2

(1 − ξ )4 (4ξ + 1)   2 (1 − ξ )6 35 3 ξ + 6ξ + 1

RBF can be classified on the basis of the type of support (global or compact) they have, meaning the domain where the chosen RBF is non zero-valued [28]. Typical RBF functions are shown in Table 2. RBF are scalar functions with the scalar variable r which, in the case of mesh morphing, can be assumed to be the Euclidean norm of the distance between two points defined in a three-dimensional space. In some cases, a polynomial corrector is added to guarantee the existence of an RBF fit. A linear system, of order equal to the number of source point introduced [29], needs to be solved for coefficients calculation. Operatively, once the RBF system coefficients have been calculated, the displacement of an arbitrary node of the mesh, either inside (interpolation) or outside (extrapolation) the domain, can be expressed as the sum of the radial contribution of each source point (if the point falls inside the influence domain). In such a way, a desired modification of the mesh nodes position (smoothing) can be rapidly applied preserving mesh topology. An interpolation function s composed by a radial basis ϕ and the aforementioned polynomial h of order m−1, where m is said to be the order of ϕ, is defined as follows if n S is the total number of contributing source points. ns        s xp = γi ϕ x p − xki  + h x p

(1)

i=1

and/or volIn Eq. (1) x p is the position of a generic node belonging  to the surface  ume mesh, xki is the i-th source node position vector and x p − xki  is the Euclidean normalized distance between two points. The RBF fitting solution consists in the evaluation of the coefficients vector γ and of the weights of the polynomial corrector β so that, at source points, the interpolant function possesses the specified (known) values of displacement. The degree of the polynomial has to be chosen depending on the kind of RBF adopted. A radial basis fit exists if the coefficients γ and the weight of the polynomial β can be found such that the desired function values are obtained at source points

136

E. Costa et al.

and the polynomial terms give zero contributions at source points, that is:   s xki = gi , 1 ≤ i ≤ C N    γi p xki = 0

(2)

i=1

for all polynomials p with a degree less or equal than that of polynomial h. The minimal degree of polynomial h depends on the choice of the RBF. A unique interpolator exists if the basis function is a conditionally positive definite function. If the basis functions are conditionally positive definite of order m ≤ 2 [30] a linear polynomial can be used:   h x p = β1 + β2 x + β3 y + β4 z

(3)

The subsequent exposition assumes that the aforementioned hypothesis is valid. A consequence of using a linear polynomial is that rigid body translations are exactly recovered. The values for the coefficients γ of RBF and the coefficients β of the linear polynomial can be obtained by solving the system:

Mint P PT 0



γ g = β 0

(4)

where g are the known values at the source points. Mint is the interpolation matrix defined calculating all the radial interactions between source points:   Mint = ϕ xki − xk j  , 1 ≤ i ≤ C, 1 ≤ j ≤ C

(5)

and P is a constraint matrix that arises imposing the orthogonality conditions: ⎛

1 ⎜1 ⎜ P =⎜. ⎝ ..

xk1 yk1 xk2 yk2 .. .. . . 1 xk N yk N

z k1 z k2 .. .

⎞ ⎟ ⎟ ⎟ ⎠

(6)

zkN

Radial basis interpolation works for scalar fields. For the smoothing problem, each component of the displacement field prescribed at the source points is interpolated as follows: ⎧ ns      ⎪ ⎪ γix ϕ x p − xki  + β1x + β2x x + β3x y + β4x z ⎪ sx x p = ⎪ ⎪ ⎪ ⎨   i=1 ns   y  y y y y γi ϕ x p − xki  + β1 + β2 x + β3 y + β4 z sy x p = (7) ⎪ i=1 ⎪ ⎪    ns  ⎪  ⎪ ⎪ γiz ϕ x p − xki  + β1z + β2z x + β3z y + β4z z. ⎩ sz x p = i=1

Unsteady FSI Analysis of a Square Array of Tubes in Water …

137

Radial basis method has several advantages that makes it very attractive in the area of mesh smoothing. The key point is that, being a meshless method, only grid points are moved regardless of element connection and it is suitable for parallel implementation. In fact, once the solution is known and shared in the memory of each calculation node of the cluster, each partition has the ability to smooth its nodes without taking care of what happens outside because the smoother is a global point function and the continuity at interfaces is implicitly guaranteed. Furthermore, despite its meshless nature, the method is able to exactly prescribe known deformations onto the surface mesh: this effect is achieved by using all the mesh nodes as RBF centres with prescribed displacements, including the zero field to guarantee that a surface is left untouched by the morphing action. In the present study we use mesh morphing to deform the Computational Fluid Dynamics (CFD) mesh according to the modal shapes computed by Finite Element Analysis (FEA). This allows to tackle the generic dynamic problem: [M]{x} ¨ + [C]{x} ˙ + [K ]{x} = {F(t)}

(8)

where M is the mass matrix, C the damping matric, K the stiffness matrix and F(t) the force vector. Exploiting the orthogonality of modal shapes, displacements can be written: {x} = [X ]{q}

(9)

in which [X ] is a matrix whose column are the eigenvectors normalized with the mass and {q} is the vector of modal coordinates. Equation (9) can be then written as: ˙ + [X ]T [K ][X ]{q} = {Q(t)} [I ]{q} ¨ + [X ]T [C][X ]{q}

(10)

in which {Q(t)} is the modal force vector and the mass normalization of modal shapes was exploited to obtain the identity matrix in Eq. (10). The modal force can be computed integrating the projection of nodal fluid forces on wetted surfaces onto the relevant mode. For dynamic systems the modal coordinate {q} can be obtained, using the Duhamel integral, as [31]:

q˙0 + ζ ωn q0 sin(ωd t) q(t) = eζ ωn t q0 cos(ωd t) + ωd t −b(t−τ ) 1 + e 2m f (τ )sin(ωd (t − τ ))dx mωd 0

(11)

138

E. Costa et al.

Fig. 3 Diagram of the general method for setting up the FSI with RBF Morph™

5 Numerical Strategy and Used Numerical Means The workflow of the proposed FSI strategy carried out in the Computer-Aided Engineering (CAE) multi-physics platform ANSYS® Workbench™ is shown in Fig. 3. Basically it can be thought being divided in two parts: a structural part in which a FEM model computes the natural modes and frequencies (vibrational characteristics) of the deformable structure to be accounted, and the CFD part to run the FSI analysis. In fact, when the FSI environment is enabled and initialized in ANSYS® Fluent® , the CFD case becomes elastic and the shape of the deformable structure(s), subjected to the flow loads, is (are) updated on-the-fly by mesh morphing using the RBF solutions reproducing the actual nodal displacements. It is worth to specify that the vibrational characteristics could be also provided by analytical models or experimental data for instance.

6 Isolated Tube Case From the operation point of view, it is worth to highlight that managing FSI studies in ANSYS® Workbench™ was quite efficient. In fact, not only such a working framework allowed us to manage both the Computer-Aided Design (CAD) and the mesh of the computational cases (CFD and FEM) through the same environment,

Unsteady FSI Analysis of a Square Array of Tubes in Water …

139

but guaranteed at the same time that the position and units of the numerical models were consistent (requirement for the correct RBF problem set-up).

6.1 CAD and FEM Case Set-up The CAD model to carry out the modal analysis of a single tube tool included both the tube itself and the steel rod. The computational mesh of the FEM model, needed to obtain the vibration characteristics of the flexible tube, was obtained extruding the surface unstructured mesh of such components. Such a model is shown in Fig. 4 The ANSYS® Mechanical® tool was employed to perform the modal analysis. As far as materials properties are concerned, the values collected in Table 1 were employed. Relating to boundary conditions, since in the experimental framework the rode was screwed into a heavy steel plate, a “fixed support” condition was assigned to the base of the steel rod while a null rotation around the vertical axis was assigned to the top of the tube leaving free the other two components of rotation. The natural modes shapes normalized with respect to masses and their corresponding frequencies for the first six modes, are shown in Fig. 5 and reported in Table 3 respectively. Each mode is associated with oscillations in the streamwise

Fig. 4 Mesh of the FEM model to perform the modal analysis

Table 3 Modal frequencies

Mode 1

Frequency (Hz) 18.03

2

18.03

3

167.39

4

167.39

5

482.43

6

482.43

140

E. Costa et al.

Fig. 5 Modal shapes

(I, III, V) and transverse direction (II, IV, VI) so there are two values for the same natural frequency. As expected, bending modes have algebraic multiplicity 2 due to the perfect symmetry of the circular cross sections. After performing the modal analysis, the nodal displacements of the external surface mesh of the tube for each mode were saved in a format readable by the morpher tool. Such data were used for setting-up the RBF solution reproducing the structural modes to be applied during the CFD computing through mesh morphing as detailed in the following section.

7 RBF Solutions Set-up The main operations that are typically needed to set up a consistent RBF solution foresee, at least, the assignment of the source nodes displacement and the definition of a delimiting volume to appropriately restrict the action of morphing. The source points, generated importing the data of FEM nodal displacements, that is global Cartesian coordinates and modal displacement vector components, assume a certain value in time dictated by the data obtained processing the modal forces during the FSI computing. Apart from the delimiting volume, which was box-shaped in the isolated tube case, a further RBF feature was needed to solve a challenging problem due to the proximity of the top surface of the tube to the boundaries of the simulation volume (ceiling). In fact, considering this geometrical aspect together with the displacement extent the tube was expected to experience when subjected to fluid loads, the computational mesh elements would be highly distorted by morphing imposing ceiling nodes as

Unsteady FSI Analysis of a Square Array of Tubes in Water …

141

Fig. 6 Detail of the mesh over a vertical cutting-plane: baseline, without and with adopting shadow area management

RBF fixed source points. The option of leaving ceiling nodes free to be deformed was considered but, even if the horizontal movement was properly accommodated and the mesh not distorted, the planarity of the ceiling was slightly violated. This effect can be appreciated in Fig. 6 that depicts the mesh around the top surfaces of the tube through a vertical cutting plane. In particular, the first image shows the grid arrangement in the baseline (undistorted) configuration, whereas the second image the one it takes after the application of the morphing of the second bending mode leaving the ceiling nodes free to be deformed (to ease the comprehension the deformation has been suitably amplified with 0.01 that corresponds to a displacement equal to half the diameter). As it can be noticed, if the ceiling is not controlled directly by employing source points on its surface, the planarity is lost following the deformation of the cylinder. Such unwanted issue was resolved adopting, for each mode, the two-step procedure of the morpher tool, according to which a first solution was designed to accurately control the surface mesh displacement while a second solution, fed by the first, was mainly intended to control the volume mesh smoothing and quality. Adopting such a technique, the first step RBF solution set-up included the source nodes generated from FEM vibrational data and, furthermore, imposed the respect of the planarity of the ceiling. This latter condition, in specific, was managed assigning to the nodes of the shadow area, namely the portion of the ceiling linked to the top surface of the tube through the structured mesh, the same displacement of the nodes of the top surface of the tube but with a null value of the vertical component. The second step of the RBF solution imposed the surface displacements ruled by the first step RBF solution and delimited the action of morphing in a box-shaped volume properly defined to lower the time requested to apply morphing and, moreover, the mesh quality deterioration with the use of the adequate radial basis and source points density. The effectiveness of the whole RBF set-up can be straightforwardly understood considering the third image of Fig. 6 in which the same morphing action and amplification of the second column one was applied. The deformations obtained amplifying the first three mass normalized modal shapes with a modal coordinate (Eq. 9) equal to 0.002 are respectively depicted in Fig. 7 to show the effect of morphing along the computational domain.

142

E. Costa et al.

Fig. 7 Preview of the morphed configuration of the bending modes

Once all RBF solutions have been verified, that is the cell quality remains satisfactory till reaching the configuration when two tubes were about to clash, it was possible to go ahead with the FSI study set-up.

7.1 CAD and CFD Case Set-up The CAD model of the isolated case simulation volume was generated using ANSYS® DesignModeler™ according to the dimensions associated with the experimental device already detailed. In specific, such geometry was separated in several zones so as to facilitate the generation of a structured mesh that guarantees the reliable control of the mesh resolution as well as a high quality of the generated cells. That framework, shown in Fig. 8 from different perspectives with dimensions expressed in function of the tube diameter D, was characterized by O-grid structure above the cylinder positioned in the middle of the transversal axis in order to better control the mesh structure and the quality of the cells as well. The distance between the tube and the outlet is of about ten times D. This latter dimension was selected according to the type of flow expected (see §3). As a matter of fact, in this portion of the domain, devoted to the wake of the cylinder, was expected to experience, at a given time, the formation and the development of several vortices and recirculation areas at both sides of the cylinder. A three-dimensional mesh composed of about 145,000 hexahedrons was generated through ANSYS® Meshing™ taking particular care in the resolution of the area around the cylinder and the cylinder wake. A height of 0.591 mm was imposed for the first cell inflated from the lateral surface of the cylinder. Such a value was set

Unsteady FSI Analysis of a Square Array of Tubes in Water …

143

Fig. 8 CAD model of the isolated tube case

to assure, in the case the maxim value of the inlet velocity to be considered in the FSI study and with the realizable k–ε turbulence model, a y+ greater than 30, namely located in the fully turbulent flow part of the law of the wall. Relating to boundary conditions, a velocity-inlet (with a zero gauge pressureoutlet with the same turbulence intensity in case of reverse flow) was set to inlet, a pressure-outlet was set to outlet, whilst wall condition without with friction was imposed for lateral, bottom and top surfaces and for the cylinder surfaces. As far as solution schemes are concerned, the SIMPLEC method was adopted for the pressure-velocity coupling, the Standard for Pressure and Second Order Upwind for Momentum and turbulence parameters. As the inclination of the cells is likely increase with the deformation of the cylinder, the Least Squares Cell Based (LSCB) was employed to discretise the Gradient.

7.2 Results of the Single Tube Case To preliminary assess, through comparison, the outputs obtainable by means of the mode-superposition approach and to verify the consistency of the RBF set-up, the isolated tube case was also run with the ANSYS System Coupling tool, whose computing, differently from the mode-superposition one, is able to include the nonlinear

144

E. Costa et al.

effects. The main settings in terms of employed models and solution schemes (e.g. time-steps, iterations per timestep, etc.) were the same for both approaches, whilst the dynamic mesh smoothing option was enabled to update the mesh when the ANSYS System Coupling was run. Figures 9 and 10 compare the data obtained by mode-superposition (RBF) and the System Coupling (SC) recording of the streamwise displacement of a monitoring point positioned in the upper part of the cylinder. The two curves are the same within four oscillations and, afterwards, a slight gap increases over time. The relative maximum difference is 2.28% and is reached after about 0.54 s, namely when 8 oscillations were done. Figure 10 gives the displacements in the transverse direction of the same monitoring point. The discrepancy between profiles appears earlier faster and it reaches an important value after 0.55 s. The relative difference between values given by each tool reaches a maximum of 47.6% after approximatively 0.54 s. From achieved signals it seemed that a slight difference in the damping of the vibration occurs. This can be especially noted in the transversal vibration plotted

Fig. 9 Isolated tube: temporal evolution of the streamwise displacement

Fig. 10 Isolated tube: temporal evolution of the transverse displacement

Unsteady FSI Analysis of a Square Array of Tubes in Water …

145

in Fig. 10. The frequency content seems to be in any case well captured as well as the magnitude of the vibrations itself. For this specific case, the magnitude of the displacement was not enough to make nonlinear effects appearing; even in the case of resonances that could occur for the full array case the displacement will be in any case limited by the clearance allowed before clashing. The required simulation time of the cases is similar. Assigning 12 processors and a simulation time of 0.3 s, the simulation is slightly faster with the RBF Morph™ (approximately 25 h 30 against 27 h 30 for the System Coupling). Because of the limited resources and time allocated to finalise the proposed method assessment, a sensitivity study was just partially performed. Even though such an investigation was not complete, several meshes ranging from 0.145 to 7 million of cells and different turbulence models were taken into account in order to evaluate their impact on the obtained results including y+ distributions. The mesh refinement was mainly performed around the tube, in its wake region and in the gap between the top and the top surface of the simulation volume. With regard to turbulence models, the k–ω Shear-Stress Transport (SST) and the Scale-Adaptive Simulation (SAS) turbulence models were not tested because of allocated schedule time. Finally, the mesh representing the best compromised judged to be such to guarantee a satisfactory accuracy in the numerical outputs (cylinder drag below 4% with respect to the finest mesh) and the respect of the activities schedule was used. The discretization level of such mesh was considered to generate the mesh of the twelve tubes case that is described hereinafter.

8 Tube Array Case 8.1 CAD Case Set-up As already introduced, the dimensions of the tubes of the twelve cylinders case are the same of the isolated tube case: length 295 mm, diameter D = 25.4 mm, gap between the top of a tubes and the ceiling of the simulation domain 5 mm. Similarly to the isolated tube case, the tubes have an O-grid structured mesh and, in addition, a larger space is devoted both to the wake evolution (15D) and the area preceding the tubes (5D) as shown in Fig. 11. The distance between tubes was the same in both streamwise and transverse directions and it was set to 0.5D = 12.7 mm. The CAD model of the case is shown in Figs. 11 and 12. This latter image, in particular, focus on the area where tubes were positioned highlighting the presence of fixed surroundings half-tubes. The dimension indicated in function of D ease the comprehension of the whole arrangement.

146

E. Costa et al.

Fig. 11 Geometry used to represent the 12 tubes case

Fig. 12 Geometry used to represent the 12 tubes case (details)

8.2 RBF Solutions Set-up The method adopted to generate the RBF solution for a mode of a single cylinder making use of the data coming from the FEM analysis, was the same of the isolated tube case. The difference basically pertained the inclusion of all (12) tubes, the fact that 4 vibration modes were used, instead of six, in accordance with the results obtained on the isolated tube case (showcasing the negligible contribution of the remaining ones). In addition, the RBF set-up of a generic flexible tube foresaw to keep both the other tubes and each side of the simulation domain including the portion of surrounding rigid tubes fixed, namely with zero movement. Since the procedure to accomplish the RBF set-up for a single tube was rather complex and envisaged that several operations needed to be repeated for each mode, exploiting the predefined geometrical distribution of the tubes and the work already done, such a process was efficiently automated employing scripts written in Python programming language.

Unsteady FSI Analysis of a Square Array of Tubes in Water …

147

Fig. 13 Mesh used to simulate the 12-tube FSI case

8.3 CFD Case Set-up Considering the experience done for the isolated case, the structured hexahedral mesh was generated so that the y+ value was beyond 30 over the portion of the surface of the tubes facing the flow. The refinement level between tubes and in the area of the wake was increased, whilst the amount cells before the tubes and in the far wake was properly reduced so as to limit the resources requests in terms of computing time. In the upper gap 9 layers of cells were created to finally achieve a total number of cells around 1.9 million (see Fig. 13). The settings of the ANSYS® Fluent® case were very similar to those of the isolated case. The only change was related to boundary condition of lateral surfaces of the simulation domain at which the condition symmetry was imposed.

9 Results of the Tube Array Configuration To describe the results gained for the 12-tube case, the numbering for columns, rows and tubes sketched in Fig. 14 was adopted. The FSI study was performed over 1.2 s starting from the configuration referring to 0.2 s of the unsteady calculation. Three input speeds were imposed and analysed: 0.32 m/s (a), 0.37 m/s (b) and 0.44 m/s (c). For each simulation, both streamwise and transverse displacement of a node located at the top of each tube, were monitored during time and saved. As example, Table 4 collects the comparison between the displacement recorded for the tube 5 and 8 for the case with low velocity (a) Vu . In case (b) and (c), the calculation stops prematurely due to the imminent impact of some tubes. As such for these cases was not possible to compare the Root Mean Square (RMS) amplitudes.

148

E. Costa et al.

Fig. 14 Numbering of the tubes of the experimental configuration

Table 4 Amplitude of the oscillations of the tube n = 5 and 8 for Vu = 0.32 m/s Tube

Streamwise displacement (mm)

Transverse displacement (mm)

Experimental

Numerical

Error (%)

Experimental

Numerical

Error (%)

5

0.94

0.857

8.8

0.43

0.288

33

8

0.90

0.922

−2.4

0.45

0.272

40

It is worth to clarify that in the experimental case not exhaustive information was given on how the RMS amplitudes were obtained. However, the authors can imagine that, on the test bench, the capture movements continues even after the collision between tubes. Checking the achievement of the regime condition by collecting the streamwise and transverse displacement history and plotting these data versus time, the following behaviours were observed: • Vu = 0.32 m/s: oscillations appeared initially in the streamwise direction. After two oscillations, the displacements in the transverse direction got importance. On the simulation time performed, no runaway was noted regardless of the direction. The tubes of the first two columns were mostly affected by the fluid, while those constituting the third column moved slightly in both directions. The last column of tubes moved of a small amount especially along the transverse direction. • Vu = 0.37 m/s: the initial behaviour was qualitatively the same of the previous one but with larger amplitudes. After 0.6 s the oscillations the tubes 1, 2, 4 and 5 in the transverse direction became very important. The calculation stopped after 1.0945 s when appearing negative volume cells in the mesh. This means that mesh deformations could no longer be absorbed correctly, and tubes 4 and 5 being too close together. • Vu = 0.44 m/s: transverse displacements increased starting from the second oscillation. The calculation stopped after 0.4405 s when negative cells were detected. Tubes 1 and 2 are the ones that got closer to each other. These two tubes, as well as tubes 4 and 5, were about to collide. The tubes in the penultimate row were

Unsteady FSI Analysis of a Square Array of Tubes in Water …

149

moving a lot in the stremwise direction before the clash. At this simulation time, the tubes of the last row still moved a little bit. In the case of a grid of in-line tubes, it was shown that the tubes lose their stability in the direction transverse to the direction of flow [1]. In the present configuration, the experimental results showed that the first elastic fluid instabilities appeared for a critical input velocity equal to Vc = 0.35 m/s. The results are consistent with this critical speed value: the calculation with Vu = 0.32 m/s did not show instability in the oscillations, and the calculation continued. With Vu = 0.37 m/s, just above Vc , the oscillations were stable for a certain duration of time and, then, the amplitude of some tubes increased till the interruption of the calculation. Finally, with Vu = 0.44 m/s, well above Vc , the interruption of the calculation appeared much more quickly. Table 5 groups the most important data from the three performed simulations. In particular, for each input velocity the simulation time, the time needed to perform the whole simulation using 12 processors as well as the vibrational behaviour of the tubes are gathered. With regard to this latter information, the tubes in red are the tubes about to collide, whilst the orange tubes oscillate strongly in the transverse direction and may collide as well. Table 5 Case 12 tubes—Summary of the results obtained Flow velocity (m/s)

Simulation time (s)

Duration of the whole simulation

0.32

1.2

6 days

0.37

1.09

5 days and 5 h

0.44

0.44

2 days and 3 h

Vibrational behaviour of tubes

150

E. Costa et al.

Fig. 15 Visualization of oscillations in the direction transverse to the flow for an input speed of 0.44 m/s

Figure 15 illustrates the movements calculated in the numerical conditions for Vu = 0.44 m/s. A strong approximation of the tubes in the direction transverse to that of the flow was found in the experimental case [1] as in the digital case.

10 Conclusions A methodology based on the use of mesh morphing, founded on radial basis functions technique, to perform transient FSI analyses according to the mode-superposition approach was assessed. Such a numerical investigation concerned the prediction of the vibrational behaviour of an array of 12 tubes arranged according to in-line configuration when subjected to a water cross-flow at different velocities. Such an assessment first foresaw the FSI analysis of an isolated tube to set-up at best the RBF solutions, the number of modes that need to be accounted so as to evaluate the effectiveness by comparing the gained results, namely the evolution of tubes deformation, with those obtained by means of the standard two-way approach. The main finding of this stage of the assessment is the consistency of the predicted vibrational behaviour of the studied system.

Unsteady FSI Analysis of a Square Array of Tubes in Water …

151

Exploiting the lessons learned, in the final stage of the assessment the tubes array was successfully simulated and results satisfactorily aligned with the expected behaviour of the system. Improvement perspectives deal with the use of other turbulence models and the inclusion of the water contribution in the calculation of the vibrational data of the tubes.

References 1. Weaver, D. S., & Abd-Rabbo, A. A. (1985). A flow visualization study of a square array of tubes in water crossflow. Journal of Fluids Engineering. 2. Herchang, A., Jiin Yuh, J., Jer-Nan, Y. (2002). Local heat transfer measurements of plate finned-tube heat exchangers by infrared thermography. International Journal of Heat and Mass Transfer. 3. Khushnood, S., et al. Cross-flow-induced-vibrations in heat exchanger tube bundles: A review. Nuclear Power Plants. Soon Heung Chang, IntechOpen. https://doi.org/10.5772/35635. 4. Thulukkanam, K. (2000). Heat exchanger design handbook. 5. Chen, S. S. (1977). Flow-induced vibrations of circular cylindrical structures—1. Stationary fluids and parallel flow. Shock and Vibration Digest, 9(10), 25–38. https://doi.org/10.1177/ 058310247700901006. 6. Chen, S. S. (1977). Flow-induced vibrations of circular cylindrical structures—2. Crossflow considerations. Shock and Vibration Digest, 9(11), 21–27. https://doi.org/10.1177/ 058310247700901106. 7. Gelbe, H., Jahr, M., & Schröder, K. (1995). Flow-induced vibrations in heat exchanger tube bundles. Chemical Engineering and Processing: Process Intensification, 34(3), 289–298. https:// doi.org/10.1016/0255-2701(94)04016-8. 8. Paidoussis, M. P. (1998). Fluid-structure interactions: Slender structures and axial flow (vol. 1). Academic Press. 9. Weaver, D. S., & Fitzpatrick, J. A. (1988). A review of cross-flow induced vibrations in heat exchanger tube arrays. Journal of Fluids and Structures, 2(1), 73–93. https://doi.org/10.1016/ S0889-9746(88)90137-5. 10. Yeung, H. C. (1984). Cross flow induced vibration of heat exchanger tubes. Hong Kong Engineer, 12(7), 33–40. 11. Chen, S. S. (1987). Flow-induced vibration of circular cylindrical structures. Washington, DC: Hemisphere Publishing Corporation. 12. Connors, H. J. (1970). An experimental investigation of the flow-induced vibration of tube arrays in cross flow (Ph.D. thesis). University of Pittsburgh. 13. Andjelic, M. (1988). Stabilittsverhalten querangestr6mter Rohrbfindel mit versetzter Dreiecksteilun (Dissertation, UniversitS.t Hannover). 14. Pettigrew, M. J., & Taylor, C. J. (1991) Fluid-elastic instability of heat exchanger tube bundles: Review and design recommendations. Int. Cot¢ Proe. Inst. Mech. Eng., Flow-induced Vibration, Brighton, paper C 416/052, pp. 349–368. 15. Lever, J. H., & Weaver, D. S. (1982). A theoretical model for fluid-elastic instability in heat exchanger tube bundles. Journal of Pressure Vessel Technology, 104(3), 147–158. https://doi. org/10.1115/1.3264196. 16. Ji, J., Ge, P., & Bi, W. (2016). Numerical analysis on shell-side flow-induced vibration and heat transfer characteristics of elastic tube bundle in heat exchanger. Applied Thermal Engineering, 107, 544–551. https://doi.org/10.1016/j.applthermaleng.2016.07.018.

152

E. Costa et al.

17. Duan, D., Ge, P., & Bi, W. (2016). Numerical investigation on heat transfer performance of planar elastic tube bundle by flow-induced vibration in heat exchanger. International Journal of Heat and Mass Transfer, 103, 868–878. https://doi.org/10.1016/j.ijheatmasstransfer.2016. 07.107. 18. Cella, U., & Biancolini, M. E. (2012). Aeroelastic analysis of aircraft wind tunnel model coupling structural and fluid dynamic computational codes. AIAA Journal of Aircraft, 49(2). 19. Groth, C., Cella, U., Costa, E., & Biancolini, M. E. (2019). Fast high fidelity CFD/CSM fluid structure interaction using RBF mesh morphing and modal superposition method. Aircraft Engineering and Aerospace Technology journal. https://doi.org/10.1108/AEAT-09-2018-0246. 20. Biancolini, M. E., Cella, U., Groth, C., & Genta, M. (2016). Static aeroelastic analysis of an aircraft wind-tunnel model by means of modal RBF mesh updating. ASCE’s Journal of Aerospace Engineering, 29(6). https://doi.org/10.1061/(asce)as.1943-5525.0000627. 21. Di Domenico, N., Groth, C., Wade, A., Berg, T., & Biancolini, M. E. (2018). Fluid structure interaction analysis: Vortex shedding induced vibrations. Procedia Structural Integrity, 8, 422– 432. https://doi.org/10.1016/j.prostr.2017.12.042. 22. Abd-Rabbo, A. A. (1984). Flow visualization and dynamics of heat exchanger tube arrays in water cross-flow (Ph.D. thesis). McMaster University. 23. IN2P3. Caractéristiques et choix des matériaux. 24. Wakeham, William A., Kestin, Joseph, & Sokolov, Mordechai. (1978). Viscosity of liquid water in the range—8 to 150 °C. Journal of Physical and Chemical Reference Data, 7(3), 944. 25. Materials data book. (2003). 26. Sumer, B. M., & Fredsøe, J. (2006). Advanced series on ocean engineering. In Hydrodynamics around cylindrical structure (Vol. 12). World Scientific Publishing Co Pte. Ltd. Singapore. 27. Hardy, R. L. (1971). Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research, 76(8), 1905–1915. https://doi.org/10.1029/JB076i008p01905. 28. De Boer, A., van der Schoot, M. S., & Bijl, H. (2007). Mesh deformation based on radial basis function interpolation. Computers and Structures, 85(11–14), 784–795. 29. Buhmann, M. D., & Functions, Radial Basis. (2003). Cambridge University Press. New York: NY, USA. 30. Beckert, A., & Wendland, H. (2011). Multivariate interpolation for fluid-structure-interaction problems using radial basis functions. Aerospace Science and Technology, 5(2), 125–134. ISSN 1270-9638. 31. Meirovitch, L. (1975). Elements of vibration analysis. International student edition, McGrawHill. URL: https://books.google.it/books?id=XBOoAAAAIAAJ.

Risk Measures Applied to Robust Aerodynamic Shape Design Optimization Domenico Quagliarella, Elisa Morales Tirado and Andrea Bornaccioni

Abstract A Robust Design Optimization (RDO) method based on the use of Conditional Value-at-Risk (CVaR) risk measure is briefly described and applied to an aerodynamic shape design problem. The technique leads to optimal design solutions resilient to production tolerances and operating conditions instabilities. The approach is illustrated through the application to an airfoil section design optimization in low transonic conditions with the flow field modeled using an Euler plus boundary layer interactive approach. The results of the robust design are compared to those obtained with a classical deterministic method, and mutual advantages and disadvantages of the two approaches are discussed.

1 Introduction Robust and reliability-based design optimization is gaining increasing favor in the industrial context, especially in those application fields where classical deterministic design approaches lead to optimal design solutions excessively sensitive to production tolerances and operating conditions stability. Unfortunately, there is no free lunch, and the robust design loop is often much more expensive if compared to classical ones. Consequently, many approaches, with varying rate of success, tried to reduce the computational load of robust and reliability-based optimization techniques without losing precision in the evaluation of statistical estimators. These techniques range from polynomial chaos [1, 2] to simplex elements stochastic collocation [3], to multi-level Montecarlo methods [4, 5]. In this work, a complementary, and not D. Quagliarella (B) · E. Morales Tirado Centro Italiano Ricerche Aerospaziali, via Maiorise 1, Capua, Italy e-mail: [email protected] E. Morales Tirado e-mail: [email protected] A. Bornaccioni Roma Tre University, via Vito Volterra, 62, Rome, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_9

153

154

D. Quagliarella et al.

conflicting, approach to robust design optimization is explored. It relies on the introduction of Conditional Value-at-Risk (CVaR) risk measure [6, 7], that originated in financial engineering, but is very well and naturally suited to reliability-based design optimization problems and represents a viable alternative to more traditional robust design approaches. The approach is illustrated through the application to a robust wing section design optimization near to transonic conditions with the flow field modeled using an Euler plus boundary layer interactive approach.

2 Risk Measures for Robust Design Optimization The adopted optimization approach is based on the introduction of risk measures [6], ρi , that depend on random variables, X , which, in turn, can be functions of deterministic design parameters, z. Following this approach we are led to define a constrained multi-objective minimization problem: min ρi (X (z))

z∈Z ⊆Rn

i = 1, . . . , p (1)

s. to: ρi (X (z)) ≤ ci i = p + 1, . . . , p + q

The constraints are also assigned considering a set of inequalities defined in terms of q further risk measure estimates. To work with this formulation, we have to be able to compare the risk function estimates for any given couple of deterministic design parameter values z1 , z2 , so that we can decide if ρi (X (z1 )) < ρi (X (z2 )) is true or not. Of course, since we are dealing with estimates of the parameter of interest based on a finite sample, we may have no assurance that the hypothesis is verified or not. So, whatever the accuracy of the estimate of the quantity of interest and whatever the sample size used, there will always be some level of random noise that will affect the behavior of the optimization algorithm. In optimization under uncertainty, therefore, the accuracy in the quantity of interest estimation and the resistance to noise of the optimization algorithm are closely linked. The robust optimization problem of this work uses the Conditional Value-at-Risk [8, 9] as the risk measure, whose definition is reported in Eq. 2. Let X be a random variable, the CVaRα of X can be thought of as the conditional expectation of losses that exceed the quantile qα . From a mathematical point of view, CVaR is given by a weighted average between qα (or Value-at-Risk VaRα or ν α ) and the losses exceeding it. The comparison of VaR and CVaR shows that the latter is more sensitive to the shape of the upper tail of the Cumulative Distribution Function (CDF). Summing up, the CVaR is expressed as: cα =

1 1−α

 α

1

ν β dβ

(2)

Risk Measures Applied to Robust Aerodynamic Shape …

155

 where ν β , the VaR of X at the confidence level β ∈ (0, 1), is defined as inf x ∈  R : FX (x) ≥ β that is the inverse of the cumulative distribution function FX of X . An in-depth description of the robust optimization approach via CVaR or, more generally, through risk measures can be found in [10]. CVaR has the advantage, compared to VaR, of being a coherent risk measure. The definition of coherency for a risk measure is a rigorous and well-defined mathematical concept that the interested reader can find in [11].

3 Optimization Algorithm The optimization algorithm selected for solving the described design problems is the “Covariance Matrix Adaptation Evolution Strategy” (CMA-ES) [12], which is a stochastic optimization algorithm based on self-adaptation of the covariance matrix of a multivariate normal distribution. It is mainly used for design optimization problems up to a few hundreds of design variables. The CMA-ES evolution strategy was developed by Hansen and Ostermeier [13], and it is suitable for nonlinear, non-convex numerical optimization problems. In the standard implementations of this Evolutionary Strategy, a population of λ ≥ 2 candidate solutions is sampled according to a multivariate normal distribution in R N , with N number of design variables:     2 k = 1, 2, . . . , λ (3) xki+1 ∼ m i N 0, C i ∼ N m i , σ i C i where xki+1 ∈ R N indicates the kth individual of the generation i + 1, m i ∈ R N is the average of the distribution at the generation i, C i ∈ R N ×N is a scaled covariance matrix of the distribution, and σ i ∈ R is the scaling parameter. A recombination operator is responsible for updating the distribution mean for each generation, and the covariance matrix represents the dependencies between the N design variables. In this context, the “Covariance Matrix Adaptation” method (CMA) is a technique to update the covariance matrix, the mean and the standard deviation of the distribution at each iteration. The CMA-ES method also exploits the concept of “accumulation” according to which, the parameters mentioned above are updated according to the whole evolutionary history and not using only the information resulting from a single generational step, with the effect of maximizing the likelihood of reproduction of the best performing individuals.

4 Flow Field Analysis Method The aerodynamic characteristics are computed using the MSES software developed by Drela [14, 15]. This choice is motivated by the need of having a reasonably fast and

156

D. Quagliarella et al.

robust solver compared to geometric changes, but at the same time able to provide a sufficiently accurate solution. It is furthermore handy the capability of running at assigned lift coefficient Cl . MSES solves the 2D Euler equations in the inviscid flow zones, while the integral boundary layer equations are used near the body and in the wake. The displacement thickness couples viscous and non-viscous flows. The laminar to turbulent boundary layer transition is predicted using the “envelope method,” a simplified version of the e N method [16–18].

5 Airfoil Shape Handling in the Presence of Uncertainties The airfoil shape is parametrized as a linear combination of an initial geometry (x0 (s), y0 (s)), and some modification functions yi (s) that may be defined analytically or by point distributions [19]. Moreover, to describe geometry uncertainties, further z j (s) modification functions are introduced. So, the airfoil shape, including uncertainties, is defined by x(s) = x0 (s),

y(s) = k y0 (s) +

n

i=1

 wi yi

+

m

Ujz j

(4)

j=1

where the airfoil shape is controlled by the design parameters wi and by the scale factor k. The U j random variables describe the uncertainty on shape and thickness of the airfoil. Therefore, each realization of the random variables U j defines a particular airfoil. It is important to note that the airfoil is rescaled to the assigned thickness before the application of the random variables that describe the uncertainty in shape.

6 Error Handling A particularly delicate problem to be addressed in the calculation of risk measurements in aerodynamic design is the treatment of those cases in which the fluiddynamic solver does not converge or goes into error. When robust optimization is faced, the proper treatment of these cases is crucial for the optimization process. Indeed, the quantity of interest to be minimized, CVaR, depends on the upper tail of the Cumulative Distribution Function. As a consequence, assigning a high penalization value to the objective function in the cases where the solver fails to converge implies a too high increase of the computed CVaR value that could be detrimental for the optimization algorithm behavior. In this particular problem, we decided to assign to the failed cases the worst objective value obtained among the properly converged cases. Numerical tests lead to conclude that this was the setup with the lowest impact on the optimization process behavior.

Risk Measures Applied to Robust Aerodynamic Shape …

157

7 Design Optimization Example A multi-point airfoil optimization problem is here used to illustrate the risk-function based approach to Robust Design Optimization (RDO). The design problem specified in the followings is first solved using a classic deterministic approach and then using the risk-based method for robust design. The results are then discussed and compared.

7.1 Aerodynamic Design Problem Description The aircraft selected to perform the airfoil aerodynamic shape optimization is a tailless blended wing body configuration (BWB). We refer to the ACFA 2020 project [20], which is a collaborative research project funded by the European Commission under the seventh research framework programme (FP7). In particular, we refer to a scaled version of the project aimed at the transportation of 320 passengers at a cruise altitude of 10,000 m and a cruise Mach number of 0.84. A planform view of the aircraft is sketched in Fig. 1, where it is also shown the section corresponding to the airfoil selected for the optimization process. The goal of this design exercise is to improve the performance of the wing section identified in Fig. 1 so that, when it replaces the original one, the characteristics of the BWB improve. y

selected section

x

Fig. 1 Schematic planform view of the blended wing body configuration

158

D. Quagliarella et al.

The translation of a 3D aerodynamic optimization problem into a two-dimensional one must be done carefully to ensure that the improvements obtained in the twodimensional case also lead to appreciable improvements when reported in the 3D configuration. In particular, we must consider that the estimation of the value of the 2D Cl obtained with the 3D simulation is approximate and that, above all, the same wing section is present in different parts of the wing. Consequently, it will be necessary to optimize the same airfoil in different operating conditions, since the wing load varies along the span. Furthermore, it is never prudent to optimize a transonic airfoil in a single design point, because a profile optimized to be shock-less at a given Cl and Mach may present sudden and intense increases in wave drag moving slightly from the design condition. Hence, we follow the above described multi-point optimization strategy to minimize the airfoil aerodynamic drag coefficient Cd at cruise conditions. Each design point is characterized by a specific value of the airfoil lift coefficient Cl . For this reason, we undertake a preliminary analysis of the three-dimensional base configuration using the Athena Vortex Lattice (AVL) software by Drela and Youngren [21]. AVL is based on the vortex lattice method, and it uses the PrandtlGlauert corrections to treat compressibility. In this way, we generate the aerodynamic loading curve at cruise conditions along the span, and we select the proper working lift coefficients for the bi-dimensional problem. This approach permits to obtain acceptable performances of the airfoil even in the three-dimensional configuration, where the airfoil works over a range of the lift coefficient. Moreover, in the bi-dimensional problem we take into account the effect of the wing sweep, for which we use the following aerodynamic corrections: M = M cos 

(5)

Cl =

Cl cos2 

(6)

t% =

t% cos 

(7)

Re = Re cos2 

(8)

where  = 55◦ is the local sweep angle with respect to the line of the leading edges. In Table 1 the parameters obtained are summarised. We use both aerodynamic and geometric constraints. The airfoil percentage thickness with  respect to the chord is fixed at the base value corrected with the sweep angle t% , while, to obtain realistic shapes, we use some constraints on the leading edge radius (LER), the trailing edge angle (TEA) and the airfoil percentual thickness with respect to the chord at x/c = 0.85 (TAT). Special attention is dedicated to the airfoil pitching moment coefficient Cm , which for this kind of configuration is a critical parameter due to the absence of the elevators.

Risk Measures Applied to Robust Aerodynamic Shape …

159

Table 1 Parameters corrections with sweep angle M Cl  t% 0.48

0.2, 0.3

Re 114 × 106

25.15

For this reason, to maintain the pitching behavior of the base airfoil as much as possible without limiting too much the minimum search process, and permitting a moderate control surfaces intervention, two constraints for the Cm are used. The first to avoid too large values of Cm in the positive α-Cm plan and the second to prevent too large values of Cm in the negative α-Cm plan. For the sake of clarity, it is pointed out that the Cm coefficient is evaluated with respect to the aerodynamic center and it is considered positive in the case of “nose down” pitching moment. Here, a synthetic description of the optimization problem is reported in mathematical terms: ⎧ min Cd1 (x) + Cd2 (x) ⎪ ⎪ x ⎪ ⎪ ⎪ subject to: ⎪ ⎪ ⎪ ⎪ t% = 26.15 ⎪ ⎪ ⎪ ⎪ LER ≥ 0.054 ⎪ ⎨ TEA ≥ 11.2◦ (9) ⎪ TAT ≥ 0.0538 ⎪ ⎪ ⎪ ⎪ Cl1 = 0.2; Cl2 = 0.3 ⎪ ⎪ ⎪ ⎪ Cm 1 , Cm 2 ≥ 0.0 ⎪ ⎪ ⎪ ⎪ C m 1 , C m 2 ≤ 0.085 ⎪ ⎩ error 1 , error 2 = 0 Note that the sub-indexes 1 and 2 denote the different design points. The penalty approach is used to translate the constrained optimization problem into an unconstrained one: min n Q 1 + Q 2 (10) x∈X ⊆R

with

and

Q 1 = Cd1 (x) + k 1 p + (LER, 0.054)+ k 2 p + (TEA, 11.2◦ ) + k 3 p + (TAT, 0.0538)+ k 4 p + (Cm 1 , 0.0) + k 4 p − (Cm 1 , 0.085) + k 5 p + (error 1 , 0)

(11)

Q 2 = Cd2 (x)+ k 4 p + (Cm 2 , 0.0) + k 4 p − (Cm 2 , 0.085) + k 5 p + (error 2 , 0)

(12)

In this case, all the constraints except those regarding the lift coefficient and the airfoil percentage thickness with respect to the chord are treated as quadratic penalties: +

p (x, y) =



0 (x − y)2

if x ≥ y if x < y

(13)

160

D. Quagliarella et al.

and p − (x, y) =



(x − y)2 0

if x ≥ y if x < y

(14)

Instead, the constraints on the Cl and the thickness do not appear because they are automatically satisfied by the computation procedure by changing the angle of attack and by re-scaling the airfoil thickness to the assigned value. The numerical values chosen for the k i coefficients are: k 1 = 5000, k 2 = 10, k 3 = 30, k 4 = 1000, k 5 = 1000. The transformation of a constrained optimization problem into an unconstrained one through the penalty approach is always a delicate process, as the choices of the weights of the penalization terms profoundly change the shape and features of the search space. Here, in particular, the violation of the constraint on leading edge radius is highly penalized as it is well known the tendency of the optimization processes to gain performance in these working condition through the reduction of the leading edge radius. Unfortunately, this often induces bad off-design stall behavior, especially near stall, and should be avoided. We also want to avoid that failed computations may deceive the optimizer, and hence we add a very high weight to the related constraints. Finally, a stiff penalty is also imposed to the pitching moment constraints, because a too marked change in this constraint would require a radical revision of the BWB’s planform. As we will see, this is one of the constraints that will have the most significant impact on robust optimization.

7.2 Deterministic Problem Solution Nineteen design variables are used to describe the shape of the wing section. The parameters used for the CMA-ES optimization algorithm are the maximum number of allowed evaluations, the population size λ, and the initial standard deviation σ . The parameters set for this problem are reported in Table 2. As shown in Fig. 2, after 5005 function evaluations a convergence is reached, and the shape in Fig. 3 is obtained. The objective function value for the optimized airfoil is 0.014886, which corresponds to a 27.176% reduction with respect to the baseline value of 0.020441. Looking at the polar curves in Fig. 4 it is clear that the new shape exhibits improved aerodynamic performances respecting the pitching moment coefficient constraints. The total number of evaluations of the objective function was 5005 and each of these required two calls to the aerodynamic solver that make up the most significant part of the computational load. The elapsed time of the optimization run was about

Table 2 CMA-ES parameters for the deterministic optimization run Maximum evaluations Population size Initial standard deviation 5005

36

0.15

Risk Measures Applied to Robust Aerodynamic Shape …

161

Objective function

0.024 0.022 0.020 0.018 0.016 0.014

0

3000

2000

1000

4000

5000

Function evaluations

y/c

Fig. 2 CMA-ES convergence history for the deterministic optimization. The black line indicates the objective function value of the baseline airfoil (—)

0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15

Deterministic design Baseline

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c Fig. 3 Airfoil shape comparison of the deterministic optimized airfoil versus the baseline airfoil

47 min on a 36-core Intel cluster node used in parallel (at the optimization code level). The processors of the node are the Intel Xeon E5-2697v4 at 2.3 GHz clock frequency.

7.3 Robust Design Optimization The classical design problem solved in the previous section produced a good improvement in the drag characteristics of the airfoil in conjunction with enhanced compliance with the pitching moment constraints. The purpose of the robust design

D. Quagliarella et al. 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0.006

1.2 1.0 0.8 0.6

Cl

Cl

162

0.0 -0.4 0.010

0.014

0.018

-0.6

-4

-2

0

2

4

6

8

10

[deg]

0.12 0.10 0.08 0.06

120 100 80 60 40 20 0 -20 -40 -60 -80 -0.6

Cl/Cd

Cm

0.2 -0.2

Cd

0.04 0.02 0.00 -0.02 -0.6

0.4

-0.2

0.2

Cl

0.6

1.0

-0.2

1.0

0.6

0.2

Cl

Fig. 4 Polar curves comparison of the deterministic optimized airfoil (- - -) versus the baseline airfoil (——)

step here described is also to improve the behavior under uncertainty in operating conditions and over the actual shape of the airfoil. The uncertainty about the operating conditions was modeled with a beta distribution (Eq. 15) on the Mach number, while the one on the airfoil shape, through Eq. 4. In particular, 12 random variables were used to represent the stochastically perturbed shape of the airfoil through the weighted application of 6 Hicks-Henne modification functions to the upper and lower airfoil surface. All the selected functions introduce a smooth bump on the airfoil at a specified position and with a given maximum height. The first two bump functions √ − (3x2 max−2x−1)x max , with a = 0.888 and x have the expression a (1 − x) xe 2xmax max = 0.034925 for the first one, and a = 0.57 and xmax = 0.082102 for the second one. The value of the parameter a is chosen to have, approximately, a maximum (unweighted) bump height equals to 0.1. The remaining four functions are   the  standard Hicks-Henne log(2)

t

bumps and have the following general expression: a sin π x − log(xmax ) , where t controls the width of the bump, a is the maximum bump magnitude, and xmax is the abscissa of the maximum point of the bump. In the present case, a is always set to 0.1, t is equal to 3, and the values for xmax are respectively 0.2, 0.4, 0.6 and 0.8.

Risk Measures Applied to Robust Aerodynamic Shape …

163

The random variables Ui have a uniform variation range between −0.15 and 0.15. The operational uncertainty related to Mach number is modeled as a four parameter beta distribution given by  f (y; α, β, a, b) =

(y−a)α−1 (b−y)β−1 (b−a)α+β−1 B(α,β)

0

if a ≤ y ≤ b otherwise

(15)

1 with B(α, β) = 0 u α−1 (1 − u)β−1 du. The parameters that define the Mach random variables are reported in Table 3. On the other hand, it was not considered useful to introduce uncertainty for the lift coefficient, given that the problem is already multi-point on it because it is necessary to guarantee good performance of the airfoil over a large region of its polar curve. The introduction of the random variables relative to the geometrical parameters and the Mach number introduces a functional dependence in Q = Q 1 + Q 2 , which is now a function of functions. The CVaR risk function, estimated with the confidence level α set to 0.9, is used to map Q into R. In order to reduce the effect on the computational cost, it was decided to take into account the effect of random perturbations only in the first design point, while for the second it was decided to adopt the nominal values of the design parameters, without taking into account the effects of random disturbances. Therefore the objective function that defines the problem of robust optimization is: (16) min n CVaR0.9 (Q 1 ) + Q 2 x∈X ⊆R

The estimate of CVaR0.9 during optimization is performed with a tiny number of samples, which is equal to 36. In fact, in this phase, we tried to reduce as much as possible the number of evaluations of Q for computational efficiency issues. However, we kept a partial check on the quality of the CVaR estimate by calculating the confidence intervals with the bootstrap method [22] and verifying that, at least on average, the error on CVaR did not introduce too much noise in the optimization procedure. The results obtained, however, were verified by recalculating CVaR with 1520 samples, to achieve a much narrower confidence interval in the result comparison step. The CMA-ES algorithm was used with parameters reported in Table 4.

Table 3 Beta distribution function parameters that define the uncertainty in Mach number α β a b 5

2

0.45

Table 4 CMA-ES parameters for the robust optimization run Maximum evaluations Population size 7001

20

0.51

Initial standard deviation 0.1

164

D. Quagliarella et al.

The convergence history of the RDO run is reported in Fig. 5, along with a line indicating the CVaR objective value of the initial airfoil. The CDF obtained by introducing uncertainties in the airfoil shape and Mach number related to the robust optimum is compared with those derived from the baseline configuration and the deterministic optimum airfoil in Fig. 6 and, in an enlarged scale along the axis of the abscissas, in Fig. 7. 0.025 0.024 0.023 0.022

CVaR

0.021 0.020 0.019 0.018 0.017 0.016 0.015 0.014

0

1000 2000 3000 4000 5000 6000 7000 8000

Function evaluations Fig. 5 CMA-ES convergence history for the robust optimization and baseline CVaR value (—)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

Baseline Deterministic design Robust design

0.1 0

0

0.05

0.1

0.15

0.2

0.25

Objective Fig. 6 Empirical CDF related to baseline, deterministic optimum and robust optimum

Risk Measures Applied to Robust Aerodynamic Shape …

165

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Baseline Deterministic design Robust design

0.2 0.1 0 0.014

0.016

0.018

0.02

0.022

0.024

Objective Fig. 7 An enlarged view of the ECDF obtained by the variation in airfoil shape and Mach number for the robust optimum, the deterministic optimum and the baseline airfoil Table 5 CVaR risk measure and related confidence intervals (CI) based on the empirical cumulative distribution functions with different sample size 36 samples 1520 samples CVaR0.9 CI low CI up CVaR0.9 CI low CI up Baseline airfoil Deterministic optimized airfoil Robust optimized airfoil

0.02213 0.02161 0.02258 0.02250 0.02237 0.02262 0.14319 0.03513 0.18786 0.10346 0.09471 0.11219 0.01498 0.01496 0.01500 0.01638 0.01588 0.01689

The CDF comparison related to the deterministic and robust optimized airfoils highlights that the robust optimal solution is less vulnerable to uncertainties in geometric shape and Mach number with respect to the deterministic and the baseline ones. In particular, it is clear that the upper tail of the deterministically optimized profile is much more sensitive to changes in pitching moment coefficient introduced by random perturbations that have a substantial impact on the penalty terms of Q. This point can also be deduced by the observation of the Conditional Value at Risk estimation with α = 0.9 provided in Table 5. The analysis of Table 5 provides significant insights. First of all, it is observed that the initial airfoil uncertainty is characterized sufficiently well even using a small number of samples. Indeed this airfoil, despite having relatively poor performance, is very internal to the area of feasibility and small perturbations do not trigger the penalties related to the constraints. In contrast, both the deterministically optimized airfoil and the one obtained by the robust procedure are much closer to the constraints.

166

D. Quagliarella et al.

y/c

In particular, the CDF of the deterministic optimum clearly shows the effect of the penalties introduced by the constraints in the high tail of the CDF. It is interesting to note that for the deterministic problem very low values of Cm 1 and Cm 2 are obtained compared to the baseline and robust optimum. This is a predictable behaviour of the deterministic optimal airfoil because the optimization algorithm tries to improve the solution as much as possible without worrying about how much the optimum is close to the Cm i constraints. The robust optimum, however, is influenced by the effect of uncertainties which automatically leads to values of Cm higher than the lower constraint. Another noteworthy point is that the estimate of CVaR obtained for the optimal robust solution with 36 samples is slightly worsened when it is analyzed with a much higher number of samples. Indeed, to avoid introducing too high oscillations in the CVaR of similar elements during the optimization process, the same sample of random variables was always used for all the candidate solutions generated during the process. As a result, the optimization algorithm adapted to sampling and introduced a bias that leads to the underestimation of CVaR. This fact, while not excessively affecting the quality of the final result, is one of the critical points of the algorithm and will be the object of further investigation and development. In Fig. 8, the airfoil shape obtained using the robust approach (solid line) is compared with the one obtained with the deterministic algorithm (dashed line). A slight camber reduction characterizes the robust design, as well as a thickness reduction around the nose. Table 6 reports the aerodynamic and geometric characteristics of the baseline and optimized nominal (unperturbed) airfoils. 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15

Deterministic design Robust design

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/c Fig. 8 Comparison of the airfoil shapes obtained in robust and deterministic optimization Table 6 Aerodynamic and geometric features of the nominal airfoils Baseline

Cd2

Cm 1

Cm 2

LER

TEA

Cl 1

Cl 2

Cd1

0.2

0.3

0.00877 0.01167 0.03229 0.08117 0.06388 22.45

TAT

objective

0.08278 0.02044

Deterministic

0.2

0.3

0.00740 0.00749 0.00006 0.00005 0.05494 21.77

0.05794 0.01489

Robust

0.2

0.3

0.00738 0.00747 0.00320 0.00282 0.05789 19.95

0.06166 0.01485

y/c

Risk Measures Applied to Robust Aerodynamic Shape …

0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15

0

0.1

0.2

0.3

0.4

0.5

167

0.6

0.7

0.8

0.9

1

x/c Fig. 9 Approximated uncertainty in shape around the optimal airfoil

Finally, Fig. 9 shows approximately the uncertainty range around the optimal airfoil, as described by the Hicks-Henne bump functions. The total number of evaluations of the objective function was 7001 and each of these required 37 calls to the aerodynamic, in particular 36 calls are used to build the coarse CDF approximation for the first design point, while the remaining one is needed for the second design point that is kept deterministic. The elapsed time of the optimization run was about ten hours on two 36-core cluster nodes used in parallel (at the optimization and CDF computation code level).

8 Conclusions Robust optimization based on advanced risk measures, such as CVaR, lends itself very well to industrial use. In this work, we have shown how this approach can be advantageous in the robust design of an airfoil with advanced features. The comparison made with the results of classical deterministic optimization shows the numerous advantages of the robust approach to aerodynamic optimization. Given these advantages, there is indeed a significant increment in the necessary computational resources. As a result, increasing the efficiency of robust optimization algorithms is one of the most current and hot research topics. The next developments we are planning for this line of research concern the introduction of reduced order models and adaptive response surfaces both in the optimization loop and in the estimation of risk functions. Furthermore, methods for closer integration of the confidence interval estimation in the design cycle and advanced uses of hypothesis testing and Bayesian inference techniques are planned to interactively improve the quality of sampling and, consequently, of the risk functions evaluation. Acknowledgements Thanks are due to the prof. M. Drela and MIT for allowing the use of MSES code in this research project.

168

D. Quagliarella et al.

References 1. Wiener, N. (1938). The homogeneous chaos. American Journal of Mathematics, 60(4), 897– 936. 2. Xiu, D. (2010). Numerical methods for stochastic computations: A spectral method approach. Princeton University Press. 3. Witteveen, J., & Iaccarino, G. (2010). Simplex elements stochastic collocation for uncertainty propagation in robust design optimization. In 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, American Institute of Aeronautics and Astronautics, January 2010. 4. Heinrich, S. (2001). Multilevel Monte Carlo methods. In S. Margenov, J. Wa´sniewski, & P. Yalamov (Eds.), Large-Scale Scientific Computing: Third International Conference, LSSC 2001, Sozopol, Bulgaria, 6–10 June 2001. Revised papers (pp. 58–67). Berlin, Heidelberg: Springer. 5. Giles, M. B. (2015). Multilevel Monte Carlo methods. Acta Numerica, 24, 259–328. 6. Quagliarella, D., Petrone, G., & Iaccarino, G. (2015). Reliability-based design optimization with the generalized inverse distribution function. In D. Greiner, B. Galván, J. Périaux, N. Gauger, K. Giannakoglou, & G. Winter (Eds.), Advances in evolutionary and deterministic methods for design, optimization and control in engineering and sciences. Computational methods in applied sciences (Vol. 36, chap. 5, pp. 77–92). Springer. ISBN 978-3-319-11540-5. 7. Quagliarella, D., & Iuliano, E. (2017). Robust design of a supersonic natural laminar flow wing-body. IEEE Computational Intelligence Magazine, 12(4), 14–27. 8. Tyrrell Rockafellar, R., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26, 1443–1471. 9. Tyrrell Rockafellar, R., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–41. 10. Quagliarella, D. (2019). Value-at-risk and conditional value-at-risk in optimization under uncertainty. In C. Hirsch, D. Wunsch, J. Szumbarski, Ł. Łaniewski-Wołłk, & J. Pons-Prats (Eds.), Uncertainty management for robust industrial design in aeronautics. Springer. 11. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228. 12. Hansen, N. (2006). The CMA evolution strategy: A comparing review. In J. A. Lozano, P. Larrañaga, I. Inza, & E. Bengoetxea (Eds.), Towards a new evolutionary computation: Advances in the estimation of distribution algorithms (pp. 75–102). Berlin, Heidelberg: Springer. 13. Hansen, N., & Ostermeier, A. (2001). Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation, 9(2), 159–195. 14. Drela, M., & Giles, M. B. (1987). Viscous-inviscid analysis of transonic and low Reynolds number airfoils. AIAA Journal, 25(10), 1347–1355. 15. Drela, M. (1996). A user’s guide to MSES 2.95. MIT Computational Aerospace Sciences Laboratory. 16. Tollmien, Walter. (1931). Grenzschichttheorie. Handbuch Experimentalphysik, 4, 241–287. 17. Schlichting, H. (1933). Zur enstehung der turbulenz bei der plattenströmung. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 181–208, 1933. 18. Van Ingen, J. (2008). The e N method for transition prediction. Historical review of work at TU Delft. In 38th Fluid Dynamics Conference and Exhibit (p. 3830). 19. Hicks, R., & Henne, P. A. (1978). Wing design by numerical optimization. Journal of Aircraft, 15(7), 407–412. 20. EADS innovation works. http://www.acfa2020.eu. 21. Drela, M., & Youngren, H. (2004). Athena vortex lattice. http://web.mit.edu/drela/Public/web/ avl. 22. Efron, B. (1992). Bootstrap methods: Another look at the jackknife. In Breakthroughs in statistics (pp. 569–593). Springer.

Aero-structural Optimization of a MALE Configuration in the AGILE MDO Framework Reinhold Maierl, Alessandro Gastaldi, Jan-Niclas Walther and Aidan Jungo

Abstract Aircraft, and in particular military aircraft, are complex systems and the demand for high-performance flying platforms is constantly growing both for civil and military purposes. The development of aircraft is inherently multidisciplinary and the exploitation of the interaction between the disciplines driving the design opens the door for new (unconventional) aircraft designs, and consequently, for novel aircraft having increased performance. In modern aircraft development processes and procedures, it is crucial to enable the engineers accessing complex design spaces, especially in the conceptual design phase where key configuration decisions are made and frozen for later development phases. Pushing more MDO and numerical analysis capabilities into the early design phase will support the decision-making process through reliable physical information for very large design spaces which can hardly be grasped and explored by humans without the support of automated numerical analysis capabilities. Therefore, from the start of the aircraft development, process computer simulations play a major role in the prediction of the physical properties and behavior of the aircraft. Recent advances in computational performance and simulation capabilities provide sophisticated physics based models, which can deliver disciplinary analysis data in a time effective manner, even for unconventional configurations. However, a major challenge arises in aircraft design as the properties from different disciplines (aerodynamics, structures, stability and control, etc.) are in constant interaction with each other. This challenge is even greater when specialized competences are provided by several multidisciplinary teams distributed among R. Maierl (B) · A. Gastaldi Airbus Defence and Space, Rechliner Strasse, 85077 Manching, Germany e-mail: [email protected] A. Gastaldi e-mail: [email protected] J.-N. Walther German Aerospace Center, c/o ZAL TechCenter Hein-Sass Weg 22, 21129 Hamburg, Germany e-mail: [email protected] A. Jungo CFS Engineering, EPFL Innovation Park, Lausanne, Switzerland e-mail: [email protected] © Springer Nature Switzerland AG 2020 M. E. Biancolini and U. Cella (eds.), Flexible Engineering Toward Green Aircraft, Lecture Notes in Applied and Computational Mechanics 92, https://doi.org/10.1007/978-3-030-36514-1_10

169

170

R. Maierl et al.

different organizations. It is therefore important to connect not only the simulation models between organizations, but also the corresponding experts to combine all competences and accelerate the design process to find the best possible solution. A multi-disciplinary study of an unmanned aerial vehicle (UAV), presented in this article, was performed by eight different partners all over Europe to show the advances during the Horizon 2020 project Aircraft 3rd Generation MDO for Innovative Collaboration of Heterogeneous Teams of Experts (AGILE). Abbreviations AGILE AVL CFD CPACS CSV DLR DoE FSI ICAS MALE MDO MLS MTOW MZFW RBF RCE SEP SU2 TSFC UAV

Aircraft 3rd Generation MDO for Innovative Collaboration of Heterogeneous Teams of Experts Athena Vortex Lattice Computational Fluid Dynamics Common Parametric Aircraft Configuration Scheme Comma Separated Value German Aerospace Center Design of Experiments Fluid Structure Interaction International Council of the Aeronautical Sciences Medium Altitude Long Endurance Multidisciplinary Design Optimization Moving Least Squares Maximum Take-Off Weight Maximum Zero Fuel Weight Radial Basis Function Remote Component Environment Specific Excess Power Stanford University Unstructured Thrust Specific Fuel Consumption Unmanned Aerial Vehicle

1 Introduction Multidisciplinary collaboration is at the core of AGILE [1], an EU-funded Horizon 2020 project, started in 2015 and finished in 2018. AGILE is developing the next generation of aircraft Multidisciplinary Design and Optimization processes, which target significant reductions in aircraft development costs and time to market, leading to cost-effective and greener aircraft solutions. AGILE has set ambitious performance targets to achieve by the end of the project in 2018: a reduction of 20% in time to converge the optimization of an aircraft and a 40% reduction in time needed to setup and solve the multidisciplinary optimization in a team of heterogeneous specialists, targeting novel configurations. This will lead to improved aircraft designs and a 40%

Aero-structural Optimization of a MALE Configuration …

171

Fig. 1 AGILE paradigm: a “blueprint for MDO”

performance gain, compared to aircraft in service today, is expected for large passenger unconventional aircraft configurations [2]. The central AGILE paradigm is illustrated in Fig. 1. It is the blueprint for all collaborative activities within the project to improve the aircraft design processes. To meet the challenges of the AGILE project a team of 19 industry, research and academia partners from Europe, Canada and Russia are collaborating together. The composition of the Consortium reflects the heterogeneous structure that is characteristic for today’s aircraft design teams. During the first year of the project (Design Campaign 1, DC-1), a reference distributed MDO system has been formulated to resolve the design of a single conventional configuration. In the second year (Design Campaign 2, DC-2), several optimization techniques have been investigated, also applying high fidelity techniques in MDO. During the last third year of the project, currently ongoing, all the developed methodologies and the AGILE MDO framework [3–5] are deployed to setup and solve multiple aircraft design and optimization problems for novel configurations, the Design Campaign 3. This study consists of six different concepts in six sub-tasks. This paper is focused on the AGILE task 4.5 which consists of the concept definition and shape optimization of a MALE UAV configuration.

2 Requirements Definition The baseline of the present design study is the OptiMALE aircraft from the German research project AeroStruct [6]. This concept is a medium altitude, long endurance unmanned aerial vehicle. The Top Level Aircraft Requirements (TLAR) of this concept were defined by Airbus Defence and Space and iterated with the Italian partners from Leonardo Company. They are listed in Table 1.

172

R. Maierl et al.

Table 1 Top level aircraft requirements Requirements Cruise above civil transport Range Runways length Cruise speed Dive speed Landing speed Payload weight Payload volume Payload power consumption Two external fuel tanks Electric powered hydraulic system Sat-Com communication system SEP SEP SEP Roll rate Sink rate Climb rate

>15 km >12,000 km 2500 m 150 m−1 180 m−1 55 m−1 800 kg 4 m3 10 kW

160 m min−1 @ 6 km 50 m min−1 @ 15 km 0 m min−1 @ 18 km 60◦ s−1 4 m s−1 160 m min−1

Fig. 2 Surveillance mission profile

Two reference missions were defined for the OptiMALE aircraft: One transfer and one surveillance mission. The latter is presented in detail in Fig. 2 and Table 2, because it was chosen as optimization reference mission.

3 Preliminary Design Workflow Following the conceptual design phase several disciplinary analyses were performed on the UAV design. The definition of the system architecture by Politecnico Torino, the design of the engine deck by CIAM, the analysis of the handling characteristics

Aero-structural Optimization of a MALE Configuration …

173

Table 2 Surveillance mission definition Start Climb Cruise flight Loitering with 1.05 g At FL 200 ÷ FL 450 Cruise flight Descent

> FL 500 > 1000 km @ FL 550 > 20 h >1000 km @ FL 550

Fig. 3 Task 4.5 – design work-flow

by TU Delft and the mission performance analysis by the DLR. The preliminary design workflow is illustrated on the left and the shape optimization is shown on the right in Fig. 3. The different partners are indicated near their responsibilities in the project. These were the first multi-disciplinary steps towards an improved aircraft design as starting point for the second part of this paper. The optimized engine deck influences the endurance objective of the consecutive shape optimization directly and the structural model is enriched by the correct distribution of the system masses.

3.1 Engine Deck Definition The engine deck design was performed by the Central Institute of Aviation Motors located in Moscow. The input required for the engine definition was provided in CPACS format by Airbus DS. For this reason the mission parameters and requirements had to be converted to the appropriate format. One major constraint to the engine design was the fixed pylon size, which allowed only a very small bypass ratio. The design decision to use two smaller engines instead of one large engine for redundancy reasons also resulted in an increased thrust specific fuel consumption

174

R. Maierl et al.

Fig. 4 Engine cycle optimization

(TSFC). According to the results shown in Fig. 4 an increase of the bypass ratio could significantly reduce the TSFC.

3.2 System Design The system design was performed by the Department of Mechanical and Aerospace engineering of Politecnico di Torino. A mass breakdown of the different sub-systems is shown in Fig. 5. It was decided to use a more electric system architecture for the OptiMALE aircraft. This is realized with a generalized electric actuation concept, excluding for flaps and landing gear actuation, which are hydraulic powered. The hydraulic pump is electrically powered and will be in operation at take-off and landing. For the anti-icing system it was decided to use an electro impulse solution. This concept provided the best trade-off between weight and power consumption for this aircraft configuration. Further information on this study can be found in [7, 8].

Aero-structural Optimization of a MALE Configuration …

175

Fig. 5 On-board system weight estimation

3.3 Handling Quality Evaluation The handling quality evaluation of the OptiMALE concept was performed by the Flight Performance and Propulsion department at TU Delft. The main required input were mass and inertia data, provided by Airbus DS, and aerodynamic data provided by AIRINNOVA. The main design recommendations expressed by the disciplinary expert are listed below. Further information concerning the tools and methodology employed by TU Delft can be gained here [9]. The following design recommendations are given after the handling quality evaluation for the OptiMALE UAV. • An assessment of the elevator deflection at the approach condition with wing flaps in the landing configuration with a high fidelity aerodynamic analysis is suggested. • The Dutch roll mode: frequency is relatively low and in some cases unstable. It is recommended to add a yaw damper to the flight control system to overcome this. • The low speed, high weight condition should be assessed with a panel method, accounting for the deformed wing shape. • At low speed and high weight, the roll mode time constant is too large and time to achieve a bank angle of 45◦ is large. • An increase of the wing span improves the roll mode and at the same time increases the moment arm of the ailerons. • Increase of the aileron size (span-wise) would be beneficial to satisfy the time to bank criterion.

176

R. Maierl et al.

4 Aero-elastic Shape Optimization The preliminary design of the OptiMALE, resulting from the described above work was the foundation for the second main task within this work package, which is presented in this chapter. The workflow of the aeroelastic shape optimization is shown in Fig. 3 on the right and consists mainly of three nested loops: An aeroelastic analysis loop, a sizing optimization loop and a shape optimization loop. Following points have to be emphasized for the here developed workflow. • The external shape and the structural sizing will be both optimized considering the aeroelastic coupling. • The maximum and the minimum load-factors define the design points for the structural optimization. • The cruise flight condition provides is the design point for the outer shape optimization. The aeroelastic loop, shown in Fig. 6, is at the heart of this process. The analysis is initialized with an undeformed aerodynamic model and an unloaded structural model. The aerodynamic pressures are converted to forces and applied on the structural model. The structural displacements from these loads are applied on the aerodynamic model in return. The process is iterative and provides a converged aeroelastic load-case. These resulting forces are used in the next step, a structural sizing optimization. Here, the optimal thickness distribution for a minimal structural weight are computed, while maintaining structural strength and stability constraints. This modifies the stiffness of the structure, therefore the aeroelastic loop must be repeated. After the structural weight is converged, the endurance of the actual configuration needs to be evaluated with an aeroelastic calculation in cruise condition. In the shape

Fig. 6 Aeroelastic loop

Aero-structural Optimization of a MALE Configuration …

177

optimization loop (outer-most loop), the geometry of the wing is updated and the analysis models are morphed thereafter.

4.1 Analysis and Tool-Chain Preparation The implementation of the central aeroelastic loop was performed by four of the partners involved in the AGILE task 4.5. The aerodynamic competence is provided by AIRINNOVA and CFSE. The structural expertise is offered by Airbus DS, and the FSI is done by DLR Hamburg. Furthermore, DLR is hosting the structural tools of Airbus DS due to industrial network security issues. The reference design configuration layout of the OptiMALE is a CPACS [10] file, enriched during the preliminary design phase of this sub task. This database was used to produce the aerodynamic and structural analysis model used in this aeroelastic shape optimization workflow.

4.1.1

Aerodynamic Model Generation

In the FSI loop, the aerodynamic calculations were performed in Lausanne by CFSE. To perform an initial Euler calculation, an unstructured mesh of 2.3 M tetrahedron cells has been created by converting the original OptiMALE CPACS file to a SUMO file [11]. This operation is fully automated and allows to rapidly generate a surface mesh of the skin of the aircraft, using the open-source software SUMO [12]. As next step a volume mesh is generated by using the software Tetgen [13]. The model generation was performed with a standard Linux workstation and is shown in Fig. 7. This mesh has been used to calculate the initial step of the FSI loop, i.e. without any

Fig. 7 Aerodynamic surface mesh

178

R. Maierl et al.

deformation. The flight condition has been chosen to represent a pull-up maneuver with a load factor of n = 3 g. For this, the required lift coefficient C L was calculated from the initial MTOW (Maximum Takeoff Weight) of the aircraft. Then, the aerodynamic solver has been set to adapt the angle of attack iteratively in order to achieve a C L of 0.552 during the calculation. The lift coefficient needs to be corrected, if the aircraft weight changes during the optimization loop, only the load factors stay constant. The aircraft was in dive condition at sea level, which corresponds to a Mach number of 0.367.

4.1.2

Structural Model Generation

The structural model was also generated from the CPACS database, with the internal Airbus DS tool Descartes [14]. It is shown in Fig. 8. The wing structure is based on a wing-box with front-, rear- and auxiliary spars in span-wise direction. The spars are supported by several ribs in the chord-wise direction, and the rib pitch increases along the span. The stability of the wing-box skin is maintained by an evenly distributed set of one-dimensional stringer elements. The T-tail empennage is essentially constructed in a similar way. The fuselage structure consists of a web of one-dimensional stringers and frames on the skin. The stiffness of the fuselage is further increased by six bulkhead elements. The structural model is additionally enriched by the results of a mass model calculation. The masses of on-board systems, obtained in the preliminary design workflow, payload and fuel are attached to the structural model, and the secondary structural

Fig. 8 Structural FEM model of the OptiMALE

Aero-structural Optimization of a MALE Configuration …

179

mass is applied to finite-element properties. A more detailed view of the automated structural mesh generation and the mass model calculation is given in [15].

4.1.3

Fluid Structure Interaction Implementation

The mapping of the calculated loads and displacements between the aerodynamic and structural domains is performed at DLR. A mesh-free approach using the radial basis function (RBF) interpolation technique described by Beckert and Wendland [16] or Rendall and Allen [17] is adopted. It allows for the computation of the point displacements on the fluid domain boundary based on the displacements in the structural domain u st using Eq. (1), where H is the interpolation matrix. Conveniently, the point forces on the aerodynamic surface points can also be mapped to loads at the structural model points using the same approach, due to the principle of virtual work in Eq. (2). u f l = H ∗ u st

(1)

f st = H T ∗ f f l

(2)

The MLS (Moving Least Squares) algorithm computes the interpolation matrix by combining a least-squares polynomial fit and a radial basis function interpolation. For the analysis of the OptiMALE, a second-order polynomial basis and the Wendland C0 radial basis function [17] are chosen after several simulation and convergence tests. Figures 9 and 10 illustrate the fluid structure interaction during the aeroelastic workflow. The force distribution from a CFD solution, as provided by CFSE, being mapped to the Airbus DS structural mesh and the resulting displacements from the structural calculation which are then mapped back to the CFD surface mesh.

4.2 Optimization Work-Flow Application The structural sizing optimization is mainly a functionality of the internal Airbus DS software Lagrange [18]. The objective function is to minimize the structural weight by altering the thicknesses and areas of the finite element properties. The optimization is constrained by stress, strain and stability allowables with safety factors applied. The optimization is then started with a converged set of forces from the aeroelastic loop. After the structural sizing optimization converged with minimal structural weight, the stiffness of the aircraft has changed and this invalidates the initial aeroelastic load-case For this reason, the aeroelastic loop is repeated with updated structural model, and the converged set of forces from the last aeroelastic loop is taken as a starting point.

180

R. Maierl et al.

After a convergence of the structural weight is achieved, an evaluation of the endurance has to be performed with Eq. (3) [19]. The actual DL ratio is taken from another run of the aeroelastic loop in cruise flight condition and the actual fuel weight fraction can be calculated with the MTOW kept constant and the converged structural weight from the second sizing optimization step. E = T S FC −1 TSFC MTOW MZFW

MT OW L ln D M Z FW

Thrust specific fuel consumption Maximum takeoff weight Maximum zero fuel weight

Fig. 9 Aerodynamic forces

(3)

Aero-structural Optimization of a MALE Configuration …

181

Fig. 10 Structural displacement

The target function for the shape optimization is to maximize this endurance value. With a set of geometric design variables of the main wing e.g. the span, the chord or the aspect ratio, the DL ratio can be increased directly or the M Z F W can be decreased indirectly.

4.3 Analysis Model Update After the geometric shape design variables are updated by the optimizer, they have to be propagated to the analysis models as well. This is the task of the internal Airbus DS tool Descartes by morphing the FEM model. The structural mesh is kept constant with respect to the number of elements and nodes, and their connectivity is preserved. Only the coordinates of the grid points are changed. The aerodynamic mesh is based on a different geometry, so it was decided to reuse the method from the FSI loop by applying a structural shape change to the aerodynamic model, as it would be a displacement value. The robustness of this approach to update the analysis model is a limited, as it is very sensitive to topological changes.

4.4 Automated Workflow Execution Enabling process automation is a major topic of the AGILE project. The inner aeroelastic loop is automated via Brics [20] and RCE [21] (Remote Component Environment) in the following steps by CFSE as shown in Fig. 11. The different tools and their contribution to the workflow are explained in the following enumeration.

182

R. Maierl et al.

1. A CSV (Comma Separated Value) file from DLR is received via Brics and RCE at CFSE. It contains a list of displacement along the x, y, z-axes for each surface node of the aerodynamic mesh (the ordering of the mesh point remains constant during the complete process). 2. Three inputs are required for the execution of the loop: The displacement data received via Brics, the aerodynamic mesh (undeformed for the first step, then with converged deformations) and a default configuration file with the updated calculation parameters. 3. Using these inputs, a Python script controls the mesh deformation process implemented in the CFD software SU2 [22]. Firstly, it applies the displacement on the surface mesh and then it uses a spring analogy method to propagate the deformation within the volume mesh, maintaining the mesh connectivity. The deformed mesh is saved in SU2 mesh format. This step usually takes less than one hour. 4. The SU2 CFD code is used to solve Euler equation in parallel on the deformed mesh with the same C L as in the initial step. The calculation parameters have been chosen to achieve convergence as quickly as possible. The convergence of the forces is checked in each loop. The aerodynamic analysis is converged in about 20h without using a high performance computing environment. 5. From the resulting pressure field given as output of the SU2 calculation, the force (in the x, y, z-direction) on each surface mesh node is calculated and saved in a CSV file. 6. The force file is sent via Brics and RCE to DLR. 7. Step 1–6 are repeated until the FSI loop convergence is achieved.

5 Workflow Results This section shows the actual advances of the high-fidelity aeroelastic shape optimization work-flow. A solution of the converged aeroelastic loop is shown in Fig. 12. Here the pressure distribution of the deformed aircraft is shown. After the conversion to nodal forces on the aerodynamic mesh, the loads are applied on the structural nodes. This force distribution is shown in Fig. 13. These loads are used to generate the first results of the structural sizing optimization of the upper wing shell are pro-

Fig. 11 Automated aeroelastic process

Aero-structural Optimization of a MALE Configuration …

183

Fig. 12 3g pull-out aeroelastic deformation and pressure distribution

Fig. 13 3g pull-out aeroelastic force distribution

vided in Fig. 14. The thickness distribution shows the solution of the optimizer to handle the wing bending moment with the lowest amount of structural weight. The next step is to run the aeroelastic loop again with the updated structural stiffness and to converge the structural weight. Now the shape optimization can be performed and the aeroelastic- and the structural sizing loop are repeated with the updated analysis models. A design of experiments (DoE) was initiated with the aim to explore the design space and determine the limits of the used tools. Five geometric parameters of the wing were chosen to perform this study: the wing span and four different chord stations along the wing. To explore the corners of the design space, four updated geometry models of the OptiMALE were generated with the Airbus in-house tool Descartes. The wing span was altered between +10 and −5% and the four chord stations were varied between ±10%. The original values of the geometric parameters are shown

184

R. Maierl et al.

Fig. 14 Optimized aircraft structural thickness distribution Table 3 Adapted wing geometry parameters Span Chord 1 Chord 2 Chord 3 Chord 4

32.4 m 2.1 m 2.0 m 1.7 m 1.0 m

in Table 3 and the updated geometry models can be found in Fig. 15. The presented shape optimization workflow from Fig. 3 was performed up to the endurance evaluation. Additionally, the C D0 was assessed for the different configurations by a turbulent skin friction method with form factor corrections [23]. The results of the design space exploration are shown in Fig. 16. The Maximum Zero Fuel Weight (MZFW) is obtained from the sizing optimization and the MTOW is calculated by adding the fuel mass which has a fixed weight fraction. This method was chosen to eliminate the effect of additional endurance by simply adding fuel. The lift to drag ration was obtained from different sources. The lift and the induced drag are calculated with SU2 Euler simulation. The friction drag is provided by an external tool as stated in the last paragraph. Finally, the endurance evaluation is performed with Eq. (3). The trend, shown in this study, is going into the expected direction. It can be clearly seen, that the maximum possible endurance will not be gained if the optimization would be exclusively aerodynamic or structural, which means that neither the minimal MZFW nor the maximal DL can guarantee the maximum endurance of the aircraft design. The objective can only be maximized if both disciplines are regarded interdependent.

Aero-structural Optimization of a MALE Configuration …

185

Fig. 15 Updated aircraft geometry

Fig. 16 OptiMALE design of experiments result

6 Conclusion and Future Prospects The ongoing AGILE project enables collaboration between tool specialists from different institutes and different countries. This leads to a work-flow of existing methods to improve the OptiMALE aircraft. It is not straight forward to find efficient solutions for interfacing tools and communicating data and expertise in a large consortium. This is particularly challenging if there are additional issues for industrial partners as e.g. network security. The present paper shows the work that was done towards overcoming this challenge, in view of achieving the desired shape optimization process. Even if the final target could not be reached within the time-frame of the AGILE project, were all necessary components developed and integrated into an optimization framework. These main components are listed as following: • Aerodynamic analysis for different flight conditions • Structural analysis for different load-cases

186

• • • • •

R. Maierl et al.

Aeroelastic fluid-structure interface Structural sizing optimization Endurance evaluation Geometry based analysis model update Shape optimization process framework

First results are obtained for key components of this process. A design space exploration was performed to test the software capabilities and the workflow robustness. The automated set-up of the aeroelastic tool chain and the additional sizing optimization is tested successfully and the last steps towards a successful shape optimization with overall geometric parameters will be part of future developments. The work presented here would not been possible without the international partners and their expertise in the AGILE project, which finished very successfully and the consortium even received the ICAS—Award for Innovation in Aeronautics [24] in 2018. Acknowledgements The research presented in this paper has been performed in the framework of the AGILE project (Aircraft 3rd Generation MDO for Innovative Collaboration of Heterogeneous Teams of Experts) and has received funding from the European Union Horizon 2020 Programme (H2020-MG-2014-2015) under grant agreement n◦ 636202. The Swiss participation in the AGILE project was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number 15.0162. The authors are grateful to the partners of the AGILE consortium for their contributions and feedback.

References 1. AGILE Project. url: www.agile-project.eu. 2. Ciampa, P. D., & Nagel, B. (2018). Streamlining cross-organizational aircraft development processes: An overview of the AGILE project. In 31th Congress of the International Council of the Aeronautical Sciences, Belo-Horizonte. 3. Ciampa, P. D., & Nagel, B. (2017). AGILE Paradigm: Developing the next generation collaborative MDO. In 18th AIAA ISSMO Multidisciplinary Analysis and Optimization Conference, Denver. 4. Van Gent, I., Ciampa, P. D., Aigner, B., Jepsen, J., La Rocca, G., & Schut, J., Knowledge architecture supporting collaborative MDO in the AGILE Paradigm. In AIAA Paper 20174139. 5. Lefebvre, T., Bartoli, N., Dubreuil, S., Panzeri, M., Lombardi, R., D’ Ippolito, R., et al. (2017). Methodological enhancements in MDO process investigated in the AGILE European project. In 18th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Denver. 6. Elssel, K., Petersson, Ö. (2016). AeroStruct - Schlussbericht, Airbus Defence and Space. 7. Fioriti, M., Boggero, L., Corpino, S., Prakasha, P. S., Ciampa, P. D., & Nagel, B. (2018). The effect of sub-systems design parameters on preliminary aircraft design in a multidisciplinary design environment. Elsevier Transportation Research Procedia, 29, 135–145. 8. Fioriti, M., Boggero, L., Corpino, S., Isyanov, A., Mirzoyan, A., Lombardi, R., & D’Ippolito, R. (2017). Automated selection of the optimal on-board systems architecture. In 18th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Denver (US-CO). 9. Voskuijl, M., La Rocca, G., & Dircken, F., (2008). Controllability of blended wing body aircraft. In 26th Congress of International Council of the Aeronautical Sciences, Anchorage, Alaska, 14–19 September 2008.

Aero-structural Optimization of a MALE Configuration …

187

10. Liersch, C., & Hepperle, M. (2011). A distributed toolbox for multidisciplinary preliminary aircraft design. CEAS Aeronautical Journal, 2, 57–68. 11. Zhang, M., Jungo, A., Gastaldi, A. A., & Melin, T. (2018). Aircraft geometry and meshing with common language schema CPACS for variable-fidelity MDO applications. Aerospace, 5, 47. 12. Tomac, M., & Eller, D. (2011). From geometry to CFD grids. An automated approach for conceptual design. Progress in Aerospace Sciences, 47(No. 11), 589–596. 13. Si, H. (2013). TetGen: A quality tetrahedral mesh generator and 3D delaunay triangulator. WIAS technical Report, No. 13. 14. Maierl, R., Petersson, Ö., & Daoud, F. (2013). Automated creation of aeroelastic optimization models from a parameterized geometry. In 15th International Forum on Aeroelasticity and Structural Dynamics, Bristol. 15. Maierl, R., Petersson, Ö., Daoud, F., & Bletzinger, K. U. (2015). Automatic generation of aeroelastic simulation models combined with a knowledge based mass prediction. 5th CEAS Air & Space Conference, Delft. 16. Beckert, A., & Wendland, H. (2001). Multivariate interpolation for fluid-structure interaction problems using radial basis functions. Aerospace Science and Technology, 5(2), 125–134. 17. Rendall, T. C. S., & Allen, C. B. (2007). Unified fluid-structure interpolation and mesh motion using radial basis functions. International Journal for Numerical Methods in Engineering, 74, 1519–1559. 18. Schuhmacher, G., Daoud, F., Petersson, Ö, & Wagner, M. (2012). Multidisciplinary airframe design optimization. In 28th International Congress of the Aeronautical Sciences, Brisbane. 19. Anderson, J. D. (2000). Introduction to flight (4th ed). McGraw-Hill. 20. Baalbergen, E., Kos, J., Louriou, C., Campguilhem, C., & Barron, J. (2016). Streamlining cross-organisation product design in aeronautics. Proceedings of the Institution of Mechanical Engineers, Part G. 21. Seider D., Fischer P., Litz M., Schreiber A., & Gerndt A. (2012). Open source software framework for applications in aeronautics and space. In IEEE Aerospace Conference, Big Sky, MT, USA. 22. Palacios, F., Colonno, M. R., Aranake, A. C., Campos, A., Copeland, S. R., Economon, T. D., et al. (2013). Stanford University Unstructured (SU2): An open-source integrated computational environment for multi-Physics simulation and design, 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA 2013–0287. Texas, USA: Grapevine. 23. Mason, W. (2002). FRICTION v3 url: http://www.dept.aoe.vt.edu/~mason/Mason_f/MRsoft. html#SkinFriction. 24. ICAS—International Council of the Aeronautical Sciences url: https://www.icas.org/.