Topology Optimization Design of Heterogeneous Materials and Structures 9781786305589, 1786305585

Table of contents :
Cover......Page 1
Half-Title Page......Page 3
Title Page......Page 5
Copyright Page......Page 6
Contents......Page 7
I.1. Background and motivations......Page 11
I.2.1. Topology optimization methods......Page 14
I.2.2. Material design and multiscale optimization......Page 17
I.2.3. Fracture resistance design......Page 21
I.3. Outline of the book......Page 23
PART 1: Multiscale Topology Optimization in the Context of Non-separated Scales......Page 25
1. Size Effect Analysis in Topology Optimization for Periodic Structures Using the Classical Homogenization......Page 27
1.1.1. Localization problem......Page 28
1.1.2. Definition and computation of the effective material properties......Page 31
1.1.3. Numerical implementation for the local problem with PER......Page 33
1.2.1. Optimization model and sensitivity number......Page 34
1.2.2. Finite element meshes and relocalization scheme......Page 36
1.2.3. Optimization procedure......Page 38
1.3. Numerical examples......Page 40
1.3.1. Doubly clamped elastic domain......Page 41
1.3.2. L-shaped structure......Page 43
1.3.3. MBB beam......Page 48
1.4. Concluding remarks......Page 49
2. Multiscale Topology Optimization of Periodic Structures Taking into Account Strain Gradient......Page 53
2.1.1. Definition of local and mesoscopic fields through the filter......Page 54
2.1.2. Microscopic unit cell calculations......Page 57
2.1.3. Mesoscopic structure calculations......Page 63
2.2.1. Model definition and sensitivity numbers......Page 65
2.2.2. Overall optimization procedure......Page 66
2.3. Validation of the non-local homogenization approach......Page 67
2.4. Numerical examples......Page 69
2.4.1. Cantilever beam with a concentrated load......Page 70
2.4.2. Four-point bending lattice structure......Page 76
2.5. Concluding remarks......Page 79
3. Topology Optimization of Meso-structures with Fixed Periodic Microstructures......Page 81
3.1. Optimization model and procedure......Page 82
3.2.1. A double-clamped beam......Page 85
3.2.2. A cantilever beam......Page 88
3.3. Concluding remarks......Page 90
PART 2: Topology Optimization for Maximizing the Fracture Resistance......Page 91
4. Topology Optimization for Optimal Fracture Resistance of Quasi-brittle Composites......Page 93
4.1.1. Phase field approximation of cracks......Page 95
4.1.2. Thermodynamics of the phase field crack evolution......Page 96
4.1.3. Weak forms of displacement and phase field problems......Page 99
4.1.4. Finite element discretization......Page 100
4.2.1. Model definitions......Page 102
4.2.2. Sensitivity analysis......Page 104
4.2.3. Extended BESO method......Page 109
4.3. Numerical examples......Page 111
4.3.1. Design of a 2D reinforced plate with one pre-existing crack notch......Page 112
4.3.2. Design of a 2D reinforced plate with two pre-existing crack notches......Page 117
4.3.3. Design of a 2D reinforced plate with multiple pre-existing cracks......Page 120
4.3.4. Design of a 3D reinforced plate with a single pre-existing crack notch surface......Page 122
4.4. Concluding remarks......Page 125
5. Topology Optimization for Optimal Fracture Resistance Taking into Account Interfacial Damage......Page 127
5.1.1. Regularized representation of a discontinuous field......Page 128
5.1.2. Energy functional......Page 130
5.1.3. Displacement and phase field problems......Page 132
5.1.4. Finite element discretization and numerical implementation......Page 135
5.2.1. Model definitions......Page 138
5.2.2. Sensitivity analysis......Page 140
5.3. Numerical examples......Page 143
5.3.1. Design of a plate with one initial crack under traction......Page 144
5.3.2. Design of a plate without initial cracks for traction loads......Page 147
5.3.3. Design of a square plate without initial cracks in tensile loading......Page 149
5.3.4. Design of a plate with a single initial crack under three-point bending......Page 152
5.3.5. Design of a plate containing multiple inclusions......Page 154
5.4. Concluding remarks......Page 157
6. Topology Optimization for Maximizing the Fracture Resistance of Periodic Composites......Page 159
6.1. Topology optimization model......Page 160
6.2.1. Design of a periodic composite under three-point bending......Page 162
6.2.2. Design of a periodic composite under non-symmetric three-point bending......Page 170
6.3. Concluding remarks......Page 175
Conclusion......Page 177
References......Page 181
Index......Page 197
Other titles from iSTE in Numerical Methods in Engineering......Page 199
EULA......Page 203

Citation preview

Topology Optimization Design of Heterogeneous Materials and Structures

Series Editor Piotr Breitkopf

Topology Optimization Design of Heterogeneous Materials and Structures

Daicong Da

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2019 The rights of Daicong Da to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019947662 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-558-9

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Part 1. Multiscale Topology Optimization in the Context of Non-separated Scales . . . . . . . . . . . . . .

1

Chapter 1. Size Effect Analysis in Topology Optimization for Periodic Structures Using the Classical Homogenization . . . . . . . . . . . . . . . . . .

3

1.1. The classical homogenization method . . . . . . . . . 1.1.1. Localization problem . . . . . . . . . . . . . . . . 1.1.2. Definition and computation of the effective material properties . . . . . . . . . . . . . . . . . . . . . 1.1.3. Numerical implementation for the local problem with PER . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Topology optimization model and procedure . . . . . 1.2.1. Optimization model and sensitivity number . . . 1.2.2. Finite element meshes and relocalization scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3. Optimization procedure . . . . . . . . . . . . . . 1.3. Numerical examples . . . . . . . . . . . . . . . . . . 1.3.1. Doubly clamped elastic domain . . . . . . . . . . 1.3.2. L-shaped structure . . . . . . . . . . . . . . . . . 1.3.3. MBB beam . . . . . . . . . . . . . . . . . . . . . 1.4. Concluding remarks . . . . . . . . . . . . . . . . . . .

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4 4

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9 10 10

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12 14 16 17 19 24 25

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Topology Optimization Design of Heterogeneous Materials and Structures

Chapter 2. Multiscale Topology Optimization of Periodic Structures Taking into Account Strain Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Non-local filter-based homogenization for non-separated scales . . . . . . . . . . . . . . . . . 2.1.1. Definition of local and mesoscopic fields through the filter . . . . . . . . . . . . . . . . . . 2.1.2. Microscopic unit cell calculations . . . . . 2.1.3. Mesoscopic structure calculations . . . . . 2.2. Topology optimization procedure . . . . . . . 2.2.1. Model definition and sensitivity numbers . 2.2.2. Overall optimization procedure . . . . . . 2.3. Validation of the non-local homogenization approach . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Numerical examples . . . . . . . . . . . . . . 2.4.1. Cantilever beam with a concentrated load 2.4.2. Four-point bending lattice structure . . . . 2.5. Concluding remarks . . . . . . . . . . . . . . .

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30 33 39 41 41 42

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43 45 46 52 55

Chapter 3. Topology Optimization of Meso-structures with Fixed Periodic Microstructures . . . . . . . . . . . .

57

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58 61 61 64 66

Part 2. Topology Optimization for Maximizing the Fracture Resistance . . . . . . . . . . . . . . . . . . . . . .

67

Chapter 4. Topology Optimization for Optimal Fracture Resistance of Quasi-brittle Composites . . . . . . . . .

69

3.1. Optimization model and procedure 3.2. Numerical examples . . . . . . . . 3.2.1. A double-clamped beam . . . . 3.2.2. A cantilever beam . . . . . . . . 3.3. Concluding remarks . . . . . . . . .

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4.1. Phase field modeling of crack propagation . . . . . . . 4.1.1. Phase field approximation of cracks . . . . . . . . . 4.1.2. Thermodynamics of the phase field crack evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 72

Contents

4.1.3. Weak forms of displacement and phase field problems . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Finite element discretization . . . . . . . . . . 4.2. Topology optimization model for fracture resistance . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Model definitions . . . . . . . . . . . . . . . . 4.2.2. Sensitivity analysis . . . . . . . . . . . . . . . 4.2.3. Extended BESO method . . . . . . . . . . . . 4.3. Numerical examples . . . . . . . . . . . . . . . . 4.3.1. Design of a 2D reinforced plate with one pre-existing crack notch . . . . . . . . . . . . . . . . 4.3.2. Design of a 2D reinforced plate with two pre-existing crack notches . . . . . . . . . . . . . . . 4.3.3. Design of a 2D reinforced plate with multiple pre-existing cracks . . . . . . . . . . . . . . . . . . . 4.3.4. Design of a 3D reinforced plate with a single pre-existing crack notch surface . . . . . . . . . . . . 4.4. Concluding remarks . . . . . . . . . . . . . . . . .

vii

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75 76

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78 78 80 85 87

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88

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93

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96

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98 101

Chapter 5. Topology Optimization for Optimal Fracture Resistance Taking into Account Interfacial Damage . . . . . . . . . . . . . . . . . . . . . . .

103

5.1. Phase field modeling of bulk crack and cohesive interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Regularized representation of a discontinuous field . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Energy functional . . . . . . . . . . . . . . . . 5.1.3. Displacement and phase field problems . . . . 5.1.4. Finite element discretization and numerical implementation . . . . . . . . . . . . . . . . . . . . . 5.2. Topology optimization method . . . . . . . . . . . 5.2.1. Model definitions . . . . . . . . . . . . . . . . 5.2.2. Sensitivity analysis . . . . . . . . . . . . . . . 5.3. Numerical examples . . . . . . . . . . . . . . . . 5.3.1. Design of a plate with one initial crack under traction . . . . . . . . . . . . . . . . . . . . . . . . . .

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104

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104 106 108

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111 114 114 116 119

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120

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Topology Optimization Design of Heterogeneous Materials and Structures

5.3.2. Design of a plate without initial cracks for traction loads . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Design of a square plate without initial cracks in tensile loading . . . . . . . . . . . . . . . . . . . . 5.3.4. Design of a plate with a single initial crack under three-point bending . . . . . . . . . . . . . . . 5.3.5. Design of a plate containing multiple inclusions . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Concluding remarks . . . . . . . . . . . . . . . . .

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123

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125

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128

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130 133

Chapter 6. Topology Optimization for Maximizing the Fracture Resistance of Periodic Composites . . . . . .

135

6.1. Topology optimization model . . . . . . . . . . . . . 6.2. Numerical examples . . . . . . . . . . . . . . . . . . 6.2.1. Design of a periodic composite under three-point bending . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Design of a periodic composite under non-symmetric three-point bending . . . . . . . . . . . . 6.3. Concluding remarks . . . . . . . . . . . . . . . . . . .

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136 138

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138

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146 151

Conclusion

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153

References

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157

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173

Index

Introduction

In this Introduction, the background and motivations of the book are presented in section I.1. A literature review on related subjects, including topology optimization methods, material design and multiscale optimization, and fracture resistance design, is presented in section I.2. The outline of the book is presented in section I.3. I.1. Background and motivations Topology optimization has been an active research topic in the last decades and has become a subject of major importance with the growing development of additive manufacturing processes, which allow the fabrication of workpieces such as lattice structures with arbitrary geometrical details. In this context, topology optimization (Bendsøe and Kikuchi 1988, Allaire 2012) aims to define the optimal structural or material geometry with regard to specific objectives (e.g. maximal stiffness, minimal mass or maximizing other physical/ mechanical properties) under mechanical constraints such as equilibrium and boundary conditions. The key merit of topology optimization over conventional size and shape optimization is that the former can provide more design freedom, consequently leading to the creation of novel and highly efficient designs. With the topology optimization technique, designers can make the best use of limited materials and guide the concept design of various practical structures, especially in automotive and aerospace engineering.

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Topology Optimization Design of Heterogeneous Materials and Structures

In recent years, there has been an increase in the use of high-performance heterogeneous materials such as fibrous composite, concrete materials and 3D printed materials. Mechanical and physical properties of complex heterogeneous materials are determined, on the one hand, by the composition of their constituents but can, on the other hand, be drastically modified at a constant volume fraction of heterogeneities, by their geometrical shape and by the presence of interfaces. Topology optimization of microstructures can help design materials with higher effective properties while maintaining the volume fraction of constituents or obtaining new properties which are not naturally available (metamaterials). Recently, the development of 3D printing techniques and additive manufacturing processes has made it possible to directly manufacture designed materials from a numerical file, opening routes for new designs, as shown in Figure I.1. It is no exaggeration to say that “additive manufacturing” and “topology optimization” are the best couple made for each other. To this end, systematic and comprehensive research on the topological design of complex heterogeneous materials is of great significance for academic research and engineering applications.

(a)

(b) Figure I.1. 3D printed lattice materials: (a) cubic and (b) cylindrical configurations (Mohammed et al. 2017)

However, in topology optimization of material modeling, the scale separation is often assumed. This assumption states that the characteristic length of microstructural details is much smaller than the dimensions of the structure, or that the characteristic wavelength of

Introduction

xi

the applied load is much larger than that of the local fluctuation of mechanical fields (Geers et al. 2010). In additive manufacturing of architectured materials such as lattice structures, the manufacturing process may induce limitations on the size of local details, which can lead to a violation of scale separation when the characteristic size of the periodic unit cell within the lattice is not much smaller than that of the structure. In such a case, classical homogenization methods may lead to inaccurate description of the effective behavior as non-local effects, or strain-gradient effects may occur within the structure. On the other hand, using a fully detailed description of the lattice structure in an optimization framework could be computationally expensive. One objective of this book is to develop multiscale topology optimization procedures not only for heterogeneous materials but also for mesoscopic structures in the context of non-separated scales.

(a)

(b)

Figure I.2. Damage phenomena in engineering: (a) macroscopic structure; (b) cracks (Nguyen 2015)

On the other hand, fatigue or failure characteristics of engineering structures are another subject of great concern, as shown in Figure I.2. Microcracking is known to be a significant factor affecting the mechanical properties and the long-term behavior of engineering facilities. The accurate modeling of these phenomena, as well as their coupled effects have received special attention. In addition, topology optimization design of composite materials accounting for fracture resistance is a rather challenging task. It is necessary to improve the fracture resistance of heterogeneous materials in terms of the required mechanical work, through an optimal placement of the inclusion phase,

xii

Topology Optimization Design of Heterogeneous Materials and Structures

taking into account the crack nucleation, propagation and interaction. However, this research remains relatively unexplored so far due to the following reasons. First, there has been a lack of robust numerical methods for fracture propagation in the presence of complex heterogeneous media until recently, especially when interface effects are presented. Second, these numerical simulation models should be formulated in a context compatible with the topology optimization scheme. For these reasons, there has been very limited research in the literature on topology optimization for maximizing the fracture resistance of heterogeneous materials before the recent works from the author and his coworkers (Xia et al. 2018a, Da et al. 2018a). I.2. Literature review In the following, section I.2.1 provides a brief literature review on the development of topology optimization methods. Section I.2.2 reviews material microstructure design and extension to multiscale topology optimization with or without scale separation. Section I.2.3 presents the newly proposed fracture resistance design framework, by combining the phase field method to take into account the heterogeneities and their interfaces in the material. I.2.1. Topology optimization methods Over the past decades, topology optimization has undergone a tremendous development since the seminal paper by Bendsøe and Kikuchi (1988). The key merit of topology optimization over conventional size and shape optimization is that the former can provide more design freedom, consequently leading to the creation of novel and highly efficient designs. Various topology optimization methods have been proposed so far, for example density-based methods (Bendsøe 1989, Zhou and Rozvany 1991, Bendsøe and Sigmund 2004), evolutionary procedures (Xie and Steven 1993, 1997), level-set method (LSM) (Sethian and Wiegmann 2000, Wang et al. 2003, Allaire et al. 2004), hybrid cellular automaton (Tovar et al. 2004) and

Introduction

xiii

phase field method (Bourdin and Chambolle 2003). All of these methods are based on finite element analysis (FEA) where the design domain is discretized into a number of finite elements. With such a setting, the optimization procedure is then to determine which points of the design domain should be full of material (solid elements) and which void (soft elements), as shown in Figure I.3. According to the update algorithm, these methods can be generally categorized into two groups: density variation and shape/boundary variation. The topology optimization technique has already become an effective tool for both academic research and engineering applications. A general review of various methods and their applications was presented by Deaton and Grandhi (2014). Regarding their strengths, weaknesses, similarities and dissimilarities, a critical review and comparison on different approaches was also given by Sigmund and Maute (2013). Design region

?

Figure I.3. Illustration for structure topology optimization

Level-set method (LSM) is a typical shape/boundary variation approach that maintains the capability of topological change. It describes the structural topology implicitly by the iso-contours of a level-set function. Using the LSM, a fixed rectilinear spatial grid and a finite element mesh of a given design domain can be constructed separately, which allows the separation of the topological description from the physical model. With the merits of the flexibility in handling complex topological changes and the smoothness of boundary representation, the LSM has been successfully applied to an increasing variety of design problems, involving, for example, multi-phase materials (Wang and Wang 2004), shell structures (Park and Youn 2008), geometric nonlinearities (Luo and Tong 2008), stress

xiv

Topology Optimization Design of Heterogeneous Materials and Structures

minimization (Allaire and Jouve 2008) and contact problems (My´sli´nski 2008). The reader can refer to the comprehensive review in van Dijk et al. (2013) for more theoretical details of different LSMs for structural topology optimization. Density-based methods are the most commonly used topology optimization approaches, such as the popular solid with isotropic material with penalization (SIMP) method. The SIMP method uses continuous design variables for topology optimization, which can be interpreted as material pseudo densities (Bendsøe 1989, Zhou and Rozvany 1991, Mlejnek 1992). The physical justification of the SIMP method was provided by Bendsøe and Sigmund (1999). A popular 99-line topology optimization Matlab code using the SIMP method was developed by Sigmund (2001) for education purposes. As a successor of the 99-line code, a more efficient 88-line Matlab code was also provided by Andreassen et al. (2011) with high computational efficiency and alternative filter implements. More details about theory, numerical methods and applications on the SIMP method can be found in (Bendsøe and Sigmund 2004). As another important branch of topology optimization, evolutionary structural optimization (ESO) (Xie and Steven 1993, 1997, Tanskanen 2002) and its later version bidirectional ESO (BESO) (Da et al. 2018c) have shown promising performance when applying to a wide range of structural design problems. ESO-type methods use a simple heuristic scheme to evolve the structural topology towards an optimum by gradually removing redundant or inefficient materials. The BESO method allows not only material removal but also material addition, showing efficient and reliable performance in various design problems (Huang and Xie 2008, 2010, Huang et al. 2011, Xia and Breitkopf 2014a,b, 2015b, Huang et al. 2015, Vicente et al. 2015, Da et al. 2017a). The early development of ESO-type methods was summarized by Xie and Steven (Xie and Steven 1997). The development of the BESO method and its various applications up to 2010 can be found in Huang and Xie (2010). A comprehensive review on the BESO method for advanced design of structures and materials was recently presented by Xia et al. (2018b).

Introduction

xv

As an extension to the original BESO method, the author and his collaborators proposed a new evolutionary topology optimization (ETO) method (Da et al. 2018c) to design continuum structures, by introducing a sensitivity-based level-set function (LSF). The proposed ETO method identifies the topology far beyond its elements, which does not involve the removal/addition of elements during the optimization process, resulting in a smoothed boundary representation and high robustness. The smooth structural topology has been extended to the robust topology optimization of continuum structures under loading and material uncertainties by Martínez-Frutos and Herrero-Pérez (2018). Inspired by the ETO method, the material removal scheme of evolutionary-type methods has been combined with the LSM to nucleate holes in the structure for optimization design of heat conduction (Xia et al. 2018c). Recently, a new computational framework for structural topology optimization based on the concept of moving morphable components has been proposed (Guo et al. 2014). The basic idea of this method is to use a set of deformable components as the basic building block of optimization structures, in order to tailor the structure topology through deformation, merge and overlap operations between components. Therefore, the design variables of the method are reduced during the topology optimization process, and the topological geometries of the structure can be presented explicitly (Zhang et al. 2016, Guo et al. 2016, Zhang et al. 2017). I.2.2. Material design and multiscale optimization Initially restricted to optimizing the geometry of structures, topology optimization techniques have now been extended to optimizing the topology of the phase within materials, for example in periodic microstructures, to design high performance materials (Sigmund 1994, Sigmund and Torquato 1997, Sigmund 2000, Yi et al. 2000, Guest and Prévost 2006, 2007, Wang et al. 2014a, Andreassen and Jensen 2014, Chen and Liu 2014, Huang et al. 2015), materials with properties not found in nature (e.g. negative Poisson’s ratio, zero

xvi

Topology Optimization Design of Heterogeneous Materials and Structures

compressibility, negative bulk modulus; see Wang et al. (2014b), Clausen et al. (2015), Da et al. (2017b), Noguchi et al. (2018)) or complex multi-physics problems (Nanthakumar et al. 2016, 2017). These techniques are based on optimizing the homogenized properties of the representative volume element and using numerical solving methods such as finite elements to compute the homogenized properties (Michel et al. 1999, Hassani and Hinton 1998a,b, Andreassen and Andreasen 2014), given one geometry of the phases and their microscopic properties, as shown in Figure I.4. A review of topology optimization of microstructures in the linear context can be found in, for example, Cadman et al. (2013).













Figure I.4. Material topologies with extreme elastic modulus and negative Poisson’s ratio: (a, b) geometries with maximum bulk modulus, (c) geometry with maximum shear modulus and (d) geometry with negative Poisson’s ratio (Da et al. 2017b)

Rather than pure material design, material microstructures have also been tailored for a fixed structure (Huang et al. 2013) to maximize macroscopic performance under specific boundary conditions, for example structural stiffness (Da et al. 2018d). In order to fully release the design freedom within multiscale optimization, Rodrigues et al. (2002) first described a hierarchical computational procedure for optimization of material distribution as well as the local material properties of mechanical elements that was later extended to 3D in Coelho et al. (2008) and to account for hyperelasticity. With this design strategy, simultaneous structure and materials design has been extensively studied, such as for composite laminate orientations

Introduction

xvii

(Setoodeh et al. 2005, 2006, Coelho et al. 2015), closed liquid cell materials (Lv et al. 2014) or multi-objective functions, for example maximum stiffness and minimum resistance to heat dissipation in de Kruijf et al. (2007) or minimum thermal expansion of the surfaces in Deng et al. (2013). Extensions to nonlinear materials (Xia and Breitkopf 2014a), multiple-phase materials (Da et al. 2017a) and optimization considering uncertainties (Guo et al. 2015, Xu et al. 2015) have been proposed recently. In the context of non-separated scales, the effectiveness of the classical homogenization-based multiscale topology optimization framework for periodic lattice structures will be first investigated here in this book. The characteristic dimensions of periodic unit cells in the lattice are comparable with the dimensions of the whole structure such that the two scales cannot be clearly separated. The dimensions of the unit cell range from large to small compared with the dimensions of the whole structure to highlight the size effect. By assuming that the material microstructures are infinitely small, the inverse homogenization designs for macroscopic structural performance were compared with the mono-scale topology optimization framework in Xie et al. (2012) and Zuo et al. (2013a). On the other hand, several computational homogenization methods modeling complex heterogeneous media when scales are not separated are available (e.g. gradient models in Peerling et al. (1996) and Kouznetsova et al. (2002), non-local elasticity theories in Eringen and Edelen (1972) and domain decomposition methods in Ladevèze et al. (2001)). Among them, the filter-based non-local homogenization technique developed in Yvonnet and Bonnet (2014a,b) and Tognevi et al. (2016) was adopted by Da et al. (2018b) to develop a topology optimization procedure for heterogeneous lattice materials in the context of non-separated scales, taking into account the strain gradient effects. The technique generalizes the homogenization theory by replacing spatial averaging operators by linear low-pass filters, and the major advantage is that it can take into account an arbitrary level of

xviii

Topology Optimization Design of Heterogeneous Materials and Structures

strain gradient without higher-order elements, in a classical finite element framework. In the case of fixed/optimized microscopic periodic cells, multiscale topological design of mesoscopic structures without scale separation is firstly proposed in this book and will be detailed in a later chapter. The idea is to use a computational homogenization method that takes into account the strain gradient effects combined with a topology optimization scheme of mesoscopic structures, allowing the topology optimization problem to be performed on a coarse mesh, instead of using the fully detailed description of the structure for computational saving, as shown in Figure I.5. In addition, other studies, for example Zhang and Sun (2006) and Alexandersen and Lazarov (2015) have also been devoted to the topology optimization of structures in the context of non-separated scales.

(a)

(b)

Figure I.5. Illustration of the two-scale optimized structure composed of the patterned microstructure periodically with (a) scale separation and (b) non-separated scales. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Introduction

xix

I.2.3. Fracture resistance design Optimization design of composite materials accounting for fracture resistance has remained relatively unexplored so far, mainly due to the lack of robust numerical methods for simulation of fracture propagation in the presence of complex heterogeneous media and interfaces until recently. In addition, these numerical simulation models should be formulated in a context compatible with topology optimization (e.g. finite elements). Gu et al. (2016) used a modified greedy optimization algorithm for composites made up of soft and stiff building blocks to improve material toughness. San and Waisman (2016) explored the optimal location of carbon black particles to maximize the rupture resistance of polymer composites using a genetic algorithm. In a recent work by the author and his coworkers (Xia et al. 2018a), topology optimization for maximizing the fracture resistance of quasi-brittle composites was introduced by combining the phase field method and a gradient-based BESO algorithm. However, in the mentioned work, the crack propagation resistance was only evaluated on the basis of phase distribution. In most heterogeneous quasi-brittle materials (e.g. ceramic matrix composites, cementitious materials), the interfacial damage plays a central role in the nucleation and propagation of microcracks (Tvergaard 1993, Lamon et al. 2000, Nguyen et al. 2016a, Narducci and Pinho 2017). Therefore, we further extended the design framework developed by Xia et al. (2018a) in order to define through topology optimization the optimal phase distribution in a quasi-brittle composite with respect to fracture resistance, taking into account crack nucleation in both the matrix and the interfaces, as shown in Figure I.6. To the author’s best knowledge, such a study is investigated in this book for the first time. Simulating interfacial damage and its interaction with matrix crack for complex heterogeneous materials is a highly challenging issue for meshing algorithms. Many numerical methods such as the eXtended Finite Element Method (XFEM) (Moes et al. 1999, Sukumar et al. 2000), the thick level-set (TLS) method (Bernard et al. 2012, Cazes and Moes 2015) and the phase field method (PFM) (Francfort and

xx

Topology Optimization Design of Heterogeneous Materials and Structures

Marigo 1998, Miehe et al. 2010a, Kuhn and Müller 2010), among the most recent popular techniques, have been introduced to investigate this topic.

Bulk cracks

Interfaces

Matrix

Inclusions

(a)

Load

Peak Load

Fracture Energy

Displacement

(b)

Figure I.6. (a) Structure containing both bulk cracks and interface cracks, possibly occurring at the interfaces, and (b) mechanical response of the damageable structure (Da et al. 2018a). For a color version of this figure, see www.iste.co.uk/da/topology.zip

Based on the pioneering work of Marigo and Francfort (Francfort and Marigo 1998), the phase field method makes use of a variational principle framework for brittle fracture (Bourdin et al. 2000) and a regularized description of the discontinuities related to the crack front (Mumford and Shah 1989, Ambrosio and Tortorelli 1990). The method was adapted to a convenient algorithmic setting by Miehe et al. (2010b). The main advantages of the method are as follows: (a) crack paths are mesh-independent; (b) initiation and propagation of multiple, complex cracks patterns can be easily handled; (c) the method is convergent with respect to the mesh size and not sensitive to the mesh regularity; (d) it is stable due to its inherent gradient-based formulation. In this book, we use the extension of the phase field to interfacial damage, as proposed by Nguyen et al. (2016b), to take into account both bulk brittle fracture and interfacial damage.

Introduction

xxi

While numerous pieces of research do exist on topology optimization, including material interface behavior (see, for example, Vermaak et al. (2014), Lawry and Maute (2015), Liu et al. (2016), Behrou et al. (2017)), with the enforcement of stress constraints (Duysinx and Bendsøe 1998, Guo et al. 2011, Xia et al. 2012, Bruggi and Duysinx 2012, Luo et al. 2013, Cai et al. 2014, Cai and Zhang 2015), local buckling constraint (Sun et al. 2017) or damage constraints (Challis et al. 2008, Amir and Sigmund 2012, Amir 2013, Jansen et al. 2013, James and Waisman 2014, Kang et al. 2016), topology optimization for maximizing the fracture resistance, taking into account interactions between interfacial damage and bulk brittle fracture for complete fracturing process, is to the best of our knowledge explored for the first time here. I.3. Outline of the book The aim of this book is to comprehensively study the topological design of complex heterogeneous materials and structures, including multiscale topology optimization without scale separation, phase field modeling and fracture resistance design. This book is organized as follows. In Part 1, we focus on multiscale topology optimization in the context of non-separated scales. In this model, we primarily present a topology optimization for periodic structures based on the classical homogenization. The dimensions of the unit cell range from large to small when compared with the dimensions of the whole structure to highlight the size effect. Within the similar established multiscale topology optimization framework without scale separation, we use a filter-based non-local homogenization method to take into account the effects of strain gradient, allowing the topology optimization problem to be performed on a coarse mesh, instead of using the fully detailed description of the structure for computational saving. In addition, the multiscale optimization model has been implemented to design the geometry of mesoscopic structures with specific microscopic unit cells. The microscopic RVE itself is defined as the design variable, and a

xxii

Topology Optimization Design of Heterogeneous Materials and Structures

stiffness interpolation is introduced to derive the sensitivities in a clear manner. In Part 2, we propose a topology optimization framework for optimizing the fracture resistance of quasi-brittle composites through a redistribution of the inclusion phases. A phase field method for fracture able to take into account initiation, propagation and interactions of complex microcrack networks is adopted. This formulation avoids the burden of remeshing problems during crack propagation and is well adapted to the topology optimization purpose. An efficient design sensitivity analysis is performed by using the adjoint method, and the optimization problem is solved by an extended BESO method. The sensitivity formulation accounts for the whole fracturing process involving crack nucleation, propagation and interaction, either from the interfaces and then through the solid phases, or the opposite. The spatial distribution of material phases is optimally designed using the extended BESO method to improve fracture resistance. We demonstrate through numerical examples that the fracture resistance of the composite can be significantly increased at a constant volume fraction of inclusions by the topology optimization process. In the Conclusion, we provide some perspectives on future research.

PART 1

Multiscale Topology Optimization in the Context of Non-separated Scales

Topology Optimization Design of Heterogeneous Materials and Structures, First Edition. Daicong Da. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

1 Size Effect Analysis in Topology Optimization for Periodic Structures Using the Classical Homogenization

In topology optimization of material modeling, the homogenization method is often adopted to link the microscopic and macroscopic scales. However, scale separation is often assumed in classical homogenization theory. This assumption states that the characteristic lengths of the microstructural details are much smaller than the dimensions of the structure, or that the characteristic wavelength of the applied load is much larger than that of the local fluctuation of mechanical fields (Geers et al. 2010). In additive manufacturing of architecture materials such as lattice structures, the manufacturing process may induce limitations on the size of local details, which can lead to a violation of scale separation when the characteristic size of the periodic unit cells within the lattice are not much smaller than that of the structure. In such a case, classical homogenization methods may lead to inaccurate description of the effective behavior as non-local effects, or strain-gradient effects, may occur within the structure. On the other hand, using a fully detailed description of the lattice structure in an optimization framework could be computationally expensive. In this chapter, we first develop a multiscale topology optimization procedure for periodic structures based on classical homogenization theory in the context of non-separated scales. The dimensions of the

4

Topology Optimization Design of Heterogeneous Materials and Structures

unit cell are not negligible with respect to the whole structure, which range from large to small compared to those of the structure to highlight the size effect. More specifically, a relocalization scheme is proposed based on the homogenization method to link the two scale fields, allowing the topology optimization problem to be solved on a coarse mesh. A corresponding reference solution is obtained by fully meshing the heterogeneities in the whole structure. The classical homogenization technique is reviewed in section 1.1, where details of the numerical computation of effective material properties as well as the relocalization scheme are also provided. The presented topology optimization model and procedure are given in section 1.2. Numerical experiments are conducted in section 1.3 to fully investigate the size effect of the periodic unit cells. Finally, section 1.4 draws the conclusion. 1.1. The classical homogenization method In this section, the computation of effective or homogenized material properties of heterogeneous composites in the context of linear elasticity is reviewed. The local problem at the microscopic scale with different types of boundary conditions is first introduced, followed by the numerical implementation in order to evaluate the effective elastic matrix using the classical finite element method (FEM). 1.1.1. Localization problem A heterogeneous structure composed of two-phase composites is considered, as shown in Figure 1.1(a). The RVE shown in Figure 1.1(b) is associated with a domain Ω and boundary ∂Ω. The ¯H. objective is to define the effective or homogenized elastic tensor C The constitutive material phases are assumed to be isotropic with constant elastic properties, and the interfaces between the different constitutive phases are assumed to be perfect. Therefore, the local problem assuming that the RVE is subject to homogeneous strains can

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

5

¯ be formulated as follows. Applying a constant macroscopic strain ε, we find the displacement field μ(x) in Ω which satisfies: ∇ · (σ(x)) = 0 in Ω

[1.1]

σ(x) = C(x) : ε(x)

[1.2]

ε = ε¯ in Ω

[1.3]

and

with

where C(x) is the constant fourth-order elasticity tensor associated with different phases, ∇ · (·) is the divergence operator and · is the space averaging over Ω. Following Yvonnet (2018), the split of the local strain field into a constant macroscopic strain field ε¯ and the remaining local fluctuation ε˜ is assumed: ¯ ˜ ε(x) = ε(x) + ε(x)

[1.4]

(b)

(c)

(a)

Figure 1.1. Illustration of a heterogeneous structure composed of periodic unit cells: (a) heterogeneous structure; (b) RVE; (c) homogenized material

6

Topology Optimization Design of Heterogeneous Materials and Structures

Averaging the above equation, we obtain  1 ¯ ˜ ¯ ˜ ε(x) = ε + ε(x) = ε + ε(x)dΩ |Ω| Ω  1 ˜ ˜ = ε¯ + {∇(u(x)) + ∇T (u(x))}dΩ 2|Ω| Ω

[1.5]

˜ is the unknown fluctuation displacement, defined as where u ˜ = u(x) − εx. ¯ Using the divergence theorem, we obtain u 1 ε(x) − ε¯ = 2|Ω|

 ∂Ω

˜ ˜ {u(x) ⊗ n + n ⊗ (u(x)}dΓ

[1.6]

Note that the condition [1.3] can be satisfied by solving the present local problem equations [1.1–1.2] with appropriate boundary conditions. In other words, the right-hand side of equation [1.6] should be equal to zero. It can be verified by the following two possible conditions: ˜ ˜ u(x) = 0 on ∂Ω or u(x) is periodic on ∂Ω

[1.7]

Following Yvonnet (2018), the corresponding two types of boundary conditions can be obtained by integrating equation [1.4] with respect to x. The first is called the kinematically uniform boundary condition (KUBC), which can be expressed as ¯ ∀x ∈ ∂Ω u(x) = εx,

[1.8]

where the displacement is directly imposed at boundary points. In the second type of boundary condition, called the periodic boundary condition (PER), the displacement field over the boundary ∂Ω takes the form ¯ + u, ˜ ∀x ∈ ∂Ω u(x) = εx

[1.9]

˜ where the fluctuation term u(x) is periodic on ∂Ω; that is, it takes the same values on two points of opposite faces over ∂Ω. It should be mentioned that an alternative localization problem assuming that the

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

7

RVE is subjected to a constant stress field can also be used to predict the effective material properties. For detailed information about the stress averaging theorem as well as the corresponding boundary conditions, we can refer to Yvonnet (2018). In this work, the second type of boundary condition is adopted to solve the present localization problem. It is also worth noting that even though only two-phase RVEs are considered here, the procedure can be straightforwardly extended to an arbitrary number of phases. 1.1.2. Definition and computation of the effective material properties According to the superposition principle, the solution of the local problem in equations [1.1–1.2] can be considered as a linear combination of three independent components of the strain tensor in two dimensions: u(x) = u(11) (x)¯ ε11 + u(22) (x)¯ ε22 + 2u(12) (x)¯ ε12

[1.10]

where u(ij) (x) is the displacement field obtained by solving the local problem (equations [1.1–1.2]) together with the PER (equation [1.9]) using 1 ε¯ = (ei ⊗ ej + ej ⊗ ei ) 2

[1.11]

where ei (i = 1, 2) are unitary basis vectors. More specifically, u(11) , u(22) and u(12) are respectively obtained by solving the local problem with       0 1/2 00 10 [1.12] , ε¯ = , ε¯ = ε¯ = 1/2 0 01 00 Setting ε(ij) (x) = ε(u(ij) (x)), we obtain: ε(x) = ε(11) (x)¯ ε11 + ε(22) (x)¯ ε22 + 2ε(12) (x)¯ ε12

[1.13]

8

Topology Optimization Design of Heterogeneous Materials and Structures

This expression can be rewritten in a compact form as ¯ ∀x ∈ Ω ε(x) = A(x) : ε,

[1.14]

where A(x) is the fourth-order localization tensor relating microscopic and macroscopic strains with (kl)

Aijkl (x) = εij (x)

[1.15]

Using the classical Hooke’s law, we obtain ε(kl) σpq (x) = Cpqij (x)Aijkl (x)¯

[1.16]

σ(x) = C(x) : Aijkl (x) : ε¯

[1.17]

or

By taking the spatial average over Ω, the constitutive relationship at the macroscopic scale can be formulated as: ¯ H : ε¯ ¯ =C σ

[1.18]

¯ H = C(x) : A(x) C

[1.19]

with

Using the classical displacement-based FEM, a matrix U containing in each row the nodal displacement solution of the above three local problems in one element is defined. Therefore, the matrix form of the localization tensor A in equation [1.14] or [1.15] can be written as A(x) = B(x)U

[1.20]

where B is the strain matrix. The matrix form of the effective elasticity ¯ H in equation [1.19] is then given by tensor C  1 H ¯ C(x)B(x)U dΩ [1.21] C = V Ω

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

9

1.1.3. Numerical implementation for the local problem with PER Following Yvonnet (2018), a technique based on Lagrange multipliers to enforce the PER in 2D cases is presented here. From equation [1.9], the displacement on the opposite faces of the RVE can be expressed as  uk+ ¯ij xk+ ˜∗i i =ε j +u [1.22] k− k− ui = ε¯ij xj + u ˜∗i where superscripts “k+” and “k−” denote nodes on the pair of opposite surfaces of the cell, and u ˜∗i is the periodic fluctuation field, which can be eliminated by comparing the difference between the displacements: k− k− k+k− uk+ ¯ij (xk+ i − ui = ε j − xj ) = R

[1.23]

Therefore, in order to enforce the periodic boundary conditions using the Lagrange multiplier method, the constraint equations can be written as: Pu − R = 0

[1.24]

The discrete form is written as Cik+k− = Pij uj − Rik+k− = 0

[1.25]

where P is a matrix that relates the whole indices of coupled nodes on opposite faces of the RVE, which is completely filled with the numbers 1, 0 and –1. It should be noted that two constraint equations exist for each pair of nodes on the boundaries in this case. The constrained minimization problem can then be stated as inf u

Ci =0, i=1,...,nc

1 T u Ku 2

[1.26]

where u is the global vector of required displacement and nc is the number of constraint equations. Introducing the vector of Lagrange

10

Topology Optimization Design of Heterogeneous Materials and Structures

multipliers Λ associated with the adopted periodicity constraints, the above equation can be rewritten as 1 L = uT Ku + Λ · (P u − R) 2 The stationary of L is expressed as  Dδu L = 0 DδΛ L = 0 Therefore, we obtain  T δu Ku + Λ · P δu = 0 δΛT P u = δΛT R

[1.27]

[1.28]

[1.29]

Because of the arbitrariness of δu and δΛ, the following linear system can be obtained:  K P

PT 0

    0 u = R Λ

[1.30]

where K is the global stiffness matrix after discretizing the elastic problem [1.1] without enforcing the Dirichlet boundary conditions, and the vector R can be trivially obtained from equation [1.23]. 1.2. Topology optimization model and procedure 1.2.1. Optimization model and sensitivity number In this chapter, the macroscopic structure is assumed to be composed of microscopic substructures/RVEs periodically. Effective material properties of the considered heterogeneous microstructures are calculated based on the above formulated homogenization method, even though the length scale of the microscopic RVE is comparable to the higher-scale structures. The topology of the RVE is tailored by means of topology optimization such that the obtained structure has the

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

11

optimal stiffness with a certain amount of materials. The final goal is to investigate the size effect of the RVE when micro- and macroscales are clearly non-separated. Therefore, this work aims to answer the following question: when the ratio between the size of the RVE and the size of the real macrostructure is taken into account, can the classical homogenization method be used as an effective tool for multiscale topology optimization of periodic structures without scale separation? This topology optimization problem can be stated mathematically as follows: Find : {ρ(1) , . . . , ρ(Ns ) } ¯T u ¯) = F ¯ Minimize : fc (ρ, u

[1.31] [1.32]

¯u = F ¯ Subject to : K¯ [1.33]  (k) : V (ρ) = ρ(k) e ve = Vreq , k = 1, . . . , Ns [1.34] Ns : ρ(1) e = · · · = ρe , e = 1, . . . , Ne

[1.35]

: ρ(k) e = ρmin or 1, e = 1, . . . , Ne , k = 1, . . . , Ns [1.36] ¯ where fc is the objective function of macrostructural compliance, F ¯ are the global load and displacement vectors respectively, Vreq is and u the required/prescribed volume of solid material in each RVE, and Ns is the number of periodic unit cells within the structure. Note that the (1) s constraint ρe = · · · = ρN e , e = 1, . . . , Ne is prescribed to ensure that the structure is composed of periodic microstructures with the existence of the RVE, which means that the pseudo densities of elements (ρmin or 1 in equation [1.36]) at the corresponding locations in each substructure are the same. Here, Ne is the number of finite elements in each RVE. During the process of evolutionary-type structural optimization, the elements are removed or added based on their sensitivity numbers. Therefore, the elements at the same locations in different substructures are removed or added simultaneously. However, the strain/stress

12

Topology Optimization Design of Heterogeneous Materials and Structures

distribution in different substructures/microscopic unit cells may not be the same in most cases. To enforce the periodic array of the microscopic unit cells, the element sensitivity numbers at the same location in each unit cell need to be consistent. They are then defined as the summation of the sensitivity of corresponding elements in all unit cells. In the conventional evolutionary structural optimization method (see, for example, Chu et al. (1996) and Huang and Xie (2007)), the element sensitivity number is defined as the change of the structural compliance or total strain energy since the removal of that element, which is then equal to the element strain energy. Therefore, the element sensitivity number in this scheme can be expressed as the variation of the overall structural compliance due to the removal of e−th elements in all substructures:  (k) Ns cla cla k cla k=1 Ωke σ (x)ε (x)dΩe , for ρe = 1 αe = [1.37] (k) 0, for ρe = ρmin . where σ cla (x) and εcla (x) are the relocalized stress and strain fields based on the classical homogenization method. Details about the computation of the relocalized stress and strain as well as the assembly ¯ in [1.33] are formulated in the next of the global stiffness matrix K section. 1.2.2. Finite element meshes and relocalization scheme ¯ in equation [1.33] at the In order to assemble the stiffness matrix K macroscopic scale, the local problem [1.1–1.3] should be solved to obtain the effective material properties at the lower scale. In this scheme, a fine mesh is adopted at the microscopic scale to account for all heterogeneous details within the RVE. However, coarse meshes are used to carry out the finite element analysis for the macrostructure in order to save computational cost. Specific finite element meshes for both macroscopic and microscopic problems are presented in the numerical examples section. It should be noted that the sensitivity number in equation [1.37] is formulated at the finest microscopic mesh; therefore, a relocalization process is required after solving the

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

13

structural problem based on coarse meshes at the higher scale to obtain the sensitivities in [1.37]. ¯ H of the RVE (see With the calculated effective material properties C ¯ can be defined in a standard section 1.1), the global stiffness matrix K finite element way as  ¯ H B(x)dΩ, ¯ B T (x)C [1.38] K= k

Ωk

¯ H are the strain matrix and the effective elastic matrix where B and C within the coarse mesh element at the macroscale respectively. With the solution of the macroscopic problem based on the coarse mesh at hand, the microscopic strain and stress fields can be reconstructed using the localization operator in each RVE as εcla (x) = A(x) : ε¯(¯ u(x)), ∀x ∈ Ω

[1.39]

σ cla (x) = C(x) : A(x) : ε¯(¯ u(x)), ∀x ∈ Ω

[1.40]

and

where the strain value ε¯(¯ u(x)) is defined as ε¯(¯ u(x)) = 12 (∇(¯ u(x)) T +∇ (¯ u(x))). The localization operator A was obtained previously by solving the RVE problem [1.1–1.3] (see section 1.1). Therefore, the sensitivity number formulated in [1.37] can be calculated to perform the topology optimization. We note that the sensitivity number can be naturally obtained from a reference solution when all heterogeneities are fully meshed. However, this could result in a huge amount of calculations, especially in topology optimization where the finite element analysis needs to be carried out in each iteration. In the examples section, the optimized topologies of the RVE as well as the resulting stiffness based on the presented method and the reference solution will be compared to investigate the size effect of the RVE, with the size ranging from large to small when compared with structure dimensions.

14

Topology Optimization Design of Heterogeneous Materials and Structures

1.2.3. Optimization procedure The BESO method starts from an initial guess of the design domain and tailors the topology according to the sensitivity numbers iteratively. In this work, the structural problem is solved on the coarse mesh to save computational cost, and a relocalized scheme based on the classical homogenization is adopted to relocalize the microscopic fields, in order to compute the sensitivity number from equation [1.37]. To avoid the common problem of a checkerboard pattern in topology optimization, the above-obtained sensitivity numbers are first smoothed by means of a filtering scheme (Bourdin 2001, Bruns and Tortorelli 2001, Sigmund 2001):  Ne

j=1 wej αj

α e =  Ne

j=1 wej

,

[1.41]

where wej is a linear weight factor defined as wej = max(0, rmin − Δ(e, j)),

[1.42]

which is determined according to the prescribed filter radius rmin and the element center-to-center distance Δ(e, j) between elements e and j. Due to the discrete nature of design variables (solid (ρe = 1) or void (ρe = ρmin ) phase) in the BESO method, the current sensitivity numbers are further smoothed along the history evolution to avoid spurious oscillations (Huang and Xie 2007) as αe(iter)

=

(iter)

(αe

(iter−1)

+ αe 2

)

,

[1.43]

where iter is the current iteration number; thus, the updated sensitivity number includes all sensitivity information in the previous iterations. With the modified sensitivity information, the considered structure can be tailored together with the target material volume at the current iteration V(iter) . The target material volume can be written as 

V(iter) = max Vreq , (1 − cer )V(iter−1) ,

[1.44]

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

15

where cer is an evolutionary volume ratio that can be set by the designer to determine the amount of material to be removed from the previous design iteration. Once the volume constraint of the solid material Vreq is reached, the volume is kept constant at the prescribed value. The optimization procedure iteratively conducts the finite element analysis (FEA) to determine the sensitivity numbers and update the topology in the RVE, and the following convergence criterion is satisfied (Huang and Xie 2007): |

Q

q=1 (fiter−q+1 − fiter−Q−q+1 ) | Q q=1 fiter−q+1

≤ τ.

[1.45]

where Q is an integer and τ is a specified small tolerance value. The overall optimization procedure of the multiscale topology optimization for periodic structures in the context of non-separated scales can be formulated using the classical homogenization scheme as follows: 1) set a coarse mesh associated with the structure such that each substructure/unit cell is meshed with the same number of coarse elements. Set another fine mesh related to the microscopic scale to discretize the RVE microstructure; 2) assign the pseudo densities (ρmin or 1) to fine mesh elements in the RVE in order to construct an initial design before optimization; 3) perform the classical homogenization method on the RVE to ¯ H , as obtain the localization tensor A and the effective elastic tensor C summarized in section 1.1; 4) solve the structure problem on the coarse mesh with the effective material property; 5) based on the solution from the structure problem, relocalize the microscopic strain and stress fields by [1.39] and [1.40] respectively; 6) compute the element sensitivity number using [1.37], and process the sensitivity number with filtering and history averaging as in equations [1.41] and [1.43] respectively;

16

Topology Optimization Design of Heterogeneous Materials and Structures

7) remove inefficient materials from the RVE according to the modified sensitivity number to satisfy the volume constraint at the current iteration as in equation [1.44]; 8) repeat steps 3–7 until the material constraint Vreq is satisfied and the convergence criterion as given in [1.45] is reached. It is worth noting that the present homogenization method allows the optimization process from a homogeneous design with ρe = 1, ∀e, since the microscale fields are relocalized at the nodes of the coarse mesh at the higher scale. As a result, there is no mandatory requirement to set one or several holes for initiating the procedure, which is the case in most inverse homogenization schemes for topological design of material microstructures (Sigmund 1994, Zuo et al. 2013b, Xia and Breitkopf 2015a, Huang et al. 2015). 1.3. Numerical examples In this section, several numerical experiments are presented to investigate the size effect of the unit cell in the classical homogenization-based topology optimization of lattice/periodic structures. The dimensions of the unit cell ranges from large to small compared with the dimensions of the structure in different cases. The examples consist of comparing the optimized topologies using the presented homogenization method, which is solved on the coarse mesh, with the reference solution where all heterogeneities are fully meshed. Regular meshes with four-node elements are adopted for all examples. Plane stress conditions are assumed. At the initiation of the topology optimization, the material distribution is homogeneous with ρe = 1, ∀e within the RVE. The material constituting the architectured structure is assumed to be isotropic, with Young’s and Poisson’s coefficients respectively given by Em = 1, 000 MPa and νm = 0.3. During the topology optimization procedure, the interior of emerged holes is meshed with a highly compliant material to maintain regular meshes, and fictitious material properties for the holes are considered as Ei = 10−6 MPa and νi = 0.3. The target volume fraction for the optimized topology of the RVE in all examples is 0.5.

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

17

1.3.1. Doubly clamped elastic domain In this first example, we investigate the topology optimization of a periodic doubly clamped square elastic domain, as shown in Figure 1.2. The horizontal and vertical displacements of both left and right ends of the beam are fixed. A concentrated force F = 100 N is loaded on the center point of this beam. The side of the square beam is L = 1, 000 mm. The structure is a lattice composed of Ns = η × η unit cells that are periodically repeated, with η representing the number of unit cells along each space direction. A coarse mesh composed of 5 × 5 elements is associated with the unit cells at the structural scale. The fine mesh on the RVE is composed of 40 × 40 elements to solve the local problem. Five cases are studied: (i) η = 2; (ii) η = 4; (iii) η = 8; (iv) η = 16; and (v) η = 20. These different cases correspond to the following coarse meshes of the structure: (i) 10 × 10 elements; (ii) 20 × 20 elements; (iii) 40 × 40 elements; (iv) 80 × 80 elements; and (v) 100 × 100 elements. The reference solution is obtained by discretizing the structure with the fine mesh for accounting for all heterogeneities, resulting in the following regular meshes for the different studied cases: (i) 80 × 80 elements; (ii) 160 × 160 elements; (iii) 320 × 320 elements; (iv) 640 × 640 elements; and (v) 800 × 800 elements. It should be recalled that the present homogenization method allows relocalization of the microscopic fields. Then, the homogenization-based topology optimization procedure only uses the values at the nodes of the coarse mesh, drastically reducing the computational time.

F L

Figure 1.2. Doubly clamped square elastic domain composed of periodic microscale unit cells: geometry and boundary conditions

18

Topology Optimization Design of Heterogeneous Materials and Structures

(a)

(b)

(c)

(d)

Figure 1.3. Optimized topologies for the double-clamped beam: columns (a) and (b) compare the global lattice topologies when using the reference solution or the homogenization method; columns (c) and (d) show the corresponding unit cell. Rows (i)–(v) correspond to the increasing number of unit cells in the lattice.

Figure 1.3 shows the different optimized topologies of the lattice structure for several numbers of unit cells along each direction to investigate the size effect of the unit cell. Figure 1.3(a) shows the final

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

19

optimized geometry of the lattice obtained from the reference solution, while Figure 1.3(b) shows the final optimized geometry of the lattice obtained from the present homogenization method. Figure 1.3(c) and 1.3(d) shows the optimized geometry of one unit cell for comparison. Along rows (i)–(v), the number of unit cells repeated along each direction is increased, and the ratio between the dimensions of the unit cells and the dimensions of the whole structure is decreased. We can observe from Figure 1.3(i)–(v) that the optimized topology converges rapidly. We can also note that, even for the case (i) where the two scales cannot be clearly separated, both methods lead to the same topology, which was not expected. To further illustrate the size effect of the unit cell, we compare the compliances for optimized geometries of the lattice when using the reference solution or the present homogenization method in Figure 1.4(a). We can see that with the decrease in the unit cell size, the gap in the resulting compliances becomes smaller and smaller. We can also note that when the number of unit cells is large, both methods lead to the same compliance. To quantify the computational saving, the numbers of degrees of freedom (DOFs) to be solved in two different solutions are compared in Figure 1.4(b). It can be observed that the number of DOFs in the homogenization method is almost negligible compared with the reference solution, especially when the number of unit cells is large. Nevertheless, the homogenization method could generate the optimized lattice structures with the same topology as well as the same stiffness as the reference solution. 1.3.2. L-shaped structure In the previous example, the boundary conditions of the macrostructure were symmetrical, resulting in the optimal topological configuration of the RVE that is orthogonal and in a fast convergence. In this example, a more complicated L-shaped structure is investigated. The geometry of the problem is shown in Figure 1.5, where the dimension of the macroscopic structure is L = 1, 000 mm. The top end

20

Topology Optimization Design of Heterogeneous Materials and Structures

of the L-shape structure is fixed, and the concentrated force is taken as F = 100 N (see Figure 1.5). The mesh used at the fine scale within the RVE is composed of 40 × 40 elements. A coarse mesh composed of 5 × 5 is associated with the unit cells at the structural scale. As in the previous example, the number of unit cells composing the beam is varied to study the size effect of the unit cell. Then, the following numbers of unit cells along each direction are investigated: (i) 2 × 2; (ii) 4 × 4; (iii) 8 × 8; (iv) 16 × 16; and (v) 20 × 20. Since the structure is not square, the number of unit cells in the x− and y−directions is not the same; for example, only three unit cells exist for case (i). Then, the following numbers of DOFs are solved respectively: (i) 3,554; (ii) 4,044; (iii) 5,924; (iv) 13,284; and (v) 18,764. For comparison, the numbers of DOFs that need to be solved using reference meshes are respectively: (i) 9,922; (ii) 39,042; (iii) 154,882; (iv) 616,962 and (v) 963,203. We can note that using the present technique, the topology optimization procedure only uses the nodal values of the coarse mesh, thus drastically reducing the computational cost. As in the first example, Figure 1.6 shows the different optimized topologies of the lattice structure for several numbers of unit cells along each direction using the reference solution and the homogenization method. We can observe from Figure 1.6(i) and 1.6(v) that the optimized topologies have a huge difference between two different solutions. However, with the large number of unit cells, the topology of the optimized structure obtained using the homogenization method is also convergent to the reference solution. We compare the compliance of the optimized geometries of the lattice using two solutions, as shown in Figure 1.7(a). In this case, the resulting compliance obtained using the homogenization method is much larger than that using the reference solution for case (i). As expected, the compliance converges to the same value with the decrease in the unit cell size. For example, the DOFs that need to be solved in two optimized models are compared in Figure 1.7(b). Here again, the homogenization topology optimization method based on coarse meshes significantly reduces the computational time.

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

21

(a)

(b) Figure 1.4. Resulting compliance and DOFs solved by topology optimization using the reference solution (blue curve) and the homogenization method (red curve). Results are plotted as a function of the number of unit cells. For a color version of this figure, see www.iste.co.uk/da/topology.zip

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Topology Optimization Design of Heterogeneous Materials and Structures

L/2

L/2 F L

Figure 1.5. L-shaped beam composed of periodic microscale unit cells: geometry and boundary conditions

(a)

(b)

(c)

(d)

Figure 1.6. Optimized topologies for the L-shaped beam: columns (a) and (b) compare the global lattice topologies when using the reference solution or the homogenization method; columns (c) and (d) show the corresponding unit cell. Rows (i)–(v) correspond to the increasing number of unit cells in the lattice

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

23

(a)

(b) Figure 1.7. Resulting compliance and DOFs solved by topology optimization using the reference solution (blue curve) and the homogenization method (red curve). Results are plotted as a function of the number of unit cells. For a color version of this figure, see www.iste.co.uk/da/topology.zip

24

Topology Optimization Design of Heterogeneous Materials and Structures

1.3.3. MBB beam In this last example, we investigate the topology optimization of a periodic Messerschmitt–Bölkow–Blohm (MBB) beam subjected to a concentrated load, where the aspect ratio of the beam is chosen as 4. The geometry of the problem is shown in Figure 1.8. The concentrated force load is applied at the center of the bottom end of the domain with a magnitude F = 100 N. The dimensions are L × H = 4, 000 × 1, 000 mm. Due to the symmetry of boundary conditions, only the right half of the MBB beam is investigated. Therefore, we assume that the half-structure consists of Ns = sx × sy unit cells repeated periodically, with sx and sy denoting the number of unit cells along the x− and y−directions respectively. In this example, we keep sx = 2sy . A total of six cases are studied: (i) sx × sy = 2 × 1; (ii) sx × sy = 4 × 2; (iii) sx × sy = 8 × 4; (iv) sx × sy = 16 × 8; (v) sx × sy = 20 × 10; (vi) sx × sy = 24 × 12. These different cases correspond to the following coarse meshes of the structure: (i) 10 × 5 elements; (ii) 20 × 10 elements; (iii) 40 × 20 elements; (iv) 80 × 40 elements; (v) 100 × 50 elements; (vi) 120 × 60 elements. As in the previous examples, the topology optimization problem of the periodic structure is solved using the reference mesh (fully accounting for the heterogeneities). In the reference model, the different cases have the following fine meshes of the structure: (i) 80 × 40 elements; (ii) 160 × 80 elements; (iii) 320 × 160 elements; (iv) 620 × 320 elements; (v) 800 × 400 elements; (vi) 960 × 480 elements respectively. Figure 1.9 shows the different optimized topologies of the lattice structure for several numbers of unit cells along each direction, using the homogenization and reference methods. Here again, the optimized topologies are different in the first few cases, while both methods lead to the same topology with the large number of unit cells. We also compare the compliance of optimized geometries of the lattice using the homogenization and reference solutions. It can be observed that the resulting compliance obtained using classical homogenization is much larger than that using the reference solution for case (i) where the scales cannot be clearly separated. However, as expected, both

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

25

methods lead to the same value of the compliance when the number of unit cells along the short side is larger than eight. We also compare the DOFs solved in two methods, as shown in Figure 1.10. It can then be suggested that the present classical homogenization method can be chosen for topology optimization of periodic structures with a large number of unit cells.

H

L F

F

Figure 1.8. MBB beam composed of periodic microscale unit cells: geometry and boundary conditions

1.4. Concluding remarks In this chapter, we presented a topology optimization framework for periodic structures based on the classical homogenization method but in the context of a non-separated scale, that is, the characteristic dimensions of periodic unit cells in the lattice are comparable with the dimensions of the whole structure. The present method uses a coarse mesh corresponding to a homogenized medium based on the classical numerical homogenization, allowing the reduction of the micro fields to perform the topology optimization. In contrast, topology optimization was performed for comparison using a fully detailed description of the heterogeneous structure. The size effect of the periodic unit cell was investigated to analyze the effectiveness of the present topology optimization based on the classical homogenization

26

Topology Optimization Design of Heterogeneous Materials and Structures

method. We showed that the present topology optimization will lead to an optimized structure with higher compliance when the scales cannot be clearly separated (a few unit cells). Furthermore, with the increase in the number of unit cells, the number of degrees of freedom to be solved can be drastically reduced when compared with the reference solution. In other words, for a large number of unit cells, the present method takes less time to obtain the optimized lattice structure without losing any stiffness.

(a)

(b)

(c)

(d)

Figure 1.9. Optimized topologies for the half-MBB beam: columns (a) and (b) compare the global lattice topologies when using the reference solution and the homogenization method; columns (c) and (d) show the corresponding unit cell. Rows (i)–(v) correspond to the increasing number of unit cells in the lattice

Size Effect Analysis in Multiscale Topology Optimization for Periodic Structures

(a)

(b) Figure 1.10. Resulting compliance and number of DOFs using the reference solution (blue curve) and the homogenization method (red curve). Results are plotted as a function of the number of unit cells. For a color version of this figure, see www.iste.co.uk/da/topology.zip

27

2 Multiscale Topology Optimization of Periodic Structures Taking into Account Strain Gradient

In contrast to the optimization scheme based on the classical homogenization in Chapter 1, this chapter presents a new multiscale topology optimization for periodic structures, using a consistent non-local filter-based homogenization scheme to account for strain gradient effects when two scales are non-separated. Although several computational homogenization methods that take into account strain gradient effects are available – see, for example, (Eringen and Edelen 1972, ?, Kouznetsova et al. 2002) – we have used the technique developed in (Yvonnet and Bonnet 2014a,b, Tognevi et al. 2016). The major advantage of this technique is that it can take into account an arbitrary level of strain gradient, without higher-order elements in a classical finite element framework. A review of the computational homogenization techniques used to take into account the strain gradient effects based on least-square polynomial filters is provided in section 2.1. The procedure combining this non-local homogenization method with the topology optimization procedure is described in section 2.2. The non-local homogenization method is validated in section 2.3, and the proposed methodology is applied to lattice structures in section 2.4, to study the gain of taking into account the strain gradient effects compared to a topology optimization combined

Topology Optimization Design of Heterogeneous Materials and Structures, First Edition. Daicong Da. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Topology Optimization Design of Heterogeneous Materials and Structures

with classical homogenization in the context of non-separated scales. Concluding remarks are drawn in section 2.5. 2.1. Non-local filter-based separated scales

homogenization

for

non-

In this section, we first briefly review the homogenization method to take into account strain gradient effects. The method, called the filter-based homogenization method, was introduced in (Yvonnet and Bonnet 2014a) and later extended in (Yvonnet and Bonnet 2014b, Tognevi et al. 2016). The main idea is to construct a mesoscopic non-local homogenized model using computations on the representative volume element (RVE) by replacing averaging operators in the homogenization theory by linear numerical filters. In this framework, a convenient numerical model based on a coarse mesh of the heterogeneous structure can be constructed, while keeping the possibility of relocalizing all microstructural mechanical fields. 2.1.1. Definition of local and mesoscopic fields through the filter We consider two scales, one is called the microscopic scale, which is associated with fine scale strain and stress fields ε(x) and σ(x), and another is called the mesoscopic scale, which is associated with strain ˆ ˆ and stress fields at the upper scale, denoted by ε(x) and σ(x). Note that the mesoscopic fields have a characteristic wavelength that is not necessarily much larger than that of the microscopic fluctuation fields. As shown in Figure 2.1, structures considered in the present work are assumed to be composed of periodic substructures or unit cells. The length scale of the microscopic unit cell is comparable to the structural length scale such that the scale separation cannot be assumed. A coarse mesh is associated with the whole structure, and each substructure/unit cell is meshed with the same number of coarse elements. In addition, the microscopic structure (unit cell) is meshed using a fine mesh related to the microscopic scale. The mesoscopic strain and stress ˆ are then approximated on the coarse mesh (see fields εˆ and σ

Non-local Homogenization for Multiscale Topology Optimization

31

Figure 2.1(a)), whereas the microscopic fields ε and σ are evaluated on the fine mesh (see Figure 2.1(b)). The size of the fine grid is assumed to be small enough to catch all the fluctuations of the microstructure at the smallest scale.

(b) (a)

Figure 2.1. (a) Coarse mesh covering the structure and unit cells; (b) fine mesh over the unit cell

Mesoscopic and microscopic fields are related by  εˆ = F{ε(x)}, ˆ = F{σ(x)}, σ

[2.1]

where F is a linear operator, acting as a low-pass filter on fine scale fluctuations. This operator is associated with a characteristic length h related to the field fluctuations observed at the mesoscopic scale. In order to construct a theory able to describe continuously mesoscopic fields from the microscale up to the macroscale, and to precisely overcome the limitations of scale separation, the following properties are required for the filter (Yvonnet and Bonnet 2014a,b, Tognevi et al. 2016): ⎧ lim F{h, ε(x)} = ε(x), ⎪ ⎪ ⎨h→0 lim F{h, ε(x)} = ε(x), ⎪ h→∞ ⎪ ⎩ F {F{ε(x)}} = F{ε(x)}, where . denotes the averaging operator.

[2.2]

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Topology Optimization Design of Heterogeneous Materials and Structures

A least-square polynomial filter introduced in (Yvonnet and Bonnet 2014b), which consists of a least-square projection of the microscopic strain field over a piecewise polynomial basis, is adopted in this work. By this filter process, the mesoscopic fields are expressed as 

 εˆij (x) = Pp=1 M p (x)ˆ εpij , P p σ ˆij (x) = p=1 M p (x)ˆ σij ,

[2.3]

where M p (x) are piecewise polynomial basis (e.g. finite element shape functions) associated with nodes p, p = 1, ..., P of a coarse mesh covering the domain where the microscopic fields are described. Given the fine scale strain field on a discrete fine mesh composed of N nodes xm , m = 1, 2, ..., N , the unknown coefficients εˆpij are required to minimize the distance between the approximation and the given fine scale strain field in the least-square sense. Let us define U so that U=

N 

⎛ ⎝

m=1

P 

⎞2 M p (xm )ˆ εpij − εij (xm )⎠ .

[2.4]

p=1

Optimality conditions give dU = 0, dˆ εqij

q = 1, 2, ..., N,

[2.5]

leading to 2

⎛

N 

M q (xm ) ⎝

m=1

P 

⎞ M p (xm )ˆ εpij − εi j(xm )⎠ = 0,

p=1

[2.6]

q = 1, 2, .., N, or P  p=1

N  m=1

 M (x )M (x ) εˆpij = p

m

q

m

 m

M q (xm )εij (xm ). [2.7]

Non-local Homogenization for Multiscale Topology Optimization

33

Then, the coefficients εpij , p = 1, 2, ..., N are found by solving the following system: Au = b,

[2.8]

where Apq =

N 

M p (xm )M q (xm ),

m=1

bq =

N 

M p (xm )εij (xm ),

[2.9]

m=1

with u = [ˆ ε1ij , εˆ2ij , ..., εˆN ij ]. 2.1.2. Microscopic unit cell calculations Let us consider a unit cell Ω ⊂ Rd , as defined in section 2.1,where d is the dimension of the space, with boundary ∂Ω. The unit cell is related to the microscopic scale, where the fields are described at the finest scale. The local problem on the unit cell for non-separated scales is defined as follows: assuming an applied non-constant mesoscopic strain ˆ field ε(x), we find ε(x) to satisfy: ∇ · (σ(x)) = 0 in Ω

[2.10]

σ(x) = C(x) : ε(x),

[2.11]

ˆ F{ε(x)} = ε(x) in Ω,

[2.12]

and

with

where C(x) is a fourth-order elasticity tensor and ∇ · (.) is the divergence operator. Instead of requiring that the spatial average of the strain fields match the mesoscopic one as in the classical homogenization, the condition [2.12] requires that the filtered part of the compatible strain field must match the given non-uniform

34

Topology Optimization Design of Heterogeneous Materials and Structures

ˆ mesoscopic strain field ε(x). This problem is then different from the local problem in classical homogenization. Following (Yvonnet and Bonnet 2014a), the microscopic strain field is split into a filtered (mesoscopic) part and the remaining fluctuation ˜ ε(x): ˆ ˜ ε(x) = ε(x) + ε(x).

[2.13]

Introducing [2.11] and [2.13] into [2.10] and using the property [2.2](c), the new localization problem is given by ∇ · (C(x) : ε˜(x)) = −∇ · (C(x) : εˆ(x)) in Ω

[2.14]

F{˜ ε(x)} = 0 in Ω.

[2.15]

with

To enforce the non-trivial condition [2.12], an auxiliary strain field ˜ e(x) is defined by ε(x) = e(x) − F(e(x)). Using again the property [2.2](c), we obtain: F{˜ ε(x)} = F {e(x) − F{e(x)}} = F{e(x)} − F {F{e(x)}} = 0.

[2.16]

The new local problem [2.14] can then be rewritten by finding e(x) to satisfy ∇ · (C(x) : [e(x) − F{e(x)}]) = −∇ · (C(x) : εˆ(x)) in Ω. [2.17] Condition [2.15] implies that ˜ ε(x) = 0

[2.18]

which is satisfied for any value of the spatial average e(x). So, we choose e(x) = 0, and this equation is classically verified for two possible sets of boundary conditions: ue (x) = 0 on ∂Ω,

[2.19]

Non-local Homogenization for Multiscale Topology Optimization

35

or ue (x) = 0 periodic on ∂Ω,

[2.20]

where ue is a compatible displacement field so that e(x) = ε(ue (x)), with ε(.) = 12 (∇(.) + ∇T (.)). In summary, the new problem is defined by equation [2.14] with boundary condition [2.19] or [2.20]. In the present work, the first set of boundary conditions is adopted. The presence of the non-local operator in the left-hand term of [2.17] induces a numerical difficulty, as the stiffness matrix associated with this linear operator is fully populated. Using (Yvonnet and Bonnet 2014b, Tognevi et al. 2016), the following iterative scheme is defined to overcome this difficulty: starting from an initialized solution e0 (x), for example e0 (x) = 0, we find the field en+1 (x) at each iteration n of the following scheme: ∇ · (C(x) : en+1 (x)) = ∇ · (C(x) : F{en (x)} − ∇ · (C(x) : εˆ(x))

[2.21]

until a convergence criterion is reached. At convergence, the strain field is recovered as follows: ˆ ε(x) = ε(x) + en+1 (x) − F{en+1 (x)}.

[2.22]

Following (Tognevi et al. 2016), we assume that the mesoscopic ˆ strain field derives from a mesoscopic displacement field u(x) related to the mesoscopic scale as follows: 1 εˆij (x) = 2



ˆj (x) ∂u ˆi (x) ∂ u + ∂xj ∂xi

 .

[2.23]

The mesoscopic displacement field is interpolated on the coarse mesh by finite element shape functions as: u ˆi (x) 

 p

M p (x)ˆ upi ,

[2.24]

36

Topology Optimization Design of Heterogeneous Materials and Structures

where M p (x) is the finite element shape function associated with the ˆ node p and u ˆpi are the nodal components of u(x) on the coarse mesh. Then, the corresponding strain field is given by εˆij (x) 

 P  1 ∂M p (x) p=1

2

∂xj

u ˆpi

 ∂M p (x) p + u ˆj , ∂xi

[2.25]

where P is the number of nodes on the coarse mesh of the unit cell. This equation can be rewritten as εˆij (x) 

 P  1 ∂M p (x) p=1

2

∂xj

δik +

 ∂M p (x) δjk u ˆpk . ∂xi

[2.26]

From the superposition principle, the solution of the local problem is then a linear combination of the components of nodal displacements components u ˆpk on the coarse mesh: εij (x) 

P 

p Dijk (x)ˆ upk .

[2.27]

p=1

Introducing the vector forms for the second-order tensors ε and σ: [ε] = [ε11 , ε22 , 2ε12 ]T , [σ] = [σ11 , σ22 , σ12 ]T , equation [2.27] can be rewritten into the matrix form in 2D as follows: ⎤ ⎡ p p (x) D12 (x)  p   D11 u ˆ p p ⎣D21 (x) D22 (x)⎦ 1p , [ε](x) = [2.28] u ˆ2 p p p D31 (x) D32 (x) p p p where the column [D11 (x), D21 (x), D31 (x)] is the strain vector ˆ obtained by solving the local problem with ε(x) given by the p expression [2.26] with u ˆp1 = 1 and u ˆp2 = 0. The column [D12 (x), p p D22 (x), D32 (x)] is the strain vector obtained by solving the local ˆ problem with ε(x) given by the expression [2.26] with u ˆp1 = 0 and u ˆp2 = 1. We show how to compute Dp (x) in the following.

Non-local Homogenization for Multiscale Topology Optimization

37

The corresponding weak form of the localization problem expressed in [2.21] can be formulated as follows: find u(x) satisfying the periodic boundary condition [2.19] so that ∀δu ∈ H 1 (Ω):  Ω

  ε [ue ]n+1 : C(x) : ε(δu(x))dΩ 

=

Ω

   F ε [ue ]n+1 (x) : C(x) : ε(δu(x))dΩ

[2.29]



−

Ω

ˆ ε(x) : C(x) : ε(δu(x))dΩ,

where H 1 (Ω) is the usual Sobolev space. Using a classical FEM discretization over the fine mesh, we obtain: KUn+1 = f n + ˆf ,

[2.30]

with  K=

Ω

BT (x)C(x)B(x)dΩ,

[2.31]

where B is the matrix of shape function derivatives, C is the matrix form associated with the fourth-order tensor C(x) in [2.11] and ˆf is the body force vector associated with the prescribed non-uniform strain εˆ(x), which corresponds to an unitary displacement of one node of the coarse mesh as follows:  ˆf = − BT (x)C(x)[ε(x)]dΩ ˆ [2.32] Ω

ˆ ˆ where [ε(x)] is the vector form associated with ε(x) and  fn =

Ω

BT (x)C(x) [F {ε ([ue ]n (x))}] dΩ.

[2.33]

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Topology Optimization Design of Heterogeneous Materials and Structures

Note that in all 2 × P problems, as well as for all iterations of the iterative procedure, the same stiffness matrix K is involved, which then needs to be computed and decomposed only once. Finally, we obtain: ⎤ ⎡ (1) p ε11 (x) D12 (x) ⎢ p D22 (x)⎦ = ⎣ε(1) 22 (x) p (1) D32 (x) ε (x)

⎡ p D11 (x) p ⎣D21 (x) p D31 (x)

12

⎤ (2) ε11 (x) ⎥ (2) ε22 (x)⎦ . (2) ε12 (x)

[2.34]

Then, we obtain the following relationships: [ε(x)] =

P 

Dp (x)ˆ up ,

[2.35]

p=1

[σ(x)] =

P 

C(x)Dp (x)ˆ up .

[2.36]

p=1

Applying the linear filter F, we obtain ˆ [σ(x)] =



ˆ p (x)ˆ up , G

[2.37]

p

with ˆ p (x) = F{C(x)Dp (x)}. G

[2.38]

As a result, the constitutive relationship obtained at the mesoscopic scale is derived by a fully microscopically-based framework without any empirical assumptions. A simple classical displacement-based finite element strategy is adopted to implement the numerical scheme without requiring higher-order elements. The overall algorithm for the microscopic unit cell computation is summarized as follows:

Non-local Homogenization for Multiscale Topology Optimization

39

For each point p of the coarse mesh covering the unit cell: 1) solve the local problem [2.17] with boundary conditions [2.19] using the FEM discretization over the fine mesh until the convergence is reached; 2) compute the microscopic strain field ε(x) using [2.22]; 3) compute Dp (x) using [2.34] and store it; ˆ p (x) using [2.38] and store it. Only the nodal values of 4) compute G ˆ p (x) at the nodes of the coarse mesh need to be stored. The full spatial G description in the unit cell  can be recovered through finite element shape i ˆ p (x) = ˆp ˆp functions as: G i M (x)[G ]i , where [G ]i are the nodal ˆ p (x) on the coarse mesh. values of G Once these tensors are computed and stored for an arbitrary ˆp distribution of nodal values of the mesoscopic displacement field u over the coarse mesh, we can compute: 1) the reconstructed local stress field σ(x) using [2.36]; 2) the mesoscopic stress field σ ˆ (x) using [2.37–2.38]. 2.1.3. Mesoscopic structure calculations ˆ ⊂ Rd Let us consider a mesoscopic structure defined in a domain Ω ˆ The structure is associated with the mesoscopic with boundary ∂ Ω. scale; that is, the strain and stress fields are described at the characteristic wavelength associated with the filter F, allowing us to define them on the coarse mesh. The structure is subdivided into periodic substructures corresponding to unit cells Ωk (k = 1, ..., Ns ), with Ns being the number of substructures, as shown in Figure 2.1. ˆ is composed of Dirichlet and Neumann parts, The boundary ∂ Ω ˆ u and ∂ Ω ˆ t , where the displacements and denoted respectively by ∂ Ω tractions are prescribed. The equilibrium equation is expressed by ˆ ˆ ∇ · (σ(x)) + ˆf = 0 in Ω

[2.39]

40

Topology Optimization Design of Heterogeneous Materials and Structures

with boundary conditions as ˆu ˆ (x) = ud on ∂ Ω u

[2.40]

ˆt ˆ · n = f d on ∂ Ω σ

[2.41]

and

completed with the mesoscopic constitutive law [2.37–2.38], where ud and f d are respectively the prescribed displacement and forces. The weak form corresponding to the mesoscopic problem ˆ ∈ H 1 (Ω) satisfying the [2.39–2.41] is given as follows: find u boundary condition [2.40] so that    ˆf · δ u ˆ · δu ˆ ext [2.42] ˆ dΩ = δ W ˆ u)] · [ε(δ ˆ u ˆ )]dΩ = ˆ dΓ + [σ(ˆ F ˆ Ω

ˆt ∂Ω

ˆ Ω

ˆ u ˆ ) is approximated on the coarse mesh using classical FEM where ε(δ shape functions: ˆ u ˆ ) = Bδ u ˆe [ε](δ

[2.43]

ˆ e are the nodal values of δ u ˆ on the coarse mesh. Then, we obtain and δ u   ˆ p (x)ˆ ˆ ext , ˆ e dΩ = δ W up · B(x)δ u [2.44] G k

ˆk Ω

ˆk p∈Ω

which leads to the linear system of equations: ˆu = F ˆ Kˆ

[2.45]

with ˆ = K

 k

ˆk p∈Ω

ˆk Ω

ˆ p (x)dΩ BT (x)G

[2.46]

Non-local Homogenization for Multiscale Topology Optimization

41

and ˆ= F



N · ˆf dΩ +



T

ˆ ∂Ω

ˆt ∂Ω

¯ NT · FdΓ.

[2.47]

The mesoscopic problem can only be solved on a coarse mesh, but the technique can provide all relocalized fine scale fields in the heterogeneous structure, as described in section 2.1.2. 2.2. Topology optimization procedure 2.2.1. Model definition and sensitivity numbers In this section, the BESO method for topology optimization is extended to strain gradient effects by incorporating the non-local model presented in the previous section, and applied to the lattice structure composed of periodic unit cells. Then, the topology of all unit cells is the same, but takes into account the response of the whole structure to maximize its stiffness. The structural stiffness maximization problem can be formulated using the design variable (k) ρe , where k and e denote the substructure number and the element number in each substructure respectively as Find : {ρ(1) , . . . , ρ(Ns ) }

[2.48]

ˆT u ˆ) = F ˆ Minimize : fc (ρ, u

[2.49]

ˆ u = F, ˆ subject to : Kˆ

[2.50]

: V (ρ) = Ns



(k) ρ(k) e ve = Vreq ,

(Ns ) , e = 1, . . . , Ne , : ρ(1) e = · · · = ρe

[2.51] [2.52]

: ρ(k) e = ρmin or 1, e = 1, . . . , Ne , k = 1, . . . , Ns [2.53] ˆ and u ˆ are respectively the where fc is the compliance functional, and F applied load and displacement vectors defined at the mesoscopic scale ˆ is assembled using in section 2.1.3. The above stiffness matrix K

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Topology Optimization Design of Heterogeneous Materials and Structures

ˆ p is obtained using [2.38]. It should be noted that the [2.46], where G ˆ p and D ˆ p in [2.38] is based on a fully computation of both tensors G microscopic framework accounting for all heterogeneities at the (k) microscale. In [2.51], ve is the volume of the e−th element in the k−th unit cell, and Ne is the number of elements in the microscale fine mesh for each substructure. Similar to the optimization model defined in section 1 using the classical homogenization method, the condition (1) (N ) ρe = · · · = ρe s , e = 1, . . . , Ne ensures that the pseudo densities (ρmin or 1) of elements at the corresponding locations in each substructure are the same. Therefore, the sensitivity number in this scheme can also be expressed as the variation of the overall structural compliance due to the removal of e−th elements in all substructures:  (k) Ns  k k=1 Ωke σ(x)ε(x)dΩe , for ρe = 1 αe = [2.54] (k) 0, for ρe = ρmin , where σ and ε are respectively microscopic stress and strain fields in the e−th element of the k−th substructure Ωke , which are directly evaluated from the obtained mesoscopic displacement field on the coarse mesh, as formulated in section 2.1. With the presented non-local homogenization scheme, the relocalized strain and stress fields can be obtained using [2.35] and [2.36]. Then, computations on the coarse mesh are only required, but all local fields can be reconstructed. As shown in section 1, this is highly advantageous by reducing the computational cost in the topology optimization procedure, using only the coarse mesh nodal values of displacements. 2.2.2. Overall optimization procedure To summarize, the objective function is computed by solving only the mesoscopic problem on a coarse mesh, and the microscopic strain and stress fields (fine scale) are relocalized by means of the localization operators calculated on the unit cells. The microstructural topology of the RVE is tailored to find the optimal material layout at the microscale so that the resulting overall structure has the maximum stiffness within a prescribed amount of material. Finally, the overall multiscale topology optimization procedure for designing the periodic

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microscopic structures without scale separation using a filter-based homogenization scheme is described as follows: 1) set a coarse mesh associated with the structure so that each substructure/unit cell is meshed with the same number of coarse elements. Set another fine mesh related to the microscopic scale to discretize the microstructure of RVE; 2) assign the pseudo densities (ρmin or 1) to fine mesh elements in the RVE to construct an initial design before optimization; 3) perform the microscopic unit cell computations as described in section 2.1.2; 4) solve the mesoscopic structure problem as summarized in section 2.1.3; 5) based on the nodal displacement solution from the mesoscopic problem on the coarse mesh, evaluate the local strain field by using [2.35] and the local stress field by using [2.36]; 6) compute the element sensitivity number by using [2.54] and modify it by using [1.41] and [1.43]; 7) update the structural topology in the RVE with [1.44]; 8) repeat steps 3–7 until the material constraint Vreq is satisfied and the convergence criterion [1.45] is reached. 2.3. Validation of the non-local homogenization approach Even though the non-local approach described in section 2.1 has been validated through various examples in previous works (see, for example, (Tognevi et al. 2016)), we present in this section a short validation test for the sake of self-consistency of the book. The validation test consists of comparing the relocalized field obtained by the present non-local homogenization method on a coarse mesh with a reference solution where all heterogeneities are fully meshed. We consider a structure shown in Figure 2.2(a), whose dimensions are L × H = 200 × 100 mm. The boundary conditions are described in Figure 2.2(a). The prescribed force is F = 2 kN. The unit cell is shown in Figure 2.2(b). The central hole radius is such that the porosity is equal to 0.6. The material constituting the architectured structure is

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Topology Optimization Design of Heterogeneous Materials and Structures

assumed to be isotropic, with Young’s and Poisson’s coefficients given respectively by Em = 1, 000 MPa and νm = 0.3. As the topology optimization is more conveniently applied with a regular mesh, the interior of the hole is meshed and associated with a fictitious, highly compliant material, with Young’s and Poisson’s coefficients given respectively by Ei = 10−6 MPa and νi = 0.3. The plain stress is assumed.

H

(a)

F (b)

L

Figure 2.2. Cantilever lattice structure composed of periodic unit cells subjected to distributed force: (a) geometry and boundary conditions of the beam; (b) coarse and fine meshes associated with the unit cell

(a)

(b)

Figure 2.3. Meshes of the whole structure for computing (a) the mesoscopic problem and (b) the reference solution

The unit cell is discretized with regular 50 × 50 four-node bilinear elements for the fine scale mesh. A 5 × 5 coarse mesh is used to construct the localization operators of the non-local homogenized model, as described in section 2.1.2. The structure is discretized by a coarse mesh, including 10 × 5 nodes. The reference solution is obtained by discretizing the structure with a regular 100 × 50 bilinear element mesh (see Figure 2.3(b). The results are presented in

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Figures 2.4 and 2.5, where the ε11 and σ11 components of the strain and stress fields are plotted for both the reference solution and the relocalized fields obtained by the homogenized model (solved on the coarse mesh). We can note a satisfying agreement between the two solutions, illustrating that the non-local homogenized model can be used for the topology optimization for taking into account strain gradients while only relying on a coarse mesh. This procedure will be described in the next section.

(a)

(b)

Figure 2.4. (a) Reference solution and (b) relocalized strain field ε11 obtained from the non-local homogenization method (computed on the coarse mesh). For a color version of this figure, see www.iste.co.uk/da/topology.zip

(a)

(b)

Figure 2.5. (a) Reference solution and (b) relocalized stress field σ11 obtained from the non-local homogenization method (computed on the coarse mesh). For a color version of this figure, see www.iste.co.uk/da/topology.zip

2.4. Numerical examples In this section, several numerical examples are presented to illustrate the capabilities of the proposed topology optimization in a strain gradient context. For all the following examples, regular meshes

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Topology Optimization Design of Heterogeneous Materials and Structures

with four-node elements are used, and the plane stress assumption is adopted. Here again, to maintain regular meshes during the topology optimization procedure, the regular mesh covers the holes and the related elements are associated with highly compliant properties. It should be noted that even though the present method was validated in the last section, the reference solution is adopted during the optimization process to cooperatively adjust the direction of topology evolution. Young’s and Poisson’s ratios for the material (ρe = 1) are respectively Em = 1, 000 MPa and νm = 0.3. The fictitious material properties for the holes (ρe = ρmin ) are taken as Ei = 10−6 MPa and νm = 0.3. At the initiation of the topology optimization, the material distribution is homogeneous with ρe = 1. 2.4.1. Cantilever beam with a concentrated load In this first example, we investigate the topology optimization of a periodic lattice structure subjected to a concentrated load. To study the influence of strain gradient effects, the dimensions of the unit cell range from large to small compared with the dimensions of the structure. The geometry of the problem is described in Figure 2.6. The x− and y−displacements of the left end of the beam are fixed.

H L F Figure 2.6. Cantilever beam lattice structure composed of periodic microscale unit cells: geometry and boundary conditions

A concentrated force load is applied on the bottom corner of the right end of the domain with a magnitude F = 100 N. The aspect ratio

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of the cantilever beam is chosen as 2 to highlight the strain gradient effects. The dimensions are L × H = 2, 000 × 1, 000 mm. The structure is a lattice composed of Ns = s1 × s2 unit cells repeated periodically along each space direction, with s1 and s2 denoting the number of substructures along the x− and y−directions respectively. A coarse mesh composed of 5 × 5 elements is associated with the unit cells at the mesoscopic scale. The fine scale mesh on the RVE is composed of 50 × 50 elements to perform the microscopic unit cell computations. In this example, we keep s1 = 2s2 (see Figure 2.7). A total of five cases are studied: (i) s1 × s2 = 2 × 1; (ii) s1 × s2 = 4 × 2; (iii) s1 × s2 = 8 × 4; (iv) s1 × s2 = 16 × 8; (v) s1 × s2 = 20 × 10 (Figure 2.7). These different cases correspond to the following coarse meshes of the structure: (i) 10 × 5 elements; (ii) 20 × 10 elements; (iii) 40 × 20 elements; (iv) 80 × 40 elements; (v) 100 × 50 elements. It is worth recalling that the present homogenization method allows the relocalization of the microscale fields. As a result, the total sensitivity number can be reduced by using the local operators in each subdomain. The target volume fraction for the optimized topology of the unit cells is 0.5. Figure 2.7 shows different optimized topologies of the lattice structure for several numbers of unit cells along each direction and using the present method to take into account strain gradient and classical homogenization. Figure 2.7(a) shows the final optimized geometry of the lattice taking into account the strain gradient, while Figure 2.7(b) shows the final optimized geometry of the lattice without taking into account the strain gradient. Figure 2.7(c) and 2.7(d) only shows the optimized geometry of a single unit cell for comparison. Along rows (i)–(v), the number of unit cells repeated along each direction is increased, and the ratio between the dimensions of the unit cells and the dimensions of the whole structure is decreased. Then, for row (i), the scales are not separated, while for row (v), the scales can be considered as separated. We can observe from Figure 2.7(i) and 2.7(v) that both topological strategies lead to different geometries of unit cells when the strain gradient is taken into account or not. We can also note

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Topology Optimization Design of Heterogeneous Materials and Structures

that when the scales are separated (row (v)), both methods lead to the same topology, as expected. To quantify the gains obtained by the present method, we compare the compliances for optimized geometries of the lattice when the strain gradient is taken into account or not in Figure 2.8. We can see that using the present non-local homogenization method, the obtained resulting compliance is lower than using the classical homogenization (then ignoring the strain gradient effects), inducing a significant gain in the resulting stiffness of the lattice. We also note that when the number of unit cells is large, both methods lead to the same compliance, which is also consistent as the scales are in this case separated.

(a)

(b)

(c)

(d)

Figure 2.7. Optimized topologies for the cantilever beam lattice structure: columns (a) and (b) compare the global lattices topologies when taking into account the strain gradient or not in the optimization procedure; columns (c) and (d) show the corresponding unit cell. Rows (i)–(v) correspond to the increasing number of unit cells in the lattice, leading to (i) non-separated scales and (v) separated scales

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Figure 2.8. Compliance obtained from topology optimization when taking into account the strain gradient effects (blue curve) and without taking into account the strain gradient (red curve). The results are plotted as a function of the number of unit cells, and large number of unit cells lead to scale separation. For a color version of this figure, see www.iste.co.uk/da/topology.zip

In this work, the standard BESO method is adopted to solve the proposed optimization problem. This method updates the topology of the RVE by removing a certain amount of material step by step, in order to finally meet the volume constraint. Normally, the evolutionary volume ratio is set to 2% (Huang and Xie 2007, 2010), which means that 2% solid elements within the design domain are deleted from the previous design iteration. This value cannot be too large to avoid accidental deletion of too many solid elements at each iteration (Huang and Xie 2007, 2010). However, if it is too small, more iterations are needed from the initial design to reach the final volume constraint. Taking case (i) in this numerical experiment as an example, when we set the evolutionary volume ratio as 1%, the different optimized topologies of the lattice structure using the present method and the classical homogenization are shown in Figure 2.9. We can see that no

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Topology Optimization Design of Heterogeneous Materials and Structures

notable difference in the topologies can be observed in Figures 2.9 and 2.7 row (i). However, the solution in Figure 2.9 requires twice the number of iterations since a small value of evolutionary volume ratio is adopted. In addition, a specified small number τ given in equation [1.45] is used to stop the optimization procedure when the volume constraint is satisfied and the objective function is kept constant. In this case, when we set a smaller τ of 0.1%, the optimized topologies for the cantilever beam lattice structure are shown in Figure 2.10. We can see no change in the topology and thus no sensitivity on τ in this case.

Figure 2.9. Optimized topologies for the cantilever beam lattice structure using a small evolutionary volume ratio cer : columns (a) and (b) compare the global lattice topologies when taking into account the strain gradient or not in the optimization procedure; columns (c) and (d) show the corresponding unit cell

Figure 2.10. Optimized topologies for the cantilever beam lattice structure using a small τ : columns (a) and (b) compare the global lattice topologies when taking into account the strain gradient or not in the optimization procedure; columns (c) and (d) show the corresponding unit cell

We show in Figure 2.11 the evolution of the microstructure topology for the case of non-separated scales (1 × 2 unit cells) using the present homogenization method when taking into account the strain gradient effects. These evolutions correspond to different iterations in the topology optimization procedure. It is worth recalling that the present optimization method is able to start from an initial guess

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without holes, which is not the case in most multiscale optimization methods currently available, which require one or several holes for initiating the procedure.

Figure 2.11. Evolution of the optimized topology of the unit cell for the cantilever beam lattice with Ns = 2 × 1 periodic cells: (a) iteration 0; (b) iteration 10; (c) iteration 20; (d) iteration 30; (e) iteration 40 (final topology)

Figure 2.12. Compliance obtained from topology optimization when taking into account the strain gradient effects (blue curve) and without taking into account the strain gradient effects (red curve). The results are plotted as a function of the number of unit cells, and large number of unit cells lead to scale separation. For a color version of this figure, see www.iste.co.uk/da/topology.zip

It should be noted that any volume fraction constraint can be selected by the designer in the present optimization procedure. In this example, when we set the volume fraction value to 0.6, the compliance of optimized geometries of the lattice is compared in Figure 2.12 when the strain gradient is taken into account or not. We can see that using

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Topology Optimization Design of Heterogeneous Materials and Structures

the present non-local homogenization method, the resulting compliance obtained in this case is also lower than that using the classical homogenization, resulting in a higher stiffness of the structure. In addition, when the number of unit cells is large, both methods lead to the same compliance, which is also consistent with the case when the volume constraint is set to 0.5. 2.4.2. Four-point bending lattice structure In this second example, a lattice structure is subjected to four-point bending. The geometry of the problem is shown in Figure 2.13. The dimensions of the lattice structure along the x− and y−directions are L = 3, 000 mm and H = 1, 000 mm respectively. The left and right bottom corner nodes are fixed in both the x− and y−directions. On the upper end, concentrated forces are applied (see Figure 2.13). F

F

L/3 H

2L/3 L

Figure 2.13. Four-point bending beam lattice structure composed of periodic unit cells: geometry and boundary conditions

The loading force is taken as F = 100 N. The mesh used for the fine scale within the RVE is composed of 40 × 40 elements. The target volume fraction for the optimized topology of unit cells is 0.6. As in the previous example, the number of unit cells composing the lattice is varied to study the effects of the scale separation. Then, the following numbers of unit cells along each direction are investigated: (i) 3 × 1; (ii) 6 × 2; (iii) 15 × 5; (iv) 30 × 10; (v) 48 × 16. The corresponding coarse meshes are composed of respectively (i) 15 × 5 elements; (ii) 30 × 10 elements; (iii) 75 × 25 elements; (iv) 150 × 50 elements; (v) 240 × 80

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elements. It is worth noting that in the last case, solving the topology optimization problem with a direct meshing of the microstructure would involve 2,462,722 degrees of freedom. Using the present technique, the topology optimization procedure only uses the nodes of the coarse mesh through the homogenized non-local model and then drastically reduces the computational cost. For the last case, the coarse mesh only contains 39,042 degrees of freedom.

(a)

(b)

(c)

(d)

Figure 2.14. Optimized topologies for the four-point bending beam lattice structure: columns (a) and (b) compare the global lattice topologies when taking into account or not the strain gradient in the optimization procedure; columns (c) and (d) show the corresponding unit cell. Rows (i)–(v) correspond to the increasing number of unit cells in the lattice, leading to (i) non-separated scales and (v) separated scales

As in the previous example, Figure 2.14 shows the different optimized topologies of the lattice structure for several numbers of unit cells along each direction and using the present method to take into account the strain gradient and the classical homogenization. We can observe from Figure 2.14(i) and 2.14(v) the different topologies when the strain gradient is taken into account or not. Here again, when scales

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Topology Optimization Design of Heterogeneous Materials and Structures

are separated (row (v)), both methods lead to the same topology. We compare the compliance of optimized geometries of the lattice in Figure 2.15 when the strain gradient is taken into account or not. Using the present non-local homogenization method, the obtained resulting compliance is again lower than that using the classical homogenization (then ignoring the strain gradient effects), inducing a significant gain in the resulting stiffness (or a decrease in the compliance) of the lattice. The evolution of the topology of the unit cell for case (v) is shown in Figure 2.16. The whole multiscale topological design process converges after 34 iterations in this case. These results show that when the scales are not separated, the present topology optimization based on a homogenization method taking into account the strain gradient brings an added value by inducing a larger stiffness of the final lattice.

Figure 2.15. Compliance obtained from topology optimization when taking into account the strain gradient effects (blue curve) and without taking into account the strain gradient (red curve) for the four-point bending beam. The results are plotted as a function of the number of unit cells, and large number of unit cells leads to scale separation. For a color version of this figure, see www.iste.co.uk/da/topology.zip

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Figure 2.16. Evolution of the optimized topology of the unit cell for the four-point bending beam with Ns = 48 × 16 periodic cells: (a) iteration 0; (b) iteration 5; (c) iteration 10; (d) iteration 20; (e) iteration 34 (final topology)

2.5. Concluding remarks In this chapter, we have presented a topology optimization method for lattice structures when scales are not separated; that is, when the characteristic dimensions of the unit cells are not much lower than the dimensions of the structure. This case can occur in many situations, for example, in additive manufacturing processes where very large times can be necessary to produce lattice structures with very small dimensions. In this context, the assumption of scale separation may be violated. We have proposed a topology optimization based on a coarse mesh to reduce the computational time while taking into account all structural details and the strain gradient effects using a non-local homogenization method. We have shown that taking into account the strain gradient effects can lead to a significant increase in the stiffness of the lattice associated with the optimized topology.

3 Topology Optimization of Meso-structures with Fixed Periodic Microstructures

This chapter extends the topology optimization to meso-structural design with fixed or given periodic cells at the microscopic scale in the context of non-separated scales. The non-local filter-based homogenization method introduced in Chapter 2 is adopted to deal with the heterogeneities of a given RVE and perform multiscale computations. The topology optimization framework that defines the microscopic RVE itself as the design variable is introduced. Therefore, the local or microscopic computation needs to be made only once over the RVE to account for full microscopic heterogeneities, and the topology of considered meso-structures is tailored in terms of stiffness. A simple stiffness interpolation scheme is introduced to simplify the sensitivity analysis of the objective function with respect to the defined design variable. The topology optimization model and the sensitivity analysis are presented in section 3.1. In section 3.2, several numerical examples are presented to validate the proposed multiscale topology optimizaiton framework for designing the mesoscopic structures. Finally, section 3.3 draws the conclusion.

Topology Optimization Design of Heterogeneous Materials and Structures, First Edition. Daicong Da. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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3.1. Optimization model and procedure In this section, the optimization problem aims to find the optimal topology of a mesoscopic structure or the layout of previously defined substructures/microscopic unit cells, in order to maximize the overall structural stiffness for a given amount of substructures. It is assumed that the stiffness matrix of each substructure is dependent on a binary design variable ρk , where ρk is equal to 1 or ρmin , corresponding to an existing or void substructure in the mesoscopic model respectively. Therefore, the substructural stiffness can be expressed as ˆ k (ρk ) = (ρk )P K ˆ0 K

[3.1]

ˆ 0 is the stiffness matrix of solid substructures, P is the where K exponent of penalization. Note that the above interpolation is similar or identical to the SIMP model (Sigmund 2001) except that the referred model is applied to isotropic materials, while the interpolation model in this work is directly defined in the substructure. This is because the substructure itself is defined as the design variable, rather than particular materials or elements. Therefore, with the volume constraint of previously defined substructures, the structural stiffness maximization problem can be formulated using the design variable ρk as Find : {ρk }

[3.2]

ˆT u ˆ) = F ˆ Minimize : fc (ρk , u

[3.3]

ˆu = F ˆ subject to : Kˆ

[3.4]

: V (ρk ) =



ρk vk = Vreq

[3.5]

k

: ρk = ρmin or 1, k = 1, . . . , Ns

[3.6]

ˆ and u ˆ are respectively the where fc is the compliance functional, and F applied load and displacement vectors defined at the mesoscopic scale. ˆ can be constructed from the stiffness of The global stiffness matrix K ˆ the substructure Kk shown in [3.1] in a standard finite element way.

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The term vk is the volume of the k−substructure, and Vreq is the target volume of the solid substructures, depending on the number of solid substructures over the whole mesoscopic structure. In order to solve the above topology optimization problem, the sensitivity of the objective function fc with respect to the design variable ρk should be made explicit. Using the stiffness interpolation scheme [3.1], the structural compliance can be derived against the design variable ρk as ˆ ∂fc ˆ T ∂u ˆT K ˆ −1 =F =F ∂ρk ∂ρk



ˆ ˆ ∂F ∂K ˆ − u ∂ρk ∂ρk



ˆ ∂K ˆ 0u ˆ Tk K ˆk ˆ = −P (ρk )P −1 u u = −ˆ uT ∂ρk

[3.7]

ˆ k is the displacement vector of the k−th substructure in the where u ˆ 0 of solid mesoscopic model. As mentioned, the stiffness matrix K subtructures is obtained by a fully microscopically based framework with heterogeneous details of the RVE (see section 2.1.2). Then, the substructural sensitivity number in [3.7] can be obtained by carrying out the mesoscopic structural computations on a coarse mesh (see also definition in section 2.1.3). In order to adopt evolutionary-type structural optimization methods, the obtained substructural sensitivity number, which denotes the ranking of substructures for updating the design variable ρk , is first modified by multiplying it by a constant − P1 : αk = −

1 ∂fc ˆ 0u ˆ Tk K ˆk = (ρk )P −1 u P ∂ρk

[3.8]

In order to avoid checkerboard patterns, the above formulated sensitivity number is smoothed by means of a filtering scheme 1.41. Due to the discrete nature of the design variable, and to avoid oscillations in evolutionary history of the design objective value in order to improve the convergence, this sensitivity number is further averaged with its historical information as in 1.43.

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Before executing the evolutionary procedure based on the computed substructural sensitivity number, the target volume of solid substructures for the current iteration needs to be assigned as in 1.44. Once the volume constraint Vreq is reached, the volume fraction of substructures will be kept constant and the optimization algorithm alters only the topology. The optimization procedure will iteratively perform the mesoscopic structural computations and update the topology of the mesoscopic structure until the volume constraint Vreq is reached and the convergence criterion 1.45 is satisfied. In summary, the overall multiscale topology optimization procedure for designing the mesoscopic heterogeneous structures with fixed microscopic unit cells in the context of non-separated scales is described as follows: 1) give the geometry of the RVE and set a fine mesh to discretize the RVE at the microscopic scale. Set another coarse mesh associated with the heterogeneous structure, such that each substructure/unit cell is meshed with the same number of coarse elements; 2) solve the microscopic/local problem only once over the RVE to establish the constitutive law and link the microscopic and mesoscopic scales as summarized in section 2.1.2; 3) solve the mesoscopic structure problem as summarized in section 2.1.3; 4) based on the stiffness matrix of solid substructures/unit cells and the nodal displacement solution from the mesoscopic problem over the coarse mesh, evaluate the sensitivity of each substructure/unit cell by [3.7]; 5) modify the substructural sensitivity number by using equation [3.8] and smooth it by using equations [1.41] and [1.43]; 6) update the topology of the considered mesoscopic structure according to [1.44]; 7) repeat 3–6 until the volume constraint Vreq is satisfied and the convergence criterion [1.45] is reached.

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3.2. Numerical examples In this section, several numerical examples are presented for designing the topology of heterogeneous meso-structures using predefined microscopic substructures/unit cells. Before starting the whole optimization procedure, the geometry of the microscopic RVE should be first provided. For all adopted microscopic cells in this work, the local solid material and void phases at the microscale are considered to be isotropic, with Young’s coefficients Es = 1000 MPa and Ei = 10−6 MPa respectively. Poisson’s coefficient is ν = 0.3. In order to ensure the consistency of all numerical examples and fairly compare the resulting designs, the volume fractions of the solid materials in all adopted RVEs are set as 60%. At a higher scale, the volume fraction of the solid microscopic cells gradually decreases from an initial 100% to target 50%, which means that the local solid isotropic material occupies 30% of the whole structural domain area for the resulting designs. The plane stress condition is assumed for all examples. 3.2.1. A double-clamped beam The boundary conditions of a mesoscopic double-clamped beam in this first example are shown in Figure 1.3 in section 1. The structure is made up of a square domain, where both the left and right ends are fixed. A concentrated force F = 1 kN is located at the center point of the double-clamped beam, and the side of the beam is L = 1,000 mm. To start the developed design optimization model, three guess designs of the microscopic RVE are considered, as shown in Figure 3.1: unit cell A is full of solid material except for a centered hole with diameter corresponding to the volume fraction of solid phases of 60%; unit cell B is an anisotropic structure with four void rectangles with the same volume fraction of solid phases; unit cell C is a simple crossed structure. For each microscopic unit cell, a 4 × 4 coarse mesh is used at the mesoscopic scale, while the microscopic mesh is composed of 40 × 40 regular four-node bilinear elements. First, the considered

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double-clamped structure is initially divided into Ns = sx × sy repeatable unit cells, where sx and sy denote the number of substructures along the x− and y−directions respectively. In this example, three cases with the following numbers of sx × sy unit cells are considered: (i) 20 × 20; (ii) 40 × 40; (iii) 80 × 80. Therefore, these configurations correspond to coarse meshes for the whole structure of 80 × 80 elements, 160 × 160 elements and 320 × 320 elements respectively. Therefore, at the end of the design optimization, the number of solid microscopic cells for the resulting structures should be 200, 800 and 3,200 respectively to meet the target volume fraction of 50%.

(a)

(b)

(c)

Figure 3.1. Illustration of three microscopic unit cells for the double-clamped beam: (a) unit cell A; (b) unit cell B; (c) unit cell C

Figure 3.2 shows the optimized double-clamped beam using specific microscopic unit cells with different coarse meshes at the mesoscopic scale, i.e. different numbers of substructures are adopted. Each row in Figure 3.2 shows the final optimized geometries of the double-clamped beam, with the increased number of unit cells repeated along each direction, while each column shows the final optimized geometries of the double-clamped beam using different microscopic unit cells. As shown in Figures 3.2 (Ai), (Bi) and (Ci), the resulting topologies are completely different since different microscopic cells are adopted. Symmetrical meso-structures are generated when microscopic cells A and C are adopted, while the resulting structure is asymmetric with unit cell B. This phenomenon is reasonable since the cell B itself is not symmetrical. With an increase in the number of

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microscopic cells, the optimized designs with unit cells A and C are the same as a cross structure. It should be noted that the final optimized geometries are completely different for cases (Ai) and (Aii), which shows the need for subdividing the meso-structure using different numbers of microscopic cells. The same effect can be observed for optimized geometries in Figures (Bi) and (Bii).

Unit cell A

(Ai)

(Aii)

(Aiii)

(Bi)

(Bii)

(Biii)

(Ci)

(Cii)

(Ciii)

Unit cell B

Unit cell C

Figure 3.2. Optimized heterogeneous double-clamped structures using specified unit cells and different coarse meshes at the mesoscopic scale

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3.2.2. A cantilever beam The second example aims to topologically design a mesoscopic cantilever beam subjected to a concentrated load with different microscopic cells. The problem geometry of the mesoscopic beam is shown in Figure 1.5. The left side of the considered beam is fixed and a concentrated force F = 100 N is located on the bottom point of the right end of the domain. The dimensions of the beam are L × H = 2000 × 1000 mm. Similar to the previous example, three RVEs are considered, as shown in Figure 3.3. Unit cells A and B have respectively a circle and a rectangle area, as the void phase to meet the volume fraction of solid phase of 60%. Unit cell C is optimally obtained by the design procedure described in section 2 for the periodic cantilever beam with the same volume constraint of the solid phase.

(a)

(b)

(c)

Figure 3.3. Illustration of three microscopic unit cells for the cantilever beam: (a) unit cell A; (b) unit cell B; (c) unit cell C

The mesh of the microscopic RVE is composed of 50 × 50 square-shaped bilinear elements at the microscale and 5 × 5 elements at the mesoscopic scale. In order to demonstrate the proposed design framework, the following numbers of substructures sx × sy are separately considered: (i) 20 × 10; (ii) 40 × 20; (iii) 80 × 40. With the successive increasing number of periodic unit cells in each direction, the above cases reduce to coarse mesh with (i) 100 × 50 elements, (ii) 200 × 100 elements and (iii) 400 × 200 elements at the mesoscopic

Topology Optimization of Meso-structures with Fixed Periodic Microstructures

65

scale respectively. The optimized geometries of the mesoscopic cantilever beam with specific microstructures for the above different cases are shown in Figure 3.4. It can be observed that the geometries of optimized meso-structures become completely different with the adoption of different microscopic RVEs, which further demonstrates the effectiveness of the proposed multiscale design framework. The final compliances of the optimized structures are also shown in Figure 3.4. As expected, with the same number of unit cells in each direction, the meso-structures composed of optimally-designed microscopic cell C always have a higher stiffness than other solutions. Figure 3.5 shows the evolution of the topology of the mesoscopic cantilever beam for the case (Cii). These evolutions correspond to different iterations in the topology optimization procedure, and the whole design optimization process converges after 40 iterations in this case.

Unit cell A

(Ai)

(Aii)

(Aiii)

(Bi)

(Bii)

(Biii)

(Ci)

(Cii)

(Ciii)

Unit cell B

Unit cell C

Figure 3.4. Optimized heterogeneous cantilever beams using specified unit cells and different coarse meshes at the mesoscopic scale

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Topology Optimization Design of Heterogeneous Materials and Structures

(a)

(d)

(b)

(e)

(c)

(f)

Figure 3.5. Evolution of the optimized cantilever beam for the case (Cii): (a) iteration 0; (b) iteration 5; (c) iteration 10; (d) iteration 20; (e) iteration 30; (f) iteration 40 (final topology)

3.3. Concluding remarks In this chapter, we extended the topology optimization framework to design the geometry of mesoscopic structures with specific microscopic unit cells. The microscopic substructure/RVE could be selected from existing materials, artificial definitions or optimal designs. Several interesting mesoscopic structures composed of customized RVEs could be obtained for specific volume constraints and boundary conditions. Numerical examples showed that structural geometries of the predefined RVE have a major influence on the optimal solutions of mesoscopic structures, demonstrating the effectiveness and significance of the proposed topological design framework. In order to further improve design freedom within the multiscale design scheme, the design model could be extended to a concurrent topology optimization framework for heterogeneous materials and structures in the context of non-separated scales, involving the topological design of microscopic unit cells simultaneously.

PART 2

Topology Optimization for Maximizing the Fracture Resistance

Topology Optimization Design of Heterogeneous Materials and Structures, First Edition. Daicong Da. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

4 Topology Optimization for Optimal Fracture Resistance of Quasi-brittle Composites

Mechanical and physical properties of complex heterogeneous materials are determined, on the one hand, by the composition of their constituents, but can, on the other hand, be drastically modified, at a constant volume fraction of heterogeneities, by their geometrical shape and by the presence of interfaces. As shown in the first part of this book, topology optimization of microstructures can help design materials with maximum structural stiffness, while satisfying the prescribed volume fraction of constituents. Recently, the development of 3D printing techniques and other additive manufacturing processes has made it possible to directly manufacture the designed materials from a numerical file, opening routes for completely new designs (see, for example, (Quan et al. 2016)). Among all properties of interest, accounting for material failure is of essential importance in the design of composite materials. As shown in Figure 4.1, it is necessary to improve the fracture resistance in terms of the required mechanical work for complete failure through an optimal placement of the inclusion phase. However, optimization design of composite materials accounting for fracture resistance has remained relatively unexplored so far, mainly due to the lack of robust numerical methods for fracture propagation until recently. One major challenge is to use topology

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Topology Optimization Design of Heterogeneous Materials and Structures

optimization to improve the fracture resistance of heterogeneous materials, taking into account the heterogeneities and their interfaces in the material.

Load

Peak Load

Matrix

Inclusion

Fracture Energy

Displacement

Figure 4.1. Illustration of a pre-cracked composite and fracture energy

In this chapter, we first propose a numerical framework for optimizing the fracture resistance of quasi-brittle composites through a modification of the topology of the inclusion phase. The phase field method for fracturing is adopted within a regularized description of discontinuities, allowing us to take into account cracking in regular meshes, which is highly advantageous for the topology optimization purpose. A computationally-efficient adjoint sensitivity formulation is derived to account for the whole fracturing process, involving crack initiation, propagation and complete failure of the specimen. The extended BESO method is formulated and used to find the optimal distribution of the inclusion phase, given a target volume fraction of inclusion and finding a maximal fracture resistance. In the following, section 4.1 first reviews the phase field method for the modeling of crack propagation, as developed by Miehe et al. (2010a, 2015a). Section 4.2 presents the topology optimization method for the design of quasi-brittle composites with fracture resistance. Section 4.3 validates the proposed design framework through a series of 2D and 3D benchmark tests. Conclusions are drawn in section 4.4.

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71

4.1. Phase field modeling of crack propagation Let Ω ∈ RD be an open domain describing a cracked solid, as shown in Figure 4.2, with D ∈ [2, 3] being the space dimension. The external boundary of Ω is denoted by ∂Ω ∈ RD−1 . Cracks that may propagate within the solid are collectively denoted by Γ . In this chapter, we adopt the framework proposed in (Mumford and Shah 1989, Ambrosio and Tortorelli 1990, Miehe et al. 2010a,b, Miehe et al. 2015a) for a regularized representation of discontinuities. In this regularized framework, the propagating cracks are approximately represented by an evolving scalar phase field d(x, t), where the diffusion is characterized by a length scale parameter .

(a)

(b)

Figure 4.2. Illustration of crack modeling by the phase field: (a) a sharp crack surface Γ embedded within the solid Ω; (b) the regularized representation of the crack by the phase field d(x)

4.1.1. Phase field approximation of cracks The scalar crack phase field d(x, t) can be determined by solving the following boundary value problem subject to Dirichlet boundary conditions (d = 1) on the crack – see Miehe et al. (2010b) for more details: ⎧ 2 2 ⎪ ⎨d(x, t) −  ∇ d(x, t) = 0, in Ω [4.1] d(x, t) = 1, on Γ ⎪ ⎩ ∇d(x, t) · n = 0, on ∂Ω,

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Topology Optimization Design of Heterogeneous Materials and Structures

where ∇2 (.) is the Laplacian operator,  is a length-scale parameter that governs the width of the regularization zone and gives for  → 0 the exact sharp crack in Γ in [4.1] and n is the unit outward normal vector to ∂Ω. In the following, we denote the crack phase field d(x, t) by d to alleviate the notations. It has been shown that the system of equations [4.1] corresponds to the Euler–Lagrange equation associated with the variational problem:  d = Arg

 inf Γ (d) ,

d∈S

 Γ (d) =

γ(d) dV,

[4.2]

Ω

where S = {d | d(x) = 1, ∀x ∈ Γ } and γ is the crack surface density function per unit volume defined by: γ(d) =

1 2  d + ∇d · ∇d. 2 2

[4.3]

The functional Γ (d) represents the total length of the crack in 2D and the total crack surface area in 3D. A detailed explanation of [5.3] can be found in Miehe et al. (2010b). 4.1.2. Thermodynamics of the phase field crack evolution The variational approach to fracture mechanics provided by Francfort and Marigo (Francfort and Marigo 1998) introduces the following energy functional for a cracked body:   J(u, Γ ) = Wu (ε(u)) dV + gc dA, [4.4] Ω

Γ

in which Wu is the energy density function, where ε = 12 (∇u + ∇T u) are the strain and u are the displacement fields. The first term on the right-hand side of [4.4] corresponds to the elastic energy stored in the cracked solid. The second term on the right-hand side of [4.4] corresponds to the energy required to create the crack according to the Griffith criterion, with gc being the critical fracture energy density (also known as Griffith’s critical energy release rate).

Fracture Resistance Design of Quasi-brittle Composites

73

In the adopted regularized framework, the phase field d(x) is introduced for the representation of cracks. Then, the above functional [4.4] is substituted by the following:   J(u, d) = Wu (ε, d) dV + gc γ(d) dV, [4.5] Ω

Ω

where γ(d) is the surface density defined in [5.3]. From [4.5], the energy potential or free energy W can be obtained: W (ε, d) = Wu (ε, d) + gc γ(d).

[4.6]

Following (Miehe et al. 2010b), the elastic energy Wu is defined in the following form that assumes the isotropic elastic behavior of the solid and accounts for damage induced only by traction, through: Wu (ε, d) = ((1 − d)2 + κ)ψ + (ε) + ψ − (ε),

[4.7]

where κ  1 is a small positive parameter introduced to prevent the singularity of the stiffness matrix due to fully broken parts, and ψ + and ψ − are the tensile and compressive strain energies: ψ ± = λtr[ε]2± /2 + μtr[ε± ]2 ,

[4.8]

with λ and μ being the Lamé coefficients of the solid. Tensile damage degradation is only taken into account in the elastic energy [4.7] through a decomposition of the elastic strain ε into tensile and compressive parts (Miehe et al. 2010b): ε = ε+ + ε −

with ε± =

3 εi ± ni ⊗ ni .

[4.9]

i=1

Here, x± = (x ± |x|)/2, and εi and ni are the eigenvalues and eigenvectors of ε respectively. The evolution of the damage variable d(x, t) can be determined by the variational derivative of the free energy W . In a rate-independent setting with the consideration of the reduced

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Topology Optimization Design of Heterogeneous Materials and Structures

Clausius–Duhem inequality, the evolution criterion is provided by the Kuhn–Tucker conditions (Miehe et al. 2010b, Nguyen et al. 2015): d˙ ≥ 0;

˙ d W ] = 0, d[−δ

[4.10]

−δd W = 2(1 − d)ψ + (ε) − gc δd γ = 0,

[4.11]

−δd W ≤ 0;

yielding

with the functional derivative (Miehe et al. 2010b) δd γ = d/ − Δd.

[4.12]

Following (Miehe et al. 2010a), the maximum tensile strain energy is stored to account for loading and unloading histories, and the damage evolution criterion [4.11] can then be expressed in the following form: gc [d − 2 ∇2 d] = 2(1 − d) max {ψ + (x, t)}.  t∈[0,T ]

[4.13]

The criterion [4.13] is a monotonously increasing function of the strain ε(x, t) that induces unnecessary stress degradation even at low strain values. To avoid this problem, an energetic damage evolution criterion with threshold was introduced in (Miehe et al. 2015a,b, Miehe and Mauthe 2016), yielding ψc [d − 2 ∇2 d] = (1 − d) max {ψ + (x, t) − ψc + }, t∈[0,T ]

[4.14]

in which ψc is a specific fracture energy density of the solid, which can be further related to a critical fracture stress σc by: ψc =

1 2 σ , 2E c

[4.15]

Fracture Resistance Design of Quasi-brittle Composites

75

where E is Young’s modulus (see more details in Miehe et al. (2015a)). The above crack evolution criterion [4.14] can be further stated as ψc [d − 2 ∇2 d] = (1 − d)H(x, t),

[4.16]

with the introduction of a strain energy history function (Miehe and Mauthe 2016) H(x, t) = max {ψ + (x, t) − ψc + }. t∈[0,T ]

[4.17]

4.1.3. Weak forms of displacement and phase field problems In the absence of body forces, the linear momentum balance equation for the solid medium can be written as ∇ · σ = 0,

[4.18]

where according to the definitions in [4.7] and [4.8], the stress tensor σ equals σ=

∂Wu = {(1 − d)2 + κ}{λtr[ε]+ 1 + 2με+ } ∂ε

[4.19]

−

+ λtr[ε]− 1 + 2με , in which 1 is the second-order identity tensor and κ  1 is a small positive parameter which is introduced to prevent the singularity of the stiffness matrix due to fully broken parts. Multiplying the governing equation [4.18] by kinematically admissible test functions for the displacement δu, integrating the resulting expression over the domain Ω and using the divergence theorem together with boundary conditions yield the associated weak form:   ¯t · δu dA, [4.20] σ : ε(δu) dV = Ω

∂Ωt

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Topology Optimization Design of Heterogeneous Materials and Structures

in which ¯t is the applied traction on the Neumann boundary ∂Ωt (see Figure 4.2). The weak form [4.20] is completed with Dirichlet boundary conditions defined on ∂Ωu . The associated weak form for the crack phase field evolution [4.16] can be obtained in a similar way:   2 {{H + ψc }dδd + ψc  ∇d · ∇(δd)} dV = Hδd dV, [4.21] Ω

Ω

in which δd ∈ H01 (Ω), d ∈ H 1 (Ω), satisfying the Dirichlet boundary conditions on Γ . 4.1.4. Finite element discretization In this work, we adopt the same finite element discretization for the approximation of the displacement field u and the crack phase field d. We can express the two finite element approximate fields (uh , dh ) as uh (x) = Nu (x)du , dh (x) = Nd (x)dd

[4.22]

and their gradients as ∇uh (x) = Bu (x)du , ∇dh (x) = Bd (x)dd ,

[4.23]

where Nu and Bu denote the matrices of shape functions and shape function derivatives associated with displacements respectively, and Nd and Bd denote the matrices of shape functions and shape function derivatives associated with the phase field variable. Here, {du , dd } denote the vectors of the nodal values of the finite element mesh for displacement and crack phase fields respectively. Introducing the above discretization into the weak form [4.20], we obtain the following discrete system of equations: Ku du = fu ,

[4.24]

Fracture Resistance Design of Quasi-brittle Composites

77

with the external force vector fu  fu = Nu¯t dA,

[4.25]

and the stiffness matrix Ku  Ku = BTu DBu dV,

[4.26]

∂Ωt

Ω

where D is the constitutive matrix corresponding to the definition in [4.19], which is given by: ∂[σ] = (1 − d)2 {λR+ [1][1]T + 2μP+ ∂[ε]

D=

−

[4.27]

−

+ {λR [1][1] + 2μP }, T

where [σ] and [ε] are the vector forms corresponding to the secondorder tensors of stress σ and strain ε. R± and P± are two operators for the decomposition of strain into the tensile and compressive parts – see, for example, (Nguyen et al. 2015). The matrices P± are such that [ε+ ] = P+ [ε] and [ε− ] = P− [ε].

[4.28]

The discretization of the phase field problem [4.21] leads to the following discrete system of equations: Kd dd = fd

[4.29]

where  Kd =

Ω

 {H + ψc } NTd Nd + ψc 2 BTd Bd dV

[4.30]

and  fd =

Ω

NTd H dV,

[4.31]

in which H is the strain energy history function defined in [5.12].

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Topology Optimization Design of Heterogeneous Materials and Structures

In the present work, a staggered solution scheme is used following (Miehe et al. 2010a), where at each time increment, the phase field problem is solved for a fixed displacement field known from the previous time step tn . The mechanical problem is then solved given the phase field at the new time step tn+1 . The overall algorithm is described as follows: 1) Set the initial fields d(t0 ), u(t0 ), and H(t0 ) at time t0 ; 2) Loop over all time increments: at each time tn+1 : a) given d(tn ), u(tn ) and H(tn ), b) compute the history function H(tn+1 ) according to [5.12], c) compute the crack phase field d(tn+1 ) by solving [4.29], d) compute u(tn+1 ) with the current crack d(tn+1 ) by solving [4.24], e) (.)n ← (.)n+1 and go to (a). 3) End. 4.2. Topology optimization model for fracture resistance The extended bidirectional evolutionary structural optimization (BESO) method developed in (Fritzen et al. 2016, Xia et al. 2017) for the design of elastoplastic structures is adopted in this work to carry out topology optimization. Composites made up of two material phases, a matrix phase and an inclusion phase, are considered. The spatial layout of the inclusion phase is optimized by the extended BESO method to yield composites with higher fracture resistance. 4.2.1. Model definitions The design domain Ω is discretized into Ne finite elements, and each element e is assigned with a topology design variable ρe . The Ne -dimensional vector containing the design variables is denoted as ρ = (ρ1 , . . . , ρNe )T . Following (Huang and Xie 2009), the design variables and the multiple material interpolation model are defined as ρe = 0 or 1,

e = 1, 2, . . . , Ne

[4.32]

Fracture Resistance Design of Quasi-brittle Composites

79

and

Ee = ρe Einc + (1 − ρe )Emat σc,e = ρe σc,inc + (1 − ρe )σc,mat ,

[4.33]

where Ee and σc,e are Young’s modulus and the critical fracture stress of the e-th element respectively. {Einc , σc,inc } and {Emat , σc,mat } are Young’s moduli and the critical fracture stresses of the inclusion and matrix phases respectively. It should be recalled that “Einc > Emat ” is assumed when carrying out topology optimization with multiple materials using the BESO method (Huang and Xie 2009). Poisson’s ratios of the two material phases are assumed to be identical. The design variables can thus be interpreted as an indicator so that the value of 1 corresponds to the inclusion phase, whereas 0 corresponds to the matrix phase. For consideration of stability, it is conventional to adopt displacement-controlled loading for nonlinear designs – e.g. (Maute et al. 1998, Schwarz et al. 2001, Huang and Xie 2008, Fritzen et al. 2016, Xia et al. 2017). For a prescribed displacement load, the fracture resistance maximization is equivalent to the maximization of the mechanical work expended during the fracturing process, as shown in Figure 4.1. In practice, the total mechanical work J is approximated by numerical integration using the trapezoidal rule, that is, nload T 1 J≈ fu(i) + fu(i−1) Δd(i) u . 2

[4.34]

i=1

(i)

Here, nload is the total number of displacement increments, Δdu (i) is the i-th increment of the nodal displacement vector and fu is the external nodal force vector (comprising surface tractions and reaction forces) at the i-th load increment.

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Topology Optimization Design of Heterogeneous Materials and Structures

During the optimization process, the volume fraction of the inclusion phase is prescribed. Then, the discretized optimization problem can be formulated as: max : J(ρ, du , dd ) ρ

(i) (i)

(i)

= fu , i = 1, . . . , nload subject to : Ku du  : V (ρ) = ρe ve = Vreq : ρe = 0 or 1, e = 1, . . . , Ne .

[4.35]

Here, ve is the volume of the e-th element, and V (ρ) and Vreq are the total and required material volumes respectively. The stiffness (i) matrix Ku at the i-th load increment is constructed following [4.26] and [4.27]. It should be recalled that with this model, the discrete topology design variable ρe ∈ {0; 1} merely indicates the associated material phase (matrix/inclusion) of the e-th element. This assumption naturally omits the definition of supplementing pseudo-relationships between intermediate densities and their constitutive behaviors, as in the case of density-based models – e.g. (Schwarz et al. 2001, Bogomolny and Amir 2012, Kato et al. 2015) – resulting in algorithmic advantages (see also (Fritzen et al. 2016, Xia et al. 2017)). 4.2.2. Sensitivity analysis In order to perform the topology optimization, the sensitivity of the objective function J with respect to topology design variables ρ needs to be provided. Following the “hard-kill” BESO procedure (Huang and Xie 2009, Xia et al. 2018b), the topology evolution is merely driven by the sensitivities of the solid phase (ρ = 1, the inclusion phase in the current context), while the sensitivities of the void phase with (ρ = 0, the matrix phase) are set to zero. The derivation of the sensitivity requires the use of the adjoint method (see, for example, (Buhl et al. 2000, Cho and Jung 2003)). Lagrange multipliers μ(i) , λ(i) of the same dimension as the vector of unknowns du are introduced in order to enforce zero residual r at

Fracture Resistance Design of Quasi-brittle Composites

81

times ti−1 and ti for each term of the quadrature rule [4.34]. Then, the objective function J can be rewritten in the following form without modifying the original objective value as nload  T T

1 (i) J = + λ r(i) fu(i) + fu(i−1) Δd(i) u 2 i=1  T

(i) (i−1) + μ r . ∗

[4.36]

Due to the asserted static equilibrium, the residuals r(i) and r(i−1) have to vanish. The objective value is thus invariant with respect to the values of the Lagrange multipliers λ(i) and μ(i) (i = 1, . . . , nload ), that is,     ∗ (i) (i) J ρ; λ , μ = J (ρ) . [4.37] i=1,...,nload

This equivalence also holds for the sensitivity with respect to changes in the topology design variable ρe on element e: ∂J ∗ ∂J = . ∂ρe ∂ρe

[4.38]

In the following, the derivative ∂J ∗ /∂ρe is computed with properly determined values of λ(i) and μ(i) , leading to certain simplifications of the derivation. To formally describe these derivations, we introduce a partitioning of all degrees of freedom (DOF) into essential (index E; associated with Dirichlet boundary conditions) and free (index F; remaining DOF) entries. For a vector w and a matrix M, we obtain  w∼

wE wF



 and M ∼

MEE MFE

MEF MFF

 [4.39]

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Topology Optimization Design of Heterogeneous Materials and Structures

In the present context, the displacements du,E on the Dirichlet boundary are prescribed, and hence they are independent of the current value of ρ. This implies that     ∂Δdu ∂ Δdu,E 0 = = ∂{Δdu,F }/∂ρe ∂ρe ∂ρe Δdu,F

[4.40] (i)

holds for arbitrary load time increments, that is, for du = du or du = (i−1) du . The components fu,F of the force vector fu vanish at all load time increments, and the only (possibly) non-zero components are the reaction forces fu,E  fu(i)

(i)



f = u,E 0

[4.41]

Equations [4.40] and [4.41] imply  T ∂Δd(i) u = 0. fu(j) ∂ρe

[4.42]

Hence, for arbitrary load increment indices i, j = 1, . . . , nload , we obtain ∂ ∂ρe



fu(j)

T



 Δd(i) u

=

(j)

∂fu ∂ρe

T Δd(i) u .

[4.43]

With the property of [4.43] at hand, the derivative of the modified design objective in [4.38] is given by ⎧   (i) (i−1) T ⎪ n load ⎨ ∂ T ∂r(i)  + f f ∗ u u ∂J 1 (i) = Δd(i) + λ u ⎪ ∂ρe 2 ∂ρe ∂ρe i=1 ⎩  + μ(i)

⎫ ⎬ T ∂r(i−1) ⎪ ∂ρe ⎪ ⎭

.

[4.44]

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83

Recalling that the equilibrium equation at each load time increment is given by [5.45], the derivatives of r(j) at the equilibrium of the j-th load increment with respect to ρe can be expanded as (j)

(j)

(j)

∂r(j) ∂du ∂fu ∂Ku (j) = − d − K(j) . u ∂ρe ∂ρe ∂ρe u ∂ρe

[4.45]

With the expression [4.45], [5.49] can be reformulated as nload ∂J ∗ 1 = ∂ρe 2



i=1

+

(i−1)

∂fu ∂ρe

(i) T

∂fu ∂ρe T



− {λ(i) }T −{μ(i) }T

  (i) Δd(i) + λ u

  (i) Δd(i) + μ u (i)

(i)

∂Ku (i) ∂du du + K(i) u ∂ρe ∂ρe (i−1)

∂Ku ∂ρe

[4.46]



(i−1)

∂du d(i−1) + K(i−1) u u ∂ρe

 .

As mentioned previously, the aim is to find proper values of the Lagrange multipliers λ(i) and μ(i) so that the sensitivities can be explicitly and efficiently computed. From the consideration of [4.41], the first two terms can be omitted by setting (i)

(i)

λE = −Δdu,E

and

(i)

(i)

μE = −Δdu,E .

[4.47]

Accounting further for the structure of the sensitivities of du in [4.40] and for the symmetry of the stiffness matrices, we obtain ⎧ nload ⎨ (i) ∂Ku (i) ∂J ∗ 1 {λ(i) }T =− d ⎩ ∂ρe 2 ∂ρe u i=1

84

Topology Optimization Design of Heterogeneous Materials and Structures (i)   (i) (i) (i) (i) T ∂Δdu,F + Ku,FE λE + Ku,FF λF ∂ρe

T ∂K(i−1)   u (i−1) (i) + μ(i) d(i−1) + Ku,FE μE u ∂ρe ⎫ T ∂Δd(i−1) ⎬ u,F (i−1) (i) . +Ku,FF μF ∂ρe ⎭

[4.48]

(i)

In order to avoid the evaluation of the unknown derivatives of du,F (i−1)

(i)

(i)

and du,F , the values of λF and μF are determined by solving the (i)

(i)

(i)

adjoint systems with the prescribed values λE = −Δdu,E and μE = (i)

−Δdu,E at the essential nodes as follows: −1  (i) (i) (i) (i) λF = Ku,FF Ku,FE Δdu,E ,

[4.49]

  (i) (i−1) −1 (i−1) (i) Ku,FE Δdu,E . μF = Ku,FF

[4.50]

and

The two relations [4.49] and [4.50] together with [4.47] completely determine the values of the Lagrange multipliers λ(i) and μ(i) . It is obvious that the first adjoint system of [4.49] is in fact self-adjoint so (i) that no additional calculation is needed and λ(i) = −Δdu . Note also that because the proportional loading is increased at a constant rate, that is, (i)

Δdu,E =

Δt(i) (i−1) Δdu,E , (i−1) Δt

[4.51]

the solution of the second linear system [4.50] can also be omitted by means of the recursion formula (i)

μF =

Δt(i) (i−1) λ . Δt(i−1) F

[4.52]

Fracture Resistance Design of Quasi-brittle Composites

85

Substituting the two Lagrange multipliers into [4.48], the objective derivative ∂J ∗ /∂ρe can be eventually computed by nload ∂J ∗ 1 = ∂ρe 2 i=1





 T ∂K(i) T ∂K(i−1)  u u λ(i) [4.53] d(i) + μ(i) d(i−1) u ∂ρe u ∂ρe

in which (j)

∂Ku (j) (j) = ke,inc − ke,mat , j = 1, 2, . . . , nload ∂ρe

[4.54]

according to the defined multiple material interpolation model in [4.33], (j) (j) where ke,inc and ke,mat are the element stiffness matrices at the j-th load time calculated using Young’s moduli Einc and Emat respectively. 4.2.3. Extended BESO method The extended BESO method recently developed in (Xia et al. 2017) augments the original proposition in (Huang and Xie 2008) through an additional damping treatment of sensitivity numbers, in order to improve the robustness and effectiveness of the method, particularly in dealing with nonlinear designs in the presence of dissipative effects. Using the extended BESO method, the target volume of the material at the current design iteration is determined by [1.44]. Once the final required material volume Vreq is reached, the optimization algorithm alters only the topology but keeps the volume fraction constant. At each design iteration, the sensitivity numbers, which denote the relative ranking of element sensitivities, are used to determine material phase exchange. When uniform meshes are used, the sensitivity number for the considered objective is defined using the element sensitivity computed from [4.53] as follows:  ∂J ∗ η { ∂ρe } , for ρe = 1 αe = [4.55] 0, for ρe = 0.

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in which η is a numerical damping coefficient (the same as the one applied in the optimality criteria method for density-based methods (Sigmund 2001)). When η = 1, we recover the conventional sensitivity numbers for linear elastic designs (Huang and Xie 2010, Xia et al. 2018b). In the presence of dissipative effects, the sensitivity numbers vary by several orders of magnitude, resulting in instabilities of the topology evolution process, especially when removing certain structural branches (see, for example, (Fritzen et al. 2016, Xia et al. 2017)). For this reason, the sensitivity numbers are dampened in this work with “η = 0.5”, as suggested in (Xia et al. 2017). In order to avoid mesh dependence and checkerboard patterns, sensitivity numbers are first smoothed by means of a filtering scheme [1.41]. It should be remembered that the filter is also responsible for material exchange from the matrix phase (ρe = 0) to the inclusion phase (ρe = 1) by attributing filtered sensitivity number values to design variables that are associated with the matrix phase. Due to the discrete nature of the BESO material model, the current sensitivity numbers need to be averaged with their historical information to improve the design convergence as in [1.43]. The update of topology design variables is realized by means of two th and αth for material removal and addition threshold parameters αdel add respectively (Huang and Xie 2008, Fritzen et al. 2016): ⎧ (k) th ⎪ ⎨ 0 if αe ≤ αdel and ρe = 1, th < α and ρ(k) = 0, = 1 if αadd [4.56] ρ(k+1) e e e ⎪ ⎩ (k) ρe otherwise. The present scheme indicates that inclusion elements are exchanged th , with matrix elements when their sensitivity numbers are less than αdel and matrix elements are reversed to inclusion elements when their th . The parameters αth and sensitivity numbers are greater than αadd del th are obtained from the following iterative algorithm, which was αadd

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first proposed in (Huang and Xie 2008) and recently adopted in (Fritzen et al. 2016, Xia et al. 2017): th th = α , where the value α 1) let αadd = αdel th th is determined iteratively so that the required material volume usage is met at the current iteration;

2) compute the admission ratio car , which is defined as the volume of the recovered elements (ρe = 0 → 1) divided by the total volume of the current design iteration. If car ≤ cmax ar , the maximum admission th and αth are re-determined ratio, then skip the next step; otherwise, αdel add in the next step; th iteratively using only the sensitivity numbers of 3) determine αadd the matrix elements (ρe = 0) until the maximum admission ratio is met, that is, car ≈ cmax ar ; th iteratively using only the sensitivity numbers of 4) determine αdel the inclusion elements (ρe = 1) until the required material volume usage is met at the current iteration.

stabilizes the topology optimization The introduction of cmax ar process by controlling the number of elements reversed from matrix to max is set to a value greater than 1% so inclusion. In the present work, car that it does not suppress the merit of the reverse procedure. 4.3. Numerical examples In this section, we show the performance of the proposed design framework through a series of 2D and 3D benchmark tests. In all 2D examples, uniform meshes of quadrilateral bilinear elements with the plane strain assumption are used. Similarly, uniform meshes of eight-node cubic elements are adopted for the 3D design case. The same finite element discretization is adopted for both displacement and crack phase fields. The characteristic length scale parameter for the phase field regularization given in [4.1] is set to be twice that of the typical finite element size  = 2he . For the sake of clear visualization, only the crack phase field with the values over 0.4 is plotted. The

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material properties of the inclusion and matrix phases are given in Table 4.1 according to Nguyen et al. (2016b). Name Young’s modulus of inclusion Young’s modulus of matrix Poisson’s ratio of both phases Critical fracture stress of inclusion Critical fracture stress of matrix

Symbol Value Unit Einc 52 GPa Emat 10.4 GPa ν 0.3 [-] σc,inc 30 MPa σc,mat 10 MPa

Table 4.1. Material properties of the inclusion and matrix phases (Nguyen et al. 2016b)

With regard to topology optimization, all parameters involved in the extended BESO method, as presented in section 4.2.3, are held constant in all the following examples. The evolutionary rate cer , which determines the relative percentage of material to be removed at each design iteration, is set to cer = 4% (except for the third example cer = 6%). The maximum admission ratio corresponding to the maximum percentage of the recovered material that is allowed per iteration is set to cmax ar = 2%. The filter radius is set at twice the typical finite element size rmin = 2he . 4.3.1. Design of a 2D reinforced plate with one pre-existing crack notch The problem setting of the 2D plate with one pre-existing crack notch is shown in Figure 4.3(a). The dimensions of the plate are 50 × 100 mm. The whole plate is uniformly discretized into 60 × 120 square-shaped bilinear elements. The plane strain assumption is adopted. The lower end of the plate is fixed vertically while free horizontally. The left bottom corner node is fixed in both directions to avoid rigid body motions. The upper end of the plate bears the weight of incremental displacement loads with Δ¯ u = 0.01 mm for the first five load increments and Δ¯ u = 0.002 mm for the following load increments are prescribed. The incremental loading process continues until the reaction force is below a prescribed criterion value, indicating that the structure is completely broken.

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Figure 4.3. A 2D plate with one pre-existing crack notch subject to incremental traction loads: (a) problem depiction and (b) initial guess design. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Figure 4.3(b) is the initial guess design with the inclusion phase occupying 10% of the domain area. Using the extended BESO method, the inclusion phase area is gradually reduced to a target area fraction, occupying 5% of the domain area. The pre-existing crack notch is simulated by prescribing Dirichlet conditions on the crack phase field with d = 1 along the crack. The surrounding area of the initial crack notch (up to two times of the length scale parameter ) is considered as a non-designable region to avoid non-physical designs with the inclusion material added within the already existing crack. The evolution of inclusion typologies, together with their final crack patterns and the design objective history, are shown in Figure 4.4. It can be observed from Figure 4.4 that the fracture resistance of the composite structure gradually improves, while the area fraction of the inclusion phase gradually decreases from initial 10% to 5%. It means that for the same fracture resistance performance, the required usage of the inclusion phase can be largely saved via an optimal spatial distribution design. Detailed propagation of the phase field crack of the optimally-designed composite structure with a single pre-existing crack notch subject to incremental traction loads is given in Figure 4.5. The crack propagates into the inner supporting structure, made up of

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the inclusion phase during the initial incremental loads. Two other cracks initiate around the upper and lower left corners of the inner supporting structure and continue to propagate horizontally until the structure is fully broken.

Figure 4.4. History of the evolution of inclusion typologies and their final crack patterns. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Figure 4.5. Crack propagation of the optimally-designed composite structure with a single pre-existing crack notch subject to incremental traction loads: ¯ = 0 mm, (b) u ¯ = 0.060 mm, (c) u ¯ = 0.076 mm, (d) u ¯ = 0.082 mm, (e) u ¯= (a) u 0.092 mm. For a color version of this figure, see www.iste.co.uk/da/topology.zip

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[kN]

The fracture resistance of the optimally-designed composite structure is validated by a comparison study. Starting from the same initial guess design (Figure 4.3(b)), topology optimization using the same parameter setting is carried out by considering only a linear elastic behavior without accounting for crack propagation, that is, linear design, yielding two parallel bars along the vertical loading direction from the left design shown in Figure 4.6. Then, a complete fracturing simulation is carried out on the linearly-designed composite structure accounting for crack propagation. In this case, the design objective values for linear and crack optimization are equal to 16.92 mJ and 19.43 mJ, respectively, and the fracture resistance is increased by 15% through crack design compared to the classical linear design.

Figure 4.6. Comparison of fracture resistance of two composite structures with a single pre-existing crack notch subject to incremental traction loads. For a color version of this figure, see www.iste.co.uk/da/topology.zip

It should be recalled that the adopted BESO method is a heuristic scheme that does not necessarily guarantee a global optimum design. In the case of linear elasticity, the method has been proved to be insensitive to the two chosen initial starting topologies for the structural stiffness maximization design (Huang and Xie 2007). However, such independence is not guaranteed when it comes to severe nonlinear problems as in the present case; that is, different starting

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topologies may lead to different local optimum designs. Meanwhile, although the BESO method allows for both material removal and addition, their efficiencies are different. As stated in section 4.2.3, sensitivity numbers are only evaluated for the inclusion elements and are set to zero for the matrix elements. It is only due to the filtering scheme [1.41] that sensitivity numbers on the matrix elements neighboring the inclusion–matrix interface are evaluated. Therefore, it is usually more efficient to perform a gradual material reduction starting from a larger initial domain. Meanwhile, the recovery/addition serves as a complementary mechanism for minor adjustment (Xia et al. 2018b).

Figure 4.7. Evolution of inclusion topologies from an initial guess design with the inclusion phase occupying 20% of the domain area, and the final crack pattern. For a color version of this figure, see www.iste.co.uk/da/topology.zip

For the purpose of comparison, we have re-designed the first example from two alternative initial topologies, as shown in Figures 4.7 and 4.8. For both cases, the resulting final inclusion topologies are local optimum designs and are different from the previous design shown in Figure 4.4. Note that in order to start with initial guess B with a lower inclusion area fraction, [1.41] needs to be th modified for material addition and the maximum admission ratio αadd should be exempted from the design. By comparing the values of the

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required work for complete fracture, a larger initial inclusion domain would result in a better design, however, with a higher computational effort.

Figure 4.8. Evolution of inclusion topologies from an initial guess design with the inclusion phase occupying 2% of the domain area, and the final crack pattern. For a color version of this figure, see www.iste.co.uk/da/topology.zip

4.3.2. Design of a 2D reinforced plate with two pre-existing crack notches A 2D plate with two pre-existing crack notches in Figure 4.9(a) is considered for design. Apart from the two pre-existing crack notches, the other problem settings are defined in the same way as in the previous example. An initial guess design with the inclusion phase occupying 15% of the domain area is assumed, as shown in Figure 4.9(b). It is expected to optimally reduce the inclusion phase area fraction from 15% to 8% by using the developed method. The pre-existing two crack notches are simulated by prescribing Dirichlet conditions with d = 1 along the crack. The surrounding area of the two initial crack notches is considered as a non-designable region to avoid non-physical designs with the inclusion material added within the already existing crack.

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Figure 4.9. Illustration of a 2D plate with two pre-existing crack notches subject to incremental traction loads: (a) problem depiction and (b) initial guess design. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Figure 4.10. History of the evolution of inclusion typologies and their final crack patterns. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Figure 4.10 shows the evolution of inclusion topologies together with their final crack patterns and the design objective history. Similar to the previous example, the fracture resistance of the composite structure gradually improves, while the area fraction of the inclusion gradually decreases from initial 14.67% to 8%. This indicates that for

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the same fracture resistance performance, the required usage of the inclusion phase material can be largely saved by using an optimal spatial distribution design. Due to the anti-symmetry of the problem setting, cracks appear anti-symmetrically in the upper and lower parts of the structure. Figure 4.11 shows the detailed propagation of the phase field crack of the optimally-designed composite structure with two pre-existing crack notches subject to incremental traction loads. The two initial cracks propagate into the inner supporting structure made up of the inclusion phase during the initial incremental loads. Then, two other cracks initiate at the upper and lower left surfaces of the inner supporting structure. All the four cracks then continue to propagate until the structure is fully broken.

Figure 4.11. Crack propagation of the optimally-designed composite structure ¯= with two pre-existing crack notches subject to incremental traction loads: (a) u ¯ = 0.060 mm, (c) u ¯ = 0.072 mm, (d) u ¯ = 0.080 mm, (e) u ¯ = 0.094 0 mm, (b) u mm. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Similar to the previous example, a comparison study is also performed to validate the performance of the fracture resistance of the optimally-designed composite structure. Topology optimization is carried out by considering only the linear elastic behavior, without accounting for crack propagation starting from the same initial guess design (Figure 4.9(b)) and using the same design parameters. Linear design without accounting for crack propagation results in two longer parallel bars compared to the linear design obtained in the previous example (due to increased inclusion usage), as shown in Figure 4.12. The linearly-designed composite structure is then subjected to a full fracturing simulation, and its load–displacement curve is compared

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[kN]

with that of the crack design. From both load–displacement curves and design objective values, the fracture resistance of the crack design is obviously increased by over 40% in comparison to the linear design.

Figure 4.12. Fracture resistance comparison of two composite structures with two pre-existing crack notches subject to incremental traction loads. For a color version of this figure, see www.iste.co.uk/da/topology.zip

4.3.3. Design of a 2D reinforced plate with multiple pre-existing cracks This example addresses a multi-objective design using the developed method to improve the fracture resistance of a 2D reinforced plate trying to accommodate the geometry of the inclusion to several different distributions of cracks and, at the same time, to deal with possible random creation of cracks within the structures. For the illustrative purposes, only three configurations are used here, as shown in Figure 4.13. All problem settings are defined in the same way as in the previous two examples, except for the use of a finer finite element discretization. Considering the simulation accuracy involving multiple cracks, a finer discretization with 100 × 200 square-shaped bilinear elements is adopted in this design. Following the same design procedure as presented in the first two examples, the area fraction of inclusion is gradually reduced from initial 28% to 8%. The initial distribution of the inclusion phase is assumed to be a square shape

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enveloping all inner cracks of three cases, as shown by the dashed lines in Figure 4.13.

Figure 4.13. Three 2D plates with multiple pre-existing cracks (cases 1–3 from left to right). For a color version of this figure, see www.iste.co.uk/da/topology.zip

The optimally designed distribution topology of the inclusion phase is presented in Figure 4.14, and their final crack patterns are shown in Figure 4.15. It can be observed that the inclusion phase is preferably distributed at places that can prevent further propagation of critical cracks. It should be recalled that due to the regularized description of cracks using the phase field method, the surrounding region of all initial cracks is assumed to be non-designable. Otherwise, the inclusion phase would fill all fictitious cracks, resulting in non-physical designs.

Figure 4.14. Optimally-designed inclusion distribution topology for three 2D plates with multiple pre-existing cracks (cases 1–3 from left to right). For a color version of this figure, see www.iste.co.uk/da/topology.zip

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[kN]

98

Figure 4.15. Final crack patterns of the three optimally designed 2D plates with multiple pre-existing cracks and their load–displacement curves. For a color version of this figure, see www.iste.co.uk/da/topology.zip

4.3.4. Design of a 3D reinforced plate with a single pre-existing crack notch surface A 3D plate with a single pre-existing crack notch surface, as shown in Figure 4.16, is considered for design to further validate the developed method. The dimensions of the 3D plate are 50 × 100 × 6.67 mm. The whole volume domain is discretized into 60 × 120 × 8 eight-node cubic elements. Similar to the 2D case in section 4.3.1, the lower end of the plate is fixed vertically while free horizontally. The central node on the right end edge is fixed in all directions to avoid rigid body motions. The upper end of the plate incremental displacement loads with Δ¯ u = 0.01 mm for the first seven load increments and Δ¯ u = 0.002 mm for the following load increments are prescribed. The incremental loading process continues until the reaction force is below a prescribed criterion value, indicating that the structure is completely broken. The pre-existing crack notch surface is simulated by prescribing Dirichlet conditions d = 1 along the crack surface. The surrounding volume of the initial crack surface (up to two times the length scale parameter ) is considered as a non-designable region to avoid non-physical designs.

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Matrix

50

Inclusion

12.5

100

6.67 50

(a)

(b)

Figure 4.16. A 3D plate with a single pre-existing crack notch subject to a traction load: (a) problem depiction and (b) initial guess design. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Figure 4.16(b) shows the initial guess design with the inclusion phase occupying 10% of the domain volume. Using the developed method, the inclusion phase area is gradually reduced to the target volume fraction, occupying 5% of the domain volume. The evolution of the spatial distribution topology of the inclusion phase, together with their final crack patterns and the design objective history are shown in Figure 4.17. The resulting hollow distribution topology design is similar to that obtained in the 2D case in section 4.3.1. It can be observed from Figure 4.17 that unlike the 2D case, the fracture resistance of the composite structure in the 3D case is obviously improved during the removal of material volume. From the 3D design, it is more obvious that for the same or even higher fracture resistance performance, the required usage of the inclusion phase can be largely saved by using an optimal spatial distribution design. This is because much more inefficient material exists in the 3D case than in the 2D case, such as within the hollow, which could be clearly removed without weakening the fracture resistance of the composite structure. Detailed phase field crack propagation of the optimally-designed 3D

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composite structure is given in Figure 4.18, where the crack propagation trajectory is similar to the corresponding 2D case in section 4.3.1. 155

External work (mJ)

154 153 152 151 150 149 148 5

10

15

Final Design

20

Design iteration number

Figure 4.17. History of the evolution of inclusion typologies and their final crack patterns. For a color version of this figure, see www.iste.co.uk/da/topology.zip

load [KN]

2.5

a

2

b

1.5

c

1

d

0.5 0

(a)

(b)

(c)

(d)

0

0.02

0.04

0.06

0.08

0.1

displacement [mm]

(e)

Figure 4.18. Crack propagation of the optimally-designed 3D composite structure with a single pre-existing crack notch subject to incremental traction ¯ = 0.084 mm, (b) u ¯ = 0.088 mm, (c) u ¯ = 0.094 mm, (d) loads: (a) u ¯ = 0.010 mm, (e) load–displacement curve. For a color version of this figure, u see www.iste.co.uk/da/topology.zip

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4.4. Concluding remarks In this chapter, we have proposed a numerical design framework for fracture resistance of composites, by the optimal design of the spatial distribution of the inclusion phase with a given target volume fraction. The phase field method model fracturing with a regularized description of discontinuity is adopted for the complete fracturing process. One particular merit of the adopted phase field method is the regularized description of discontinuous fields. This avoids the burden of remeshing during the crack propagation, which is fully adapted to topology optimization. The optimal design of the spatial distribution of the inclusion phase is realized by means of topology optimization using an extended bidirectional evolutionary structural optimization method. Both 2D and 3D benchmark tests demonstrated that significant improvement of the fracture resistance of composites can be achieved for designs accounting for full failure when compared to conventional linear designs. Compared to previous studies on this subject in the literature, this work provides a much more efficient alternative for the design of high fracture-resistant composites. Two-fold merits of the developed design framework exist: on the one hand, the adoption of topology optimization provides an uttermost design freedom, yielding higher fracture-resistant designs; on the other hand, limited number of iterations are required for the design as a result of gradient information, which is of essential importance in computationally-demanding fracturing simulation.

5 Topology Optimization for Optimal Fracture Resistance Taking into Account Interfacial Damage

As observed in the previous chapter, crack propagation resistance was only evaluated on the basis of phase distribution. In most heterogeneous quasi-brittle materials (e.g. ceramic matrix composites, cementitious materials), the interfacial damage plays a central role in the nucleation and propagation of microcracks (Lamon et al. 2000, Tvergaard 1993, Nguyen et al. 2016a, Narducci and Pinho 2017). In this chapter, the main objective is to extend the framework developed in Chapter 4 for using topology optimization to define the optimal phase distribution in a two-phase composite with respect to fracture resistance, taking into account crack nucleation both in the matrix and in the interfaces. A phase field method for fracture which is capable of describing interactions between bulk brittle fracture and interfacial damage is adopted within a diffuse approximation of discontinuities. Efficient design sensitivity analysis is performed by using the adjoint method, and the optimization problem is solved by an extended BESO method. The sensitivity formulation accounts for the whole fracturing process involving crack nucleation, propagation and interaction, either from the interfaces and then through the solid phases, or in the opposite direction. The spatial distribution of material phases is optimally designed to improve the fracture resistance. Section 5.1

Topology Optimization Design of Heterogeneous Materials and Structures, First Edition. Daicong Da. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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provides the detailed phase field framework incorporating bulk fracture and cohesive interface. Numerical details and FEM discretization details are also presented. In section 5.2, we formulate the topology optimization model, containing a detailed sensitivity derivation, as well as the updating scheme with a damping on sensitivity numbers. In section 5.3, several numerical benchmark tests are presented to demonstrate the potential of the proposed method. Conclusions and perspectives are drawn in section 5.4. 5.1. Phase field modeling of bulk crack and cohesive interfaces In this section, we describe the numerical method used by the topology optimization algorithm to obtain the fracture energy of the sample. This technique is based on the phase field model for fracture extended to interfacial damage by Nguyen et al. (2016b), and allows simulation of the initiation and propagation of a multiple-cracks network in heterogeneous microstructures. The main concepts and details are recalled in the following sections. 5.1.1. Regularized representation of a discontinuous field Let Ω ∈ Rd be an open domain describing a solid with the external boundary ∂Ω. The solid contains internal material interfaces between different phases, collectively denoted by Γ I . During the loading, cracks may propagate within the solid and can pass through the material interfaces, as shown in Figure 5.1(a). The crack surfaces are denoted by Γ c . In this chapter, we adopt smeared representations of both cracks and material interfaces; that is, the cracks are approximated by an evolving phase field d(x, t). Interfaces between different material phases are described by a fixed scalar phase field β(x). The material interfaces do not evolve during the loading. The regularized parameters describing the actual widths of the smeared cracks and material interfaces are respectively denoted by d and β . In the following, the same regularization length  = β = d is adopted for cracks and material interfaces for the sake of simplicity.

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Crack Matrix

Inclusion

Interface

(a)

(b)

(c)

Figure 5.1. Illustration of the regularized representation of cracks and interfaces: (a) a solid containing interfaces and cracks; (b) regularized representation of the interfaces; (c) regularized representation of the cracks. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Given a non-evolving sharp crack defined on a surface Γ c , a regularized (smeared) representation of the corresponding damage d(x) (see Figure 5.1(c)) can be obtained by solving equation [4.1]. In Nguyen et al. (2016b), an interface phase field was introduced to describe in the same way the discontinuities related to the damage of interfaces, obtained by solving the problem: ⎧ 2 2 ⎪ ⎨β(x) −  ∇ β(x) = 0, in Ω β(x) = 1, on Γ I ⎪ ⎩ ∇β(x) · n = 0, on ∂Ω.

[5.1]

Equation [5.1] corresponds to the Euler–Lagrange equation associated with the variational problem 

 β(x, t) = Arg

β

inf Γ (β) ,

β∈Sβ

 β

γβ (β) dΩ

Γ (β) =

[5.2]

Ω

where Sβ = {β | β(x) = 1, ∀x ∈ Γ I }, Γ β represents the total interface length and γβ is defined by γβ (β) =

1  β(x)2 + ∇β(x) · ∇β(x). 2l 2

[5.3]

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For  → 0, the above variation principle leads to a sharp interface description. This function will be used as an indicator to particularize the damage model to the interfaces or the bulk in the formulation described in the following sections. In addition, it is necessary to introduce an approximation for the displacement jump at the interfaces, in order to associate a damage model specific to the interfaces and different from the bulk. For this purpose, the following approximation is proposed (Nguyen et al. 2016b) using the Taylor expansion of the displacement field around a point x located on the interface:



h I h I [[u(x)]]  w(x) = u x + n − u x − n 2 2 = h∇(u(x))nI ,

[5.4]

where w(x) is the smoothed displacement jump approximation and nI is an approximation of the normal to the interface Γ I at a point x. Several techniques can be used to define this normal. For example, in (Nguyen et al. 2016b), a level-set technique was proposed (for more details, see the paper mentioned above). In Verhoosel and de Borst (2013), another definition was introduced using the possible modification of the interface by the damage related to bulk cracks. 5.1.2. Energy functional The following total energy functional is introduced for the solid body related to both cracks and interfaces:   e e Wu (ε (u, β), d)dΩ + [1 − β(x)]gc γd (d)dΩ E= Ω

Ω



[5.5]

I

+

ψ (w)γβ (β)dΩ, Ω

where gc is the toughness and ψ I is a strain density function depending on the displacement jump across the interface Γ I . The term εe is the

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bulk part of the infinitesimal strain tensor ε, which satisfies the following relationship: ¯ ε = εe + ε,

[5.6]

where ε¯ is the strain part induced by the smoothed jump at the interfaces such that ε¯ → 0 away from the interfaces (see Verhoosel and de Borst (2013)), in which case we recover the energy functional for a cracked body without considering the interface behavior:   E= Wue (εe (u), d)dΩ + gc γd (d)dΩ. [5.7] Ω

Ω

From equation [5.5], the free energy W can be identified as W = Wue (εe (u, β), d) + [1 − β(x)]gc γd (d) + ψ I (w, α)γβ (β).

[5.8]

The principle of maximum dissipation requires that dissipation Ad˙ reaches maximum under the constraint; that is, d˙ > 0, F = 0. Therefore, F =−

∂Wue ∂W =− − (1 − β)gc δγ(d) = 0 ∂d ∂d

[5.9]

with the functional derivative (Miehe et al. 2010a) δγ(d) =

d − Δd. 

[5.10]

It follows that if d˙ > 0, then −2(1 − d)ψe+ + (1 − β)gc δγ(d) = 0,

[5.11]

To handle loading and unloading, the strain history function adopted in Miehe et al. (2010a) and Nguyen et al. (2016b) is used as follows: H(x, t) = max {ψe+ (x, t)} τ ∈[0,t]

[5.12]

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and [5.11] is substituted by −2(1 − d)H + (1 − β)gc δγ(d) = 0.

[5.13]

5.1.3. Displacement and phase field problems Using [5.10], the evaluation of the crack field d(x, t) can be determined by solving the following phase field problem: ⎧ gc 2 2 ⎪ ⎨2(1 − d)H − (1 − β)  (d −  ∇ d) = 0, in Ω d(x) = 1, on Γ c ⎪ ⎩ ∇d(x) · n = 0, on ∂Ω.

[5.14]

The associated weak form can be obtained as (see Nguyen et al. (2016b)):    gc dδd + [1 − β]gc ∇d · ∇(δd) dΩ 2H + [1 − β]  Ω [5.15]  = 2HδddΩ. Ω

Using the variational principle for minimizing the total energy E with respect to the displacement u, the weak form associated with the displacement problem can be formulated as 

∂Wue : εe (δu)dΩ + e ∂ε Ω   = f · δudΩ + Ω

 Ω

∂ψ I (w) · ∂wγβ (β)dΩ δw

[5.16]

¯ · δudΓ = δW ext . F

∂ΩF

In the absence of body forces, equation [5.16] can be rewritten as  σ e : εe (δu)dΩ + t(w) · δwγβ (β)dΩ Ω



−

[5.17] σ : ∇ δudΩ = 0, e

Ω

s

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109

e where σ e = ∂w ∂εe is the Cauchy stress, and t(w) is the traction vector acting on the interface Γ I oriented by nI and δw = h∇(δu)nI . Using σ e n = t, the above equation can be rewritten further as  σ e : {εe (δu) + n ⊗ δwγβ (β) − ∇s δu} dΩ [5.18]

Ω

which is satisfied for an admissible strain field: εe = ∇s u − n⊗S wγβ

[5.19]

where (∇s u)ij = (ui,j + uj,i )/2 and (n⊗S w)ij = (ni wj + wi nj ). ε¯ can be identified as ε¯ = n⊗S wγβ . With the above description of the strain energy function, the Cauchy stress can now be written as ∂ψe+ ∂ψe− [g(d) + k] + ∂εe ∂εe   = [(1 − d)2 + k] λtrεe + 1 + 2μεe+

σe =

[5.20]

+ λtrεe − 1 + 2μεe− . The general form of the traction vector t(w) in equation [5.17] is given by t(w) = [tn , tt ]

T

[5.21]

where tn and tt denote respectively normal and tangential parts of the traction vector t across the interface oriented by its normal nI . In the presented work, a simplified nonlinear elastic cohesive model is used. We have shown that the framework proposed in Nguyen et al. (2016b) makes it possible to avoid the use of internal variables related to the interface damage, in order to handle loading and unloading by exploiting the damage phase field itself as the history variable. In addition, by only taking into account the normal traction, that is, t(w) · nI = tn , the cohesive law can be written as n



w wn n I t = gc exp − n [5.22] δn δ

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Topology Optimization Design of Heterogeneous Materials and Structures

where wn = w · nI . For this model, we obtain KI = ∂t(w) ∂w . The relationship between δ n , the toughness gcI and the fracture strength tu is given by δ n = gcI /(tu e), with e = exp(1) (see Figure 5.2).

Figure 5.2. Illustration of the cohesive model for the interfaces

Even though the phase field problem is linear in the staggered scheme, that is, for a fixed value u, it should be mentioned that for a fixed crack phase field value d, the mechanical problem [5.16] is nonlinear because it involves the computation of eigenvalues of εe and the interface cohesive model in [5.22]. A linear procedure to solve this nonlinear problem by the Newton method is introduced in the following. From [5.16] and [5.20], the balance equation can be rewritten as   e e R= σ : ε (δu)dΩ + γβ (x)t(w, α) · δwdΩ Ω



−



[5.23]

¯ · udΓ = 0, F

f · δudΩ − Ω

Ω

∂ΩF

where εe (δu) = ∇s δu − n⊗S wγβ . In a standard Newton method, the displacements are updated for each loading by solving the following tangent equation: DΔu R(uk , d) = −R(uk , d) = 0,

[5.24]

Topology Optimization for Optimal Fracture Resistance

111

where uk is the displacement solution from the k-th iteration. The displacements at the current iteration are given by uk+1 = uk + Δu.

[5.25]

From [5.24], we obtain  ∂σ e k DΔu R(u ) = : εe (Δε) : εe (δε) e Ω ∂ε  ∂t(w) : Δw : δwdΩ + Ω ∂w

[5.26]

with Δw(x) = h∇Δu(x)

∇φ(x) ||∇φ(x)||

[5.27]

and   ∂[σ e ] T 2 + + = C(u, d) = [(1 − d) + k] λR [1] [1] + 2μP ∂[εe ]   [5.28] + λR− [1]T [1] + 2μP− where [σ e ] and [εe ] are the vector forms corresponding to the second-order tensors σ e and εe respectively, and C is the matrix form corresponding to the fourth-order tensor C. 5.1.4. Finite element implementation

discretization

and

numerical

A staggered solution procedure is adopted in this work, where the phase field and the mechanical problems are solved alternatively. At each increment, given the displacement field from the mechanical problem, the phase field problem is linear using a shift algorithm (for more details, see Nguyen et al. (2015)). Using the FEM, the phase field and phase field gradient in one element are approximated by d(x) = Nd (x)de , ∇d(x) = Bd (x)de ,

[5.29]

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Topology Optimization Design of Heterogeneous Materials and Structures

where de are nodal phase field values in one element, and Nd (x) and Bd (x) are respectively matrices of shape functions and shape function derivatives associated with the phase field variable. Using the same FEM discretization in the weak form [5.15], the following linear discrete system of equations can be obtained: ˜ = Fd Kd d

[5.30]

where    gc T Kd = ( (1 − β) + 2H)NT N + (1 − β)g B B c d d d d dΩ [5.31]  Ω and  Fd = Ω

2NTd H(un )dΩ.

[5.32]

Similarly, the displacement field and the incremental displacement field can be expressed using the FEM approximations u = Nue , Δu = NΔue

[5.33]

where N denotes the matrix of shape functions associated with displacement variables, and ue and Δue are respectively nodal displacement components and nodal incremental displacement components in one element. Furthermore, we obtain [ε](Δu) = Bu Δue

[5.34]

where Bu is a matrix of shape function derivatives. From [5.4], the diffuse jump approximation vector and its incremental counterparts can be discretized as ˜ u ue , Δw = hNB ˜ u Δue w = hNB

[5.35]

where 

 n 1 n2 0 0 N= , 0 0 n1 n2

[5.36]

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113

and n1 and n2 are the x− and y−components of the normal vector. The smoothed jump strain at the interfaces is defined by ⎤ ⎤ ⎡ ω 1 n1 ε¯11 ⎦. ω2 n2 ¯ = ⎣√ε¯22 ⎦ = γβ (x) ⎣ [ε] 1 √ (ω n + ω n ) 2ε¯12 1 2 2 1 2 ⎡

[5.37]

Then ˜ u Δue ¯ [ε(Δu)] = hγβ (x)MB

[5.38]

with ⎡

n21 M=⎣ 0 √1 n1 n2 2

n1 n2 0 √1 n2 2 2

⎤ 0 n22 ⎦ . 1 √ n1 n2 2

0 n1 n2 √1 n2 2 1

[5.39]

Using the above FEM discretization, the tangent problem reduces to the following linear system of algebraic equations: Ktan Δ˜ u = −R(˜ uk ),

[5.40]

where  Ktan = Ω

˜ T MT ]C(x)[Bu − hγβ (x)B ˜u M]dΩ [BTu − hγβ (x)B u



[5.41] ˜uT NT KI NB ˜u dΩ, h γβ (x)B 2

+ Ω

and  R= Ω

 + Ω

˜ T MT ]C(x)[Bu − hγβ (x)B ˜u M](ue )k dΩ [BTu − hγβ (x)B u

˜ T NT t(wk )dΩ + hγβ (x)B u



 ¯ T dΓ. FN

T

fN dΩ + Ω

Ω

[5.42]

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Topology Optimization Design of Heterogeneous Materials and Structures

The overall algorithm is described as follows: 1) Set the initial displacement field u0 (x), the phase field d0 (x) and the strain history function H0 ; 2) Compute the phase field β(x); 3) For all loading increments: (at each time tn+1 ), given dn , un and Hn (x): a) compute the history function H(tn+1 ) according to [5.12], b) compute the crack phase field dn+1 (x) by solving linear problem [5.30], c) compute un+1 (x): i) initialize uk = un , While ||Δuk+1 || > ,   1: ii) compute Δuek+1 by [5.40], iii) update uk+1 = uk + Δuek+1 , iv) (.)n+1 → (.)n and go to (a). End End. 5.2. Topology optimization method In this section, we present the topology optimization method based on bidirectional evolutionary structural optimization (BESO) (Huang and Xie 2007). In the proposed procedure, the geometry of the constant volume of inclusion phases is optimized to maximize the fracture resistance of the sample. This procedure involves the evaluation of the sensitivity of the whole fracturing process (initiation of multiple cracks, propagation and complete failure of the sample) with respect to changes in the geometry. 5.2.1. Model definitions Similar to section 4.2, the total number of finite elements in the considered domain Ω is denoted by Ne and each element e is assigned a topology design variable ρe . Following the multiple material

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115

interpolation model in (Huang and Xie 2009, Da et al. 2017a), we obtain Ee = ρe Einc + (1 − ρe )Emat

[5.43]

where Einc and Emat are Young’s moduli of the inclusion and matrix phases respectively. The density value takes either 0 or 1, corresponding to the matrix and inclusion phases respectively. Following equation [4.34], the objective function J equivalent to the total mechanical work during the fracturing process is calculated by using numerical integration, that is, nload

1  (n) (n−1) T J≈ fext + fext Δu(n) , 2

[5.44]

n=1

and will be used as our definition of fracture resistance in the following. Here, nload is the total number of displacement increments, Δu(n) is the n is the external nodal force n-th nodal displacement component and fext at the n-th load increment. During the design optimization, the material volume fractions of the matrix and inclusion phases are prescribed. Then, the topology optimization problem subjected to the balance equation and the inclusion volume constraint can be formulated as max : J(ρ, u, d, β) ρ

subjected to : R = 0  : V (ρ) = ρe ve = Vreq : ρe = 0 or 1, e = 1, . . . , Ne .

[5.45]

Here, ve is the volume of the eth element, V (ρ) and Vreq are the total and required material volumes respectively, and R is the nodal residual force: R = fext − fint .

[5.46]

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Topology Optimization Design of Heterogeneous Materials and Structures

In [5.46], fint is defined in each element as the internal force vector given in terms of the associated topology design variable ρe and the Cauchy stress as fint =

Ne  e=1

 BT σ e dΩe .

ρe

[5.47]

Ωe

5.2.2. Sensitivity analysis In order to compute the sensitivity of the objective function J with respect to topology design variables ρ, two Lagrangian multipliers μ(n) , λ(n) are introduced to enforce zero residual R at time tn−1 and tn for each term of the total mechanical work [5.44]. Similar to section 4.2, the two Lagrangian multipliers have the same dimension as the vector of unknowns u. Therefore, the objective function J can be rewritten in the following form without modifying the original objective value as nload 

T 1  (n) (n−1) T  J= Δu(n) + λ(n) R(n) fext + fext 2 n=1 

T (n) (n−1) + μ R .

[5.48]

Introducing a partitioning of all degrees of freedom (DOF) into essential (index E; associated with Dirichlet boundary conditions) and free (index F; remaining DOF) entries and using the same procedure in section 4.2, the derivative of the modified design objective in [5.48] can be rewritten as ⎧  nload ⎨ (n) (n−1) T ∂ J ∂fext 1  ∂fext = + Δu(n) ⎩ ∂ρe ∂ρe 2 ∂ρe n=1 [5.49] ⎫

T ∂R(n)

T ∂R(n−1) ⎬ . + λ(n) + μ(n) ∂ρe ∂ρe ⎭

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117

Recalling the balance equation at each load time increment in [5.23], the derivatives of R(m) at the equilibrium of the m-th load increment with respect to ρe can be expanded as ∂Rm ∂f m = ext − ∂ρe ∂ρe



(m) ∂Δu

Ωe

BT (σ e )(m) dΩe − Ktan

(m)

[5.50]

∂ρe

where (m)

Ktan = −

∂R(m) ∂u(m)

[5.51]

is the tangent stiffness matrix of the nonlinear mechanical system at the balance equation of the m-th load increment. Using the expression [5.50], [5.49] can be reformulated as ⎧  n (n) T load ⎨

 ∂J ∂fext 1 Δu(n) + λ(n) = ⎩ ∂ρe ∂ρe 2 n=1

 +

(n−1)

∂fext ∂ρe

− λ

(n)

T

T



Δu(n) + μ(n)

 T

e (n)

B (σ )

dΩe +

Ωe



− μ(n)

T



(n) (n) ∂Δu Ktan ∂ρe



(n−1) (n−1) ∂Δu BT (σ e )(n−1) dΩi + Ktan ∂ρe Ωe

[5.52] ⎫ ⎬ ⎭

.

The first two terms in [5.53] can be omitted by setting (n)

(n)

λE = −ΔuE

and

(n)

(n)

μE = −ΔuE .

[5.53]

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Topology Optimization Design of Heterogeneous Materials and Structures

With the symmetry of the stiffness matrices, we obtain nload 

T  ∂ J 1  (n) = BT (σ e )(n) dΩe − λ ∂ρe 2 Ω e n=1

T  (n) − μ BT (σ e )(n−1) dΩi Ωi (n)

(n) (n) (n) (n) T ∂ΔuF − Ktan,FE λE + Ktan,FF λF ∂ρe

−



(n−1) (n) Ktan,FE μE

+

(n−1) (n) T Ktan,FF μF

(n−1)

∂ΔuF ∂ρe

[5.54]  .

(n)

To avoid the evaluation of the unknown derivatives of uF and (n−1) uF , that is, eliminating the last two lines of [5.55], the values of (n) (n) λF and μF can be found as follows by solving the adjoint systems (n) (n) (n) (n) with the prescribed values λE = −ΔuE and μE = −ΔuE at the essential nodes:

−1 (n) (n) (i) (n) Ktan,FE ΔuE , λF = Ktan,FF

[5.55]

(n) (n−1) −1 (n−1) (n) Ktan,FE ΔuE . μF = Ktan,FF

[5.56]

and

The two relations [5.55] and [5.56] together with [5.53] fully determine the values of the Lagrange multipliers λ(n) and μ(n) .  e can be computed by Finally, the objective function gradient ∂ J/∂ρ nload 

T  ∂ J 1  (n) =− BT (σ e )(n) dΩi λ ∂ρe 2 Ωi n=1 

T  T e (n−1) (n) B (σ ) dΩi . + μ Ωi

[5.57]

Topology Optimization for Optimal Fracture Resistance

119

As also shown in section 4.2, in order to improve the robustness and efficiency of the method, especially in dealing with nonlinear designs, the sensitivity numbers obtained above are first modified using an additional damping treatment as in equation [4.55]. To avoid checkerboard patterns, sensitivity numbers are further smoothed by means of a filtering scheme as in equation [1.41]. Finally, due to the discrete nature of the BESO material model, it is necessary to average the current sensitivity numbers with their historical information to improve the design convergence as in equation [1.43]. It should be noted that the material volume fraction of reinforced inclusion phases in this work is kept constant during the optimization process. Interfaces between different material phases or even the topology of the reinforced inclusion materials will be tailored through the redistribution of the quantitative inclusion phases. The extended BESO method is adopted to solve the optimization problem using the modified sensitivity numbers, which account for the whole fracturing process involving crack nucleation, propagation and interaction until the complete failure of the considered heterogeneous materials, in order to improve the fracture resistance of the samples. 5.3. Numerical examples In this section, several numerical examples are presented to demonstrate the potential of the proposed topology optimization framework. In all tests, regular meshes using quadrilateral bilinear elements are adopted, and the plane strain condition is assumed. The same finite element discretization is adopted for both displacement and crack phase fields. The regularization parameter  describing the width of the smeared crack and interface is chosen as two times the finite element size  = 2e . The material parameters of each phase are taken as Ei = 52 GPa, Em = 10.4 GPa and vi = vm = 0.3, where the indices i and m correspond to the matrix and inclusion materials respectively. These parameters are those of a mortar composed of a cement paste (matrix) and sand (inclusion). The toughness is gc = gcI = 1 × 10−4 kN/mm, and the interface fracture strength is chosen as tu = 10−2 GPa.

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Topology Optimization Design of Heterogeneous Materials and Structures

5.3.1. Design of a plate with one initial crack under traction The geometry of this example is shown in Figure 5.3(a). The dimensions of the plate are 50 × 100 mm, and the domain is uniformly discretized into 60 × 120 square-shaped bilinear elements. The boundary conditions are as follows: on the lower end, the vertical displacements are fixed while the horizontal displacements are free, and the left bottom corner node is fixed in both directions. On the upper end, the horizontal displacements are free, while the vertical displacements are prescribed with monotonic displacement increments ¯ = 0.005 mm during the simulation. The incremental loading process U continues until the reaction force is below a prescribed value, indicating that the structure is completely broken. It is worth noting that during the crack propagation, interfacial damage can occur and interact with the propagation of the pre-existing matrix crack.

Matrix Inclusion

50

15.83

20

Crack

12.5

100

(a)

50

25

(b)

(c)

Figure 5.3. A plate with one pre-existing crack notch subjected to incremental traction loads: (a) problem geometry; (b) initial guess design; (c) final design

Figure 5.3(b) shows the initial guess design and consists of a single square inclusion occupying a volume fraction of 10% of the sample. In all of the following examples, the material volume fraction of the inclusion phase is maintained constant during the optimization process. Using the extended BESO method, the inclusion phase will be redistributed based on sensitivity numbers, in order to improve the fracture resistance of the considered structure. The pre-existing crack

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121

notch is simulated by applying Dirichlet conditions on the crack phase field with d = 1 along the crack. The surrounding area of the initial crack notch (up to two times the length scale parameter ) is considered as a non-designable region to avoid non-physical designs with the inclusion material added within the already existing crack (for a discussion, see Xia et al. (2018a)). The final structural topology of the inclusion phase is shown in Figure 5.3(c). It can be observed that the material on the right-hand side of the reinforcement inclusion moves up and down on the left-hand side, and the holes are generated to tailor the topology of the inclusion phases. The improvement of fracture resistance of the resulting composite structure is evaluated by comparing the initial and new design responses in Figure 5.4. It is worth noting that in the present work, a staggered procedure is used for solving the coupled displacement–phase field problems formulated in section 4.1. Then, for one load increment, the number of sub-iterations, that is, the number of times the displacement and phase field problems are solved alternatively, has an effect on the solution. It is worth noting that most authors (see, for example, Miehe et al. (2010a)), also our own previous works, Nguyen et al. (2015), Nguyen et al. (2016b), among many others) only use one sub-iteration by assuming that the load increments are small, but this requires a preliminary convergence study. In the following, we study the effects of using two, three, four and five sub-iterations in the staggered scheme on the load–displacement response of the structure. In Figure 5.4, we denote the final and initial designs by “FD” and “ID” respectively, and “s-it” means sub-iterations. We can observe from Figure 5.4(a) that the number of sub-iterations in the whole optimization procedure has an effect on the final load–displacement curve but almost no effect on the final optimized shape (see Figure 5.4(b)–(i)). We note that for four sub-iterations, the load–displacement curve is roughly converged. With these observations in mind, we then conduct the whole optimization procedure for obtaining the final design with only two sub-iterations to reduce the computational cost, as it has been seen that the number of sub-iterations has a small influence on the final shape of the inclusion.

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Topology Optimization Design of Heterogeneous Materials and Structures

[kN]

Then, we use the obtained final design and recompute the load–displacement curve with four sub-iterations to avoid underestimating the fracture energy. In the present example, we can see from Figure 5.4 that the total fracture energy required for complete failure is 3.97 mJ for the initial design and 6.04 mJ for the final design, which means that the final structure is 52% more resistant to fracture than the initial guess design.


  




   










  






   






  
 



   




  




   
 










Figure 5.4. Comparison of fracture resistance of two composite structures with one initial crack subjected to incremental traction loads: (a) load– displacement curves with different numbers of sub-iteration; (b) ID with 2 s-it; (c) FD with 2 s-it; (d) ID with 3 s-it; (e) FD with 3 s-it; (f) ID with 4 s-it; (g) FD with 4 s-it; (h) ID with 5 s-it; (i) FD with 5 s-it (ID/FD: initial/final design, s-it: subiteration). For a color version of this figure, see www.iste.co.uk/da/topology.zip

Detailed propagation of the phase field crack of the final design composite structure with a single pre-existing crack notch subjected to incremental traction loads is given in Figure 5.5. The initial crack propagates through the inner supporting structure and is blocked by the

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123

reinforced inclusion phase during the previous incremental loads. Then, two interface cracks nucleate and propagate along the upper and lower material interfaces. Inclusion materials, which are redistributed up and down on the left-hand side, try to prevent the vertical propagation of the matrix crack. Finally, the interface and matrix cracks intersect and propagate horizontally until the structure is fully broken.

(a)

(b)

(c)

(d)

(e)

Figure 5.5. Crack propagation of the final design composite structure with a ¯ = 0 mm; single initial crack subjected to incremental traction loads: (a) U ¯ ¯ ¯ ¯ (b) U = 0.05 mm; (c) U = 0.06 mm; (d) U = 0.08 mm; (e) U = 0.095 mm. For a color version of this figure, see www.iste.co.uk/da/topology.zip

5.3.2. Design of a plate without initial cracks for traction loads The problem setting of this example is the same as in the last section except that there is no initial crack. The geometry of the plate is shown in Figure 5.6(a), where the inclusion phase occupies a volume fraction of 5% of the sample. The optimized geometry of the inclusion phase is shown in Figure 5.6(b). Detailed propagations of the phase field cracks of the initially and finally designed composite structures subjected to incremental traction load are shown in Figures 5.7 and 5.8 respectively. The cracks are first generated around the upper and lower material interfaces for the initial guess design, while they nucleate in the middle of the inner inclusion phase as well as in the interfaces for the new design. Then, similar interface cracks are generated around the upper and lower material interfaces. Finally, cracks propagate

124

Topology Optimization Design of Heterogeneous Materials and Structures

horizontally until the structure is fully broken in both cases. The final crack patterns, the total fracture energy required for complete failure and the load–displacement curves for the initial and optimized designs are given in Figure 5.9. In this example, the structure with optimal design is 15% more resistant to fracture than the initial composite structure.

Matrix Inclusion 12.5 15.83

20

100

50

(a)

(b)

Figure 5.6. A plate without initial cracks subjected to incremental traction loads: (a) geometry of the initial design and (b) final design

(a)

(b)

(c)

(d)

(e)

Figure 5.7. Crack propagation of the initial design composite structure without ¯ = 0 mm; initial cracks subjected to incremental traction loads: (a) U ¯ ¯ ¯ ¯ = 0.105 (b) U = 0.025 mm; (c) U = 0.065 mm; (d) U = 0.085 mm; (e) U mm. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Topology Optimization for Optimal Fracture Resistance

(a)

(b)

(c)

(d)

125

(e)

[kN]

Figure 5.8. Crack propagation of the final design composite structure without ¯ = 0 mm; initial cracks subjected to incremental traction loads: (a) U ¯ = 0.035 mm; (c) U ¯ = 0.055 mm; (d) U ¯ = 0.075 mm; (e) U ¯ = 0.105 (b) U mm. For a color version of this figure, see www.iste.co.uk/da/topology.zip

 
 



 




  





Figure 5.9. Comparison of the fracture resistance of two composite structures without initial cracks subjected to incremental traction loads. For a color version of this figure, see www.iste.co.uk/da/topology.zip

5.3.3. Design of a square plate without initial cracks in tensile loading This example aims to design a square composite plate without initial cracks subjected to uniaxial tension. The problem geometry of the square plate is shown in Figure 5.10(a). The dimensions of the plate are 100 × 100 mm2 , and the domain is uniformly discretized into

126

Topology Optimization Design of Heterogeneous Materials and Structures

120 × 120 square-shaped bilinear elements. The boundary conditions are as follows: on the upper and lower ends, the vertical displacements are fixed, while the horizontal displacements are free. On the left and right ends, the horizontal displacements are prescribed by an ¯ = 0.005 mm during the simulation. The increasing uniform value of U incremental loading process continues until the reaction force is below a prescribed value, indicating that the structure is fully broken. The initial guess involves an inclusion representing 5% of the volume fraction, as shown in Figure 5.10(a). The final design of the inclusion phase is shown in Figure 5.10(b). The width of the final design is larger than that of the initial guess to resist the x-directional tension, and the inclusion phase topology is changed. Detailed propagation of the phase field crack in the composite structures with optimal design is shown in Figure 5.11. Here, the interface cracks first initiate at the interface (see Figure 5.11(b)). Then, the left and right interface cracks propagate vertically and merge. The complete fracture patterns and design objective values are shown in Figure 5.12(b) for the initial design, and in Figure 5.12(c) for the optimal design. Both responses are compared in Figure 5.12(a). Here, the fracture resistance of the final design structure is increased by 33% when compared to the initial design.

Matrix

15

Inclusion

100

33.33

100

(a)

(b)

Figure 5.10. A square plate without initial cracks subjected to uniaxial tension: (a) geometry of the initial design and (b) final design

Topology Optimization for Optimal Fracture Resistance

(a)

(b)

(d)

(c)

[kN]

Figure 5.11. Crack propagation of the initial design composite structure ¯ = 0 mm; without initial cracks subjected to uniaxial tension: (a) U ¯ ¯ ¯ (b) U = 0.025 mm; (c) U = 0.04 mm; (d) U = 0.08 mm. For a color version of this figure, see www.iste.co.uk/da/topology.zip

 

 
 
 



  



 



Figure 5.12. Comparison of fracture resistance of two composite structures without initial cracks subjected to uniaxial tension. For a color version of this figure, see www.iste.co.uk/da/topology.zip

127

128

Topology Optimization Design of Heterogeneous Materials and Structures

5.3.4. Design of a plate with a single initial crack under three-point bending The purpose of this example is to design a plate subjected to three-point bending with one initial crack. The problem geometry of the square plate is shown in Figure 5.13(a). The dimensions of the plate are 50 × 100 mm. The domain is uniformly discretized into 60 × 120 square-shaped bilinear elements. The load consists of a prescribed displacement at the center of the beam on the top edge. The left bottom corner node is fixed, while the node at the right bottom corner is fixed in the y-displacement and free in the x-displacement. For this case, an initial pre-existing crack is shown in Figure 5.13(b), as well as the initial guess design with the inclusion phase occupying a volume fraction of 10% of the domain area. The computation is ¯ = 0.01 mm performed with monotonic displacement increments of U until the reaction force is below a prescribed criterion value. The displacements are prescribed along the y-direction, while the displacement along the x-direction is free. Matrix

25 20

50

Inclusion Crack

12.5

(a)

100

(b)

(c)

Figure 5.13. A plate with one initial crack subjected to three-point bending: (a) geometry and boundary condition; (b) geometry of the initial design and crack; (c) final design. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Topology Optimization for Optimal Fracture Resistance

129

The pre-existing crack is simulated by applying Dirichlet conditions with d = 1 along the crack. The surrounding area of the initial crack notch is considered as a non-design region. Figure 5.13(c) shows the final design of the inclusion topology. Detailed propagation of the phase field crack of the finally designed composite structure with a single pre-existing crack notch subjected to three-point bending with monotonic displacement increments is shown in Figure 5.14. The initial matrix crack first propagates vertically and is blocked by the reinforced inclusion phase materials. During the following steps, it tries to spread along the horizontal direction but is blocked by the inclusion material redistributed by the proposed topology optimization method. Eventually, the matrix crack propagates along the loading direction until the structure is fully broken. The two load–displacement curves are compared in Figure 5.15(a). The final crack patterns as well as the total fracture energy required for complete failure for the initial and final designs are shown in Figure 5.15(b) and (c) respectively. In this example, the fracture resistance of the final design is increased by 76% when compared to the initial design.

(a)

(b)

(c)

(d) Figure 5.14. Crack propagation of the initial design composite structure ¯ = 0 mm; without initial cracks subjected to three-point bending: (a) U ¯ = 0.11 mm; (c) U ¯ = 0.14 mm; (d) U ¯ = 0.16. For a color version (b) U of this figure, see www.iste.co.uk/da/topology.zip

Topology Optimization Design of Heterogeneous Materials and Structures

[kN]

130





 
  










 








Figure 5.15. Comparison of fracture resistance of two composite structures with one initial crack subjected to three-point bending. For a color version of this figure, see www.iste.co.uk/da/topology.zip

5.3.5. Design of a plate containing multiple inclusions Finally, a plate containing 20 periodically distributed square inclusions is considered. The plate is modeled as a square domain of side 100 mm. The length of the inclusions is computed such that the

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131

volume fraction of the inclusion phase is equal to 20%. The boundary conditions are the same as described in section 4.3 and are shown in Figure 5.16(a): on the lower and upper ends, the y-displacements are fixed, while the x-displacements are free. On the left and right ends, the y-displacements are free, while the x-displacements are prescribed, ¯ during the simulation. The with a uniform increasing value of U computation is performed with monotonic displacement increments ¯ = 0.005 mm until the structure is completely broken. Here again, U the surrounding region of the initial crack is assumed to be non-designable to avoid non-physical designs. The final design is presented in Figure 5.16(c). Detailed propagation of the phase field crack of the final design of the considered composite structure containing multiple inclusions and a single initial crack subjected to tensile loads is shown in Figure 5.17. It can be observed that the initial crack propagates vertically and new cracks are generated around intermediate interfaces of the structure. The inclusion materials are redistributed to prevent further propagation of critical cracks. In the subsequent loading steps, the pre-existing crack propagates through the reinforced inclusion materials and is blocked at the beginning. Then, both matrix and interface cracks propagate towards the vertical direction as well as along the material interfaces until the structure is fully broken. The comparisons of final crack paths, the total fracture energy required for complete failure and load–displacement curves between the initial and optimal designs are provided in Figure 5.18. In this example, the fracture resistance is increased by 53% in comparison with the initial design. It is worth noting that in such an optimization process, the crack pattern can involve multiple cracks, the trajectories and number of which fully depend on the geometry of the inclusion phases. Thus, it becomes very difficult to guess a priori the optimal geometry of inclusions to increase the fracture resistance. Finally, the main advantages of the phase field method in the topology optimization process are summarized as follows: (a) it is easy to couple the phase field method with a topology optimization algorithm, as a fixed mesh

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can be used for the fracture simulation; (b) initiation of cracks can be included in the analysis, which would not be possible with techniques such as XFEM (Moes et al. 1999); (c) it is simple to include interfacial damage in the present framework, using, for example, extensions of the phase field as proposed in Nguyen et al. (2016b). Using cohesive elements in this case would make the analysis much more complex when defining the sensitivity with respect to the material density in the topological algorithm. Thus, the present framework seems to be very promising to design new composite materials with enhanced fracture resistance. 100 Matrix Inclusion

100

10

Crack

10

25

(a)

(b)

(c)

Figure 5.16. A plate containing multiple inclusions and a single initial crack subjected to tensile loads: (a) geometry and boundary condition; (b) geometry of the initial design and crack; (c) final design. For a color version of this figure, see www.iste.co.uk/da/topology.zip

(a)

(b)

(c)

(d)

Figure 5.17. Crack propagation of the final design composite structure ¯ = 0.01 mm; with one initial crack subjected to tensile loads: (a) U ¯ = 0.02 mm; (c) U ¯ = 0.05 mm; (d) U ¯ = 0.09 mm. For a color (b) U version of this figure, see www.iste.co.uk/da/topology.zip

133

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Figure 5.18. Comparison of fracture resistance of two composite structures containing multiple inclusions and one initial crack subjected to tensile loads. For a color version of this figure, see www.iste.co.uk/da/topology.zip

5.4. Concluding remarks In this chapter, we have developed the topology optimization framework to improve the fracture resistance of composites through a redistribution of constant inclusion phases, considering interactions between bulk brittle fracture and interfacial damage. The sole and unique phase field is used to describe both bulk brittle fracture and interface cracking, thus allowing the interaction between the two types of cracks. Before performing the topology optimization, a computationally efficient adjoint sensitivity formulation is derived to

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account for the whole fracturing process involving crack nucleation, propagation and interaction, either from the interfaces and then through the solid phases, or from the composite. Based on the determined sensitivity numbers, the constant amount of reinforced inclusion materials is redistributed using the extended BESO method. Several benchmark tests were presented to demonstrate the potential of the proposed design framework. It was shown that significant improvement of the fracture resistance of the considered composite structures can be achieved for final designs accounting for full failure when compared to the initial guess. To the best of our knowledge, the topology optimization for fracture resistance taking into account the interactions between interfacial damage and bulk brittle fracture for the complete fracturing process has been done for the first time in this chapter. The presented method thus provides a very promising design tool to improve the fracture resistance of heterogeneous materials where both interfacial damage and matrix crack propagation occur, and could constitute a basis for improving other physical properties involving interfacial damage.

6 Topology Optimization for Maximizing the Fracture Resistance of Periodic Composites

This chapter extends the topology optimization framework of optimal fracture resistance presented in the previous chapter to two-phase periodic composites, by considering the heterogeneities and their interfaces in the material. The phase field method presented in section 5.1 is adopted for modeling fracture propagation in two-phase composites. The composite is assumed to be composed of the substructure or the representative volume element (RVE) periodically. Therefore, the optimization is carried out only on the RVE, but takes into account the response of the whole composite specimen to maximize its fracture resistance. The extended BESO method is adopted to redistribute the inclusion phase in the RVE with constant volume fraction. In the following, the topology optimization model for maximizing the fracture resistance of periodic composites is proposed in section 6.1. The element sensitivity number accounting for the complete fracturing process of the whole periodic structure is defined. Several numerical examples are presented in section 6.2 to show significant improvement of the fracture resistance of the optimized periodic composites, compared to initial designs. In addition, the optimized inclusion phase is applied to larger samples, which contain a

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larger number of unit cells, to further validate the proposed numerical design framework. Finally, section 6.3 draws the conclusion. 6.1. Topology optimization model In this section, the optimization problem is solved to find the optimal material distribution or the topology of the inclusion phase within the RVE, in order to maximize the fracture resistance of the resulting periodic composites. It is assumed that the total number of substructures/RVEs in the composite is Ns . The fracture resistance maximization problem can then be formulated using the design variable ρke , where k and e denote the substructure number and the element number in each substructure respectively: Find : {ρ(1) , . . . , ρ(Ns ) } Maximize : J(ρ, u, d, β) subjected to : R = 0 : V (ρ) = Ns



[6.1] [6.2] [6.3]

(k) ρ(k) e ve = Vreq ,

(Ns ) , e = 1, . . . , Ne , : ρ(1) e = · · · = ρe

[6.4] [6.5]

: ρ(k) e = 0 or 1, e = 1, . . . , Ne , k = 1, . . . , Ns . [6.6] where J is the fracture energy, which is calculated using numerical (k) integration [5.44]. R is the nodal residual force [5.46]. In [6.4], ve is the volume of the e−th element in the k−th unit cell, and Ne is the number of elements in each cell. Similar to the optimization model defined in sections 1 and 2 in Chapter 4, the condition (1) (N ) ρe = · · · = ρe s , e = 1, . . . , Ne ensures that the pseudo densities (0 or 1) of elements at the corresponding locations in each substructure are the same. From equation [5.43] derived in section 5.2, we obtain the same material interpolation model: Ee = ρe Einc + (1 − ρe )Emat

[6.7]

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where ρe takes the value of 0 and 1 which correspond to the matrix and the inclusion phase respectively. Based on the sensitivity analysis formulated in section 5.2, the derivative of the total mechanical work J with respect to the topology design variable ρe can be stated as: nload   T  T  ∂ J 1  =− BT (σ e )(n) dΩe + μ(n) λ(n) ∂ρe 2 Ωi n=1   BT (σ e )(n−1) dΩi . [6.8] Ωe

However, since the considered composites are periodic in this scheme, the optimization process is carried out only within the substructure/RVE. Therefore, the element sensitivity numbers at the same location in each substructure need to be consistent to enforce the periodic array of the substructures. They are then defined as the summation of the sensitivity of corresponding elements in all substructures; that is, the sensitivity number αe is formulated as: αe =

Ns  ∂ J (k)

[6.9]

k=1 ∂ρe

As a result, the above sensitivity information takes into account the fracture response of the whole periodic composite in order to maximize its fracture resistance. Following section 5.2, the sensitivity numbers associated with the relative ranking of the element sensitivities are treated with the damping as in [4.55]. In order to avoid checkerboard patterns, the above formulated sensitivity numbers are then smoothed by means of a filtering scheme as described in [1.41]. Due to the discrete nature of the design variable of the adopted method and in order to avoid oscillations in evolutionary history of the design objective value, the current sensitivity number is further averaged with its historical information as in [1.43].

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It should be recalled that the material volume fraction of reinforced inclusion phases is kept constant during the optimization process. However, the geometry of inclusions is optimized in the substructure/RVE in order to maximize the fracture resistance of the resulting periodic composite sample. The extended BESO method formulated in section 5.2 is adopted to solve the optimization problem [6.1–6.6] using the above modified sensitivity numbers, which account for the whole fracturing process involving crack nucleation, propagation and interaction until complete failure of the considered periodic composite. 6.2. Numerical examples In this section, several numerical examples are presented to demonstrate the proposed topology optimization framework for maximizing the fracture resistance of periodic composites. In all tests, regular meshes using quadrilateral bilinear elements are adopted, and the plane strain assumption is adopted. The same finite element discretization is used for both displacement and crack phase fields. The regularization parameter  describing the width of smeared crack and interface is chosen as twice the finite element size  = 2e . In addition, material properties of both matrix and inclusion phases are the same, as shown in section 5.3, and so are the toughness and the interface fracture strength. 6.2.1. Design of a periodic composite under three-point bending The problem geometry of the first example is shown in Figure 6.1(a), where the composite is composed of Ns = 7 × 3 unit cells periodically repeated along the x− and y−directions respectively. In the initial guess design of the RVE, as shown in Figure 6.1(a), the inclusion phase covers the centered circle domain with a diameter corresponding to the volume fraction of 30%. The dimensions of the whole composite structure are 350 × 150 mm, and the domain is

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uniformly discretized into 350 × 150 square-shaped bilinear elements. The boundary conditions are as follows: the left bottom corner node is fixed, while at the right bottom corner node, the y-displacement is fixed and the x-displacement is free. On the upper end, the load ¯ at the center point. The consists of a prescribed displacement U displacement is prescribed along the y-direction while the displacement along the x−direction is free. The computation is ¯ = 0.01 mm performed with monotonic displacement increments of U until the reaction force is below a prescribed criterion value, indicating that the structure is completely broken.

((a))

(b) ( )

(a)

(c)

(b)

Figure 6.1. A periodic composite without initial crack subjected to three-point bending: geometry of (a) the initial design and (b) the final design

During the optimization process, the material volume fraction of the inclusion phase remains constant in all the next examples. Based on their sensitivity numbers, the inclusion phase will be redistributed within the periodic cell by the extended BESO method in order to improve the fracture resistance of the whole periodic composite. Figure 6.1(b) shows the final design of inclusion topology and the resulting composite structure. It can be observed that the material on the bottom side of the reinforcement inclusion phase moves left and right, and the final design inclusion phases are interconnected in horizontal cells to form a “wave” structure. Detailed propagation of the phase field crack of the initial and final design periodic composites subjected to three-point bending with

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monotonic displacement increments is shown in Figures 6.2 and 6.3 respectively. We can see that the trajectories, numbers as well as propagation paths of the phase field cracks are completely different since the geometry of the inclusion phases has changed. However, both crack patterns are symmetric since we use the symmetric boundary conditions and initial cracks are generated at the interface between different phases. Subsequently, one bulk crack initiates and propagates along the force direction for the initial design, until the periodic structure is broken. For an optimal design, more interface cracks initiate and propagate along the phase interfaces, resulting in an increase of interface damage of the composite. Finally, bulk cracks propagate vertically as loading continues, until the periodic structure is fully broken. The final crack patterns, the total required fracture energies for complete failure and the load–displacement curves for the initial and optimal designs are shown in Figure 6.4. In this example, the periodic composite with optimal design is 124% more resistant to fracture than the initial composite structure.

(a)

(b)

(c)

(d)

Figure 6.2. Crack propagation in the initial periodic composite without initial ¯ = 0 mm; (b) U ¯ = 0.11 mm; cracks subjected to three-point bending: (a) U ¯ = 0.20 mm; (d) U ¯ = 0.28 mm. For a color version of this figure, see (c) U www.iste.co.uk/da/topology.zip

Maximizing the Fracture Resistance of Periodic Composites

(a)

(b)

(c)

(d)

(e)

(f)

141

Figure 6.3. Crack propagation of the final design periodic composite without ¯ = 0 mm; (b) U ¯ = 0.17 mm; initial cracks subjected to three-point bending: (a) U ¯ = 0.23 mm; (d) U ¯ = 0.28 mm; (e) U ¯ = 0.32 mm; (f) U ¯ = 0.37 mm. For a (c) U color version of this figure, see www.iste.co.uk/da/topology.zip

In the following, we use the optimized geometry of the inclusion phase for a larger periodic sample that contains a much larger number of unit cells. The geometries of the initial and optimal design periodic composites are shown in Figure 6.5(a) and Figure 6.5(b) respectively. The dimensions of the two composite structures are 700 × 300 mm and are both uniformly discretized into 700 × 300 square-shaped bilinear elements. The boundary conditions are the same as in Figure 6.1. However, the composites are herein composed of Ns = 14 × 6 unit cells periodically, along the x− and y−directions respectively. The computation is performed with monotonic displacement increments of ¯ = 0.01 mm until the reaction force is below a prescribed criterion U value.

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[kN]













Figure 6.4. Comparison of fracture resistance of two periodic composites without initial cracks subjected to three-point bending. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Maximizing the Fracture Resistance of Periodic Composites

143

300

(a)

700

Matrix Inclusion

300

(b)

700

Figure 6.5. A larger periodic composite without initial crack subjected to three-point bending: geometry of (a) the initial design and (b) the final design

In this case, the detailed propagation of the phase field crack in the initial and optimal periodic composites is shown in Figures 6.6 and 6.7 respectively. Again, crack pattern involves multiple cracks, and its trajectory and number fully depend on the geometry of the inclusion phases. The composite structure with an optimized inclusion geometry is more damaged, requiring larger mechanical energy for complete failure. The comparisons of final crack paths, the total fracture energy required for complete failure and load–displacement curves between initial and optimal designs are shown in Figure 6.8. For a larger sample, the fracture resistance of the final periodic composite increases by 82.5% in comparison with the initial design.

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 6.6. Crack propagation in the initial periodic composite without initial ¯ = 0 mm; (b) U ¯ = 0.25 mm; cracks subjected to three-point bending: (a) U ¯ = 0.35 mm; (d) U ¯ = 0.41 mm; (e) U ¯ = 0.44 mm; (f) U ¯ = 0.59 mm. For a (c) U color version of this figure, see www.iste.co.uk/da/topology.zip

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6.7. Crack propagation in the optimized design periodic composite ¯ = 0 mm; (b) U ¯ = without initial cracks subjected to three-point bending: (a) U ¯ = 0.52 mm; (d) U ¯ = 0.57 mm; (e) U ¯ = 0.61 mm; (f) U ¯ = 0.72 0.39 mm; (c) U mm. For a color version of this figure, see www.iste.co.uk/da/topology.zip

Maximizing the Fracture Resistance of Periodic Composites










[kN]







 





Figure 6.8. Comparison of fracture resistance of two periodic composites with a larger number of cells subjected to three-point bending. For a color version of this figure, see www.iste.co.uk/da/topology.zip

145

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Topology Optimization Design of Heterogeneous Materials and Structures

6.2.2. Design of a periodic composite under non-symmetric three-point bending This example aims to design a periodic composite subjected to non-symmetric three-point bending without initial cracks. Compared with the boundary condition shown in Figure 6.1, the load in this case consists of a prescribed displacement at the center point of the right half-beam on the top edge, as shown in Figure 6.9(a). The vertical displacement is prescribed by an increasing uniform value of ¯ = 0.01 mm during the simulation, until the considered periodic U composite is fully broken. The initial guess design of the inclusion geometry is the same as in the last example, which means that the inclusion phase occupies a volume fraction of 30% of the sample. The considered composite is composed of Ns = 9 × 3 unit cells periodically repeated along each space direction. The dimension of the composite is 450 × 150 mm, and the domain is uniformly discretized into 450 × 150 square-shaped bilinear elements. The redistribution of the inclusion phase is carried out in a single unit cell based on the sensitivity information, which accounts for the fracture resistance response of the whole periodic structure.

(a)

(b)

Figure 6.9. A periodic composite without initial crack subjected to non-symmetric three-point bending: geometry of (a) the initial design and (b) the final design

Maximizing the Fracture Resistance of Periodic Composites

(a)

(b)

(c)

(d)

147

Figure 6.10. Crack propagation in the initial periodic composite without ¯ = initial cracks subjected to non-symmetric three-point bending: (a) U ¯ = 0.17 mm; (c) U ¯ = 0.24 mm; (d) U ¯ = 0.30 mm. For a 0 mm; (b) U color version of this figure, see www.iste.co.uk/da/topology.zip

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6.11. Crack propagation in the optimized design periodic composite without initial cracks subjected to non-symmetric three-point bending: (a) ¯ = 0 mm; (b) U ¯ = 0.17 mm; (c) U ¯ = 0.24 mm; (d) U ¯ = 0.33 mm; U ¯ = 0.39 mm; (f) U ¯ = 0.44 mm. For a color version of this figure, see (e) U www.iste.co.uk/da/topology.zip

The optimized geometry of the inclusion phase as well as the resulting periodic composites are shown in Figure 6.9(b). Detailed propagations of the phase field cracks of the initial and final design composite structures subjected to incremental traction load are given in Figures 6.10 and 6.11 respectively. It can be observed that both cracks are first generated around the material interface below the applied force. Then, only one bulk crack along the force direction is generated

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for the initial guess design, while more complex cracks nucleate for the new design. Finally, cracks propagate vertically until the structure is fully broken in both cases. We can see that the new design periodic composite is more damaged, resulting in larger fracture energy. The final crack patterns, the total fracture energy required for complete failure and the load–displacement curves for the initial and final designs are shown in Figure 6.12. In this example, the periodic composite with an optimal inclusion geometry is 100.6% more resistant to fracture than the initial composite.






 



[kN]






 





Figure 6.12. Comparison of fracture resistance of two periodic composites without initial cracks subjected to non-symmetric three-point bending. For a color version of this figure, see www.iste.co.uk/da/topology.zip

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149

Finally, we apply the optimized geometry of the inclusion phase obtained in the first example for a larger periodic sample but subjected to non-symmetric three-point bending. The dimensions as well as the boundary conditions of the initial and optimal periodic composites are shown in Figure 6.13(a) and Figure 6.13(b) respectively. Both the structures are uniformly discretized into 900 × 300 square-shaped bilinear elements. The force is applied to the top edge at a distance of 250 mm from the apex of the upper right corner. The vertical displacement is also prescribed as an increasing uniform value of ¯ = 0.01 mm during the simulation until the reaction force is below a U prescribed value, indicating that the periodic composite is fully broken. The complete fracture patterns and design objective values for the initial and optimal designs are shown in Figure 6.14(a) and Figure 6.14(b) respectively. Both responses are compared in Figure 6.14(c). Here, the fracture resistance of the optimized design periodic composite increases by 135.4% compared to the initial design.

300

(a)

900

250

Matrix Inclusion

300

(b)

900

250

Figure 6.13. A larger periodic composite without initial crack subjected to non-symmetric three-point bending: geometry of (a) the initial design and (b) the final design

Topology Optimization Design of Heterogeneous Materials and Structures






 








 



[kN]

150



Figure 6.14. Comparison of fracture resistance of two periodic composites with a larger number of cells subjected to non-symmetric three-point bending. For a color version of this figure, see www.iste.co.uk/da/topology.zip

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151

6.3. Concluding remarks This chapter has extended the topology optimization framework for maximizing the fracture resistance to periodic two-phase composites. The optimization design domain is defined in a single cell since the considered composite structure is periodic. Element sensitivity number in the cell is then defined as the summation of the sensitivity of corresponding elements in all cells, in order to account for the fracture response of the whole periodic composite during the fracturing process. As a result, the topology optimization is carried out in one single cell, and the fracture resistance of the whole composite structure is improved. Several numerical examples have demonstrated that the fracture resistance of the considered periodic composite can be improved significantly for final designs accounting for full failure when compared to the initial guess designs. In addition, the optimized geometry of the inclusion phase can be effectively applied to periodic samples with a larger number of unit cells. In conclusion, the presented topology optimization framework has a big potential to design new periodic composites with enhanced fracture resistance.

Conclusion

In this book, we have made contributions to the modeling of heterogeneous materials and structures by topology optimization, with special emphasis on (a) multiscale topology optimization in the context of non-separated scales and (b) maximizing the fracture resistance of heterogeneous materials. In Part 1, we first used a coarse mesh corresponding to a homogenized medium based on the classical numerical homogenization, allowing the reduction of micro fields to perform the topology optimization. The size effect of the periodic unit cell was investigated to show that the present framework leads to an optimized structure with higher compliance when the scales cannot clearly be separated. To this end, we developed a new multiscale topology optimization procedure using a non-local filter-based homogenization scheme, in order to take into account all structural details and strain gradient effects. We showed that the strain gradient effects can lead to a significant increase in the stiffness of the lattice associated with the optimized topology. Finally, we extended the multiscale topology optimization framework to the design of the geometry of mesoscopic structures with fixed microscopic unit cells. As expected, periodic unit cells had a major influence on the optimal solutions of mesoscopic structures. In summary, the topology optimization and the numerical homogenization scheme without scale separation were combined in this part, allowing the optimization problem to be performed on a

Topology Optimization Design of Heterogeneous Materials and Structures, First Edition. Daicong Da. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

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coarse mesh, instead of using the fully detailed description of the structure for computational saving. In Part 2, we proposed to investigate the use of topology optimization for maximizing the fracture resistance of heterogeneous structures and materials. The objective here was to maximize the required mechanical work until the complete failure of heterogeneous materials, which involves crack nucleation, propagation and interaction. To our best knowledge, this part constituted the first contribution to proposing a numerical design framework for fracture resistance of quasi-brittle composites, by an optimal design of the spatial distribution of the inclusion phase with a given target volume fraction. In order to model the complete fracturing process, the phase field method for fracturing is adopted with a regularized description of discontinuity. Second, we extended the design framework by taking into account crack nucleation in both the matrix and the interfaces. The material volume fraction of reinforced inclusion phases is kept constant during the optimization process. Finally, we extended the topology optimization framework by taking into account interfacial damage to periodic composites. Numerical results showed that it is easy to combine the phase field method with a topology optimization algorithm, as a fixed mesh can be used for the simulation of fracture. Moreover, interfacial damage can be included in the present framework quite simply by extending phase field modeling. A series of numerical examples demonstrated that the fracture resistance of the (periodic) composite can be significantly increased by the proposed topology optimization process, compared with the initial designs. Therefore, the present topology optimization framework seems to be very promising for designing new composite materials with enhanced fracture resistance. In general, the design of multiscale structures without scale separation and fracture resistance optimization are both new fields that have remained relatively unexplored until recently. Combining (new) topology optimization techniques and additive manufacturing processes, many potential developments can be identified for

Conclusion

155

engineering problems. In the following, we provide some perspectives on this book: 1) In the context of non-separated scales, improvements of connectivity between optimized periodic cells (see, for example, Figure 2.7) can be investigated. In addition, material microstructures are assumed to be periodic in this work, even though this is not mandatory. Optimization design of non-periodic materials to fully explore the design freedom within the multiscale framework and achieve higher performance structures could constitute an interesting future study. Furthermore, the development and verification of an advanced lattice structure by additive manufacturing processes, while considering large deformation problems and multi-functional and multi-physics behavior, would also have great research prospects. 2) Through Part 2, we initiated a wide field of study concerning topology optimization for maximizing the fracture resistance. The preliminary results presented in this book could be further investigated along the following lines: (a) reduction of the associated computational costs; (b) experimental validations of the predicted fracture resistance (using, for example, 3D printing, 3D microtomography and in situ testing); (c) extension of such studies to other mechanical behaviors (fracture in elastoplastic and viscoelastic materials) and (d) development of multiscale analysis in this context to apply this methodology at the structural scale. Recent works (Nguyen et al. 2018) could provide a promising basis for such explorations.

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Index

B, C, D

L, M, N

bulk fracture, 71 cantilever beam, 46 coarse meshes, 12 crack, 69 interaction, 119, 138 nucleation, 119, 138 propagation, 87, 119, 138 discontinuities, 104 doubly-clamped structure, 16, 61

L-shaped structure, 16 Lagrange multipliers, 9 localization problem, 4, 33 low-pass filter, 30 mesoscopic scale, 30 structures, 39, 58 Messerschmitt-Bölkow-Blohm (MBB) beam, 16 multiscale topology optimization mesoscopic structural design, 57 non-separated scales, 29 scale separation, 3 nonlocal homogenization, 30

E, F, H effective material properties, 7 extended phase field modeling, 104 failure, 87, 119, 138 fine meshes, 12 four-point bending beam, 52, 64 fracture energy, 119, 138 fracture resistance design Interfacial damage, 103 Periodic composites, 135 heterogeneous materials, 16, 45, 61 structures, 87, 119, 138 homogenization method, 4

P, Q, R periodic boundary conditions, 9 phase field modeling, 71 quasi-brittle materials, 87 regularized representation, 71, 104 relocalization scheme, 12 representative volume element (RVE), 33, 58, 136

Topology Optimization Design of Heterogeneous Materials and Structures, First Edition. Daicong Da. © ISTE Ltd 2019. Published by ISTE Ltd and John Wiley & Sons, Inc.

174

Topology Optimization Design of Heterogeneous Materials and Structures

S, T size effects, 16, 58 staggered scheme, 75, 108 stiffness interpolation, 58 strain energy history function, 72, 106 three-point bending, 138 non-symmetric, 138

topology optimization fracture resistance design, 69 model definition, 10, 41, 58, 78, 114, 136 optimization procedure, 14, 42, 58, 85, 136 sensitivity analysis, 10, 58, 80, 116, 136

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2011 CHINESTA Francisco, CESCOTTO Serge, CUETO Elias, LORONG Philippe Natural Element Method for the Simulation of Structures and Processes DAVIM Paulo J. Finite Element Method in Manufacturing Processes POMMIER Sylvie, GRAVOUIL Anthony, MOËS Nicolas, COMBESCURE Alain Extended Finite Element Method for Crack Propagation

2010 SOUZA DE CURSI Eduardo, SAMPAIO Rubens Modeling and Convexity

2008 BERGHEAU Jean-Michel, FORTUNIER Roland Finite Element Simulation of Heat Transfer

EYMARD Robert Finite Volumes for Complex Applications V: Problems and Perspectives FREY Pascal, GEORGE Paul Louis Mesh Generation: Application to finite elements – 2nd edition GAY Daniel, GAMBELIN Jacques Modeling and Dimensioning of Structures MEUNIER Gérard The Finite Element Method for Electromagnetic Modeling

2005 BENKHALDOUN Fayssal, OUAZAR Driss, RAGHAY Said Finite Volumes for Complex Applications IV: Problems and Perspectives

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