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The homogenization method for topology optimization of structures: old and new. (English) Zbl 1452.74096

Summary: Topology optimization of structures is nowadays a well developed field with many different approaches and a wealth of applications. One of the earliest methods of topology optimization was the so-called homogenization method, introduced in the early eighties. It became extremely popular in its over-simplified version, called SIMP (Solid Isotropic Material with Penalisation), which retains only the notion of material density and forgets about true composite materials with optimal (possibly non isotropic) microstructures. However, the appearance of mature additive manufacturing technologies which are able to build finely graded microstructures (sometimes called lattice materials) drastically change the picture and one can see a resurrection of the homogenization method for such applications. Indeed, homogenization is the right technique to deal with microstructured materials where anisotropy plays a key role, a feature which is absent from SIMP. Homogenization theory allows to replace the microscopic details of the structure (typically a complex networks of bars, trusses and plates) by a simpler effective elasticity tensor describing the mesoscopic properties of the structure. The goal of these lecture notes is to review the necessary mathematical tools of homogenization theory and apply them to topology optimization of mechanical structures. The ultimate application, targeted here, is the topology optimization of structures built with lattice materials. Practical and numerical exercises are given, based on the finite element free software FreeFem++.

MSC:

74P15 Topological methods for optimization problems in solid mechanics
74Q05 Homogenization in equilibrium problems of solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74K99 Thin bodies, structures
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids

Software:

SQPlab; FreeFem++; PLCP
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Allaire, G., “Homogenization and two-scale convergence,” SIAM J. Math. Anal., 23: 1482-1518 (1992). · Zbl 0770.35005
[2] Allaire, G., Shape Optimization by the Homogenization Method, Vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York (2002). · Zbl 0990.35001
[3] Allaire, G., Conception Optimale de Structures, Vol. 58 of Mathématiques et Applications, Springer, Heidelberg (2007). · Zbl 1132.49033
[4] Allaire, G., Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation. Translated from the French by Alan Craig, Numer. Math. Sci. Comput., Oxford University Press, Oxford, UK (2007). · Zbl 1120.65001
[5] Allaire, G., Geoffroy-Donders, P., and Pantz, O., Topology Optimization of Modulated and Oriented Periodic Microstructures by the Homogenization Method, Computers & Mathematics with Applications, special issue SimAM (2019). · Zbl 1443.74246
[6] Allaire, G., and Pantz, O., “Structural optimization with FreeFem++,” Struct. Multidiscip. Optim., 32: 173-181 (2006). · Zbl 1245.74049
[7] Bendsøe, M., and Kikuchi, N., “Generating optimal topologies in structual design using a homogenization method,” Comput. Methods Appl. Mech. Engrg., 71(2): 197-224 (1988). · Zbl 0671.73065
[8] Bendsøe, M., and Sigmund, O., Topology Optimization, Springer-Verlag, Berlin (2003). · Zbl 1059.74001
[9] Bensoussan, A., Lions, J.-L., and Papanicolaou, G., Asymptotic Analysis for Periodic Structures, Studies in Mathematics and Its Applications, North-Holland, Amsterdam (1978). · Zbl 0404.35001
[10] Bonnans, J. F., Gilbert, C., Lemaréchal, C., and Sagastizábal, C., Numerical Optimization: Theoretical and Practical Aspects (Universitext), Second edition, Springer-Verlag, Berlin (2006). · Zbl 1108.65060
[11] Cherkaev, A., Variational Methods for Structural Optimization, Springer Verlag, New York (2000). · Zbl 0956.74001
[12] Cioranescu, D., and Donato, P., An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Applications, 17, Oxford (1999). · Zbl 0939.35001
[13] Ekeland, I., and Témam, R., Convex Analysis and Variational Problems, Classics in Applied Mathematics, 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999). · Zbl 0939.49002
[14] Geihe, B., Lenz, M., Rumpf, M., and Schultz, R., “Risk averse elastic shape optimization with parametrized fine scale geometry,” Math. Program., Ser. A, 141: 383-403 (2013). · Zbl 1280.90090
[15] Geoffroy-Donders, P., Homogenization method for topology optimization of structures built with lattice materials, PhD thesis, Ecole Polytechnique, Université Paris-Saclay (2018).
[16] Geoffroy-Donders, P., Allaire, G., and Pantz, O., 3-d topology optimization of modulated and oriented periodic microstructures by the homogenization method, submitted. HAL preprint: hal-01939201 (November 2018).
[17] Gibson, I., Rosen, D., and Stucker, B., Additive Manufacturing Technologies, Springer, New York (2015).
[18] Groen, J. P., and Sigmund, O., “Homogenization based topology optimization for high resolution manufacturable microstructures,” International Journal for Numerical Methods in Engineering, 113: 1148-1163 (2018).
[19] Hashin, Z., and Shtrikman, S., “A variational approach to the theory of the elastic behavior of multiphase materials,” J. Mech. Phys. Solids, 11: 127-140 (1963). · Zbl 0108.36902
[20] Haslinger, J., and Mäkinen, R., Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, Philadelphie (2003). · Zbl 1020.74001
[21] Hecht, F., “New development in freefem++,” J. Numer. Math., 20: 251-265 (2012). · Zbl 1266.68090
[22] Henrot, A., and Pierre, M., Shape Variation and Optimization: A Geometrical Analysis. English Version of the French Publication with Additions and Updates, EMS Tracts in Mathematics, 28. European Mathematical Society (EMS), Zürich (2018). · Zbl 1392.49001
[23] Hornung, U., Editor, Homogenization and Porous Media, Springer Verlag (1996). · Zbl 0885.35010
[24] Jikov, V., Kozlov, S., and Oleinik, O., Homogenization of Differential Operators and Integral Functionals, Springer, Berlin (1995).
[25] Kawohl, B., Pironneau, O., Tartar, L., and Zolésio, J.-P., Optimal shape design. Lectures given at the Joint C.I.M./C.I.M.E. Summer School held in Tróia, June 1-6, 1998. Edited by Cellina, A., and Ornelas, A. Lecture Notes in Mathematics, 1740. Fondazione CIME/CIME Foundation Subseries. Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence (2000).
[26] Katsikadelis, J. T., The Boundary Element Method for Engineers and Scientists: Theory and Applications, Elsevier (2016).
[27] Milton, G., The Theory of Composites, Cambridge University Press (2001).
[28] Murat, F., “Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients,” Annali Mat. Pura Appli., 112: 49-68 (1977). · Zbl 0349.49005
[29] Murat, F., and Tartar, L., H-Convergence, in Topics in the Mathematical Modeling of Compositic Materials, Progress in Nonlinear Differential Equations and their Applieations., ed. Cherkaev, A., and Kohn, R., Birkhäuser, Boston, 31 (1997), 21-43. · Zbl 0920.35019
[30] Nguetseng, G., “A general convergence result for a functional related to the theory of homogenization,” SIAM J. Math. Anal., 20: 608-623 (1989). · Zbl 0688.35007
[31] Nocedal, J., and Wright, S., Numerical Optimization, Springer Science (1999). · Zbl 0930.65067
[32] Pantz, O., and Trabelsi, K., “A post-treatment of the homogenization method for shape optimization,” SIAM J. Control Optim., 47: 1380-1398 (2008). · Zbl 1161.49042
[33] Pedersen, P., “On optimal orientation of orthotropic materials,” Structural Optimization, 1(2): 101-106 (1989).
[34] Rozvany, G., Structural Design via Optimality Criteria, Kluwer Academic Publishers, Dordrecht (1989). · Zbl 0687.73079
[35] Sokolowski, J., and Zolésio, J.-P., Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16, Springer, Berlin (1992). · Zbl 0761.73003
[36] Tartar, L., An introduction to the Homogenization Method in Optimal Design, in Optimal Shape Design (Tróia, 1998), Cellina, A., and Ornelas, A. eds., Lecture Notes in Mathematics 1740, pp. 47-156, Springer, Berlin (2000). · Zbl 1040.49022
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