Optimal design in small amplitude homogenization
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 3, pp. 543-574.

This paper is concerned with optimal design problems with a special assumption on the coefficients of the state equation. Namely we assume that the variations of these coefficients have a small amplitude. Then, making an asymptotic expansion up to second order with respect to the aspect ratio of the coefficients allows us to greatly simplify the optimal design problem. By using the notion of H-measures we are able to prove general existence theorems for small amplitude optimal design and to provide simple and efficient numerical algorithms for their computation. A key feature of this type of problems is that the optimal microstructures are always simple laminates.

DOI : https://doi.org/10.1051/m2an:2007026
Classification : 15A15,  15A09,  15A23
Mots clés : optimal design, H-measures, homogenization
     author = {Allaire, Gr\'egoire and Guti\'errez, Sergio},
     title = {Optimal design in small amplitude homogenization},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {543--574},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {3},
     year = {2007},
     doi = {10.1051/m2an:2007026},
     zbl = {pre05289384},
     mrnumber = {2355711},
     language = {en},
     url = {www.numdam.org/item/M2AN_2007__41_3_543_0/}
Allaire, Grégoire; Gutiérrez, Sergio. Optimal design in small amplitude homogenization. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 3, pp. 543-574. doi : 10.1051/m2an:2007026. http://www.numdam.org/item/M2AN_2007__41_3_543_0/

[1] G. Allaire, Shape Optimization by the Homogenization Method. Springer-Verlag (2002). | MR 1859696 | Zbl 0990.35001

[2] G. Allaire and S. Gutiérrez, Optimal design in small amplitude homogenization (extended version). Preprint available at http://www.cmap.polytechnique.fr/preprint/repository/576.pdf (2005).

[3] G. Allaire and F. Jouve, Optimal design of micro-mechanisms by the homogenization method. Eur. J. Finite Elements 11 (2002) 405-416. | Zbl 1120.74710

[4] G. Allaire, F. Jouve and H. Maillot, Topology optimization for minimum stress design with the homogenization method. Struct. Multidiscip. Optim. 28 (2004) 87-98.

[5] J.C. Bellido and P. Pedregal, Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing. Discr. Contin. Dyn. Syst. 8 (2002) 967-982. | Zbl 1035.49008

[6] M.P. Bendsøe and O. Sigmund, Topology Optimization. Theory, Methods, and Applications. Springer-Verlag, New York (2003). | MR 2008524 | Zbl 1059.74001

[7] A. Cherkaev, Variational Methods for Structural Optimization. Springer Verlag, New York (2000). | MR 1763123 | Zbl 0956.74001

[8] A. Donoso and P. Pedregal, Optimal design of 2D conducting graded materials by minimizing quadratic functionals in the field. Struct. Multidiscip. Optim. 30 (2005) 360-367.

[9] P. Duysinx and M.P. Bendsøe, Topology optimization of continuum structures with local stress constraints. Int. J. Num. Meth. Engng. 43 (1998) 1453-1478. | Zbl 0924.73158

[10] P. Gérard, Microlocal defect measures. Comm. Partial Diff. Equations 16 (1991) 1761-1794. | Zbl 0770.35001

[11] Y. Grabovsky, Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals. Adv. Appl. Math. 27 (2001) 683-704. | Zbl 1001.49002

[12] F. Hecht, O. Pironneau and K. Ohtsuka, FreeFem++ Manual. Downloadable at http://www.freefem.org

[13] L. Hörmander, The analysis of linear partial differential operators III. Springer, Berlin (1985). | MR 781536 | Zbl 0601.35001

[14] R.V. Kohn, Relaxation of a double-well energy. Cont. Mech. Thermodyn. 3 (1991) 193-236. | Zbl 0825.73029

[15] R. Lipton, Relaxation through homogenization for optimal design problems with gradient constraints. J. Optim. Theory Appl. 114 (2002) 27-53. | Zbl 1005.49005

[16] R. Lipton, Stress constrained G closure and relaxation of structural design problems. Quart. Appl. Math. 62 (2004) 295-321. | Zbl 1075.74066

[17] R. Lipton and A. Velo, Optimal design of gradient fields with applications to electrostatics. Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XIV, Stud. Math. Appl. 31 (2002) 509-532. | Zbl 1080.78003

[18] G. Milton, The theory of composites. Cambridge University Press (2001). | Zbl 0993.74002

[19] F. Murat and L. Tartar, Calcul des Variations et Homogénéisation, Les Méthodes de l'Homogénéisation Théorie et Applications en Physique, Coll. Dir. Études et Recherches EDF, 57, Eyrolles, Paris (1985) 319-369. English translation in Topics in the mathematical modelling of composite materials, A. Cherkaev and R. Kohn Eds., Progress in Nonlinear Differential Equations and their Applications 31, Birkhäuser, Boston (1997).

[20] U. Raitums, The extension of extremal problems connected with a linear elliptic equation. Soviet Math. 19 (1978) 1342-1345. | Zbl 0428.49002

[21] L. Tartar, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. Royal Soc. Edinburgh 115A (1990) 93-230. | Zbl 0774.35008

[22] L. Tartar, Remarks on optimal design problems. Calculus of variations, homogenization and continuum mechanics (Marseille, 1993), World Sci. Publishing, River Edge, NJ, Ser. Adv. Math. Appl. Sci. 18 (1994) 279-296. | Zbl 0884.49015

[23] L. Tartar, An introduction to the homogenization method in optimal design, in Optimal shape design (Tróia, 1998), A. Cellina and A. Ornelas Eds., Springer, Berlin, Lect. Notes Math. 1740 (2000) 47-156. | Zbl 1040.49022