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Optimal design in small amplitude homogenization

Published online by Cambridge University Press:  02 August 2007

Grégoire Allaire  and
Sergio Gutiérrez
Grégoire Allaire
Affiliation:
Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique, 91128 Palaiseau, France. gregoire.allaire@polytechnique.fr
Sergio Gutiérrez
Affiliation:
Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique, 91128 Palaiseau, France. gregoire.allaire@polytechnique.fr : Departamento de Ingeniería Estructural y Geotécnica, Pontificia Universidad Católica de Chile, Santiago Chile, Chile. sgutierr@ing.puc.cl
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Abstract

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.

Keywords

Optimal designH-measureshomogenization.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 3 , May 2007 , pp. 543 - 574
Copyright
© EDP Sciences, SMAI, 2007

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References

G. Allaire, Shape Optimization by the Homogenization Method. Springer-Verlag (2002).
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).
Allaire, G. and Jouve, F., Optimal design of micro-mechanisms by the homogenization method. Eur. J. Finite Elements 11 (2002) 405–416.
Allaire, G., Jouve, F. and Maillot, H., Topology optimization for minimum stress design with the homogenization method. Struct. Multidiscip. Optim. 28 (2004) 87–98.
Bellido, J.C. and Pedregal, P., Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing. Discr. Contin. Dyn. Syst. 8 (2002) 967–982. CrossRef
M.P. Bendsøe and O. Sigmund, Topology Optimization. Theory, Methods, and Applications. Springer-Verlag, New York (2003).
A. Cherkaev, Variational Methods for Structural Optimization. Springer Verlag, New York (2000).
Donoso, A. and Pedregal, P., Optimal design of 2D conducting graded materials by minimizing quadratic functionals in the field. Struct. Multidiscip. Optim. 30 (2005) 360–367. CrossRef
Duysinx, P. and Bendsøe, M.P., Topology optimization of continuum structures with local stress constraints. Int. J. Num. Meth. Engng. 43 (1998) 1453–1478. 3.0.CO;2-2>CrossRef
Gérard, P., Microlocal defect measures. Comm. Partial Diff. Equations 16 (1991) 1761–1794. CrossRef
Grabovsky, Y., Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals. Adv. Appl. Math. 27 (2001) 683–704. CrossRef
F. Hecht, O. Pironneau and K. Ohtsuka, FreeFem++ Manual. Downloadable at http://www.freefem.org
L. Hörmander, The analysis of linear partial differential operators III. Springer, Berlin (1985).
Kohn, R.V., Relaxation of a double-well energy. Cont. Mech. Thermodyn. 3 (1991) 193–236. CrossRef
Lipton, R., Relaxation through homogenization for optimal design problems with gradient constraints. J. Optim. Theory Appl. 114 (2002) 27–53. CrossRef
Lipton, R., Stress constrained G closure and relaxation of structural design problems. Quart. Appl. Math. 62 (2004) 295–321. CrossRef
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.
G. Milton, The theory of composites. Cambridge University Press (2001).
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).
Raitums, U., The extension of extremal problems connected with a linear elliptic equation. Soviet Math. 19 (1978) 1342–1345.
Tartar, L., H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. Royal Soc. Edinburgh 115A (1990) 93–230.
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.
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.