Vector variational problems and applications to optimal design
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 357-381.

We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical simulation leads to approximated optimal configurations. We include several such simulations in 2d and 3d.

DOI: 10.1051/cocv:2005010
Classification: 49J45,  74P10,  74Q15
Keywords: effective, homogenized or relaxed integrand, gradient Young measures, laminates
@article{COCV_2005__11_3_357_0,
author = {Pedregal, Pablo},
title = {Vector variational problems and applications to optimal design},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {357--381},
publisher = {EDP-Sciences},
volume = {11},
number = {3},
year = {2005},
doi = {10.1051/cocv:2005010},
zbl = {1089.49022},
mrnumber = {2148849},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2005010/}
}
TY  - JOUR
AU  - Pedregal, Pablo
TI  - Vector variational problems and applications to optimal design
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2005
DA  - 2005///
SP  - 357
EP  - 381
VL  - 11
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2005010/
UR  - https://zbmath.org/?q=an%3A1089.49022
UR  - https://www.ams.org/mathscinet-getitem?mr=2148849
UR  - https://doi.org/10.1051/cocv:2005010
DO  - 10.1051/cocv:2005010
LA  - en
ID  - COCV_2005__11_3_357_0
ER  - 
%0 Journal Article
%A Pedregal, Pablo
%T Vector variational problems and applications to optimal design
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2005
%P 357-381
%V 11
%N 3
%I EDP-Sciences
%U https://doi.org/10.1051/cocv:2005010
%R 10.1051/cocv:2005010
%G en
%F COCV_2005__11_3_357_0
Pedregal, Pablo. Vector variational problems and applications to optimal design. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 357-381. doi : 10.1051/cocv:2005010. http://www.numdam.org/articles/10.1051/cocv:2005010/

[1] G. Allaire, Shape optimization by the homogenization method. Springer (2002). | MR | Zbl

[2] S. Antman, Nonlinear Problems of Elasticity. Springer (1995). | MR | Zbl

[3] E. Aranda and P. Pedregal, Constrained envelope for a general class of design problems. DCDS-A, Supplement Volume 2003 (2002) 30-41. | Zbl

[4] E.J. Balder, Lectures on Young Measures. Cahiers de Mathématiques de la Décision No. 9517, CEREMADE, Université Paris IX (1995).

[5] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63 (1977) 337-403. | Zbl

[6] J.M. Ball, A version of the fundamental theorem for Young measures, PDE's and continuum models of phase transitions, M. Rascle, D. Serre and M. Slemrod Eds. Springer. Lect. Notes Phys. 344 (1989) 207-215. | Zbl

[7] J.M. Ball, Some open problems in elasticity, in Geometry, Mechanics and Dynamics, P. Newton, P. Holmes, A. Weinstein Eds. Springer (2002) 3-59. | Zbl

[8] J.M. Ball and R.D. James, Finephase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52. | Zbl

[9] J.M. Ball and F. Murat, Remarks on Chacon'sbiting lemma. Proc. AMS 107 (1989) 655-663. | Zbl

[10] K. Battacharya and G. Dolzmann, Relaxation of some multi-well problems. Proc. Roy. Soc. Edinb. 131A (2001) 279-320. | Zbl

[11] J.C. Bellido, Explicit computation of the relaxed density coming from a three-dimensional optimal design problem. Non-Lin. Anal. TMA 52 (2002) 1709-1726. | Zbl

[12] J.C. Bellido and P. Pedregal, Explicit quasiconvexification of some cost functionals depending on derivatives of the state in optimal design. Disc. Cont. Dyn. Syst. A 8 (2002) 967-982. | Zbl

[13] M.P. Bendsoe, Optimization of structural topology, shape and material. Springer (1995). | MR | Zbl

[14] M. Bousselsal and M. Chipot, Relaxation of some functionals of the calculus of variations. Arch. Math. 65 (1995) 316-326. | Zbl

[15] M. Bousselsal and R. Le Dret, Remarks on the quasiconvex envelope of some functions depending on quadratic forms. Boll. Union. Mat. Ital. Sez. B 5 (2002) 469-486.

[16] M. Bousselsal and R. Le Dret, Relaxation of functionals involving homogeneous functions and invariance of envelopes. Chinese Ann. Math. Ser. B 23 (2002) 37-52. | Zbl

[17] L. Carbone and R. De Arcangelis, Unbounded functionals in the Calculus of Variations, Representation, Relaxation and Homogenization, Chapman and Hall. CRC, Monographs and Surveys in Pure and Applied Mathematics. Boca Raton, Florida 125 (2002) | MR | Zbl

[18] P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity. North-Holland, Amsterdam (1987). | MR | Zbl

[19] B. Dacorogna, Direct methods in the Calculus of Variations. Springer (1989). | MR | Zbl

[20] G. Dolzmann, B. Kirchheim, S. Muller and V. Sverak, The two-well problem in three dimensions. Calc. Var. 10 (2000) 21-40. | Zbl

[21] A. Donoso and P. Pedregal, Optimal design of 2-d conducting graded materials by minimizing quadratic functionals in the field. Struct. Opt. (in press) (2004). | MR

[22] A. Donoso, Optimal design modelled by Poisson's equation in the presence of gradients in the objective. Ph.D. Thesis, Univ. Castilla-La Mancha (2004).

[23] A. Donoso, Numerical simulations in 3-d heat conduction: minimizing the quadratic mean temperature gradient (2004), submitted.

[24] D. Faraco, Beltrami operators and microstructure. Ph.D. Thesis, University of Helsinki (2002).

[25] I. Fonseca, D. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition. Calc. Var. 2 (1994) 283-313. | Zbl

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

[27] D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59-90. | Zbl

[28] R. Kohn, The relaxation of a double-well energy. Cont. Mech. Thermodyn. 3 (1991) 193-236. | Zbl

[29] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems, I, II and III. CPAM 39 (1986) 113-137, 139-182 and 353-377. | Zbl

[30] R. Lipton and A. Velo, Optimal design of gradient fields with applications to electrostatics, in Nonlinear Partial Differential Equations Appl., College de France Seminar, D. Cioranescu, F. Murat and J.L Lions Eds. Chapman and Hall/CRCResearch Notes in Mathematics (2000). | Zbl

[31] Ch.B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. | Zbl

[32] Ch.B. Morrey, Multiple Integrals in the Calculus of Variations. Berlin, Springer (1966). | Zbl

[33] P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997). | MR | Zbl

[34] P. Pedregal, Variational methods in nonlinear elasticity. SIAM, Philadelphia (2000). | MR | Zbl

[35] P. Pedregal, Constrained quasiconvexification of the square of the gradient of the state in optimal design. QAM 62 (2004) 459-470. | Zbl

[36] P. Pedregal, Optimal design in 2-d conductivity for quadratic functionals in the field, in Proc. NATO Advan. Meeting Non-lin. Homog., Warsaw, Poland, Kluwer (2004) 229-246.

[37] P. Pedregal, Optimal design in two-dimensional conductivity for a general cost depending on the field. Arch. Rat. Mech. Anal. (2004) (in press). | MR | Zbl

[38] Y. Reshetnyak, General theorems on semicontinuity and on convergence with a functional. Sibir. Math. 8 (1967) 801-816. | Zbl

[39] L. Tartar, Remarks on optimal design problems, in Calculus of Variations, Homogenization and Continuum Mechanics, G. Buttazzo, G. Bouchitte and P. Suquet Eds. World Scientific, Singapore (1994) 279-296. | Zbl

[40] L. Tartar, An introduction to the homogenization method in optimal design, Springer. Lect. Notes Math. 1740 (2000) 47-156. | Zbl

Cited by Sources: