An introduction to the homogenization method in optimal design

Tartar, Luc

doi:10.1007/bfb0106742

Cited by 99 publications

(106 citation statements)

References 18 publications

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“…A common feature of most of the recently developed methods is to try to circumvent the inceptive ill-posedness of shape optimization problems which manifests itself, in numerical practice, by the occurrence of many local minima, possibly far from being global. Probably the most successful approach is the homogenization method [1,6,7,19,23]: it allows to find a global minimizer in most instances, at the price of introducing composite materials in the optimal shape (a tricky penalization procedure is required for extracting a classical shape out of it). Unfortunately, the rigorous derivation of the homogenized or relaxed formulation of shape optimization is complete only for a few, albeit important, choices of the objective function (mostly self-adjoint problems like compliances or eigenvalues optimization).…”

Section: Introductionmentioning

confidence: 99%

Optimal design in small amplitude homogenization

Allaire

Gutiérrez

2007

ESAIM: M2AN

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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.Mathematics Subject Classification. 15A15, 15A09, 15A23.

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Section: Introductionmentioning

confidence: 99%

Optimal design in small amplitude homogenization

Allaire

Gutiérrez

2007

ESAIM: M2AN

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“…We shall not give here a detailed presentation of the many problems and results in this very wide field, but we limit ourselves to discuss some model problems. We refer the reader interested in a deeper knowledge and analysis of this fascinating field to one of the several books on the subject ( [3], [114], [142], [146]), to the notes by L. Tartar [149], or to the recent collection of lecture notes by D. Bucur and G. Buttazzo [37].…”

Section: Introductionmentioning

confidence: 99%

Optimal Shapes and Masses, and Optimal Transportation Problems

Buttazzo

Pascale

2003

Optimal Transportation and Applications

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Summary. We present here a survey on some shape optimization problems that received particular attention in the last years. In particular, we discuss a class of problems, that we call mass optimization problems, where one wants to find the distribution of a given amount of mass which optimizes a given cost functional. The relation with mass transportation problems will be discussed, and several open problems will be presented.

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“…Theorem 1.1 provides explicit formulas for the derivatives of the G-limit in terms of the microscopic problems (11). The identification of the suitable microscopic problems together with the local formulas (12)(13)(14) are necessary for numerical solution schemes based on relaxation through homogenization, see (Ref. 12).…”

Section: Introductionmentioning

confidence: 99%

“…General differentiability properties of G-limits obtained without the use of local formulas are presented in (Ref. 13). Differentiability properties for the the effective dielectric tensor in the contexts of periodic homogenization and statistically homogeneous random media were presented in (Ref.…”

Section: Introductionmentioning

confidence: 99%