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Convergence relative to a microstructure: properties, optimal bounds and application. (English) Zbl 06850619

Summary: In this work, we study a new notion involving convergence of microstructures represented by matrices \(B^\epsilon\) related to the classical \(H\)-convergence of \(A^\epsilon\). It incorporates the interaction between the two microstructures. This work is about its effects on various aspects : existence, examples, optimal bounds on emerging macro quantities, application etc. Five among them are highlighted below : \((1)\) The usual arguments based on translated inequality, \(H\)-measures, Compensated Compactness etc for obtaining optimal bounds are not enough. Additional compactness properties are needed. \((2)\) Assuming two-phase microstructures, the bounds define naturally four optimal regions in the phase space of macro quantities. The classically known single region in the self-interacting case, namely \(B^\epsilon= A^\epsilon\) can be recovered from them, a result that indicates we are dealing with a true extension of the \(\mathcal{G}\)-closure problem. \((3)\) Optimality of the bounds is not immediate because of (a priori) non-commutativity of macro-matrices, an issue not present in the self-interacting case. Somewhat surprisingly though, commutativity follows a posteriori, in certain cases. \((4)\) From the application to “Optimal Oscillation-Dissipation Problems”, it emerges that oscillations and dissipation can co-exist optimally and the microstructures behind them need not be the same though they are closely linked. Furthermore, optimizers are found among \(N\)-rank laminates with interfaces. This is a new feature. \((5)\) Explicit computations in the case of canonical microstructures are performed, in which we make use of \(H\)-measure in a novel way.

MSC:

62K05 Optimal statistical designs
78M40 Homogenization in optics and electromagnetic theory
74Q20 Bounds on effective properties in solid mechanics
78A48 Composite media; random media in optics and electromagnetic theory
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