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Journal Articles Duke Mathematical Journal Year : 2023

Shape optimization of light structures and the vanishing mass conjecture

Abstract

This work proves rigorous results about the vanishing-mass limit of the classical problem to find a shape with minimal elastic compliance. Contrary to all previous results in the mathematical literature, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results are the first to establish the convergence of approximately optimal shapes of (exact) size $\e \todown 0$ to a limit generalized shape represented by a (possibly diffuse) probability measure. This limit generalized shape is a minimizer of the limit compliance, which involves a new integrand, namely the one conjectured by Bouchitt\'e in 2001 and predicted heuristically before in works of Allaire \& Kohn and Kohn \& Strang from the 1980s and 1990s. This integrand gives the energy of the limit generalized shape understood as a fine oscillation of (optimal) lower-dimensional structures. Its appearance is surprising since the integrand in the original compliance is just a quadratic form and the non-convexity of the problem is not immediately obvious. In fact, it is the interaction of the mass constraint with the requirement of attaining the loading (in the form of a divergence-constraint) that gives rise to this new integrand. We also present connections to the theory of Michell trusses, first formulated in 1904, and show how our results can be interpreted as a rigorous justification of that theory on the level of functionals in both two and three dimensions, finally settling this long-standing open problem. Our proofs rest on compensated compactness arguments applied to an explicit family of (symmetric) $\dive$-quasiconvex quadratic forms, computations involving the Hashin--Shtrikman bounds for the Kohn--Strang integrand, and the characterization of limit minimizers due to Bouchitt\'e \& Buttazzo.
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Dates and versions

hal-03147028 , version 1 (23-02-2021)
hal-03147028 , version 2 (06-01-2022)

Identifiers

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Jean-François Babadjian, Flaviana Iurlano, Filip Rindler. Shape optimization of light structures and the vanishing mass conjecture. Duke Mathematical Journal, inPress, 172 (1), pp.43-103. ⟨10.1215/00127094-2022-0031⟩. ⟨hal-03147028v2⟩
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