# 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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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03147028
Submitted on : Tuesday, February 23, 2021 - 2:18:45 PM
Last modification on : Thursday, February 25, 2021 - 3:27:21 AM

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LightStructures.pdf
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• HAL Id : hal-03147028, version 1

### Citation

Jean-François Babadjian, Flaviana Iurlano, Filip Rindler. Shape optimization of light structures and the vanishing mass conjecture. 2021. ⟨hal-03147028⟩

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