Calculus of Variations and Geometric Measure Theory

Shape optimization of light structures and the vanishing mass conjecture

Filip Rindler (University of Warwick)

created by gelli on 25 Mar 2021
modified on 06 Apr 2021

7 apr 2021 -- 17:00   [open in google calendar]

Dipartimento di Matematica di Pisa online

Abstract.

It is a classical problem in the theory of shape optimization to find a shape with minimal (linear) elastic compliance (or, equivalently, maximal stiffness) for a given amount of mass and prescribed external forces. It is an intriguing question with a long history, going back to Michell's seminal 1904 work on trusses, to determine what happens in the limit of vanishing mass. Contrary to all previous results, which utilize a soft mass constraint by introducing a Lagrange multiplier, we here consider the hard mass constraint. Our results establish the convergence of approximately optimal shapes of (exact) size tending to zero 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 Bouchitte in 2001 and predicted heuristically before in works of Allaire-Kohn (80's) and Kohn-Strang (90's). 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. I will also present connections to the theory of Michell trusses 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. This is joint work with J.F. Babadjian (Paris-Saclay) and F. Iurlano (Paris-Sorbonne).

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