## F. Maestre - A. Münch - P. Pedregal

# Optimal design under the 1d wave equation

created by maestre on 30 Apr 2008

modified on 03 May 2008

[

BibTeX]

*Preprint*

**Inserted:** 30 apr 2008

**Last Updated:** 3 may 2008

**Year:** 2007

**Abstract:**

An optimal design problem governed by the wave equation is examined in
detail. Specically, we seek the time-dependent optimal layout of two isotropic
materials on a 1-d domain by minimizing a functional depending quadratically
on the gradient of the state with coecients that may depend on space, time
and design. As it is typical in this kind of problems, they are ill-posed in the
sense that there is not an optimal design. We therefore examine relaxation
by using the representation of two-dimensional ($(x; t) \inR^2$) divergence free
vector elds as rotated gradients. By means of gradient Young measures, we
transform the original optimal design problem into a non-convex vector variational
problem, for which we can compute an explicit form of the \'constrained
quasiconvexifcation\' of the cost density. Moreover, this quasiconvexication
is recovered by first or second-order laminates which give us the optimal microstructure
at every point. Finally, we analyze the relaxed problem and some
numerical experiments are performed. The perspective is similar to the one developed
in previous papers for linear elliptic state equations. The novelty here
lies in the state equation (the wave equation), and our contribution consists in
understanding the dierences with respect to elliptic cases.

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