Optimal design under the one-dimensional wave equation
|Author||Maestre Caballero, Faustino
Pedregal Tercero, Pablo
|Department||Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico|
|Published in||Interfaces and Free Boundaries, 10 (1), 87-117.|
|Abstract||An optimal design problem governed by the wave equation is examined in
detail. Specifically, we seek the time-dependent optimal layout of two isotropic materials on a 1-d domain by minimizing a functional depending ...
An optimal design problem governed by the wave equation is examined in detail. Specifically, 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 coefficients 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) ∈ IR 2) divergence free vector fields 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 quasiconvexification ” of the cost density. Moreover, this quasiconvexification 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 differences with respect to elliptic cases.
|Cite||Maestre Caballero, F., Münch, A. y Pedregal Tercero, P. (2008). Optimal design under the one-dimensional wave equation. Interfaces and Free Boundaries, 10 (1), 87-117.|