Relaxation of an optimal design problem for the heat equation
2008 - A. Munch, P. Pedregal, F. Peragio
Journal de Mathématiques Pures et Appliquées 89, 225-247 (2008)
authors IMACI
We consider the heat equation in (0,T)×? , ??RN , N?1 , and address the nonlinear optimal design problem which consists in finding the distribution in ? of two given isotropic materials which minimizes a suitable cost functional depending on the heat flux. Both the case of a time-independent design and of the time-dependent one are analyzed. Well-posed relaxations of the two problems are obtained by using two well-known approaches: the homogenization method and the classical tools of non-convex, vector, variational problems. We also implement several numerical experiments based on these relaxed formulations to support the theoretical results. Finally, we point out some differences and analogies of the two proposed methods.