Optimal design of 2D conducting graded materials by minimizing quadratic functionals in the field

Abstract

We explore a typical optimal design problem in 2D conductivity: given fixed amounts of two isotropic dielectric materials, decide how we are to mix them in a 2D domain so as to minimize a certain cost functional depending on the underlying electric field. We rely on a reformulation of the optimal design problem as a vector variational problem and examine its relaxation, taking advantage of the explicit formulae for the relaxed integrands recently computed in Pedregal (2003). We provide numerical evidence, based on our relaxation, that Tartar’s result Tartar (1994) is verified when the target field is zero (also for divergence-free fields) and optimal solutions are given by first-order laminates. This same evidence also holds for a general quadratic functional in the field.