Homogenization & optimal shape design-based approach in optimal material distribution problems. Part I: the scalar case

摘要

The paper considers the problem of optimum distribution of two materials with a linear scalar elliptic PDE as the state problem and a general objective functional, not only compliance. One derives the relaxed form of the problem, develops a numerical approximation theory and presents an embedded optimal shape design approach: The composite materials used in the relaxation are constructed by the conventional boundary variation technique. This approach is useful when the G_0 closure is not known explicitly, as in the case of the system of equations of linear elasticity. Also, an alternative approach to relaxation of the problem with the objective functional involving gradients of the state problem solution (the functional being only lower semicontinuous, not continuous) is contained in the paper. Main ideas of the numerical realization are given. The theory is illustrated by numerical examples.