| Popis: |
In optimal design problems the goal is to find an optimal distribution of materials in a fixed domain which optimizes a criterion, computed through the solution of a partial differential equation modelling the involved physics. We consider optimal design problems in the setting of linearized elasticity, and restrict ourselves to domains filled with two isotropic elastic phases. Since a classical solution usually does not exist, we use the homogenization method in order to get a proper relaxation of the original problem. Unfortunately, the related G-closure problem in linearized elasticity is not solved. However, in the compliance minimization problem, G-closure can be replaced by a smaller subset made of sequential laminates, which is explicitly known. Moreover, for the compliance minimization the necessary conditions of optimality are easily derived, which enables a development of optimality criteria method for the numerical solution. We explicitly calculate the lower Hashin-Shtrikman bound on the complementary energy in two space dimensions, and necessary conditions of optimality. We develop an optimality criteria method for the single state compliance minimization problems. |
