Shape Optimization and Free Boundary Problems with Grid Adaptation (Optimal Design).

Abstract

Shape optimal design has been applied to linearly elastic bodies, and to the two dimensional seepage flow through porous media. The objective of the structural shape design is to minimize the weight within a bound on local stress measure, or to minimize the maximum local stress measure within a bound on the weight. In both design problems, a derived optimality condition requires that the unit mutual energy has constant value along the design boundary. However, instead of using the value of mutual energy, the condition for constant stress on the design boundary was used as the basis for computational procedures. A two step iteration scheme associated with the domain adaptation was provided to show the effectiveness of the computational procedure. A remeshing procedure was necessary in order to eliminate a non-smooth result for the predicted shape. The disruption of smoothness usually occurs near the constrained region, or near the border between the different materials. Diverse examples were solved and analyzed to interpret the validity of the combination of domain and grid adaptive methods. The choice of an appropriate form of sequencing between shape and mesh adaptation steps is an important factor in the success of the solution procedure.