Adaptive finite element methods for shape optimization of linearly elastic structures

Abstract

Grid adaptive methods combined with means for automatic remeshing are applied to problems in shape optimal design of linearly elastic structures. The quantitative effect of element distortion near the design boundaries is identified in terms of interpolation error associated with the finite element discretization. The grid adaptation is itself formulated as a structural optimization problem, with an objective function that reflects the discretization error. A ‘necessary condition’ from this formulation provides the basis for a computational procedure to predict the modified grid. To avoid the sometimes drastic distortion of the FEM grid that might otherwise occur in conjunction with design change, remeshing must be performed at intermediate stages of the overall solution process. In order to produce results for the optimal shape design without interruption in this process, the computer program combines numerical grid generation and automatic remeshing with the grid adaptation and design change. Results for several shape design problems obtained with the use of grid adaptation are compared to computational results predicted from a fixed grid. Both ‘ r-’ and ‘ h-adaptation’ are tested.