Variable-topology shape optimization of linear elastic structures

Significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems. We study the optimal shape design of a two-dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material without any restrictions on the design topology.The macroscopic version of this problem is not well-posed. A well-posed, relaxed form of the optimization problem is obtained by introducing microstructure to the material model. We evaluate the relaxation approach to topology design using a rank-2 layered composite. A combined analytical-computational approach is proposed to solve the relaxed optimization problem. We develop a series of reduced problems by analytically optimizing the microstructural design fields. This process results in an optimization problem that only involves the bulk density distribution.Finite element optimization procedures based on the reduced displacement formulation are developed and numerical solutions are presented. The numerical examples show that stable solutions which are convergent with respect to mesh refinement can be obtained to the relaxed problem.An improper choice of the discrete function spaces for the density design field and the displacement response field can lead to grid-scale anomalies, since the reduced displacement formulation is a two-field, mixed variational problem. We present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models for the variable-density design problem, which is analogous, but simpler, than the optimized rank-2 model. Stability results for specific mixed finite element models are presented, and we propose a patch test that is useful in identifying unstable elements.Relaxed designs are, in general, impractical to manufacture since they contain regions comprised of material with microscopic perforations. Optimization problems with penalties on perforated material revert to the original ill-posed macroscopic design problem leading to designs which are non-convergent with respect to mesh refinement. We propose a new formulation based on perimeter control that yields a well-posed problem; so there is no need to introduce microstructure. Finite element solutions based on the new formulation are convergent with respect to mesh refinement and give the designer some control over the number and size of holes in the design.U of I OnlyETDs are only available to UIUC Users without author permissio