TOPOLOGY DESIGN WITH OPTIMIZED, SELF-ADAPTIVE MATERIALS

Significant performance improvements can be obtained if the topology o f an elastic structure is allowed to vary in shape optimization proble ms. We study the optimal shape design of a two-dimensional elastic con tinuum for minimum compliance subject to a constraint on the total vol ume of material. The macroscopic version of this problem is not well-p osed if no restrictions are placed on the structure topology; relaxati on of the optimization problem via quasiconvexification or homogenizat ion methods is required. The effect of relaxation is to introduce a pe rforated microstructure that must be optimized simultaneously with the macroscopic distribution of material. A combined analytical-computati onal approach is proposed to solve the relaxed optimization problem. B oth stress and displacement analysis methods are presented. Since rank -2 layered composites are known to achieve optimal energy bounds, we r estrict the design space to this class of microstructures whose effect ive properties can easily be determined in explicit form. We develop a series of reduced problems by sequentially interchanging extremizatio n operators and analytically optimizing the microstructural design fie lds. This results in optimization problems involving the distribution of an adaptive material that continuously optimizes its microstructure in response to the current state of stress or strain. A further reduc ed problem, involving only the response field, can be obtained in the stress-based approach, but the requisite interchange of extremization operators is not valid in the case of the displacement-based model. Fi nite element optimization procedures based on the reduced displacement formulation are developed and numerical solutions are presented. Care must be taken in selecting the discrete function spaces for the desig n density and displacement response, since the reduced problem is a tw o-field, mixed variational problem. An improper choice for the solutio n space leads to instabilities in the optimal design similar to those encountered in mixed formulations of the Stokes problem.