Developments in topology and shape optimization

Our contribution consists of three parts: a gradient-based parameter-free shape optimization method; a stress-constrained topology optimization; and a wave tailoring topology optimization. In shape optimization, the independent node movement approach, wherein finite element node coordinates are used directly as design variables, allows the most freedom for shape change and avoids the time-consuming parameterization process. However, this approach lacks length scale control that is necessary to ensure a well-posed shape optimization problem and avoid numerical instability. Motivated by the success of filtering techniques that impose minimum length scales in topology optimization, we introduce a consistent filtering scheme to provide length scale control and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach. In topology optimization we propose an effective algorithm to incorporate local stress constraints. To generate a well-posed problem we use the restriction method whereby we utilize a density filter for length scale control. The solid isotropic material with penalization (SIMP) is incorporated to generate black-and-white designs. To resolve the stress singularity phenomenon, we introduce a SIMP-motivated stress definition and a global/regional stress measure combined with an adaptive normalization scheme to control the local stress level. Lastly, we apply topology optimization to tailor the stress wave propagation in a two-phase composite plate. To generate a well-posed topology optimization problem we use the relaxation approach which requires homogenization theory to relate the macroscopic material properties to the microstructure, here a sequentially ranked laminate. We introduce an algorithm whereby the laminate volume fractions and orientations are optimized at each material point. To resolve numerical instabilities associated with the dynamic simulation and constrained optimization problem, we filter the laminate parameters. This also has the effect of generating smoothly varying microstructures