PROJECTED BARYCENTRIC DENSITIES FOR MULTI-MATERIAL TOPOLOGY OPTIMIZATION USING SHAPE FUNCTION WITH PENALIZATION APPROACH

The paper proposes and implements modifications to the Shape Function with Penalization (SFP) approach to solve multi-material topology optimization (MMTO) problems. The SFP approach employs barycentric shape functions from the Finite Element literature, to evaluate material densities allowing one to solve MMTO requiring two or three variables per cell. Barycentricity, i.e., non-negativity and partition of unity amongst densities is the key for a cell to be assigned one of the specified materials and therefore no other. We show that applying filtering to SFP evaluated material densities leads to barycentric filtered density fields. Applying projection further to these filtered fields leads to loss of this property. We demonstrate next that applying filtering and projection to the design variables directly helps retain barycentricity in the resulting material densities. The corresponding MMTO problem formulation is developed followed by sensitivity analysis. The obtained formulation is applied on some well known topology optimization problems demonstrating the method's ability to yield solutions with up to seven materials employing only 3 design variables per cell. Solutions elucidate that the modified projection operator successfully generates crisp transition regions between material (and void) phases.