Maximum length scale control in density-based multi-material topology optimization

In this work, the method of the maximum length scale control is proposed for density-based multi-material topology optimization. The three-field approach of multi-material topology optimization is presented, which includes the density filter, the projection with Heaviside function, and the uniform multiphase materials interpolation (UMMI) scheme. Then, the local constraints are built by introducing porosity and aggregated by p-mean function to achieve maximum length scale control for the solid phase. Besides, three control schemes are studied and compared. The maximum length scale constraint for single solid phase (MaxLSC-S) and for entire solid phases (MaxLSC-U) are proposed. Based on them, the maximum length scale constraint with hybrid control scheme (MaxLSC-H) are presented. The proposed schemes realize the independent maximum length scale control of a certain material, the simultaneous control of multiple materials, and the maximum length scale control of the joints between two candidate materials. The optimization formulations and the sensitivity analysis of the related optimization responses are subsequently given. Numerical tests demonstrate that the proposed method can contribute to improving the manufacturability of length scale constrained designs and provides possibilities to achieve the desired properties on the design.