Overlap-free multi-material topology optimization for minimum compliance in two and three dimensions by level-set-based negative-mapping interpolation

Grey elements and material overlaps at the interfaces limit density-based multi-material topology optimization. This article extends the level-set-based negative-mapping interpolation to the multi-material proportional topology optimization of macro-scale structures in two and three dimensions. An alternating active-phase algorithm decomposes M-phase problems into two-phase subproblems described by level-set functions, an evolutionary strategy removes the need for sensitivity analysis, and the negative-mapping interpolation eliminates the overlaps of the phases in the interface elements. Cantilever and half-Messerschmitt–Bölkow–Blohm (MBB) beams in two dimensions and a cantilever beam in three dimensions, with three and four material phases, give smooth boundaries, no grey elements and overlap-free designs. The compliance lies below the solid isotropic material with penalization (SIMP) value in four of the eight two-dimensional cases, and 0.3%, 0.4%, 4.8% and 12.7% above it in the other four. The influences of the material properties, interface treatment and algorithmic parameters are also examined.