Multi-material topology optimization based on enhanced alternating active-phase algorithm

The alternating active-phase algorithm (AAPA) is widely recognized as being efficient for multi-material topology optimization. It has the advantages of easy implementation and broad applicability, but faces shortcomings, such as a uniform punishment intensity for all elements, excessive gray elements, and slow optimization convergence. To overcome these shortcomings, proposed here is an enhanced AAPA (EAAPA), the core ideas of which are as follows. (1) A new equal-scale Heaviside projection function (EHPF) is proposed, with the scale factor introduced to adjust the physical density adaptively to overcome the challenges of the invariance of the density sum and the uniqueness of projecting faced by the traditional Heaviside projection function. This adaptation reduces the number of gray elements and accelerates the convergence. (2) An innovative solid isotropic material with exponential sigmoid function (SIMESF) is constructed, with a penalty threshold and a steepness factor introduced to divide different penalty intensity regions and control their transition steepness. This improves the applicability and flexibility of the interpolation model, addressing the issue of a single punishment intensity in the solid isotropic material with penalization. The parameter values of the SIMESF model are explored via the design of a bridge, and the performances of AAPA and EAAPA are evaluated via the designs of simply supported and cantilever beams. The results show clearly that EAAPA has significant advantages in terms of optimization effectiveness and convergence speed.