A generalized shape function approach for multimaterial topology optimization

2026-07-22

AI Summary

This work addresses key challenges in multi-material topology optimization—namely, limitations on the number of candidate materials, redundant design spaces, and difficulties in ensuring physically valid material interpolation—by introducing a generalized shape function (gSF) method. The approach employs an n-dimensional linear shape function to map the multi-material simplex domain onto a compact design space, using natural coordinates as design variables to determine material densities. By integrating density filtering with a barycentric-projection strategy, the method rigorously enforces barycentric coordinate properties. For the first time, it establishes a generalized n-linear shape function applicable to arbitrary dimensions, thereby overcoming conventional constraints on material count and enabling a highly scalable optimization framework. The method successfully optimizes 2D and 3D structures—including compliant mechanisms—with up to 24 and 15 materials, respectively, achieving smooth convergence of objective functions and demonstrating its efficiency, generality, and engineering applicability.

Abstract

This paper presents a generalized shape function (gSF) approach for multi-material topology optimization that utilizes a compact design space to produce optimized configurations featuring a large number of materials. Building upon 1D (linear), 2D (bilinear), and 3D (trilinear) shape functions, generalized nD (n-linear) shape functions are conceptualized to map the multi-material simplex domain. Natural coordinates of these shape functions are considered the design variables used to determine the material densities. These densities are mathematically proven to satisfy the essential barycentric properties, guaranteeing a physically valid material interpolation space. Furthermore, we demonstrate that applying density filtering directly to the natural coordinates is mathematically equivalent to filtering the densities themselves, and that the tailored projection scheme preserves these vital barycentric properties in the projected states. The versatility, efficacy, and success of the gSF approach are demonstrated across various 2D and 3D stiff-structure (SS) and compliant-mechanism (CM) design problems. Strain energy is minimized for SS, whereas a multicriteria objective is minimized for CMs with given volume constraints. Sensitivity analysis is performed using the adjoint variable method, and the optimization problem is solved using the method of moving asymptotes. Results for SSs and CMs in 2D and 3D, respectively, up to 24 and 15 different materials, are presented. Objective history plots indicate smooth convergence. The proposed approach removes practical restrictions on the number of candidate materials, offering excellent scalability for large-scale engineering applications.

Problem

• multimaterial topology optimization
• design space
• material interpolation
• barycentric properties
• scalability

Innovation

• generalized shape function
• multi-material topology optimization
• barycentric coordinates
• density filtering
• n-linear interpolation