AI 中文总结
该研究提出广义形状函数方法用于多材料拓扑优化，基于1D、2D、3D形状函数概念化广义nD形状函数，以自然坐标为设计变量确定材料密度，经多种验证。通过不同设计问题验证其有效性，消除材料数量限制，为大规模工程应用提供良好扩展性。
AI 中文摘要
本文提出一种用于多材料拓扑优化的广义形状函数（gSF）方法，利用紧凑设计空间生成含大量材料的优化构型。基于1D、2D和3D形状函数概念化广义nD形状函数来映射多材料单纯形域，将其自然坐标作为设计变量确定材料密度，证明其满足重心性质，保证物理有效的材料插值空间。还表明对自然坐标直接密度滤波与对密度本身滤波等效，且定制投影方案能保持投影状态下的重心性质。通过2D和3D的刚性结构及柔顺机构设计问题验证了gSF方法的通用性、有效性和成功性。在给定体积约束下，刚性结构使应变能最小，柔顺机构使多准则目标最小。用伴随变量法进行灵敏度分析，用移动渐近线法求解优化问题。给出了2D和3D中分别含多达24和15种不同材料的刚性结构及柔顺机构的结果，目标历史图显示收敛平稳。该方法消除了对候选材料数量的实际限制，为大规模工程应用提供了良好的可扩展性。
英文摘要
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.
Comments36 figures
Journal refInternational Journal for Numerical Methods in Engineering, 2026
