Multi-material topology optimization that is based on homogenization schemes has evolved from the conventional optimization methods to the general multi-material solution methodology. While the former interpolation schemes restrict the solution to the isotropic material with constant Poisson’s ratio, the latter generalizes the application to both isotropic and anisotropic materials, as well as a combination of them. Within density-based multi-material topology optimization, the discrete material optimization scheme is a well-known tool to solve the mixture of isotropic and anisotropic materials; nevertheless, most of its applications are based on open-source finite element codes. An alternative to the discrete material optimization scheme is the element duplication method that relies on the idea of stacking multiple elements and assigning one candidate material to each stacked element, thus avoiding the need-to-know important information from open-source finite element engines to compute the first-order sensitivities. Besides its simple implementation along with commercial finite element solvers, the element duplication procedure diminishes computational efficiency due to the element stacking process. In this paper, a solution for this process is proposed for the multi-material topology optimization problem by considering the mixture of isotropic and anisotropic materials without the need for stacking elements in commercial finite element engines, improving the numerical efficiency of element duplication methods as well as being an alternative to compute the sensitivities in the discrete material optimization scheme. Traditional topology optimization response sensitivities are thoroughly discussed, and several numerical examples are presented demonstrating the effectiveness of the proposed approach. In addition, a new approach to compute displacement sensitivities is presented.

基于同质化方案的多材料拓扑优化已从传统的优化方法演变为通用的多材料解决方案方法。前者的插值方案将解限于具有恒定泊松比的各向同性材料，而后者则将其适用于各向同性和各向异性材料，以及它们的组合。在基于密度的多材料拓扑优化中，离散材料优化方案是解决各向同性和各向异性材料混合问题的众所周知的工具。但是，它的大多数应用程序都是基于开源有限元代码的。离散材料优化方案的另一种替代方法是元素复制方法，该方法依赖于堆叠多个元素并将一种候选材料分配给每个堆叠元素的想法，从而避免了开源有限元引擎需要了解的重要信息。计算一阶灵敏度。除了其与商业有限元求解器的简单实现外，由于元素堆叠过程，元素复制过程还会降低计算效率。在本文中，通过考虑各向同性和各向异性材料的混合，而无需在商用有限元引擎中堆叠元素，提出了针对多材料拓扑优化问题的此过程的解决方案，提高了元素复制方法的数值效率，并且是计算离散材料优化方案中灵敏度的一种替代方法。彻底讨论了传统的拓扑优化响应敏感性，并给出了几个数值示例，证明了该方法的有效性。此外，提出了一种计算位移敏感度的新方法。