This paper provides an extended level set (X-LS) based topology optimiza- tion method for multi material design. In the proposed method, each zero level set of a level set function {\phi}ij represents the boundary between materials i and j. Each increase or decrease of {\phi}ij corresponds to a material change between the two materials. This approach reduces the dependence of the initial configuration in the optimization calculation and simplifies the sensitivity analysis. First, the topology optimization problem is formulated in the X-LS representation. Next, the reaction-diffusion equation that updates the level set function is introduced, and an optimization algorithm that solves the equilibrium equations and the reaction-diffusion equation using the fi- nite element method is constructed. Finally, the validity and utility of the proposed topology optimization method are confirmed using two- and three- dimensional numerical examples.

中文翻译：

---

扩展水平集方法：具有完美对称性质的多相表示及其在多材料拓扑优化中的应用

本文为多材料设计提供了一种基于扩展水平集（X-LS）的拓扑优化方法。在所提出的方法中，水平集函数 {\phi}ij 的每个零水平集表示材料 i 和 j 之间的边界。{\phi}ij 的每次增加或减少对应于两种材料之间的材料变化。这种方法减少了优化计算中初始配置的依赖性并简化了敏感性分析。首先，拓扑优化问题在 X-LS 表示中被公式化。接下来，引入更新水平集函数的反应扩散方程，并构建使用有限元方法求解平衡方程和反应扩散方程的优化算法。最后，