This paper proposes a new level set based multi-material topology optimization method, where a difference-set-based multi-material level set (DS-MMLS) model is developed for topology description and an alternating active-phase algorithm is implemented. Based on the alternating active-phase algorithm, a multi-material topology optimization problem with N + 1 phases is split into N(N + 1)/2 binary-phase topology optimization sub-problems. Compared with the initial multi-material problem, each sub-problem involves fewer design variables and volume constraints. In the DS-MMLS model, N + 1 phases are represented by the sequential difference set of N level set functions. Based on this model, the topological evolution of two active phases can be easily achieved by updating a single level set function in a fixed domain, which contributes a great convenience to the implementation of the alternating active-phase algorithm with level set method. Therefore, the proposed method can be easily extended to topology optimization problems with more material phases. To demonstrate its effectiveness, some 2D and 3D numerical examples with different material phases are presented. The results reveal that the proposed method is effective for multi-material topology optimization problems.

本文提出了一种新的基于水平集的多材料拓扑优化方法，建立了基于差异集的多材料水平集（DS-MMLS）模型用于拓扑描述，并实现了交替的有源相算法。基于交替活动相算法，将具有N + 1个相的多材料拓扑优化问题分解为N（N + 1）/ 2个二元相拓扑优化子问题。与最初的多材料问题相比，每个子问题都涉及较少的设计变量和体积约束。在DS-MMLS模型中，N + 1个相由N的顺序差集表示水平设置功能。在此模型的基础上，通过在固定域中更新单个水平集函数，可以很容易地实现两个激活相的拓扑演化，这极大地方便了采用水平集方法实现交替激活相算法的实现。因此，所提出的方法可以容易地扩展到具有更多材料相的拓扑优化问题。为了证明其有效性，给出了一些具有不同材料相的2D和3D数值示例。结果表明，该方法对于多材料拓扑优化问题是有效的。