In this paper, a simultaneous shape and topology optimization method is presented for designing multi-material structures. The whole shape and the layer’s material distributions of a laminated shell structure composed of multi-materials are optimized. The free-form optimization method for shells and the generalized solid isotropic material with penalization (GSIMP) method are respectively employed and combined effectively for shape and topology optimization. Shape along with fictitious homogenized-density variations are used as design variables and simultaneously determined. In other words, the optimal topology is determined in the variable design surface optimized by shape optimization. Compliance is used as the objective functional and minimized under the volume and the area constraints for each material. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined with the unified H¹ gradient method, where the sensitivity functions are respectively applied as the Robin condition, to the design surface and the domain in order to determine the optimal shape and topology simultaneously. Several numerical results including a comparison with the non-simultaneous methods are presented to show the effectiveness of the proposed method. With the proposed method, the optimal lighter and stiffer multi-material laminated shell structure can be obtained without any design parameterization, free of numerical instabilities such as checkerboard pattern and zigzag shape problems.

在本文中，提出了一种用于设计多材料结构的同步形状和拓扑优化方法。优化了由多种材料组成的叠层壳结构的整体形状和层的材料分布。分别采用壳的自由形式优化方法和带惩罚的广义固体各向同性材料（GSIMP）方法并有效结合进行形状和拓扑优化。形状以及虚构的均质密度变化被用作设计变量并同时确定。换句话说，最优拓扑是在通过形状优化优化的可变设计面中确定的。合规性被用作目标函数，并在每种材料的体积和面积约束下最小化。优化设计问题被表述为分布参数优化问题，并从理论上推导出关于形状和密度变化的灵敏度函数。最佳形状和密度变化均由统一的 H 确定¹梯度方法，其中灵敏度函数分别作为罗宾条件应用于设计表面和域，以便同时确定最佳形状和拓扑。一些数值结果，包括与非同时方法的比较，展示了所提出方法的有效性。使用所提出的方法，无需任何设计参数化，即可获得最佳的更轻、更硬的多材料叠层壳结构，没有棋盘格图案和锯齿形等数值不稳定性问题。