This paper presents the topology optimization of hierarchical microtextures for wetting behavior in the Cassie-Baxter mode, considering a structural unit of the hierarchical microtexture composed of base and secondary structures. The geometrical configuration of the considered structural unit can be described as a fiber bundle composed of an external surface of the base structure and the pattern of the secondary structures. Thus, two design variables are defined, one for the external surface of the base structure, and the other for the pattern of the secondary structures. The Young-Laplace equation, including a term depending on the mean curvature of the external surface, is used to describe the liquid/vapor interface imposed with a surface tension in the Cassie-Baxter mode. To overcome the difficulty of numerically computing the second-order derivative of the external surface, two partial differential equation filters are sequentially applied to the design variable of the base structure to ensure the numerical accuracy and feasibility of using an efficient linear-element-based finite element method to solve the Young-Laplace equation. To improve the performance of the hierarchical microtextures, the volume of the liquid bulges suspended at the liquid/vapor interface in the Cassie-Baxter mode, before the transition into the Wenzel mode, is minimized to optimize the match between the external surface of the base structure and the pattern of the secondary structures. In the topology optimization process, penalization of the material density of the surface tension is achieved by an artificial Marangoni phenomenon. In numerical examples, solid surfaces are tiled into textures with axial symmetry, radial symmetry, chirality, and quasiperiodicity; and structural units are derived consisting of base structures with peak shapes and dense secondary structures surrounding the crests of the peaks. The optimized performance of the derived structural units has been confirmed by comparisons.

本文考虑了由基础结构和二级结构组成的分层微纹理的结构单元，提出了在Cassie-Baxter模式下用于润湿行为的分层微纹理的拓扑优化。所考虑的结构单元的几何构造可以描述为由基础结构的外表面和次级结构的图案组成的纤维束。因此，定义了两个设计变量，一个用于基本结构的外表面，另一个用于辅助结构的图案。包括取决于外表面平均曲率的项的Young-Laplace方程用于描述在Cassie-Baxter模式下施加表面张力的液/气界面。为克服数值计算外表面二阶导数的困难，两个偏微分方程滤波器顺序应用于基础结构的设计变量，以确保数值精度和使用基于线性元素的有效有限元的可行性。求解Young-Laplace方程的有限元方法。为了提高分层微纹理的性能，在过渡到Wenzel模式之前，要最小化在Cassie-Baxter模式下悬浮在液/气界面处的液体凸起的体积，以优化底座外表面之间的匹配度。结构和二级结构的模式。在拓扑优化过程中，表面张力的材料密度的损失是通过人为的Marangoni现象实现的。在数值示例中，将实体表面平铺为具有轴向对称，径向对称，手性和准周期性的纹理；得出的结构单元由具有峰形的基础结构和围绕峰顶的致密二级结构组成。通过比较确认了派生结构单元的优化性能。