A new topology optimization method for free-form surfaces is developed, significantly reducing the dimensions of the design problem, providing smooth structural boundary descriptions and inherently avoiding checkerboard patterns. First, the basic idea is to represent the topology of free-form surfaces with a manifold-based material field function with the predefined spatial correlation. Then we employ a series expansion and truncation technique on the material field function to reduce the control coefficients of the topology description down to a small set. The spatial correlation of the manifold-based material field is formulated in an exponential form expressed with the geodesic distance. Herein, the geodesic distance is evaluated based on the heat method, which is convenient to implement and can achieve high computational efficiency and accuracy. The sensitivity analysis procedures are provided, and the gradient-based optimization algorithm is utilized to solve the proposed optimization model without requiring special filtering techniques. Several numerical examples are presented to illustrate the validity and applicability of the present topology optimization method on free-form surface structures and its potential for gradient-free topology optimization.

开发了一种用于自由曲面的新拓扑优化方法，显着降低了设计问题的维度，提供了平滑的结构边界描述，并在本质上避免了棋盘图案。首先，基本思想是用具有预定义空间相关性的基于流形的材料场函数来表示自由曲面的拓扑结构。然后我们对材料场函数采用级数展开和截断技术，将拓扑描述的控制系数降低到一个小集合。基于流形的材料场的空间相关性以指数形式表示，用测地线距离表示。这里，测地线距离是基于热法评估的，实现方便，计算效率高，精度高。提供了灵敏度分析程序，并利用基于梯度的优化算法来求解所提出的优化模型，无需特殊的滤波技术。给出了几个数值例子来说明本拓扑优化方法在自由曲面结构上的有效性和适用性及其在无梯度拓扑优化中的潜力。