Manifold-based material field series expansion method for topology optimization on free-form surfaces

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.