In this paper, the authors propose a dimension reduction level set method (DR-LSM) for shape and topology optimization of heat conduction problems on general free-form surfaces utilizing the conformal geometry theory. The original heat conduction optimization problem defined on a free-form surface embedded in the 3D space can be equivalently transferred and solved on a 2D parameter domain utilizing the conformal invariance of the Laplace equation along with the extended level set method (X-LSM). Reducing the dimension can not only significantly reduce the computational cost of finite element analysis but also overcome the hurdles of dynamic boundary evolution on free-form surfaces. The equivalence of this dimension reduction method rests on the fact that the covariant derivatives on the manifold can be represented by the Euclidean gradient operators multiplied by a scalar with the conformal mapping. The proposed method is applied to the design of conformal thermal control structures on free-form surfaces. Specifically, both the Hamilton–Jacobi equation and the heat equation, the two governing PDEs for boundary evolution and thermal conduction phenomena, are transformed from the manifold in 3D space to the 2D rectangular domain using conformal parameterization. The objective function, constraints, and the design velocity field are also computed equivalently with FEA on the 2D parameter domain with properly modified forms. The effectiveness and efficiency of the proposed method are systematically demonstrated through five numerical examples of heat conduction problems on the manifolds.

在本文中，作者提出了一种降维水平集方法（DR-LSM），用于利用共形几何理论对一般自由曲面上的热传导问题进行形状和拓扑优化。利用拉普拉斯方程的共形不变性以及扩展水平集方法 (X-LSM)，可以在 2D 参数域上等效地传递和求解嵌入在 3D 空间中的自由曲面上的原始热传导优化问题。减小尺寸不仅可以显着降低有限元分析的计算成本，还可以克服自由曲面上动态边界演化的障碍。这种降维方法的等价性取决于以下事实：流形上的协变导数可以由欧几里得梯度算子乘以具有保形映射的标量来表示。所提出的方法适用于自由曲面上保形热控制结构的设计。具体来说，Hamilton-Jacobi 方程和热方程，边界演化的两个控制PDE 和热传导现象，使用保形参数化从 3D 空间中的流形转换到 2D 矩形域。目标函数、约束和设计速度场也可以在 2D 参数域上使用 FEA 等效地计算，并具有适当修改的形式。所提出的方法的有效性和效率通过在歧管上的热传导问题的五个数值例子系统地证明。