This paper aims to establish a novel and efficient topology optimization method for the thermal-fluid. To adaptively design the fluid–solid coupling structure and make the objective energy to dissipate, the proposed method considers several constraints, such as the volume conservation, inlet and outlet flow velocity field and fluid–solid boundary constraints. The governing system includes the phase-field model, the steady state Darcy equation and the heat transfer equation. Under the constraints of multiple physical fields, we prove the existence of minimal solutions to the optimization problem. We use a Crank–Nicolson (CN) type scheme to discretize the governing system. The multigrid method is used to solve the resulting system of discrete equations. We prove the boundedness and unconditional stability of the original energy, which implies that a large time step can be used. The proposed discrete system is both spatially and temporally second-order accurate. Various computational tests have been performed to demonstrate that the numerical approach is efficient in designing the complicated structures of thermal fluid flows.

本文旨在建立一种新颖高效的热流体拓扑优化方法。为了自适应地设计流固耦合结构并使目标能量消散，该方法考虑了几个约束条件，例如体积守恒、入口和出口流速场以及流固边界约束。控制系统包括相场模型、稳态达西方程和传热方程。在多个物理场的约束下，我们证明了优化问题的最小解的存在性。我们使用 Crank-Nicolson (CN) 类型的方案来离散化管理系统。多重网格方法用于求解所得到的离散方程组。我们证明了原始能量的有界性和无条件稳定性，这意味着可以使用大的时间步长。所提出的离散系统在空间和时间上都是二阶精确的。已经进行了各种计算测试以证明数值方法在设计热流体流动的复杂结构方面是有效的。