A Duality-Preserving Adjoint Method for Segregated Navier–Stokes Solvers
A Duality-Preserving Adjoint Method for Segregated Navier–Stokes Solvers
复制标题
分离纳维斯托克斯求解器的对偶保持伴随法
DOI：
10.1016/j.jcp.2024.112860
复制
发表时间：
2024
影响因子：
4.1
通讯作者：
He, Ping
中科院分区：
文献类型：
--
作者：
Fang, Lean;He, Ping
伴随方法可以有效地计算具有多个输入的系统的梯度，并已广泛用于流体力学中基于梯度的大规模优化。为了确保优化的数值稳健性，我们需要开发一种伴随求解算法，该算法在每次优化迭代时具有与原始流求解器相似（如果不相同）的收敛速度。这种一致的原始伴随收敛行为也称为对偶保持（DP）。现有的 DP 伴随方法主要适用于具有可压缩流的全耦合纳维-斯托克斯 (NS) 求解器。然而，针对分离 NS 求解器提出的 DP 伴随函数很少，例如，广泛用于不可压缩流动模拟的压力关联方程半隐式方法 (SIMPLE) 算法求解器。这项研究将通过推导一个定点分离伴随公式来填补这一空白，该公式完全保留了原始求解器的收敛行为。我们首先将稳态分离 NS 原始求解过程重写为完全耦合的左预处理 Richardson 格式。然后，我们转置预处理器矩阵以获得DP迭代伴随公式。除了代数推导之外，我们还创建了一种新的图形表示，可以显着简化新分离求解器的 DP 伴随开发。我们证明所提出的图表示等效于上述预条件子转置方法并保证对偶性。我们使用复杂性不断增加的三种情况来评估我们提出的算法：微型问题、机翼和机翼。为了量化对偶性，我们比较了原始求解器和伴随求解器之间的特征值，并观察到了良好的一致性。我们还评估各种数值设置（例如，内部迭代容差和非对偶预处理器）对特征值分布的影响程度。我们提出的伴随算法以具有竞争力的速度实现了机器精度的精确梯度计算。最后，我们将伴随求解器纳入大规模基于梯度的优化框架，并展示其机翼空气动力学形状优化的能力。所提出的定点伴随方法有可能使伴随解对于任何分离的 NS 求解器都更加稳健和高效。
Adjoint methods efficiently compute gradients for systems with many inputs and have been widely used for large-scale gradient-based optimization in fluid mechanics. To ensure optimization's numerical robustness, we need to develop an adjoint solution algorithm that has a similar, if not the same, convergence rate as the primal flow solver at each optimization iteration. This consistent primal-adjoint convergence behavior is also called duality-preserving (DP). Existing DP adjoint methods are mostly for fully coupled Navier–Stokes (NS) solvers with compressible flows. However, few DP adjoints have been proposed for segregated NS solvers, e.g., the semi-implicit method for pressure-linked equations (SIMPLE) algorithm solvers widely used for incompressible flow simulations. This study will fill this gap by deriving a fixed-point segregated adjoint formulation that fully preserves the convergence behavior of primal solvers. We first rewrite the steady-state segregated NS primal solution process into a fully coupled left-preconditioned Richardson format. Then, we transpose the preconditioner matrix to obtain a DP iterative adjoint formulation. In addition to algebraic derivation, we create a new graph representation that significantly streamlines the DP adjoint development for new segregated solvers. We prove that the proposed graph representation is equivalent to the above preconditioner transpose approach and guarantees duality. We evaluate our proposed algorithm using three cases with increasing complexity: a miniature-sized problem, an airfoil, and a wing. To quantify duality, we compare the eigenvalues between the primal and adjoint solvers and observe excellent agreements. We also evaluate to what extent various numerical settings (e.g., inner iteration tolerances and non-dual preconditioner) impact the eigenvalue distributions. Our proposed adjoint achieves machine-precision accurate gradient computation with competitive speed. Finally, we incorporate our adjoint solver into a large-scale gradient-based optimization framework and demonstrate its capability for wing aerodynamic shape optimization. The proposed fixed-point adjoint approach has the potential to make the adjoint solution more robust and efficient for any segregated NS solvers.