论文标题
用于计算三维流体拓扑优化的多个溶液的预处理
Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization
论文作者
论文摘要
拓扑优化问题通常支持多个局部最小值，而现实世界的应用通常是三维的。在先前的工作中[I。 P. A. Papadopoulos，P。E。Farrell和T. M. Surowiec，计算拓扑优化问题的多个解决方案，Siam on Scientific Computing杂志，（2021）]，作者开发了一种放气屏障方法，该算法可以系统地计算多个拓扑优化问题的算法。在这项工作中，我们开发了用于应用这种方法在stokes流中产生的线性系统的预处理，使其可在三个维度中使用。特别是，我们开发了一种嵌套的块预处理方法，该方法将线性系统降低以求解两个对称的阳性原始矩阵和增强的动量块。增强的拉格朗日术语用于控制最内向的Schur补体，我们采用了使用核心捕获弛豫方法的几何多机方法来进行增强动量块。我们在使用拟议的迭代求解器计算的三维示例中介绍了多个解决方案。
Topology optimization problems generally support multiple local minima, and real-world applications are typically three-dimensional. In previous work [I. P. A. Papadopoulos, P. E. Farrell, and T. M. Surowiec, Computing multiple solutions of topology optimization problems, SIAM Journal on Scientific Computing, (2021)], the authors developed the deflated barrier method, an algorithm that can systematically compute multiple solutions of topology optimization problems. In this work we develop preconditioners for the linear systems arising in the application of this method to Stokes flow, making it practical for use in three dimensions. In particular, we develop a nested block preconditioning approach which reduces the linear systems to solving two symmetric positive-definite matrices and an augmented momentum block. An augmented Lagrangian term is used to control the innermost Schur complement and we apply a geometric multigrid method with a kernel-capturing relaxation method for the augmented momentum block. We present multiple solutions in three-dimensional examples computed using the proposed iterative solver.
