论文标题
用于流体固定相互作用问题的空间自适应高阶方法的多移民预处理
A Multigrid Preconditioner for Spatially Adaptive High-order Meshless Method on Fluid-solid Interaction Problems
论文作者
论文摘要
我们提出了一个单层的几何多裂式预处理，用于解决Stokes限制中的流体 - 固定相互作用问题。这些问题是通过空间自适应高阶网离散化的，即具有自适应$ h $ finement的广义移动最小二乘（GML）。在Stokes限制中，固体运动学可以由管理润滑作用的奇异性主导。通过自适应$ h $ fifinement解决这些奇异性会导致方程式不良的线性系统。为了构建插值和限制性操作员 - 多式预科器的关键成分，我们利用了自适应$ h $ refinement中生成的GMLS节点的层次结构集的几何信息。我们通过基于物理的分裂来构建脱钩的Smoothers，然后通过乘积重叠的Schwarz方法将它们组合在一起。通过数值示例，包括固体体的不同数字和形状，我们证明了性能并评估设计的预处理的可扩展性。随着总自由度和固体数量$ n_s $的增加，提出的单片几何多物种预处理器可以确保使用Krylov迭代方法求解从空间适应GMLS扫描中产生的方程式的线性系统时，可以确保收敛性和良好的可扩展性。更具体地说，对于固定数量的固体，随着离散化分辨率的逐步精制，可以将线性求解器的迭代次数保持在相同的水平上，这表明我们的预处理程序几乎相对于总自由度的线性可伸缩性。当$ n_s $增加时，迭代次数几乎与$ \ sqrt {n_s} $成比例，这意味着相对于固体数量的sublrinear最优性。
We present a monolithic geometric multigrid preconditioner for solving fluid-solid interaction problems in Stokes limit. The problems are discretized by a spatially adaptive high-order meshless method, the generalized moving least squares (GMLS) with adaptive $h$-refinement. In Stokes limit, solid kinematics can be dominated by the singularities governing the lubrication effects. Resolving those singularities with adaptive $h$-refinement can lead to an ill-conditioned linear system of equations. For constructing the interpolation and restriction operators - the key ingredients of the multigrid preconditioner, we utilize the geometric information of hierarchical sets of GMLS nodes generated in adaptive $h$-refinement. We build decoupled smoothers through physics-based splitting and then combine them via a multiplicative overlapping Schwarz approach. Through numerical examples with the inclusion of different numbers and shapes of solid bodies, we demonstrate the performance and assess the scalability of the designed preconditioner. As the total degrees of freedom and the number of solid bodies $N_s$ increase, the proposed monolithic geometric multigrid preconditioner can ensure convergence and good scalability when using the Krylov iterative method for solving the linear systems of equations generated from the spatially adaptive GMLS discretization. More specifically, for a fixed number of solid bodies, as the discretization resolution is incrementally refined, the number of iterations of the linear solver can be maintained at the same level, indicating nearly linear scalability of our preconditioner with respect to the total degrees of freedom. When $N_s$ increases, the number of iterations is nearly proportional to $\sqrt{N_s}$, implying the sublinear optimality with respect to the number of solid bodies.
