A multigrid preconditioner for spatially adaptive high-order meshless method on fluid–solid interaction problems
A multigrid preconditioner for spatially adaptive high-order meshless method on fluid–solid interaction problems
复制标题
用于解决流固相互作用问题的空间自适应高阶无网格方法的多重网格预处理器
DOI：
10.1016/j.cma.2022.115506
复制
发表时间：
2022
影响因子：
7.2
通讯作者：
Pan, Wenxiao
中科院分区：
文献类型：
--
作者：
Ye, Zisheng;Hu, Xiaozhe;Pan, Wenxiao
提出了一种求解Stokes极限下流固耦合问题的整体几何多重网格预处理器。采用一种空间自适应高阶无网格法--带自适应h加密的广义移动最小二乘法(GMLS)对问题进行离散。为了解决流固耦合问题，我们需要处理一个由流场和固体体组成的紧耦合系统，从而得到一个具有块状结构的线性方程组。在斯托克斯极限下，固体运动学可以由控制润滑效果的奇点所支配。用自适应h-细化解决这些奇点可能导致病态的线性方程组。多重网格预处理器的关键部分包括内插算子和限制算子以及光滑器。在构造内插和限制算子时，我们利用了自适应h-精化过程中生成的GMLS节点层次集的几何信息。我们通过基于物理的分裂来构建去耦合的平滑器，然后通过乘法重叠的Schwarz方法将它们组合在一起。通过包含不同数目和形状的实体的数值算例，我们展示了所设计的预处理器的性能，并评估了其可扩展性。对于空间自适应GMLS离散化产生的线性方程组，当采用Krylov迭代方法求解时，随着总自由度和实体个数的增加，本文提出的整体式几何多重网格预处理器能够保证收敛和良好的可扩展性。更具体地说，对于固定数量的实体，随着离散化分辨率的递增细化，线性求解器的迭代次数可以保持在相同的水平，这表明我们的预条件相对于总自由度几乎是线性可伸缩的。当实体数目N S增加时，迭代次数与S近似成正比，这意味着关于实体数目的次线性最优性。
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. For solving fluid–solid interaction problems, we need to deal with a tightly coupled system consisting of the flow field and solid bodies, resulting in a linear system of equations with a block structure. 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. The key ingredients of the multigrid preconditioner include the interpolation and restriction operators and the smoothers. For constructing the interpolation and restriction operators, 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 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 the number of solid bodies N s increases, the number of iterations is nearly proportional to N s, implying the sublinear optimality with respect to the number of solid bodies.