求解三维流固耦合问题的一种全隐全耦合区域分解并行算法

A FULLY IMPLICIT AND MONOLITHIC PARALLEL DECOMPOSITION METHOD FOR 3D FLUID-SOLID INTERACTION PROBLEMS

三维流固耦合问题的非结构网格数值算法在很多工程领域都有重要应用,目前现有的数值方法主要基于分区算法,即流体和固体区域分别进行求解,因此存在收敛速度较慢以及附加质量导致的稳定性问题,此外,该类算法的并行可扩展性不高,在大规模应用计算方面也受到一定限制.本文针对三维非定常流固耦合问题,提出一种基于区域分解的全隐全耦合可扩展并行算法.首先基于任意拉格朗日-欧拉框架建立流固耦合控制方程,然后时间方向采用二阶向后差分隐式格式、空间方向采用非结构稳定化有限元方法进行离散.对于大规模非线性离散系统,构造一种结合非精确Newton法、Krylov子空间迭代法与区域分解Schwarz预条件子的Newton-Krylov-Schwarz(NKS)并行求解算法,实现流体、固体和动网格方程的一次性整体求解.采用弹性障碍物绕流的标准测试算例对数值方法的准确性进行了验证,数值性能测试结果显示本文构造的全隐全耦合算法具有良好的稳定性,在不同的物理参数下具有良好的鲁棒性,在“天河二号”超级计算机上,当并行规模从192增加到3072个处理器核时获得了91%的并行效率.性能测试结果表明本文构造的NKS算法有望应用于复杂区域流固耦合问题的大规模数值模拟研究中.

Numerical methods based on unstructured meshes for the three-dimensional fluid-solid interaction problems have many applications in science and engineering. Most of the existing algorithms are based on the partitioned approach that the equations for the fluid and solid are solved separately using existing solvers by enabling them to share interface data with one another. The convergence of the partitioned approach is sometimes difficult to achieve because the method is basically a Gauss-Seidel type process and it may encounter the instability problem of the so-called added mass effect. Moreover, the parallel scalability of the solution algorithm is also an important issue when solving the large-scale problem. In contrast, the monolithic approach shows a more robust convergence and also eliminates the added mass effect even for complicated problems. In this work, a fully implicit and monolithic scalable parallel algorithm based on domain decomposition method is developed for the three-dimensional unsteady fluid-solid interaction problem. The governing equations are established based on the arbitrary Lagrangian-Eulerian framework,and a stabilized unstructured finite element method is employed for the discretization in space and a second-order fully implicit backward differentiation formula in time. An inexact Newton-Krylov method together with a restricted additive Schwarz preconditioner is constructed to solve the large, sparse system of nonlinear algebraic equations resulted from the discretization. The accuracy of the numerical method is verified by a benchmark problem of flows around an elastic obstacle. The numerical performance tests show that the fully implicit and monolithic method has good stability with large time step sizes and good robustness under different physical parameters, and a parallel efficiency of 91% was achieved for 3072 processor cores on the “Tianhe 2” supercomputer. The experimental results show that the proposed numerical method is expected to be applied for the numerical simulation of large-scale fluid-structure interaction problems in complex regions.