A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid
A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid
复制标题
使用几何多重网格解决有限应变超弹性问题的无矩阵方法
DOI：
10.1002/nme.6336
复制
发表时间：
2020
影响因子：
2.9
通讯作者：
Davydov D
中科院分区：
文献类型：
--
作者：
Davydov D
本文研究了准静态有限应变超弹性问题的无矩阵算法。在流体力学和波传播领域，具有无矩阵算子计算的迭代求解器已成为稀疏矩阵的一种有吸引力的替代方案，因为它们显著减少了经典有限元求解器中的限制因素--内存通信量。具体地说，我们研究了有限元正切算子的不同的无矩阵实现，并确定了包含复杂本构行为的广义方法是否可行。为了改善迭代求解器的收敛性能，我们还提出了一种构造水平正切算子的方法，并将其用于定义几何多重网格预条件算子。对于二维和三维非均匀超弹性材料的典型数值算例，比较了无矩阵算子和几何多重网格预处理器与基于矩阵的单节点代数多重网格预处理器的性能。我们发现，有限应变固体力学的无矩阵方法是非常有前途的，它的性能是基于线性矩阵的方法的两到五倍，并且有可能开发出与超弹性本构关系无关的高效的数值实现。
This work investigates matrix‐free algorithms for problems in quasi‐static finite‐strain hyperelasticity. Iterative solvers with matrix‐free operator evaluation have emerged as an attractive alternative to sparse matrices in the fluid dynamics and wave propagation communities because they significantly reduce the memory traffic, the limiting factor in classical finite element solvers. Specifically, we study different matrix‐free realizations of the finite element tangent operator and determine whether generalized methods of incorporating complex constitutive behavior might be feasible. In order to improve the convergence behavior of iterative solvers, we also propose a method by which to construct level tangent operators and employ them to define a geometric multigrid preconditioner. The performance of the matrix‐free operator and the geometric multigrid preconditioner is compared to the matrix‐based implementation with an algebraic multigrid (AMG) preconditioner on a single node for a representative numerical example of a heterogeneous hyperelastic material in two and three dimensions. We find that matrix‐free methods for finite‐strain solid mechanics are very promising, outperforming linear matrix‐based schemes by two to five times, and that it is possible to develop numerically efficient implementations that are independent of the hyperelastic constitutive law.