The performance of finite element solvers on modern computer architectures is typically memory bound for sufficiently large problems. The main cause for this is that loading matrix elements from RAM into CPU cache is significantly slower than performing the arithmetic operations when solving the problem. In order to improve the performance of iterative solvers within the high-performance computing context, so-called matrix-free methods are widely adopted in the fluid mechanics community, where matrix-vector products are computed on-the-fly. To date, there have been few (if any) assessments into the applicability of the matrix-free approach to problems in solid mechanics. In this work, we perform an initial investigation on the application of the matrix-free approach to problems in quasi-static finite-strain hyperelasticity to determine whether it is viable for further extension. Specifically, we study different numerical implementations 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 preconditioner on a single node for a representative numerical example of a heterogeneous hyperelastic material in two and three dimensions. We conclude that the application of matrix-free methods to finite-strain solid mechanics is promising, and that is it possible to develop numerically efficient implementations that are independent of the hyperelastic constitutive law.

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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

对于足够大的问题，有限元求解器在现代计算机体系结构上的性能通常受内存限制。造成这种情况的主要原因是，在解决问题时，将矩阵元素从 RAM 加载到 CPU 缓存中的速度明显慢于执行算术运算。为了在高性能计算环境中提高迭代求解器的性能，流体力学社区广泛采用所谓的无矩阵方法，其中矩阵向量积是动态计算的。迄今为止，几乎没有（如果有的话）评估无矩阵方法对固体力学问题的适用性。在这项工作中，我们对无矩阵方法在准静态有限应变超弹性问题中的应用进行了初步调查，以确定它是否适合进一步扩展。具体来说，我们研究了有限元切线算子的不同数值实现，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行为，我们还提出了一种构造水平切线算子并使用它们来定义几何多重网格预处理器的方法。将无矩阵算子和几何多重网格预处理器的性能与在单个节点上使用代数多重网格预处理器的基于矩阵的实现进行比较，以获得二维和三维异质超弹性材料的代表性数值示例。我们得出结论，无矩阵方法在有限应变固体力学中的应用是有前景的，并且可以开发独立于超弹性本构定律的数值有效的实现。