Parallel computing is omnipresent in today's scientific computer landscape, starting at multicore processors in desktop computers up to massively parallel clusters. While domain decomposition methods have a long tradition in computational mechanics to decompose spatial problems into multiple subproblems that can be solved in parallel, advancing solution schemes for dynamics or quasi-statics are inherently serial processes. For quasi-static simulations, however, there is no accumulating 'time' discretization error, hence an alternative approach is required. In this paper, we present an Adaptive Parallel Arc-Length Method (APALM). By using a domain parametrization of the arc-length instead of time, the multi-level error for the arc-length parametrization is formed by the load parameter and the solution norm. By applying local refinements in the arc-length parameter, the APALM refines solutions where the non-linearity in the load-response space is maximal. The concept is easily extended for bifurcation problems. The performance of the method is demonstrated using isogeometric Kirchhoff-Love shells on problems with snap-through and pitch-fork instabilities. It can be concluded that the adaptivity of the method works as expected and that a relatively coarse approximation of the serial initialization can already be used to produce a good approximation in parallel.

翻译：并行计算在当今的科学计算领域中无处不在，从台式计算机中的多核处理器到大规模并行集群均有所应用。尽管区域分解方法在计算力学中有着悠久的历史，能够将空间问题分解为多个可并行求解的子问题，但动力学或准静态问题的推进求解方案本质上是串行过程。然而，对于准静态模拟而言，不存在累积的"时间"离散误差，因此需要一种替代方法。本文提出了一种自适应并行弧长法（APALM）。通过使用弧长的域参数化替代时间参数化，弧长参数化的多级误差由载荷参数和解的范数构成。通过在弧长参数上进行局部细化，APALM能够对载荷-响应空间中非线性程度最大的区域进行解的精化。该概念可轻松推广至分岔问题。我们采用等几何Kirchhoff-Love壳单元，针对存在跳跃失稳和叉形失稳的问题验证了该方法的性能。结论表明，该方法的自适应性符合预期，且串行初始化的相对粗略近似已可用于在并行计算中产生良好的近似解。