分布式内存下的聚合非适应有限元方法
Distributed-memory parallelization of the aggregated unfitted finite element method
AI总结：
本文研究了聚合非适应有限元方法在分布式内存平台上的实现与性能，通过引入精心设计的约束去除 badly cut cells 的基函数，从而得到良好设定的线性代数方程组，同时保持底层有限元空间的最优逼近阶数，并展示了该方法在大规模问题求解中的高效性。
摘要：
聚合非适应有限元方法（AgFEM）是一种最近提出的方法，旨在解决嵌入式、非适应或扩展有限元方法中出现的条件和稳定性问题。该方法基于通过引入精心设计的约束来移除与 badly cut cells 相关的基函数，从而得到良好设定的线性代数方程组，同时保持底层有限元空间的最优逼近阶数。本文的具体目标是展示该方法在分布式内存平台上的实现与性能，旨在高效求解大规模问题。特别是，我们展示通过考虑 AgFEM，所得到的线性代数方程组可以有效地使用标准代数多网格预条件器求解。这与之前的工作形成对比，之前的工作考虑了高度定制化的预条件器，以便允许使用迭代求解器与非适应技术结合。另一个与文献中现有方法的不同之处是所提出方法能够处理的问题规模。虽然大多数先前讨论非适应方法线性求解器的参考都是基于串行不可扩展的算法，我们提出了一种并行分布式内存方法，能够高效求解大规模问题。这通过在复杂3D域上定义的弱标度测试来展示，该测试在Marenostrum-IV平台上处理高达3亿自由度和十亿个单元的规模。AgFEM方法的并行实现可在大规模有限元包FEMPAR中找到。
英文摘要：
The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR.
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