The extended finite element method (XFEM) has been successfully implemented in solving crack propagation problems by enriching the polynomial basis functions of standard finite elements with specialized non-smooth functions. The resulting approximation space can be used to solve problems with moving discontinuities, such as cracks, without the need of remeshing in the vicinity of the crack. The enrichment of the displacement field in XFEM inflicts a substantial increase in the ellipticity of the discretized problem. As a consequence, the resulting algebraic systems become strongly ill-conditioned, rendering the convergence of iterative solvers very slow. On the other hand, direct solvers may become inefficient in 3D problems, due to the increased bandwidth of the system matrix. In this paper, two of the most efficient domain decomposition solvers, namely the FETI-DP and P-FETI-DP, are proposed for solving the linear systems resulting from XFEM crack propagation analysis in large-scale 3D problems. Both solvers are amenable to parallelization and can be implemented in modern parallel computing environments, with multicore processors and distributed memory systems, following appropriate modifications, to achieve a drastic reduction of memory and computing time in computationally intensive problems.

扩展有限元法 (XFEM) 通过用专门的非光滑函数丰富标准有限元的多项式基函数，成功地解决了裂纹扩展问题。由此产生的近似空间可用于解决移动不连续性问题，例如裂缝，而无需在裂缝附近重新划分网格。XFEM 中位移场的丰富导致离散问题的椭圆度显着增加。结果，得到的代数系统变得非常病态，使得迭代求解器的收敛非常缓慢。另一方面，由于带宽增加的系统矩阵。在本文中，提出了两个最有效的域分解求解器，即 FETI-DP 和 P-FETI-DP，用于求解大规模 3D 问题中 XFEM 裂纹扩展分析产生的线性系统。这两种求解器都适合并行化，并且可以在现代并行计算环境中实现，使用多核处理器和分布式内存系统，经过适当的修改，在计算密集型问题中实现内存和计算时间的大幅减少。