A parallel scalable domain decomposition preconditioner for elastic crack simulation using XFEM

Abstract

In this article, a parallel overlapping domain decomposition preconditioner is proposed to solve the linear system of equations arising from the extended finite element discretization of elastic crack problems. The algorithm partitions the computational mesh into two types of subdomains: the regular subdomains and the crack tip subdomains based on the observation that the crack tips have a significant impact on the convergence of the iterative method while the impact of the crack lines is not that different from those of regular mesh points. The tip subdomains consist of mesh points at crack tips and all neighboring points where the branch enrichment functions are applied. The regular subdomains consist of all other mesh points, including those on the crack lines. To overcome the mismatch between the number of subdomains and the number of processor cores, the proposed method is divided into two steps: solve the crack tip problem and then the regular subdomain problem during each iteration. The proposed method was used to develop a parallel XFEM package which is able to test different types of iterative methods. To achieve good parallel efficiency, additional methods were introduced to reduce communication and to maintain the load balance between processors. Numerical experiments indicate that the proposed method significantly reduces the number of iterations and the total computation time compared to the classical methods. In addition, the method scales up to 8192 processor cores with over 70% parallel efficiency to solve problems with more than 2 × 10 8 degrees of freedom.

A multi-point constraint unfitted finite element method

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Over the last two decades unfitted finite element methods (UFEM), that allow the use of relatively simple background meshes, have proved to be useful tools for solving partial differential equations (PDE) on domains that may be highly complex and may evolve with time. Under the umbrella of unfitted finite element methods, a range of approaches and techniques have been developed including the generalised finite element method (GFEM) [1-11], extended finite element method (XFEM) [12][13][14][15][16][17][18] and cut finite element method (CutFEM) [19][20][21][22][23][24][25].The generalised and extended finite element methods (GFEM and XFEM) are partition of unity methods [26,27] that employ additional functions, or enrichments, to capture solution features, such as strong and weak discontinuities, internal to the elements. The enrichment functions are often chosen a priori based on the physics being simulated, though they may be calculated on the fly numerically, as seen in the global local GFEM [5].

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“…Over the last two decades unfitted finite element methods (UFEM), that allow the use of relatively simple background meshes, have proved to be useful tools for solving partial differential equations (PDE) on domains that may be highly complex and may evolve with time. Under the umbrella of unfitted finite element methods, a range of approaches and techniques have been developed including the generalised finite element method (GFEM) [1-11], extended finite element method (XFEM) [12][13][14][15][16][17][18] and cut finite element method (CutFEM) [19][20][21][22][23][24][25].…”