The element free Galerkin method (EFGM) has been proven to be effective in the analysis of certain types of mechanical problems such as modeling of crack growth. However, the EFGM requires more computational time than usual finite element method. To overcome the drawback of the EFGM, the sub-domain technique is an attractive alternative. For an example of modeling crack growth, the computational cost would be reduced greatly if the domain in the vicinity of the crack path is modeled by the EFGM, and the other domains are discretized by the FEM. The sub-domain technique can be also applied to a parallel-processing algorithm to solve larger problems such as the analysis of 3-dimensional crack growth. Since integration cells are defined independently to the interpolation function of displacement, the sub-domain method used in the FEM can not be applied to the EFGM. This paper presents a sub-domain method based on the Lagrange multiplier approach for the application to the EFGM. The displacement field of each sub-domain is defined within a sub-domain without use of the information of the other sub-domain. The compatibility and equilibrium conditions are imposed on the interface between sub-domains using the Lagrange multiplier approach. Since both displacement and traction are unknown on the interface, a mixed formulation is derived in the proposed method. Instabilities of solutions may be triggered in mixed formulation unless the interpolation functions of the unknowns are carefully chosen. To avoid instabilities in mixed formulations, the continuous displacement field is used on the interfaces while the traction field is interpolated by discontinuous function. To demonstrate validity and efficiency of the proposed method, two numerical examples are presented. In the first example fatigue crack growth is simulated using a coupled FEM-EFGM based on the proposed sub-domain technique. The second example shows applicability of the proposed method to parallel algorithms. A 2-dimensional problem with 120,000 DOF is divided into 4 sub-domains, and solved on a cluster system with four 2.4 GHz Intel Zeon processors within 920 seconds. The proposed method yields very accurate results, and requires less computation time compared to the original EFGM.
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