Numerical simulations of fracture problems by coupling the FEM and the direct method of lines

In this paper, we propose a method for calculating stress intensity factors of plane fracture problems with cracks by coupling the finite element method (FEM) and the direct method of lines. We introduce polygonal artificial boundaries and divide the physical domain into two parts: a large domain without any crack tips and small polygonal domains surrounding the crack tips. We solve the problem defined in the small domains with crack tips by using the direct method of lines and design discrete nonlocal artificial boundary conditions on the polygonal artificial boundaries by imposing the continuity of displacement and normal stress. Then the original problem is reduced to a boundary value problem defined in a domain without any crack tips. The finite element approximation of the reduced problem is considered and we can prove that the finite element approximation is well posed. Numerical examples and results of a fracture problem with exact solution and two typical fracture problems demonstrate the efficiency and accuracy of the present method.

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