IMPROVED EXTENDED SCALED BOUNDARY FINITE ELEMENT METHODS

Combining the main advantages of the extended finite element methods (XFEM) and the scaled boundary finite element methods (SBFEM), improved extended scaled boundary finite element methods (iXSBFEM) are proposed. The proposed methods can provide a new way for the simulation of fracture problems. Similar to XFEM, two orthogonal level set functions are used to characterize the internal crack surface in materials, and how the element is partitioned by a crack can be judged by level set functions. These elements partitioned by the crack are treated as a subdomain of SBFEM, and then the element stiffness matrix of these discontinuous elements can be directly solved by SBFEM, thus avoiding the need for further element subdivision to the solution of the discontinuous element stiffness matrix in XFEM. At the same time, with the help of the main idea of XFEM, the real displacement of the intersection point between the crack and the element boundary is considered as the additional degrees of freedom of the element nodes, thereby it gives explicit physical meaning of additional degrees of freedom. For the element containing the crack tip, several layers of elements around the crack tip are selected as a super element, and the super element is used as a subdomain of SBFE to solve the stiffness matrix. The node displacement inside the super element can be obtained by the SBFE displacement approximation. The stress intensity factor can be directly obtained based on the singular displacement or stress at the crack tip, without the need of other numerical methods. Finally, several numerical examples are given to verify the effectiveness of the proposed iXSBFEM. Compared with the standard XFEM, the relative error convergence of the iXSBFEM based on the displacement norm is better, and the stress intensity factors computed by stress based and displacement based method in iXSBFEM both are in good agreement with the analytical solution.