Schur decomposition in the scaled boundary finite element method in elastostatics

Abstract

The scaled boundary finite element method (SBFEM) was originally proposed for modelling elastodynamics in bounded and unbounded media. The method has demonstrated its superiority to the finite element method and the boundary element method when dealing with problems involving unbounded computational domains, or with difficulties of irregular frequencies and sharp corners. The SBFEM transforms the governing equations from partial differential equations to ordinary differential equations (ODEs). In addition, only the boundary of the study domain needs to be discretised which significantly reduces the computational cost. In the existing solution procedure, an eigenvalue problem of the Hamiltonian matrix, formulated from the coefficient matrices of ODEs, needs to be solved. Subsequently, the eigenvectors of the Hamiltonian matrix are arranged in a matrix form for the stiffness matrix in the nodal force-displacement relationship. However, the matrix formulated by the eigenvectors is close to singular when multiple eigenvalues with parallel eigenvectors exist, which leads to an inaccurate solution. In the present study, this problem is eliminated by using the Schur decomposition instead of the eigenvalue decomposition. A three-dimensional study of a cylindrical pile subjected to uniformly distributed load, is carried out. The performance and efficiency of the Schur decomposition are discussed in some detail for achieving more accurate solutions in using the SBFEM.