The application of simultaneous elimination and back-substitution method (SEBSM) in finite element method

Abstract

Purpose The purpose of this paper is to introduce a new method called simultaneous elimination and back-substitution method (SEBSM) to solve a system of linear equations as a new finite element method (FEM) solver. Design/methodology/approach In this paper, a new technique assembling the global stiffness matrix will be proposed and meanwhile the direct method SEBSM will be applied to solve the equations formed in FEM. Findings The SEBSM solver for FEM with the present assembling technique has distinct advantages in both computational time and memory space occupation over the conventional methods, such as the Gauss elimination and LU decomposition methods. Originality/value The developed solver requires less memory space no matter the coefficient matrix is a typical sparse matrix or not, and it is applicable to both symmetric and unsymmetrical linear systems of equations. The processes of assembling matrix and dealing with constraints are straightforward, so it is convenient for coding. Compared to the previous solvers, the proposed solver has favorable universality and good performances.

Newton-2EGSOR Method for Unconstrained Optimization Problems with a Block Diagonal Hessian

“…Solving Eq. (7) with a coefficient matrix (9) using a direct method such as Gauss elimination method [22] or simultaneous method [23] needs more memory space and computing time since the coefficient matrix (9) is sparse and large scale. Furthermore, the implementations of the Newton method that use a direct method to obtain the Newton direction are not suitable for many large problems because of their cost in terms of computing the first and second partial derivative.…”