A new fast method for solving finite element equations iteratively based on Gauss-Seidel

Abstract

Solving large equations systems is the most consuming part of finite element modeling – iterative techniques are favored for models with numerous degrees of freedom where direct techniques have high storage requirements. Classical iterative techniques such as Gauss-Seidel (GS) are robust due to guaranteed convergence and algorithmic simplicity. Performance of iterative techniques chiefly depends on system scale and stiffness matrix properties – which are influenced by structural configuration. However, it is possible to adjust an iterative algorithm such that its speed is greatly enhanced for a certain class of structural configurations. This paper presents an adjusted GS solver, the “Constrained Gauss-Seidel” (CGS), which has been formulated to solve typical multi-story structures with enhanced speed. The innovation in CGS comes from the adoption of a diaphragmatic relaxation mechanism which results in dividing the equations into two groups to optimally deal with the different unknown types. In this paper, the concept and algorithm of the newly developed CGS method are elucidated. Then, 16 practical examples are solved to assess the solving speed of CGS against other iterative methods – GS and Modified Gauss-Seidel (MGS, MGS*). The convergence speed of CGS reached 33 times, 3.7 times, and 2 times those of GS, MGS, and MGS* respectively.