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

Abstract

Solving large equation systems is the most time-consuming part of finite-element modelling – iterative techniques are favoured 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, ‘constrained Gauss–Seidel’ (CGS), which has been formulated to solve typical multi-storey structures with an enhanced speed. The innovation in CGS originates from the adoption of a diaphragmatic relaxation mechanism that results in dividing 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 attained 33 times, 3.7 times and 2 times those of GS, MGS and MGS*, respectively.