Convergence improvement of the simultaneous relaxation method used in the finite element analysis of incompressible fluid flows

Abstract

Purpose This paper aims to present an improved finite element method used for achieving faster convergence in simulations of incompressible fluid flows. For stable computations of incompressible fluid flows, it is important to ensure that the flow field satisfies the equation of continuity in each element of a generally distorted mesh. The study aims to develop a numerical approach that satisfies this requirement based on the highly simplified marker-and-cell (HSMAC) method and increases computational speed by introducing a new algorithm into the simultaneous relaxation of velocity and pressure. Design/methodology/approach First, the paper shows that the classical HSMAC method is equivalent to a Jacobi-type method in terms of the simultaneous relaxation of velocity and pressure. Then, a Gauss–Seidel or successive over-relaxation (SOR)-type method is introduced in the Newton–Raphson iterations to take into account all the derivative terms in the first-order Taylor series expansion of a nodal-averaged error explicitly. Here, the nine-node quadrilateral (Q2–Q1) elements are used. Findings The new finite element approach based on the improved HSMAC algorithm is tested on fluid flow problems including the lid-driven square cavity flow and the flow past a circular cylinder. The results show significant improvement of the convergence property with the accuracy of the numerical solutions kept unchanged even on a highly distorted mesh. Originality/value To the best of the author’s knowledge, the idea of using the Gauss–Seidel or SOR method in the simultaneous relaxation procedure of the HSMAC method has not been proposed elsewhere.