Free element method and its application in CFD

Abstract

Purpose The purpose of this paper is to propose a new strong-form numerical method, called the free element method, for solving general boundary value problems governed by partial differential equations. The main idea of the method is to use a locally formed element for each point to set up the system of equations. The proposed method is used to solve the fluid mechanics problems. Design/methodology/approach The proposed free element method adopts the isoparametric elements as used in the finite element method (FEM) to represent the variation of coordinates and physical variables and collocates equations node-by-node based on the newly derived element differential formulations by the authors. The distinct feature of the method is that only one independently formed individual element is used at each point. The final system of equations is directly formed by collocating the governing equations at internal points and the boundary conditions at boundary points. The method can effectively capture phenomena of sharply jumped variables and discontinuities (e.g. the shock waves). Findings a) A new numerical method called the FEM is proposed; b) the proposed method is used to solve the compressible fluid mechanics problems for the first time, in which the shock wave can be naturally captured; and c) the method can directly set up the system of equations from the governing equations. Originality/value This paper presents a completely new numerical method for solving compressible fluid mechanics problems, which has not been submitted anywhere else for publication.