Pseudo Arc-Length Numerical Algorithm for Computationaldynamics

Differential equations are often used to describe the dynamic problems. Classical approaches are always hard to solve it in engineering practice due to its characteristics of strong discontinuity, rigidity and shock singularity, among which singularity problem is one of the most difficult and hot issues among scholars. Pseudo arc-length numerical algorithm is proposed for singularity problems in computational dynamics, whose basic idea is to introduce a pseudo arc-length parameter in the original governing equations so that a constraint equation is added. Under the action of a pseudo arc-length parameter, the original discrete elements are distorted to achieve the goal of eliminating or weakening singularity. Firstly, this paper gave an introduction about the pseudo arc-length method for solving the singularity problem in steady diffusion-convection equations. Then the local pseudo arc-length algorithm is proposed for hyperbolic conservation laws. This algorithm has two steps. The first step is to determine the location of the strong discontinuity and the second one aims to eliminate or reduce the singularity by reconstructing the local mesh. Secondly, a global pseudo arc-length algorithm is put forward for high dimensional problems, which can reconstruct the mesh in whole area. Since the reconstructed mesh can adaptively catch the singularity points, the singularity is greatly reduced. Thirdly, the threedimensional global pseudo arc-length algorithm and its computational difficulties in numerical algorithm convergence caused by large grid distortion in three-dimensional space are presented. Then the combination of block reconstruction and integral calculation strategy is adopted in the algorithm design process to achieve the pseudo arc-length numerical algorithm in three-dimensional space. Finally, numerical examples were employed to verify the validity of the pseudo arc-length algorithm.