计算动力学中的伪弧长方法研究

PSEUDO ARC-LENGTH NUMERICAL ALGORITHM FOR COMPUTATIONAL DYNAMICS

动力学问题通常采用微分方程来描绘,但由于工程实际问题的复杂性,微分方程模型常伴随着解的不连续性、刚性或激波间断奇异性特点,传统方法很难求解,奇异性问题是计算动力学难点,同时也是国内外学者研究的热点.伪弧长数值算法是针对计算动力学中的奇异性问题所提出的,其基本思想为通过在解曲线上引入伪弧长参数,并增加一个约束方程,在伪弧长参数作用下,使得原始离散单元发生扭曲形变,从而达到消除或减弱奇异性的目的.本文首先介绍伪弧长方法求解定常对流-扩散方程的奇异性问题,并提出针对双曲守恒定律的局部伪弧长算法,其思想在于首先通过间断解的梯度变换来确定强间断所处位置,进而通过局部网格点重构以及数值修正来达到强间断处奇异性消除与降低的目的.针对高维问题,提出全局伪弧长方法,通过对整个计算区域内的网格点进行重构,使得所有网格点向奇异间断点处移动,从而降低间断点的影响域,达到降低奇异性的目的.重点讨论了三维全局伪弧长算法问题的计算难点,即三维空间网格扭曲大变形导致的数值算法不收敛,并提出在算法设计过程中采用分块重构与整体计算相结合的策略,实现了三维空间中的伪弧长数值算法,最后通过数值实验来验证伪弧长算法对于奇异性问题的有效性.

Di_erential 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 di_cult 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 di_usion-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 di_culties 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.