计算动力学中的伪弧长方法研究

发布时间：2018-04-16 11:03

本文选题：计算动力学 + 伪弧长　；参考：《力学学报》2017年03期

【摘要】：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.
[Abstract]:Dynamic problems are usually described by differential equations, but because of the complexity of practical problems in engineering, the model of differential equations is often accompanied by discontinuity, rigidity or shock wave singularity, so the traditional method is difficult to solve.Singularity is a difficult problem in computational dynamics, and it is also a hot topic for scholars at home and abroad.The pseudo-arc length numerical algorithm is proposed to solve the singularity problem in computational dynamics. The basic idea is to introduce pseudo-arc length parameters into the solution curve and add a constraint equation under the action of pseudo-arc length parameters.The original discrete element is distorted and the singularity is eliminated or weakened.In this paper, we first introduce the pseudo-arc length method to solve the singularity problem of steady convection-diffusion equation, and propose a local pseudo-arc length algorithm for hyperbolic conservation law. The idea is to determine the position of strong discontinuity by gradient transformation of discontinuous solution.Then the singularity of strong discontinuity is eliminated and reduced by local mesh reconstruction and numerical correction.A global pseudo-arc length method is proposed to solve the high dimensional problem. By reconstructing the grid points in the whole computing area, all the grid points move to the singular discontinuity points, thus reducing the influence region of the discontinuous points and reducing the singularity.In this paper, the computational difficulty of 3D global pseudo-arc length algorithm is discussed, that is, the numerical algorithm does not converge due to the distortion and deformation of three-dimensional mesh, and the strategy of combining block reconstruction and global computation in the design process of the algorithm is proposed.The pseudo arc length numerical algorithm in three dimensional space is implemented. Finally, the validity of the pseudo arc length algorithm for singularity problem is verified by numerical experiments.
【作者单位】：北京理工大学爆炸科学与技术国家重点实验室;
【基金】：国家自然科学基金资助项目(11390363,11532012)
【分类号】：O175

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