求解接触问题鞍点系统的一类两网格预处理方法

A Two-Level Preconditioning Method Designed for Solving Saddle Point System in Contact Problems

  • 摘要: 在结构力学接触问题的数值模拟中,使用Lagrange乘子法施加接触约束会产生大规模离散鞍点线性方程组. 由于鞍点矩阵的不定性,其求解难度较大,针对二维绑定接触问题,提出了一种新型插值方式的两网格预处理方法,其核心思想是对任意光滑算法,在精确求解粗网格系统时实现单次收敛. 此外,构造的粗网格算子具有对称正定性,能有效反映接触约束条件. 数值实验结果表明, 该方法仅需要较少的迭代次数即可收敛, 在多种接触模型中表现出较高的计算效率, 其计算时间相比于SIMPLE方法最多可减少80%. 同时, 该方法在不同的接触约束条件下仍保持较强的健壮性, 且适用于大规模计算.

     

    Abstract: Generally, imposing contact constraints with Lagrange multiplier method on the numerical simulation of contact problems in structural mechanics, it can cause a large-scale discrete saddle point system. Due to the indefiniteness of the saddle point matrix and the resolve difficulty, a two-level preconditioning method with novel interpolation approach was proposed to solve the two-dimensional tied contact problem. It was arranged that single-step convergence could be achieved with any smooth method when the coarse grid system was solved accurately. Furthermore, the constructed coarse grid operator should be symmetric positive definite, reflect the contact constraints effectively. The results of numerical experiments show that this method can achieve convergence with a relatively small number of iterations and high computational efficiency in various contact models, can reduce the computation time up to 80% compared with the SIMPLE method. Furthermore, this method presents strong robustness under different contact constraint conditions, being suitable for large-scale computation.

     

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