【摘要】 针对参考节点分别为q=3和q=4的网格结构模型,设计了两种预处理方法:以块对角逆为预条件子的共轭梯度法(BPCG)及以块下三角逆为预条件子的PGMRES法。数值结果表明,BPCG法对q=3具有很好的求解效率和鲁棒性,但对q=4的情形,特别是当α很小时,其求解效率将变得很差。当α很小时,以块下三角逆为预条件子的PGMRES法对求解q=4的蜂窝状结构在计算CPU和算法稳定性等方面均全面占优。在这两种预处理方法中,利用了基于标量椭圆问题的GAMG法求各个子块矩阵的逆,以提高内迭代运算效率。近似连续方程的建立为内迭代方法的合理性提供了有效的理论支撑。更多还原

【Abstract】 In this paper,we propose two kinds of preconditioning methods for the discrete models of lattice structures with 3 or 4 reference nodes.One is the block preconditioned conjugate gradient method(BPCG) with a block diagonal inverse as a preconditioner,and another is the preconditioning GMRES method(PGMRES) based on the block lower triangular inverse as a preconditioner.The numerical results show that the BPCG method is of good efficiency and robustness for the lattice structures with 3 reference nodes,but for honeycomb lattices,especially when α is very small,the efficiency of this method becomes poor.In this case,the PGMRES method is best both in the calculation of the stability and CPU times.In order to improve the efficiency of the inner iterations,we use the PCG method with GAMG preconditioner based on scalar elliptic problems to find the inverse of each sub-block matrix.In addition,the approximately continuous models of lattice structure provide some theoretical basis for the reliability of the inner iterations.更多还原