基于分区加速和总体共轭梯度法的耦合界面数据传递问题研究

Research of data transfer on coupling interface based on partition acceleration and global CG algorithm

对于耦合动力学问题的分析过程,在界面上需频繁进行数据交换。为此,基于紧支径向基函数和多项式基函数推导了界面数据传递的插值算法,给出了传递矩阵的具体形式。通过分析时间复杂度,找出该算法在大节点量时效率不高的原因在于径向基矩阵的构造和传递矩阵的计算。为加快径向基矩阵的构造速度,提出分区加速处理以提高相关节点的搜索效率;为避免传递矩阵求解过程中的求逆运算,将其转化为多右端项的大型稀疏对称线性方程组问题,引入多右端项的总体共轭梯度迭代方法求解,并讨论了初始估计矩阵的选取方法。数值算例结果表明,结合使用分区加速原理和总体共轭梯度迭代方法,可在不损失插值精度的前提下显著提高求解效率。

Frequent data exchanges take place on the interface in the solution of coupled problems. For this problem,a kind of interpolation algorithm was derived from the compactly supported radial basis functions and the polynomial basis functions;and the specific form of the transfer matrix was given as well. By analyzing the complexity of the algorithm,it was found that constructing the radial basis matrix and computing the transfer matrix consumed most CPU time, causing computational inefficiency when the number of nodes on the coupling interface was large. The partition acceleration principle was pro- posed to accelerate the construction of the radial basis matrix by improving the efficiency of searching the related nodes. To avoid matrix inversion, the process of computing the transfer matrix is converted to solve a multi-right-hand sparse symmetric linear system of equations,and the global CG iterative method was introduced to deal with it. Finally how to select the initial guess matrix was discussed. Numerical ex- amples show that a combination of the partition acceleration treatment and the global conjugate gradient iterative method can improve numerial efficiency greatly without accuracy loss.