薄板弯曲问题边界元法分析中预条件GMRES算法

Preconditioned GMRES algorithm for the boundary element analysis of the thin plate bending problem

为探讨预条件GMRES算法在求解弹性力学问题边界元法方程时的数值特性,以不同约束条件的薄板弯曲问题边界元分析为例,讨论了Jacobi、块Jacobi和对称Gauss-Sediel迭代作为预条件的三种预条件形式对GMRES收敛性的影响,并分析了其与约束条件的关系。分析表明:右块Jacobi和Jacobi预条件对该问题加速收敛效果更好,且对同类边界条件简支比固支收敛快,这与边界元法系数矩阵计算中的刚体位移原理有关。并且随问题规模的增大,预条件的GMRES算法效率比Gauss消去优势更明显;当问题自由度接近1万时,后者计算时间为前者的60倍。

To investigate the numerical properties of the preconditioning generalized minimum residual algorithm(GMRES)in solving equations formed in the boundary element(BE)analysis of elastic problems,the BE analysis of a thin plate bending problem was considered with different boundary conditions.A sample,the effects of a Jacobi preconditioner,block Jacobi preconditioner,and Gauss-Seidel iteration preconditioner on the convergence behavior of GMRES algorithm are investigated.The relationship between their performance and the boundary conditions of the studied problem is explored.The analysis shows that right block Jacobi and Jacobi preconditioners both exhibit desirable performance in accelerating convergence of the GMRES algorithm.In addition,their accelerating performance turns out to be better in problems with simply supported boundaries than fixed boundaries,which may result from the rigid displacement principle in the calculation of the BEM coefficient matrix.In addition,with the degree of the problems increasing,the advantage of GMRES algorithm becomes more obvious compared with the Gauss elimination method.When the degree of freedom of the studied problem reaches 10,000,the computation time of the latter is 60 times of the former.