Enhancing the performance of the FETI method with preconditioning ¶techniques implemented on clusters of networked computers

 The FETI domain decomposition method for solving large-scale problems in computational structural mechanics involves the solution of an interface problem, which is handled by a Preconditioned Conjugate Projected Gradient (PCPG) algorithm. Two preconditioners are widely used to accelerate the convergence of the iterative PCPG algorithm: the optimal Dirichlet preconditioner and the economical lumped preconditioner. The Dirichlet preconditioner is computationally more efficient than the lumped preconditioner for ill-conditioned problems, but needs additional storage for the stiffness matrices of the subdomains' internal degrees of freedom (d.o.f.). In this study a new set of PCPG preconditioners is presented by providing approximate expressions to the inverse iteration matrix of the PCPG algorithm. The resulting approximate Dirichlet preconditioners are obtained by using instead of the whole stiffness matrix of the internal d.o.f. in each subdomain the following alternatives: a diagonal scaling matrix, a SSOR type matrix or an incomplete Cholesky factorization matrix. The computational behavior and performance of the proposed PCPG preconditioners is evaluated using an implementation of the FETI method on a cluster of ethernet-networked PCs running the message passing software PVM. It is demonstrated that the FETI method equipped with the approximate Dirichlet preconditioners leads for a number of large-scale problems to faster and less storage demanding overall solutions than with either Dirichlet or lumped preconditioner. Received: 28 December 2001 / Accepted: 22 August 2002     

基于FETI的2维阵列电磁特性分析

应用有限元分裂互连方法（FETI）对2维阵列的电磁散射特性进行分析,将原求解区域划分成若干个不重叠的子区域,并根据其特性划分成9种类型。在相邻区域公共面上采用电场连续性条件,保证电场分量的连续性。对每个子域进行独立的有限元分析,通过公共面上的电流进行区域信息交换,确保相邻面上的区域耦合能够准确地被表示。此算法在保持有限元法计算精度的同时,大大减少了计算时间和内存消耗,拓展了有限元方法在周期结构上的应用。     

Inexact FETI‐DP methods

Inexact FETI‐DP domain decomposition methods are considered. Preconditioners based on formulations of FETI‐DP as a saddle point problem are used which allow for an inexact solution of the coarse problem. A positive definite reformulation of the preconditioned saddle point problem, which also allows for approximate solvers, is considered as well. In the formulation that iterates on the original FETI‐DP saddle point system, it is also possible to solve the local Neumann subdomain problems inexactly. Given good approximate solvers for the local and coarse problems, convergence bounds of the same quality as for the standard FETI‐DP methods are obtained. Numerical experiments which compare the convergence of the inexact methods with that of standard FETI‐DP are shown for 2D and 3D elasticity using GMRES and CG as Krylov space methods. Based on parallel computations, a comparison of one variant of the inexact FETI‐DP algorithms and the standard FETI‐DP method is carried out and similar parallel performance is achieved. Parallel scalability of the inexact variant is also demonstrated. It is shown that for a very large number of subdomains and a very large coarse problem, the inexact method can be superior. Copyright © 2006 John Wiley & Sons, Ltd.