A parallel hp‐FEM infrastructure for three‐dimensional structural acoustics

This paper describes a parallel three‐dimensional numerical infrastructure for the solution of a wide range of time‐harmonic problems in structural acoustics and vibration. High accuracy and rate of error‐convergence, in the mid‐frequency regime,is achieved by the use of hp‐finite and infinite element approximations. The infrastructure supports parallel computation in both single and multi‐frequency settings. Multi‐frequency solves utilize concurrent factoring of the frequency‐dependent linear algebraic systems and are naturally scalable. Scalability of large‐scale single‐frequency problems is realized by using FETI‐DP—an iterative domain‐decomposition scheme. Numerical examples are presented to cover applications in vibratory response of fluid‐filled elastic structures as well as radiation and scattering from elastic structures submerged in an infinite acoustic medium. We demonstrate both the numerical accuracy as well as parallel scalability of the infrastructure in terms of problem parameters that include wavenumber and number of frequencies, polynomial degree of finite/infinite element approximations as well as the number of processors. Scalability and accuracy is evaluated for both single and multiple frequency sweeps on four high‐performance parallel computing platforms: SGI Altix, SGI Origin, IBM p690 SP and Linux‐cluster. Results show good performance on shared as well as distributed‐memory architecture. Copyright © 2006 John Wiley & Sons, Ltd.     

FETI‐DP,BDDC, and block Cholesky methods

The FETI‐DP and BDDC algorithms are reformulated using Block Cholesky factorizations, an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations. Instead of introducing Lagrange multipliers to enforce the coarse level, primal continuity constraints in these algorithms, a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom. With the new formulation of these algorithms, a simplified proof is provided that the spectra of a pair of FETI‐DP and BDDC algorithms, with the same set of primal constraints, are essentially the same. Numerical experiments for a two‐dimensional Laplace's equation also confirm this result. Copyright © 2005 John Wiley & Sons, Ltd.     

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。     

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     

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.     

A dual‐primal FETI method for solving a class of fluid–structure interaction problems in the frequency domain

The dual‐primal finite element tearing and interconnecting method (FETI‐DP) is extended to systems of linear equations arising from a finite element discretization for a class of fluid–structure interaction problems in the frequency domain. A preconditioned generalized minimal residual method is used to solve the linear equations for the Lagrange multipliers introduced on the subdomain boundaries to enforce continuity of the solution. The coupling between the fluid and the structure on the fluid–structure interface requires an appropriate choice of coarse level degrees of freedom in the FETI‐DP algorithm to achieve fast convergence. Several choices are proposed and tested by numerical experiments on three‐dimensional fluid–structure interaction problems in the mid‐frequency regime that demonstrate the greatly improved performance of the proposed algorithm over the standard FETI‐DP method. Copyright © 2011 John Wiley & Sons, Ltd.     

A unified dual‐primal finite element tearing and interconnecting approach for incompressible Stokes equations

A unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd.     

New implementations for the Simultaneous-FETI method

In this work, we present alternative implementations for the Simultaneous-FETI (S-FETI) method. Developed in recent years, this method has shown to be very robust for highly heterogeneous problems. However, the memory cost in S-FETI is greatly increased and can be a limitation to its use. Our main objective is to reduce this memory usage without losing significant time performance. The algorithm is based on the exploitation of the sparsity patterns found on the block of search directions, allowing to store less vectors per iteration in comparison to a full storage scheme. In addition, different variations for the S-FETI method are also proposed, including a new treatment for the possible dependencies between directions and the use of the Lumped preconditioner. Several tests are performed in order to establish the impact of the modifications presented in this work compared to the original S-FETI algorithm.     

Automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms

We introduce spectral coarse spaces for the balanced domain decomposition and the finite element tearing and interconnecting methods. These coarse spaces are specifically designed for the two‐level methods to be scalable and robust with respect to the coefficients in the equation and the choice of the decomposition. We achieve this by solving generalized eigenvalue problems on the interfaces between subdomains to identify the modes that slow down convergence. Theoretical bounds for the condition numbers of the preconditioned operators, which depend only on a chosen threshold, and the maximal number of neighbors of a subdomain are presented and proved. For the finite element tearing and interconnecting method, there are two versions of the two‐level method: one based on the full Dirichlet preconditioner and the other on the, cheaper, lumped preconditioner. Some numerical tests confirm these results. Copyright © 2013 John Wiley & Sons, Ltd.