Approximate factorizations are probably the most powerful preconditioners at the present time in the context of iterative solution methods for FE structural analysis. In this contribution we focus on some aspects of the reduction method proposed previously, which allow the use of perturbed approximate factorizations, In particular, we show that it is not suitable for systems arising from discretizations with plate or shell elements. In contrast, corrected incomplete Cholesky preconditioners are shown to exhibit a much better convergence for such systems. (C) 1999 Civil-Comp Ltd and Elsevier Science Ltd. All rights reserved.

This contribution describes how iterative solvers can meet specific requirements of industrial FE analyses, focusing on the case frequently met where the unknowns are subject to linear equality constraints. Standard iterative methods designed to deal with that kind of problem suffer from a significant overhead with respect to the CPU times involved in the solution of unconstrained problems, whereas the performance of direct solvers traditionally used is not affected. Here we propose a subspace projection method that allows the use of any iterative scheme able to solve the unconstrained problem, with the same preconditioner. We also highlight that there is no loss of efficiency due to the presence of linear constraints.

The preconditioned conjugate gradient algorithm is a well-known and powerful method used to solve large sparse symmetric positive definite linear systems. Such systems are generated by the finite element discretization in structural analysis but users of finite elements in this context generally still rely on direct methods. It is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods.

The purpose of this contribution is to show the performances of three parallel preconditioners developped for stress analysis. The problem is the solution of large linear systems of algebraic equations, which arise in the finite element discretizations of linear elastic structures, the computation Ing made by a parallel iterative method. During the last W years, a general theory has been developped for the study of additive and multiplicative Schwarz methods. We apply the additive Schwarz theory, as a preconditioner which is by nature parallel. Other widely used preconditioners are based on incomplete Cholesky factorizations (IC), but for the problems treated here a reduction stage to a Stieltjes form is required in order to obtain an IC factorizable matrix [1]. The parallelization of the IC preconditioner is performed by replication of boundary unknowns on two or more subdomains. The last preconditioner is based on the Schur complement n :thod which consists in solving the boundary problem obtained after elimination of the intermal unknowns. These three preconditioners are based on domain decomposition schemes. Numerical analyses on regular and irregular meshes show the performances of the three methods and the influences of different parameters such as the amount of overlap, the mber of subdomains or the decomposition method.

During the last few years, a general theory has been developped for the study of additive and multiplicative Schwarz methods. We apply the additive Schwarz theory, as preconditioner, to the solution of large linear systems of algebraic equations, which arise in the finite element discretizations of linear elastic equations. This preconditioner is by nature parallel and other widely used preconditioners are based on incomplete Cholesky factorization (IC) for which a reduction stage to Stieltjes matrix is required in order to obtain an IC factorizable matrix [5]. The parallelization of the IC preconditioner is performed by replication of boundary unknowns on two or more subdomains. These two preconditioners are based on domain decomposition schemes.

A quasi-3D calculation program based on finite elements is presented in the spirit of Wu’s approach. In this work, however, the flow along the S2 surface is replaced by the calculation of the exact mass averaged-pitch averaged flow in a meridional plane. Extra terms appear in this equation which result from the deviations from axisymmetry and which can be calculated from the knowledge of the blade-to-blade flows. Due to the mass-averaging, these terms represent the only interaction from blade-to-blade S1 surfaces to the meridional flow. The complete program is integrated in a single package requiring only ten percent more computer storage than each of the composing S1 or S2 codes taken alone. The various parts of the program are described as well as the interaction process and specific approximations. Example of calculations compared with experimental data are given, showing good agreement with experimental data.

AbstractThe equations for the meridional through‐flow in a turbomachine are formulated as a quasi‐harmonic non‐linear equation. This equation is then solved iteratively by the finite element method and in order to obtain convergence an under‐relaxation factor has to be introduced. Comparisons with experimental results in single and multistage axial compressors show very good agreement. The finite element method is thus shown to be an efficient tool for this type of problem.

A new method for the numerical solution of the meridional through-flow equations in an axial flow machine is presented based on the finite-element method. A rigorous derivation of the pitch-averaged flow equations is presented and the assumption of axisymmetric flow leads, with the introduction of a stream function, to the equation to be solved. A description is given of the finite-element technique which is applied in this problem. The method of solution allows the calculation of transonic stages. Numerical results are compared with experimental data and show very satisfactory agreement. This method appears, therefore, to compare very favorably with the other methods used up to now. Although the present results pertain to axial flow machines, the method is easily applicable to radial flow machines as well and the way of solution for this case is indicated.

The basic equations of electromagnetism are written in the form of a quasi-harmonic equation. The application of the weighted residual process leads to a non-linear system of algebraic equations which is solved by a full Newton-Raphson procedure. The iteration scheme is developed and applied to numerical examples.

The transient heat conduction problem can be solved by application of Galerkin's method to space as well as time discretization. The formulation corresponds to the procedure known as finite elements in time and space. A linear time expansion leads to a step by step technique which is convergent, consistent and absolutely stable. Several numerical examples are presented using two-dimensional isoparametric elements.

The time dependence of temperatures as solutions of transient heat conduction problems, may be obtained by different numerical techniques. Three procedures are presented. The step-by-step methods, based (i) on finite-elements and (ii) finite-differences in time are briefly reviewed, (iii) The application of the numerical Laplace transform is extensively discussed and its introduction in a finite element program is presented. The accuracy and convergence of the numerical results are discussed and a practical engineering problem is solved for which the computer expenses are compared.

This paper investigates the performance of two methods based on the preconditioned conjugate gradient for the solution of large sparse systems of linear equations which arise from the p-version of the finite element method. The suitability of those methods to the p- version of the finite element method is demonstrated through numerical tests in three dimensions.