Guy Warzée

HIGH-PERFORMANCE PCG SOLVERS FOR FEM STRUCTURAL ANALYSIS

The preconditioned conjugate gradient algorithm is a well-known and powerful method used to solve large sparse symmetric positive definite linear systems Such system are generated by the finite element discretisation in structural analysis but users of finite elements in this contest 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.

Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix

Incomplete factorizations are popular preconditioning techniques for solving large and sparse linear systems. In the case of highly indefinite complex–symmetric linear systems, the convergence of Krylov subspace methods sometimes degrades with increasing level of fill-in. The reasons for this disappointing behaviour are twofold. On the one hand, the eigenvalues of the preconditioned system tend to 1, but the ‘convergence’ is not monotonous. On the other hand, the eigenvalues with negative real part, on their move towards 1 have to cross the origin, whence the risk of clustering eigenvalues around 0 while ‘improving’ the preconditioner. This makes it risky to predict any gain when passing from a level to a higher one. We examine a remedy which consists in slightly moving the spectrum of the original system matrix along the imaginary axis. Theoretical analysis that motivates our approach and experimental results are presented, which displays the efficiency of the new preconditioning techniques. Copyright

Problem-dependent preconditioners for iterative solvers in FE elastostatics

Abstract 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.

Finite element analysis of transient heat conduction application of the weighted residual process

Abstract The transient heat conduction problem can be solved by application of Galerkins 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.

Superconvergent patch recovery technique for the finite element method in acoustics

A posteriori error estimation has become very popular, mainly in linear elasticity. A robust implementation of the superconvergent patch recovery technique of O. C. Zienkiewicz and J. Z. Zhu is presented for acoustic finite element analyses: the original concepts are extended to complex variables, and both local and global behaviours of the recovery procedure and the error estimation are studied. The numerical tests confirm the improvement of the rates of convergence for the recovered solution and also show the reliability of the error estimator except at frequencies corresponding either to the analytical or to the finite element eigenfrequencies.

Efficient iterative solution of constrained finite element analyses

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.

Finite element method and laplace transform—Comparative solutions of transient heat conduction problems

Abstract 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.

Finite element analysis of nonlinear magnetic fields

Abstract 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.