Ingrid Lenhardt

Krylov subspace methods for structural finite element analysis

Nonlinear problems from structural finite element analysis usually are tackled with Newton-like methods. In every step a linear system of equations has to be solved. For these problems iterative Krylov subspace methods with several preconditioning strategies and an extended Lanczos method for nonlinear computations with arc length methods are presented. For a parallel implementation domain decomposition methods are used. Each processor works on a local grid and computes a partial stiffness matrix. The preconditioned parallel iterative methods only work with these partial stiffness matrices. The range of applications and the speedups for the treated methods is shown by examples from structural analysis (without mathematical proofs).

Nonlinear structural finite element analysis using the preconditioned Lanczos method on serial and parallel computers

The application of the Lanczos algorithm in Newton-like methods for solving non-linear systems of equations arising in nonlinear structural finite element analysis is presented. It is shown that with appropriate preconditioners iterative methods can be developed which are robust and efficient even for ill conditioned problems. Though the real advantage of iterative solvers seems to exist on distributed memory machines, even on serial machines the performance can be improved compared with direct solvers while saving memory capacity. With a specific modification of the Lanczos algorithm in combination with arc-length procedures a further speed-up of the nonlinear analysis can be achieved.For parallel implementations domain decomposition methods are used. A parallel preconditioning strategy based on an incomplete factorisation method is presented. An example is taken and the quality and efficiency of two different domain decomposition methods are discussed for a large shell structure.

On applications of parallel solution techniques for highly nonlinear problems involving static and dynamic buckling

Within this contribution the efficient finite element analysis of shell structures with highly nonlinear behavior is presented. The coupled nonlinear system of equations resulting from the FE discretization is solved using Newton-like procedures, thus the solution of a linear system of equations is needed in each Newton iteration. For fine discretizations the resulting linear systems of equations become very large and their solution dominates the computational effort. Consequently, parallel computers offer major capabilities to reduce the CPU time needed. A geometrical approach for parallelization is used, standard methods for the graph partitioning are employed. It is well known that iterative methods for the solution of linear equation systems are much more suitable for parallelizing compared to direct methods. Therefore in a first step the use of such methods is investigated for the application to badly conditioned problems typical for shell problems in particular in failure situations with almost singular matrices. In the analysis of shell structures with tendencies to buckle a static and a dynamic approach are discussed considering both physical and computational aspects.

Parallele Verfahren für partielle Differentialgleichungen

In diesem Kapitel betrachten wir zur Einfuhrung — und aus historischen Grundendie klassischen Iterationsverfahren zur Auflosung von linearen Gleichungssystemen: Jacobi-Verfahren, Gaus-Seidel-Verfahren, SOR-Verfahren. Diese haben heute als eigenstandige Verfahren kaum mehr Bedeutung, man kann jedoch mit ihnen einige grundlegende Unterschiede und Prinzipien bei der Parallelisierung aufzeigen. Die Darstellungen in Abschnitt 2.1 und Abschnitt 2.2 lehnen sich eng an [26] und [39] an.

Der Gauß-Algorithmus — Anwendung bei Integralgleichungen

In diesem Kapitel betrachten wir eine parallele Implementierung des Gaus- Algorithmus, die fur die Losung von linearen Gleichungssystemen mit vollbesetzten unsymmetrischen Matrizen geeignet ist. Als ein Anwendungsfall, bei dem Matrizen dieses Typs auftreten, stellen wir das Nystromverfahren vor, das zur numerischen Losung linearer Fredholmscher Integralgleichungen zweiter Art dient. Zuvor aber geben wir einige grundlegende Eigenschaften dieser Integralgleichung an.

GrundsÄtzliches über Parallelrechner

In diesem Kapitel besprechen wir einleitend den Aufbau und die Leistungsbeurteilung von Parallelrechnern. Es schliest sich eine Diskussion von Programmiermodellen an.

Die Methode der konjugierten Gradienten

Im vorliegenden Kapitel werden wir die Methode der konjugierten Gradienten zur Losung des linearen Gleichungssystems

Das symmetrische Eigenwert-Problem

Ax = b mit A sysmmetrisch positiv definit