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Speedup of an iterative FE-Solver due to an optimal preconditioning having regard to the imposition of constraints

Titelangaben

Billenstein, Daniel ; Zimmermann, Markus ; Diwisch, Pascal ; Rieg, Frank:
Speedup of an iterative FE-Solver due to an optimal preconditioning having regard to the imposition of constraints.
In: Proceedings NAFEMS World Congress 2017. - Stockholm , 2017 . - S. 14-15
ISBN 978-1-910643-37-2

Abstract

Evaluating mechanical parts and assemblies using standardized calculations often lacks depth regarding the obtained solution, which is why more sophisticated numerical simulations have to be consulted. The widely-used finite element analysis (FEA) is a matrix-based algorithm, the main runtime of which is determined by solving one or more linear system of equations. Amongst the number of different numeric algorithms to solve this problem the iterative method of conjugated gradients (CG method) has proven itself as a viable approach. Attempts to decrease the runtime of this algorithm are based on optimizing the condition number of the matrix, which results in decreasing the number of iteration steps necessary to solve the linear system. This can be achieved by preconditioning the matrix, for example by using a relaxation parameter. This Symmetric-Successive-Over-Relaxation-Preconditioning (SSOR) yields the big advantage over other methods (shifted incomplete cholesky, jacobi over relaxation etc.) that no additional memory is needed. While the optimal relaxation parameter can in theory be calculated using the eigenvalue of the matrix with the biggest magnitude, obtaining the eigenvalues of the matrix is often more time-consuming (especially with large linear systems) than solving the linear system using an unfavourable relaxation parameter. Due to this immense calculatory effort, finding the optimal relaxation parameter has already been subject of research, which yielded that the part geometry, mesh size and boundary conditions have a big influence on the relaxation parameter. On the other hand, material parameters and the magnitude of external forces did not show any dependencies in this regard. Since both the part geometry and the mesh size are very model-specific and often predefined parameters, further investigations in this direction are considered unrewarding. Boundary conditions and thus constraints in general, can be taken into account regarding method of imposition and the specific setting parameters. The imposition of constraints modifies the matrix shape badly, so the condition number is heavily dependent very addicted to this. There are three widely used imposition procedures, namely, the Master-slave method, the (perturbed) Lagrange multiplier method and the Penalty method. The latter two introduce an additional factor with direct influence on the condition of the matrix and consequently on the number of iterations. Bearing this in mind, this approach includes determining the ideal relaxation parameter in each case. This reveals, among others, that it is possible to determine a certain range for the relaxation parameter for any of the three methods, which results in a minimal number of iterations. This study and the consequent insights provide the calculation engineer with assistance in actuating the solver to maintain low calculation times.

Weitere Angaben

Publikationsform: Aufsatz in einem Buch
Begutachteter Beitrag: Ja
Keywords: FEA; Solver
Institutionen der Universität: Fakultäten > Fakultät für Ingenieurwissenschaften > Lehrstuhl Konstruktionslehre/CAD
Fakultäten > Fakultät für Ingenieurwissenschaften > Lehrstuhl Konstruktionslehre/CAD > Lehrstuhl Konstruktionslehre/CAD - Univ.-Prof. Dr.-Ing. Frank Rieg
Fakultäten
Fakultäten > Fakultät für Ingenieurwissenschaften
Titel an der UBT entstanden: Ja
Themengebiete aus DDC: 600 Technik, Medizin, angewandte Wissenschaften > 620 Ingenieurwissenschaften
Eingestellt am: 17 Jul 2017 11:12
Letzte Änderung: 16 Jul 2018 08:55
URI: https://eref.uni-bayreuth.de/id/eprint/38467