Tetrahedral Grid Generators and the Eigenvalue Calculation with Edge Elements
Generadores de malla tetraédricos y el cálculo de Eigenvalores con elementos de contorno
Gerardo Mario Ortigoza Capetillo
Facultad de Ingeniería Universidad Veracruzana Calzada Adolfo Ruiz Cortines s/n, Fracc. Costa Verde Boca del Río Ver, México. E–mail: gortigoza@uv.mx
]]> Article received on September 03, 2008
Abstract
In this work we investigate some computational aspects of the eigenvalue calculation with edge elements; those include: the importance of the grid generator and node–edge numbering. As the examples show, the sparse structure of the mass and stiffness matrices is highly influenced by the edge numbering.
Tetrahedral grid generators are mainly designed for nodal based finite elements so an edge numbering is required. Two different edge numbering schemes are tested with six different grid generators. Significant bandwidth reduction can be obtained by the proper combination of the edge numbering scheme with the grid generator method. Moreover, an ordering algorithm such as the Reverse Cuthill McKee can improve the bandwidth reduction which is necessary to reduce storage requirements.
Keywords: Tetrahedral grid generators, edge elements, RCM ordering, generalized eigenproblem.
Resumen
En este trabajo se investigan algunos aspectos computacionales del cálculo de eigenvalores con elementos de contorno tales como la importancia del generador de mallas y la numeración de nodos y lados. Como muestran los ejemplos, la estructura esparcida de las matrices de masa y momentos es altamente influenciada por la numeración de los lados.
]]> Generadores de mallas en tetraedros son diseñados principalmente para elementos finitos basados en los nodos, así una numeración de los lados es requerida. Se realizaron pruebas con dos esquemas de enumeración de los lados con seis generadores de mallas distintos. Una reducción de banda significante puede obtenerse con una combinación apropiada de esquema de numeración de los lados con el método empleado por el generador de malla. Más aún un algoritmo de reordenamiento como el RCM puede mejorar la reducción de ancho de banda lo cual es necesario para reducir los requerimientos de almacenamiento.Palabras Clave: Generadores de mallas en tetraedros, elementos de contorno, reordenamientos RCM, valores propios generalizados.
DESCARGAR ARTÍCULO EN FORMATO PDF
References
1. Jianming J. (2002).The Finite Element Method in Electromagnetics (2nd ed.). New York: John Wiley and Sons. [ Links ]
2. Volakis, J. L., Chatterjee, A., & Kempel, L. C. (1998). Finite Element Method for electromagnetics: to antennas, microwave circuits, and scattering applications. New York: IEEE Press. [ Links ]
]]>3. Reddy, C.J., Deshpande, M.D., Cockrell, C. R., & Beck, F. B. (1998). Finite element method for eigenvalue problems in electromagnetics. (NASA Technical Paper 3485). Hampton, Virginia: NASA Center for AeroSpace Information. [ Links ]
4. Multiphysics Modeling and Simulation Software– COMSOL. (s.f.). Retrieved from http://www.comsol.com/ [ Links ]
5. Hang, S., & Gartner, K. (2005). Meshing piecewise linear complexes by constrained Delaunay tetrahedralizations. 14th International Meshing Roundtable, San Diego, CA, USA, 147–164. [ Links ]
6. Persson, P.O., & Strang, G. (2004). A simplemesh generator in matlab. SIAM Review, 46(2), 329–345. [ Links ]
7. Qmg 2.0 (s.f.) Retrieved from http://www.cs.cornell.edu/home/vavasis/qmg2.0/qmg2_0_home.html [ Links ]
8. CFD Flow Modeling Software & Solutions from Fluent (s.f.). Retrieved from http://www.fluent.com. [ Links ]/
]]>9. Ansys Simulation Driven Product Development (s.f.). Retrieved from http://www.ansys.com/ [ Links ]
10. Hoole, S.R.H., Jayakumaran, S., & Yoganathan S. (1986). Tetrahedrons, edges and nodes in 3d finite element meshes. Electronic Letters, 22 (14), 735–737. [ Links ]
11. Ortigoza, G. (2009). Triangular grid generators for the eigenvalue calculation with edge elements. Revista Mexicana de Física, 55 (2), 154–160. [ Links ]
12. Shires D., & Mohan R. (2003). Optimization and performance of a FORTRAN 90 mpi–based unstructured code on large–scale parallel systems. The Journal of Supercomputing, 25 (2), 131–141. [ Links ]
13. A survey of unstructured mesh generation technology retrieved from http://www.andrew.cmu.edu/user/sowen/survey/ [ Links ]
14. Hansen, P. C. T., Ostromsky, A., Basermann, P., (1994). Weidner., Reordering of sparse matrices for parallel processing. APPARC PaA3a Technical report. [ Links ]
]]>15. Burgess D. A., & Giles, M. B. (1997). Renumbering unstructured grids to improve performance of codes on hierarchical memory machines. Advances in Engineering Software, 28 (3), 189–201. [ Links ]
16. Kaveh, A. & Rahimi Bondarabady, H. A. (2002). An hybrid method for finite element ordering. Computers & Structures, 80 (3–4), 219–225. [ Links ]
17. Paulino, G. H., Meneses, I.F., Gattass M., & Mukherjee, S. (1994). Node and element resequencing using the Laplacian of a finite element graph: Part I – General concepts and algorithm. International Journal for Numerical Methods in Engineering, 37(9), 1511–1530. [ Links ]
18. Cuthill, E., & McKee, J. (1969). Reducing the bandwidth of sparse symmetric matrices. ACM Annual Conference 24th national conference, New York, USA, 157–172. [ Links ]
19. Lang, B. & Aachen, R. (2000). Direct solvers for symmetric eigenvalue problems. In Johannes Grotendorst (Ed.) Modern methods and algorithms of quantum chemistry proceedings, Jülich, Alemania, 3, 231–259. [ Links ]
]]>20. Bai, Z., Demmel, J., Dongarra, J., Ruhe, A. & Van der Vorst, H. (2000). Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Philadelphia: Society for Industrial and Applied Mathematics. [ Links ]
21. Hernandez, V., Román, J.E. Tomas, A., & Vidal, V. (2007). A survey of software for sparse eigenvalue problems. (SLEPc Technical Report STR–6). España: Universidad Politécnica de Valencia. [ Links ]
22. Baglama, J., Calvetti, D., & Reichel, L. (2003). Algorithm 827: irbleigs: a matlab program for computing a few eigenpairs of a large sparse hermitian matrix. ACM transactions on mathematical software, 29 (3), 337–348. [ Links ]
23. Lehoucq, R. B., Sorensen, D. C., & Yang, C. (1998). Arpack: user's guide: Solution of large–scale eigenvalue problems with implicit restarted arnoldi methods. Philadelphia : SIAM. [ Links ]
24. Fokkema D., Sleijpen G., & Van der, H. (1998). Jacobi–Davidson style qr and qz algorithms for the reduction of matrix pencils. SIAM Journal on Scientific Computing, 20 (1), 94–125. [ Links ]
]]>25. Knyazev, A. (2001) Towards the optimal preconditioned eigensolver: locally conditioned conjugate gradient method. SIAM Journal of Scientific Computation, 23 (2), 517–541. [ Links ]
26. Money J. H. & Ye Q. (2005). Algorithm 845: Eigifp: a matlab program for solving large symmetric generalized eigenvalue problems. ACM transactions on mathematical software, 31 (2), 270–279. [ Links ]
27. Kaufman L. (2000). Band reduction algorithms revisited. ACM Transactions on Mathematical Software, 26 (4), 551–567. [ Links ]
]]>