Abstract
A mesh smoothing method based on Riemannian metric comparison is presented in this paper. This method minimizes a cost function constructed from a measure of metric non-conformity that compares two metrics: the metric that transforms the element into a reference element and a specified Riemannian metric, that contains the target size and shape of the elements. This combination of metrics allows to cast the proposed mesh smoothing method in a very general frame, valid for any dimension and type of element. Numerical examples show that the proposed method generates high quality meshes as measured both in terms of element characteristics and also in terms of orthogonality at the boundary and overall smoothness, when compared to other known methods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
1. S. R. Mathur and N. K. Madavan, “Solution-adaptive structured-unstructured grid method for unsteady turbomachinery analysis, Part I: Methodology,” Journal of Propulsion and Power, vol. 10, pp. 576–584, July and August 1994.
2. M. K. Patel, K. A. Pericleous, and S. Baldwin, “The development of a structured mesh grid adaption technique for resolving shock discontinuities in upwind Navier-Stokes codes,” International Journal for Numerical Methods in Fluids, vol. 20, pp. 1179–1197, 1995.
3. D. Aït-Ali-Yahia, G. Baruzzi, W. G. Habashi, M. Fortin, J. Dompierre, and M.-G. Vallet, “Anisotropic mesh adaptation: Towards user-independent, meshindependent and solver-independent CFD. Part II: Structured grids,” International Journal for Numerical Methods in Fluids, vol. 39, pp. 657–673, July 2002.
4. A. Tam, D. Aït-Ali-Yahia, M. P. Robichaud, M. Moore, V. Kozel, and W. G. Habashi, “Anisotropic mesh adaptation for 3D flows on structured and unstructured grids,” Computer Methods in Applied Mechanics and Engineering, vol. 189, pp. 1205–1230, May 2000.
5. F. Bossen, “Anisotropic mesh generation with particles,” Master's thesis, Carnegie Mellon University, Pittsburgh, PA, May 1996. CMU-CS-96-134.
6. J. F. Thompson, Z. U. A. Warsi, and C. W. Mastin, Numerical Grid Generation, Foundations and Applications. New York: North-Holland, 1985.
7. P. R. Eiseman, “Alternating direction adaptive grid generation,” American Institute of Aeronautics and Astronautics Journal, p. 1937, 1983.
8. S. P. Spekreijse, “Elliptic grid generation based on Laplace equations and algebraic transformations,” Journal of Computational Physics, vol. 118, pp. 38–61, Apr. 1995.
9. B. K. Soni, R. Koomullil, D. S. Thompson, and H. Thornburg, “Solution adaptive grid strategies based on point redistribution,” Computer Methods in Applied Mechanics and Engineering, vol. 189, pp. 1183–1204, 2000.
10. P. R. Eiseman, “Adaptive grid generation,” Computer Methods in Applied Mechanics and Engineering, vol. 64(1–3), pp. 321–376, 1987.
11. S. McRae, “r-refinement grid adaptation algorithms and issues,” Computer Methods in Applied Mechanics and Engineering, vol. 189, pp. 1161–1182, 2000.
12. L. A. Freitag and C. Ollivier-Gooch, “Tetrahedral mesh improvement using swapping and smoothing,” International Journal for Numerical Methods in Engineering, vol. 40, pp. 3979–4002, 1997.
13. L. A. Freitag, “On combining Laplacian and optimization-based mesh smoothing techniques,” AMD Trends in Unstructured Mesh Generation, ASME, vol. 220, pp. 37–43, 1997.
14. A. Oddy, J. Goldak, M. McDill, and M. Bibby, “A distorsion metric for isoparametric finite elements,” Transactions of the CSME, vol. 12, no. 4, pp. 213–218, 1988.
15. S. A. Cannan, J. R. Tristano, and M. L. Staten, “An approach to combined Laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes,” in 7th Internation Meshing Roundtable, (Detroit, Michigan, USA), pp. 479–494, 1998.
16. P. M. Knupp, “Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II - a framework for volume mesh optimization and the condition number of the Jacobian matrix,” International Journal for Numerical Methods in Engineering, vol. 48, pp. 1165–1185, 2000.
17. M.-G. Vallet, Génération de maillages éléments finis anisotropes et adaptatifs. PhD thesis, Université Pierre et Marie Curie, Paris VI, France, 1992.
18. P.-L. George and H. Borouchaki, Delaunay Triangulation and Meshing. Applications to Finite Elements. Paris: Hermès, 1998.
19. P. J. Frey and P.-L. George, Mesh Generation. Application to Finite Elements. Paris: Hermès, 2000.
20. P. Labbé, J. Dompierre, M.-G. Vallet, F. Guibault, and J.-Y. Trépanier, “A universal measure of the conformity of a mesh with respect to an anisotropic metric field,” International Journal for Numerical Methods in Engineering, vol. 61, pp. 2675–2695, 2004.
21. C. L. Bottasso, “Anisotropic mesh adaption by metric-driven optimization,” International Journal for Numerical Methods in Engineering, vol. 60, pp. 597–639, May 2004.
22. Y. Sirois, J. Dompierre, M.-G. Vallet, and F. Guibault, “Measuring the conformity of non-simplicial elements to an anisotropic metric field,” International Journal for Numerical Methods in Engineering, vol. 64, no. 14, pp. 1944–1958, 2005.
23. Y. Sirois, J. Dompierre, M.-G. Vallet, P. Labbé, and F. Guibault, “Progress on vertex relocation schemes for structured grids in a metric space,” in 8th International Conference on Numerical Grid Generation, (Honolulu, USA), pp. 389–398, June 2002.
24. é. Seveno, Génération automatique de maillages tridimensionnels isotropes par une méthode frontale. PhD thesis, Université Pierre et Marie Curie, Paris VI, Mar. 1998.
25. Y. Sirois, J. Dompierre, M.-G. Vallet, and F. Guibault, “Using a Riemannian metric as a control function for generalized elliptic smoothing,” in Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, (San Jose, CA), June 2005.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer
About this paper
Cite this paper
Sirois, Y., Dompierre, J., Vallet, MG., Guibault, F. (2006). Mesh Smoothing Based on Riemannian Metric Non-Conformity Minimization. In: Pébay, P.P. (eds) Proceedings of the 15th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34958-7_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-34958-7_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34957-0
Online ISBN: 978-3-540-34958-7
eBook Packages: EngineeringEngineering (R0)