Abstract
Mesh adaptation is considered here as the research of an optimum that minimizes the P 1 interpolation error of a function υ of ℝn given a number of vertices. A continuous modeling is described by considering classes of equivalence between meshes which are analytically represented by a metric tensor field. Continuous metrics are exhibited for Lp error model and mesh order of convergence are analyzed. Numerical examples are provided in two and three dimensions.
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References
1. Almeida, R., Feijóo, R., Galeao, A., Padra, C., and Silva, R. Adaptive finite element computational fluid dynamics using an anisotropic error estimator. Comput. Methods Appl. Mesh. Engrg. 182 (2000), 379–400.
2. Biswas, R., and Strawn, R. A new procedure for dynamic adaption of threedimensional unstructured grids. Applied Numerical Mathematics 13 (1994), 437–452.
3. Buscaglia, G., and Dari, E. Anisotropic mesh optimization and its application in adaptivity. Int. J. Numer. Meth. Engng 40 (1997), 4119–4136.
4. Castro-Diaz, M., Hecht, F., Mohammadi, B., and Pironneau, O. Anisotropic unstructured mesh adaptation for flow simulations. Int. J. Numer. Meth. Fluids 25 (1997), 475–491.
5. Chen, L., Sun, P., and Xu, J. Optimal anisotropic simplicial meshes for minimizing interpolation errors in L p-norms. Math. Comp. 0 (2006). to appear.
6. Courty, F., Leservoisier, D., George, P.-L., and Dervieux, A. Continuous metrics and mesh adaptation. Applied Numerical Mathematics 56 (2006), 117–145.
7. Dervieux, A., Leservoisier, D., George, P.-L., and Coudiere, Y. About theoretical and practical impact of mesh adaptations on approximation of functions and of solution of pde. Int. J. Numer. Meth. Fluids 43 (2003), 507–516.
8. Dobrzynski, C. Adaptation de maillage anisotrope 3D et application à l'aérothermique des bâtiments. PhD thesis, Université Pierre et Marie Curie, Paris VI, Paris, France, 2005. (in French).
9. Dompierre, J., Vallet, M., Fortin, M., Bourgault, Y., and Habashi, W. Anisotropic mesh adaptation: towards a solver and user independent cfd. AIAA paper 97–0861 (1997).
10. Formaggia, L., Micheletti, S., and Perotto, S. Anisotropic mesh adaptation in computational fluid dynamics: Application to the advection-diffusionreaction and the Stokes problem. Applied Numerical Mathematics 51 (2004), 511–533.
11. Frey, P. About surface remeshing. In Proc. of 9th Int. Meshing Rountable (New Orleans, LO, USA, 2000), pp. 123–136.
12. Frey, P., and Alauzet, F. Anisotropic mesh adaptation for transient flows simulations. In Proc. of 12th Int. Meshing Rountable (Santa Fe, New Mexico, USA, 2003), pp. 335–348.
13. Frey, P., and Alauzet, F. Anisotropic mesh adaptation for CFD computations. Comput. Methods Appl. Mech. Engrg. 194, 48–49 (2005), 5068–5082.
14. Frey, P., and George, P.-L. Mesh generation. Application to finite elements. Hermès Science, Paris, Oxford, 2000.
15. George, P.-L., and Borouchaki, H. Back to edge flips in 3 dimensions. In Proc. of 12th Int. Meshing Rountable (Santa Fe, NM, USA, 2003), pp. 393–402.
16. Gruau, C., and Coupez, T. 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric. Comput. Methods Appl. Mech. Engrg. 194, 48–49 (2005), 4951–4976.
17. Huang, W. Metric tensors for anisotropic mesh generation. J. Comput. Phys. 204 (2005), 633–665.
18. Kallinderis, Y., and Vijayan, P. Adaptive refinement-coarsening scheme for three-dimensional unstructured meshes. AIAA Journal 31, 8 (1993), 1440–1447.
19. Leservoisier, D., George, P.-L., and Dervieux, A. Métrique continue et optimisation de maillage. RR-4172, INRIA, Apr. 2001. (in French).
20. Li, X., Shephard, M., and Beall, M. 3D anisotropic mesh adaptation by mesh modification. Comput. Methods Appl. Mech. Engrg. 194, 48–49 (2005), 4915–4950.
21. Löhner, R., and Baum, J. Adaptive h-refinement on 3D unstructured grids for transient problems. Int. J. Numer. Meth. Fluids 14 (1992), 1407–1419.
22. Mavriplis, D. Adaptive meshing techniques for viscous flow calculations on mixed-element unstructured meshes. AIAA paper 97–0857 (1997).
23. Pain, C., Humpleby, A., de Oliveira, C., and Goddard, A. Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations. Comput. Methods Appl. Mech. Engrg. 190 (2001), 3771–3796.
24. Peraire, J., Peiro, J., and Morgan, K. Adaptive remeshing for threedimensional compressible flow computations. J. Comput. Phys. 103 (1992), 269–285.
25. Rausch, R., Batina, J., and Yang, H. Spatial adaptation procedures on tetrahedral meshes for unsteady aerodynamic flow calculations. AIAA Journal 30 (1992), 1243–1251.
26. Schall, E., Leservoisier, D., Dervieux, A., and Koobus, B. Mesh adaptation as a tool for certified computational aerodynamics. Int. J. Numer. Meth. Fluids 45 (2004), 179–196.
27. Speares, W., and Berzins, M. A 3d unstructured mesh adaptation algorithm for time-dependent shock-dominated problems. Int. J. Numer. Meth. Fluids 25 (1997), 81–104.
28. Tam, A., Ait-Ali-Yahia, D., Robichaud, M., Moore, M., Kozel, V., and Habashi, W. Anisotropic mesh adaptation for 3d flows on structured and unstructured grids. Comput. Methods Appl. Mech. Engrg. 189 (2000), 1205–1230.
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Alauzet, F., Loseille, A., Dervieux, A., Frey, P. (2006). Multi-Dimensional Continuous Metric for Mesh Adaptation. 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_12
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DOI: https://doi.org/10.1007/978-3-540-34958-7_12
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