Verification of three‐dimensional anisotropic adaptive processes

A verification methodology for adaptive processes is devised. The mathematical claims made during the process are identified and measures are presented in order to verify that the mathematical equations are solved correctly. The analysis is based on a formal definition of the optimality of the adaptive process in the case of the control of the L∞-norm of the interpolation error. The process requires a reconstruction that is verified using a proper norm. The process also depends on mesh adaptation toolkits in order to generate adapted meshes. In this case, the non-conformity measure is used to evaluate how well the adapted meshes conform to the size specification map at each iteration. Finally, the adaptive process should converge toward an optimal mesh. The optimality of the mesh is measured using the standard deviation of the element-wise value of the L∞-norm of the interpolation error. The results compare the optimality of an anisotropic process to an isotropic process and to uniform refinement on highly anisotropic 2D and 3D test cases. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  Patrick M. Knupp,et al.  Local Anisotropic Interpolation Error Estimates Based on Directional Derivatives Along Edges , 2008, SIAM J. Numer. Anal..

[2]  Youssef Belhamadia,et al.  Anisotropic mesh adaptation for the solution of the Stefan problem , 2004 .

[3]  Yuri V. Vassilevski,et al.  Minimization of gradient errors of piecewise linear interpolation on simplicial meshes , 2010 .

[4]  D. Ait-Ali-Yahia,et al.  Anisotropic mesh adaptation for 3D flows on structured and unstructured grids , 2000 .

[5]  Michel Fortin,et al.  Certifiable Computational Fluid Dynamics Through Mesh Optimization , 1998 .

[6]  C.R.E. de Oliveira,et al.  Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations , 2001 .

[7]  Weizhang Huang,et al.  Metric tensors for anisotropic mesh generation , 2005 .

[8]  François Guibault,et al.  A universal measure of the conformity of a mesh with respect to an anisotropic metric field , 2004 .

[9]  Thierry Coupez,et al.  Génération de maillage et adaptation de maillage par optimisation locale , 2000 .

[10]  Mark S. Shephard,et al.  3D anisotropic mesh adaptation by mesh modification , 2005 .

[11]  R. B. Simpson,et al.  On optimal interpolation triangle incidences , 1989 .

[12]  Arnd Meyer,et al.  A new methodology for anisotropic mesh refinement based upon error gradients , 2004 .

[13]  Pascal Frey,et al.  Anisotropic mesh adaptation for CFD computations , 2005 .

[14]  E. F. D'Azevedo,et al.  On optimal triangular meshes for minimizing the gradient error , 1991 .

[15]  Marco Picasso,et al.  Adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives , 2006 .

[16]  Carlo L. Bottasso,et al.  Anisotropic mesh adaption by metric‐driven optimization , 2004 .

[17]  J. Remacle,et al.  A mesh adaptation framework for dealing with large deforming meshes , 2010 .

[18]  F. Alauzet,et al.  Continuous Mesh Model and Well-Posed Continuous Interpolation Error Estimation , 2009 .

[19]  C. Dobrzynski,et al.  Anisotropic Delaunay Mesh Adaptation for Unsteady Simulations , 2008, IMR.

[20]  F. Courty,et al.  Continuous metrics and mesh adaptation , 2006 .

[21]  J. Dompierre,et al.  Numerical comparison of some Hessian recovery techniques , 2007 .

[22]  E. F. D’Azevedo,et al.  Optimal Triangular Mesh Generation by Coordinate Transformation , 1991, SIAM J. Sci. Comput..

[23]  Youssef Belhamadia,et al.  Three-dimensional anisotropic mesh adaptation for phase change problems , 2004 .

[24]  Long Chen,et al.  Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm , 2007, Math. Comput..