×

Fast three dimensional r-adaptive mesh redistribution. (English) Zbl 1349.65419

Summary: This paper describes a fast and reliable method for redistributing a computational mesh in three dimensions which can generate a complex three dimensional mesh without any problems due to mesh tangling. The method relies on a three dimensional implementation of the parabolic Monge-Ampère (PMA) technique, for finding an optimally transported mesh. The method for implementing PMA is described in detail and applied to both static and dynamic mesh redistribution problems, studying both the convergence and the computational cost of the algorithm. The algorithm is applied to a series of problems of increasing complexity. In particular very regular meshes are generated to resolve real meteorological features (derived from a weather forecasting model covering the UK area) in grids with over \(2 \times 10^7\) degrees of freedom. The PMA method computes these grids in times commensurate with those required for operational weather forecasting.

MSC:

65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35K96 Parabolic Monge-Ampère equations

Software:

PFFT; FFTW; MOVCOL
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Piccolo, Chiara; Cullen, Mike, A new implementation of the adaptive mesh transform in the Met Office 3D-Var System, Q. J. R. Meteorol. Soc., 138, 667, 1560-1570 (July 2012)
[2] Gullbrand, Jessica; Chow, Fotini Katopodes, The effect of numerical errors and turbulence models in large-eddy simulations of channel flow, with and without explicit filtering, J. Fluid Mech., 495, 323-341 (November 2003) · Zbl 1077.76032
[3] Huang, Weizhang; Russell, Robert D., A moving collocation method for solving time dependent partial differential equations, Appl. Numer. Math., 20, 101-116 (1996) · Zbl 0859.65112
[4] Piccolo, Chiara; Cullen, Mike, Adaptive mesh method in the Met Office variational data assimilation system, Q. J. R. Meteorol. Soc., 137, 656, 631-640 (April 2011)
[5] Pain, C. C.; Piggott, M. D.; Goddard, A. J.H.; Fang, F.; Gorman, G. J.; Marshall, D. P.; Eaton, M. D.; Power, P. W.; de Oliveira, C. R.E., Three-dimensional unstructured mesh ocean modelling, Ocean Model., 10, 1-2, 5-33 (January 2005)
[6] Chacón, L.; Delzanno, G. L.; Finn, J. M., Robust, multidimensional mesh-motion based on Monge-Kantorovich equidistribution, J. Comput. Phys., 230, 1, 87-103 (January 2011) · Zbl 1205.65260
[7] Behrens, Jörn, Atmospheric and ocean modeling with an adaptive finite element solver for the shallow-water equations, Appl. Numer. Math., 26, 1-2, 217-226 (January 1998) · Zbl 0897.76046
[8] Weller, Hilary, Predicting mesh density for adaptive modelling of the global atmosphere, Philos. Trans. R. Soc., Math. Phys. Eng. Sci., 367, 1907, 4523-4542 (November 2009) · Zbl 1192.86014
[9] Ainsworth, Mark; Senior, Bill, Aspects of an adaptive hp-finite element method: adaptive strategy, conforming approximation and efficient solvers, Comput. Methods Appl. Mech. Eng., 150, 1-4, 65-87 (December 1997) · Zbl 0906.73057
[10] Behrens, Jörn, Adaptive Atmospheric Modeling: Key Techniques in Grid Generation, Data Structures, and Numerical Operations with Applications (2006), Springer · Zbl 1138.86002
[11] Budd, C. J.; Williams, J. F., Moving mesh generation using the parabolic Monge-Ampère equation, SIAM J. Sci. Comput., 31, 5, 3438-3465 (2009) · Zbl 1200.65099
[12] Budd, Chris J.; Huang, Weizhang; Russell, Robert D., Adaptivity with moving grids, Acta Numer., 18, 1-131 (May 2009)
[13] Huang, Weizhang; Russell, Robert D., Adaptive Moving Mesh Methods, Appl. Math. Sci., vol. 174 (2011), Springer · Zbl 1227.65090
[14] Delzanno, G. L.; Chacón, L.; Finn, J. M.; Chung, Y.; Lapenta, G., An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization, J. Comput. Phys., 227, 23, 9841-9864 (December 2008) · Zbl 1155.65394
[15] Finn, John M.; Delzanno, Gian Luca; Chacón, Luis, Grid generation and adaptation by Monge-Kantorovich optimization in two and three dimensions, (Proceedings of the 17th International Meshing Roundtable (2008), Springer), 551-568
[16] Budd, C. J.; Cullen, M. J.P.; Walsh, E. J., Monge-Ampére based moving mesh methods for numerical weather prediction, with applications to the Eady problem, J. Comput. Phys., 236, 247-270 (March 2013) · Zbl 1286.65178
[17] Cao, Weiming, On the error of linear interpolation and the orientation, aspect ratio, and internal angles of a triangle, SIAM J. Numer. Anal., 43, 1, 19-40 (January 2005) · Zbl 1092.65006
[18] Huang, Weizhang; Ren, Yuhe; Russell, Robert D., Moving Mesh Partial Differential Equations (MMPDES) based on the equidistribution principle, SIAM J. Numer. Anal., 31, 3, 709-730 (1994) · Zbl 0806.65092
[19] Brenier, Yann, Polar factorization and monotone rearrangement of vector-valued functions, Commun. Pure Appl. Math., XLIV, 375-417 (1991) · Zbl 0738.46011
[20] Sewell, M. J., Some applications of transformation theory in mechanics, Large Scale Atmosphere. Large Scale Atmosphere, Ocean Dyn., 2, 143-223 (2002)
[21] Froese, Brittany Dawn, Numerical methods for the elliptic Monge-Ampère equation and optimal transport (2012), Simon Fraser University, PhD · Zbl 1252.65180
[22] Cullen, M. J.P., A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow (2006), Imperial College Press
[23] Chynoweth, S.; Sewell, J., A concise derivation of the semi-geostrophic equations, Q. J. R. Meteorol. Soc., 117, 502, 1109-1128 (1991)
[24] Ceniceros, Hector D.; Hou, Thomas Y., An efficient dynamically adaptive mesh for potentially singular solutions, J. Comput. Phys., 172, 2, 609-639 (September 2001) · Zbl 0986.65087
[25] Budd, C. J.; Galaktionov, V. A., On self-similar blow-up in evolution equations of Monge-Ampère type, IMA J. Appl. Math., 78, 2, 338-378 (2013) · Zbl 1267.35045
[26] Frigo, Matteo; Johnson, Steven G., The design and implementation of FFTW3, Special issue on “Program Generation, Optimization, and Platform Adaptation”. Special issue on “Program Generation, Optimization, and Platform Adaptation”, Proc. IEEE, 93, 2, 216-231 (2005)
[27] Pippig, Michael, PFFT: an extension of FFTW to massively parallel architectures, SIAM J. Sci. Comput., 35, 3, 213-236 (2013) · Zbl 1275.65098
[28] Sulman, Mohamed; Williams, J. F.; Russell, R. D., Optimal mass transport for higher dimensional adaptive grid generation, J. Comput. Phys., 230, 9, 3302-3330 (May 2011) · Zbl 1218.65065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.