Computing the Distance between Two Finite Element Solutions Defined on Different 3D Meshes on a GPU
Abstract
This article introduces a new method to efficiently compute the distance (i.e., $L^p$ norm of the difference) between two functions supported by two different meshes of the same 3D domain. The functions that we consider are typically finite element solutions discretized in different function spaces supported by meshes that are potentially completely unrelated. Our method computes an approximation of the distance by resampling both fields over a set of parallel 2D regular grids. By leveraging the parallel horse power of computer graphics hardware (graphics processing unit (GPU)), our method can efficiently compute distances between meshes with multimillion elements in seconds. We demonstrate our method applied to different problems (distance between known functions, Poisson solutions, and linear elasticity solutions) using different function spaces (Lagrange polynomials from order one to seven) and different meshes (tetrahedral and hexahedral, with linear or quadratic geometry).
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
Supplementary Material
PLEASE NOTE: These supplementary files have not been peer-reviewed.
Index of Supplementary Materials
Title of paper: Computing the distance between two finite element solutions defined on different 3D meshes on a GPU
Authors: Maxence Reberol, Bruno Levy
File: ffes_source.zip
Type: Compressed code files
Contents: Source code of the implementation referenced in the paper and used to generate the results.
Justification: The source code of our implementation is useful to anyone who would like to use our program or to reproduce and extend our results.
References
1.
Geogram: A Programming Library of Geometric Algorithms, http://alice.loria.fr/software/geogram/doc/html/index.html.
2.
MFEM: Modular Finite Element Methods, http://mfem.org.
3.
D. Akio and A. Koide, An efficient method of triangulating equi-valued surfaces by using tetrahedral cells, IEICE Trans. Inform. Syst., 74 (1991), pp. 214--224.
4.
F. Alauzet, A parallel matrix-free conservative solution interpolation on unstructured tetrahedral meshes, Comput. Methods Appl. Mech. Engrg., 299 (2016), pp. 116--142, https://doi.org/10.1016/j.cma.2015.10.012.
5.
M. Brasher and R. Haimes, Rendering planar cuts through quadratic and cubic finite elements, in IEEE Visualization, 2004, IEEE, Washington, DC, 2004, pp. 409--416, https://doi.org/10.1109/VISUAL.2004.91.
6.
P. Bussetta, R. Boman, and J.-P. Ponthot, Efficient 3D data transfer operators based on numerical integration, Internat. J. Numer. Methods Engrg., 102 (2015), pp. 892--929, https://doi.org/10.1002/nme.4821.
7.
P. E. Farrell, The addition of fields on different meshes, J. Comput. Phys., 230 (2011), pp. 3265--3269, https://doi.org/10.1016/j.jcp.2011.01.028.
8.
P. E. Farrell, M. D. Piggott, C. C. Pain, G. J. Gorman, and C. R. G. Wilson, Conservative interpolation between unstructured meshes via supermesh construction, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 2632--2642, https://doi.org/10.1016/j.cma.2009.03.004.
9.
Y. Feng, D. Perić, and D. Owen, A non-nested Galerkin multi-grid method for solving linear and nonlinear solid mechanics problems, Comput. Methods Appl. Mech. Engrg., 144 (1997), pp. 307--325, https://doi.org/10.1016/s0045-7825(96)01183-8.
10.
C. Geuzaine, B. Meys, F. Henrotte, P. Dular, and W. Legros, A Galerkin projection method for mixed finite elements, IEEE Trans. Magnetics, 35 (1999), pp. 1438--1441, https://doi.org/10.1109/20.767236.
11.
B. Haasdonk, M. Ohlberger, M. Rumpf, A. Schmidt, and K. G. Siebert, Multiresolution visualization of higher order adaptive finite element simulations, Computing, 70 (2003), pp. 181--204.
12.
X. Jiao and M. T. Heath, Common-refinement-based data transfer between non-matching meshes in multiphysics simulations, Internat. J. Numer. Methods Engrg., 61 (2004), pp. 2402--2427, https://doi.org/10.1002/nme.1147.
13.
A. O. Leone, P. Marzano, E. Gobbetti, R. Scateni, and S. Pedinotti, Discontinuous finite element visualization, in Proceedings of the 8th International Symposium on Flow Visualization, Sorrento, Italy, 1998.
14.
M. Livesu, A. Sheffer, N. Vining, and M. Tarini, Practical hex-mesh optimization via edge-cone rectification, ACM Trans. Graphics, 34 (2015), 141, https://doi.org/10.1145/2766905.
15.
S. Menon and D. P. Schmidt, Conservative interpolation on unstructured polyhedral meshes: An extension of the supermesh approach to cell-centered finite-volume variables, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 2797--2804, https://doi.org/10.1016/j.cma.2011.04.025.
16.
B. Nelson and R. M. Kirby, Ray-tracing polymorphic multidomain spectral/hp elements for isosurface rendering, IEEE Trans. Vis. Comput. Graphics, 12 (2006), pp. 114--125, https://doi.org/10.1109/tvcg.2006.12.
17.
B. Nelson, R. M. Kirby, and R. Haimes, GPU-based interactive cut-surface extraction from high-order finite element fields, IEEE Trans. Vis. Comput. Graphics, 17 (2011), pp. 1803--1811, https://doi.org/10.1109/tvcg.2011.206.
18.
J. Peraire, J. Peiro, and K. Morgan, Multigrid solution of the 3-D compressible Euler equations on unstructured tetrahedral grids, Internat. J. Numer. Methods Engrg., 36 (1993), pp. 1029--1044, https://doi.org/10.1002/nme.1620360610.
19.
K. Perlin, An image synthesizer, SIGGRAPH Comput. Graph., 19 (1985), pp. 287--296, https://doi.org/10.1145/325165.325247.
20.
S. Plimpton, B. Hendrickson, and J. Stewart, A parallel rendezvous algorithm for interpolation between multiple grids, in Proceedings of the 1998 ACM/IEEE Conference on Supercomputing, SC '98, ACM, New York, IEEE Computer Society, Washington, DC, 1998, pp. 1--8, https://doi.org/10.1109/sc.1998.10032.
21.
J.-F. Remacle, N. Chevaugeon, É. Marchandise, and C. Geuzaine, Efficient visualization of high-order finite elements, Internat. J. Numer. Methods Engrg., 69 (2007), pp. 750--771, https://doi.org/10.1002/nme.1787.
22.
H. Si, TetGen, a Delaunay-based quality tetrahedral mesh generator, ACM Trans. Math. Software, 41 (2015), 11, https://doi.org/10.1145/2629697.
23.
S. R. Slattery, P. P. H. Wilson, and R. P. Pawlowski, The data transfer kit: A geometric rendezvous-based tool for multiphysics data transfer, in Proceedings of the 2013 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (Sun Valley, ID), American Nuclear Society, La Grange Park, IL, 2013, pp. 1262--1272.
24.
G. M. Treece, R. W. Prager, and A. H. Gee, Regularised marching tetrahedra: Improved iso-surface extraction, Computers & Graphics, 23 (1999), pp. 583--598, https://doi.org/10.1016/s0097-8493(99)00076-x.
25.
M. Üffinger, S. Frey, and T. Ertl, Interactive high-quality visualization of higher-order finite elements, in Computer Graphics Forum, 29 (2010), pp. 337--346, https://doi.org/10.1111/j.1467-8659.2009.01603.x.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: C131 - C155
DOI: 10.1137/17M1115976
ISSN (online): 1095-7197
Copyright
© 2018, Society for Industrial and Applied Mathematics.
History
Submitted: 13 February 2017
Accepted: 14 December 2017
Published online: 20 February 2018
Keywords
MSC codes
Authors
Funding Information
H2020 European Research Council https://doi.org/10.13039/100010663 : StG-2012-307877
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- MFEM: A modular finite element methods libraryComputers & Mathematics with Applications, Vol. 81 | 1 Jan 2021