Abstract
We present an algorithm called discrete mesh optimization (DMO), a greedy approach to topology-consistent mesh quality improvement. The method requires a quality metric for all element types that appear in a given mesh. It is easily adaptable to any mesh and metric as it does not rely on differentiable functions. We give examples for triangle, quadrilateral, and tetrahedral meshes and for various metrics. The method improves quality iteratively by finding the optimal position for each vertex on a discretized domain. We show that DMO outperforms other state of the art methods in terms of convergence and runtime.
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V. Aizinger, C. Dawson, A discontinuous galerkin method for two-dimensional flow and transport in shallow water. Adv. Water Res. 25(1), 67–84 (2002)
N. Amenta, M. Bern, D. Eppstein, Optimal point placement for mesh smoothing. J. Algorithms 30(2), 302–322 (1999)
I. Babuška, A.K. Aziz, On the angle condition in the finite element method. SIAM J. Numer. Anal. 13(2), 214–226 (1976)
T.J. Baker, Mesh movement and metamorphosis. Eng. Comput. 18(3), 188–198 (2002)
R.E. Bank, R.K. Smith, Mesh smoothing using a posteriori error estimates. SIAM J. Numer. Anal. 34(3), 979–997 (1997)
R.E. Bank, A Software Package for Solving Elliptic Partial Differential Equations–Users Guide 7.0. Frontiers in Applied Mathematics, vol. 15 (SIAM, Philadelphia, 1998)
T.D. Blacker, M.B. Stephenson, Paving: a new approach to automated quadrilateral mesh generation. Int. J. Numer. Methods Eng. 32(4), 811–847 (1991)
T.D. Blacker, M.B. Stephenson, S. Canann, Analysis automation with paving: a new quadrilateral meshing technique. Adv. Eng. Softw. Work. 13(5–6), 332–337 (1991)
F.J. Blom, Considerations on the spring analogy. Int. J. Numer. Methods Fluids 32(6), 647–668 (2000)
M.L. Brewer, L.F. Diachin, P.M. Knupp, T. Leurent, D.J. Melander, The mesquite mesh quality improvement toolkit, in IMR (2003)
S.A. Canann, Y.-C. Liu, A.V. Mobley, Automatic 3d surface meshing to address today’s industrial needs. Finite Elem. Anal. Des. 25(1–2), 185–198 (1997)
S.A. Canann, J.R. Tristano, M.L. Staten et al., An approach to combined laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes, in IMR (1998), pp. 479–494. Citeseer
V.F. De Almeida, Domain deformation mapping: application to variational mesh generation. SIAM J. Sci. Comput. 20(4), 1252–1275 (1999)
Ch. Farhat, C. Degand, B. Koobus, M. Lesoinne, Torsional springs for two-dimensional dynamic unstructured fluid meshes. Comput. Methods Appl. Mech. Eng. 163(1–4), 231–245 (1998)
D.A. Field, Laplacian smoothing and delaunay triangulations. Int. J. Numer. Methods Biomed. Eng. 4(6), 709–712 (1988)
M.S. Floater, Parametrization and smooth approximation of surface triangulations. Comput. Aided Geom. Des. 14(3), 231–250 (1997)
L.A. Freitag, On combining laplacian and optimization-based mesh smoothing techniques. ASME Applied mechanics division-publications-amd, vol. 220 (1997), pp. 37–44
L.A. Freitag, P.M. Knupp, Tetrahedral mesh improvement via optimization of the element condition number. Int. J. Numer. Methods Eng. 53(6), 1377–1391 (2002)
L. Freitag, P. Plassmann, M Jones, An efficient parallel algorithm for mesh smoothing. Technical report, Argonne National Laboratory, IL (1995)
L. Freitag, M. Jones, P. Plassmann, A parallel algorithm for mesh smoothing. SIAM J. Sci. Comput. 20(6), 2023–2040 (1999)
L.A. Freitag, P. Plassmann et al., Local optimization-based simplicial mesh untangling and improvement. Int. J. Numer. Methods Eng. 49(1), 109–125 (2000)
P.-L. George, H. Borouchaki, Delaunay Triangulation and Meshing: Application to Finite Elements (Hermés Science, Paris, 1998)
C. Georgiadis, P.-A. Beaufort, J. Lambrechts, J.-F. Remacle, High quality mesh generation using cross and asterisk fields: application on coastal domains. arXiv preprint arXiv:1706.02236 (2017)
L.R. Herrmann, Laplacian-isoparametric grid generation scheme. J. Eng. Mech. Div. 102(5), 749–907 (1976)
T.R. Jensen, B. Toft, Graph Coloring Problems, vol. 39 (Wiley, New York, 2011)
R.E. Jones, Qmesh: a self-organizing mesh generation program. Technical report, Sandia Laboratories, Albuquerque, NM (1974)
J. Kim, A multiobjective mesh optimization algorithm for improving the solution accuracy of pde computations. Int. J. Comput. Methods 13(01), 1650002 (2016)
P.M. Knupp, Winslow smoothing on two-dimensional unstructured meshes. Eng. Comput. 15(3), 263–268 (1999)
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. Int. J. Numer. Methods Eng. 48(8), 1165–1185 (2000)
P.M. Knupp, Algebraic mesh quality metrics. SIAM J. Sci. Comput. 23(1), 193–218 (2001)
P. Knupp, Updating meshes on deforming domains: an application of the target-matrix paradigm. Int. J. Numer. Methods Biomed. Eng. 24(6), 467–476 (2008)
M. Křížek, On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29(2), 513–520 (1992)
J. Park, S.M. Shontz, Two derivative-free optimization algorithms for mesh quality improvement. Procedia Comput. Sci. 1(1), 387–396 (2010)
P.-O. Persson, Mesh size functions for implicit geometries and pde-based gradient limiting. Eng. Comput. 22(2), 95–109 (2006)
P.-O. Persson, G. Strang, A simple mesh generator in matlab. SIAM Rev. 46(2), 329–345 (2004)
R. Rangarajan, On the resolution of certain discrete univariate max–min problems. Comput. Optim. Appl. 68(1), 163–192 (2017)
R. Rangarajan, A.J. Lew, Provably robust directional vertex relaxation for geometric mesh optimization. SIAM J. Sci. Comput. 39(6), A2438–A2471 (2017)
M. Rumpf, A variational approach to optimal meshes. Numer. Math. 72(4), 523–540 (1996)
J. Shewchuk, What is a good linear finite element? interpolation, conditioning, anisotropy, and quality measures. Preprint. University of California at Berkeley, 73:137 (2002)
Shewchuk. Stellar: A tetrahedral mesh improvement program, 05-23-2018. Available from: https://people.eecs.berkeley.edu/~jrs/stellar/input_meshes.zip
S.M. Shontz, S.A. Vavasis, A mesh warping algorithm based on weighted laplacian smoothing, in IMR (2003), pp. 147–158
H. Xu, T.S. Newman, 2D FE quad mesh smoothing via angle-based optimization, in International Conference on Computational Science (Springer, Berlin, 2005), pp. 9–16
K. Xu, X. Gao, G. Chen, Hexahedral mesh quality improvement via edge-angle optimization. Comput. Graph. 70, 17–27 (2018)
P.D. Zavattieri, E.A. Dari, G.C. Buscaglia, Optimization strategies in unstructured mesh generation. Int. J. Numer. Methods Eng. 39(12), 2055–2071 (1996)
T. Zhou, K. Shimada, An angle-based approach to two-dimensional mesh smoothing, in IMR (2000), pp. 373–384
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Zint, D., Grosso, R. (2019). Discrete Mesh Optimization on GPU. In: Roca, X., Loseille, A. (eds) 27th International Meshing Roundtable. IMR 2018. Lecture Notes in Computational Science and Engineering, vol 127. Springer, Cham. https://doi.org/10.1007/978-3-030-13992-6_24
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DOI: https://doi.org/10.1007/978-3-030-13992-6_24
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