×

A unified framework for the computational comparison of adaptive mesh refinement strategies for all-quadrilateral and all-hexahedral meshes: locally adaptive multigrid methods versus h-adaptive methods. (English) Zbl 07505904

Summary: This paper provides a detailed comparison in a solids mechanics context of adaptive mesh refinement methods for all-quadrilateral and all-hexahedral meshes. The adaptive multigrid Local Defect Correction method and the well-known hierarchical \(h\)-adaptive refinement techniques are placed into a generic algorithmic setting for an objective numerical comparison. Such a comparison is of great interest as local multigrid AMR approaches are from now rarely employed to adaptively solve implicit systems in solid mechanics. However they present various interesting features mainly related to their intrinsic idea of partitioning the degrees of freedom on different mesh levels. For this study, we rely on a fully-automatic mesh refinement algorithm providing the desired refined mesh directly from the user-prescribed accuracy. The refinement process is driven by an a posteriori error estimator combined to mesh optimality criteria. In this study, the most efficient strategy based on mesh optimality criterion and refinement ratio is identified for all-quadrilateral and all-hexahedral finite elements meshes. The quality of refined meshes is finally appreciated in term of number of nodes but also through the verification of final solution’s accuracy. A special attention is devoted to the fulfillment of local precisions which are of great importance from an engineering point of view. Numerical \(2D\) and \(3D\) experiments of different complexities revealing local phenomena enable to highlight the essential features of the considered mesh refinement methods within an elastostatic framework. This study points out the great potentialities of locally adaptive multigrid method, which clearly appears to be the most powerful strategy in terms of standard metrics of efficiency (dimension of systems to be solved, storage requirements, CPU time).

MSC:

65Nxx Numerical methods for partial differential equations, boundary value problems
74Sxx Numerical and other methods in solid mechanics
35Jxx Elliptic equations and elliptic systems

Software:

Cast3M; Isabelle
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Demkowicz, L.; Devloo, Ph.; Oden, J. T., On an h-type mesh-refinement strategy based on minimization of interpolation errors, Comput. Methods Appl. Mech. Eng., 53, 1, 67-89 (1985) · Zbl 0556.73081
[2] Babuška, I.; Szabo, B., On the rates of convergence of the finite element method, Int. J. Numer. Methods Eng., 18, 3, 323-341 (1982) · Zbl 0498.65050
[3] Ghosh, S.; Manna, S. K., R-adapted arbitrary Lagrangian-Eulerian finite element method in metal-forming simulation, J. Mater. Eng. Perform., 2, 2, 271-282 (1993)
[4] Fish, J., The s-version of the finite element method, Comput. Struct., 43, 3, 539-547 (1992) · Zbl 0775.73247
[5] Babuška, I.; Guo, B. Q., The h, p and h-p version of the finite element method; basis theory and applications, Adv. Eng. Softw., 15, 3, 159-174 (1992) · Zbl 0769.65078
[6] Belytschko, T.; Tabbara, M., H-adaptive finite element methods for dynamic problems, with emphasis on localization, Int. J. Numer. Methods Eng., 36, 24, 4245-4265 (1993) · Zbl 0794.73071
[7] Dìez, P.; Huerta, A., A unified approach to remeshing strategies for finite element h-adaptivity, Comput. Methods Appl. Mech. Eng., 176, 1, 215-229 (1999) · Zbl 0942.74071
[8] Babuška, I.; Miller, A.; Vogelius, M., Adaptive methods and error estimation for elliptic problems of structural mechanics, (Adaptive Computational Methods for Partial Differential Equations (1983)), 20 · Zbl 0581.73080
[9] D. Davydov, J-P. Pelteret, D. Arndt, M. Kronbichler, P. Steinmann, A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid, Int. J. Numer. Methods Eng. n/a(n/a).
[10] Brandt, A., Rigorous quantitative analysis of multigrid I: constant coefficients two-level cycle with l2-norm, SIAM J. Numer. Anal., 31, 6, 1695-1730 (1994) · Zbl 0817.65126
[11] Bai, D.; Brandt, A., Local mesh refinement multilevel techniques, SIAM J. Sci. Stat. Comput., 8, 2, 109-134 (mar 1987) · Zbl 0619.65091
[12] Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 3, 484-512 (1984) · Zbl 0536.65071
[13] Khadra, K.; Angot, Ph.; Caltagirone, J. P., A comparison of locally adaptive multigrid methods: L.D.C., F.A.C., and F.I.C., (Melson, N. D.; McCormick, S. F.; Manteuffel, T. A., NASA Conference Publication 3224, 6th Copper Mountain Conference on Multigrid Methods, vol. 1 (1993)), 275-292
[14] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comput., 31, 138, 333-390 (1977) · Zbl 0373.65054
[15] Biotteau, E.; Gravouil, A.; Lubrecht, A. A.; Combescure, A., Multigrid solver with automatic mesh refinement for transient elastoplastic dynamic problems, Int. J. Numer. Methods Eng., 84, 8, 947-971 (2010) · Zbl 1202.74068
[16] Biotteau, E.; Gravouil, A.; Lubrecht, T.; Combescure, A., Three dimensional automatic refinement method for transient small strain elastoplastic finite element computations, Comput. Mech., 49, 123-136 (2012) · Zbl 1356.74031
[17] Barbié, L.; Ramière, I.; Lebon, F., Strategies around the local defect correction multi-level refinement method for three-dimensional linear elastic problems, Comput. Struct., 130, 73-90 (2014)
[18] Barbié, L.; Ramière, I.; Lebon, F., An automatic multilevel refinement technique based on nested local meshes for nonlinear mechanics, Comput. Struct., 147, 14-25 (2015)
[19] Liu, H.; Ramière, I.; Lebon, F., On the coupling of local multilevel mesh refinement and ZZ methods for unilateral frictional contact problems in elastostatics, Comput. Methods Appl. Mech. Eng., 323, 1-26 (2017) · Zbl 1439.74219
[20] Hackbusch, W., Local defect correction method and domain decomposition techniques, (Defect Correction Methods: Theory and Applications (1984), Springer: Springer Vienna), 89-113 · Zbl 0552.65070
[21] Ferket, P. J.J.; Reusken, A. A., Further analysis of the local defect correction method, Computing, 56, 117-139 (1996) · Zbl 0843.65072
[22] (2020), AMReX - a software framework designed for building massively parallel block-structured adaptive mesh refinement (amr) applications
[23] (2020), ExaHyPE- an Exascale Hyperbolic PDE Engine
[24] Nicolas, G.; Fouquet, T.; Geniaut, S.; Cuvilliezam, S., Improved adaptive mesh refinement for conformal hexahedral meshes, Adv. Eng. Softw., 102, 14-28 (2016)
[25] Ramière, I.; Angot, Ph.; Belliard, M., A fictitious domain approach with spread interface for elliptic problems with general boundary conditions, Comput. Methods Appl. Mech. Eng., 196, 4, 766-781 (2007) · Zbl 1121.65364
[26] Düster, A.; Parvizian, J.; Yang, Z.; Rank, E., The finite cell method for three-dimensional problems of solid mechanics, Comput. Methods Appl. Mech. Eng., 197, 45, 3768-3782 (2008) · Zbl 1194.74517
[27] Belytschko, T.; Parimi, Ch.; Moës, N.; Sukumar, N.; Usui, Sh., Structured extended finite element methods for solids defined by implicit surfaces, Int. J. Numer. Methods Eng., 56, 4, 609-635 (2003) · Zbl 1038.74041
[28] Wang, E.; Nelson, Th.; Rauch, R., Back to elements - tetrahedra vs. hexahedra, (Proceedings of the 2004 International ANSYS Conference (2004))
[29] Biswas, R.; Strawn, R. C., Tetrahedral and hexahedral mesh adaptation for cfd problems, Appl. Numer. Math., 26, 1, 135-151 (1998) · Zbl 0905.76044
[30] Huo, S. H.; Li, Y. S.; Duan, S. Y.; Han, X.; Liu, G. R., Novel quadtree algorithm for adaptive analysis based on cell-based smoothed finite element method, Eng. Anal. Bound. Elem., 106, 541-554 (2019) · Zbl 1464.74180
[31] Michel, B.; Nonon, C.; Sercombe, J.; Michel, F.; Marelle, V., Simulation of pellet-cladding interaction with the pleiades fuel performance software environment, Nucl. Technol., 182, 124 (2013)
[32] Zhu, J. Z.; Hinton, E.; Zienkiewicz, O. C., Mesh enrichment against mesh regeneration using quadrilateral elements, Commun. Numer. Methods Eng., 9, 7, 547-554 (1993) · Zbl 0796.65125
[33] Ródenas, J.; Albelda, J.; Tur, M.; Fuenmayor, F., A hierarchical h-adaptivity methodology based on element subdivision, Rev. UIS Ing., 16, 263-280 (2017)
[34] Schneiders, R., Refining quadrilateral and hexahedral element meshes, (Fifth International Meshing Roundtable, vol. 1 (1998))
[35] Tchon, K.; Dompierre, J.; Camarero, R., Conformal refinement of all-quadrilateral and all-hexahedral meshes according to an anisotropic metric, (11th International Meshing Roundtable (2002)), 231-242
[36] Ledoux, F.; Shepherd, J. F., Topological modifications of hexahedral meshes via sheet operations: a theoretical study, Eng. Comput., 26, 433-447 (2009)
[37] Melander, D. J.; Benzley, S. E.; Tautges, T. J., Generation of Multi-Million Element Meshes for Solid Model-Based Geometries: The Dicer Algorithm (1997)
[38] Merkley, K.; Ernst, C.; Shepherd, J. F.; Borden, M. J., Methods and applications of generalized sheet insertion for hexahedral meshing, (Brewer, Michael L.; Marcum, David, Proceedings of the 16th International Meshing Roundtable (2008), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 233-250 · Zbl 1134.65321
[39] Staten, M. L.; Shepherd, J. F.; Shimada, K., Mesh matching – creating conforming interfaces between hexahedral meshes, (Garimella, Rao V., Proceedings of the 17th International Meshing Roundtable (2008), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 467-484
[40] Mitchell, W. F.; McClain, M. A., A comparison of hp-adaptive strategies for elliptic partial differential equations, ACM Trans. Math. Softw., 41, 1, 1-39 (2014) · Zbl 1369.65148
[41] Hennig, P.; Kästner, M.; Morgenstern, Ph.; Peterseim, D., Adaptive mesh refinement strategies in isogeometric analysis — a computational comparison, Comput. Methods Appl. Mech. Eng., 316, 424-448 (2017), Special Issue on Isogeometric Analysis: Progress and Challenges · Zbl 1439.65169
[42] Ehlers, W.; Ammann, M.; Diebels, S., H-adaptive FE methods applied to single- and multiphase problems, Int. J. Numer. Methods Eng., 54, 2, 219-239 (2002) · Zbl 1098.74689
[43] Zienkiewicz, O. C.; Zhu, J. Z., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Eng., 24, 2, 337-357 (1987) · Zbl 0602.73063
[44] Oñate, E.; Bugeda, G., Mesh optimality criteria for adaptive finite element computations, Eng. Comput., 307-321 (1993)
[45] Ramière, I.; Liu, H.; Lebon, F., Original geometrical stopping criteria associated to multilevel adaptive mesh refinement for problems with local singularities, Comput. Mech. (2019) · Zbl 1470.74076
[46] Babuška, I.; Strouboulis, T.; Gangaraj, S. K.; Upadhyay, C. S., Pollution error in the h-version of the finite element method and the local quality of the recovered derivatives, Comput. Methods Appl. Mech. Eng., 140, 1, 1-37 (1997) · Zbl 0896.73055
[47] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Method (1991), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0788.73002
[48] Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O., Toward a universal h-p adaptive finite element strategy, part 1. Constrained approximation and data structure, Comput. Methods Appl. Mech. Eng., 77, 1, 79-112 (1989) · Zbl 0723.73074
[49] Babuška, I.; Zlamal, M., Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal., 10, 5, 863-875 (1973) · Zbl 0237.65066
[50] Bernardi, C.; Maday, Y.; Patera, A. T., Domain decomposition by the mortar element method, (Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters (1993), Springer: Springer Netherlands, Dordrecht), 269-286 · Zbl 0799.65124
[51] Červený, J.; Dobrev, V.; Kolev, T., Nonconforming mesh refinement for high-order finite elements, SIAM J. Sci. Comput., 41, 4, C367-C392 (2019) · Zbl 1471.65210
[52] Graziadei, M.; Mattheij, R. M.M.; ten Thije Boonkkamp, J. H.M., Local defect correction with slanting grids, Numer. Methods Partial Differ. Equ., 20, 1, 1-17 (2004) · Zbl 1038.65122
[53] Koliesnikova, D.; Ramière, I.; Lebon, F., Analytical comparison of two multiscale coupling methods for nonlinear solid mechanics, J. Appl. Mech., 87, 9, Article 094501 pp. (2020)
[54] Strouboulis, T.; Haque, K. A., Recent experiences with error estimation and adaptivity. Part II: error estimation for h-adaptive approximations on grids of triangles and quadrilaterals, Comput. Methods Appl. Mech. Eng., 100, 3, 359-430 (1992) · Zbl 0782.65127
[55] Grätsch, T.; Bathe, Kl.-J., Review: a posteriori error estimation techniques in practical finite element analysis, Comput. Struct., 83, 4-5, 235-265 (2005)
[56] Özakça, M., Comparison of error estimation methods and adaptivity for plane stress/strain problems, Struct. Eng. Mech., 15, 579-608 (2003)
[57] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique, Int. J. Numer. Methods Eng., 33, 7, 1331-1364 (1992) · Zbl 0769.73084
[58] Fournier, D.; Herbin, R.; Tellier, R. Le., Discontinuous Galerkin discretization and \(hp\)-refinement for the resolution of the neutron transport equation, SIAM J. Sci. Comput., 35, 2, 936-956 (2013) · Zbl 1267.74119
[59] Di Pietro, D. A.; Vohralik, M.; Yousef, S., An a posteriori-based, fully adaptive algorithm with adaptive stopping criteria and mesh refinement for thermal multiphase compositional flows in porous media, Comput. Math. Appl., 68, 12, Part B, 2331-2347 (2014), Advances in Computational Partial Differential Equations · Zbl 1362.65097
[60] Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33, 3, 1106-1124 (1996) · Zbl 0854.65090
[61] Morin, P.; Siebert, K. G.; Veeser, A., A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 18, 05, 707-737 (2008) · Zbl 1153.65111
[62] Bugeda, G., A comparison between new adaptive remeshing strategies based on point wise stress error estimation and energy norm error estimation, Commun. Numer. Methods Eng., 18, 7, 469-482 (2002) · Zbl 1051.74042
[63] Ladevèze, P.; Pelle, J. P.; Rougeot, P., Error estimation and mesh optimization for classical finite elements, Eng. Comput., 8, 69-80 (1991)
[64] Bugeda, G.; Oliver, J., A general methodology for structural shape optimization problems using automatic adaptive remeshing, Int. J. Numer. Methods Eng., 36, 18, 3161-3185 (1993) · Zbl 0780.73049
[65] Fournier, D.; Herbin, R.; Le Tellier, R., Discontinuous Galerkin discretization and \(h - p\)-refinement for the resolution of the neutron transport equation, SIAM J. Sci. Comput., 35, A936-A956 (2013) · Zbl 1267.74119
[66] Faucher, V.; Casadei, F.; Valsamosand, G.; Larcher, M., High resolution adaptive framework for fast transient fluid-structure interaction with interfaces and structural failure – application to failing tanks under impact, Int. J. Impact Eng., 127, 62-85 (2018)
[67] Ciarlet, Ph. G., Finite Element Method for Elliptic Problems (2002), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA
[68] Woodbury, A. C.; Staten, M. L.; Shepherd, J. F.; Benzley, S. E., Localized coarsening of conforming all-hexahedral meshes, Eng. Comput., 27, 1, 95-104 (2011)
[69] Bank, R.; Sherman, A., Some refinement algorithms and data structures for regular local mesh refinement, (Applications of Mathematics and Computing to the Physical Sciences (1999))
[70] Nochetto, R. H.; Veeser, A., Primer of Adaptive Finite Element Methods (2012), Springer Berlin Heidelberg · Zbl 1252.65192
[71] CEA, Cast3m (2020)
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.