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Distributed-memory parallelization of the aggregated unfitted finite element method. (English) Zbl 1442.65280

Summary: The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65Y05 Parallel numerical computation

Software:

FEMPAR
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Full Text: DOI arXiv

References:

[1] Sukumar, N.; Moës, N.; Moran, B.; Belytschko, T., Extended finite element method for three-dimensional crack modelling, Internat. J. Numer. Methods Engrg., 48, 11, 1549-1570 (2000) · Zbl 0963.74067
[2] Massing, A.; Larson, M. G.; Logg, A.; Rognes, M. E., A Nitsche-based cut finite element method for a fluid-structure interaction problem, Commun. Appl. Math. Comput. Sci., 10, 2, 97-120 (2015) · Zbl 1326.74122
[3] Sauerland, H.; Fries, T. P., The extended finite element method for two-phase and free-surface flows: A systematic study, J. Comput. Phys., 230, 9, 3369-3390 (2011) · Zbl 1316.76050
[4] Burman, E.; Elfverson, D.; Hansbo, P.; Larson, M. G.; Larsson, K., Shape optimization using the cut finite element method, Comput. Methods Appl. Mech. Engrg., 328, 242-261 (2018) · Zbl 1439.74316
[5] Chiumenti, M.; Neiva, E.; Salsi, E.; Cervera, M.; Badia, S.; Moya, J.; Chen, Z.; Lee, C.; Davies, C., Numerical modelling and experimental validation in selective laser melting, Addit. Manuf., 18, 171-185 (2017)
[6] Nguyen, L.; Stoter, S.; Baum, T.; Kirschke, J.; Ruess, M.; Yosibash, Z.; Schillinger, D., Phase-field boundary conditions for the voxel finite cell method: surface-free stress analysis of CT-based bone structures, Int. J. Numer. Methods Biomed. Eng., 33, 12, Article e2880 pp. (2017)
[7] Burman, E.; Claus, S.; Hansbo, P.; Larson, M. G.; Massing, A., Cutfem: Discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg., 104, 7, 472-501 (2015) · Zbl 1352.65604
[8] Elfverson, D.; Larson, M. G.; Larsson, K., Cutiga with basis function removal, Adv. Model. Simul. Eng. Sci., 5, 1, 6 (2018)
[9] Schillinger, D.; Ruess, M., The finite cell method: A review in the context of higher-order structural analysis of CAD and image-based geometric models, Arch. Comput. Methods Eng., 22, 3, 391-455 (2015) · Zbl 1348.65056
[10] Badia, S.; Verdugo, F.; Martín, A. F., The aggregated unfitted finite element method for elliptic problems, Comput. Methods Appl. Mech. Engrg., 336, 533-553 (2018) · Zbl 1440.65175
[11] Nadal, E.; Ródenas, J. J.; Albelda, J.; Tur, M.; Tarancón, J. E.; Fuenmayor, F. J., Efficient finite element methodology based on Cartesian grids: application to structural shape optimization, Abstr. Appl. Anal., 2013, 1-19 (2013) · Zbl 1328.74081
[12] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 1, 239-261 (2005) · Zbl 1117.76049
[13] Sanches, R. A.K.; Bornemann, P. B.; Cirak, F., Immersed b-spline (i-spline) finite element method for geometrically complex domains, Comput. Methods Appl. Mech. Engrg., 200, 13-16, 1432-1445 (2011) · Zbl 1228.74097
[14] Sukumar, N.; Chopp, D. L.; Moës, N.; Belytschko, T., Modeling holes and inclusions by level sets in the extended finite-element method, Comput. Methods Appl. Mech. Engrg., 190, 46-47, 6183-6200 (2001) · Zbl 1029.74049
[15] de Prenter, F.; Verhoosel, C. V.; van Zwieten, G. J.; van Brummelen, E. H., Condition number analysis and preconditioning of the finite cell method, Comput. Methods Appl. Mech. Engrg., 316, 297-327 (2017) · Zbl 1439.65137
[16] Davis, T. A., Direct Methods for Sparse Linear Systems, 217 (2006), Society for Industrial and Applied Mathematics · Zbl 1119.65021
[17] Saad, Y., Iterative Methods for Sparse Linear Systems, Second Edition (2003), Society for Industrial and Applied Mathematics · Zbl 1031.65046
[18] Briggs, W. L.; Henson, V. E.; McCormick, S. F., A multigrid tutorial, second edition (2000), Society for Industrial and Applied Mathematics · Zbl 0958.65128
[19] Chow, E.; Falgout, R. D.; Hu, J. J.; Tuminaro, R. S.; Yang, U. M., 10. a survey of parallelization techniques for multigrid solvers, (Parallel Processing for Scientific Computing (2006), Society for Industrial and Applied Mathematics), 179-201
[20] Wesseling, P.; Oosterlee, C. W., Geometric multigrid with applications to computational fluid dynamics, J. Comput. Appl. Math., 128, 1-2, 311-334 (2001) · Zbl 0989.76069
[21] Ruge, J. W.; Stüben, K., 4. algebraic multigrid, (Multigrid Methods (1987), Society for Industrial and Applied Mathematics), 73-130 · Zbl 0659.65094
[22] Vanek, P.; Brezina, M.; Mandel, J., Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math., 88, 3, 559-579 (2001) · Zbl 0992.65139
[23] Badia, S.; Martín, A. F.; Principe, J., Multilevel balancing domain decomposition at extreme scales, SIAM J. Sci. Comput., 38, 1, C22-C52 (2016) · Zbl 1334.65217
[24] Toselli, A.; Widlund, O. B., Domain Decomposition Methods — Algorithms and Theory, Springer Series in Computational Mathematics (2005), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg · Zbl 1069.65138
[25] Berger-Vergiat, L.; Waisman, H.; Hiriyur, B.; Tuminaro, R.; Keyes, D., Inexact Schwarz-algebraic multigrid preconditioners for crack problems modeled by extended finite element methods, Internat. J. Numer. Methods Engrg., 90, 3, 311-328 (2012) · Zbl 1242.74094
[26] Hiriyur, B.; Tuminaro, R.; Waisman, H.; Boman, E.; Keyes, D., A quasi-algebraic multigrid approach to fracture problems based on extended finite elements, SIAM J. Sci. Comput., 34, 2, A603-A626 (2012) · Zbl 1390.74181
[27] J.N. Jomo, F.D. Prenter, M. Elhaddad, D.D. Angella, C.V. Verhoosel, S. Kollmannsberger, J.S. Kirschke, E.H.V. Brummelen, E. Rank, Robust and parallel scalable iterative solutions for large-scale finite cell analyses, Arxiv, pages 1-32, 2018.; J.N. Jomo, F.D. Prenter, M. Elhaddad, D.D. Angella, C.V. Verhoosel, S. Kollmannsberger, J.S. Kirschke, E.H.V. Brummelen, E. Rank, Robust and parallel scalable iterative solutions for large-scale finite cell analyses, Arxiv, pages 1-32, 2018.
[28] Menk, A.; Bordas, S. P.A., A robust preconditioning technique for the extended finite element method, Internat. J. Numer. Methods Engrg., 85, 13, 1609-1632 (2011) · Zbl 1217.74128
[29] Badia, S.; Verdugo, F., Robust and scalable domain decomposition solvers for unfitted finite element methods, J. Comput. Appl. Math., 344, 740-759 (2018) · Zbl 1462.65216
[30] Mandel, J.; Dohrmann, C. R., Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10, 7, 639-659 (2003) · Zbl 1071.65558
[31] Heroux, M. A.; Phipps, E. T.; Salinger, A. G.; Thornquist, H. K.; Tuminaro, R. S.; Willenbring, J. M.; Williams, A.; Stanley, K. S.; Bartlett, R. A.; Howle, V. E.; Hoekstra, R. J.; Hu, J. J.; Kolda, T. G.; Lehoucq, R. B.; Long, K. R.; Pawlowski, R. P., An overview of the trilinos project, ACM Trans. Math. Software, 31, 3, 397-423 (2005) · Zbl 1136.65354
[32] S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, D. May, L.C. McInnes, R.T. Mills, T. Munson, K. Rupp, P. Sanan, B. Smith, S. Zampini, H. Zhang, H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc; S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, D. May, L.C. McInnes, R.T. Mills, T. Munson, K. Rupp, P. Sanan, B. Smith, S. Zampini, H. Zhang, H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc
[33] Balay, S.; Abhyankar, S.; Adams, M.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Dener, A.; Eijkhout, V.; Gropp, W.; Kaushik, D.; Knepley, M.; May, D.; McInnes, L. C.; Mills, R. T.; Munson, T.; Rupp, K.; Sanan, P.; Smith, B.; Zampini, S.; Zhang, H.; Zhang, H., PETSc Users Manual, Technical Report ANL-95/11 - Revision 3.10 (2018), Argonne National Laboratory
[34] Badia, S.; Martín, A. F.; Verdugo, F., Mixed aggregated finite element methods for the unfitted discretization of the Stokes problem, SIAM J. Sci. Comput., 40, 6, B1541-B1576 (2018) · Zbl 1412.65184
[35] Badia, S.; Martín, A. F.; Principe, J., FEMPAR: An object-oriented parallel finite element framework, Arch. Comput. Methods Eng., 25, 2, 195-271 (2018) · Zbl 1392.65005
[36] Helzel, C.; Berger, M. J.; Leveque, R. J., A high-resolution rotated grid method for conservation laws with embedded geometries, SIAM J. Sci. Comput., 26, 3, 785-809 (2005) · Zbl 1074.35071
[37] Johansson, A.; Larson, M. G., A high order discontinuous Galerkin nitsche method for elliptic problems with fictitious boundary, Numer. Math., 123, 4, 607-628 (2013) · Zbl 1269.65126
[38] Kummer, F., Extended discontinuous Galerkin methods for two-phase flows: the spatial discretization, Internat. J. Numer. Methods Engrg., 109, 2, 259-289 (2017)
[39] GAMG online documetnation, https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/PC/PCGAMG.html; GAMG online documetnation, https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/PC/PCGAMG.html
[40] Barcelona Supercomputing Center home page, https://www.bsc.es/; Barcelona Supercomputing Center home page, https://www.bsc.es/
[41] Burstedde, C.; Wilcox, L. C.; Ghattas, O., P4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees, SIAM J. Sci. Comput., 33, 3, 1103-1133 (2011) · Zbl 1230.65106
[42] Marco, O.; Sevilla, R.; Zhang, Y.; Ródenas, J. J.; Tur, M., Exact 3D boundary representation in finite element analysis based on Cartesian grids independent of the geometry, Internat. J. Numer. Methods Engrg., 103, 6, 445-468 (2015) · Zbl 1352.65592
[43] M. Olm, S. Badia, A.F. Martín, On a general implementation of \(h\)- and \(p\)-adaptive curl-conforming finite elements 2018.; M. Olm, S. Badia, A.F. Martín, On a general implementation of \(h\)- and \(p\)-adaptive curl-conforming finite elements 2018.
[44] Shephard, M. S., Linear multipoint constraints applied via transformation as part of a direct stiffness assembly process, Internat. J. Numer. Methods Engrg., 20, 11, 2107-2112 (1984) · Zbl 0547.73069
[45] S. Badia, A.F. Martín, E. Neiva, F. Verdugo, A generic finite element framework on parallel tree-based adaptive meshes, Arxiv, 2019.; S. Badia, A.F. Martín, E. Neiva, F. Verdugo, A generic finite element framework on parallel tree-based adaptive meshes, Arxiv, 2019.
[46] Bader, M., Space-filling curves: An introduction with applications in scientific computing, ((Texts in Computational Science and Engineering) (2012)), 285
[47] Karypis, G., METIS And parmetis, (Padua, D., Encyclopedia of Parallel Computing (2011), Springer US: Springer US Boston, MA), 1117-1124 · Zbl 1231.68001
[48] Bangerth, W.; Burstedde, C.; Heister, T.; Kronbichler, M., Algorithms and data structures for massively parallel generic adaptive finite element codes, ACM Trans. Math. Software, 38, 2, 14:1-14:28 (2012) · Zbl 1365.65247
[49] Badia, S.; Martín, A. F.; Principe, J., Implementation and scalability analysis of balancing domain decomposition methods, Arch. Comput. Methods Eng., 20, 3, 239-262 (2013) · Zbl 1354.65261
[50] MareNostrum4 User’s Guide. Technical report. Barcelona Supercomputing Centre, 2018.; MareNostrum4 User’s Guide. Technical report. Barcelona Supercomputing Centre, 2018.
[51] Becker, R., Mesh adaptation for Dirichlet flow control via nitsche’s method, Commun. Numer. Methods. Eng., 18, 9, 669-680 (2002) · Zbl 1073.76582
[52] Nitsche, J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hambg, 36, 1, 9-15 (1971) · Zbl 0229.65079
[53] Massing, A.; Larson, M. G.; Logg, A.; Rognes, M. E., A stabilized nitsche fictitious domain method for the Stokes problem, J. Sci. Comput., 61, 3, 604-628 (2014) · Zbl 1417.76028
[54] May, D. A.; Sanan, P.; Rupp, K.; Knepley, M. G.; Smith, B. F., Extreme-scale multigrid components within PETSc, Proceedings of the Platform for Advanced Scientific Computing Conference on - PASC ’16, 1-12 (2016)
[55] Hu, J.; Tuminaro, R.; Adams, M. F.; Brezina, M., Parallel multigrid smoothing: Polynomial versus Gauss-seidel, J. Comput. Phys., 188, 2, 593-610 (2003) · Zbl 1022.65030
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