Robust high-order unfitted finite elements by interpolation-based discrete extension. (English) Zbl 1524.65749
Summary: In this work, we propose a novel formulation for the solution of partial differential equations using finite element methods on unfitted meshes. The proposed formulation relies on the discrete extension operator proposed in the aggregated finite element method. This formulation is robust with respect to the location of the boundary/interface within the cell. One can prove enhanced stability results, not only on the physical domain, but on the whole active mesh. However, the stability constants grow exponentially with the polynomial order being used, since the underlying extension operators are defined via extrapolation. To address this issue, we introduce a new variant of aggregated finite elements, in which the extension in the physical domain is an interpolation for polynomials of order higher than two. As a result, the stability constants only grow at a polynomial rate with the order of approximation. We demonstrate that this approach enables robust high-order approximations with the aggregated finite element method. The proposed method is consistent, optimally convergent, and with a condition number that scales optimally for high order approximation.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74B10 | Linear elasticity with initial stresses |
Keywords:
embedded methods; immersed methods; unfitted finite elements; high-order finite elements; aggregated finite elementsReferences:
| [1] | Waisman, H.; Berger-Vergiat, L., An adaptive domain decomposition preconditioner for crack propagation problems modeled by XFEM, Int. J. Multiscale Comput. Eng., 11, 6, 633-654 (2013) |
| [2] | Alauzet, F.; Fabrèges, B.; Fernández, M. A.; Landajuela, M., Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures, Comput. Methods Appl. Mech. Eng., 301, 300-335 (2016) · Zbl 1423.76201 |
| [3] | 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 |
| [4] | Kirchhart, M.; Gross, S.; Reusken, A., Analysis of an XFEM discretization for Stokes interface problems, SIAM J. Sci. Comput., 38, 2, A1019-A1043 (2016) · Zbl 1381.76182 |
| [5] | Badia, S.; Caicedo, M. A.; Martín, A. F.; Principe, J., A robust and scalable unfitted adaptive finite element framework for nonlinear solid mechanics, Comput. Methods Appl. Mech. Eng., 386, Article 114093 pp. (2021) · Zbl 1507.74446 |
| [6] | Burman, E.; Elfverson, D.; Hansbo, P.; Larson, M. G.; Larsson, K., Shape optimization using the cut finite element method, Comput. Methods Appl. Mech. Eng., 328, 242-261 (2018) · Zbl 1439.74316 |
| [7] | Neiva, E.; Chiumenti, M.; Cervera, M.; Salsi, E.; Piscopo, G.; Badia, S.; Martín, A. F.; Chen, Z.; Lee, C.; Davies, C., Numerical modelling of heat transfer and experimental validation in powder-bed fusion with the virtual domain approximation, Finite Elem. Anal. Des., 168, Article 103343 pp. (2020) |
| [8] | Carraturo, M.; Jomo, J.; Kollmannsberger, S.; Reali, A.; Auricchio, F.; Rank, E., Modeling and experimental validation of an immersed thermo-mechanical part-scale analysis for laser powder bed fusion processes, Addit. Manuf., 36, Article 101498 pp. (2020) |
| [9] | Badia, S.; Hampton, J.; Principe, J., Embedded multilevel Monte Carlo for uncertainty quantification in random domains, Int. J. Uncertain. Quantificat., 11, 1, 119-142 (2021) · Zbl 1498.65004 |
| [10] | 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. Eng., 316, 297-327 (2017) · Zbl 1439.65137 |
| [11] | Badia, S.; Verdugo, F.; Martín, A. F., The aggregated unfitted finite element method for elliptic problems, Comput. Methods Appl. Mech. Eng., 336, 533-553 (2018) · Zbl 1440.65175 |
| [12] | Neiva, E.; Badia, S., Robust and scalable h-adaptive aggregated unfitted finite elements for interface elliptic problems, Comput. Methods Appl. Mech. Eng., 380, Article 113769 pp. (2021) · Zbl 1506.65221 |
| [13] | 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 |
| [14] | Kummer, F., Extended discontinuous Galerkin methods for two-phase flows: the spatial discretization, Int. J. Numer. Methods Eng., 109, 2, 259-289 (2017) · Zbl 07874355 |
| [15] | Lehrenfeld, C., High order unfitted finite element methods on level set domains using isoparametric mappings, Comput. Methods Appl. Mech. Eng., 300, 716-733 (2016) · Zbl 1425.65168 |
| [16] | Guzmán, J.; Sánchez, M. A.; Sarkis, M., A finite element method for high-contrast interface problems with error estimates independent of contrast, J. Sci. Comput., 73, 1, 330-365 (2017) · Zbl 1380.65369 |
| [17] | Li, K.; Atallah, N. M.; Main, G. A.; Scovazzi, G., The shifted interface method: a flexible approach to embedded interface computations, Int. J. Numer. Methods Eng., 121, 3, 492-518 (2020) · Zbl 07843207 |
| [18] | Elhaddad, M.; Zander, N.; Bog, T.; Kudela, L.; Kollmannsberger, S.; Kirschke, J.; Baum, T.; Ruess, M.; Rank, E., Multi-level hp-finite cell method for embedded interface problems with application in biomechanics, Int. J. Numer. Methods Biomed. Eng., 34, 4, Article e2951 pp. (2018) |
| [19] | Xu, F.; Schillinger, D.; Kamensky, D.; Varduhn, V.; Wang, C.; Hsu, M.-C., The tetrahedral finite cell method for fluids: immersogeometric analysis of turbulent flow around complex geometries, Comput. Fluids, 141, 135-154 (2016) · Zbl 1390.76372 |
| [20] | Jomo, J. N.; Prenter, F. D.; Elhaddad, M.; Angella, D. D.; Verhoosel, C. V.; Kollmannsberger, S.; Kirschke, J. S.; Brummelen, E. H.V.; Rank, E., Robust and parallel scalable iterative solutions for large-scale finite cell analyses, Finite Elem. Anal. Des., 163, 14-30 (2019) |
| [21] | Hubrich, S.; Düster, A., Numerical integration for nonlinear problems of the finite cell method using an adaptive scheme based on moment fitting, Comput. Math. Appl., 77, 7, 1983-1997 (2019) · Zbl 1442.65024 |
| [22] | 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 |
| [23] | Dauge, M.; Düster, A.; Rank, E., Theoretical and numerical investigation of the finite cell method, J. Sci. Comput., 65, 3, 1039-1064 (2015) · Zbl 1331.65160 |
| [24] | Larsson, K.; Kollmannsberger, S.; Rank, E.; Larson, M. G., The finite cell method with least squares stabilized Nitsche boundary conditions, Comput. Methods Appl. Mech. Eng., 393, Article 114792 pp. (2022) · Zbl 1507.65244 |
| [25] | Elfverson, D.; Larson, M. G.; Larsson, K., CutIGA with basis function removal, Adv. Model. Simul. Eng. Sci., 5, 1, 6 (2018) |
| [26] | Burman, E., Ghost penalty, C. R. Math., 348, 21-22, 1217-1220 (2010) · Zbl 1204.65142 |
| [27] | Burman, E.; Claus, S.; Hansbo, P.; Larson, M. G.; Massing, A., CutFEM: discretizing geometry and partial differential equations, Int. J. Numer. Methods Eng., 104, 7, 472-501 (2015) · Zbl 1352.65604 |
| [28] | Hoang, T.; Verhoosel, C. V.; Qin, C.-Z.; Auricchio, F.; Reali, A.; van Brummelen, E. H., Skeleton-stabilized immersogeometric analysis for incompressible viscous flow problems, Comput. Methods Appl. Mech. Eng., 344, 421-450 (2019) · Zbl 1440.76069 |
| [29] | 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 |
| [30] | Helzel, C.; Berger, M.; Leveque, R., A high-resolution rotated grid method for conservation laws with embedded geometries, SIAM J. Sci. Comput., 26, 3, 785-809 (2005) · Zbl 1074.35071 |
| [31] | Müller, B.; Krämer-Eis, S.; Kummer, F.; Oberlack, M., A high-order discontinuous Galerkin method for compressible flows with immersed boundaries, Int. J. Numer. Methods Eng., 110, 1, 3-30 (2017) · Zbl 1380.65384 |
| [32] | Chu, B.-D.; Martin, F.; Reif, U., Stabilization of spline bases by extension, Adv. Comput. Math., 48, 3, 23 (2022) · Zbl 1504.41007 |
| [33] | 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 |
| [34] | Verdugo, F.; Martín, A. F.; Badia, S., Distributed-memory parallelization of the aggregated unfitted finite element method, Comput. Methods Appl. Mech. Eng., 357, Article 112583 pp. (2019) · Zbl 1442.65280 |
| [35] | Badia, S.; Martín, A. F.; Neiva, E.; Verdugo, F., The aggregated unfitted finite element method on parallel tree-based adaptive meshes, SIAM J. Sci. Comput., 43, 3, C203-C234 (2021) · Zbl 1472.65141 |
| [36] | Burman, E.; Hansbo, P.; Larson, M. G., Explicit time stepping for the wave equation using CutFEM with discrete extension, SIAM J. Sci. Comput., 44, 3, A1254-A1289 (2022) · Zbl 1493.65146 |
| [37] | Badia, S.; Neiva, E.; Verdugo, F., Linking ghost penalty and aggregated unfitted methods, Comput. Methods Appl. Mech. Eng., 388, Article 114232 pp. (2022) · Zbl 1507.65231 |
| [38] | Burman, E.; Cicuttin, M.; Delay, G.; Ern, A., An unfitted hybrid high-order method with cell agglomeration for elliptic interface problems, SIAM J. Sci. Comput., 43, 2, A859-A882 (2021) · Zbl 1475.65186 |
| [39] | Kudela, L.; Zander, N.; Kollmannsberger, S.; Rank, E., Smart octrees: accurately integrating discontinuous functions in 3D, Comput. Methods Appl. Mech. Eng., 306, 406-426 (2016) · Zbl 1436.65022 |
| [40] | Saye, R., Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid-structure interaction, and free surface flow: part I, J. Comput. Phys., 344, 647-682 (2017) · Zbl 1380.76045 |
| [41] | Hansbo, P.; Larson, M. G.; Larsson, K., Cut finite element methods for linear elasticity problems, (Geometrically Unfitted Finite Element Methods and Applications (2017), Springer), 25-63 · Zbl 1390.74180 |
| [42] | Larson, M. G.; Zahedi, S., Stabilization of high order cut finite element methods on surfaces, IMA J. Numer. Anal., 40, 3, 1702-1745 (2020) · Zbl 1466.65195 |
| [43] | Duprez, M.; Lozinski, A., ϕ-fem: a finite element method on domains defined by level-sets, SIAM J. Numer. Anal., 58, 2, 1008-1028 (2020) · Zbl 1435.65199 |
| [44] | Atallah, N. M.; Canuto, C.; Scovazzi, G., The high-order shifted boundary method and its analysis, Comput. Methods Appl. Mech. Eng., 394, Article 114885 pp. (2022) · Zbl 1507.65230 |
| [45] | Höllig, K.; Reif, U.; Wipper, J., Weighted extended b-spline approximation of Dirichlet problems, SIAM J. Numer. Anal., 39, 2, 442-462 (2002) · Zbl 0996.65119 |
| [46] | Karniadakis, G.; Sherwin, S., Spectral/HP Element Methods for Computational Fluid Dynamics (2013), Oxford University Press · Zbl 1256.76003 |
| [47] | Szabó, B.; Babuška, I., Finite Element Analysis (1991), John Wiley & Sons · Zbl 0792.73003 |
| [48] | 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. Eng., 190, 46-47, 6183-6200 (2001) · Zbl 1029.74049 |
| [49] | Freund, J.; Stenberg, R., On weakly imposed boundary conditions for second order problems, (Finite Elements in Fluids. Finite Elements in Fluids, Italia, 15-21.10.1995 (1995), Padovan Yliopisto), 327-336 |
| [50] | Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods (1994), Springer: Springer New York · Zbl 0804.65101 |
| [51] | Hansbo, A.; Hansbo, P., An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., 191, 47-48, 5537-5552 (2002) · Zbl 1035.65125 |
| [52] | Elman, H.; Silvester, D.; Wathen, A., Finite Elements and Fast Iterative Solvers (2014), Oxford University Press · Zbl 1304.76002 |
| [53] | Düster, A.; Rank, E.; Szabó, B., The p-Version of the Finite Element and Finite Cell Methods, 1-35 (2017), John Wiley & Sons, Ltd |
| [54] | Requicha, A. A.; Voelcker, H. B., Constructive solid geometry (1977), University of Rochester, Technical report Production Automation Project TM-25 |
| [55] | Divi, S. C.; Verhoosel, C. V.; Auricchio, F.; Reali, A.; van Brummelen, E. H., Error-estimate-based adaptive integration for immersed isogeometric analysis, Comput. Math. Appl., 80, 11, 2481-2516 (2020) · Zbl 1455.74087 |
| [56] | Saye, R. I., High-order quadrature on multi-component domains implicitly defined by multivariate polynomials, J. Comput. Phys., 448, Article 110720 pp. (2022) · Zbl 1537.65022 |
| [57] | Lorensen, W. E.; Cline, H. E., Marching cubes: a high resolution 3D surface construction algorithm, (Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH ’87 (1987), ACM Press) |
| [58] | Müller, B.; Kummer, F.; Oberlack, M., Highly accurate surface and volume integration on implicit domains by means of moment-fitting, Int. J. Numer. Methods Eng., 96, 8, 512-528 (2013) · Zbl 1352.65083 |
| [59] | Sudhakar, Y.; Wall, W. A., Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods, Comput. Methods Appl. Mech. Eng., 258, 39-54 (2013) · Zbl 1286.65037 |
| [60] | Mousavi, S. E.; Xiao, H.; Sukumar, N., Generalized Gaussian quadrature rules on arbitrary polygons, Int. J. Numer. Methods Eng., 82, 1, 99-113 (2009) · Zbl 1183.65026 |
| [61] | Chin, E. B.; Lasserre, J. B.; Sukumar, N., Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra, Comput. Mech., 56, 6, 967-981 (2015) · Zbl 1336.65020 |
| [62] | Pardo, D.; Álvarez-Aramberri, J.; Paszynski, M.; Dalcin, L.; Calo, V., Impact of element-level static condensation on iterative solver performance, Comput. Math. Appl., 70, 10, 2331-2341 (2015) · Zbl 1443.65299 |
| [63] | Babuška, I.; Craig, A.; Mandel, J.; Pitkäranta, J., Efficient preconditioning for the p-version finite element method in two dimensions, SIAM J. Numer. Anal., 28, 3, 624-661 (1991) · Zbl 0754.65083 |
| [64] | Casarin, M. A., Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids (1996), New York University |
| [65] | Arnold, D. N.; Awanou, G., The serendipity family of finite elements, Found. Comput. Math., 11, 3, 337-344 (2011) · Zbl 1218.65125 |
| [66] | Babuška, I.; Griebel, M.; Pitkäranta, J., The problem of selecting the shape functions for a p-type finite element, Int. J. Numer. Methods Eng., 28, 8, 1891-1908 (1989) · Zbl 0705.73246 |
| [67] | Ern, A.; Guermond, J.-L., Finite Elements I: Approximation and Interpolation, vol. 72 (2021), Springer Nature |
| [68] | Sherwin, S. J.; Casarin, M., Low-energy basis preconditioning for elliptic substructured solvers based on unstructured spectral/hp element discretization, J. Comput. Phys., 171, 1, 394-417 (2001) · Zbl 0985.65143 |
| [69] | Badia, S.; Verdugo, F., Gridap: an extensible finite element toolbox in Julia, J. Open Sour. Softw., 5, 52, 2520 (2020) |
| [70] | Verdugo, F.; Badia, S., The software design of Gridap: a finite element package based on the Julia JIT compiler, Comput. Phys. Commun., 276, Article 108341 pp. (2022) · Zbl 1516.65162 |
| [71] | Verdugo, F.; Neiva, E.; Badia, S., GridapEmbedded (2021), Version 0.8., available at |
| [72] | Lehoucq, R. B.; Sorensen, D. C.; Yang, C., ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (1998), SIAM · Zbl 0901.65021 |
| [73] | Intel MKL PARDISO - parallel direct sparse solver interface |
| [74] | Hu, N.; Guo, X.-Z.; Katz, I., Bounds for eigenvalues and condition numbers in the p-version of the finite element method, Math. Comput., 67, 224, 1423-1450 (1998) · Zbl 0907.65112 |
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.
