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High order unfitted finite element methods on level set domains using isoparametric mappings. (English) Zbl 1425.65168
Summary: We introduce a new class of unfitted finite element methods with high order accurate numerical integration over curved surfaces and volumes which are only implicitly defined by level set functions. An unfitted finite element method which is suitable for the case of piecewise planar interfaces is combined with a parametric mapping of the underlying mesh resulting in an isoparametric unfitted finite element method. The parametric mapping is constructed in a way such that the quality of the piecewise planar interface reconstruction is significantly improved allowing for high order accurate computations of (unfitted) domain and surface integrals. We present the method, discuss implementational aspects and present numerical examples which demonstrate the quality and potential of this method.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D17 Computer-aided design (modeling of curves and surfaces)
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[1] Babuška, I., The finite element method with penalty, Math. Comp., 27, 122, 221-228, (1973) · Zbl 0299.65057
[2] Barrett, J. W.; Elliott, C. M., Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer. Math., 49, 4, 343-366, (1986) · Zbl 0614.65116
[3] Glowinski, R.; Pan, T.-W.; Periaux, J., A fictitious domain method for Dirichlet problem and applications, Comput. Methods Appl. Mech. Engrg., 111, 3-4, 283-303, (1994) · Zbl 0845.73078
[4] Burman, E.; Hansbo, P., Fictitious domain finite element methods using cut elements: II. a stabilized Nitsche method, Appl. Numer. Math., 62, 4, 328-341, (2012) · Zbl 1316.65099
[5] Peskin, C. S.; McQueen, D. M., A three-dimensional computational method for blood flow in the heart I. immersed elastic fibers in a viscous incompressible fluid, J. Comput. Phys., 81, 2, 372-405, (1989) · Zbl 0668.76159
[6] Bastian, P.; Engwer, C., An unfitted finite element method using discontinuous Galerkin, Internat. J. Numer. Methods Engrg., 79, 12, 1557-1576, (2009) · Zbl 1176.65131
[7] Fries, T.-P.; Belytschko, T., The extended/generalized finite element method: an overview of the method and its applications, Internat. J. Numer. Methods Engrg., 84, 3, 253-304, (2010) · Zbl 1202.74169
[8] Hansbo, A.; Hansbo, P., An unfitted finite element method, based on nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191, 47, 5537-5552, (2002) · Zbl 1035.65125
[9] Groß, S.; Reichelt, V.; Reusken, A., A finite element based level set method for two-phase incompressible flows, Comput. Vis. Sci., 9, 239-257, (2006) · Zbl 1119.76042
[10] Massjung, R., An unfitted discontinuous Galerkin method applied to elliptic interface problems, SIAM J. Numer. Anal., 50, 6, 3134-3162, (2012) · Zbl 1262.65178
[11] Becker, R.; Burman, E.; Hansbo, P., A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Engrg., 198, 41-44, 3352-3360, (2009) · Zbl 1230.74169
[12] Olshanskii, M. A.; Reusken, A.; Grande, J., A finite element method for elliptic equations on surfaces, SIAM J. Numer. Anal., 47, 5, 3339-3358, (2009) · Zbl 1204.58019
[13] Lorensen, W. E.; Cline, H. E., Marching cubes: A high resolution 3d surface construction algorithm, (ACM SIGGRAPH Computer Graphics, vol. 21, (1987), ACM), 163-169
[14] Nærland, T. A., Geometry decomposition algorithms for the Nitsche method on unfitted geometries, (2014), University of Oslo, (Master’s thesis)
[15] Mayer, U. M.; Gerstenberger, A.; Wall, W. A., Interface handling for three-dimensional higher-order XFEM-computations in fluid-structure interaction, Internat. J. Numer. Methods Engrg., 79, 7, 846-869, (2009) · Zbl 1171.74447
[16] Lehrenfeld, C., The Nitsche XFEM-DG space-time method and its implementation in three space dimensions, SIAM J. Sci. Comput., 37, 1, A245-A270, (2015) · Zbl 1326.65162
[17] Lehrenfeld, C., On a space-time extended finite element method for the solution of a class of two-phase mass transport problems, (February, 2015), RWTH Aachen
[18] Gross, S., DROPS package for simulation of two-phase flows, (2015)
[19] Engwer, C.; Heimann, F., Dune-UDG: a cut-cell framework for unfitted discontinuous Galerkin methods, (Advances in DUNE, (2012), Springer), 89-100
[20] Burman, E.; Claus, S.; Hansbo, P.; Larson, M. G.; Massing, A., Cutfem: discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg., (2014), URL http://dx.doi.org/10.1002/nme.4823 · Zbl 1352.65604
[21] Renard, Y.; Pommier, J., Getfem++, an open-source finite element library, (2014)
[22] T. Carraro, S. Wetterauer, On the implementation of the eXtended finite element method (XFEM) for interface problems, arXiv preprint arXiv:1507.04238.
[23] Chernyshenko, A. Y.; Olshanskii, M. A., An adaptive octree finite element method for pdes posed on surfaces, Comput. Methods Appl. Mech. Engrg., 291, 146-172, (2015) · Zbl 1425.65155
[24] Müller, B.; Kummer, F.; Oberlack, M., Highly accurate surface and volume integration on implicit domains by means of moment-Fitting, Internat. J. Numer. Methods Engrg., 96, 8, 512-528, (2013) · Zbl 1352.65083
[25] 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. Engrg., 258, 39-54, (2013) · Zbl 1286.65037
[26] Saye, R., High-order quadrature method for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput., 37, 2, A993-A1019, (2015) · Zbl 1328.65070
[27] E. Burman, P. Hansbo, M.G. Larson, A cut finite element method with boundary value correction, arXiv preprint arXiv:1507.03096. · Zbl 1380.65362
[28] Grande, J.; Reusken, A., A higher order finite element method for partial differential equations on surfaces, tech. rep. 403, (2014), Institut für Geometrie und Praktische Mathematik, RWTH Aachen
[29] Cheng, K. W.; Fries, T.-P., Higher-order XFEM for curved strong and weak discontinuities, Internat. J. Numer. Methods Engrg., 82, 5, 564-590, (2010) · Zbl 1188.74052
[30] Dréau, K.; Chevaugeon, N.; Moës, N., Studied X-FEM enrichment to handle material interfaces with higher order finite element, Comput. Methods Appl. Mech. Engrg., 199, 29, 1922-1936, (2010) · Zbl 1231.74406
[31] T.-P. Fries, S. Omerovi, Higher-order accurate integration of implicit geometries, Internat. J. Numer. Methods Engrg., http://dx.doi.org/10.1002/nme.5121.
[32] Basting, S.; Weismann, M., A hybrid level set—front tracking finite element approach for fluid-structure interaction and two-phase flow applications, J. Comput. Phys., 255, 228-244, (2013) · Zbl 1349.76176
[33] Oswald, P., On a BPX-preconditioner for \(\mathbb{P}_1\) elements, Computing, 51, 125-133, (1993) · Zbl 0787.65018
[34] Schöberl, J., C++11 implementation of finite elements in ngsolve, tech. rep. ASC-2014-30, (2014), Institute for Analysis and Scientific Computing, September
[35] Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35, 230-254, (1995) · Zbl 0915.65121
[36] C. Lehrenfeld, A. Reusken, Optimal preconditioners for Nitsche-XFEM discretizations of interface problems, arXiv preprint arXiv:1408.2940. · Zbl 1360.65281
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