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Higher-order extended FEM for weak discontinuities – level set representation, quadrature and application to magneto-mechanical problems. (English) Zbl 1352.74377

Summary: In this article, we present the application of bilinear and biquadratic extended FEM (XFEM) formulations to model weak discontinuities in magnetic and coupled magneto-mechanical boundary value problems. For properly resolving the location of curved interfaces and the discontinuous physical behaviour, the major part of the contribution is devoted to review and develop methods for level set representation of curved interfaces and numerical integration of the weak form in higher-order XFEM formulations. In order to reduce the complexity of the representation of curved interfaces, an element local approach that allows for an automated computation of the level set values and also improves the compatibility between the level set representation and the integration subdomains is proposed. Integration rules for polygons and strain smoothing are applied in conjunction with biquadratic elements and compared with curved integration subdomains. Eventually, a coupled magneto-mechanical demonstration problem is modelled and solved by XFEM. For demonstration purposes, a magneto-mechanical coupling due to magnetic stresses is considered. Errors and convergence rates are analysed for the different level set representations and numerical integration procedures as well as their dependence on the ratio of material parameters at an interface.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory

Software:

XFEM
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Full Text: DOI

References:

[1] Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) · Zbl 0955.74066 · doi:10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J
[2] Belytschko, Arbitrary discontinuities in finite elements, International Journal for Numerical Methods in Engineering 50 pp 993– (2001) · Zbl 0981.74062 · doi:10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M
[3] Daux, Arbitrary branched and intersecting cracks with the extended finite element method, International Journal for Numerical Methods in Engineering 48 pp 1741– (2000) · Zbl 0989.74066 · doi:10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L
[4] Sukumar, Modeling holes and inclusions by level sets in the extended finite element method, Computer Methods in Applied Mechanics and Engineering 190 pp 6183– (2001) · Zbl 1029.74049 · doi:10.1016/S0045-7825(01)00215-8
[5] Moës, A computational approach to handle complex microstructure geometries, Computer Methods in Applied Mechanics and Engineering 192 pp 3163– (2003) · Zbl 1054.74056 · doi:10.1016/S0045-7825(03)00346-3
[6] Kästner, Multiscale XFEM-modelling and simulation of the inelastic material behaviour of textile-reinforced polymers, International Journal for Numerical Methods in Engineering 86 pp 477– (2011) · Zbl 1216.74024 · doi:10.1002/nme.3065
[7] Legrain, An X-FEM and level set computational approach for image-based modelling: application to homogenization, International Journal for Numerical Methods in Engineering 86 pp 915– (2011) · Zbl 1235.74297 · doi:10.1002/nme.3085
[8] Rochus, Electrostatic simulation using XFEM for conductor and dielectric interfaces, International Journal for Numerical Methods in Engineering 85 pp 1207– (2011) · Zbl 1217.78061
[9] Linder, New finite elements with embedded strong discontinuities for the modeling of failure in electromechanical coupled solids, Computer Methods in Applied Mechanics and Engineering 200 pp 141– (2011) · Zbl 1225.74095 · doi:10.1016/j.cma.2010.07.021
[10] Rojas-Díaz, Fracture in magnetoelectroelastic materials using the extended finite element method, International Journal for Numerical Methods in Engineering 88 pp 1238– (2011) · Zbl 1242.74157 · doi:10.1002/nme.3219
[11] Fries, The extended/generalized finite element method: an overview of the method and its applications, International Journal for Numerical Methods in Engineering 84 pp 253– (2010) · Zbl 1202.74169
[12] Cheng, Higher-order XFEM for curved strong and weak discontinuities, International Journal for Numerical Methods in Engineering 82 pp 564– (2010) · Zbl 1188.74052
[13] Fries, A corrected XFEM approximation without problems in blending elements, International Journal for Numerical Methods in Engineering 75 pp 503– (2008) · Zbl 1195.74173 · doi:10.1002/nme.2259
[14] Belytschko, Structured extended finite element methods for solids defined by implicit surfaces, International Journal for Numerical Methods in Engineering 56 pp 609– (2003) · Zbl 1038.74041 · doi:10.1002/nme.686
[15] Moës, Imposing Dirichlet boundary conditions in the extended finite element method, International Journal for Numerical Methods in Engineering 67 pp 1641– (2006) · Zbl 1113.74072 · doi:10.1002/nme.1675
[16] Mousavi, Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons, Computational Mechanics 47 pp 535– (2011) · Zbl 1221.65078 · doi:10.1007/s00466-010-0562-5
[17] Bordas, On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM), International Journal for Numerical Methods in Engineering 86 pp 637– (2011) · Zbl 1216.74019 · doi:10.1002/nme.3156
[18] Stolarska, Modelling crack growth by level sets in the extended finite element method, International Journal for Numerical Methods in Engineering 51 pp 943– (2001) · Zbl 1022.74049 · doi:10.1002/nme.201
[19] Osher, Level set methods: an overview and some recent results, Journal of Computational Physics 169 pp 463– (2001) · Zbl 0988.65093 · doi:10.1006/jcph.2000.6636
[20] Dréau, Studied X-FEM enrichment to handle material interfaces with higher order finite element, Computer Methods in Applied Mechanics and Engineering 199 pp 1922– (2010) · Zbl 1231.74406 · doi:10.1016/j.cma.2010.01.021
[21] Moumnassi, Finite element analysis on implicitly defined domains: an accurate representation based on arbitrary parametric surfaces, Computer Methods in Applied Mechanics and Engineering 200 pp 774– (2011) · Zbl 1225.65111 · doi:10.1016/j.cma.2010.10.002
[22] Legrain, High Order X-FEM and Levelsets for Complex Microstructures: Uncoupling Geometry and Approximation, Computer Methods in Applied Mechanics and Engineering 241-244 pp 172– (2012) · Zbl 1353.74071 · doi:10.1016/j.cma.2012.06.001
[23] Kästner M Skalenübergreifende Modellierung und Simulation des mechanischen Verhaltens von textilverstärktem Polypropylen unter Nutzung der XFEM 2009 http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-27515
[24] Pereira, Hp-generalized FEM and crack surface representation for non-planar 3-D cracks, International Journal for Numerical Methods in Engineering 77 pp 601– (2009) · Zbl 1156.74383 · doi:10.1002/nme.2419
[25] Ventura, On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method, International Journal for Numerical Methods in Engineering 66 pp 761– (2006) · Zbl 1110.74858 · doi:10.1002/nme.1570
[26] Natarajan, Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping, International Journal of Numerical Methods in Engineering 80 pp 103– (2009) · Zbl 1176.74190 · doi:10.1002/nme.2589
[27] Natarajan, Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework, International Journal for Numerical Methods in Engineering 83 pp 269– (2010) · Zbl 1193.74153
[28] Mousavi, Generalized Duffy transformation for integrating vertex singularities, Computational Mechanics 45 pp 127– (2010) · Zbl 05662215 · doi:10.1007/s00466-009-0424-1
[29] Park, Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems, International Journal for Numerical Methods in Engineering 78 pp 1220– (2009) · Zbl 1183.74305 · doi:10.1002/nme.2530
[30] Mousavi, Generalized Gaussian quadrature rules on arbitrary polygons, International Journal for Numerical Methods in Engineering 82 pp 99– (2010) · Zbl 1183.65026
[31] Mousavi, Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method, Computer Methods in Applied Mechanics and Engineering 199 pp 3237– (2010) · Zbl 1225.74099 · doi:10.1016/j.cma.2010.06.031
[32] Bordas, Strain smoothing in FEM and XFEM, Computers and Structures 88 pp 1419– (2010) · doi:10.1016/j.compstruc.2008.07.006
[33] Chen, Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth, Computer Methods in Applied Mechanics and Engineering 209-212 pp 250– (2012) · Zbl 1243.74170 · doi:10.1016/j.cma.2011.08.013
[34] Lasserre, Integration on a convex polytope, Proceedings of the American Mathematical Society 126 pp 2433– (1998) · Zbl 0901.65012 · doi:10.1090/S0002-9939-98-04454-2
[35] Lasserre, Integration and homogeneous functions, Proceedings of the American Mathematical Society 127 pp 813– (1999) · Zbl 0913.65015 · doi:10.1090/S0002-9939-99-04930-8
[36] Liu, A smoothed finite element method for mechanics problems, Computational Mechanics 39 pp 859– (2007) · Zbl 1169.74047 · doi:10.1007/s00466-006-0075-4
[37] Eringen, Electrodynamics of Continua (1990) · doi:10.1007/978-1-4612-3226-1
[38] Groot, Foundations of Electrodynamics (1972)
[39] Engel, On the electromagnetic force on a polarizable body, American Journal of Physics 70 pp 428– (2002) · doi:10.1119/1.1432971
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