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The \(hp\)-\(d\)-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. (English) Zbl 1242.74161

Summary: The finite cell method (FCM) combines the fictitious domain approach with the p-version of the finite element method and adaptive integration. For problems of linear elasticity, it offers high convergence rates and simple mesh generation, irrespective of the geometric complexity involved. This article presents the integration of the FCM into the framework of nonlinear finite element technology. However, the penalty parameter of the fictitious domain is restricted to a few orders of magnitude in order to maintain local uniqueness of the deformation map. As a consequence of the weak penalization, nonlinear strain measures provoke excessive stress oscillations in the cells cut by geometric boundaries, leading to a low algebraic rate of convergence. Therefore, the FCM approach is complemented by a local overlay of linear hierarchical basis functions in the sense of the \(hp\)-\(d\) method, which synergetically uses the h-adaptivity of the integration scheme. Numerical experiments show that the \(hp\)-\(d\) overlay effectively reduces oscillations and permits stronger penalization of the fictitious domain by stabilizing the deformation map. The \(hp\)-\(d\)-adaptive FCM is thus able to restore high convergence rates for the geometrically nonlinear case, while preserving the easy meshing property of the original FCM. Accuracy and performance of the present scheme are demonstrated by several benchmark problems in one, two, and three dimensions and the nonlinear simulation of a complex foam sample.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
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[1] Bathe, Finite Element Procedures (1996)
[2] Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2000) · Zbl 1191.74002
[3] Felippa CA Introduction to Finite Element Methods University of Colorado Boulder http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/Home.html
[4] Cottrell, Isogeometric Analysis: Toward Integration of CAD and FEA (2008) · Zbl 1378.65009
[5] Hughes, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 pp 4135– (2005) · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008
[6] Bazilevs, Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering 199 pp 229– (2010) · Zbl 1227.74123 · doi:10.1016/j.cma.2009.02.036
[7] Del Pino, Numerical Methods for Scientific Computing: Variational Problems and Applications (2003)
[8] Glowinski, Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems, Computer Methods in Applied Mechanics and Engineering 196 pp 1498– (2007) · Zbl 1173.65369 · doi:10.1016/j.cma.2006.05.013
[9] Ramière, A fictitious domain approach with spread interface for elliptic problems with general boundary conditions, Computer Methods in Applied Mechanics and Engineering 196 pp 766– (2007) · Zbl 1121.65364 · doi:10.1016/j.cma.2006.05.012
[10] Bishop, Rapid stress analysis of geometrically complex domains using implicit meshing, Computational Mechanics 30 pp 460– (2003) · Zbl 1038.74626 · doi:10.1007/s00466-003-0424-5
[11] Löhner, Adaptive embedded and immersed unstructured grid techniques, Computer Methods in Applied Mechanics and Engineering 197 pp 2173– (2008) · Zbl 1158.76408 · doi:10.1016/j.cma.2007.09.010
[12] Baaijens, A fictitious domain/mortar element method for fluid-structure interaction, International Journal for Numerical Methods in Fluids 35 pp 743– (2001) · Zbl 0979.76044 · doi:10.1002/1097-0363(20010415)35:7<743::AID-FLD109>3.0.CO;2-A
[13] Farhat, A fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems. Part I: Dirichlet boundary conditions, International Journal for Numerical Methods in Engineering 54 pp 1309– (2002) · Zbl 1008.76039 · doi:10.1002/nme.461
[14] Haslinger, Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type, Computational Optimization and Applications 26 pp 231– (2003) · Zbl 1077.49030 · doi:10.1023/A:1026095405906
[15] Burman, Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method, Computer Methods in Applied Mechanics and Engineering 199 pp 2680– (2010) · Zbl 1231.65207 · doi:10.1016/j.cma.2010.05.011
[16] Burman, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method, Applied Numerical Mathematics (2010) · Zbl 1231.65207
[17] 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
[18] Gerstenberger, An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction, Computer Methods in Applied Mechanics and Engineering 197 pp 1699– (2008) · Zbl 1194.76117 · doi:10.1016/j.cma.2007.07.002
[19] Haslinger, A new fictitious domain approach inspired by the extended finite element method, SIAM Journal on Numerical Analysis 47 pp 1474– (2009) · Zbl 1205.65322 · doi:10.1137/070704435
[20] Becker, A hierarchical NXFEM for fictitious domain simulations, International Journal for Numerical Methods in Engineering 86 pp 549– (2011) · Zbl 1216.74018 · doi:10.1002/nme.3093
[21] Bastian, An unfitted finite element method using discontinuous Galerkin, International Journal for Numerical Methods in Engineering 79 pp 1557– (2009) · Zbl 1176.65131 · doi:10.1002/nme.2631
[22] Lui, Spectral domain embedding for elliptic PDEs in complex domains, Journal of Computational and Applied Mathematics 225 pp 541– (2009) · Zbl 1160.65346 · doi:10.1016/j.cam.2008.08.034
[23] Parussini, Fictitious domain approach with hp-finite element approximation for incompressible fluid flow, Journal of Computational Physics 228 pp 3891– (2009) · Zbl 1169.76037 · doi:10.1016/j.jcp.2009.02.019
[24] Allaire, Structural optimization using sensitivity analysis and a level-set method, Journal of Computational Physics 194 pp 363– (2004) · Zbl 1136.74368 · doi:10.1016/j.jcp.2003.09.032
[25] Parvizian, Finite cell method: h- and p-extension for embedded domain methods in solid mechanics, Computational Mechanics 41 pp 122– (2007) · Zbl 1162.74506 · doi:10.1007/s00466-007-0173-y
[26] Düster, The finite cell method for three-dimensional problems of solid mechanics, Computer Methods in Applied Mechanics and Engineering 197 pp 3768– (2008) · Zbl 1194.74517 · doi:10.1016/j.cma.2008.02.036
[27] Szabó, Finite Element Analysis (1991)
[28] Szabó, Encyclopedia of Computational Mechanics 1 pp 119– (2004)
[29] Rank, Computational Structural Engineering pp 87– (2009) · doi:10.1007/978-90-481-2822-8_9
[30] Ruess, The finite cell method for bone simulations: verification and validation, Biomechanics and Modeling in Mechanobiology (2011)
[31] Vos, A comparison of fictitious domain methods appropriate for spectral /hp element discretisations, Computer Methods in Applied Mechanics and Engineering 197 pp 2275– (2008) · Zbl 1158.76357 · doi:10.1016/j.cma.2007.11.023
[32] Parvizian, Topology optimization using the finite cell method, Optimization and Engineering (2011)
[33] Rank, Shell finite cell method: a high order fictitious domain approach for thin-walled structures, Computer Methods in Applied Mechanics and Engineering (2011) · Zbl 1230.74232 · doi:10.1016/j.cma.2011.06.005
[34] Düster, p-FEM applied to finite isotropic hyperelastic bodies, Computer Methods in Applied Mechanics and Engineering 192 pp 5147– (2003) · Zbl 1053.74043 · doi:10.1016/j.cma.2003.07.003
[35] Bonet, Nonlinear Continuum Mechanics for Finite Element Analysis (2008) · Zbl 1142.74002 · doi:10.1017/CBO9780511755446
[36] de Souza Neto, Computational Methods for Plasticity: Theory and Applications (2008) · doi:10.1002/9780470694626
[37] Rank, Adaptive remeshing and h-p domain decomposition, Computer Methods in Applied Mechanics and Engineering 101 pp 299– (1992) · Zbl 0782.65145 · doi:10.1016/0045-7825(92)90027-H
[38] Rank, A multiscale finite-element-method, Computers & Structures 64 pp 139– (1997) · Zbl 0918.73222 · doi:10.1016/S0045-7949(96)00149-6
[39] Krause, Multiscale computations with a combination of the h- and p-versions of the finite-element method, Computer Methods in Applied Mechanics and Engineering 192 pp 3959– (2003) · Zbl 1037.74047 · doi:10.1016/S0045-7825(03)00395-5
[40] Düster, Applying the hp-d version of the FEM to locally enhance dimensionally reduced models, Computer Methods in Applied Mechanics and Engineering 196 pp 3524– (2007) · Zbl 1173.74413 · doi:10.1016/j.cma.2006.10.018
[41] Schillinger, The finite cell method for geometrically nonlinear problems of solid mechanics, IOP Conference Series: Material Science and Engineering 10 pp 012170– (2010) · doi:10.1088/1757-899X/10/1/012170
[42] Schillinger, An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry, Computer Methods in Applied Mechanics and Engineering 200 (47-48) pp 3358– (2011) · Zbl 1230.74197 · doi:10.1016/j.cma.2011.08.002
[43] Gerstenberger, An embedded Dirichlet formulation for 3D continua, International Journal for Numerical Methods in Engineering 82 pp 537– (2010) · Zbl 1188.74056
[44] Embar, Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements, International Journal for Numerical Methods in Engineering 83 pp 877– (2010) · Zbl 1197.74178
[45] Samet, The design and analysis of spatial data structures (1989)
[46] Zohdi, Aspects of the computational testing of the mechanical properties of microheterogeneous material samples, International Journal for Numerical Methods in Engineering 50 pp 2573– (2001) · Zbl 1098.74721 · doi:10.1002/nme.146
[47] Zienkiewicz, The Finite Element Method: The Basis 1 (2000) · Zbl 0991.74002
[48] Belytschko, Nonlinear Finite Elements for Continua and Structures (2006)
[49] Wriggers, Nonlinear Finite Element Methods (2008)
[50] Ibrahimbegovic, Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods (2009) · Zbl 1168.74002
[51] MATLAB, MATLAB User’s Guide Version 7.6 (2008)
[52] Nübel, An RP-adaptive finite element method for elastoplastic problems, Computational Mechanics 39 pp 557– (2007) · Zbl 1163.74046 · doi:10.1007/s00466-006-0111-4
[53] Krause R Multiscale computations with a combined h - and p -version of the finite-element method PhD thesis 1996
[54] Düster A High order finite elements for three-dimensional, thin-walled nonlinear continua PhD thesis 2001
[55] Yserantant, On the multi-level splitting of finite element spaces, Numerische Mathematik 49 pp 379– (1986) · Zbl 0608.65065 · doi:10.1007/BF01389538
[56] Krysl, Natural hierarchical refinement for finite element methods, International Journal for Numerical Methods in Engineering 56 pp 1109– (2003) · Zbl 1078.74660 · doi:10.1002/nme.601
[57] Bungartz, Sparse grids, Acta Numerica 13 pp 147– (2004) · Zbl 1118.65388 · doi:10.1017/S0962492904000182
[58] Krause, hp-Version finite elements for geometrically non-linear problems, Communications in Numerical Methods in Engineering 11 pp 887– (1995) · Zbl 0835.73075 · doi:10.1002/cnm.1640111103
[59] Düster, p-FEM applied to finite isotropic hyperelastic bodies, Computer Methods in Applied Mechanics and Engineering 192 pp 5147– (2003)
[60] Dong, A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems, Computers & Structures 87 pp 59– (2009) · doi:10.1016/j.compstruc.2008.08.008
[61] Ibrahimbegovic, A consistent finite element formulation of nonlinear membrane shell theory with particular reference to elastic rubberlike material, Finite Element in Analysis and Design 12 pp 75– (1993) · Zbl 0776.73065 · doi:10.1016/0168-874X(93)90008-E
[62] Gharzeddine, Incompatible mode method for finite deformation quasi-incompressible elasticity, Computational Mechanics 24 pp 419– (2000) · Zbl 0962.74058 · doi:10.1007/s004660050001
[63] Suri, Analytical and computational assessment of locking in the hp finite element method, Computer Methods in Applied Mechanics and Engineering 133 pp 347– (1996) · Zbl 0893.73070 · doi:10.1016/0045-7825(95)00947-7
[64] Chilton, On the selection of a locking-free hp element for elasticity problems, International Journal for Numerical Methods in Engineering 40 pp 2045– (1997) · Zbl 0886.73061 · doi:10.1002/(SICI)1097-0207(19970615)40:11<2045::AID-NME158>3.0.CO;2-Z
[65] Heisserer, On volumetric locking-free behaviour of p-version finite elements under finite deformations, Communications in Numerical Methods in Engineering 24 pp 1019– (2008) · Zbl 1153.74043 · doi:10.1002/cnm.1008
[66] Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Computer Methods in Applied Mechanics and Engineering 191 pp 537– (2002) · Zbl 1035.65125 · doi:10.1016/S0045-7825(02)00524-8
[67] Fernandez-Mendez, Imposing essential boundary conditions in mesh-free methods, Computer Methods in Applied Mechanics and Engineering 193 pp 1257– (2004) · Zbl 1060.74665 · doi:10.1016/j.cma.2003.12.019
[68] Fritz, A comparison of mortar and Nitsche techniques for linear elasticity, Calcolo 41 pp 115– (2004) · Zbl 1099.65123 · doi:10.1007/s10092-004-0087-4
[69] Mok, Algorithmic aspects of deformation dependent loads in non-linear static finite element analysis, Engineering Computations 16 pp 601– (1999) · Zbl 0986.74071 · doi:10.1108/02644409910277951
[70] Schweizerhof, Displacement dependent pressure loads in nonlinear finite element analyses, Computers & Structures 18 pp 1099– (1984) · Zbl 0554.73069 · doi:10.1016/0045-7949(84)90154-8
[71] Simo, A note on finite-element implementation of pressure boundary loading, Communications in Applied Numerical Methods 7 pp 513– (1991) · Zbl 0735.73080 · doi:10.1002/cnm.1630070703
[72] Yosibash, Axisymmetric pressure boundary loading for finite deformation analysis using p-FEM, Computer Methods in Applied Mechanics and Engineering 196 pp 1261– (2007) · Zbl 1173.74451 · doi:10.1016/j.cma.2006.09.006
[73] Heisserer U 2008 High-order finite elements for material and geometrical nonlinear finite strain problems PhD thesis
[74] Yosibash, A CT-based high-order finite element analysis of the human proximal femur compared to in-vitro experiments, Journal of Biomedical Engineering 129 pp 297– (2007)
[75] Fiedler, Computed tomography based finite element analysis of the thermal properties of cellular aluminium, Materialwissenschaften und Werkstofftechnik 40 pp 139– (2009) · doi:10.1002/mawe.200900419
[76] Wenisch P Wenisch O Fast octree-based voxelization of 3D boundary representation-objects Technical report 2004
[77] Sandia National Laboratories, Trilinos Version 10.2 (2010)
[78] Schenk O Gärtner K Karypis G Luce R Carbonetto P 2010 http://www.pardiso-project.org
[79] Bonet J Wood R 2008 http://www.flagshyp.com
[80] Sorger C 2010
[81] Schöberl J Netgen version 4.9.13 2010 http://sourceforge.net/projects/netgen-mesher
[82] Kitware Inc., ParaView Version 3.8.1. Open-Source Scientific Visualization Package (2010)
[83] Šolín, Higher-Order Finite Element Methods (2004)
[84] Demkowicz, Computing with hp-Adaptive Finite Elements: One and Two Dimensional Elliptic and Maxwell Problems 1 (2006) · Zbl 1111.65103 · doi:10.1201/9781420011685
[85] Banhart, Manufacture, characterization and application of cellular metals and metal foams, Progress in Material Science 46 pp 559– (2001) · doi:10.1016/S0079-6425(00)00002-5
[86] Zohdi, Introduction to Computational Micromechanics (2005) · Zbl 1143.74002 · doi:10.1007/978-3-540-32360-0
[87] Sehlhorst, Numerical investigations of foam-like materials by nested high-order finite element methods, Computational Mechanics 45 pp 45– (2009) · Zbl 1398.74402 · doi:10.1007/s00466-009-0414-3
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