The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models

Abstract

The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
Fig. 43
Fig. 44
Fig. 45
Fig. 46
Fig. 47
Fig. 48
Fig. 49
Fig. 50
Fig. 51
Fig. 52
Fig. 53
Fig. 54
Fig. 55
Fig. 56
Fig. 57
Fig. 58
Fig. 59
Fig. 60
Fig. 61
Fig. 62
Fig. 63
Fig. 64
Fig. 65
Fig. 66
Fig. 67
Fig. 68
Fig. 69
Fig. 70
Fig. 71
Fig. 72
Fig. 73
Fig. 74
Fig. 75
Fig. 76
Fig. 77
Fig. 78
Fig. 79
Fig. 80
Fig. 81
Fig. 82
Fig. 83
Fig. 84
Fig. 85
Fig. 86
Fig. 87
Fig. 88
Fig. 89
Fig. 90
Fig. 91
Fig. 92
Fig. 93
Fig. 94
Fig. 95
Fig. 96
Fig. 97
Fig. 98
Fig. 99
Fig. 100
Fig. 101
Fig. 102
Fig. 103
Fig. 104
Fig. 105

Notes

  1. 1.

    Index N.

  2. 2.

    Index S.

  3. 3.

    Courtesy of IZFP Fraunhofer Institute for Non-Destructive Testing, Saarbrücken, Germany; http://www.izfp.fraunhofer.de.

  4. 4.

    Surface Tesselation Language.

  5. 5.

    On a Intel(R) Core(TM)2 P8800 @ 2.66 GHz.

  6. 6.

    Representative volume element.

  7. 7.

    Digital Imaging and Communications in Medicine.

  8. 8.

    Cell index \(\{.\}_{c}\).

  9. 9.

    Sub-cell index \(\{.\}_{sc}\).

  10. 10.

    http://fcmlab.cie.bgu.tum.de.

References

  1. 1.

    Abedian A, Parvizian J, Düster A, Khademyzadeh H, Rank E (2013) Performance of different integration schemes in facing discontinuities in the finite cell method. Int J Comput Methods 10(3):1–24

    Google Scholar 

  2. 2.

    Abedian A, Parvizian J, Düster A, Rank E (2013) The finite cell method for the J\(_2\) flow theory of plasticity. Finite Elem Anal Des 69:37–47

    Google Scholar 

  3. 3.

    Agoston MK (2005) Computer graphics and geometric modeling, vol 2. Springer, Berlin

    Google Scholar 

  4. 4.

    Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Annavarapu C, Hautefeuille M, Dolbow JE (2012) A robust Nitsche’s formulation for interface problems. Comput Methods Appl Mech Eng 225:44–54

    MathSciNet  Google Scholar 

  6. 6.

    Apostolatos A, Schmidt R, Wüchner R, Bletzinger K-U (2014) A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis. Int J Numer Methods Eng 97(7):473–504

    Google Scholar 

  7. 7.

    McNeel & Associates (2013) Rhinoceros-accurate freeform modeling for Windows. http://www.rhino3d.com

  8. 8.

    Babuška I, Banerjee U, Osborn JE (2003) Meshless and generalized finite element methods: a survey of some major results. In: Griebel M, Schweitzer MA (eds) Meshfree methods for partial differential equations. Springer, Berlin, pp 1–20

  9. 9.

    Babuška I (1972) The finite element method with penalty. Math Comput 27(122):221–228

    Google Scholar 

  10. 10.

    Baiges J, Codina R (2010) The fixed-mesh ALE approach applied to solid mechanics and fluid-structure interaction problems. Int J Numer Methods Eng 81:1529–1557

    MATH  MathSciNet  Google Scholar 

  11. 11.

    Baiges J, Codina R, Henke F, Shahmiri S, Wall WA (2012) A symmetric method for weakly imposing Dirchlet boundary conditions in embedded finite element meshes. Int J Numer Methods Eng 90:636–658

    MATH  MathSciNet  Google Scholar 

  12. 12.

    Banhart J (2001) Manufacture, characterization and application of cellular metals and metal foams. Prog Mater Sci 46:559–632

    Google Scholar 

  13. 13.

    Bastian P, Engwer C (2009) An unfitted finite element method using discontinuous Galerkin. Int J Numer Methods Eng 79:1557–1576

    MATH  MathSciNet  Google Scholar 

  14. 14.

    Bathe K-J (1996) Finite element procedures. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  15. 15.

    Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2010) Isogeometric analysis using T-splines. Comput Methods Appl Mech Eng 199:229–263

    MATH  MathSciNet  Google Scholar 

  16. 16.

    Bazilevs Y, Hsu MC, Scott MA (2012) Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Comput Methods Appl Mech Eng 249–252:28–41

    MathSciNet  Google Scholar 

  17. 17.

    Bazilevs Y, Hughes TJR (2007) Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput Fluids 36:12–26

    MATH  MathSciNet  Google Scholar 

  18. 18.

    Bazilevs Y, Michler CM, Calo VM, Hughes TJR (2010) Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly-enforced boundary conditions on unstretched meshes. Comput Methods Appl Mech Eng 199:780–790

    MATH  MathSciNet  Google Scholar 

  19. 19.

    Beirão da Veiga L, Buffa A, Cho D, Sangalli G (2012) Analysis-suitable T-splines are dual-compatible. Comput Methods Appl Mech Eng 249:42–51

    Google Scholar 

  20. 20.

    Belytschko T, Liu WK, Moran B (2006) Nonlinear finite elements for continua and structures. Wiley, NewYork

    Google Scholar 

  21. 21.

    Belytschko T, Parimi C, Moës N, Sukumar N, Usui S (2003) Structured extended finite element methods for solids defined by implicit surfaces. Int J Numer Methods Eng 56(4):609–635

    MATH  Google Scholar 

  22. 22.

    Bindick S, Stiebler M, Krafczyk M (2011) Fast kd-tree-based hierarchical radiosity for radiative heat transport problems. Int J Numer Methods Eng 86(9):1082–1100

    MATH  Google Scholar 

  23. 23.

    Bishop J (2003) Rapid stress analysis of geometrically complex domains using implicit meshing. Comput Mech 30:460–478

    MATH  Google Scholar 

  24. 24.

    Bonet J, Wood R (2008) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press, Cambridge

    Google Scholar 

  25. 25.

    Borden MJ, Scott MA, Evans JA, Hughes TJR (2011) Isogeometric finite element data structures based on Bézier extraction of NURBS. Int J Numer Methods Eng 87:15–47

    MATH  MathSciNet  Google Scholar 

  26. 26.

    Bornemann B, Cirak F (2013) A subdivision-based implementation of the hierarchical b-spline finite element method. Comput Methods Appl Mech Eng 253:584–598

    MATH  MathSciNet  Google Scholar 

  27. 27.

    Bungartz H-J, Griebel M (2004) Sparse grids. Acta Numer 13(1):147–269

    MathSciNet  Google Scholar 

  28. 28.

    Bungartz H-J, Griebel M, Zenger C (2004) Introduction to computer graphics. Charles River Media Inc, Prague

  29. 29.

    Burman E, Hansbo P (2010) Fictitious domain finite element methods using cut elements: a stabilized lagrange multiplier method. Comput Methods Appl Mech Eng 62(4):2680–2686

    MathSciNet  Google Scholar 

  30. 30.

    Burman E, Hansbo P (2012) Fictitious domain finite element methods using cut elements: a stabilized Nitsche method. Appl Numer Math 62(4):328–341

    MATH  MathSciNet  Google Scholar 

  31. 31.

    Canuto C, Hussaini MY, Quarteroni A, Zang TA (2006) Spectral methods: fundamentals in single domains. Springer, Berlin

    Google Scholar 

  32. 32.

    Canuto C, Hussaini MY, Quarteroni A, Zang TA (2007) Spectral methods: evolution to complex geometries and applications to fluid dynamics. Springer, Berlin

    Google Scholar 

  33. 33.

    Chapman B, Jost G, Van Der Pas R (2008) Using OpenMP: portable shared memory parallel programming. The MIT Press, Cambridge

    Google Scholar 

  34. 34.

    Chilton L, Suri M (1997) On the selection of a locking-free hp element for elasticity problems. Int J Numer Methods Eng 40(11):2045–2062

    MATH  MathSciNet  Google Scholar 

  35. 35.

    Cohen E, Martin T, Kirby RM, Lyche T, Riesenfeld RF (2010) Analysis-aware modeling: understanding quality considerations in modeling for isogeometric analysis. Comput Methods Appl Mech Eng 199:334–356

    MATH  MathSciNet  Google Scholar 

  36. 36.

    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: towards integration of CAD and FEA. Wiley, New York

    Google Scholar 

  37. 37.

    de Souza Neto EA, Perić D, Owen DRJ (2008) Computational methods for plasticity: theory and applications. Wiley, New York

    Google Scholar 

  38. 38.

    Dede’ L, Borden MJ, Hughes TJR (2012) Isogeometric analysis for topology optimization with a phase field model. Arch Comput Methods Eng 19:427–465

    MathSciNet  Google Scholar 

  39. 39.

    Del Pino S, Pironneau O (2003) A fictitious domain based general PDE solver. In: Kuznetsov Y, Neittanmaki P, Pironneau O (eds) Numerical methods for scientific computing: variational problems and applications. CIMNE, Barcelona

    Google Scholar 

  40. 40.

    Demkowicz L, Kurtz J, Pardo D, Paszynski M, Rachowicz W, Zdunek A (2007) Computing with Hp-adaptive finite elements, vol 2: frontiers three-dimensional elliptic and Maxwell problems with applications. Chapman & Hall/CRC, London.

  41. 41.

    Demkowicz LF (2006) Computing with Hp-adaptive finite elements, vol 1: one and two dimensional elliptic and Maxwell problems. Chapman & Hall/CRC, London.

  42. 42.

    Dokken T, Lyche T, Pettersen KF (2013) Polynomial splines over locally refined box-partitions. Comput Aided Geom Des 30(21):331–356

    MATH  MathSciNet  Google Scholar 

  43. 43.

    Dolbow J, Harari I (2009) An efficient finite element method for embedded interface problems. Int J Numer Methods Eng 78:229–252

    MATH  MathSciNet  Google Scholar 

  44. 44.

    Dong S, Yosibash Z (2009) A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems. Comput Struct 87(1):59–72

    Google Scholar 

  45. 45.

    Düster A (2001) High order finite elements for three-dimensional, thin-walled nonlinear continua. Dissertation, Technische Universität München.

  46. 46.

    Düster A, Bröker H, Rank E (2001) The \(p\)-version of the finite element method for three-dimensional curved thin walled structures. Int J Numer Methods Eng 52:673–703

    MATH  Google Scholar 

  47. 47.

    Düster A, Hartmann S, Rank E (2003) p-fem applied to finite isotropic hyperelastic bodies. Comput Methods Appl Mech Eng 192(47):5147–5166

    MATH  Google Scholar 

  48. 48.

    Düster A, Niggl A, Rank E (2007) Applying the hp-d version of the fem to locally enhance dimensionally reduced models. Comput Methods Appl Mech Eng 196(37):3524–3533

    MATH  Google Scholar 

  49. 49.

    Düster A, Parvizian J, Yang Z, Rank E (2010) The finite cell method for three-dimensional problems of solid mechanics. Comput Methods Appl Mech Eng 197:3768–3782

    Google Scholar 

  50. 50.

    Elguedj T, Bazilevs Y, Calo VM, Hughes TJR (2008) \(\bar{B}\) and \(\bar{F}\) projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements. Comput Methods Appl Mech Eng 197:2732–2762

    MATH  Google Scholar 

  51. 51.

    Embar A, Dolbow J, Harari I (2010) Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. Int J Numer Methods Eng 83:877–898

    MATH  MathSciNet  Google Scholar 

  52. 52.

    Evans JA, Bazilevs Y, Babuška I, Hughes TJR (2009) n-widths, sup-infs, and optimality ratios for the \(k\)-version of the isogeometric finite element method. Comput Methods Appl Mech Eng 198(21–26):1726–1741

    MATH  Google Scholar 

  53. 53.

    Evans JA, Hughes TJR (2013) Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations. Math Models Methods Appl Sci 23:1421

    MATH  MathSciNet  Google Scholar 

  54. 54.

    Farin G (2002) Curves and surfaces for computer aided geometric design. Morgan Kaufmann Publishers, Los Altos

    Google Scholar 

  55. 55.

    Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193:1257–1275

    MATH  Google Scholar 

  56. 56.

    Flemisch B, Wohlmuth BI (2007) Stable lagrange multipliers for quadrilateral meshes of curved interfaces in 3d. Comput Methods Appl Mech Eng 196(8):1589–1602

    MATH  MathSciNet  Google Scholar 

  57. 57.

    Franke D, Düster A, Nübel V, Rank E (2010) A comparison of the h-, p-, hp-, and rp-version of the FEM for the solution of the 2d Hertzian contact problem. Comput Mech 45(5):513–522

    MATH  Google Scholar 

  58. 58.

    Gerstenberger A, Wall WA (2008) Enhancement of fixed-grid methods towards complex fluid-structure interaction applications. Int J Numer Methods Fluids 57:1227–1248

    MATH  MathSciNet  Google Scholar 

  59. 59.

    Gerstenberger A, Wall WA (2010) An embedded Dirichlet formulation for 3D continua. Int J Numer Methods Eng 82:537–563

    MATH  MathSciNet  Google Scholar 

  60. 60.

    Giannelli C, Jüttler B, Speleers H (2012) THB-splines: the truncated basis for hierarchical splines. Comput Aided Geom Des 29(7):485–498

    MATH  Google Scholar 

  61. 61.

    Glowinski R, Kuznetsov Y (2007) Distributed lagrange multipliers based on fictitious domain method for second order elliptic problems. Comput Methods Appl Mech Eng 196:1498– 1506

  62. 62.

    Griebel M, Schweitzer MA (2004) A particle-partition of unity method. Part V: boundary conditions. In: Hildebrandt S, Karcher H (eds) Geometric analysis and nonlinear partial differential equations. Springer, Berlin, pp 519–542

    Google Scholar 

  63. 63.

    Grossmann D, Jüttler B, Schlusnus H, Barner J, Vuong AH (2012) Isogeometric simulation of turbine blades for aircraft engines. Comput Aided Geom Des 29(7):519–531

    MATH  Google Scholar 

  64. 64.

    Hansbo A, Hansbo P (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Methods Appl Mech Eng 191:537–552

    Google Scholar 

  65. 65.

    Hansbo P (2005) Nitsche’s method for interface problems in computational mechanics. GAMM Mitteilungen 28(2):183–206

    MATH  MathSciNet  Google Scholar 

  66. 66.

    Harari I, Dolbow J (2010) Analysis of an efficient finite element method for embedded interface problems. Comput Mech 46:205–211

    MATH  MathSciNet  Google Scholar 

  67. 67.

    Harari I, Shavelzon E (2012) Embedded kinematic boundary conditions for thin plate bending by Nitsche’s approach. Int J Numer Methods Eng 92(1):99–114

    MathSciNet  Google Scholar 

  68. 68.

    Haslinger J, Renard Y (2009) A new fictitious domain approach inspired by the extended finite element method. SIAM J Numer Anal 47:1474–1499

    MATH  MathSciNet  Google Scholar 

  69. 69.

    Hautefeuille M, Annavarapu C, Dolbow JE (2012) Robust imposition of Dirichlet boundary conditions on embedded surfaces. Int J Numer Methods Eng 90:40–64

    MATH  MathSciNet  Google Scholar 

  70. 70.

    Heisserer U, Hartmann S, Düster A, Yosibash Z (2008) On volumetric locking-free behaviour of p-version finite elements under finite deformations. Commun Numer Methods Eng 24(11):1019–1032

    MATH  Google Scholar 

  71. 71.

    Hesthaven JS, Gottlieb S, Gottlieb D (2007) Spectral methods for time-dependent problems. Cambridge University Press, Cambridge

    Google Scholar 

  72. 72.

    Höllig K (2003) Finite element methods with B-Splines. Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  73. 73.

    Höllig K, Hörner J, Hoffacker A (2012) Finite element analysis with b-splines: weighted and isogeometric methods. Curves and surfaces, vol 6920, Lecture Notes in Computer ScienceSpringer, Berlin, pp 330–350.

  74. 74.

    Höllig K, Reif U, Wipper J (2001) Weighted extended b-spline approximation of Dirichlet problems. SIAM J Numer Anal 39:442–462

    MATH  MathSciNet  Google Scholar 

  75. 75.

    Holzapfel GA (2000) Nonlinear solid mechanics. A continuum approach for engineering, Wiley, New York

    Google Scholar 

  76. 76.

    Hsu MC, Akkerman I, Bazilevs Y (2012) Wind turbine aerodynamics using ALE-VMS: validation and the role of weakly enforced boundary conditions. Comput Mech 50:499–511

    MATH  MathSciNet  Google Scholar 

  77. 77.

    Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, New York

    Google Scholar 

  78. 78.

    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    MATH  MathSciNet  Google Scholar 

  79. 79.

    Hughes TJR, Evans JA, Reali A (2013) Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. ICES REPORT 13–24, The Institute for Computational Engineering and Sciences, The University of Texas at Austin.

  80. 80.

    Ibrahimbegović A (2009) Nonlinear solid mechanics: theoretical formulations and finite element solution methods. Springer, Berlin

  81. 81.

    Johannessen KA, Kvamsdal T, Dokken T (2014) Isogeometric analysis using LR B-splines. Comput Methods Appl Mech Eng 269:471–514

    MATH  MathSciNet  Google Scholar 

  82. 82.

    Joulaian M, Düster A (2013) Local enrichment of the finite cell method for problems with material interfaces. Comput Mech 52:741–762

    MATH  Google Scholar 

  83. 83.

    Juntunen J, Stenberg R (2009) Nitsche’s method for general boundary conditions. Math Comput 78:1353–1374

    MATH  MathSciNet  Google Scholar 

  84. 84.

    Kagan P, Fischer A (2000) Integrated mechanically based CAE system using B-spline finite elements. Comput Aided Des 32(8–9):539–552

    MATH  Google Scholar 

  85. 85.

    Keyak JH, Falkinstein Y (2003) Comparison of in situ and in vitro CT scan-based finite element model predictions of proximal femoral fracture load. Med Eng Phys 25(9):781–787

    Google Scholar 

  86. 86.

    Kim H-J, Seo Y-D, Youn S-K (2009) Isogeometric analysis for trimmed CAD surfaces. Comput Methods Appl Mech Eng 198:2982–2995

    MATH  Google Scholar 

  87. 87.

    Kim H-J, Seo Y-D, Youn S-K (2010) Isogeometric analysis with trimming technique for problems of arbitrary complex topology. Comput Methods Appl Mech Eng 199:45–48

    Google Scholar 

  88. 88.

    Kopriva DA (2009) Implementing spectral methods for partial differential equations. Springer, Berlin

    Google Scholar 

  89. 89.

    Krause R, Rank E (2003) Multiscale computations with a combination of the h-and p-versions of the finite element method. Comput Methods Appl Mech Eng 192(35):3959–3983

    MATH  Google Scholar 

  90. 90.

    Kreikemeier J (2012) Modelling of phase boundaries via the GAUSS-Point Method. Technische Mechanik 32(6):658–666

    Google Scholar 

  91. 91.

    Krysl P, Grinspun E, Schröder P (2003) Natural hierarchical refinement for finite element methods. Int J Numer Methods Eng 56:1109–1124

    MATH  Google Scholar 

  92. 92.

    Kudela L (2013) Highly Accurate Subcell Integration in the Context of The Finite Cell Method. Master Thesis, Technische Universität München.

  93. 93.

    Legay A, Wang HW, Belytschko T (2005) Strong and weak arbitrary discontinuities in spectral finite elements. Int J Numer Methods Eng 64:991–1008

    MATH  MathSciNet  Google Scholar 

  94. 94.

    Legrain G (2013) A NURBS enhanced extended finite element approach for unfitted CAD analysis. Comput Mech 1:34

    Google Scholar 

  95. 95.

    Legrain G, Cartraud P, Perreard I, Moës N (2011) An X-FEM and level set computational approach for image-based modelling: application to homogenization. Int J Numer Methods Eng 86(7):915–934

    MATH  Google Scholar 

  96. 96.

    Legrain G, Chevaugeon N, Dréau K (2012) High order X-FEM and levelsets for complex microstructures: uncoupling geometry and approximation. Comput Methods Appl Mech Eng 241:172–189

    Google Scholar 

  97. 97.

    Lew AJ, Buscaglia GC (2008) A discontinuous Galerkin-based immersed boundary method. Int J Numer Methods Eng 76:427–454

    MATH  MathSciNet  Google Scholar 

  98. 98.

    Lew AJ, Negri M (2011) Optimal convergence of a discontinuous-galerkin-based immersed boundary method. ESAIM Math Model Numer Anal 45(04):651–674

    MATH  MathSciNet  Google Scholar 

  99. 99.

    Li Z, Ito K (2006) The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains. Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  100. 100.

    Lipton S, Evans JA, Bazilevs Y, Elguedj T, Hughes TJR (2010) Robustness of isogeometric structural discretizations under severe mesh distortion. Comput Methods Appl Mech Eng 199(5):357–373

    MATH  Google Scholar 

  101. 101.

    Intact Solutions LLC (2009) Scan&solve\(^{TM}\): Fea without meshing (white paper). http://www.intact-solutions.com/Scan&Solve.pdf

  102. 102.

    Löhner R, Cebral RJ, Camelli FE, Appanaboyina S, Baum JD, Mestreau EL, Soto OA (2008) Adaptive embedded and immersed unstructured grid techniques. Comput Methods Appl Mech Eng 197:2173–2197

    MATH  Google Scholar 

  103. 103.

    Lui SH (2009) Spectral domain embedding for elliptic PDEs in complex domains. J Comput Appl Math 225(2):541–557

    MATH  MathSciNet  Google Scholar 

  104. 104.

    Mergheim J, Steinmann P (2006) A geometrically nonlinear FE approach for the simulation of strong and weak discontinuities. Comput Methods Appl Mech Eng 195(37):5037–5052

    MATH  MathSciNet  Google Scholar 

  105. 105.

    Mittal R, Iaccarino G (2005) Immersed boundary methods. Annu Rev Fluid Mech 37:239–261

    MathSciNet  Google Scholar 

  106. 106.

    Moës N, Cloirec M, Cartraud P, Remacle J-F (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192:3163–3177

    MATH  Google Scholar 

  107. 107.

    Moumnassi M, Belouettar S, Béchet E, Bordas SPA, Quoirin D, Potier-Ferry M (2011) Finite element analysis on implicitly defined domains: an accurate representation based on arbitrary parametric surfaces. Comput Methods Appl Mech Eng 200(5):774–796

    MATH  Google Scholar 

  108. 108.

    Mousavi SE, Sukumar N (2010) Generalized gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method. Comput Methods Appl Mech Eng 199(49):3237–3249

    MATH  MathSciNet  Google Scholar 

  109. 109.

    Mousavi SE, Sukumar N (2011) Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons. Comput Mech 47(5):535–554

    MATH  MathSciNet  Google Scholar 

  110. 110.

    Nagy A, Benson DJ (2014) On the numerical integration of trimmed isogeometric elements. Preprint.

  111. 111.

    Neittaanmäki P, Tiba D (1995) An embedding domains approach in free boundary problems and optimal design. SIAM J Control Optim 33(5):1587–1602

    MATH  MathSciNet  Google Scholar 

  112. 112.

    Netz T, Düster A, Hartmann S (2013) High-order finite elements compared to low-order mixed element formulations. ZAMM J Appl Math Mech 93(2–3):163–176

    MATH  Google Scholar 

  113. 113.

    Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wüchner R, Bletzinger KU, Bazilevs Y, Rabczuk T (2011) Rotation free isogeometric thin shell analysis using PHT-splines. Comput Methods Appl Mech Eng 200(47):3410–3424

    MATH  Google Scholar 

  114. 114.

    Nguyen-Thanh N, Nguyen-Xuan H, Bordas SPA, Rabczuk T (2011) Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids. Comput Methods Appl Mech Eng 200(21):1892–1908

    MATH  MathSciNet  Google Scholar 

  115. 115.

    Nitsche JA (1970) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36:9– 15

    MathSciNet  Google Scholar 

  116. 116.

    Noel AT, Szabó BA (1997) Formulation of geometrically non-linear problems in the spatial reference frame. Int J Numer Methods Eng 40(7):1263–1280

    MATH  Google Scholar 

  117. 117.

    Nübel V, Düster A, Rank E (2007) An rp-adaptive finite element method for the deformation theory of plasticity. Comput Mech 39(5):557–574

    MATH  Google Scholar 

  118. 118.

    Parussini L, Pediroda V (2009) Fictitious domain approach with hp-finite element approximation for incompressible fluid flow. J Comput Phys 228(10):3891–3910

    MATH  MathSciNet  Google Scholar 

  119. 119.

    Parvizian J, Düster A, Rank E (2007) Finite cell method: h- and p- extension for embedded domain methods in solid mechanics. Comput Mech 41:122–133

    Google Scholar 

  120. 120.

    Parvizian J, Düster A, Rank E (2012) Toplogy optimization using the finite cell method. Optim Eng 13:57–78

    MATH  MathSciNet  Google Scholar 

  121. 121.

    Peskin C (2002) The immersed boundary method. Acta Numer 11:479–517

    MATH  MathSciNet  Google Scholar 

  122. 122.

    Pham DL, Xu C, Prince JL (2000) A survey of current methods in medical image segmentation. Annu Rev Biomed Eng 2(1):315–337

    Google Scholar 

  123. 123.

    Piegl L, Tiller W (1997) The NURBS book. Springer, Berlin

    Google Scholar 

  124. 124.

    Ramière I, Angot P, Belliard M (2007) A general fictitious domain method with immersed jumps and multilevel nested structured meshes. J Comput Phys 225:1347–1387

    MATH  MathSciNet  Google Scholar 

  125. 125.

    Ramière I, Angot P, Belliard M (2007) A fictitious domain approach with spread interface for elliptic problems with general boundary conditions. Comput Methods Appl Mech Eng 196:766–781

    MATH  Google Scholar 

  126. 126.

    Rangarajan R, Lew AJ, Buscaglia GC (2009) A discontinuous-galerkin-based immersed boundary method with non-homogeneous boundary conditions and its application to elasticity. Comput Methods Appl Mech Eng 198(17):1513–1534

    MATH  MathSciNet  Google Scholar 

  127. 127.

    Ranjbar M, Mashayekhi M, Parvizian J, Düster A, Rank E (2014) Using the finite cell method to predict crack initiation in ductile materials. Comput Mater Sci 82:427–434

    Google Scholar 

  128. 128.

    Rank E (1992) Adaptive remeshing and h-p domain decomposition. Comput Methods Appl Mech Eng 101:299–313

    MATH  Google Scholar 

  129. 129.

    Rank E (1993) A zooming-technique using a hierarchical hp-version of the finite element method. In: Whiteman J (ed) The mathematics of finite elements and applications. John Wiley & Sons, Chichester.

  130. 130.

    Rank E, Düster A, Nübel V, Preusch K, Bruhns OT (2005) High order finite elements for shells. Comput Methods Appl Mech Eng 194:2494–2512

    MATH  Google Scholar 

  131. 131.

    Rank E, Kollmannsberger S, Sorger C, Düster A (2011) Shell finite cell method: a high order fictitious domain approach for thin-walled structures. Comput Methods Appl Mech Eng 200(45):3200–3209

    MATH  Google Scholar 

  132. 132.

    Rank E, Krause R (1997) A multiscale finite element method. Comput Struct 64(1):139–144

    MATH  Google Scholar 

  133. 133.

    Rank E, Ruess M, Kollmannsberger S, Schillinger D, Düster A (2012) Geometric modeling, isogeometric analysis and the finite cell method. Comput Methods Appl Mech Eng 249–250: 104–115

  134. 134.

    Richter T, Wick T (2010) Finite elements for fluid-structure interaction in ale and fully eulerian coordinates. Comput Methods Appl Mech Eng 199:2633–2642

    MATH  MathSciNet  Google Scholar 

  135. 135.

    Rogers DF (2001) An introduction to NURBS with historical perspective. Morgan Kaufmann Publishers, Los Altos

    Google Scholar 

  136. 136.

    Rueberg T, Cirak F (2012) Subdivision-stabilised immersed B-spline finite elements for moving boundary flows. Comput Methods Appl Mech Eng 209–212:266–283

    Google Scholar 

  137. 137.

    Ruess M, Schillinger D, Bazilevs Y, Varduhn V, Rank E (2013) Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. Int J Numer Methods Eng 95(10):811–846

    MathSciNet  Google Scholar 

  138. 138.

    Ruess M, Schillinger D, Özcan AI, Rank E (2014) Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries. Comput Methods Appl Mech Eng 269:46–71

    MATH  Google Scholar 

  139. 139.

    Ruess M, Tal D, Trabelsi N, Yosibash Z, Rank E (2012) The finite cell method for bone simulations: verification and validation. Biomech Model Mechanobiol 11(3):425–437

    Google Scholar 

  140. 140.

    Ruess M, Varduhn V, Yosibash Z, Rank E (2012) A parallel high-order fictitious domain approach for biomechanical applications. In: Parallel and distributed computing, international symposium, pp 279–285.

  141. 141.

    Rvachev VL, Sheiko TL, Shapiro V, Tsukanov I (2000) On completeness of rfm solution structures. Comput Mech 25:305–316

    MATH  MathSciNet  Google Scholar 

  142. 142.

    Rvachev VL, Sheiko TL, Shapiro V, Tsukanov I (2001) Transfinite interpolation over implicitly defined sets. Comput Aided Geom Des 18(3):195–220

    MATH  MathSciNet  Google Scholar 

  143. 143.

    Sadd MH (2009) Elasticity, theory, applications, and numerics. Academic Press, London

    Google Scholar 

  144. 144.

    Samet H (1990) The design and analysis of spatial data structures, vol 199. Addison-Wesley, Reading.

  145. 145.

    Samet H (2006) Foundations of multidimensional and metric data structures. Morgan Kaufmann Publishers, Los Altos

    Google Scholar 

  146. 146.

    Sanches R, Bornemann P, Cirak F (2011) Immersed B-spline (i-spline) finite element method for geometrically complex domains. Comput Methods Appl Mech Eng 200:1432–1445

    MATH  MathSciNet  Google Scholar 

  147. 147.

    Sanders JD, Laursen TA, Puso MA (2012) A Nitsche embedded mesh method. Comput Mech 49(2):243–257

    MATH  MathSciNet  Google Scholar 

  148. 148.

    Sauerland H, Fries TP (2011) The extended finite element method for two-phase and free-surface flows: a systematic study. J Comput Phys 230:3369–3390

    MATH  MathSciNet  Google Scholar 

  149. 149.

    Schileo E, Dall’Ara E, Taddei F, Malandrino A, Schotkamp T, Baleani M, Viceconti M (2008) An accurate estimation of bone density improves the accuracy of subject-specific finite element models. J Biomech 41(11):2483–2491

    Google Scholar 

  150. 150.

    Schillinger D (2012) The \(p\)- and B-spline versions of the geometrically nonlinear finite cell method and hierarchical refinement strategies for adaptive isogeometric and embedded domain analysis. Dissertation, Technische Universität München, http://d-nb.info/103009943X/34

  151. 151.

    Schillinger D, Cai Q, Mundani R-P, Rank E (2013) Nonlinear structural analysis of complex CAD and image based geometric models with the finite cell method. In: Bader M (ed) Lecture notes in computational science and engineering, vol 93. Springer, Berlin

  152. 152.

    Schillinger D, Dede’ L, Scott MA, Evans JA, Borden MJ, Rank E, Hughes TJR (2012) An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput Methods Appl Mech Eng 249–250:116– 150

    MathSciNet  Google Scholar 

  153. 153.

    Schillinger D, Düster A, Rank E (2012) The hp-d adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int J Numer Methods Eng 89:1171–1202

  154. 154.

    Schillinger D, Evans JA, Frischmann F, Hiemstra RR, Hsu M-C, Hughes TJR (2014) Collocation on standard hp finite element meshes: reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics. ICES REPORT 14–01, The University of Texas at Austin

  155. 155.

    Schillinger D, Evans JA, Reali A, Scott MA, Hughes TJR (2013) Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput Methods Appl Mech Eng 267:170–232

    MATH  MathSciNet  Google Scholar 

  156. 156.

    Schillinger D, Hossain SJ, Hughes TJR (2014) Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis. Comput Methods Appl Mech Eng 277:1–45

  157. 157.

    Schillinger D, Kollmannsberger S, Mundani R-P, Rank E (2010) The finite cell method for geometrically nonlinear problems of solid mechanics. IOP Conf Ser Mater Sci Eng 10:012170

    Google Scholar 

  158. 158.

    Schillinger D, Rank E (2011) An unfitted \(hp\) adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry. Comput Methods Appl Mech Eng 200(47–48):3358–3380

    MATH  MathSciNet  Google Scholar 

  159. 159.

    Schillinger D, Ruess M, Düster A, Rank E (2011) The Finite Cell Method for large deformation analysis. PAMM 11(1):271–272

    Google Scholar 

  160. 160.

    Schillinger D, Ruess M, Zander N, Bazilevs Y, Düster A, Rank E (2012) Small and large deformation analysis with the \(p\)- and B-spline versions of the finite cell method. Comput Mech 50(4):445–478

    MATH  MathSciNet  Google Scholar 

  161. 161.

    Schmidt R, Kiendl J, Bletzinger KU, Wüchner R (2010) Realization of an integrated structural design process: analysis-suitable geometric modelling and isogeometric analysis. Comput Vis Sci 13(7):315–330

    MATH  Google Scholar 

  162. 162.

    Scott MA, Li X, Sederberg TW, Hughes TJR (2012) Local refinement of analysis-suitable T-splines. Comput Methods Appl Mech Eng 213–216:206–222

    MathSciNet  Google Scholar 

  163. 163.

    Scott MA, Simpson RN, Evans JA, Lipton S, Bordas SPA, Hughes TJR, Sederberg TW (2013) Isogeometric boundary element analysis using unstructured T-splines. Comput Methods Appl Mech Eng 254:197–221

    MATH  MathSciNet  Google Scholar 

  164. 164.

    Scott MA, Thomas DC, Evans EJ (2014) Isogeometric spline forests. Comput Methods Appl Mech Eng 269:222–264

    MATH  MathSciNet  Google Scholar 

  165. 165.

    Sehlhorst H-G, Jänicke J, Düster A, Rank E, Steeb H, Diebels S (2009) Numerical investigations of foam-like materials by nested high-order finite element methods. Comput Mech 45:45–59

    MATH  Google Scholar 

  166. 166.

    Seo Y-D, Kim H-J, Youn S-K (2010) Isogeometric topology optimization using trimmed spline surfaces. Comput Methods Appl Mech Eng 199:3270–3296

    MATH  MathSciNet  Google Scholar 

  167. 167.

    Shahmiri S, Gerstenberger A, Wall WA (2011) An xfem-based embedding mesh technique for incompressible viscous flows. Int J Numer Methods Fluids 65:166–190

    MATH  MathSciNet  Google Scholar 

  168. 168.

    Shepherd JF, Johnson CR (2008) Hexahedral mesh generation constraints. Eng Comput 24(3):195–213

    Google Scholar 

  169. 169.

    Simpson RN, Scott MA, Taus M, Thomas DC, Lian H (2014) Acoustic isogeometric boundary element analysis. Comput Methods Appl Mech Eng 269:265–290

    MATH  MathSciNet  Google Scholar 

  170. 170.

    Stavrev A (2012) The role of higher-order geometry approximation and accurate quadrature in NURBS based immersed boundary methods. Master Thesis, Technische Universität München.

  171. 171.

    Stenberg R (1998) Mortaring by a method of J.A. Nitsche. In: Idelshon SR, Oñate E, Dvorkin EN (eds) Computational mechanics: new trends and applications. CIMNE, Barcelona, Spain, pp 47–83

  172. 172.

    Sukumar N, Chopp DL, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite-element method. Comput Methods Appl Mech Eng 190:6183–6200

    MATH  Google Scholar 

  173. 173.

    Süli E, Mayers DF (2003) An introduction to numerical analysis. Cambridge University Press, Cambridge

    Google Scholar 

  174. 174.

    Suri M (1996) Analytical and computational assessment of locking in the hp finite element method. Comput Methods Appl Mech Eng 133(3–4):347–371

    MATH  MathSciNet  Google Scholar 

  175. 175.

    Szabó B, Babuška I (1991) Finite element analysis. Wiley, New York

    Google Scholar 

  176. 176.

    Szabó BA, Düster A, Rank E (2004) The p-version of the finite element method. In: Stein E, de Borst R, and Hughes TJR (eds) Encyclopedia of computational mechanics, vol 1, chapter 5. Wiley, New York, pp 119–139.

  177. 177.

    Taddei F, Pani M, Zovatto L, Tonti E, Viceconti M (2008) A new meshless approach for subject-specific strain prediction in long bones: evaluation of accuracy. Clin Biomech 23(9):1192–1199

    Google Scholar 

  178. 178.

    Trabelsi N, Yosibash Z, Milgrom C (2009) Validation of subject-specific automated p-fe analysis of the proximal femur. J Biomech 42(3):234–241

    Google Scholar 

  179. 179.

    Tsukanov I, Shapiro V (2005) Meshfree modeling and analysis of physical fields in heterogeneous media. Adv Comput Math 23:95–124

    MATH  MathSciNet  Google Scholar 

  180. 180.

    Ventura G (2002) An augmented Lagrangian approach to essential boundary conditions in meshless methods. Int J Numer Methods Eng 53(4):825–842

    Google Scholar 

  181. 181.

    Vinci C (2009) Application of Dirichlet boundary conditions in the finite cell method. Master Thesis, Technische Universität München.

  182. 182.

    Šolín P, Segeth K, Doležel I (2004) Higher-order finite element methods. Chapman & Hall/CRC, London

    Google Scholar 

  183. 183.

    Vuong AV, Giannelli C, Jüttler B, Simeon B (2011) A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput Methods Appl Mech Eng 200(49–52):3554–3567

    MATH  Google Scholar 

  184. 184.

    Wall WA, Gamnitzer P, Gerstenberger A (2008) Fluid-structure interaction approaches on fixed grids based on two different domain decomposition ideas. Int J Comput Fluid Dyn 22:411–427

    MATH  MathSciNet  Google Scholar 

  185. 185.

    Wang W, Zhang Y, Scott MA, Hughes TJR (2011) Converting an unstructured quadrilateral mesh to a standard T-spline surface. Comput Mech 48(4):477–498

    MATH  MathSciNet  Google Scholar 

  186. 186.

    Wick T (2013) Fully Eulerian fluid-structure interaction for time-dependent problems. Comput Methods Appl Mech Eng 255:14–26

  187. 187.

    Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin

    Google Scholar 

  188. 188.

    Yang Z, Kollmannsberger S, Düster A, Ruess M, Garcia EG, Burgkart E, Rank E (2012) Non-standard bone simulation: interactive numerical analysis by computational steering. Comput Vis Sci 14:207–216

    Google Scholar 

  189. 189.

    Yang Z, Ruess M, Kollmannsberger S, Düster A, Rank E (2012) An efficient integration technique for the voxel-based finite cell method. Int J Numer Methods Eng 91:457–471

    Google Scholar 

  190. 190.

    Yosibash Z, Padan R, Joskowicz L, Milgrom C (2007) A CT-based high-order finite element analysis of the human proximal femur compared to in-vitro experiments. ASME J Biomech Eng 129:297

  191. 191.

    Yosibash Z, Trabelsi N, Milgrom C (2007) Reliable simulations of the human proximal femur by high-order finite element analysis validated by experimental observations. J Biomech 40(16):3688–3699

    Google Scholar 

  192. 192.

    Yserantant H (1986) On the multi-level splitting of finite element spaces. Numer Math 49:379–412

    MathSciNet  Google Scholar 

  193. 193.

    Zander N (2011) The finite cell method for linear thermoelasticity. Master Thesis, Technische Universität München.

  194. 194.

    Zander N, Bog T, Elhaddad M, Espinoza R, Hu H, Joly AF, Wu C, Zerbe P, Düster A, Kollmannsberger S, Parvizian J, Ruess M, Schillinger D, Rank E (2014) FCMLab: a finite cell research toolbox for MATLAB. Advances in engineering software, submitted.

  195. 195.

    Zander N, Kollmannsberger S, Ruess M, Yosibash Z, Rank E (2012) The finite cell method for linear thermoelasticity. Comput Math Appl 64(11):3527–3541

    MATH  MathSciNet  Google Scholar 

  196. 196.

    Zhang L, Gerstenberger A, Wang X, Liu WK (2004) Immersed finite element method. Comput Methods Appl Mech Eng 193:2051–2067

    MATH  MathSciNet  Google Scholar 

  197. 197.

    Zhang Y, Wang W, Hughes TJR (2012) Solid T-spline construction from boundary representations for genus-zero geometry. Comput Methods Appl Mech Eng 249–252:185–197

  198. 198.

    Zhang Y, Wang W, Hughes TJR (2013) Conformal solid T-spline construction from boundary T-spline representations. Comput Mech 51:1051–1059

    MATH  MathSciNet  Google Scholar 

  199. 199.

    Zhu T, Atluri SN (1998) A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Comput Mech 21:211–222

    MATH  MathSciNet  Google Scholar 

  200. 200.

    Zienkiewicz OC, Taylor RL (2005) The finite element method-solid mechanics, vol 2, 6th edn. Butterworth-Heinemann, London

  201. 201.

    Zienkiewicz OC, Taylor RL (2005) The finite element method-the basis, vol 1, 6th edn. Butterworth-Heinemann, London

  202. 202.

    Zohdi TI, Wriggers P (2001) Aspects of the computational testing of the mechanical properties of microheterogeneous material samples. Int J Numer Methods Eng 50(11):2573–2599

    MATH  Google Scholar 

  203. 203.

    Zohdi TI, Wriggers P (2008) An introduction to computational micromechanics. Springer, Berlin

    Google Scholar 

  204. 204.

    Zorin D, Schröder P, DeRose T, Kobbelt L, Levin A, Sweldens W (2000) Subdivision for modeling and animation. Tech rep.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dominik Schillinger.

Additional information

Dedicated to Ernst Rank on the occasion of his 60th birthday.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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 Computat Methods Eng 22, 391–455 (2015). https://doi.org/10.1007/s11831-014-9115-y

Download citation