Skip to main content
SpringerLink
Log in

Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

The Finite Cell Method (FCM) is an embedded domain method, which combines the fictitious domain approach with high-order finite elements, adaptive integration, and weak imposition of unfitted Dirichlet boundary conditions. For smooth problems, FCM has been shown to achieve exponential rates of convergence in energy norm, while its structured cell grid guarantees simple mesh generation irrespective of the geometric complexity involved. The present contribution first unhinges the FCM concept from a special high-order basis. Several benchmarks of linear elasticity and a complex proximal femur bone with inhomogeneous material demonstrate that for small deformation analysis, FCM works equally well with basis functions of the p-version of the finite element method or high-order B-splines. Turning to large deformation analysis, it is then illustrated that a straightforward geometrically nonlinear FCM formulation leads to the loss of uniqueness of the deformation map in the fictitious domain. Therefore, a modified FCM formulation is introduced, based on repeated deformation resetting, which assumes for the fictitious domain the deformation-free reference configuration after each Newton iteration. Numerical experiments show that this intervention allows for stable nonlinear FCM analysis, preserving the full range of advantages of linear elastic FCM, in particular exponential rates of convergence. Finally, the weak imposition of unfitted Dirichlet boundary conditions via the penalty method, the robustness of FCM under severe mesh distortion, and the large deformation analysis of a complex voxel-based metal foam are addressed.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bathe K-J (1996) Finite element procedures. Prentice Hall, Upper Saddle River

    Google Scholar 

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

    MATH  Google Scholar 

  3. 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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  5. 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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Peskin C (2002) The immersed boundary method. Acta Numer 11: 1–39

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. 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

    Article  MATH  Google Scholar 

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

    Google Scholar 

  11. 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

    Article  MathSciNet  MATH  Google Scholar 

  12. 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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Baaijens FPT (2001) A fictitious domain/mortar element method for fluid-structure interaction. Int J Numer Methods Fluids 35: 743–761

    Article  MathSciNet  MATH  Google Scholar 

  15. Farhat C, Hetmaniuk U (2002) A fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems. Part I: Dirichlet boundary conditions. Int J Numer Methods Eng 54: 1309–1332

    Article  MathSciNet  MATH  Google Scholar 

  16. Haslinger J, Kozubek T, Kunisch K, Peichl G (2003) Shape optimization and fictitious domain approach for solving free boundary problems of bernoulli type. Comput Optim Appl 26: 231–251

    Article  MathSciNet  MATH  Google Scholar 

  17. Burman E, Hansbo P (2010) Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Comput Methods Appl Mech Eng 199: 2680–2686

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Burman E, Hansbo P (2011) Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl Numer Math. doi:10.1016/j.apnum.2011.01.008

  20. 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

    Article  MATH  Google Scholar 

  21. Gerstenberger A, Wall WA (2008) An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction. Comput Methods Appl Mech Eng 197: 1699–1714

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Becker R, Burman E, Hansbo P (2011) A hierarchical NXFEM for fictitious domain simulations. Int J Numer Methods Eng 86(4–5): 54–559

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. 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

    Article  Google Scholar 

  29. Düster A, Parvizian J, Yang Z, Rank E (2008) The Finite Cell Method for three-dimensional problems of solid mechanics. Comput Methods Appl Mech Eng 197: 3768–3782

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  31. Szabó B, Düster A, Rank E (2004) The p-version of the finite element method. In: Stein E, Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 1. Wiley, Chichester, pp 119–139

    Google Scholar 

  32. Rank E, Düster A, Schillinger D, Yang Z (2009) The Finite Cell Method: high order simulation of complex structures without meshing. In: Yuan Y, Cui J, Mang H (eds) Computational structural engineering. Springer, Heidelberg, pp 87–92

    Chapter  Google Scholar 

  33. Ruess M, Tal D, Trabelsi N, Yosibash Z, Rank E (2011) The Finite Cell Method for bone simulations: verification and validation. Biomech Model Mechanobiol. doi:10.1007/s10237-011-0322-2

  34. Schillinger D, Düster A, Rank E (2011) The hp-d adaptive Finite Cell Method for geometrically nonlinear problems of solid mechanics. Int J Numer Methods Eng. doi:10.1002/nme.3289

  35. Vos PEJ, van Loon R, Sherwin SJ (2008) A comparison of fictitious domain methods appropriate for spectral/hp element discretisations. Comput Methods Appl Mech Eng 197: 2275–2289

    Article  MATH  Google Scholar 

  36. Parvizian J, Düster A, Rank E (2011) Topology optimization using the Finite Cell Method. Opt Eng. doi:10.1007/s11081-011-9159-x

  37. 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: 3200–3209

    Article  MATH  Google Scholar 

  38. Vinci C (2009) Application of Dirichlet boundary conditions in the Finite Cell Method. Master thesis, Lehrstuhl für Computation in Engineering, Technische Universität München

  39. Zander N (2011) The Finite Cell Method for linear thermoelasticity. Master thesis, Lehrstuhl für Computation in Engineering, Technische Universität München

  40. Abedian A, Parvizian J, Düster A, Khademyzadeh H, Rank E (2010) The Finite Cell Method for elasto-plastic problems. In: Proceedings of the 10th international conference on computational structures technology, Valencia, 2010

  41. Cai Q, Kollmannsberger S, Mundani RP, Rank E (2011) The Finite Cell Method for solute transport problems in porous media. In: Proceedings of the 16th international conference on finite elements in flow problems, Munich, 2011

  42. Düster A, Sehlhorst HG, Rank E (2012) Numerical homogenization of heterogeneous and cellular materials utilizing the Finite Cell Method. Comp Mech. doi:10.1007/s00466-012-0681-2

  43. 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 (accepted)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  47. Samet H (2006) Foundations of multidimensional and metric data structures. Morgan Kaufmann Publishers, San Francisco

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  49. Düster A (2001) High order finite elements for three-dimensional, thin-walled nonlinear continua. PhD thesis, Lehrstuhl für Bauinformatik, Technische Universität München, Shaker, Aachen

  50. 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

    Article  MATH  Google Scholar 

  51. 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

  52. 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

    Google Scholar 

  53. Piegl LA, Tiller W (1997) The NURBS book (monographs in visual communication). Springer, New York

    Book  Google Scholar 

  54. Rogers DF (2001) An introduction to NURBS with historical perspective. Morgan Kaufmann Publishers, San Francisco

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Google Scholar 

  57. 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

    Article  MathSciNet  MATH  Google Scholar 

  58. Babuška I, Banerjee U, Osborn JE (2002) Meshless and generalized finite element methods: a survey of some major results. In: Griebel M, Schweitzer MA (eds) Meshfree methods for partial differential equations. Lecture notes in computational science and engineering, vol 26. Springer, Berlin

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

    Article  MathSciNet  MATH  Google Scholar 

  60. Zienkiewicz OC, Taylor RL (2005) The finite element method vol 1: the basis. Butterworth-Heinemann, Oxford

    Google Scholar 

  61. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  63. 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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  65. Sadd MH (2009) Elasticity: theory, applications, and numerics. Academic Press, Oxford

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  68. Bungartz HJ, Griebel M, Zenger C (2004) Introduction to computer graphics. Charles River Media, Hingham

    Google Scholar 

  69. Novelline R (1997) Squire’s fundamentals of radiology. Harvard University Press, Cambridge

    Google Scholar 

  70. 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. J Biomech Eng 129: 297–309

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  78. Sehlhorst HG, Jänicke R, 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

    Article  MATH  Google Scholar 

  79. Krause R, Mücke R, Rank E (1995) hp-version finite elements for geometrically non-linear problems. Commun Numer Methods Eng 11: 887–897

    Article  MATH  Google Scholar 

  80. Bertsekas DP (1996) Constrained optimization and lagrange multiplier methods. Athina Scientific, Nashua

    Google Scholar 

  81. Trilinos Version 10.4, Sandia National Laboratories, Los Alamos, NM. http://trilinos.sandia.gov/

  82. MKL PARDISO. http://www.intel.com/software/products/mkl. 16 January 2012

  83. FlagShyp 08 Version 2.30, Nonlinear FE solver developed by J. Bonet and R. Wood. http://www.flagshyp.com

  84. Visual DoMesh 2008 Version 1.1, Mesh generator developed by C. Sorger, Chair for Computation in Engineering, Technische Universität München

  85. Netgen Version 4.9.13, Tetrahdral mesh generator developed by J. Schöberl. http://sourceforge.net/projects/netgen-mesher

  86. ParaView Version 3.8.1, Open-source scientific visualization package. Kitware Inc., Clifton Park, NY. http://www.paraview.org

  87. Dong S, Yosibash Z (2009) A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems. Comput Struct 87(1–2): 59–72

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  89. 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: 357–373

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  91. Zohdi T, Wriggers P (2005) Introduction to computational micromechanics. Springer, Berlin

    Book  MATH  Google Scholar 

  92. Lapack Version 3.2.2. (2010) Linear algebra package. http://www.netlib.org/lapack/

  93. Chapman B, Jost G, van der Pas R (2008) Using OpenMP. Portable shared memory parallel programming. MIT Press, Cambridge

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl für Computation in Engineering, Department of Civil Engineering and Surveying, Technische Universität München, Arcisstr. 21, 80333, Munich, Germany

    Dominik Schillinger, Martin Ruess, Nils Zander & Ernst Rank

  2. Department of Structural Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, USA

    Yuri Bazilevs

  3. Numerische Strukturanalyse mit Anwendungen in der Schiffstechnik (M-10), Technische Universität Hamburg-Harburg, Schwarzenbergstr. 95c, 21073, Hamburg, Germany

    Alexander Düster

Authors
  1. Dominik Schillinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Ruess
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nils Zander
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yuri Bazilevs
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Alexander Düster
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ernst Rank
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dominik Schillinger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schillinger, D., Ruess, M., Zander, N. et al. Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method. Comput Mech 50, 445–478 (2012). https://doi.org/10.1007/s00466-012-0684-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-012-0684-z

Keywords

Search

Navigation