The Finite Cell Method: A Higher Order Fictitious Domain Approach for Wave Propagation Analysis in Heterogeneous Structures

  • S. DuczekEmail author
  • U. GabbertEmail author
Part of the Research Topics in Aerospace book series (RTA)


In this chapter a recently developed novel approach to simulate the propagation of ultrasonic guided waves in heterogeneous, especially cellular lightweight structures is presented. One of the most important drawbacks of traditional finite element-based approaches is the need for a geometry-conforming discretization. It is generally acknowledged that the mesh generation process constitutes the bottleneck in the current simulation pipeline. Therefore, different measures have been taken to at least alleviate the meshing bruden. One of these attempts is the finite cell method (FCM). It combines the advantages known from higher order finite element methods (FEM; exponential convergence rates) with those of fictitious domain methods (FDM; automatic mesh generation using Cartesian grids). In the framework of the FCM we do not rely on body-fitted discretizations and therefore shift the effort typically required for the mesh generation to the numerical integration of the system matrices which is performed by means of an adaptive Gaussian quadrature. The advantage of such a procedure in the context of wave propagation analysis is seen in the fully automated analysis process. Consequently, hardly any user input is required making the simulation very robust. Notwithstanding, it can be mathematically proven that the optimal rates of convergence are retained.


  1. 1.
    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 69:37–47CrossRefzbMATHGoogle Scholar
  2. 2.
    Abedian A, Parvizian J, Düster A, Rank E (2014) The FCM compared to the h-version FEM for elasto-plastic problems. Appl Math Mech 35:1239–1248CrossRefGoogle Scholar
  3. 3.
    Almeida JPM, Pereira OJBA (1996) A set of hybrid equilibrium finite element models for the analysis of three-dimensional solids. Int J Numer Methods Eng 39:2789–2802CrossRefzbMATHGoogle Scholar
  4. 4.
    Belytschko T, Chen JS (2009) Meshfree and particle methods. Wiley, ChichesterGoogle Scholar
  5. 5.
    Blyth MG, Pozrikidis C (2006) A Lobatto interpolation grid over the triangle. IMA J Appl Math 71:153–169MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cattani C, Paoluzzi A (1990) Boundary integration over linear polyhedra. Comput Aided Des 22:130–135CrossRefzbMATHGoogle Scholar
  7. 7.
    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 199:334–356MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, HobokenCrossRefGoogle Scholar
  9. 9.
    Dasgupta G (2003) Integration within polygonal finite elements. J Aerosp Eng 16:9–18CrossRefGoogle Scholar
  10. 10.
    Dauge M, Düster A, Rank E (2015) Theoretical and numerical investigation of the finite cell method. J Sci Comput 65:1039–1064MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Duarte C, Babuška I, Oden J (2000) Generalized finite element method for three-dimensional structural mechanics problems. Comput Struct 77(2):215–232MathSciNetCrossRefGoogle Scholar
  12. 12.
    Duczek S (2014) Higher order finite elements and the fictitious domain concept for wave propagation analysis. VDI Fortschritt-Berichte Reihe 20 Nr. 458Google Scholar
  13. 13.
    Duczek S, Gabbert U (2015) Efficient integration method for fictitious domain approaches. Comput Mech 56:725–738MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Duczek S, Gabbert U (2016) The finite cell method for polygonal meshes: polygonal-FCM. Comput Mech 58:587–618MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duczek S, Joulaian M, Düster A, Gabbert U (2014) Numerical analysis of Lamb waves using the finite and spectral cell methods. Int J Numer Methods Eng 99:26–53MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Duczek S, Liefold S, Gabbert U (2015) The finite and spectral cell methods for smart structure applications: transient analysis. Acta Mech 226:845–869MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Duczek S, Duvigneau F, Gabbert U (2016) The finite cell method for arbitrary tetrahedral meshes. Finite Elem Anal Des 121:18–32MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dumonet D (2014) Towards efficient and accurate 3d cut cell integration in the context of the finite cell method. Master’s thesis, Technical University MunichGoogle Scholar
  19. 19.
    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 196:3524–3533MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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 197:3768–3782MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Düster A, Sehlhorst HG, Rank E (2012) Numerical homogenization of heterogeneous and cellular materials utilizing the finite cell method. Comput Mech 1:1–19MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gao XW (2002) The radial integration method for evaluation of domain integrals with boundary-only discretization. Eng Anal Bound Elem 26:905–916CrossRefzbMATHGoogle Scholar
  23. 23.
    Gonzales-Ochoa C, McCammon S, Peters J (1998) Computing moments of objects enclosed by piecewise polynomial surfaces. ACM Trans Graph 17:143–157CrossRefGoogle Scholar
  24. 24.
    Hematiyan MR (2007) A general method for evaluation of 2d and 3d domain integrals without domain discretization and its application in BEM. Comput Mech 39:509–520CrossRefzbMATHGoogle Scholar
  25. 25.
    Hubrich S, Joulaian M, Düster A (2015) Numerical integration in the finite cell method based on moment-fitting. In: Proceedings of the 3rd ECCOMAS young investigators conference and 6th GACM colloquium – YIC GACM 2015Google Scholar
  26. 26.
    Joulaian M, Düster A (2013) Local enrichment of the finite cell method for problems with material interfaces. Comput Mech 52:741–762CrossRefzbMATHGoogle Scholar
  27. 27.
    Joulaian M, Duczek S, Gabbert U, Düster A (2014) Finite and spectral cell method for wave propagation in heterogeneous materials. Comput Mech 54:661–675MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kaufmann P, Martin S, Botsch M, Gross M (2009) Flexible simulation of deformable models using discontinuous Galerkin FEM. Graph Model 71:153–167CrossRefGoogle Scholar
  29. 29.
    Komatitsch D, Tromp J (2002) Spectral-element simulations of global seismic wave propagation I. – Validation. Int J Geophys 149:390–412CrossRefGoogle Scholar
  30. 30.
    Komatitsch D, Tromp J (2002) Spectral-element simulations of global seismic wave propagation II. – Three-dimensional models, oceans, rotation and self-gravitation. Int J Geophys 150:303–318CrossRefGoogle Scholar
  31. 31.
    Kudela L (2013) Highly accurate subcell integration in the context of the finite cell method. Master’s thesis, Technical University MunichGoogle Scholar
  32. 32.
    Kudela L, Zander N, Bog T, Kollmannsberger S, Rank E (2015) Efficient and accurate numerical quadrature for immersed boundary methods. Adv Model Simul Eng Sci 2–10:1–22Google Scholar
  33. 33.
    Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55:1–34CrossRefGoogle Scholar
  34. 34.
    Luo H, Pozrikidis C (2006) A Lobatto interpolation grid in the tetrahedron. IMA J Appl Math 71:298–313MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mirtich B (1996) Fast and accurate computation of polyhedral mass properties. J Graph Tools 1:31–50CrossRefGoogle Scholar
  36. 36.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 64:131–150CrossRefzbMATHGoogle Scholar
  37. 37.
    Mousavi SE, Sukumar N (2011) Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons. Comput Mech 47:535–554MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mousavi SE, Xiao H, Sukumar N (2010) Generalized Gaussian quadrature rules on arbitrary polygons. Int J Numer Methods Eng 82:99–113MathSciNetzbMATHGoogle Scholar
  39. 39.
    Müller B, Kummer F, Oberlack M (2013) Highly accurate surface and volume integration on implicit domains by means of moment-fitting. Int J Numer Methods Eng 96:512–528MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ostachowicz W, Kudela P, Krawczuk M, \(\dot{\text{Z}}\) ak A (2011) Guided waves in structures for SHM: the time-domain spectral element method. Wiley, HobokenGoogle Scholar
  41. 41.
    Parvizian J, Düster A, Rank E (2007) Finite cell method: h- and p-extension for embedded domain problems in solid mechanics. Comput Mech 41:121–133MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Parvizian J, Düster A, Rank E (2012) Topology optimization using the finite cell method. Optim Eng 13:57–78MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Patera AT (1984) A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J Comput Phys 54:468–488CrossRefzbMATHGoogle Scholar
  44. 44.
    Persson PO (2005) Mesh generation for implicit geometries. PhD thesis, Massachusetts Institute of TechnologyGoogle Scholar
  45. 45.
    Pozrikidis C (2005) Introduction to finite and spectral methods using MATLAB. Chapman and Hall, Boca RatonzbMATHGoogle Scholar
  46. 46.
    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 82:427–434CrossRefGoogle Scholar
  47. 47.
    Rathod HT, Govinda Rao HS (1995) Integration of polynomials over linear polyhedra in Euclidean three-dimensional space. Comput Methods Appl Mech Eng 126:373–392MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Ruess M, Tal D, Trabelsi N, Yosibash Z, Rank E (2012) The finite cell method for bone simulations: verification and validation. Biomech Model Mechanobiol 11:425–437CrossRefGoogle Scholar
  49. 49.
    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 95:811–846MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Schillinger D, Ruess M (2015) The finite cell method: a review in the context of high-order structural analysis of CAD and image-based geometric models. Arch Comput Methods Eng 22:391–455MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    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 89:1171–1202MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    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 50:445–478MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Schillinger D, Cai Q, Mundani RP, Rank E (2013) A Review of the Finite Cell Method for Nonlinear Structural Analysis of Complex CAD and Image-Based Geometric Models. In: Advanced computing lecture notes in computational science and engineering, vol 93. Springer, Dordrecht, pp 1–23Google Scholar
  54. 54.
    Sehlhorst HG (2011) Numerical homogenization strategies for cellular materials with applications in structural mechanics. VDI Fortschritt-Berichte Reihe 18 Nr. 333Google Scholar
  55. 55.
    Sommariva A, Vianello M (2007) Product Gauss cubature over polygons based on Green’s integration formula. BIT Numer Math 47:441–453MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Sudhakar Y, Moitinho de Almeida JP, Wall WA (2014) An accurate, robust, and easy-to-implement method for integration over arbitrary polyhedra: application to embedded interface methods. J Comput Phys 273:393–415CrossRefzbMATHGoogle Scholar
  57. 57.
    Szabó B, Babuška I (1991) Finite element analysis. Wiley, New YorkzbMATHGoogle Scholar
  58. 58.
    Szabó B, Babuška I (2011) Introduction to finite element analysis: formulation, verification and validation. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  59. 59.
    Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45:309–328MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Trimmer HG, Stern JM (1980) Computation of global geometric properties of solid objects. Comput Aided Des 12:301–304CrossRefGoogle Scholar
  61. 61.
    Varduhn V, Hsu MC, Ruess M, Schillinger D (2016) The tetrahedral finite cell method: high-order immersogeometric analysis on adaptive non-boundary-fitted meshes. Int J Numer Methods Eng online:1–26Google Scholar
  62. 62.
    Verhoosel CV, van Zwieten G J, van Rietbergen B, de Borst R (2015) Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone. Comput Methods Appl Mech Eng 284:138–164MathSciNetCrossRefGoogle Scholar
  63. 63.
    Wassermann B, Kollmannsberger S, Bog T, Rank E (2017) From geometric design to numerical analysis: a direct approach using the finite cell method on constructive solid geometry. Comput Math Appl 24. doi: [Online]
  64. 64.
    Wassouf Z (2010) Die Mortar Methode für Finite Elemente hoher Ordnung. PhD thesis, Technical University MunichGoogle Scholar
  65. 65.
    Wenisch P, Wenisch O (2004) Fast octree-based voxelization of 3d boundary representation-objects. Tech. rep., Technical University MunichGoogle Scholar
  66. 66.
    Xu F, Schillinger D, Kamensky D, Varduhn V, Wang C, Hsu MC (2015) The tetrahedral finite cell method for fluids: immersogeometric analysis of turbulent flow around complex geometries. Comput Fluids online:1–20Google Scholar
  67. 67.
    Yang Z (2011) The finite cell method for geometry-based structural simulation. PhD thesis, Technical University MunichGoogle Scholar
  68. 68.
    Yang Z, Kollmannsberger S, Düster A, Ruess M, Grande Garcia E, Burgkart R, Rank E (2011) Non-standard bone simulation: interactive numerical analysis by computational steering. Comput Vis Sci 14:207–216MathSciNetCrossRefGoogle Scholar
  69. 69.
    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 91:457–471CrossRefGoogle Scholar
  70. 70.
    Zander N, Kollmannsberger S, Ruess M, Yosibash Z, Rank E (2012) The finite cell method for linear thermoelasticity. Comput Math Appl 64:3527–3541MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute of Mechanics, Otto von Guericke University MagdeburgMagdeburgGermany

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