Some Use Cases for Composite Finite Elements in Image Based Computing

  • Lars Ole SchwenEmail author
  • Torben Pätz
  • Tobias Preusser
Conference paper


Many bio-medical simulations involve structures of complicated shape. Frequently, the geometry information is given by radiological images. A particular challenge for model discretization in this context is generating appropriate computational meshes.One efficient approach for Finite Element simulations avoiding meshing is the Composite Finite Element approach that has been developed and implemented for image based simulations during the past decade. In the present paper, we provide an overview of previous own work in this field, summarizing the method and showing selected applications: simulation of radio-frequency ablation including vaporization, simulation of elastic deformation of trabecular bone, and numerical homogenization of material properties for the latter.


Trabecular Bone Representative Volume Element Finite Element Space Discontinuous Coefficient Multigrid Solver 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We acknowledge Martin Rumpf, Stefan Sauter, and Uwe Wolfram for their collaboration and many fruitful and inspiring discussions regarding CFE and their applications.


  1. 1.
    G. Allaire, Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, vol. 146 (Springer, New York, 2002)Google Scholar
  2. 2.
    E.J. Berjano, Theoretical modeling for radiofrequency ablation: state-of-the-art and challenges for the future. BioMed. Eng. OnLine 5, 24 (2006)CrossRefGoogle Scholar
  3. 3.
    A. Bonito, R.A. DeVore, R.H. Nochetto, Adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Numer. Anal. 51(6), 3106–3134 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Cazzani, M. Rovati, Extrema of Young’s modulus for cubic and transversely isotropic solids. Int. J. Solids Struct. 40, 1713–1744 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    C. Dick, J. Georgii, R. Westermann, A real-time multigrid finite hexahedra method for elasticity simulation using CUDA. Simul. Model. Pract. Theory 19, 801–816 (2011)CrossRefGoogle Scholar
  6. 6.
    Y. Efendiev, J. Galvis, T.Y. Hou, Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    S. Frei, T. Richter, A locally modified parametric finite element method for interface problems. SIAM J. Numer. Anal. 52(5), 2315–2334 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    L.J. Gibson, Biomechanics of cellular solids. J. Biomech. 38(3), 377–399 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    W. Hackbusch, S.A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math. 75(4), 447–472 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    T.P. Harrigan, M. Jasty, R.W. Mann, W.H. Harris, Limitations of the continuum assumption in cancellous bone. J. Biomech. 21(4), 269–275 (1988)CrossRefGoogle Scholar
  11. 11.
    Q.C. He, A. Curnier, A more fundamental approach to damaged elastic stress-strain relations. Int. J. Solids Struct. 32(10), 1433–1457 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    J.L. Hellrung Jr., L. Wang, E. Sifakis, J.M. Teran, A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions. J. Comput. Phys. 231(4), 2015–2048 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S.J. Hollister, D.P. Fyhrie, K.J. Jepsen, S.A. Goldstein, Application of homogenization theory to the study of trabecular bone mechanics. J. Biomech. 24(9), 825–839 (1991)CrossRefGoogle Scholar
  14. 14.
    S.J. Hollister, N. Kikuchi, A comparison of homogenization and standard mechanics analyses for periodic porous composites. Comput. Mech. 10(2), 73–95 (1992)CrossRefzbMATHGoogle Scholar
  15. 15.
    T. Kröger, I. Altrogge, T. Preusser, P.L. Pereira, D. Schmidt, A. Weihusen, H.O. Peitgen, Numerical simulation of radio frequency ablation with state dependent material parameters in three space dimensions, in MICCAI (2), ed. by R. Larsen, M. Nielsen, J. Sporring. Lecture Notes in Computer Science, vol. 4191 (Springer, New York, 2006), pp. 380–388Google Scholar
  16. 16.
    G. Legrain, P. Cartraud, I. Perreard, N. Moës, An X-FEM and level set computational approach for image-based modelling: application to homogenization. Int. J. Numer. Methods Eng. 86(7), 915–934 (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    X. Li, J. Lowengrub, A. Rätz, A. Voigt, Solving PDEs in complex geometries: a diffuse domain approach. Commun. Math. Sci. 7(1), 81 (2009)Google Scholar
  18. 18.
    F. Liehr, T. Preusser, M. Rumpf, S. Sauter, L.O. Schwen, Composite finite elements for 3D image based computing. Comput. Vis. Sci. 12(4), 171–188 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    T. Pätz, T. Preusser, Composite finite elements for a phase change model. SIAM J. Sci. Comput. 34(5), B672–B691 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    T. Preusser, M. Rumpf, S. Sauter, L.O. Schwen, 3D composite finite elements for elliptic boundary value problems with discontinuous coefficients. SIAM J. Sci. Comput. 33(5), 2115–2143 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    T. Preusser, M. Rumpf, L.O. Schwen, Finite element simulation of bone microstructures, in Proceedings of the 14th Workshop on the Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields, University of Ulm, July 2007, pp. 52–66Google Scholar
  22. 22.
    A.G. Rumpf, Institute for Numerical Simulation, University of Bonn: Quocmesh software library.
  23. 23.
    M. Rumpf, L.O. Schwen, H.J. Wilke, U. Wolfram, Numerical homogenization of trabecular bone specimens using composite finite elements, in The International Journal of Multiphysics, Special Edition: Multiphysics Simulations – Advanced Methods for Industrial Engineering. Selected Contributions from 1st Fraunhofer Multiphysics Conference, 2010, pp. 127–143Google Scholar
  24. 24.
    S.A. Sauter, R. Warnke, Composite finite elements for elliptic boundary value problems with discontinuous coefficients. Computing 77(1), 29–55 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    D. Schillinger, E. Rank, 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), 3358–3380 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    L.O. Schwen, Composite finite elements for trabecular bone microstructures. Ph.D. thesis, University of Bonn, 2010Google Scholar
  27. 27.
    L.O. Schwen, T. Pätz, T. Preusser, Composite finite element simulation of radio frequency ablation and bone elasticity, in Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), Vienna, ed. by J. Eberhardsteiner, et al., September 2012Google Scholar
  28. 28.
    L.O. Schwen, T. Preusser, M. Rumpf, Composite finite elements for 3D elasticity with discontinuous coefficients, in Proceedings of the 16th Workshop on the Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields, University of Ulm, 2009Google Scholar
  29. 29.
    L.O. Schwen, U. Wolfram, Validation of composite finite elements efficiently simulating elasticity of trabecular bone. Comput. Methods Biomech. Biomed. Eng. 17(6), 652–660 (2014)CrossRefGoogle Scholar
  30. 30.
    L.O. Schwen, U. Wolfram, H.J. Wilke, M. Rumpf, Determining effective elasticity parameters of microstructured materials, in Proceedings of the 15th Workshop on the Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields, University of Ulm, July 2008, pp. 41–62Google Scholar
  31. 31.
    J. Stefan, Ueber die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Ann. Phys. 42, 269–286 (1891)CrossRefzbMATHGoogle Scholar
  32. 32.
    T.Stein, Untersuchungen zur Dosimetrie der hochfrequenzstrominduzierten interstitiellen Thermotherapie in bipolarer Technik. Ecomed (2000)Google Scholar
  33. 33.
    K. Stüben, A review of algebraic multigrid. J. Comput. Appl. Math. 128(1–2), 281–309 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    K. Ün, G. Bevill, T.M. Keaveny, The effects of side-artifacts on the elastic modulus of trabecular bone. J. Biomech. 39, 1955–1963 (2006)CrossRefGoogle Scholar
  35. 35.
    U. Wolfram, L.O. Schwen, U. Simon, M. Rumpf, H.J. Wilke, Statistical osteoporosis models using composite finite elements: a parameter study. J. Biomech. 42(13), 2205–2209 (2009)CrossRefGoogle Scholar
  36. 36.
    U. Wolfram, H.J. Wilke, P.K. Zysset, Rehydration of vertebral trabecular bone: Influences on its anisotropy, its stiffness and the indentation work with a view to age, gender and vertebral level. Bone 46, 348–354 (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Lars Ole Schwen
    • 1
    Email author
  • Torben Pätz
    • 1
  • Tobias Preusser
    • 1
    • 2
  1. 1.Fraunhofer MEVISBremenGermany
  2. 2.Jacobs University BremenBremenGermany

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