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The role of weakly imposed Dirichlet boundary conditions for numerically stable computations of swelling phenomena

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Abstract

It is still a challenge to model swelling phenomena occurring in charged hydrated porous media. This is not only due to the overall complexity of the model but also to the fact that boundary conditions occur, which depend on internal variables. In the present contribution, a multi-component model based on the Theory of Porous Media (TPM) is presented. The advantage of this model is that it is thermodynamically consistent and it consists of only three primary variables. As a result of the boundary conditions depending on internal variables, the numerical treatment within the finite element method (FEM) by use of the mixed finite element scheme reveals artificial oscillations in the numerical results. To overcome these oscillations, we propose to fulfil boundary conditions weakly.

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References

  1. Acartürk A, Ehlers W, Abbas I (2004) Modelling of swelling phenomena in charged hydrated porous media. PAMM 4: 296–297

    Article  Google Scholar 

  2. Bathe K-J (1996) Finite element procedures, 2nd edn. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  3. Boer R (2000) Theory of porous media. Springer, Berlin

    MATH  Google Scholar 

  4. de Boer R, Ehlers W (1986) Theorie der Mehrkomponentenkontinua mit Anwendungen auf bodenmechanische Probleme. Forschungsberichte aus dem Fachbereich Bauwesen Heft 40, Universität-GH-Essen

  5. Bowen RM (1976) Theory of mixtures. In: Eringen AC(eds) Continuum physics, vol III, mixtures and EM field theories. Academic Press, London, pp 1–127

    Google Scholar 

  6. Bowen RM (1980) Incompressible porous media models by use of the theory of mixtures. Int J Eng Sci 18: 1129–1148

    Article  MATH  Google Scholar 

  7. Chapelle D, Bathe KJ (1993) The inf-sup test. Comput Struct 47: 537–545

    Article  MATH  MathSciNet  Google Scholar 

  8. Donnan FG (1911) Theorie der Membrangleichgewichte und Membranpotentiale bei Vorhandensein von nicht dialysierenden Elektrolyten. Ein Beitrag zur physikalisch-chemischen Physiologie. Zeitschrift für Elektrochemie und Angewandte Physikalische Chemie 17: 572–581

    Google Scholar 

  9. Dowell EH, Hall KC (2001) Modeling of fluid–structure interaction. Ann Rev Fluid Mech 33: 445–490

    Article  Google Scholar 

  10. Ehlers W (1989) Poröse Medien—ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie. Forschungsberichte aus dem Fachbereich Bauwesen, Heft 47, Universität-GH-Essen

  11. Ehlers W (2002) Foundations of multiphasic and porous materials. In: Ehlers W, Bluhm J(eds) Porous media: theory, experiments and numerical applications. Springer, Berlin, pp 3–86

    Google Scholar 

  12. Ehlers W, Eipper G (1999) Finite elastic deformations in liquid-saturated and empty porous solids. Transp Porous Media 34: 179–191

    Article  MathSciNet  Google Scholar 

  13. Ehlers W, Ellsiepen P (2001) Theoretical and numerical methods in environmental continuum mechanics based on the Theory of Porous Media. In: Schrefler BA (ed) Environmental geomechanics. Springer, Wien, CISM Courses and Lectures No. 417, pp 1–81

  14. Ehlers W, Karajan N, Markert B (2006) A porous media model describing the inhomogeneous behaviour of the human intervertebral disc. Materialwissenschaften und Wekstofftechnik 37: 546–551

    Article  Google Scholar 

  15. Ehlers W, Karajan N, Markert B (2008) An extended biphasic model for charged hydrated tissues with application to the intervertebral disc. Biomech Model Mechanobiol. doi:10.1007/s10237-008-0129-y

  16. Ehlers W, Markert B, Acartürk A (2005a) Swelling phenomena of hydrated porous materials. In: Abousleiman YN, Cheng AH-D, Ulm FJ(eds) Poromechanics III, Proceedings of the 3rd Biot Conference on Poromechanics. Balkema, Leiden, pp 781–786

    Google Scholar 

  17. Ehlers W, Markert B, Karajan N, Acartürk A (2005b) A coupled FE analysis of the intervertebral disc based on a multiphasic TPM formulation. In: Holzapfel GA, Ogden RW(eds) IUTAM symposium on mechanics of biological tissue. Springer, Wien, pp 373–386

    Google Scholar 

  18. Felippa CA, Park KC (1980) Staggered transient analysis procedures for coupled mechanical systems: formulation. Comput Methods Appl Mech Eng 24: 61–111

    Article  MATH  Google Scholar 

  19. Felippa CA, Park KC, Farhat C (2001) Partitioned analysis of coupled mechanical systems. Comput Methods Appl Mech Eng 190: 3247–3270

    Article  MATH  Google Scholar 

  20. Frijns AJH, Huyghe JM, Janssen JD (1997) A validation of the quadriphasic mixture theory for intervertebral disc tissue. Int J Eng Sci 35: 1419–1429

    Article  MATH  Google Scholar 

  21. Frijns AJH, Huyghe JM, Kaasschieter EF, Wijlaars MW (2003) Numerical simulation of deformations and electrical potentials in a cartilage substitute. Biorheology 40: 123–131

    Google Scholar 

  22. Gu WY, Lai WM, Mow VC (1997) A triphasic analysis of negative osmotic flows through charged hydrated soft tissues. J Biomech 30: 71–78

    Article  Google Scholar 

  23. Hansbo P (1995) Lagrangian incompressible flow computations in three dimensions by use of space-time finite elements. Int J Numer Methods Fluids 20: 989–1001

    Article  MATH  MathSciNet  Google Scholar 

  24. Hansbo P, Hermansson J (2003) Nitsche’s method for coupling non-matching meshes in fluid-structure vibration problems. Comput Mech 32: 134–139

    Article  MATH  Google Scholar 

  25. Hirt CW, Amsden AA, Cook JL (1974) An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J Comput Phys 14: 227–253

    Article  MATH  Google Scholar 

  26. Hirt CW, Cook JL, Butler TD (1970) A Lagrangian method for calculating the dynamics of an incompressible fluid with free surface. J Comput Phys 5: 103–124

    Article  MATH  Google Scholar 

  27. Huyghe JM, Janssen JD (1997) Quadriphasic mechanics of swelling incompressible porous media. Int J Eng Sci 35: 793–802

    Article  MATH  Google Scholar 

  28. Kaasschieter EF, Frijns AJH, Huyghe JMRJ (2003) Mixed finite element modelling of cartilaginous tissues. Math Comput Simul 61: 549–560

    Article  MATH  MathSciNet  Google Scholar 

  29. Lai WM, Hou JS, Mow VC (1991) A triphasic theory for the swelling and deformation behaviours of articular cartilage. ASME J Biomech Eng 113: 245–258

    Article  Google Scholar 

  30. Lai WM, Mow VC, Sun DD, Ateshian GA (2000) On the electric potentials inside a charged soft hydrated biological tissue: Streaming potential vs. diffusion potential. ASME J Biomech Eng 122: 336–346

    Article  Google Scholar 

  31. Lanir Y (1987) Biorheology and fluidflux in swelling tissues. I. Bicomponent theory for small deformations, including concentration effects. Biorheology 24: 173–187

    Google Scholar 

  32. van Loon R, Huyghe JM, Wijlaars MW, Baaijens FPT (2003) 3D FE implementation of an incompressible quariphasic mixture model. Int J Numer Methods Eng 57: 1243–1258

    Article  MATH  Google Scholar 

  33. Mow VC, Ateshian GA, Lai WM, Gu WY (1998) Effects of fixed charges on the stress-relaxation behavior of hydrated soft tissues in a confined compression problem. Int J Solids Struct 35: 4945–4962

    Article  MATH  Google Scholar 

  34. Mow VC, Kuei SC, Lai WM, Armstrong CG (1980) Biphasic creep and relaxation of articular cartilage in compression: theory and experiments. ASME J Biomech Eng 102: 73–84

    Article  Google Scholar 

  35. Mow VC, Ratcliffe A (1997) Structure and function of articular cartilage and meniscus. In: Mow VC, Hayes WC(eds) Basic orthopaedic biomechanics, 2nd edn.. Lippincott-Raven, Philadelphia, pp 113–176

    Google Scholar 

  36. Radovitzky R, Ortiz M (1998) Lagrangian finite element analysis of newtonian fluid flows. Int J Numer Methods Eng 43: 607–619

    Article  MATH  MathSciNet  Google Scholar 

  37. Ramaswamy B, Kawahara M (1987) Lagrangian finite element analysis applied to viscous free surface fluid flow. Int J Numer Methods Fluids 7: 953–984

    Article  MATH  Google Scholar 

  38. Scardovelli R, Zaleski S (1999) Direct numerical simulation of free-surface and interfacial flow. Ann Rev Fluid Mech 31: 567–603

    Article  MathSciNet  Google Scholar 

  39. Snijders H, Huyghe JM, Janssen JD (1995) Triphasic finite element model for swelling porous media. Int J Numer Methods Fluids 20: 1039–1046

    Article  MATH  Google Scholar 

  40. Sun DN, Gu WY, Guo XE, Mow WMLVC (1999) A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues. Int J Numer Methods Eng 45: 1375–1402

    Article  MATH  Google Scholar 

  41. Tezduyar TE (2001) Finite element methods for flow problems with moving boundaries and interfaces. Arch Comput Methods Eng 8: 83–130

    Article  MATH  Google Scholar 

  42. Wall WA (1999) Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen. Dissertation, Institut für Baustatik, Universität Stuttgart

  43. Wall WA, Genkinger S, Ramm E (2007) A strong coupling partitioned approach for fluid–structure interaction with free surfaces. Comput Fluids 36: 169–183

    Article  MATH  Google Scholar 

  44. Wilson W, van Donkelaar CC, Huyghe JM (2005) A comparison between mechano-electrochemical and biphasic swelling theories for soft hydrated tissue. ASME J Biomech Eng 127: 158–165

    Article  Google Scholar 

  45. Zienkiewicz OC, Qu S, Taylor RL, Nakazawa S (1986) The patch test for mixed formulations. Int J Numer Methods Eng 23: 1873–1883

    Article  MATH  MathSciNet  Google Scholar 

  46. Zienkiewicz OC, Taylor RL (2000) The finite element method, vol 1, 5th edn. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  47. Zienkiewicz OC, Taylor RL, Sherwin SJ, Peiró J (2003) On discontinuous galerkin methods. Int J Numer Methods Eng 58: 1119–1148

    Article  MATH  Google Scholar 

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  1. Institute of Applied Mechanics (CE), University of Stuttgart, Pfaffenwaldring 7, 70 569, Stuttgart, Germany

    W. Ehlers & A. Acartürk

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  1. W. Ehlers
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  2. A. Acartürk
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Ehlers, W., Acartürk, A. The role of weakly imposed Dirichlet boundary conditions for numerically stable computations of swelling phenomena. Comput Mech 43, 545–557 (2009). https://doi.org/10.1007/s00466-008-0329-4

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