Archive for Rational Mechanics and Analysis

, Volume 222, Issue 3, pp 1445–1519 | Cite as

Analysis of Nonlinear Poro-Elastic and Poro-Visco-Elastic Models

  • Lorena Bociu
  • Giovanna Guidoboni
  • Riccardo Sacco
  • Justin T. Webster
Article

Abstract

We consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid–solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain. In particular, we study the problem of the existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present. The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis. Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory.

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References

  1. 1.
    Araujo R.P., Sean McElwain, Sean McElwain: A mixture theory for the genesis of residual stresses in growing tissues I: a general formulation. SIAM J. Appl. Math. 65(4), 1261–1284 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arnold D.N., Brezzi F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. Modeling and Numer. Anal. 19(1), 7–32 (1985)MathSciNetMATHGoogle Scholar
  3. 3.
    Biot M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)ADSCrossRefMATHGoogle Scholar
  4. 4.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York, 1991Google Scholar
  5. 5.
    Canic S., Tambaca J., Guidoboni G., Mikelic A., Hartley C.J., Rosenstrauch D.: Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow. SIAM J. Appl. Math. 67(1), 164–193 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cao Y., Chen S., Meir A.J.: Analysis and numerical approximations of equations of nonlinear poroelasticity. DCDS-B 18, 1253–1273 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cao Y., Chen S., Meir A.J.: Steady flow in a deformable porous medium. Math. Meth. Appl. Sci. 37, 1029–1041 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cao, Y., Chen, S. Meir, A.J.: Quasilinear poroelasticity: analysis and hybrid finite element approximation. Num. Meth. PDE. (2014). doi:10.1002/num.21940
  9. 9.
    Causin P., Guidoboni G., Harris A., Prada D., Sacco R., Terragni S.: A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head. Math. Biosci. 257, 33–41 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Causin P., Sacco R.:: A discontinuous Petrov–Galerkin method with Lagrangian multipliers for second order elliptic problems. SIAM J. Numer. Anal. 43(1), 280–302 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chapelle D., Sainte-Marie J., Gerbeau J.-F., Vignon-Clementel I.: Aporoelastic model valid in large strains with applications to perfusion in cardiac modeling. Comput. Mech. 46(1), 91–101 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ciarlet, P.G.: Three-Dimensional Elasticity, vol. 1. Elsevier, New York, 1988Google Scholar
  13. 13.
    Cockburn B., Dong B., Guzmán J., Restelli M., Sacco R.: A Hybridizable discontinuous galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput. 31(5), 3827–3846 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cockburn B., Gopalakrishnan J.: A characterization of hybridized mixed methods for second order elliptic problems. SIAM J. Numer. Anal. 42(1), 283–301 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cockburn B., Gopalakrishnan J., Lazarov R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cowin S.C.: Bone poroelasticity. J. Biomech. 32(3), 217–238 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Detournay, E., Cheng, A.H.-D.: Fundamentals of poroelasticity. In: Fairhurst, C. (ed.) Chapter 5 in Comprehensive Rock Engineering: Principles, Practice and Projects, Vol. II, Analysis and Design Method, pp. 113–171. Pergamon Press, 1993Google Scholar
  18. 18.
    Evans, L.: Partial Differential Equations, vol. 19, 2nd edn. AMS, Graduate Studies in Mathematics, 2010Google Scholar
  19. 19.
    Farhloul M.: A mixed finite element method for the Stokes equations. Numer. Methods Partial Differ. Equ. 10(5), 591–608 (1994)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Farhloul M., Fortin M.: A New mixed finite element for the stokes and elasticity problems. SIAM J. Numer. Anal. 30(4), 971–990 (1993)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Frijns, A.J.H.: A Four-Component Mixture Theory Applied to Cartilaginous Tissues: Numerical Modelling and Experiments. Thesis (Dr.ir.)–Technische Universiteit Eindhoven (The Netherlands), 2000Google Scholar
  22. 22.
    Fung, Y.C.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York, 1993Google Scholar
  23. 23.
    Gaspar F.J., Lisbona F.J., Vabishchevich P.N.: Finite difference schemes for poro-elastic problems. Comput. Methods Appl. Math. 2, 132–142 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gaspar F.J., Lisbona F.J., Vabishchevich P.N.: A finite difference analysis of Biot’s consolidation model. Appl. Numer. Math. 44, 487–506 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Herrmann L.R.: Elasticity equations for incompressible and nearly incompressible materials by a variational theorem. AIAA J. 3(10), 1896–1900 (1965)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Hsu C.T., Cheng P.: Thermal dispersion in a porous medium. Int. J. Heat Mass Transf. 33(8), 1587–1597 (1990)CrossRefMATHGoogle Scholar
  27. 27.
    Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Upper Saddle River, 1987Google Scholar
  28. 28.
    Huyghe J.M., Arts T., van Campen D.H., Reneman R.S.: Porous medium finite element model of the beating left ventricle. Am. J. Physiol. 262, 1256–1267 (1992)Google Scholar
  29. 29.
    Kesavan, S.: Topics in Functional Analysis and Applications. New Age International Publishers, 1989Google Scholar
  30. 30.
    Klisch S.M.: Internally constrained mixtures of elastic continua. Math. Mech. Solids 4, 481–498 (1999)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Korsawe J., Starke G., Wang W., Kolditz O.: Finite element analysis of poro-elastic consolidation in porous media: standard and mixed approaches. Comput. Methods Appl. Mech. Eng. 195(9–12), 1096–1115 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Lai W.M., Hou J.S., Mow V.C.: A triphasic theory for the swelling and deformation behaviors of articular cartilage. ASME J. Biomech. Eng. 113, 245–258 (1991)CrossRefGoogle Scholar
  33. 33.
    Langer R.: Perspectives and challenges in tissue engineering and regenerative medicine. Adv. Mater. 21(32–33), 3235–3236 (2009)CrossRefGoogle Scholar
  34. 34.
    Lemon G., King J.R., Byrne H.M., Jensen O.E., Shakesheff K.M.: Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory. J. Math. Biol. 52, 571–594 (2006)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lewis, R.W., Schrefler, B.A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. Wiley, New York, 1998Google Scholar
  36. 36.
    Mader T.H., Gibson C.R., Pass A.F., Kramer L.A., Lee A.G., Fogarty J., Tarver W.J., Dervay J.P., Hamilton D.R., Sargsyan A.E., Phillips J.L., Tran D., Lipsky W., Choi J., Stern C., Kuyumjian R., Polk J.D.: Optic Disc edema, globe flattening, choroidal folds, and hyperopic shifts observed in astronauts after long-duration space flight. Opthalmology 118(10), 2058–2069 (2011)CrossRefGoogle Scholar
  37. 37.
    Mazzucato A.L., Nistor V.: Well-posedness and regularity for the elasticity equations with mixed boundary conditions on polyhedral domains and domains with cracks. Arch. Ration. Mech. Anal. 195, 25–73 (2010)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mow V.C., Kuei S.C., Lai W.M., Armstrong C.G.: Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. ASME J. Biomech. Eng. 102, 73–84 (1980)CrossRefGoogle Scholar
  39. 39.
    Nicaise S.: About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate equation i: regularity of solutions.. Annali della Scuola Normale Superiore di Pisa Classe di Scienze 4e série 19, 327–361 (1992)MathSciNetMATHGoogle Scholar
  40. 40.
    Owczarek S.: A Galerkin method for Biot consolidation model. Math. Mech. Solids 15, 42–56 (2010)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Phillips P.J., Wheeler M.F.: A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity. Comput. Geosci. 12(4), 417–435 (2008)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Phillips, P.J., Wheeler, M.F.: Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosci. 13(1), 5–12 2009Google Scholar
  43. 43.
    Preziosi L., Tosin A.: Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58, 625–656 (2009)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics. Texts in Applied Mathematics, vol. 37. Springer, Berlin, 2007 Google Scholar
  45. 45.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations Springer, New York, 1994Google Scholar
  46. 46.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of Finite Element Methods, I. Springer, Berlin 1977Google Scholar
  47. 47.
    Rempel S., Schulze B.-W.: Mixed boundary value problems for Lamé’s system in three dimensions. Math. Nachr. 119, 265–290 (1984)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Roberts, J.E., Thomas, J.M.: Mixed and hybrid methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Finite Element Methods, Part I, vol. 2. North-Holland, Amsterdam, 1991Google Scholar
  49. 49.
    Savaré, G.: Regularity and perturbation results for mixed second order elliptic problems. Commun. PDE 22 (1997)Google Scholar
  50. 50.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49. AMS, Mathematical Surveys and Monographs, 1996Google Scholar
  51. 51.
    Showalter R.E.: Diffusion in poro-elastic media. J. Math. Anal. Appl. 251, 310–340 (2000)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Showalter R.E.: Diffusion in poro-platic media. Math. Methods Appl. Sci. 27, 2131–2151 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Stewart, D.E.: Dynamics with Inequalities: Impacts and Hard Constraints. SIAM, Philadelphia, 2011Google Scholar
  54. 54.
    Su N., Showalter R.E.: Partially saturated flow in a poroelastic medium. DCDS-B 1, 403–420 (2001)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Zenisek A.: The existence and uniqueness theorem in Biot’s consolidation theory. Appl. Math. 29, 194–211 (1984)MathSciNetMATHGoogle Scholar
  56. 56.
    Zienkiewicz, O.C., Taylor R.L.: The Finite Element Method, 5th edn. Wiley-VCH, Weinheim, 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Lorena Bociu
    • 1
  • Giovanna Guidoboni
    • 2
  • Riccardo Sacco
    • 3
  • Justin T. Webster
    • 4
  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.Department of Mathematical SciencesIndiana University Purdue University IndianapolisIndianapolisUSA
  3. 3.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  4. 4.Department of MathematicsCollege of CharlestonCharlestonUSA

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