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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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)
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)
Biot M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York, 1991
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)
Cao Y., Chen S., Meir A.J.: Analysis and numerical approximations of equations of nonlinear poroelasticity. DCDS-B 18, 1253–1273 (2013)
Cao Y., Chen S., Meir A.J.: Steady flow in a deformable porous medium. Math. Meth. Appl. Sci. 37, 1029–1041 (2014)
Cao, Y., Chen, S. Meir, A.J.: Quasilinear poroelasticity: analysis and hybrid finite element approximation. Num. Meth. PDE. (2014). doi:10.1002/num.21940
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)
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)
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)
Ciarlet, P.G.: Three-Dimensional Elasticity, vol. 1. Elsevier, New York, 1988
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)
Cockburn B., Gopalakrishnan J.: A characterization of hybridized mixed methods for second order elliptic problems. SIAM J. Numer. Anal. 42(1), 283–301 (2004)
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)
Cowin S.C.: Bone poroelasticity. J. Biomech. 32(3), 217–238 (1999)
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, 1993
Evans, L.: Partial Differential Equations, vol. 19, 2nd edn. AMS, Graduate Studies in Mathematics, 2010
Farhloul M.: A mixed finite element method for the Stokes equations. Numer. Methods Partial Differ. Equ. 10(5), 591–608 (1994)
Farhloul M., Fortin M.: A New mixed finite element for the stokes and elasticity problems. SIAM J. Numer. Anal. 30(4), 971–990 (1993)
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), 2000
Fung, Y.C.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York, 1993
Gaspar F.J., Lisbona F.J., Vabishchevich P.N.: Finite difference schemes for poro-elastic problems. Comput. Methods Appl. Math. 2, 132–142 (2002)
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)
Herrmann L.R.: Elasticity equations for incompressible and nearly incompressible materials by a variational theorem. AIAA J. 3(10), 1896–1900 (1965)
Hsu C.T., Cheng P.: Thermal dispersion in a porous medium. Int. J. Heat Mass Transf. 33(8), 1587–1597 (1990)
Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Upper Saddle River, 1987
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)
Kesavan, S.: Topics in Functional Analysis and Applications. New Age International Publishers, 1989
Klisch S.M.: Internally constrained mixtures of elastic continua. Math. Mech. Solids 4, 481–498 (1999)
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)
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)
Langer R.: Perspectives and challenges in tissue engineering and regenerative medicine. Adv. Mater. 21(32–33), 3235–3236 (2009)
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)
Lewis, R.W., Schrefler, B.A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. Wiley, New York, 1998
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)
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)
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)
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)
Owczarek S.: A Galerkin method for Biot consolidation model. Math. Mech. Solids 15, 42–56 (2010)
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)
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 2009
Preziosi L., Tosin A.: Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58, 625–656 (2009)
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics. Texts in Applied Mathematics, vol. 37. Springer, Berlin, 2007
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations Springer, New York, 1994
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 1977
Rempel S., Schulze B.-W.: Mixed boundary value problems for Lamé’s system in three dimensions. Math. Nachr. 119, 265–290 (1984)
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, 1991
Savaré, G.: Regularity and perturbation results for mixed second order elliptic problems. Commun. PDE 22 (1997)
Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49. AMS, Mathematical Surveys and Monographs, 1996
Showalter R.E.: Diffusion in poro-elastic media. J. Math. Anal. Appl. 251, 310–340 (2000)
Showalter R.E.: Diffusion in poro-platic media. Math. Methods Appl. Sci. 27, 2131–2151 (2004)
Stewart, D.E.: Dynamics with Inequalities: Impacts and Hard Constraints. SIAM, Philadelphia, 2011
Su N., Showalter R.E.: Partially saturated flow in a poroelastic medium. DCDS-B 1, 403–420 (2001)
Zenisek A.: The existence and uniqueness theorem in Biot’s consolidation theory. Appl. Math. 29, 194–211 (1984)
Zienkiewicz, O.C., Taylor R.L.: The Finite Element Method, 5th edn. Wiley-VCH, Weinheim, 2002
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Bociu, L., Guidoboni, G., Sacco, R. et al. Analysis of Nonlinear Poro-Elastic and Poro-Visco-Elastic Models. Arch Rational Mech Anal 222, 1445–1519 (2016). https://doi.org/10.1007/s00205-016-1024-9
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DOI: https://doi.org/10.1007/s00205-016-1024-9