291
Views
0
CrossRef citations to date
0
Altmetric
Articles

Optimal control in poroelasticity

&
Pages 1774-1796 | Received 07 Mar 2021, Accepted 11 Nov 2021, Published online: 26 Nov 2021

References

  • Biot MA. General theory of three-dimensional consolidation. J Appl Phys. 1941;12(2):155–164.
  • Detournay E, Cheng AH-D. Poroelastic response of a borehole in non- hydrostatic stress field. Int J Rock Mech Mining Sci. 1988;25:171–182.
  • Dusseault MB, Bruno MS, Barrera J. Casing shear: causes, cases, cures. SPE Dril Comp. 2001;16(2):98–107.
  • Garagash D, Detournay E. An analysis of the influence of the pressurization rate on the borehole breakdown pressure. J Solids and Struct. 1997;34:3099–3118.
  • Kim J-M, Parizek R. Numerical simulation of the Noordbergum effect resulting from groundwater pumping in a layered aquifer system. J Hydrol (Amst). 1997;202:231–243.
  • Langford T. Northwest Houston sinking faster than coastal areas. Reporter-News.com, Aug. 28, 1997.
  • Lubick N. Modeling complex, multiphase porous media systems. SIAM News, Apr. 3, 2002.
  • Rutqvist J, Tsang C-F. Analysis of thermal-hydrologic-mechanical behavior near an emplacement drift at Yucca mountain. J Contam Hydrol. 2003;62–63:637–652.
  • Settari A, Walters DA. Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction. Houston, TX: SPE Reservoir Simulation Symposium; Tech. Report. 1999.
  • Wang HF. Theory of linear poroelasticity with applications to geomechanics and hydrogeology. Princeton (NJ): Princeton University Press; 2000.
  • Wang Y, Dusseault M. A coupled conductive-convective thermo-poroelastic solution and implications for wellbore stability. J Petroleum Sci Eng. 2003;38:187–198.
  • DiSilvestro MR, Suh J-KF. Biphasic poroviscoelastic characteristics of proteoglycan-depleted articular cartilage: simulation of degeneration. Ann Biomed Eng. 2002;30:792–800. Available from: https://doi.org/https://doi.org/10.1114/1.1496088.
  • Mak AF. The apparent viscoelastic behavior of articular cartilage – the contributions from the intrinsic matrix viscoelasticity and interstitial fluid flows. J Biomech Eng. 1986;108:123–130.
  • Setton LA, Zhu W, Mow VC. The biphasic poroviscoelastic behavior of articular cartilage: role of the surface zone in governing the compressive behavior. J Biomech. 1993;26:581–592.
  • Soltz MA, Ateshian GA. Experimental verification and theoretical prediction of cartilage interstitial fluid pressurization at an impermeable contact interface in confined compression. J Biomech. 1998;31:927–934.
  • Suh J-K, Bai S. Finite element formulation of biphasic poroviscoelastic model for articular cartilage. J Biomech Eng. 1998;120:195–201.
  • Causin P, Guidoboni G, Harris A, et al. A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head. Math Biosci. 2014;257:33–41.
  • Roose T, Netti PA, Munn L, et al. Solid stress generated by spheroid growth estimated using a linear poroelastic model. Microvasc Res. 2003;66:204–212.
  • Smillie A, Sobey I, Molnar Z. A hydro-elastic model of hydrocephalus. Oxford University Computing Laboratory: Numerical Analysis Group; 2004. Tech. report.
  • Swan CC, Lakes RS, Brand RA, et al. Micromechanically based poroelastic modeling of fluid flow in Haversian bone. J Biomech Eng. 2003;125(1):25–37.
  • Terzaghi K. Erdbaumechanik auf bodenphysikalischer Grundlage. Wien: Deuticke; 1925.
  • Auriault J-L, Sanchez-Palencia E. A study of the macroscopic behavior of a deformable saturated porous medium. Journal de Mecanique. 1977;10(4):575–603.
  • Banks HT, Bekele-Maxwell K, Bociu L, et al. Local sensitivity via the complex-step derivative approximation for 1-D poro-elastic and poro-visco-elastic models. Math Control Related Fields. 2019;9(4):623–642.
  • Banks HT, Bekele-Maxwell K, Bociu L, et al. Sensitivity analysis in poro-elastic and poro-visco-elastic models with respect to boundary data. Q Appl Math. 2017;75:697–735.
  • Bociu L, Guidoboni G, Sacco R, et al. Analysis of nonlinear poro-elastic and poro-visco-elastic models. Arch Ration Mech Anal. 2016;222:1145–1519.
  • Bociu L, Noorman M. Poro-visco-elastic models in biomechanics: sensitivity analysis. Commun Appl Anal. 2019;23(1):61-–77.
  • Bociu L, Guidoboni G, Sacco R, et al. On the role of compressibility in poroviscoelastic models. Math Biosci Eng. 2019;16(5):6167–6208.
  • Bociu L, Webster J. Nonlinear quasi-static poroelasticity. J Differ Equ. 2021;296:242–278.
  • Cao Y, Chen S, Meir AJ. Analysis and numerical approximations of equations of nonlinear poroelasticity. DCDS-B. 2013;18:1253–1273.
  • Coussy O. Poromechanics. John Wiley & Sons, Ltd; 2004.
  • Owczarek S. A Galerkin method for Biot consolidation model. Math Mech Solids. 2010;15:42–56.
  • Phillips PJ, Wheeler MF. A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity I: the continuous in time case. Comput Geosci. 2007;11(2):131–144.
  • Phillips PJ, Wheeler MF. A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity II: the continuous in time case. Comput Geosci. 2007;11(2):145–158.
  • Phillips PJ, Wheeler MF. A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity. Comput Geosci. 2008;12(4):417–435.
  • Phillips PJ, Wheeler MF. Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput Geosci. 2009;13(1):5–12.
  • Showalter RE. Diffusion in poro-elastic media. J Math Anal Appl. 2000;251:310–340.
  • Su N, Showalter RE. Partially saturated flow in a poroelastic medium. DCDS-B. 2001; 1:403.
  • Verri M, Guidoboni G, Bociu L, et al. The role of structural viscosity in deformable porous media with applications in biomechanics. Math Biosci Eng. 2018;15(4):933–959.
  • Zenisek A. The existence and uniqueness theorem in Biot's consolidation theory. Appl Math. 1984;29:194–211.
  • Lions JL. Optimal control of systems governed by partial differential equations. Berlin: Springer; 1971.
  • Tröltzsch F. Optimal control of partial differential equations, theory, methods, and applications. Vol. 112. Providence (RI): AMS; 2010.
  • Araujo RP, Sean McElwain DL. A mixture theory for the genesis of residual stresses in growing tissues I: a general formulation. SIAM J Appl Math. 2005;65(4):1261–1284.
  • Frijns AJH. A four-component mixture theory applied to cartilaginous tissues: numerical modelling and experiments [Thesis (Dr.ir.)]. The Netherlands: Technische Universiteit Eindhoven; 2000.
  • Klisch SM. Internally constrained mixtures of elastic continua. Math Mech Solids. 1999;4:481–498.
  • Lemon G, King JR, Byrne HM, et al. Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory. J Math Biol. 2006;52:571–594.
  • Preziosi L, Tosin A. Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J Math Biol. 2009;58:625–656.
  • Detournay E, Cheng AH-D. Fundamentals of poroelasticity. In: Fairhurst C, editor. Comprehensive rock engineering: principles, practice and projects. Vol. ii, Chapter 5, analysis and design method. Pergamon Press; 1993. p. 113–171.
  • 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 4 e série. 1992;19:327–361.
  • Showalter RE. Degenerate evolution equations and applications. Indiana University Math J. 1974;23(8):655–677.
  • Evans L. Partial differential equations. 2nd ed. AMS; 2010. (Graduate Studies in Mathematics; 19).
  • Leske MC, Heijl A, Hyman L, et al. Predictors of long-term progression in the early manifest glaucoma trial. Ophthalmology. 2007;114:1965–1972.
  • Guidoboni G, Harris A, Carichino L, et al. Effect of intraocular pressure on the hemodynamics of the central retinal artery: a mathematical model. Math Biosci Eng. 2014;11(3):523–546.
  • Morgan-Davies J, Taylor N, Hill AR, et al. Three dimensional analysis of the lamina cribrosa in glaucoma. Br J Ophthalmol. 2004;88(10):1299–1304.
  • Hollows R, Graham P. Intraocular pressure, glaucoma, and glaucoma suspects in a defined population. Br J Ophthalmol. 1996;50:570–586.
  • Shah R, Wormald R. Glaucoma. Clin. Evid. 9, online.
  • Harris A, Guidoboni G, Siesky B, et al. Ocular blood flow as a clinical observation: value, limitations and data analysis. Progress in Retinal and Eye Res. 2020;78:100841.
  • Weinreb RN, Harris A. Ocular blood flow in glaucoma. Amsterdam, The Netherlands: Kugler Publications; 2009. (World Glaucoma Aassociation Consensus Series).

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.