Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Sensitivity analysis in poro-elastic and poro-visco-elastic models with respect to boundary data

Authors: H. T. Banks, K. Bekele-Maxwell, L. Bociu, M. Noorman and G. Guidoboni
Journal: Quart. Appl. Math. 75 (2017), 697-735
MSC (2010): Primary 49K40, 49Q12, 74B20, 35Q92, 46N60, 62P10
DOI: https://doi.org/10.1090/qam/1475
Published electronically: July 28, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we consider poro-elastic and poro-visco-elastic models inspired by problems in medicine and biology, and we perform sensitivity analysis on the solutions of these fluid-solid mixture problems with respect to the imposed boundary data, which are the main drivers of the system. Moreover, we compare the results obtained in the elastic case vs. visco-elastic case, as it is known that structural viscosity of biological tissues decreases with age and disease. Sensitivity analysis is the first step towards optimization and control problems associated with these models, which is our ultimate goal.

References [Enhancements On Off] (What's this?)

  • [1] W. Kyle Anderson and Eric J. Nielsen,Sensitivity analysis for Navier-Stokes equations on unstructured meshes using complex variables, AIAA Journal, 39 (2001).
  • [2] W. Kyle Anderson, Eric J. Nielsen, and D.L. Whitfield, Multidisciplinary sensitivity derivatives using complex variables, Technical report, Engineering Research Center Report, Missisipi State University, mSSU-COE-ERC-98-08, July, 1998.
  • [3] Robyn P. Araujo and D. L. Sean McElwain, A mixture theory for the genesis of residual stresses in growing tissues. I. A general formulation, SIAM J. Appl. Math. 65 (2005), no. 4, 1261–1284. MR 2147327, https://doi.org/10.1137/040607113
  • [4]
    H.T. Banks, K. Bekele-Maxwell, L. Bociu, M. Noorman and K. Tillman,
    The complex-step method for sensitivity analysis of non-smooth problems arising in biology,
    Eurasian Journal of Mathematical and Computer Applications 3 (2015), 15-68.
  • [5] H. T. Banks, Kidist Bekele-Maxwell, Lorena Bociu, and Chuyue Wang, Sensitivity via the complex-step method for delay differential equations with non-smooth initial data, Quart. Appl. Math. 75 (2017), no. 2, 231–248. MR 3614496, https://doi.org/10.1090/qam/1458
  • [6]
    H.T. Banks, K. Bekele-Maxwell, L. Bociu, M. Noorman, and G. Guidoboni,
    Sensitivity analysis in poro-elastic and poro-visco-elastic models,
    CRSC-TR17-01, Center for Research in Scientific Computation, N. C. State University, Raleigh, NC, February 2017,
  • [7] H. T. Banks, S. Dediu, and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems, J. Inverse Ill-Posed Probl. 15 (2007), no. 7, 683–708. MR 2374978, https://doi.org/10.1515/jiip.2007.038
  • [8] H. T. Banks, Shuhua Hu, and W. Clayton Thompson, Modeling and inverse problems in the presence of uncertainty, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2014. MR 3203115
  • [9] H. T. Banks and H. T. Tran, Mathematical and experimental modeling of physical and biological processes, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2009. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2488750
  • [10] G.A. Behie, A. Settari, and D.A. Walters, Use of coupled reservoir and geomechanical modeling for integrated reservoir analysis and management, Technical Report, Canadian International Petroleum Conference, Calgary, Canada, 2000.
  • [11]
    M.A. Biot,
    General theory of three-dimensional consolidation,
    J. Appl. Phys., 12(2) (1941), 155-164.
  • [12] Lorena Bociu, Giovanna Guidoboni, Riccardo Sacco, and Justin T. Webster, Analysis of nonlinear poro-elastic and poro-visco-elastic models, Arch. Ration. Mech. Anal. 222 (2016), no. 3, 1445–1519. MR 3544331, https://doi.org/10.1007/s00205-016-1024-9
  • [13]
    Jeff Borggaard, Vitor Leite Nunes
    Fréchet sensitivity analysis for partial differential equations with distributed parameters,
    Proceedings of the 2011 American Control Conference. IEEE, 2011.
  • [14] M.S. Bruno, Geomechanical analysis and decision analysis for mitigating compaction related casing damage, Society of Petroleum Engineers, 2001.
  • [15] Yanzhao Cao, Song Chen, and A. J. Meir, Analysis and numerical approximations of equations of nonlinear poroelasticity, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), no. 5, 1253–1273. MR 3038752, https://doi.org/10.3934/dcdsb.2013.18.1253
  • [16] Yanzhao Cao, Song Chen, and A. J. Meir, Quasilinear poroelasticity: analysis and hybrid finite element approximation, Numer. Methods Partial Differential Equations 31 (2015), no. 4, 1174–1189. MR 3343603, https://doi.org/10.1002/num.21940
  • [17] Paola Causin, Giovanna Guidoboni, Alon Harris, Daniele Prada, Riccardo Sacco, and Samuele Terragni, A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head, Math. Biosci. 257 (2014), 33–41. MR 3272403, https://doi.org/10.1016/j.mbs.2014.08.002
  • [18] D. Chapelle, J.-F. Gerbeau, J. Sainte-Marie, and I. E. Vignon-Clementel, A poroelastic model valid in large strains with applications to perfusion in cardiac modeling, Comput. Mech. 46 (2010), no. 1, 91–101. MR 2644400, https://doi.org/10.1007/s00466-009-0452-x
  • [19] O. Coussy, Poromechanics, Wiley, 2004.
  • [20]
    S.C. Cowin,
    Bone poroelasticity,
    J. Biomech., 32(3) (1999), 217-238.
  • [21]
    E. Detournay and A.H.-D. Cheng,
    Fundamentals of poroelasticity, Chapter 5 in Comprehensive Rock Engineering: Principles, Practice and Projects, Vol. II, Analysis and Design Method, ed. C. Fairhurst,
    Pergamon Press, (1993), 113-171.
  • [22] E. Detournay and A.H.-D. Cheng, Poroelastic response of a borehole in non- hydrostatic stress field, International Journal of Rock Mechanics and Mining Sciences, 25, 1988, 171-182.
  • [23] M.B. Dusseault, M.S. Bruno, and J. Barrera, Casing shear: Causes, cases, cures, Society of Petroleum Engineers, 2001.
  • [24] Arnoldus Joannes Hubertus Frijns, A four-component mixture theory applied to cartilaginous tissues: Numerical modelling and experiments, ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Dr.ir.)–Technische Universiteit Eindhoven (The Netherlands). MR 2715421
  • [25] D. Garagash and E. Detournay, An analysis of the influence of the pressurization rate on the borehole breakdown pressure, Journal of Solids and Structures, 1997.
  • [26] Giovanna Guidoboni, Alon Harris, Lucia Carichino, Yoel Arieli, and Brent A. Siesky, Effect of intraocular pressure on the hemodynamics of the central retinal artery: a mathematical model, Math. Biosci. Eng. 11 (2014), no. 3, 523–546. MR 3153556, https://doi.org/10.3934/mbe.2014.11.523
  • [27]
    C.T. Hsu and P. Cheng,
    Thermal dispersion in a porous medium,
    Int. J. Heat Mass Tran., 33 (8) (1990), pp. 1587-1597.
  • [28] J. Hudson, O. Stephansson, J. Andersson, C.-F. Tsang, and L. Ling, Coupled T-H-M issues related to radioactive waste repository design and performance, International Journal of Rock Mechanics and Mining Sciences, 38:143-161, 2001.
  • [29]
    J.M. Huyghe, T. Arts, D.H. van Campen and R.S. Reneman,
    Porous medium finite element model of the beating left ventricle,
    Am. J. Physiol., 262 (1992), 1256-1267.
  • [30] J.-M. Kim and R. Parizek, Numerical simulation of the Noordbergum effect resulting from groundwater pumping in a layered aquifer system, Journal of Hydrology, 202:231-243, 1997.
  • [31] Stephen M. Klisch, Internally constrained mixtures of elastic continua, Math. Mech. Solids 4 (1999), no. 4, 481–498. MR 1723007, https://doi.org/10.1177/108128659900400405
  • [32]
    W.M. Lai, J.S. Hou and V.C. Mow,
    A triphasic theory for the swelling and deformation behaviors of articular cartilage,
    ASME J. Biomech. Eng., 113 (1991), 245-258.
  • [33] Terri Langford, Northwest Houston sinking faster than coastal areas, Reporter-News.com, Aug. 28 1997.
  • [34] Greg Lemon, John R. King, Helen M. Byrne, Oliver E. Jensen, and Kevin M. Shakesheff, Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory, J. Math. Biol. 52 (2006), no. 5, 571–594. MR 2235518, https://doi.org/10.1007/s00285-005-0363-1
  • [35] N. Lubick, Modeling complex, multiphase porous media systems, SIAM News, 5(3), 2002.
  • [36] J. N. Lyness, Numerical algorithms based on the theory of complex variables, Proc. ACM 22nd Nat. Conf., 4 (1967), 124-134.
  • [37] J. N. Lyness and C. B. Moler, Numerical differentiation of analytic functions, SIAM J. Numer. Anal. 4 (1967), 202–210. MR 0214285, https://doi.org/10.1137/0704019
  • [38] Joaquim R. R. A. Martins, Ilan M. Kroo, and Juan J. Alonso. An automated method for sensitivity analysis using complex variables, AIAA Paper 2000-0689 (Jan.), 2000.
  • [39] Joaquim R. R. A. Martins, Peter Sturdza, and Juan J. Alonso, The complex-step derivative approximation, ACM Trans. Math. Software 29 (2003), no. 3, 245–262. MR 2002731, https://doi.org/10.1145/838250.838251
  • [40]
    Misra S, Macura KJ, Ramesh KT, Okamura AM,
    The Importance of Organ Geometry and Boundary Constraints for Planning of Medical Interventions,
    Medical engineering and physics. 2009;31(2):195-206. doi:10.1016/j.medengphy.2008.08.002.
  • [41]
    V.C. Mow, S.C. Kuei, W.M. Lai and C.G. Armstrong,
    Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments,
    ASME J. Biomech. Eng., 102 (1980), 73-84.
  • [42] Sebastian Owczarek, A Galerkin method for Biot consolidation model, Math. Mech. Solids 15 (2010), no. 1, 42–56. MR 2848828, https://doi.org/10.1177/1081286508090966
  • [43] Phillip Joseph Phillips and Mary F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case, Comput. Geosci. 11 (2007), no. 2, 131–144. MR 2327964, https://doi.org/10.1007/s10596-007-9045-y
  • [44] Phillip Joseph Phillips and Mary F. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. II. The discrete-in-time case, Comput. Geosci. 11 (2007), no. 2, 145–158. MR 2327966, https://doi.org/10.1007/s10596-007-9044-z
  • [45] Phillip Joseph Phillips and Mary F. Wheeler, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity, Comput. Geosci. 12 (2008), no. 4, 417–435. MR 2461315, https://doi.org/10.1007/s10596-008-9082-1
  • [46] Luigi Preziosi and Andrea Tosin, Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications, J. Math. Biol. 58 (2009), no. 4-5, 625–656. MR 2471305, https://doi.org/10.1007/s00285-008-0218-7
  • [47] R. Rajapakse, Stress analysis of borehole in poroelastic medium, Journal of Engineering Mechanics, 119(6):1205-1227, 1993.
  • [48] T. Roose, P.A. Netti, L. Munn, Y. Boucher, and R. Jain, Solid stress generated by spheroid growth estimated using a linear poroelastic model, Microvascular Research, 66:204-212, 2003.
  • [49] J. Rutqvist and C.-F. Tsang, Analysis of thermal-hydrologic-mechanical behavior near an emplacement drift at Yucca mountain, Journal of Contaminant Hydrology, 62-63:637-652, 2003.
  • [50] A. Settari and D.A. Walters, Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction, Technical Report, SPE Reservoir Simulation Symposium, Houston, TX, 1999.
  • [51] R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl. 251 (2000), no. 1, 310–340. MR 1790411, https://doi.org/10.1006/jmaa.2000.7048
  • [52] Alan Smillie, Ian Sobey, and Zoltan Molnar, A hydroelastic model of hydrocephalus, J. Fluid Mech. 539 (2005), 417–443. MR 2262053, https://doi.org/10.1017/S0022112005005707
  • [53] William Squire and George Trapp, Using complex variables to estimate derivatives of real functions, SIAM Rev. 40 (1998), no. 1, 110–112. MR 1612506, https://doi.org/10.1137/S003614459631241X
  • [54] R. E. Showalter and Ning Su, Partially saturated flow in a poroelastic medium, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 4, 403–420. MR 1876882, https://doi.org/10.3934/dcdsb.2001.1.403
  • [55]
    Sun W, Sacks MS, Scott MJ,
    Effects of Boundary Conditions on the Estimation of the Planar Biaxial Mechanical Properties of Soft Tissues,
    ASME. J Biomech Eng. 2005;127(4):709-715. doi:10.1115/1.1933931.
  • [56] C.C. Swan, R.S. Lakes, R.A. Brand, and K.J. Stewart, Micromechanically based poroelastic modeling of fluid flow in haversian bone, Journal of Biomechanical Engineering, 125(1):25-37, Feb. 2003.
  • [57] K. Terzaghi, Principle of soil mechanics, Eng. News Record, A Series of Articles, 1925.
  • [58]
    Michael D Vahey, Daniel A Fletcher,
    The biology of boundary conditions: cellular reconstitution in one, two, and three dimensions, Current Opinion in Cell Biology, Volume 26, February 2014, Pages 60-68, ISSN 0955-0674, http://dx.doi.org/10.1016/j.ceb.2013.10.001. (http://www.sciencedirect.com/science/article/pii/S0955067413001543)
  • [59]
    Stephen D. Waldman, Michael Lee,
    Boundary conditions during biaxial testing of planar connective tissues. Part 1: Dynamic Behavior,
    J. Mater. Sci. Mater. Med. 13.10 (Oct 2002), 933-938.
  • [60] H. F. Wang, Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press, Princeton, N.J., 2000.
  • [61] Y. Wang and M. Dusseault, A coupled conductive-convective thermo-poroelastic solution and implications for wellbore stability, Journal of Petroleum Science and Engineering, 38 (2003), 187-198.
  • [62] Alexander Ženíšek, The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat. 29 (1984), no. 3, 194–211 (English, with Russian and Czech summaries). MR 747212

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 49K40, 49Q12, 74B20, 35Q92, 46N60, 62P10

Retrieve articles in all journals with MSC (2010): 49K40, 49Q12, 74B20, 35Q92, 46N60, 62P10

Additional Information

H. T. Banks
Affiliation: Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212
Email: htbanks@ncsu.edu

K. Bekele-Maxwell
Affiliation: Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212
Email: ktzeleke@ncsu.edu

L. Bociu
Affiliation: Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212
Email: lvbociu@ncsu.edu

M. Noorman
Affiliation: Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212
Email: mjnoorma@ncsu.edu

G. Guidoboni
Affiliation: Indiana University Purdue University Indianapolis, 402 N. Blackford St, LD270, Indianapolis, Indiana 46202-3267
Email: gguidobo@iupui.edu

DOI: https://doi.org/10.1090/qam/1475
Keywords: Sensitivity, poro-elastic, poro-visco-elastic, biological tissues, complex-step method
Received by editor(s): April 28, 2017
Published electronically: July 28, 2017
Article copyright: © Copyright 2017 Brown University

American Mathematical Society