December  2019, 9(4): 623-642. doi: 10.3934/mcrf.2019044

Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models

1. 

Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8212, USA

2. 

Department of Electrical Engineering and Computer Science, Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

* Corresponding author: H. Thomas Banks

Received  November 2017 Revised  April 2018 Published  November 2019

Poro-elastic systems have been used extensively in modeling fluid flow in porous media in petroleum and earthquake engineering. Nowadays, they are frequently used to model fluid flow through biological tissues, cartilages, and bones. In these biological applications, the fluid-solid mixture problems, which may also incorporate structural viscosity, are considered on bounded domains with appropriate non-homogeneous boundary conditions. The recent work in [12] provided a theoretical and numerical analysis of nonlinear poro-elastic and poro-viscoelastic models on bounded domains with mixed boundary conditions, focusing on the role of visco-elasticity in the material. Their results show that higher time regularity of the sources is needed to guarantee bounded solution when visco-elasticity is not present. Inspired by their results, we have recently performed local sensitivity analysis on the solutions of these fluid-solid mixture problems with respect to the boundary source of traction associated with the elastic structure [3]. Our results show that the solution is more sensitive to boundary datum in the purely elastic case than when visco-elasticity is present in the solid matrix. In this article, we further extend this work in order to include local sensitivities of the solution of the coupled system to the boundary conditions imposed on the Darcy velocity. Sensitivity analysis is the first step in identifying important parameters to control or use as control terms in these poro-elastic and poro-visco-elastic models, which is our ultimate goal.

Citation: H. Thomas Banks, Kidist Bekele-Maxwell, Lorena Bociu, Marcella Noorman, Giovanna Guidoboni. Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models. Mathematical Control and Related Fields, 2019, 9 (4) : 623-642. doi: 10.3934/mcrf.2019044
References:
[1]

W. Kyle Anderson and E. J. Nielsen, Sensitivity analysis for Navier-Stokes equations on unstructured meshes using complex variables, AIAA Journal, 39 (2001).

[2]

W. Kyle Anderson, E. 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]

H. T. BanksK. Bekele-MaxwellL. BociuM. Noorman and G. Guidoboni, Sensitivity analysis in poro-elastic and poro-visco-elastic models, Quart. Appl. Math., 75 (2017), 697-735.  doi: 10.1090/qam/1475.

[4]

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.

[5]

H. T. BanksK. Bekele-MaxwellL. BociuM. 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. 

[6]

H. T. BanksK. Bekele-MaxwellL. Bociu and C. Wang, Sensitivity via the complex-step method for delay differential equations with non-smooth initial data, Quart. Appl. Math., 75 (2017), 231-248.  doi: 10.1090/qam/1458.

[7]

H. T. BanksS. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problem, J. Inverse and Ill-Posed Problems, 15 (2007), 683-708.  doi: 10.1515/jiip.2007.038.

[8] H. T. BanksS. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty, Chapman & Hall/CRC Press, Boca Raton, FL, 2014. 
[9] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, Boca Raton, FL, July, 2008. 
[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 (1941), 155-164.  doi: 10.1063/1.1712886.

[12]

L. BociuG. GuidoboniR. Sacco and J. Webster, Analysis of nonlinear poro-elastic and poro-viscoelastic models, Archive for Rational Mechanics and Analysis, 222 (2016), 1445-1519.  doi: 10.1007/s00205-016-1024-9.

[13]

M. S. Bruno, Geomechanical Analysis and Decision Analysis for Mitigating Compaction Related Casing Damage, Society of Petroleum Engineers, 2001. doi: 10.2118/71695-MS.

[14]

Y. CaoS. Chen and A. J. Meir, Analysis and numerical approximations of equations of nonlinear poroelasticity, DCDS-B, 18 (2013), 1253-1273.  doi: 10.3934/dcdsb.2013.18.1253.

[15]

P. CausinG. GuidoboniA. HarrisD. PradaR. Sacco and S. Terragni, A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head, Mathematical Biosciences, 257 (2014), 33-41.  doi: 10.1016/j.mbs.2014.08.002.

[16]

D. ChapelleJ. Sainte-MarieJ.-F. Gerbeau and I. Vignon-Clementel, A poroelastic model valid in large strains with applications to perfusion in cardiac modeling, Comput. Mech., 46 (2010), 91-101.  doi: 10.1007/s00466-009-0452-x.

[17]

O. Coussy, Poromechanics, Wiley, 2004. doi: 10.1002/0470092718.

[18]

S. C. Cowin, Bone poroelasticity, J. Biomech., 32 (1999), 217-238. 

[19]

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.

[20]

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.  doi: 10.1016/0148-9062(88)92299-1.

[21]

M. B. Dusseault, M. S. Bruno and J. Barrera, Casing Shear: Causes, Cases, Cures, Society of Petroleum Engineers, 2001. doi: 10.2118/48864-MS.

[22]

D. Garagash and E. Detournay, An analysis of the influence of the pressurization rate on the borehole breakdown pressure, Journal of Solids and Structures, 34 (1997), 3099-3118.  doi: 10.1016/S0020-7683(96)00174-6.

[23]

G. GuidoboniA. HarrisL. CarichinoY. Arieli and B. A. Siesky, Effect of intraocular pressure on the hemodynamics of the central retinal artery: A mathematical model, Mathematical Biosciences and Engineering, 11 (2014), 523-546.  doi: 10.3934/mbe.2014.11.523.

[24]

C. T. Hsu and P. Cheng, Thermal dispersion in a porous medium, Int. J. Heat Mass Tran., 33 (1990), 1587-1597.  doi: 10.1016/0017-9310(90)90015-M.

[25]

J. HudsonO. StephanssonJ. AnderssonC.-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 (2001), 143-161. 

[26]

J. M. HuygheT. ArtsD. H. van Campen and R. S. Reneman, Porous medium finite element model of the beating left ventricle, Am. J. Physiol., 262 (1992), 1256-1267.  doi: 10.1152/ajpheart.1992.262.4.H1256.

[27]

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 (1997), 231-243.  doi: 10.1016/S0022-1694(97)00067-X.

[28]

W. M. LaiJ. 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.  doi: 10.1115/1.2894880.

[29]

T. Langford, Northwest Houston Sinking Faster than Coastal Areas, Reporter-News.com, Aug. 28, 1997.

[30]

N. Lubick, Modeling Complex, Multiphase Porous Media Systems, SIAM News, 2002.

[31]

J. R. R. A. Martins, I. M. Kroo and J. J. Alonso., An Automated Method for Sensitivity Analysis Using Complex Variables, AIAA Paper 2000-0689 (Jan.), 2000. doi: 10.2514/6.2000-689.

[32]

J. R. R. A. MartinsP. Sturdza and J. J. Alonso., The complex-step deriva- tive approximation, Journal ACM Transactions on Mathematical Software (TOMS), 29 (2003), 245-262.  doi: 10.1145/838250.838251.

[33]

V. C. MowS. C. KueiW. 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.  doi: 10.1115/1.3138202.

[34]

S. Owczarek, A Galerkin method for Biot consolidation model, Math. Mech. Solids, 15 (2010), 42-56.  doi: 10.1177/1081286508090966.

[35]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity I: The continuous in time case, Computational Geosciences, 11 (2007), 131-144.  doi: 10.1007/s10596-007-9045-y.

[36]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity II: The continuous in time case, Computational Geosciences, 11 (2007), 145-158.  doi: 10.1007/s10596-007-9044-z.

[37]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity, Computational Geosciences, 12 (2008), 417-435.  doi: 10.1007/s10596-008-9082-1.

[38]

R. Rajapakse, Stress analysis of borehole in poroelastic medium, Journal of Engineering Mechanics, 119 (1993), 1205-1227.  doi: 10.1061/(ASCE)0733-9399(1993)119:6(1205).

[39]

T. RooseP. A. NettiL. MunnY. Boucher and R. Jain, Solid stress generated by spheroid growth estimated using a linear poroelastic model, Microvascular Research, 66 (2003), 204-212. 

[40]

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 (2003), 637-652.  doi: 10.1016/S0169-7722(02)00184-5.

[41]

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. doi: 10.2118/51927-MS.

[42]

R. E. Showalter, Diffusion in poro-elastic media, JMAA, 251 (2000), 310-340.  doi: 10.1006/jmaa.2000.7048.

[43]

A. SmillieI. Sobey and Z. Molnar, A hydro-elastic model of hydrocephalus, J. Fluid Mech., 539 (2005), 417-443.  doi: 10.1017/S0022112005005707.

[44]

W. Squire and G. Trapp, Using complex variables to estimate derivatives of real functions, SIAM Review, 40 (1998), 100-112.  doi: 10.1137/S003614459631241X.

[45]

N. Su and R. E. Showalter, Partially saturated flow in a poroelastic medium, DCDS-B, 1 (2001), 403-420.  doi: 10.3934/dcdsb.2001.1.403.

[46]

C. C. SwanR. S. LakesR. A. Brand and K. J. Stewart, Micromechanically based poroelastic modeling of fluid flow in haversian bone, Journal of Biomechanical Engineering, 125 (2003), 25-37.  doi: 10.1115/1.1535191.

[47]

K. Terzaghi, Erdbaumechanik auf bodenphysikalischer Grundlage, Deuticke, Wien, 1925.

[48] H. F. Wang, Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press, Princeton, N.J., 2001.  doi: 10.1515/9781400885688.
[49]

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.  doi: 10.1016/S0920-4105(03)00032-9.

[50]

A. Zenisek, The existence and uniqueness theorem in Biot's consolidation theory, Appl. Math., 29 (1984), 194-211. 

show all references

References:
[1]

W. Kyle Anderson and E. J. Nielsen, Sensitivity analysis for Navier-Stokes equations on unstructured meshes using complex variables, AIAA Journal, 39 (2001).

[2]

W. Kyle Anderson, E. 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]

H. T. BanksK. Bekele-MaxwellL. BociuM. Noorman and G. Guidoboni, Sensitivity analysis in poro-elastic and poro-visco-elastic models, Quart. Appl. Math., 75 (2017), 697-735.  doi: 10.1090/qam/1475.

[4]

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.

[5]

H. T. BanksK. Bekele-MaxwellL. BociuM. 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. 

[6]

H. T. BanksK. Bekele-MaxwellL. Bociu and C. Wang, Sensitivity via the complex-step method for delay differential equations with non-smooth initial data, Quart. Appl. Math., 75 (2017), 231-248.  doi: 10.1090/qam/1458.

[7]

H. T. BanksS. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problem, J. Inverse and Ill-Posed Problems, 15 (2007), 683-708.  doi: 10.1515/jiip.2007.038.

[8] H. T. BanksS. Hu and W. C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty, Chapman & Hall/CRC Press, Boca Raton, FL, 2014. 
[9] H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, Boca Raton, FL, July, 2008. 
[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 (1941), 155-164.  doi: 10.1063/1.1712886.

[12]

L. BociuG. GuidoboniR. Sacco and J. Webster, Analysis of nonlinear poro-elastic and poro-viscoelastic models, Archive for Rational Mechanics and Analysis, 222 (2016), 1445-1519.  doi: 10.1007/s00205-016-1024-9.

[13]

M. S. Bruno, Geomechanical Analysis and Decision Analysis for Mitigating Compaction Related Casing Damage, Society of Petroleum Engineers, 2001. doi: 10.2118/71695-MS.

[14]

Y. CaoS. Chen and A. J. Meir, Analysis and numerical approximations of equations of nonlinear poroelasticity, DCDS-B, 18 (2013), 1253-1273.  doi: 10.3934/dcdsb.2013.18.1253.

[15]

P. CausinG. GuidoboniA. HarrisD. PradaR. Sacco and S. Terragni, A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head, Mathematical Biosciences, 257 (2014), 33-41.  doi: 10.1016/j.mbs.2014.08.002.

[16]

D. ChapelleJ. Sainte-MarieJ.-F. Gerbeau and I. Vignon-Clementel, A poroelastic model valid in large strains with applications to perfusion in cardiac modeling, Comput. Mech., 46 (2010), 91-101.  doi: 10.1007/s00466-009-0452-x.

[17]

O. Coussy, Poromechanics, Wiley, 2004. doi: 10.1002/0470092718.

[18]

S. C. Cowin, Bone poroelasticity, J. Biomech., 32 (1999), 217-238. 

[19]

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.

[20]

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.  doi: 10.1016/0148-9062(88)92299-1.

[21]

M. B. Dusseault, M. S. Bruno and J. Barrera, Casing Shear: Causes, Cases, Cures, Society of Petroleum Engineers, 2001. doi: 10.2118/48864-MS.

[22]

D. Garagash and E. Detournay, An analysis of the influence of the pressurization rate on the borehole breakdown pressure, Journal of Solids and Structures, 34 (1997), 3099-3118.  doi: 10.1016/S0020-7683(96)00174-6.

[23]

G. GuidoboniA. HarrisL. CarichinoY. Arieli and B. A. Siesky, Effect of intraocular pressure on the hemodynamics of the central retinal artery: A mathematical model, Mathematical Biosciences and Engineering, 11 (2014), 523-546.  doi: 10.3934/mbe.2014.11.523.

[24]

C. T. Hsu and P. Cheng, Thermal dispersion in a porous medium, Int. J. Heat Mass Tran., 33 (1990), 1587-1597.  doi: 10.1016/0017-9310(90)90015-M.

[25]

J. HudsonO. StephanssonJ. AnderssonC.-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 (2001), 143-161. 

[26]

J. M. HuygheT. ArtsD. H. van Campen and R. S. Reneman, Porous medium finite element model of the beating left ventricle, Am. J. Physiol., 262 (1992), 1256-1267.  doi: 10.1152/ajpheart.1992.262.4.H1256.

[27]

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 (1997), 231-243.  doi: 10.1016/S0022-1694(97)00067-X.

[28]

W. M. LaiJ. 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.  doi: 10.1115/1.2894880.

[29]

T. Langford, Northwest Houston Sinking Faster than Coastal Areas, Reporter-News.com, Aug. 28, 1997.

[30]

N. Lubick, Modeling Complex, Multiphase Porous Media Systems, SIAM News, 2002.

[31]

J. R. R. A. Martins, I. M. Kroo and J. J. Alonso., An Automated Method for Sensitivity Analysis Using Complex Variables, AIAA Paper 2000-0689 (Jan.), 2000. doi: 10.2514/6.2000-689.

[32]

J. R. R. A. MartinsP. Sturdza and J. J. Alonso., The complex-step deriva- tive approximation, Journal ACM Transactions on Mathematical Software (TOMS), 29 (2003), 245-262.  doi: 10.1145/838250.838251.

[33]

V. C. MowS. C. KueiW. 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.  doi: 10.1115/1.3138202.

[34]

S. Owczarek, A Galerkin method for Biot consolidation model, Math. Mech. Solids, 15 (2010), 42-56.  doi: 10.1177/1081286508090966.

[35]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity I: The continuous in time case, Computational Geosciences, 11 (2007), 131-144.  doi: 10.1007/s10596-007-9045-y.

[36]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity II: The continuous in time case, Computational Geosciences, 11 (2007), 145-158.  doi: 10.1007/s10596-007-9044-z.

[37]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity, Computational Geosciences, 12 (2008), 417-435.  doi: 10.1007/s10596-008-9082-1.

[38]

R. Rajapakse, Stress analysis of borehole in poroelastic medium, Journal of Engineering Mechanics, 119 (1993), 1205-1227.  doi: 10.1061/(ASCE)0733-9399(1993)119:6(1205).

[39]

T. RooseP. A. NettiL. MunnY. Boucher and R. Jain, Solid stress generated by spheroid growth estimated using a linear poroelastic model, Microvascular Research, 66 (2003), 204-212. 

[40]

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 (2003), 637-652.  doi: 10.1016/S0169-7722(02)00184-5.

[41]

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. doi: 10.2118/51927-MS.

[42]

R. E. Showalter, Diffusion in poro-elastic media, JMAA, 251 (2000), 310-340.  doi: 10.1006/jmaa.2000.7048.

[43]

A. SmillieI. Sobey and Z. Molnar, A hydro-elastic model of hydrocephalus, J. Fluid Mech., 539 (2005), 417-443.  doi: 10.1017/S0022112005005707.

[44]

W. Squire and G. Trapp, Using complex variables to estimate derivatives of real functions, SIAM Review, 40 (1998), 100-112.  doi: 10.1137/S003614459631241X.

[45]

N. Su and R. E. Showalter, Partially saturated flow in a poroelastic medium, DCDS-B, 1 (2001), 403-420.  doi: 10.3934/dcdsb.2001.1.403.

[46]

C. C. SwanR. S. LakesR. A. Brand and K. J. Stewart, Micromechanically based poroelastic modeling of fluid flow in haversian bone, Journal of Biomechanical Engineering, 125 (2003), 25-37.  doi: 10.1115/1.1535191.

[47]

K. Terzaghi, Erdbaumechanik auf bodenphysikalischer Grundlage, Deuticke, Wien, 1925.

[48] H. F. Wang, Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press, Princeton, N.J., 2001.  doi: 10.1515/9781400885688.
[49]

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.  doi: 10.1016/S0920-4105(03)00032-9.

[50]

A. Zenisek, The existence and uniqueness theorem in Biot's consolidation theory, Appl. Math., 29 (1984), 194-211. 

Figure 1.  Linear spline approximations of $ g_1 $ and $ \psi_1 $
Figure 2.  Sensitivity of $ u $ with respect to $ g_{1h} $ in the direction $ \phi_4 $.
Figure 3.  Sensitivity of $ u $ with respect to $ \psi_{1h} $ in the direction of $ \phi_4 $
Figure 4.  Sensitivity of $ p $ with respect to $ g_{1h} $ in the direction of $ \phi_4 $.
Figure 5.  Sensitivity of $ p $ with respect to $ \psi_{1h} $ in the direction of $ \phi_4 $.
Figure 6.  Sensitivity of $ v $ with respect to $ g_{1h} $ in the direction of $ \phi_4 $.
Figure 7.  Sensitivity of $ v $ with respect to $ \psi_{1h} $ in the direction of $ \phi_4 $.
Figure 8.  Linear spline approximations of $ g_2 $ and $ \psi_2 $
Figure 9.  Sensitivity of $ u $ with respect to $ g_{2h} $ in the direction of $ \phi_4 $.
Figure 10.  Sensitivity of $ u $ with respect to $ \psi_{2h} $ in the direction of $ \phi_4 $.
Figure 11.  Sensitivity of $ p $ with respect to $ g_{2h} $ in the direction of $ \phi_4 $.
Figure 12.  Sensitivity of $ p $ with respect to $ \psi_{2h} $ in the direction of $ \phi_4 $.
Figure 13.  Sensitivity of $ v $ with respect to $ g_{2h} $ in the direction of $ \phi_4 $.
Figure 14.  Sensitivity of $ v $ with respect to $ \psi_{2h} $ in the direction of $ \phi_4 $
Figure 15.  Sensitivity of $ v $ with respect to $ \psi_{2h} $ in the direction of $ \bar{\psi} = \sum\limits_{i = 1}^{13} \phi_i $
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