# American Institute of Mathematical Sciences

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 & Related Fields, 2019, 9 (4) : 623-642. doi: 10.3934/mcrf.2019044
##### References:

show all references

##### References:
Linear spline approximations of $g_1$ and $\psi_1$
Sensitivity of $u$ with respect to $g_{1h}$ in the direction $\phi_4$.
Sensitivity of $u$ with respect to $\psi_{1h}$ in the direction of $\phi_4$
Sensitivity of $p$ with respect to $g_{1h}$ in the direction of $\phi_4$.
Sensitivity of $p$ with respect to $\psi_{1h}$ in the direction of $\phi_4$.
Sensitivity of $v$ with respect to $g_{1h}$ in the direction of $\phi_4$.
Sensitivity of $v$ with respect to $\psi_{1h}$ in the direction of $\phi_4$.
Linear spline approximations of $g_2$ and $\psi_2$
Sensitivity of $u$ with respect to $g_{2h}$ in the direction of $\phi_4$.
Sensitivity of $u$ with respect to $\psi_{2h}$ in the direction of $\phi_4$.
Sensitivity of $p$ with respect to $g_{2h}$ in the direction of $\phi_4$.
Sensitivity of $p$ with respect to $\psi_{2h}$ in the direction of $\phi_4$.
Sensitivity of $v$ with respect to $g_{2h}$ in the direction of $\phi_4$.
Sensitivity of $v$ with respect to $\psi_{2h}$ in the direction of $\phi_4$
Sensitivity of $v$ with respect to $\psi_{2h}$ in the direction of $\bar{\psi} = \sum\limits_{i = 1}^{13} \phi_i$
 [1] Maria-Magdalena Boureanu, Andaluzia Matei, Mircea Sofonea. Analysis of a contact problem for electro-elastic-visco-plastic materials. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1185-1203. doi: 10.3934/cpaa.2012.11.1185 [2] Khalid Addi, Oanh Chau, Daniel Goeleven. On some frictional contact problems with velocity condition for elastic and visco-elastic materials. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1039-1051. doi: 10.3934/dcds.2011.31.1039 [3] Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016 [4] Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks & Heterogeneous Media, 2008, 3 (3) : 651-673. doi: 10.3934/nhm.2008.3.651 [5] Tuan Anh Dao, Hironori Michihisa. Study of semi-linear $\sigma$-evolution equations with frictional and visco-elastic damping. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1581-1608. doi: 10.3934/cpaa.2020079 [6] Linglong Du. Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks & Heterogeneous Media, 2018, 13 (4) : 549-565. doi: 10.3934/nhm.2018025 [7] Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387 [8] Ming Chen, Chongchao Huang. A power penalty method for the general traffic assignment problem with elastic demand. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1019-1030. doi: 10.3934/jimo.2014.10.1019 [9] Michael Eden, Michael Böhm. Homogenization of a poro-elasticity model coupled with diffusive transport and a first order reaction for concrete. Networks & Heterogeneous Media, 2014, 9 (4) : 599-615. doi: 10.3934/nhm.2014.9.599 [10] J. A. Barceló, M. Folch-Gabayet, S. Pérez-Esteva, A. Ruiz, M. C. Vilela. Elastic Herglotz functions in the plane. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1495-1505. doi: 10.3934/cpaa.2010.9.1495 [11] Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems & Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103 [12] Zhijian Yang, Ke Li. Longtime dynamics for an elastic waveguide model. Conference Publications, 2013, 2013 (special) : 797-806. doi: 10.3934/proc.2013.2013.797 [13] Kewei Zhang. On equality of relaxations for linear elastic strains. Communications on Pure & Applied Analysis, 2002, 1 (4) : 565-573. doi: 10.3934/cpaa.2002.1.565 [14] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [15] Jonatan Lenells. Traveling waves in compressible elastic rods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 151-167. doi: 10.3934/dcdsb.2006.6.151 [16] Giovanni Alberti, Giuseppe Buttazzo, Serena Guarino Lo Bianco, Édouard Oudet. Optimal reinforcing networks for elastic membranes. Networks & Heterogeneous Media, 2019, 14 (3) : 589-615. doi: 10.3934/nhm.2019023 [17] Micol Amar, Daniele Andreucci, Paolo Bisegna, Roberto Gianni. Homogenization limit and asymptotic decay for electrical conduction in biological tissues in the high radiofrequency range. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1131-1160. doi: 10.3934/cpaa.2010.9.1131 [18] Timothy J. Healey. A rigorous derivation of hemitropy in nonlinearly elastic rods. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 265-282. doi: 10.3934/dcdsb.2011.16.265 [19] David Russell. Structural parameter optimization of linear elastic systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1517-1536. doi: 10.3934/cpaa.2011.10.1517 [20] David L. Russell. Trace properties of certain damped linear elastic systems. Evolution Equations & Control Theory, 2013, 2 (4) : 711-721. doi: 10.3934/eect.2013.2.711

2018 Impact Factor: 1.292

## Tools

Article outline

Figures and Tables