# 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
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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 $\bar{\psi} = \sum\limits_{i = 1}^{13} \phi_i$
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