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

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doi:10.3934/mcrf.2019044

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2019, Volume 9, Issue 4: 623-642. Doi: 10.3934/mcrf.2019044

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

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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
^* Corresponding author: H. Thomas Banks

Received: October 31, 2017

Revised: March 31, 2018

Published: November 2019

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Abstract

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.

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Mathematics Subject Classification: Primary: 49K40, 49Q12, 74B20; Secondary: 35Q92, 46N60, 62P10.

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Figure 1. Linear spline approximations of $ g_1 $ and $ \psi_1 $

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Figure 2. Sensitivity of $ u $ with respect to $ g_{1h} $ in the direction $ \phi_4 $.

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Figure 3. Sensitivity of $ u $ with respect to $ \psi_{1h} $ in the direction of $ \phi_4 $

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Figure 4. Sensitivity of $ p $ with respect to $ g_{1h} $ in the direction of $ \phi_4 $.

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Figure 5. Sensitivity of $ p $ with respect to $ \psi_{1h} $ in the direction of $ \phi_4 $.

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Figure 6. Sensitivity of $ v $ with respect to $ g_{1h} $ in the direction of $ \phi_4 $.

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Figure 7. Sensitivity of $ v $ with respect to $ \psi_{1h} $ in the direction of $ \phi_4 $.

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Figure 8. Linear spline approximations of $ g_2 $ and $ \psi_2 $

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Figure 9. Sensitivity of $ u $ with respect to $ g_{2h} $ in the direction of $ \phi_4 $.

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Figure 10. Sensitivity of $ u $ with respect to $ \psi_{2h} $ in the direction of $ \phi_4 $.

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Figure 11. Sensitivity of $ p $ with respect to $ g_{2h} $ in the direction of $ \phi_4 $.

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Figure 12. Sensitivity of $ p $ with respect to $ \psi_{2h} $ in the direction of $ \phi_4 $.

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Figure 13. Sensitivity of $ v $ with respect to $ g_{2h} $ in the direction of $ \phi_4 $.

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Figure 14. Sensitivity of $ v $ with respect to $ \psi_{2h} $ in the direction of $ \phi_4 $

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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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