Boundary Sensitivity via the Complex-Step Derivative Approximation for 1D Poro-Elastic and Poro-Visco-Elastic Models
Thursday, September 7, 2017 - 9:00am - 9:35am
Poro-elastic systems have been used extensively in modeling fluid flow in porous media in petroleum and earthquake engineering. Currently, 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. In this presentation we consider the Lamina Cribrosa, a thin, mesh-like tissue at the base of the optic nerve head. The LC maintains the difference between the intraocular pressure (IOP) (inside the eye) and the retrolaminar tissue pressure (RLTp) (in the optic nerve head). It is suspected that the LC and its biomechanics plays an important role in glaucoma. 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. We use the complex-step method to compute and compare relative sensitives for the analysis on the solutions of these fluid-solid mixture problems with respect to the boundary source of traction associated with the elastic structure. The associated problems lack smoothness (lack analyticity) but we use recent efforts for delay differential equations (DDEs) with non-smooth (discontinuous and even distributional) history function to motivate our development of the complex-step method in problems that lack analyticity. Our results demonstrate that the complex-step method performs quite well even when the underlining smoothness assumptions for the traditional Cauchy-Riemann derivation of the the method do not hold. This represents joint efforts with Kidist Bekele-Maxwell, Lorena Bociu, Marcella Noorman, Giovanna Guidoboni, and Monica Wang.