108 research outputs found
A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model
We study a finite element computational model for solving the coupled problem
arising in the interaction between a free fluid and a fluid in a poroelastic
medium. The free fluid is governed by the Stokes equations, while the flow in
the poroelastic medium is modeled using the Biot poroelasticity system.
Equilibrium and kinematic conditions are imposed on the interface. A mixed
Darcy formulation is employed, resulting in continuity of flux condition of
essential type. A Lagrange multiplier method is employed to impose weakly this
condition. A stability and error analysis is performed for the semi-discrete
continuous-in-time and the fully discrete formulations. A series of numerical
experiments is presented to confirm the theoretical convergence rates and to
study the applicability of the method to modeling physical phenomena and the
sensitivity of the model with respect to its parameters
Robust Discretization of Flow in Fractured Porous Media
Flow in fractured porous media represents a challenge for discretization
methods due to the disparate scales and complex geometry. Herein we propose a
new discretization, based on the mixed finite element method and mortar
methods. Our formulation is novel in that it employs the normal fluxes as the
mortar variable within the mixed finite element framework, resulting in a
formulation that couples the flow in the fractures with the surrounding domain
with a strong notion of mass conservation. The proposed discretization handles
complex, non-matching grids, and allows for fracture intersections and
termination in a natural way, as well as spatially varying apertures. The
discretization is applicable to both two and three spatial dimensions. A priori
analysis shows the method to be optimally convergent with respect to the chosen
mixed finite element spaces, which is sustained by numerical examples
Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach
We develop a computational model to study the interaction of a fluid with a
poroelastic material. The coupling of Stokes and Biot equations represents a
prototype problem for these phenomena, which feature multiple facets. On one
hand it shares common traits with fluid-structure interaction. On the other
hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical
simulation of the Stokes-Biot coupled system is a challenging task. The need of
large memory storage and the difficulty to characterize appropriate solvers and
related preconditioners are typical shortcomings of classical discretization
methods applied to this problem. The application of loosely coupled time
advancing schemes mitigates these issues because it allows to solve each
equation of the system independently with respect to the others. In this work
we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot
equations. The scheme is based on Nitsche's method for enforcing interface
conditions. Once the interface operators corresponding to the interface
conditions have been defined, time lagging allows us to build up a loosely
coupled scheme with good stability properties. The stability of the scheme is
guaranteed provided that appropriate stabilization operators are introduced
into the variational formulation of each subproblem. The error of the resulting
method is also analyzed, showing that splitting the equations pollutes the
optimal approximation properties of the underlying discretization schemes. In
order to restore good approximation properties, while maintaining the
computational efficiency of the loosely coupled approach, we consider the
application of the loosely coupled scheme as a preconditioner for the
monolithic approach. Both theoretical insight and numerical results confirm
that this is a promising way to develop efficient solvers for the problem at
hand
Mathematical and Numerical Modeling of Inflammation
When the body is attacked by a bacterial infection, it initiates a series of events designed to eradicate the infection while causing minimal damage to the body. Our goal is to investigate the defenses of the organ walls to the spread of infection. To do this we have chosen to model a volume of the body that includes the organ wall, the lumen outside of it and the blood and tissue within it. We have also taken into account the varied responses of the body, and our model includes many interacting agents that are part of the infection and defense processes, including the agents that attempt to prevent the infection from breaching the organ wall. The mathematical model is based on a system of nonlinear transient partial differential equations. The numerical model is based on cell-centered finite differences in space and implicit Euler in time. The model is implemented in MATLAB, and has many visualization options to better see the progression of the infection. It is hoped that this model will help in better understanding the failure of the body’s defenses in such situations as Necrotizing Enterocolitis (NEC), and eventually lead to the development of a method of prevention
Mathematical and Numerical Modeling of Inflammation
When the body is attacked by a bacterial infection, it initiates a series of events designed to eradicate the infection while causing minimal damage to the body. Our goal is to investigate the defenses of the organ walls to the spread of infection. To do this we have chosen to model a volume of the body that includes the organ wall, the lumen outside of it and the blood and tissue within it. We have also taken into account the varied responses of the body, and our model includes many interacting agents that are part of the infection and defense processes, including the agents that attempt to prevent the infection from breaching the organ wall. The mathematical model is based on a system of nonlinear transient partial differential equations. The numerical model is based on cell-centered finite differences in space and implicit Euler in time. The model is implemented in MATLAB, and has many visualization options to better see the progression of the infection. It is hoped that this model will help in better understanding the failure of the body’s defenses in such situations as Necrotizing Enterocolitis (NEC), and eventually lead to the development of a method of prevention
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