286 research outputs found
Two-field finite element solver for linear poroelasticity, A
Includes bibliographical references.2020 Summer.Poroelasticity models the interaction between an elastic porous medium and the fluid flowing in it. It has wide applications in biomechanics, geophysics, and soil mechanics. Due to difficulties of deriving analytical solutions for the poroelasticity equation system, finite element methods are powerful tools for obtaining numerical solutions. In this dissertation, we develop a two-field finite element solver for poroelasticity. The Darcy flow is discretized by a lowest order weak Galerkin (WG) finite element method for fluid pressure. The linear elasticity is discretized by enriched Lagrangian () elements for solid displacement. First order backward Euler time discretization is implemented to solve the coupled time-dependent system on quadrilateral meshes. This poroelasticity solver has some attractive features. There is no stabilization added to the system and it is free of Poisson locking and pressure oscillations. Poroelasticity locking is avoided through an appropriate coupling of finite element spaces for the displacement and pressure. In the equation governing the flow in pores, the dilation is calculated by taking the average over the element so that the dilation and the pressure are both approximated by constants. A rigorous error estimate is presented to show that our method has optimal convergence rates for the displacement and the fluid flow. Numerical experiments are presented to illustrate theoretical results. The implementation of this poroelasticity solver in deal.II couples the Darcy solver and the linear elasticity solver. We present the implementation of the Darcy solver and review the linear elasticity solver. Possible directions for future work are discussed
A mixed finite element method for nearly incompressible multiple-network poroelasticity
In this paper, we present and analyze a new mixed finite element formulation
of a general family of quasi-static multiple-network poroelasticity (MPET)
equations. The MPET equations describe flow and deformation in an elastic
porous medium that is permeated by multiple fluid networks of differing
characteristics. As such, the MPET equations represent a generalization of
Biot's equations, and numerical discretizations of the MPET equations face
similar challenges. Here, we focus on the nearly incompressible case for which
standard mixed finite element discretizations of the MPET equations perform
poorly. Instead, we propose a new mixed finite element formulation based on
introducing an additional total pressure variable. By presenting energy
estimates for the continuous solutions and a priori error estimates for a
family of compatible semi-discretizations, we show that this formulation is
robust in the limits of incompressibility, vanishing storage coefficients, and
vanishing transfer between networks. These theoretical results are corroborated
by numerical experiments. Our primary interest in the MPET equations stems from
the use of these equations in modelling interactions between biological fluids
and tissues in physiological settings. So, we additionally present
physiologically realistic numerical results for blood and tissue fluid flow
interactions in the human brain
Stabilized lowest order finite element approximation for linear three-field poroelasticity
A stabilized conforming mixed finite element method for the three-field
(displacement, fluid flux and pressure) poroelasticity problem is developed and
analyzed. We use the lowest possible approximation order, namely piecewise
constant approximation for the pressure and piecewise linear continuous
elements for the displacements and fluid flux. By applying a local pressure
jump stabilization term to the mass conservation equation we ensure stability
and avoid pressure oscillations. Importantly, the discretization leads to a
symmetric linear system. For the fully discretized problem we prove existence
and uniqueness, an energy estimate and an optimal a-priori error estimate,
including an error estimate for the divergence of the fluid flux. Numerical
experiments in 2D and 3D illustrate the convergence of the method, show the
effectiveness of the method to overcome spurious pressure oscillations, and
evaluate the added mass effect of the stabilization term.Comment: 25 page
Stability and Monotonicity for Some Discretizations of the Biot's Model
We consider finite element discretizations of the Biot's consolidation model
in poroelasticity with MINI and stabilized P1-P1 elements. We analyze the
convergence of the fully discrete model based on spatial discretization with
these types of finite elements and implicit Euler method in time. We also
address the issue related to the presence of non-physical oscillations in the
pressure approximation for low permeabilities and/or small time steps. We show
that even in 1D a Stokes-stable finite element pair fails to provide a monotone
discretization for the pressure in such regimes. We then introduce a
stabilization term which removes the oscillations. We present numerical results
confirming the monotone behavior of the stabilized schemes
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