11 research outputs found
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
A nonconforming finite element method for the Biotâs consolidation model in poroelasticity
A stable finite element scheme that avoids pressure oscillations for a three-field Biotâs model in poroelasticity is considered. The involved variables are the displacements, fluid flux (Darcy velocity), and the pore pressure, and they are discretized by using the lowest possible approximation order: CrouzeixâRaviart finite elements for the displacements, lowest order RaviartâThomas-NĂ©dĂ©lec elements for the Darcy velocity, and piecewise constant approximation for the pressure. Mass-lumping technique is introduced for the RaviartâThomas-NĂ©dĂ©lec elements in order to eliminate the Darcy velocity and, therefore, reduce the computational cost. We show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass-lumping. Finally, numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the RaviartâThomas-NĂ©dĂ©lec elements is used
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