A Study of Two Modes of Locking in Poroelasticity

Yi, Son-Young

doi:10.1137/16m1056109

Cited by 69 publications

(50 citation statements)

References 17 publications

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“…Several recent (and not so recent) studies, see e.g. [29,6,4,17,31,38], focus on a three-field formulation of Biot's model, involving the elastic displacement, fluid pressure and fluid velocity. Four-field formulations where also the elasticity equation is in mixed form, designed to provide robust numerical methods for nearly incompressible materials, have also been studied [37,20,21].…”

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confidence: 99%

A Mixed Finite Element Method for Nearly Incompressible Multiple-Network Poroelasticity

Lee¹,

Piersanti²,

Mardal³

et al. 2019

SIAM J. Sci. Comput.

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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.

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mentioning

confidence: 99%

A Mixed Finite Element Method for Nearly Incompressible Multiple-Network Poroelasticity

Lee¹,

Piersanti²,

Mardal³

et al. 2019

SIAM J. Sci. Comput.

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“…The author of a recent work [38] has pointed out that if ker (aupT)=0, one can remove spurious pressure oscillations which arise when c 0 = 0 and K → 0. Since we use standard mixed finite element spaces scriptVh={v∈H0,normalΓd(div;Ω):boldnormalv|K∈BDMk(K)} and scriptPh={w∈L2(Ω):w|K∈Pk−1(K)} for the displacement and pressure variables, there naturally holds (aupT)=0 .…”

Section: A Semi-discrete Schemementioning

confidence: 99%

“…The advantages of adopting such a discretization are two-fold: On one hand, the normal components of displacement across elements are continuous and therefore are locally conservative; On the other hand, the tangential components are discretized through an interior penalty discontinuous Galerkin method. As it is discontinuous Galerkin approximation, such a discretization enables us to overcome the locking phenomenon and the pressure oscillation [18,31,38]. We comment here that applying H(div)-conforming finite elements in a DG framework was initially proposed in [11] (see also [12,33]) for solving Stokes equations in fluid mechanics.…”

Section: Introductionmentioning

confidence: 99%

“…This work can be regarded as a further development of H(div)-conforming finite element methods for solving Biot’s problem. By using the framework presented in [28,29,36,38], we give a detailed analysis of our method. In particular, for both the semi-discrete and the fully discrete schemes for Biot’s model, we prove the existence and uniqueness theorems of the approximate solutions and derive the optimal convergence rate for each variable.…”

Section: Introductionmentioning

confidence: 99%

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An H(div)-Conforming Finite Element Method for the Biot Consolidation Model

Zeng

Cai

Wang

2019

EAJAM

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In this paper, we develop an H(div)-conforming finite element method for Biot’s consolidation model in poroelasticity. In our method, the flow variables are discretized by an H(div)-conforming mixed finite elements. For relaxing the H1-conformity of the displacement, we approximate the displacement by using an H(div)-conforming finite element method, in which the tangential components are discretized in the interior penalty discontinuous Galerkin framework. For both the semi-discrete and the fully discrete schemes, we prove the existence and uniqueness theorems of the approximate solutions and derive the optimal convergence rate for each variable.

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“…Moreover, many different numerical schemes have been proposed for this formulation with varying success, e.g. [12,24,32,45,47,51,[61][62][63][64][65] and references therein. The main difficulties encountered when developing numerical methods for this model are volumetric locking and spurious, nonphysical pressure oscillations.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model

et al. 2021

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We present an a priori and a posteriori error analysis of a conforming finite element method for a four-field formulation of the steady-state Biot’s consolidation model. For the a priori error analysis we provide suitable hypotheses on the corresponding finite dimensional subspaces ensuring that the associated Galerkin scheme is well-posed. We show that a suitable choice of subspaces is given by the Raviart–Thomas elements of order k ≥ 0 for the fluid flux, discontinuous polynomials of degree k for the fluid pressure, and any stable pair of Stokes elements for the solid displacements and total pressure. Next, we develop a reliable and efficient residual-based a posteriori error estimator. Both the reliability and efficiency estimates are shown to be independent of the modulus of dilatation. Numerical examples in 2D and 3D verify our analysis and illustrate the performance of the proposed a posteriori error indicator.

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A Study of Two Modes of Locking in Poroelasticity

Cited by 69 publications

References 17 publications

A Mixed Finite Element Method for Nearly Incompressible Multiple-Network Poroelasticity

A Mixed Finite Element Method for Nearly Incompressible Multiple-Network Poroelasticity

An H(div)-Conforming Finite Element Method for the Biot Consolidation Model

Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model

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