Staggered grid discretizations for the quasi-static Biot's consolidation problem

Gaspar, Francisco J.; Lisbona, F. J.; Вабищевич, П. Н.

doi:10.1016/j.apnum.2005.07.002

Cited by 39 publications

(36 citation statements)

References 12 publications

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“…A reason for this instability has been identified in [13,14], where the use of a staggered grid discretization for poroelasticity problems was proposed, leading to oscillation-free solutions for any value of the discretization parameters. Stabilization techniques, based on either a problem reformulation or on the addition of artificial terms to the original equations, can be found in [16] and [17], respectively.…”

Section: Discrete Formulationmentioning

confidence: 99%

“…So, pressure points in the staggered grid should be located at the physical boundary, and the displacement points are then defined at the cell faces. The staggered discretization in space has been described in various publications [12,14], including a proof of convergence. For (15), we also consider a staggered grid in time.…”

Section: Discrete Formulationmentioning

confidence: 99%

“…We focus on a staggered grid arrangement for discretization, for which the convergence of discrete schemes has been proved rigorously [13,14]. We mention that the well-known Local Fourier Analysis, LFA, [8,36] can also successfully be applied to develop efficient multigrid methods for systems of partial differential equations with constant (or frozen) coefficients.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Distributive smoothers in multigrid for problems with dominating grad–div operators

Gaspar

Gracia

Lisbona

et al. 2008

Numerical Linear Algebra App

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SUMMARYIn this paper we present efficient multigrid methods for systems of partial differential equations that are governed by a dominating grad-div operator. In particular, we show that distributive smoothing methods give multigrid convergence factors that are independent of problem parameters and of the mesh sizes in space and time. The applications range from model problems to secondary consolidation Biot's model. We focus on the smoothing issue, and mainly solve academic problems on Cartesian staggered grids.

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Section: Discrete Formulationmentioning

confidence: 99%

Section: Discrete Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Distributive smoothers in multigrid for problems with dominating grad–div operators

Gaspar

Gracia

Lisbona

et al. 2008

Numerical Linear Algebra App

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“…In one dimension, however, convergence of the discrete solution has been shown in [12] for a staggered arrangement of the poroelasticity unknowns in the nonlinear case. Convergence results of discrete schemes for the (multidimensional) linear system of poroelasticity equations, on a staggered and on a collocated grid, are available, see, for example, [13,14].…”

Section: Introductionmentioning

confidence: 99%

Multigrid method for nonlinear poroelasticity equations

Luo

Rodrigo

Gaspar

et al. 2015

Comput. Visual Sci.

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In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the "full approximation scheme" and "Newton multigrid" for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies. B P. Luo p.luo@tudelft.nl

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“…In addition, stabilised finite difference methods using central differences on staggered grids are used by Gaspar et al (2003Gaspar et al ( , 2006 to solve Biot's model. However, the numerical solution of the two-dimensional poroelasticity equations is usually approached using finite element methods (Bause et al 2017;Lewis and Schrefler 1978;Phillips and Wheeler 2007).…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

Rahrah

Vermolen

2018

Transp Porous Med

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Stress and water injection induce deformations and changes in pore pressure in the soil. The interaction between the mechanical deformations and the flow of water induces a change in porosity and permeability, which results in nonlinearity. To investigate this interaction and the impact of mechanical vibrations and pressure pulses on the flow rate through the pores of a porous medium under a pressure gradient, a poroelastic model is proposed. In this paper, a Galerkin finite element method is applied for solving the quasi-static Biot's consolidation problem for poroelasticity, considering nonlinear permeability. Space discretisation using Taylor-Hood elements is considered, and the implicit Euler scheme for time stepping is used. Furthermore, Monte Carlo simulations are performed to quantify the impact of variation in the parameters on the model output. Numerical results show that pressure pulses and soil vibrations in the direction of the flow increase the amount of water that can be injected into a deformable fluid-saturated porous medium.

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Staggered grid discretizations for the quasi-static Biot's consolidation problem

Cited by 39 publications

References 12 publications

Distributive smoothers in multigrid for problems with dominating grad–div operators

Distributive smoothers in multigrid for problems with dominating grad–div operators

Multigrid method for nonlinear poroelasticity equations

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

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