Convergence of a Cell-Centered Finite Volume Discretization for Linear Elasticity

Nordbotten, Jan Martin

doi:10.1137/140972792

Cited by 48 publications

(60 citation statements)

References 26 publications

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“…However, due to the lack of simple finite element methods for elasticity and the Biot equations, of relevance for the current work is the analysis within the hybrid FV framework [14] and the recent analysis of the MPFA [3] and MPSA methods [29].…”

Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

“…Herein, we will expand on the FV framework as developed in [29] to establish a coupled discretization for the Biot equations directly. This approach leads us to a new discretization, which we refer to as MPFA/MPSA-FV.…”

Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

“…In order to analyze locking, a term needs to be introduced accounting for the divergence of displacement, ∇ · u 0 . We choose not to include this in our analysis for four reasons: discretization obtained herein is in itself locking-free on most grids [29]. (c) For the Biot equations, locking-free schemes can be achieved at no additional complexity by introducing a solid pressure in the formulation [24].…”

Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

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Stable Cell-Centered Finite Volume Discretization for Biot Equations

Nordbotten¹

2016

SIAM J. Numer. Anal.

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Abstract. In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate subproblems. The coupled discretization has the following key properties, the combination of which is novel: (1) The variables for the pressure and displacement are co-located and are as sparse as possible (e.g., one displacement vector and one scalar pressure per cell center). (2) With locally computable restrictions on grid types, the discretization is stable with respect to the limits of incompressible fluid and small time-steps. (3) No artificial stabilization term has been introduced. Furthermore, due to the finite volume structure embedded in the discretization, explicit local expressions for both momentum-balancing forces and mass-conservative fluid fluxes are available. We prove stability of the proposed method with respect to all relevant limits. Together with consistency, this proves convergence of the method. Finally, we give numerical examples verifying both the analysis and the convergence of the method. 1. Introduction. Deformable porous media are becoming increasingly important in applications. In particular, the emergence of strongly engineered geological systems such as CO 2 storage [31, 33], geothermal energy [35], and shale-gas extraction all require analysis of the coupling of fluid flow and deformation. Beyond the subsurface, physiological processes are increasingly simulated, exploiting the Biot models [8].With this motivation, we consider the following model problem poroelastic media [9]:

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Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

Section: Definition 1 We Denote a Discretization Of Equations (1) Asmentioning

confidence: 99%

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Stable Cell-Centered Finite Volume Discretization for Biot Equations

Nordbotten¹

2016

SIAM J. Numer. Anal.

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“…with the constraints given by (16). The weights w K,K can be chosen as the harmonic mean of the largest eigenvalue of the stiffness tensor C of the adjacent cells K and K .…”

Section: 2mentioning

confidence: 99%

“…The stress continuity condition (16) implies that t K,K ,s,σ whenever σ = K ∩ K . The system of equations for linear elasticity are then given by the discrete conservation of momentum (13), the definition of the force on faces (12) and the multi-point approximation of sub-face forces given by (18).…”

Section: 2mentioning

confidence: 99%

Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

2017

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Abstract. In this paper we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regards to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular we look how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics.

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Finite volume methods for elasticity with weak symmetry

Keilegavlen

Nordbotten

2017

Numerical Meth Engineering

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We introduce a new cell-centered finite volume discretization for elasticity with weakly enforced symmetry of the stress tensor. The method is motivated by the need for robust discretization methods for deformation and flow in porous media and falls in the category of multi-point stress approximations (MPSAs). By enforcing symmetry weakly, the resulting method has flexibility beyond previous MPSA methods. This allows for a construction of a method that is applicable to simplexes, quadrilaterals, and most planar-faced polyhedral grids in both 2D and 3D, and in particular, the method amends a convergence failure in previous MPSA methods for certain simplex grids. We prove convergence of the new method for a wide range of problems, with conditions that can be verified at the time of discretization. We present the first set of comprehensive numerical tests for the MPSA methods in three dimensions, covering Cartesian and simplex grids, with both heterogeneous and nearly incompressible media. The tests show that the new method consistently is second order convergent in displacement, despite being the lowest order, with a rate that mostly is between 1 and 2 for stresses. The results further show that the new method is more robust and computationally cheaper than previous MPSA methods.

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Convergence of a Cell-Centered Finite Volume Discretization for Linear Elasticity

Cited by 48 publications

References 26 publications

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Stable Cell-Centered Finite Volume Discretization for Biot Equations

Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

Finite volume methods for elasticity with weak symmetry

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