Discrete Duality Finite Volume Method Applied to Linear Elasticity

Martin, Bernice M.; Pascal, Frédéric

doi:10.1007/978-3-642-20671-9_70

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…Teng et al [425] and Chen et al [426] developed a finite volume method to simulate the draping of woven fabrics, where the governing nonlinear equations were solved using a single-step full Newton-Raphson method. Martin and Pascal [431,434] proposed a novel discrete duality finite volume method for solving linear elasticity problems on unstructured meshes; the main characteristic of the discretisation is the integration of the governing equations over two meshes: the given primal mesh and also over a dual mesh built from the primal one. Pietro et al [432] proposed a novel discretisation scheme for linear elasticity with only one degree of freedom per control-volume face, corresponding to the normal component of the displacement.…”

Section: Other Approachesmentioning

confidence: 99%

Thirty years of the finite volume method for solid mechanics

Cardiff,

Demirdžić

2018

Preprint

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Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, including "cell-centred", "staggered", "vertex-centred", "periodic heterogenous microstructural", "Godunov-type", "matrix-free", "meshless", as well as others. This article gives an overview, historical perspective, comparison and critical analysis of the different approaches, including their relative strengths, weaknesses, similarities and dissimilarities, where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finishes with a look towards future research directions and steps required for finite volume solid mechanics to achieve widespread acceptance.

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Section: Other Approachesmentioning

confidence: 99%

Thirty years of the finite volume method for solid mechanics

Cardiff,

Demirdžić

2018

Preprint

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“…The mimetic finite difference methods [19,20,21] can also be applied to the elasticity problems in the nearly incompressible case as well as to Stokes problems in the incompressible case on general 3D meshes. Moreover, the 3D discrete duality finite volume method [22] has been constructed for linear elasticity problems. However, it is subtle to preserve the discrete duality property, which is used derive optimal estimates of the discrete error.…”

Section: Introductionmentioning

confidence: 99%

A low-order finite element method for three dimensional linear elasticity problems with general meshes

Hoang¹,

Vo²,

Ong³

2017

Computers & Mathematics with Applications

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The paper is concerned with a low-order finite element method, namely the staggered cellcentered finite element method (SC-FEM), which has been proposed and analyzed in [1] for two-dimensional compressible and nearly incompressible linear elasticity problems. In this work, we extend the results to the three-dimensional case and focus on the creating of the meshes, including a dual mesh and its tetrahedral submesh from a general primal mesh. The displacement is approximated by piecewise trilinear functions on the subdual mesh and the pressure by piecewise constant functions on the dual mesh. As for two-dimensional case, such construction of the meshes and approximation spaces satisfies the macroelement condition, which implies stability and convergence of the scheme. Numerical experiments are carried out to investigate the performance of the proposed method on various mesh types.

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“…This includes also recent work on cellcentered methods [23,10]. When additional variables are introduced, convergence of finite volume methods has been established [20]. Similarly, convergence has recently been established for face-valued finite volume methods [19].…”

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confidence: 99%

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

Nordbotten¹

2015

SIAM J. Numer. Anal.

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Abstract. We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.

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Discrete Duality Finite Volume Method Applied to Linear Elasticity

Cited by 3 publications

References 11 publications

Thirty years of the finite volume method for solid mechanics

Thirty years of the finite volume method for solid mechanics

A low-order finite element method for three dimensional linear elasticity problems with general meshes

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

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