Hybrid Finite Volume Discretization of Linear Elasticity Models on General Meshes

Abstract

This paper presents a new discretization scheme for linear elasticity models using only one degree of freedom per face corresponding to the normal component of the displacement. The scheme is based on a piecewise constant gradient construction and a discrete variational formulation for the displacement field. The tangential components of the displacement field are eliminated using a second order linear interpolation. Our main motivation is the coupling of geomechanical models and porous media flows arising in reservoir or CO2 storage simulations. Our scheme guarantees by construction the compatibility condition between the displacement discretization and the usual cell centered finite volume discretization of the Darcy flow model. In addition it applies on general meshes possibly non conforming such as Corner Point Geometries commonly used in reservoir and CO2 storage simulations.

A two-stage preconditioner for multiphase poromechanics in reservoir simulation

“…Therefore, formulations that combine FE approaches for mechanics with a Finite Volume (FV) method for fluid flow and transport are often adopted by modelers [32,36,48,49]. Some authors have recently started investigating facecentered and cell-centered finite-volume methods also for the mechanical subproblem [50][51][52][53][54] to account for mechanical effects when flow processes are of predominant importance. Also, in the context of single-phase poromechanics, locally mass-conservative approaches have been proposed using mixed three-field (displacement-velocity-pressure) or four-field (stress tensor-displacement-velocity-pressure) formulations, respectively [e.g., 27,29,30,[55][56][57][58][59][60][61][62][63][64][65][66].…”

Thirty years of the finite volume method for solid mechanics

“…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.…”

Network element methods for linear elasticity

“…The objective of the present article is the extension of the network element method to the more complex system of linear elasticity. As the network underlying the NEM generalizes in some sense the notion of mesh, if applied on a mesh it will also offer alternatives to existing first order methods for linear elasticity on mesh with general cell geometries, such as the first order virtual element method (see [32,33]) or hybrid finite volume like methods (see [34][35][36]). Without any attempt at being exhaustive, let us nevertheless mention that linear elasticity problems have been of course investigated with both strong and weak form meshless methods (see [9,[37][38][39]), up to considering fretting [40] or fracture [41,42] problems.…”