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Jan M Nordbotten - One of the best experts on this subject based on the ideXlab platform.
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finite Volume Discretization for poroelastic media with fractures modeled by contact mechanics
International Journal for Numerical Methods in Engineering, 2020Co-Authors: Runar Lie Berge, Jan M Nordbotten, Inga Berre, Eirik Keilegavlen, Barbara WohlmuthAbstract:A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law while slip is described by a Coulomb-type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complimentary function, and a semismooth Newton method is used to solve the system of equations. For the Discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite Volume Discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2d and 3d, showing that the convergence rates of the finite Volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.
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A finite-Volume Discretization for deformation of fractured media
Computational Geosciences, 2018Co-Authors: Eren Ucar, Eirik Keilegavlen, Inga Berre, Jan M NordbottenAbstract:Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-Volume approach, termed the multi-point stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (face pairs in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-Volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation, considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate our proposed approach using both problems, for which analytical solutions are available, and more complex benchmark problems, including comparison with a finite-element Discretization.
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stable cell centered finite Volume Discretization for biot equations
SIAM Journal on Numerical Analysis, 2016Co-Authors: Jan M NordbottenAbstract: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 conve...
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convergence of a cell centered finite Volume Discretization for linear elasticity
SIAM Journal on Numerical Analysis, 2015Co-Authors: Jan M NordbottenAbstract: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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stable cell centered finite Volume Discretization for biot equations
arXiv: Numerical Analysis, 2015Co-Authors: Jan M NordbottenAbstract: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 sub-problems. 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 as well as 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 convergence of the method.
Pascal Omnes - One of the best experts on this subject based on the ideXlab platform.
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numerical results for a discrete duality finite Volume Discretization applied to the navier stokes equations
International Conference on Finite Volumes for Complex Applications, 2017Co-Authors: Sarah Delcourte, Pascal OmnesAbstract:We present an application of the discrete duality finite Volume method to the numerical approximation of the 2D Stokes or (unsteady) Navier–Stokes equations associated to Dirichlet boundary conditions. The finite Volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical MAC scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. Different numerical examples over triangular, cartesian, quadrangular and locally refined meshes are led in order to illustrate the possibilities and weaknesses of the method.
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An a Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Stokes Equations
ESAIM: Mathematical Modelling and Numerical Analysis, 2015Co-Authors: Pascal OmnesAbstract:We derive an a posteriori error estimation for the discrete duality finite Volume (DDFV) Discretization of the stationary Stokes equations on very general twodimensional meshes, when a penalty term is added in the incompressibility equation to stabilize the variational formulation. Two different estimators are provided: one for the error on the velocity and one for the error on the pressure. They both include a contribution related to the error due to the stabilization of the scheme, and a contribution due to the Discretization itself. The estimators are globally upper as well as locally lower bounds for the errors of the DDFV Discretization. They are fully computable as soon as a lower bound for the inf-sup constant is available. Numerical experiments illustrate the theoretical results and we especially consider the influence of the penalty parameter on the error for a fixed mesh and also of the mesh size for a fixed value of the penalty parameter. A global error reducing strategy that mixes the decrease of the penalty parameter and adaptive mesh refinement is described.
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a discrete duality finite Volume Discretization of the vorticity velocity pressure stokes problem on almost arbitrary two dimensional grids
Numerical Methods for Partial Differential Equations, 2015Co-Authors: Sarah Delcourte, Pascal OmnesAbstract:We present an application of the discrete duality finite Volume method to the numerical approximation of the vorticity-velocity-pressure formulation of the two-dimensional Stokes equations, associated to various nonstandard boundary conditions. The finite Volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical Marker and Cell scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. The efficiency of the scheme is illustrated by numerical examples over unstructured triangular and locally refined nonconforming meshes, which confirm the theoretical convergence analysis led in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1–30, 2015
Jerome Droniou - One of the best experts on this subject based on the ideXlab platform.
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Total-velocity-based finite Volume Discretization of two-phase Darcy flow in highly heterogeneous media with discontinuous capillary pressure
2020Co-Authors: Konstantin Brenner, Roland Masson, Jerome Droniou, El Houssaine QuenjelAbstract:This work proposes a finite Volume scheme for two-phase Darcy flow in heterogeneous porous media with different rock types. The fully implicit Discretization is based on cell centered as well as face centered degrees of freedom in order to capture accurately the nonlinear transmission conditions at different rock type interfaces. These conditions play a major role in the flow dynamics. The scheme is formulated with natural physical unknowns avoiding the cumbersome use of the global pressure. It combines a Two-Point Flux Approximation of the gradient normal fluxes with a Hybrid Upwinding approximation of the transport terms. The convergence of the scheme to a weak solution is established taking into account discontinuous capillary pressure at different rock type interfaces and the degeneracy of the phase mobilities. Numerical experiments show the additional robustness of the proposed Discretization compared with the classical Phase Potential Upwinding approach.
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Error estimates for the convergence of a finite Volume Discretization of convection-diffusion equations
Journal of Numerical Mathematics, 2003Co-Authors: Jerome DroniouAbstract:We study error estimates for a finite Volume Discretization of an elliptic equation. We prove that, for $s\\in [0,1]$, if the exact solution belongs to $H^{1+s}$ and the right-hand side is $f+\\div(G)$ with $f\\in L^2$ and $G\\in (H^s)^N$, then the solution of the finite Volume scheme converges in discrete $H^1$-norm to the exact solution, with a rate of convergence of order $h^s$ (where $h$ is the size of the mesh).
Kai Stemmer - One of the best experts on this subject based on the ideXlab platform.
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finite Volume Discretization for dynamic viscosities on voronoi grids
Physics of the Earth and Planetary Interiors, 2008Co-Authors: C Huttig, Kai StemmerAbstract:We present a new formulation to discretize the viscous term in the momentum equation of the Navier–Stokes set. A technique based on the finite Volume method enables thermal convection models to utilize spatially varying viscosity on a collocated variable arrangement. This technique can be applied to various grids in two or three dimensions with Voronoi properties, either irregular as the spiral grid or regular like the cubed sphere grid, icosahedral or simple boxes. A model for mantle convection implements this Discretization and is compared to other published models. Further computational aspects are illuminated to efficiently reduce required resources.
M Hohmeyer - One of the best experts on this subject based on the ideXlab platform.
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hybrid finite element finite Volume Discretization of complex geologic structures and a new simulation workflow demonstrated on fractured rocks
Geofluids, 2007Co-Authors: Adriana Paluszny, Stephan Matthai, M HohmeyerAbstract:The generation of computational meshes of complex geological objects is a challenge: their shape needs to be retained, resolution has to adapt to local detail, and variations of material properties in the objects have to be represented. Also mesh refinement and adaptation must be sufficient to resolve variations in the computed variable(s). Here, we present an unstructured hybrid finite element, node-centred finite-Volume Discretization suitable for solving fluid flow, reactive transport, and mechanical partial differential equations on a complex geometry with inhomogeneous material domains. We show that resulting meshes accurately capture free-form material interfaces as defined by non-uniform rational B-spline curves and surfaces. The mesh Discretization error is analysed for the elliptic pressure equation and an error metric is introduced to guide mesh refinement. Finite elements and finite Volumes are represented in parametric space and integrations are conducted numerically. Subsequently, integral properties are mapped to physical space using Jacobian transformations. This method even retains its validity when the mesh is deformed. The resulting generic formulation is demonstrated for a transport calculation performed on a complex discrete fracture model.
