scholarly journals Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

2012 ◽  
Vol 236 (18) ◽  
pp. 4827-4839 ◽  
Author(s):  
D. De Santis ◽  
G. Geraci ◽  
A. Guardone
Keyword(s):  
Finite Element ◽  
Euler Equation ◽  
Finite Volume ◽  
Spherical Coordinates ◽  
Node Pair ◽  
Finite Element Schemes

scholarly journals Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2972
Author(s):  
Saray Busto ◽  
Michael Dumbser ◽  
Laura Río-Martín
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Yield Stress ◽  
Finite Volume ◽  
Power Law ◽  
Source Terms ◽  
Cartesian Grids ◽  
Rans Equations ◽  
Edge Based ◽  
Finite Element Schemes

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

scholarly journals A UNIFIED APPROACH TO MIMETIC FINITE DIFFERENCE, HYBRID FINITE VOLUME AND MIXED FINITE VOLUME METHODS

2010 ◽  
Vol 20 (02) ◽  
pp. 265-295 ◽  
Author(s):  
JÉRÔME DRONIOU ◽  
ROBERT EYMARD ◽  
THIERRY GALLOUËT ◽  
RAPHAÈLE HERBIN
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Finite Volume ◽  
Mixed Finite Element ◽  
Finite Volume Methods ◽  
Finite Volume Scheme ◽  
Volume Scheme ◽  
Common Framework ◽  
Finite Element Schemes ◽  
Unified Method

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the methods (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.

Forward modeling of gravity data using finite-volume and finite-element methods on unstructured grids

Geophysics ◽  
2013 ◽  
Vol 78 (3) ◽  
pp. G69-G80 ◽  
Author(s):  
Hormoz Jahandari ◽  
Colin G. Farquharson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Volume ◽  
Gravitational Potential ◽  
Unstructured Grids ◽  
Gravity Data ◽  
Forward Modeling ◽  
The Finite Element Method ◽  
Finite Element Schemes ◽  
Element Method

Minimum-structure inversion is one of the most effective tools for the inversion of gravity data. However, the standard Gauss-Newton algorithms that are commonly used for the minimization procedure and that employ forward solvers based on analytic formulas require large memory storage for the formation and inversion of the involved matrices. An alternative to the analytical solvers are numerical ones that result in sparse matrices. This sparsity suits gradient-based minimization methods that avoid the explicit formation of the inversion matrices and that solve the system of equations using memory-efficient iterative techniques. We have developed several numerical schemes for the forward modeling of gravity data using the finite-element and finite-volume methods on unstructured grids. In the finite-volume method, a Delaunay tetrahedral grid and its dual Voronoï grid are used to find the primary solution (i.e., gravitational potential) at the centers and vertices of the tetrahedra, respectively (cell-centered and vertex-centered schemes). In the finite-element method, Delaunay tetrahedral grids are used to develop linear and quadratic finite-element schemes. Different techniques are used to recover the vertical component of gravitational acceleration from the gravitational potential. In the finite-volume scheme, a differencing method is used; in the finite-element method, basis functions are used. The capabilities of the finite-volume and finite-element schemes were tested on simple and realistic synthetic examples. The results showed that the quadratic finite-element scheme is the most accurate but also the most computationally demanding scheme. The best trade-offs between accuracy and computational resource requirement were achieved by the linear finite-element and vertex-centered finite-volume schemes.

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