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 Construction of Multivariate Interpolation Hermite Polynomials for Finite Element Method

2020 ◽  
Vol 226 ◽  
pp. 02007
Author(s):  
Galmandakh Chuluunbaatar ◽  
Alexander A. Gusev ◽  
Ochbadrakh Chuluunbaatar ◽  
Vladimir P. Gerdt ◽  
Sergue I. Vinitsky ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Hermite Polynomials ◽  
Analytical Form ◽  
Multivariate Interpolation ◽  
Partial Derivatives ◽  
Helmholtz Problem ◽  
Finite Element Schemes ◽  
Element Method

A new algorithm for constructing multivariate interpolation Hermite polynomials in analytical form in a multidimensional hypercube is presented. These polynomials are determined from a specially constructed set of values of the polynomials themselves and their partial derivatives with continuous derivatives up to a given order on the boundaries of the finite elements. The effciency of the finite element schemes, algor thms and programs is demonstrated by solving the Helmholtz problem for a cube.

scholarly journals Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping

2007 ◽  
Vol 17 (3) ◽  
pp. 375-393 ◽  
Author(s):  
Sébastien Jund ◽  
Stéphanie Salmon
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Finite Elements ◽  
Taylor Expansion ◽  
High Order ◽  
Computational Time ◽  
Mass Lumping ◽  
Stiffness Matrices ◽  
High Order Time ◽  
Finite Element Schemes

Arbitrary High-Order Finite Element Schemes and High-Order Mass LumpingComputers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elements of orderkand show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also construct in a systematic way a high-order quadrature which is optimal in terms of the number of points, which enables the use of mass lumping, up toP5elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.

scholarly journals Strategy to Improve Edge-Based Smoothed Finite Element Solutions Using Enriched 2D Solid Finite Elements

2021 ◽  
Vol 11 (8) ◽  
pp. 3476
Author(s):  
Min-Han Oh ◽  
San Kim
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Estimation Method ◽  
Piecewise Constant ◽  
Quadrilateral Elements ◽  
Solid Finite Elements ◽  
Edge Based ◽  
Solution Accuracy ◽  
Enriched Finite Elements ◽  
Error Estimation Method

In this paper, we present an automatic procedure that enhances the solution accuracy of edge-based smoothed 2D solid finite elements (three-node triangular and four-node quadrilateral elements). To obtain an enhanced solution, an adaptive enrichment scheme that uses enriched 2D solid finite elements and can effectively improve solution accuracy by applying cover functions adaptively without mesh-refinement is adopted in this procedure. First, the error of the edge-based finite element solution is estimated using a devised error estimation method, and appropriate cover functions are assigned for each node. While the edge-based smoothed finite elements provide piecewise constant strain fields, the proposed enrichment scheme uses the enriched finite elements to obtain a higher order strain field within the finite elements. Through various numerical examples, we demonstrate the accuracy improvement and efficiency achieved.

scholarly journals Bounds on Amplification Factors, Element-Node and Node-Node Equivalence for Square Finite Elements in Kinematic Shallow Water Equations

2021 ◽  
Author(s):  
Jagadeesh Anmala ◽  
Rabi H Mohtar
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Mass Matrix ◽  
Upper And Lower Bounds ◽  
Nodal Solutions ◽  
Lumped Mass ◽  
Amplification Factors ◽  
Finite Element Schemes

Abstract The upper and lower bounds of amplification factors of lumped finite element schemes are compared with nodal (integer or half-integer multiple of) eigen-value solutions of consistent finite element scheme at element and node levels of error analysis. The closeness or proximity between bounds on solutions of amplification factors and eigen-solutions reveals that the two methods, consistent and lumped finite element schemes are equivalent. The element error solutions of lumped mass matrix assumption and consistent nodal solution denotes the element-node error equivalence and the nodal solutions of all of the finite element schemes denote the node-node error equivalence for square finite elements in kinematic wave shallow water equations. The comparison plots of lumped and consistent finite element schemes are presented in this paper for illustration.

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.

scholarly journals Finite elements in computational electromagnetism

Acta Numerica ◽  
2002 ◽  
Vol 11 ◽  
pp. 237-339 ◽  
Author(s):  
R. Hiptmair
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Linear Model ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Differential Forms ◽  
Asymptotic Convergence ◽  
Discrete Differential Forms ◽  
Computational Electromagnetism ◽  
Finite Element Schemes

This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

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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