Finite Volume Element Research Articles

Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage

Abstract. New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

Finite volume element approximation for nonlinear diffusion problems with degenerate diffusion coefficients

Some standard numerical methods, such as mimetic finite difference method, finite volume method and mixed finite element method, often fail in solving nonlinear diffusion problems with degenerate diffusion coefficients, since a harmonic average of diffusion coefficients is involved in these methods. To avoid such problem, we present some finite volume element schemes to solve a class of 1D degenerate nonlinear parabolic equations in this paper. Some fully discrete schemes are given by using linear and quadratic finite volume elements in space and a backward difference formulation in time. To deal with unphysical numerical oscillation, we apply two nonnegativity-preserving repair techniques based on a posteriori corrections to finite volume element solutions. One is a local approach, in which any negative energy corresponding to some mesh node is absorbed by the positive values near the current node. The other is a global strategy, in which the total negative energy is redistributed to each positive value in the numerical solution in proportion to its value. In addition, some monotonous finite volume element schemes with lumped-mass strategy are presented. Numerical examples are included to demonstrate the effectiveness and competitive behavior of the proposed methods.

The CFD Analysis of the Combustion Chamber in Common Rail Engines

This paper reports on the development of the geometrical model of the combustion chamber. The results obtained from test were applied to numerical simulations – performed on the Farymann 18WM engine. The analysis was carried out in ANSYS Fluent environment using the finite volume element method. Based on the results, it can be stated that: (1) differences in test and simulation results result from: grid limitations, accuracy of the calculation models, simplifications and limitations of the models, (2) numerical simulations are helpful in determining the parameters of test objects without the need for employing a test set-up, finally, (3) unrestrained adjustment of simulation parameters enables modification of technical parameters of devices to assess their impact on the particular model.

Interior Estimates of Finite Volume Element Methods Over Quadrilateral Meshes for Elliptic Equations

In this paper, we study the interior error estimates of a class of finite volume element methods (FVEMs) over quadrilateral meshes for elliptic equations. We first derive the global $H^1$- and $L^2...

A Crank–Nicolson discontinuous finite volume element method for a coupled non-stationary Stokes–Darcy problem

In this paper, we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a Crank–Nicolson discretization in time, and we obtain the solution at the first time step by applying a first order backward Euler method to solve the coupling of time-dependent Stokes–Darcy problem. The proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by the piecewise constant elements. Then, the unique solvability of the approximate solution for the discrete problem is derived. In addition, the optimal error estimates of the full discretization in standard L2− norm and broken H1− norm are obtained. Finally, a series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, adaptive mesh refinement, mass conservation, capability to conveniently deal with complicated geometries with different types of boundary conditions, and the applicability to the problems with realistic parameters. In particular, the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain, it encompasses both fractures with high permeability which act as the prior flow conduit, and fractures with low permeability as a flow barrier.

Multiscale finite volume method for finite-volume-based simulation of poroelasticity

We propose a multiscale finite volume method (MSFV) for simulation of coupled flow-deformation in heterogeneous porous media under elastic deformation (i.e., poroelastic model). The fine-scale fully resolved system of equations is obtained based on a conservative finite-volume method in which the displacement and pore pressure unknowns are located in a staggered configuration. The coupling is treated through a fully-coupled fully-implicit formulation. On this fully-coupled finite-volume system, coarse-scale grids for flow and deformation are imposed. Local basis functions for scalar pore pressure and vectorial displacement unknowns are then solved over their respective local domains at the beginning of the simulation, and reused for the rest of the time-dependent simulations. These local basis functions are then clustered to form the prolongation operator. As for the finite-volume nature of the proposed multiscale system, finite-volume restriction operators for poroelastic systems are utilised. Once the coarse-scale system is solved, its solution is prolonged back to the original fine-scale resolution, providing approximate fine-scale solution. The finite-volume multiscale formulation provides conservative stress and mass flux both at fine and coarse scale. Several numerical test cases are provided first to validate the fine-scale finite-volume discrete fully-implicit simulation, and then to investigate the accuracy of the proposed multiscale formulation. Moreover we compare our fully implicit MSFV method with hybrid multiscale Finite Element-Finite Volume (h-MSFE-FV). Our multiscale method allows for quantification of the elastic geomechanical behaviour with using only a fraction of the fine-scale grid cells, even for highly heterogeneous time-dependent models. As such, it casts a promising approach for field-scale quantification of the mechanical deformation and stress field due to injection and production in a subsurface formation.

3D simulation and parametric analysis of polymer melt flowing through spiral mandrel die for pipe extrusion

AbstractWith the increasing demands for large scale and high productivity, polymer pipes are recently produced using the advanced spiral mandrel dies. However, the fundamental research related to polymer melt flow mechanism in the spiral mandrel die for pipe extrusion is lagging behind. In the present study, the mathematical model for such a complex three‐dimensional non‐isothermal viscous flow of polymer melts obeying power law model was developed based on computational fluid dynamics theory. Finite volume element method was applied to predict the rheological behaviors of polymer melt flowing through the complex flow channel. The essential flow characteristics including velocity, pressure drop, wall shear stress, and temperature were investigated. The effects of both mandrel structure parameters and mass flow rate upon the flow patterns were further discussed. Some recommendations on spiral mandrel die design for pipe production were put forward.

Numerical investigation of flow regimes in T‐shaped micromixers: Benchmark between finite volume and spectral element methods

AbstractComputational fluid dynamics (CFD) is very appealing to investigate mixing and reaction in microdevices, as it allows easily investigating different operating conditions as well as mixer geometries. This latter aspect is very important as the flow in microdevices is laminar so the mixing between reactants should be promoted by a clever mixer design, aimed at breaking the flow symmetries. Recently time periodic motions that improve mixing have been observed to take place in a T‐junction at low Reynolds numbers. In this case the numerical modelling should be based on direct numerical simulations (DNS), thus involving high computational resources. In this work, two different CFD approaches, i.e., finite volume and spectral element methods, are applied and compared for the analysis of the mixing process in the well known T‐shaped micromixer. Spectral elements methods are particularly suited for DNS; however, they have been scarcely applied to study micromixers, while plenty of works can be found with finite volume methods. The analysis is carried out using both ideal and non‐ideal liquid binary mixtures, the latter presenting a negative fluidity of mixing (i.e., the viscosity of the mixture is higher than that of the pure components). Moreover the numerical results are validated with simple flow visualization experiments.

Finite Volume Element Approximation for the Elliptic Equation with Distributed Control

In this paper, we consider a priori error estimates for the finite volume element schemes of optimal control problems, which are governed by linear elliptic partial differential equation. The variational discretization approach is used to deal with the control. The error estimation shows that the combination of variational discretization and finite volume element formulation allows optimal convergence. Numerical results are provided to support our theoretical analysis.

Mixed displacement–rotation–pressure formulations for linear elasticity

We propose a new locking-free family of mixed finite element and finite volume element methods for the approximation of linear elastostatics, formulated in terms of displacement, rotation vector, and pressure. The unique solvability of the three-field continuous formulation, as well as the well-definiteness and stability of the proposed Galerkin and Petrov–Galerkin methods, is established thanks to the Babuška–Brezzi theory. Optimal a priori error estimates are derived using norms robust with respect to the Lamé constants, turning these numerical methods particularly appealing for nearly incompressible materials. We exemplify the accuracy (in a suitably weighted norm), as well the applicability of the new formulation and the mixed schemes by conducting a number of computational tests in 2D and 3D, also including cases not covered by our theoretical analysis.

A non-intrusive reduced basis approach for parametrized heat transfer problems

Computation Fluid Dynamics (CFD) simulation has become a routine design tool for i) predicting accurately the thermal performances of electronics set ups and devices such as cooling system and ii) optimizing configurations. Although CFD simulations using discretization methods such as finite volume or finite element can be performed at different scales, from component/board levels to larger system, these classical discretization techniques can prove to be too costly and time consuming, especially in the case of optimization purposes where similar systems, with different design parameters have to be solved sequentially. The design parameters can be of geometric nature or related to the boundary conditions. This motivates our interest on model reduction and particularly on reduced basis methods. As is well documented in the literature, the offline/online implementation of the standard RB method (a Galerkin approach within the reduced basis space) requires to modify the original CFD calculation code, which for a commercial one may be problematic even impossible. For this reason, we have proposed in a previous paper, with an application to a simple scalar convection diffusion problem, an alternative non-intrusive reduced basis approach (NIRB) based on a two-grid finite element discretization. Here also the process is two stages: offline, the construction of the reduced basis is performed on a fine mesh; online a new configuration is simulated using a coarse mesh. While such a coarse solution, can be computed quickly enough to be used in a rapid decision process, it is generally not accurate enough for practical use. In order to retrieve accuracy, we first project every such coarse solution into the reduced space, and then further improve them via a rectification technique. The purpose of this paper is to generalize the approach to a CFD configuration.

Studies of Resistances of Natural Liquid Flow in Helical and Curved Pipes

Abstract The main aim of this research was to determine in three ways, i.e. experimentally, analytically and by means of numerical modelling, the resistances of the flow of a natural liquid in a helical pipe and in curved pipes. The analyses were carried out for three pipes: one helical pipe and two curved pipes. Each of the pipes was 2 m long and its inside diameter was 4 mm. The experiment was carried out on a test stand making it possible to measure the rate of the flow of the liquid, the temperature at the pipe’s inlet and outlet and the pressure at the pipe’s inlet and outlet. The resistances of the flow of the liquid were calculated from analytical or empirical formulas found in the literature on the subject. Moreover, numerical modelling was performed using the finite volume element method.

A multi-physics framework model towards coupled fire-structure interaction for Flax/PP composite beams

In this paper, a coupled fire-structure model combining the finite volume and finite element methods, which captures the essential physics of the problem has been developed. The model is based on a multi-physics framework where the essential physics pertaining to the combustion process of the fire and resultant thermo-mechanical response of the structure, in particular, of natural fibre reinforced composites, has been incorporated. In addition, a relatively new concept of adiabatic surface temperature has been introduced as a practical means to transfer data between fire and thermal/structural models at the gas-solid interface. The model predicts the temperature, stress distribution and deformation behaviour of composite beams under combined thermal and mechanical loads.

Modelling stress-dependent single and multi-phase flows in fractured porous media based on an immersed-body method with mesh adaptivity

This paper presents a novel approach for hydromechanical modelling of fractured rocks by linking a finite-discrete element solid model with a control volume-finite element fluid model based on an immersed-body approach. The adaptive meshing capability permits flow within/near fractures to be accurately captured by locally-refined mesh. The model is validated against analytical solutions for single-phase flow through a smooth/rough fracture and reported numerical solutions for multi-phase flow through intersecting fractures. Examples of modelling single- and multi-phase flows through fracture networks under in situ stresses are further presented, illustrating the important geomechanical effects on the hydrological behaviour of fractured porous media.

Modeling hemodynamics in intracranial aneurysms: Comparing accuracy of CFD solvers based on finite element and finite volume schemes.

Image-based computational fluid dynamics (CFD) has shown potential to aid in the clinical management of intracranial aneurysms, but its adoption in the clinical practice has been missing, partially because of lack of accuracy assessment and sensitivity analysis. To numerically solve the flow-governing equations, CFD solvers generally rely on 2 spatial discretization schemes: finite volume (FV) and finite element (FE). Since increasingly accurate numerical solutions are obtained by different means, accuracies and computational costs of FV and FE formulations cannot be compared directly. To this end, in this study, we benchmark 2 representative CFD solvers in simulating flow in a patient-specific intracranial aneurysm model: (1) ANSYS Fluent, a commercial FV-based solver, and (2) VMTKLab multidGetto, a discontinuous Galerkin (dG) FE-based solver. The FV solver's accuracy is improved by increasing the spatial mesh resolution (134k, 1.1m, 8.6m, and 68.5m tetrahedral element meshes). The dGFE solver accuracy is increased by increasing the degree of polynomials (first, second, third, and fourth degree) on the base 134k tetrahedral element mesh. Solutions from best FV and dGFE approximations are used as baseline for error quantification. On average, velocity errors for second-best approximations are approximately 1cm/s for a [0,125]cm/s velocity magnitude field. Results show that high-order dGFE provides better accuracy per degree of freedom but worse accuracy per Jacobian nonzero entry as compared with FV. Cross-comparison of velocity errors demonstrates asymptotic convergence of both solvers to the same numerical solution. Nevertheless, the discrepancy between underresolved velocity fields suggests that mesh independence is reached following different paths.

Development of a high-order compact finite-difference total Lagrangian method for nonlinear structural dynamic analysis

• A compact finite-difference total Lagrangian method (CFDTLM) is developed for nonlinear structural dynamics analysis. • The large deformation of elastic structures for Hookean and neo-Hookean materials is accurately simulated using the CFDTLM. • The CFDTLM is extended and assessed for simulating thermo-elastodynamic problems. • The numerical solution procedure adopted is simpler than high-order finite volume and finite element formulations . • The solution methodology proposed is accurate for simulating nonlinear structural dynamics problems. A high-order compact finite-difference total Lagrangian method (CFDTLM) is developed and applied to nonlinear structural dynamic analysis. The two-dimensional simulation of thermo-elastodynamics is numerically performed in generalized curvilinear coordinates by taking into account the geometric and material nonlinearities. The spatial discretization is carried out by a fourth-order compact finite-difference scheme and an implicit second-order accurate dual time-stepping method is applied for the time integration. The accuracy and capability of the proposed solution methodology for the nonlinear structural analysis is investigated through simulating different static and dynamic benchmark problems including large deformations, large displacements and Hookean/neo-Hookean materials. The solution method is demonstrated to be free of shear-locking behavior. The results obtained by the present solution algorithm are compared with the analytical solution and the numerical results of the finite element and finite volume methods to examine the accuracy and robustness of the solution method proposed. A grid study is also performed to investigate the grid size effect on the accuracy and performance of the solution. Indications are that the solution methodology proposed is accurate for simulating nonlinear structural dynamics problems.

A Linear Energy-Preserving Finite Volume Element Method for the Improved Korteweg−de Vries Equation

In this paper, we introduce a linearized energy-preserving scheme which preserves the discrete global energy of solutions to the improved Korteweg−deVries equation. The method presented is based on the finite volume element method, by resorting to the variational derivative to transform the improved Korteweg−deVries equation into a new form, and then designing energy-preserving schemes for the transformed equation. The proposed scheme is much more efficient than the standard nonlinear scheme and has good stability. To illustrate its efficiency and conservative properties, we also compare it with other nonlinear schemes. Finally, we verify the efficiency and conservative properties through numerical simulations.

Two-Level Least Squares Methods in Krylov Subspaces

Two-level least squares acceleration approaches are applied to the Chebyshev acceleration method and the restarted conjugate residual method in solving systems of linear algebraic equations with sparse unsymmetric coefficient matrices arising from finite volume or finite element approximations of boundary-value problems on irregular grids. Application of the proposed idea to other iterative restarted processes also is considered. The efficiency of the algorithms suggested is investigated numerically on a set of model Dirichlet problems for the convection-diffusion equation.

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

In this article we introduce a family of discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid mixed/discontinuous approximation) measured in suitable norms are derived, showing optimal orders of convergence.

A Sparse Grid Stochastic Collocation and Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Elliptic Equations

A Sparse Grid Stochastic Collocation and Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Elliptic Equations