Finite Volume Element Research Articles

Consistency, convergence and error estimates for a mixed finite element–finite volume scheme to compressible Navier–Stokes equations with general inflow/outflow boundary data

Abstract We study convergence of a mixed finite element-finite volume scheme for the compressible Navier–Stokes equations in the isentropic regime under the full range $1<\gamma <\infty $ of the adiabatic coefficients $\gamma $ for the problem with general nonzero inflow–outflow boundary conditions. We propose a modification of Karper’s scheme (2013, A convergent FEM-DG method for the compressible Navier–Stokes equations. Numer. Math., 125, 441–510) in order to accommodate the nonzero boundary data, prove existence of its solutions, establish the stability and uniform estimates, derive a convenient consistency formulation of the balance laws and use it to show the weak convergence of the numerical solutions to a dissipative solution with the Reynolds defect introduced in Abbatiello et al. (2021, Generalized solutions to models of compressible viscous fluids. Discrete Contin. Dyn. Syst., 41, 1--28). If the target system admits a strong solution then the convergence is strong towards the strong solution. Moreover, we establish the convergence rate of the strong convergence in terms of the size of the space discretization $h$ (which is supposed to be comparable with the time step $\varDelta t$). In the case of the nonzero inflow–outflow boundary data all results are new. The latter result is new also for the no-slip boundary conditions and adiabatic coefficients $\gamma $ less than the threshold $3/2$.

Bilinear immersed finite volume element method for solving matrix coefficient elliptic interface problems with non-homogeneous jump conditions

In this paper, a new bilinear immersed finite volume element method based on rectangular mesh is presented to solve the elliptic interface problems with non-homogeneous jump conditions and sharp-edged interfaces. This method is capable of dealing with the case when the interface passes through grid points and when the solutions are oscillating. Plenty of numerical experiments show that our method is nearly second-order accuracy for the solution and is first-order accuracy for the gradient of the solution in the L∞ norm.

Hybrid finite volume-finite element methods for hydro-mechanical analysis in highly heterogeneous porous media

In this study, two classes of advanced finite volume schemes, including Multi-Point Flux Approximation (MPFA) and Dual Discrete Finite Volume (DDFV) method, have been employed in conjunction with the finite element (FE) geomechanics simulator to model the coupled fluid-solid system. Fully saturated porous media with poroelastic behavior, random field permeability and elastic modulus are considered as parameters. The performance of the proposed hydro-mechanical models, including MPFA O-FEM and DDFV-FEM, is examined through different test cases. First, the models are validated and compared with the closed-form solutions in the homogeneous domain. Second, the methods' stability and convergence properties are analyzed in highly heterogeneous porous media with different permeability and elastic modulus realizations. Finally, implementing the models in a large-size realistic problem of a synthetic reservoir accomplishes this comprehensive study. Although both methods agree well with the analytical solutions in homogeneous porous media, the DDFV-FEM yields higher accuracy. In the cases where the random field of permeability varies over several orders of magnitude (4th and more), the MPFA O-FEM suffers from significant oscillations and even instability. The analysis illustrates that MPFA O-FEM exhibits oscillation in the cases with high anisotropy ratio, while the DDFV-FEM demonstrates superior performance for handling anisotropy and heterogeneity.

The finite volume element method on the Shishkin mesh for a singularly perturbed reaction–diffusion problem

In this paper, we construct a finite volume element method (FVEM) on the Shishkin mesh for solving a singularly perturbed reaction–diffusion problem. The stability of the method is established in energy norm. Further, we derive the optimal error estimate in energy norm, under the decomposition of solution. Some numerical experiments are provided to confirm our theoretical results.

Error estimates of finite volume method for Stokes optimal control problem

In this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal L^{2}-norm error estimates. The approximate orders for the state, costate, and control variables are O(h^{2}) in the sense of L^{2}-norm. Furthermore, we derive H^{1}-norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

Two-Level Iterative Methods for Solving the Saddle Point Problems

Iterative processes in the Krylov subspaces for solving large ill conditioned saddle-type SLAEs with sparse matrices arising in finite difference, finite volume, and finite element approximations of multidimensional boundary value problems with complex geometric and functional properties of the initial data, characteristic of many relevant applications are studied. Combined two-level iterative algorithms using efficient Chebyshev acceleration and variational the conjugate directions methods, as well as the Golub-Kahan bi-diagonalization algorithms in the Krylov subspaces are considered. Examples of two-dimensional and three-dimensional filtration problems are used to study the resource consumption and computational performance of the proposed algorithms, as well as their scalable parallization on the multiprocessor systems with distributed and hierarchical shared memory.

Conservation Laws for Free-Boundary Fluid Layers

Time-dependent models of fluid motion in thin layers, subject to signed source terms, represent important sub-problems within climate dynamics. Examples include ice sheets, sea ice, and even shallow oceans and lakes. We address these problems as discrete-time sequences of continuous-space weak formulations, namely (monotone) variational inequalities or complementarity problems, in which the conserved quantity is the layer thickness. Free boundaries wherein the thickness and mass flux both go to zero at the margin of the fluid layer generically arise in such models. After showing these problems are well-posed in several cases, we consider the limitations to discrete conservation in numerical schemes. A free boundary in a region of negative source -- an ablation-caused margin -- turns out to be a barrier to exact conservation in either a continuous- or discrete-space sense. We then propose computable a posteriori quantities which allow conservation-error bounds in finite volume and finite element schemes.

Numerical simulation of a composite shell for underwater application using CFD and FEA

In the present investigation, an attempt has been made to use fiber-reinforced composite material in place of metal for a shell used for underwater applications which is majorly subjected to hydro-static pressure. The study comprises the use of popular numerical techniques such as finite volume and finite element methods. CFD tool ‘ANSYS fluent’ which works based on finite volume method is used for the fluid flow simulation to get the forces acting on the structure. Structural analysis is performed for the imported loads from CFD result onto the structure with one-way fluid-structure coupling. FEM tool ‘ANSYS static structural which works based on finite element method is used for obtaining deformations and stresses in the shell. A sufficient number of iterations are made to get convergence of the numerical solution. Alternatives such as increasing the shell thickness and/or providing stiffeners to the shell are suggested to replace the metallic structure with FRP composites.

Numerical simulation and optimisation of a reinforced steel plate against underwater explosions

Optimum arrangement of reinforcing blades can increase the resistance of marine structures and reduce potential losses in the case of explosive events. In this research, to find the most appropriate arrangement, six different geometries were modelled in 3D (one base case and five cases with different arrangements of reinforcing blades). By using Lagrangian and Eulerian methods for grid generation, the effects of the explosion on these structures in the water environment were simulated, and the variation of the Von-Mises stress, damage parameter, strain, displacement, and displacement rate were predicted for all cases. The simulations were based on finite element-finite volume methods. The accuracy of the numerical approaches was confirmed by analysing benchmark problems and comparing the obtained results with analytical and experimental data. It was found that maximum damage takes place in Case A (plate without any blades) with maximum displacement and displacement rate of 521 mm and 151 mm/ms, respectively. Also, the results of parametric simulations revealed that Case F with six perpendicular reinforcing blades was the best arrangement. In this case, the displacement and displacement rate of the plate relative to the base case (without any blades) were reduced by 40% and 64%, respectively.

Detailed vibrational analysis of unbalanced morning glory spillways using coupled finite volume-finite element method

In this research, the structural behavior of a morning glory spillway has been evaluated when there is an unbalanced construction in the spillway crest. The unbalanced construction is defined as the ratio of vertical unbalance in construction to the diameter of the spillway crest. The spillway structure has been modeled using finite element method and the reservoir fluid has been simulated using finite volume method. The behavior of morning glory spillways with unbalanced crests has been investigated in two scenarios: (a) the crest without any anti-vortex structure, and (b) the crest with anti-vortex structures. Both scenarios have been simulated using dynamic linear analysis method. In addition, in each scenario, three amounts of unbalanced constructions of 0.017, 0.035 and 0.052 have been considered. The analysis results have been compared with the structural response of a symmetric spillway without unbalanced construction. Based on the results, the most critical condition occurs during the water flow over the unbalanced spillway with no anti-vortex structure. On the other hand, the presence of anti-vortex piers causes asymmetric flow on the spillway crest and subsequently asymmetric flow pressure at one side of the piers.

Two-Grid Finite Volume Element Method Combined with Crank-Nicolson Scheme for Semilinear Parabolic Equations

Two-Grid Finite Volume Element Method Combined with Crank-Nicolson Scheme for Semilinear Parabolic Equations

Numerical Solutions of the Improved Boussinesq Equation with Stokes Damping

In this paper, we propose new finite volume element schemes to numerically solve the improved Boussinesq equation with Stokes damping. The new schemes can inherit characteristic properties of the conservation of mass and the decrease of total energy from the improved Boussinesq equation with Stokes damping. Numerical experiments illustrate that the proposed schemes are second-order accuracy in space and time.

Mixed Finite Volume Element Method for Vibration Equations of Beam with Structural Damping

In this paper, for the initial and boundary value problem of beams with structural damping, by introducing intermediate variables, the original fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed. Numerical examples are provided to confirm the theoretical results. In the end, we test the value of δ to observe its influence on the model.

Quadratic Finite Volume Element Schemes over Triangular Meshes for a Nonlinear Time-Fractional Rayleigh-Stokes Problem

In this article, we study a 2D nonlinear time-fractional Rayleigh-Stokes problem, which has an anomalous sub-diffusion term, on triangular meshes by quadratic finite volume element schemes. Time-fractional derivative, defined... | Find, read and cite all the research you need on Tech Science Press

An adjoint-based a posteriori analysis of numerical approximation of Richards equation

<p style='text-indent:20px;'>This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.</p>

Error estimates in $ L^2 $ and $ L^\infty $ norms of finite volume method for the bilinear elliptic optimal control problem

<abstract> This paper discusses some a priori error estimates of bilinear elliptic optimal control problems based on the finite volume element approximation. A case-based numerical example serves to discuss with optimal $ L^2 $-norm error estimates and $ L^{\infty} $-norm error estimates, and supports two key insights. First, the approximate orders for the state, costate and control variables are $ O(h^2) $ in the sense of $ L^{2} $-norm. Second, the approximate orders for the state, costate and control variables are $ O(h^2\sqrt{|lnh|}) $ in the sense of $ L^{\infty} $-norm. </abstract>

New immersed finite volume element method for elliptic interface problems with non-homogeneous jump conditions

In this paper, we propose a new immersed finite volume element method to solve elliptic problems with discontinuous diffusion coefficient and sharp-edged interfaces on Cartesian mesh. The method uses the non-traditional immersed finite volume element method together with additional immersed finite element function on interface element. It can deal with the case when the solution or its normal derivative is discontinuous. Extensive numerical experiments for various problems show that the new method is approximate second-order convergence in the L∞ norm for piecewise smooth solutions, and more than 1.65th order accuracy is observed for solution with singularity.

A hybrid finite volume/finite element method for shallow water waves by static deformation on seabeds

Purpose The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear one-dimensional shallow water equations, including friction terms for the water free-surface and the two-dimensional second-order solid elastostatic equations for the bed deformation. Second, to develop a robust hybrid finite element/finite volume method for solving free-surface runup flow problems over deformable beds. The authors combine the finite volume for free-surface flows and the finite element method for bed elasticity. Design/methodology/approach The authors propose a new model for wave runup by static deformation on seabeds. The model consists of the depth-averaged shallow water system for the water free-surface coupled to the second-order elastostatic formulation for the bed deformation. At the interface between the water flow and the seabed, transfer conditions are implemented. Here, hydrostatic pressure and friction forces are considered for the elastostatic equations, whereas bathymetric forces are accounted for in the shallow water equations. As numerical solvers, the authors propose a well-balanced finite volume method for the flow system and a stabilized finite element method for elastostatics. Findings The developed coupled depth-averaged shallow water system and second-order solid elastostatic system is well suited for modeling wave runup by deformation on seabeds. The derived coupling conditions at the interface between the water flow and the bed topography resolve well the condition transfer between the two systems. The proposed hybrid finite volume element method is accurate and efficient for this class of models. The novel technique used for wet/dry treatment accurately captures the moving fronts in the computational domain without generating nonphysical oscillations. The presented numerical results demonstrate the high performance of the proposed methods. Originality/value Enhancing modeling and computations for wave runup problems is at an early stage in the literature, and it is a new and exciting area of research. To the best of our knowledge, solving wave runup problems by static deformation on seabeds using a hybrid finite volume element method is presented for the first time. The results of this research study, and the research methodologies, will have an important influence on a range of other scientists carrying out research in related fields.

New superconvergent structures developed from the finite volume element method in 1D

New superconvergence structures are introduced by the finite volume element method (FVEM), which gives us the freedom to choose the superconvergent points of the derivative (for odd order schemes) and the superconvergent points of the function value (for even order schemes) for $k\geq 3$. The general orthogonal condition and the modified M-decomposition (MMD) technique are established to prove the superconvergence properties of the new structures. In addition, the relationships between the orthogonal condition and the convergence properties for the FVE schemes are carried out in Table 1. The above results are also valid to $k=1,2$. For these cases, the new superconvergence structures are the same with the Gauss-Lobatto structure, which yields to the FVE schemes with Gauss-points-dependent dual meshes. Numerical results are given to illustrate the theoretical results.

Simulation of Temperature Evolution of Cork Composites During Moulding Process

Cork-based composites result from a combination of cork granules with different materials – like thermosets or thermoplastics – and its manufacture involves a thermal process. In order to simulate the manufacturing process, of these types of composites, a new methodology was applied. A material composed of cork and a thermoplastic served as a case study. A model for the prediction of a cork composite mixture properties and a simulation methodology were developed for studying the variation of temperature during the moulding process of cork composites. Density, thermal conductivity, and specific heat were determined based on the formulation of the composite and the properties of cork and the agglutinant agent. Numerical analyses were carried out and compared to experimental results obtained from a moulding process. Three types of simulations, according to the model of the chosen properties were developed using finite volume and finite element methods. In general, the results from the simulations were in good agreement with experimental results.