Mixed Hybrid Finite Element Method Research Articles

Efficient geometric nonlinear analyses of circular plate bending problems

In this paper, a hybrid/mixed nonlinear shell element is developed in polar coordinate system based on Hellinger/Reissner variational principle and the large-deflection theory of plate. A numerical solution scheme is formulated using the hybrid/mixed finite element method (HMFEM), in which the nodal values of bending moments and the deflection are the unknown discrete parameters. Stability of the present element is studied. The large-deflection analyses are performed for simple supported and clamped circular plates under uniformly distributed and concentrated loads using HMFEM and the traditional displacement finite element method. A parametric study is also conducted in the research. The accuracy of the shell element is investigated using numerical computations. Comparisons of numerical solutions are made with theoretical results, finite element analysis and the available numerical results. Excellent agreements are shown.

A mixed-hybrid finite element method for convection–diffusion problems

A mixed-hybrid finite element method for scalar diffusion–convection problems is presented. In this method, the lowest degree Raviart–Thomas finite element is used for the diffusive term. The convective term is treated by means of a Lagrange multiplier which is introduced in the hybrid form. The numerical results show that this method produces very accurate solutions for high Peclet numbers.

Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods

We present adaptive mixed hybrid finite element discretizations of the Richards equation, a nonlinear parabolic partial differential equation modeling the flow of water into a variably saturated porous medium. The approach simultaneously constructs approximations of the flux and the pressure head in Raviart–Thomas spaces. The resulting nonlinear systems of equations are solved by a Newton method. For the linear problems of the Newton iteration a multigrid algorithm is used. We consider two different kinds of error indicators for space adaptive grid refinement: superconvergence and residual based indicators. They can be calculated easily by means of the available finite element approximations. This seems attractive for computations since no additional (sub-)problems have to be solved. Computational experiments conducted for realistic water table recharge problems illustrate the effectiveness and robustness of the approach.

The Edge-Based Face Element Method for 3D-Stream Function and Flux Calculations in Porous Media Flow

We present a velocity-oriented discrete analog of the partial differential equations governing porous media flow: the edge-based face element method. Conventional finite element techniques calculate pressures in the nodes of the grid. However, such methods do not satisfy the requirement of flux continuity at the faces. In contrast, the edge-based method calculates vector potentials along the edges, leading to continuity of fluxes. The method is algebraically equivalent with the popular block-centered finite difference method and with the mixed-hybrid finite element method, but is algorithmically different and has the same robustness as the more conventional node-based velocity-oriented method. The numerical examples are computed analytically and may, therefore, be considered as an 'heuristic proof' of the theory and its practical applicability for reservoir engineering and geohydrology. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.

Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods

We present adaptive mixed hybrid finite element discretizations of the Richards equation, a nonlinear parabolic partial differential equation modeling the flow of water into a variably saturated porous medium. The approach simultaneously constructs approximations of the flux and the pressure head in Raviart–Thomas spaces. The resulting nonlinear systems of equations are solved by a Newton method. For the linear problems of the Newton iteration a multigrid algorithm is used. We consider two different kinds of error indicators for space adaptive grid refinement: superconvergence and residual based indicators. They can be calculated easily by means of the available finite element approximations. This seems attractive for computations since no additional (sub-)problems have to be solved. Computational experiments conducted for realistic water table recharge problems illustrate the effectiveness and robustness of the approach.

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.

The Andra Couplex 1 Test Case: Comparisons Between Finite-Element, Mixed Hybrid Finite Element and Finite Volume Element Discretizations

We present here results for the Andra Couplex 1 test case, obtained with the code Cast3m. This code is developped at the CEA (Commissariat р l'щnergie atomique) and is used mainly to solve problems of solid mechanics, fluid mechanics and heat transfers. Different types of discretization are available, among them finite element, finite volume and mixed hybrid finite element method. Cast3m is also a componant of the platteform Alliances (co-developped by Andra, CEA), which will be used by Andra for the safety calculation of an underground waste disposal in year 2004. We solve the Darcy equation for the water flow and a convection–diffusion transport equation for the Iodine 129 which escapes from a repository cave into the water. The water flow is calculated with a MHFE discretization. It is shown that this method provides sharp results even on relatively coarse grids. The convection–diffusion transport equation is discretized with FE (Finite Element), MHFE (Mixed Hybrid Finite Element) and FV (Finite Volume) methods. In our comparison, we point out the differences of these methods in term of accuracy, respect of the maximum principle and calculations cost. Neither the finite element nor the mixed hybrid finite element approach respects the maximum principle. This results in the presence of negative concentrations near the repository cave, whereas FV calculations respect the monotonicity. We show that mass lumping techniques suppress this problem but with strong restrictions on the grid. FE and MHFE approaches are more accurate than FV for the diffusion equation, but the overall results are equivalent since the advective terms are dominant in the far field and are discretized with centered schemes. We conclude by studying the influence of the grid: a very fine grid near the repository solves almost all the problems of monotonicity, without employing mass lumping techniques. We also observed a very important increase of the accuracy on a structured grid made up of rectangles.

Iterative Solutions of a Mixed Hybrid Finite Element Scheme for the Signorini Problem

AbstractA mixed hybrid finite element method of the lowest order is studied for the Signorini problem. An iterative method with a preconditioner being a classical finite element approximation of the Laplace operator is constructed. A multistage iterative procedure for the mixed hybrid finite element scheme is constructed, the rate of convergence and the complexity of this method are analysed.

Experimental investigation on criterion of three-dimensional mixed-mode fracture for concrete

An investigation was carried out to establish a criterion for three-dimensional (3-D) mixed-mode fracture of concrete. Experiments on 3-D mixed-mode fracture were conducted to obtain the critical load P cr. The stress intensity factor of the specimens per unit load with different combinations of K I , , K II and K III was calculated by the mixed hybrid finite element method. Based on the stress intensity factor per unit load and the critical load, 25 groups of critical stress intensity factors were calculated, from which a criterion for the 3-D mixed-mode fracture of concrete was derived.

Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow

Richards’ equation (RE) is commonly used to model flow in variably saturated porous media. However, its solution continues to be difficult for many conditions of practical interest. Among the various time discretizations applied to RE, the method of lines (MOL) has been used successfully to introduce robust, accurate, and efficient temporal approximations. At the same time, a mixed-hybrid finite element method combined with an adaptive, higher order time discretization has shown benefits over traditional, lower order temporal approximations for modeling single-phase groundwater flow in heterogeneous porous media. Here, we extend earlier work for single-phase flow and consider two mixed finite element methods that have been used previously to solve RE using lower order time discretizations with either fixed time steps or empirically based adaption. We formulate the two spatial discretizations within a MOL context for the pressure head form of RE as well as a fully mass-conservative version. We conduct several numerical experiments for both spatial discretizations with each formulation, and we compare the higher order, adaptive time discretization to a first-order approximation with formal error control and adaptive time step selection. Based on the numerical results, we evaluate the performance of the methods for robustness and efficiency.

Regularization of parameter estimation by adaptive discretization using refinement and coarsening indicators

Abstract - This paper is concerned with the ill-posed problem of identifying a hydraulic transmissivity in an isotropic and confined aquifer in two space dimensions. To define a regularization by adaptive discretization of the parameter, we use refinement and coarsening indicators, which give the first order effect on the optimal data misfit of adding or removing degrees of freedom to a current set of parameters. The direct problem is discretized by a mixed hybrid finite element method. A combination of the direct problem discretization with the adaptive discretrization for the transmissivity allows us to prove convergence of the algorithm as the mesh size goes to zero, stability of the finite-dimensional approximation, and convergence for noisy data with an appropriate stopping rule.

The maximum principle violations of the mixed‐hybrid finite‐element method applied to diffusion equations

AbstractThe abundant literature of finite‐element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial–boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time‐dependent solutions for such problems during small time duration obtained by using a non‐conforming mixed‐hybrid finite‐element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite‐difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non‐physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial–boundary value problem will be presented. One of the propositions given to avoid any non‐physical oscillations is to use the mass‐lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd.

Mixed-finite element and finite volume discretization for heavy brine simulations in groundwater

Recently, a new theory of high-concentration brine transport in groundwater has been developed. This approach is based on two nonlinear mass conservation equations, one for the fluid (flow equation) and one for the salt (transport equation), both having nonlinear diffusion terms. In this paper, we present and analyze a numerical technique for the solution of such a model. The approach is based on the mixed hybrid finite element method for the discretization of the diffusion terms in both the flow and transport equations, and a high-resolution TVD finite volume scheme for the convective term. This latter technique is coupled to the discretized diffusive flux by means of a time-splitting approach. A commonly used benchmark test (Elder problem) is used to verify the robustness and nonoscillatory behavior of the proposed scheme and to test the validity of two different formulations, one based on using pressure head ψ and concentration c as dependent variables, and one using pressure p and mass fraction ω as dependent variables. It is found that the latter formulation gives more accurate and reliable results, in particular, at large times. The numerical model is then compared against a semi-analytical solution and the results of a laboratory test. These tests are used to verify numerically the performance and robustness of the proposed numerical scheme when high-concentration gradients (i.e., the double nonlinearity) are present.

Multiscale temporal integration

We study the application of the variational multiscale method to the problem of temporal integration, with the final goal of designing integration schemes that go beyond the classical notion of upwinding. We develop a formulation based on a mixed hybrid finite element method, where the fine scale mode problems automatically decouple at the element level without the need to resort to a localization assumption. We give general orthogonality conditions for the trial and test spaces that allow to construct hierarchical p methods. We test some simple ideas for the modeling of the unresolved scales. The resulting algorithms are analyzed using classical analytical measures.

Hydrogeologic behavior of an alluvial aquifer, Salta Province, Argentina: Simulations of hydraulic conductivity field, groundwater flow, and chloride migration

This work was designed to analyze the hydrogeological behavior of an alluvial aquifer in the River Mojotoro basin site in the Province of Salta, Argentina. The study area presents coarse-grained sediments with high infiltration capacity. The hydraulic conductivity field is affected by the physical heterogeneity of the medium and a geostatistical method, kriging, was used to construct this field from local measurements. The groundwater flow velocity field is computed using a hybrid mixed finite-element method and small variations in the flow directions were determined. Given the existence of a point source of chloride in the region under study, the spatial distribution of the generated plume in the saturated zone is analyzed. The advection-diffusion equation describing the transport of chloride in vertical and horizontal cross-sectional planes of the aquifer is discretized employing a method of characteristics combined with a mixed finite-element procedure, which provides an efficient method to eliminate spurious numerical oscillations and handle the convective term in the equation. The migration of chloride shows patterns, which are affected by the litologic heterogeneities of the aquifer and the regional groundwater flow.

Infiltration of Salt Solute in Homogeneous and Saturated Porous Media – An Analytical Solution Evaluated by Numerical Simulations

In many cases various land disposal activities (e.g. infiltration, injection wells) constitute an important potential source of groundwater contamination. Using a 2D physical model, the behaviour of the infiltration of a salt solute, locally injected in a homogeneous and saturated porous medium, has been analysed. Under various experimental conditions (density effects, injection flow rate) the salt solute penetrates the porous media and leads to a steady-state regime inside the mixing zone. By using experimental observations, the basic equations describing the flow and transport phenomena can be simplified and an analytical solution obtained. Its validity is subject to numerical verification. The numerical model, based on the development of the mass balance equation expressed by its conservative form, uses a combination of the mixed hybrid finite element (MHFE) and discontinuous finite element (DFE) methods. The efficiency of this numerical model was previously verified on standard benchmarks, for example Elder's problem and Henry's problem. In the first step, the qualitative good agreement between the experimental and numerical results enabled us to use the numerical model in order to verify some hypotheses resulting from visual observations. Thus, the numerical results reveal the existence of a steady-state regime inside the mixing zones. Nevertheless, both its vertical and longitudinal extensions are less than those observed in the physical model. In the second step, the numerical results enable to establish the validity domain as well as the accuracy of the proposed analytical solution.

Numerical Reliability for Mixed Methods Applied to Flow Problems in Porous Media

This paper is devoted to the numerical reliability and time requirements of the Mixed Finite Element (MFE) and Mixed-Hybrid Finite Element (MHFE) methods. The behavior of these methods is investigated under the influence of two factors: the mesh discretization and the medium heterogeneity. We show that, unlike the MFE, the MHFE suffers with the presence of badly shaped discretized elements. Thereat, a numerical reliability analyzing software (Aquarels) is used to detect the instability of a matrix-inversion code generated automatically by a symbolic manipulator. We also show that the spectral condition number of the algebraic systems furnished by both methods in heterogeneous media grows up linearly according to the smoothness of the hydraulic conductivity. Furthermore, it is found that the MHFE could accumulate numerical errors if large jumps in the tensor of conductivity take place. Finally, we compare running-times for both algorithms by giving various numerical experiments.

Complementary finite element methods applied to the numerical homogenization of 3D absolute permeability

AbstractTo be able to use a limited number of relatively large grid cells in numerical oil reservoir simulators and groundwater models, upscaling of the absolute permeability is frequently applied. The spatial fine‐scale permeability distribution, which is generally obtained from geological and geostatistical models, is incorporated in the relatively large grid cells of the numerical model. If the porous medium may be approximated as a periodic medium, upscaling can be performed by the homogenization method. Numerical homogenization gives rise to an approximation error. The complementarity between the conformal–nodal finite element method and the mixed–hybrid finite element method has been used to quantify this error. The two methods yield, respectively, upper and lower bounds for the eigenvalues of the coarse‐scale permeability tensor. Results of numerical experiments obtained using tetrahedral meshes are shown both in the far field and in the near well region. Copyright © 2001 John Wiley & Sons, Ltd.

Review and complements on mixed-hybrid finite element methods for fluid flows

Mixed and hybrid finite element methods for the resolution of a wide range of linear and nonlinear boundary value problems (linear elasticity, Stokes problem, Navier–Stokes equations, Boussinesq equations, etc.) have known a great development in the last few years. These methods allow simultaneous computation of the original variable and its gradient, both of them being equally accurate. Moreover, they have local conservation properties (conservation of the mass and the momentum) as in the finite volume methods. The purpose of this paper is to give a review on some mixed finite elements developed recently for the resolution of Stokes and Navier–Stokes equations, and the linear elasticity problem. Further developments for a quasi-Newtonian flow obeying the power law are presented.

Approximation par les éléments finis mixtes d'une équation de diffusion non linéaire modélisant un écoulement diphasique en milieu poreux

We study the counter-current displacement of oil by water in porous media. The nonlinear diffusion equation modeling this flow problem is discretized in space with the mixed hybrid finite element method. We propose to evaluate the diffusion coefficient on each element by using the mean value of the saturation on the element's edges. Comparisons done with the analytical solution of this problem show that the new approach is more accurate than the classical one which is based on the mean value of the saturation on the element.