Mixed Hybrid Finite Element Research Articles

Numerical modelling of adsorption and desorption of water vapor in zeolite 13X using a two-temperature model and mixed-hybrid finite element method numerical solver

In this paper, we present a new two-temperature mathematical model for a thermo-chemical energy storage based on the adsorption and desorption of the water vapor in zeolite 13X. The adsorption process is modelled using the Linear Driving Force (LDF) model and the Langmuir-Freundlich isotherms. A numerical solver based on the mixed-hybrid finite element method and operator splitting technique is proposed. Furthermore, we present a computational study of the charging and discharging processes of the thermo-chemical energy storage emphasising the behaviour of the two temperatures: fluid temperature and zeolite temperature.

An alternative model of multicomponent diffusion based on a combination of the Maxwell-Stefan theory and continuum mechanics

We present a theory of multicomponent mixtures which does not employ any splitting of component fluxes into convective and diffusive parts. Instead, momentum balance is formulated individually for each component in which both 1) viscous friction within a component, and 2) momentum exchange among different components, are taken into account. While the viscous friction is described using the Newtonian stress tensor, the Maxwell-Stefan theory is used to describe the momentum exchange among different components. When the viscosity is neglected, the model of ideal mixture of ideal gases leads to a hyperbolic system of conservation laws. For the non-ideal mixtures, we obtain a first-order system in a non-conservative form. A simplified version of the model is discretized using a combination of the finite volume method and the mixed-hybrid finite element method. Numerical examples are provided to show typical behavior of the solution of the model equations.

Energetic Convergence of a New Hybrid Mixed Finite Element

Energetic Convergence of a New Hybrid Mixed Finite Element

Multiscale Hybrid-Mixed Finite Element Method for Flow Simulation in Fractured Porous Media

The multiscale hybrid-mixed (MHM) method is applied to the numerical approximation of two-dimensional matrix fluid flow in porous media with fractures. The two-dimensional fluid flow in the reservoir and the one-dimensional f... | Find, read and cite all the research you need on Tech Science Press

Multidimensional mixed–hybrid finite element method for compositional two-phase flow in heterogeneous porous media and its parallel implementation on GPU

A general multidimensional numerical scheme, primarily designed to simulate two-phase compositional flow in porous media, is presented with serial and parallel implementations suitable for solving problems with degenerate (capillary) diffusion or capillary barrier effect in heterogeneous porous materials. The numerical scheme is based on the mixed–hybrid finite element method with the semi-implicitapproach for the time discretization in order to obtain a system of linear equations in each time step. The scheme is implemented in serial for CPU and in parallel for CPU and GPU using TNL that provides an efficient abstraction layer for accessing various parallel hardware architectures. In order to demonstrate the applicability of the numerical scheme, a numerical analysis is presented for problems of two-phase flow in 1D, 2D, and 3D for which exact (semi-analytical) solutions are known and a series of benchmark problems for two-phase flow in heterogeneous porous media is discussed to show correct simulation of the capillary barrier effect. The efficiency and accuracy of the implementations on CPU and GPU are discussed. Moreover, we construct an analytical solution and use it to demonstrate convergence of the numerical scheme for two-phase compositional flow problems in porous media.

A coupling of hybrid mixed and continuous Galerkin finite element methods for poroelasticity

A coupling finite element method for Biot’s model in poroelasticity is considered. The method is based on a coupling of a hybrid mixed finite element method for the pressure and velocity of the fluid phase with a continuous Galerkin finite element method for the displacement of the solid phase. The subproblem for pressure and Darcy velocity are solved at element level and these variables are eliminated in favor of the Lagrange multiplier, identified as pressure trace at the element interfaces. The method is consistent and locally mass conservative. By introducing the energy norm, we can obtain the stability of this system. The optimal error estimates are derived for both semi discrete and fully discrete schemes. Finally, numerical results illustrate the accuracy of the method.

A hybrid mixed finite element method for miscible displacement problem with MCC procedure

A new combined method is developed for solving incompressible miscible displacement in porous media. In this method, a hybrid mixed element method is constructed for the pressure and velocity equation, while a mass-conservative characteristic (MCC) finite element method is presented for concentration equation. The introduction of these two numerical algorithms not only makes the coefficient matrixes symmetric positive definite, but also keeps the mass balance. The stability and consistence of this method are analyzed and the optimal error estimates in L2-norm for velocity, concentration and pressure are derived. Finally, some numerical results are presented.

A mixed hybrid finite element framework for the simulation of swelling ionized hydrogels

Ionized hydrogels, as osmoelastic media, swell enormously (1000 times its original volume in unionized water) due to the osmotic pressure difference caused by the presence of the negatively charged ion groups attached to the solid matrix (polymer chains). The coupling between the extremely large deformations (induced by swelling) and fluid permeation is a field of application that regular poroelasticity formulations cannot handle. In this work, we present a mixed hybrid finite element (MHFE) computational framework featuring a three-field (deformation-chemical potential-flux) formulation. This formulation guarantees that mass conservation is preserved both locally and globally. The impact of such a property on the swelling simulations is demonstrated by four numerical examples in 2D. This paper focuses on the implementation aspects of the MHFE model and shows that it stays robust and accurate for a volume increase of more than 3000%.

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

SummaryFinite element formulations for second‐order elliptic problems, including the classic H1‐conforming Galerkin method, dual mixed methods, a discontinuous Galerkin method, and two primal hybrid methods, are implemented and numerically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static condensation at the element level, aiming at reducing the size of the global system of equations. For a three‐dimensional test problem with smooth solution, the simulations are performed with h‐refinement, for hexahedral and tetrahedral meshes, and uniform polynomial degree distribution up to four. For a singular two‐dimensional problem, the results are for approximation spaces based on given sets of hp‐refined quadrilateral and triangular meshes adapted to an internal layer. The different formulations are compared in terms of L2‐convergence rates of the approximation errors for the solution and its gradient, number of degrees of freedom, both with and without static condensation. Some insights into the required computational effort for each simulation are also given.

A stabilized hybrid mixed DGFEM naturally coupling Stokes–Darcy flows

The coupling of free fluid and porous medium flows, governed by the Stokes and Darcy systems in their mixed velocity–pressure formulations, is addressed in two space dimensions. A stabilized mixed hybrid finite element method is proposed using discontinuous finite element spaces for the velocity and pressure approximations. Lagrange multipliers, associated with the traces of the velocity fields, are introduced to weakly impose the transmission conditions on the edges of the elements in both domains. The Beavers–Joseph–Saffman interface conditions on the interface between Stokes and Darcy domains also are naturally imposed by the same Lagrange multipliers. By condensing the velocity and pressure degrees-of-freedom at the element level the global system is assembled involving only the degrees-of-freedom associated with the multipliers. Stability of the proposed formulation is proved for ample choices of the finite element spaces including equal order of approximations for all fields on triangle and quadrilateral elements. Numerical experiments are performed illustrating flexibility, robustness and optimal rates of convergence.

An Efficient ELLAM Implementation for Modeling Solute Transport in Fractured Porous Media

The goal of this study is to introduce an adaptation of the Eulerian-Lagrangian localized adjoint method (ELLAM) for the simulation of mass transport in fractured porous media, and to evaluate the performance of ELLAM in such a case. The fractures are represented explicitly using the discrete fracture model. The velocity field was calculated using the mixed hybrid finite element method. A sound ELLAM implementation is developed to address numerical artifacts (spurious oscillations and numerical dispersion) arising from the presence of fractures. The efficiency of the developed ELLAM implementation was further improved by taking advantage of the parallel computing on shared memory architecture for the tasks related to particles tracking and linear system resolving. The performance of ELLAM was tested by comparing its results with the Eulerian discontinuous Galerkin method based on several benchmark problems dealing with different fracture configurations. The results highlight the robustness and accuracy of ELLAM, as it allows the use of large time steps, and overcomes the Courant-Friedrichs-Lewy (CFL) restriction. The outcome also shows that ELLAM is more efficient when fracture density is increased.

A three-dimensional transient mixed hybrid finite element model for superabsorbent polymers with strain-dependent permeability.

A hydrogel is a cross-linked polymer network with water as solvent. Industrially widely used superabsorbent polymers (SAP) are partially neutralized sodium polyacrylate hydrogels. The extremely large degree of swelling is one of the most distinctive characteristics of such hydrogels, as the volume increase can be about 30 times its original volume when exposed to physiological solution. The large deformation resulting from the swelling demands careful numerical treatment. In this work, we present a biphasic continuum-level swelling model using the mixed hybrid finite element method (MHFEM) in three dimensions. The hydraulic permeability is highly dependent on the swelling ratio, resulting in values that are orders of magnitude apart from each other. The property of the local mass conservation of MHFEM contributes to a more accurate calculation of the deformation as the permeability across the swelling gel in a transient state is highly non-uniform. We show that the proposed model is able to simulate the free swelling of a random-shaped gel and the squeezing of fluid out of a swollen gel. Finally, we make use of the proposed numerical model to study the onset of surface instability in transient swelling.

The Multiscale Hybrid-Mixed method for the Maxwell Equations in Heterogeneous Media

In this work, we address time dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high performance numerical methods suitable to the simulation of such multiscale problems. Particularly, we extend the Multiscale Hybrid-Mixed finite element method (MHM for short) for the two-and three-dimensional time-dependent Maxwell equations with heterogeneous coefficients. The MHM method arises from the decomposition of the exact electric and magnetic fields in terms of the solutions of locally independent Maxwell problems tied together with a one-field formulation on top of a coarse-mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are driven by local Maxwell problems with tangential component of the magnetic field prescribed on faces. A high-order discontinuous Galerkin method in space combined with a second-order explicit leapfrog scheme in time discretizes the local problems. This makes the MHM method effective and yields a staggered algorithm within a divide-and-conquer framework. Several numerical tests assess the optimal convergence of the MHM method and its capacity to preserve the energy principle, as well as its accuracy to solving heterogeneous media problems on coarse meshes.

Three-dimensional hybrid-mixed stress elements for free vibration analysis

Hexahedral hybrid-mixed finite elements are proposed for free vibration analysis of three-dimensional solids. The element formulation relies on the simultaneous and independent approximations of stress and displacement in the element domain as well as the displacement on their boundary. Sets of complete and linearly independent non-nodal Legendre polynomials used for the field variables lead to symmetric, highly sparse and well conditioned solving systems. Numerical tests show that the elements yield accurate results, even in the presence of high stress gradients, and they seem to be free of shear and volumetric locking and have low sensitivity to mesh distortion. The hierarchical p-refinement strategy is exploited.

The Multiscale Robin Coupled Method for flows in porous media

A multiscale mixed method aiming at the accurate approximation of velocity and pressure fields in heterogeneous porous media is proposed. The procedure is based on a new domain decomposition method in which the local problems are subject to Robin boundary conditions. The domain decomposition procedure is defined in terms of two independent spaces on the skeleton of the decomposition, corresponding to interface pressures and fluxes, that can be chosen with great flexibility to accommodate local features of the underlying permeability fields. The well-posedness of the new domain decomposition procedure is established and its connection with the method of Douglas et al. (1993) [12], is identified, also allowing us to reinterpret the known procedure as an optimized Schwarz (or Two-Lagrange-Multiplier) method. The multiscale property of the new domain decomposition method is indicated, and its relation with the Multiscale Mortar Mixed Finite Element Method (MMMFEM) and the Multiscale Hybrid-Mixed (MHM) Finite Element Method is discussed. Numerical simulations are presented aiming at illustrating several features of the new method. Initially we illustrate the possibility of switching from MMMFEM to MHM by suitably varying the Robin condition parameter in the new multiscale method. Then we turn our attention to realistic flows in high-contrast, channelized porous formations. We show that for a range of values of the Robin condition parameter our method provides better approximations for pressure and velocity than those computed with either the MMMFEM and the MHM. This is an indication that our method has the potential to produce more accurate velocity fields in the presence of rough, realistic permeability fields of petroleum reservoirs.

Fully implicit mixed-hybrid finite-element discretization for general purpose subsurface reservoir simulation

We present a new fully-implicit, mixed-hybrid, finite-element (MHFE) discretization scheme for general-purpose compositional reservoir simulation. The locally conservative scheme solves the coupled momentum and mass balance equations simultaneously, and the fluid system is modeled using a cubic equation-of-state. We introduce a new conservative flux approach for the mass balance equations for this fully-implicit approach. We discuss the nonlinear solution procedure for the proposed approach, and we present extensive numerical tests to demonstrate the convergence and accuracy of the MHFE method using tetrahedral elements. We also compare the method to other advanced discretization schemes for unstructured meshes and tensor permeability. Finally, we illustrate the applicability and robustness of the method for highly heterogeneous reservoirs with unstructured grids.

Monolithic modeling and finite element analysis of piezoelectric energy harvesters

This paper is devoted to monolithic modeling of piezoelectric energy harvesting devices. From a modeling perspective, piezoelectric energy harvesting is a strongly coupled phenomenon with two-way coupling between the electromechanical effect of the piezoelectric material and the harvesting circuit. Even in applications related to shunt damping, where the attached electrical circuit is passive, accurate modeling of the strong coupling is crucial for proper evaluation of the relevant parameters. The article proposes a monolithic mixed-hybrid finite element formulation for the predictive modeling and simulation of piezoelectric energy harvesting devices. The governing equations of the coupled electromechanical problem are converted into a single integral form with six independent unknown fields. Such a holistic approach provides consistent solution to the coupled field equations which involve structural dynamics, electromechanical effect of the piezoelectric patches and the dynamics of the attached harvesting circuit. This allows accurate computation of the eigenvalues and corresponding mode shapes of a harvester for any finite resistive load coupled to the harvester. The fully three-dimensional mixed-hybrid formulation is capable of analyzing structures with non-uniform geometry and varying material properties. The results of the finite element model are verified against the analytical results of a bimorph harvester with tip mass reported in the literature.

Preconditioners with projectors for mixed hybrid finite element methods

Abstract We propose and investigate numerically two new preconditioners for the matrices, which arise in the mixed-hybrid finite element methods for diffusion equation in strongly heterogeneous media. Both preconditioners include special projectors on the vector spaces orthogonal to the vectors with constant components. We give general description of the preconditioners and discuss numerical results which demonstrate their efficiency compared to the classical diagonal preconditioner.

A parallel mixed–hybrid finite element method for two phase flow problems in porous media using MPI

A parallel mixed–hybrid finite element method for two phase flow problems in porous media using MPI

Implicit finite volume and discontinuous Galerkin methods for multicomponent flow in unstructured 3D fractured porous media

We present a new implicit higher-order finite element (FE) approach to efficiently model compressible multicomponent fluid flow on unstructured grids and in fractured porous subsurface formations. The scheme is sequential implicit: pressures and fluxes are updated with an implicit Mixed Hybrid Finite Element (MHFE) method, and the transport of each species is approximated with an implicit second-order Discontinuous Galerkin (DG) FE method. Discrete fractures are incorporated with a cross-flow equilibrium approach. This is the first investigation of all-implicit higher-order MHFE-DG for unstructured triangular, quadrilateral (2D), and hexahedral (3D) grids and discrete fractures. A lowest-order implicit finite volume (FV) transport update is also developed for the same grid types. The implicit methods are compared to an Implicit-Pressure-Explicit-Composition (IMPEC) scheme. For fractured domains, the unconditionally stable implicit transport update is shown to increase computational efficiency by orders of magnitude as compared to IMPEC, which has a time-step constraint proportional to the pore volume of discrete fracture grid cells. However, when lowest-order Euler time-discretizations are used, numerical errors increase linearly with the larger implicit time-steps, resulting in high numerical dispersion. Second-order Crank–Nicolson implicit MHFE-DG and MHFE-FV are therefore presented as well. Convergence analyses show twice the convergence rate for the DG methods as compared to FV, resulting in two to three orders of magnitude higher computational efficiency. Numerical experiments demonstrate the efficiency and robustness in modeling compressible multicomponent flow on irregular and fractured 2D and 3D grids, even in the presence of fingering instabilities.