Mixed Hybrid Finite Element Research Articles

Solution of the Elliptic Interface Problem by a Hybrid Mixed Finite Element Method

This paper addresses the elliptic interface problem involving jump conditions across the interface. We propose a hybrid mixed finite element method on the triangulation where the interfaces are aligned with the mesh. The second-order elliptic equation is initially decomposed into two equations by introducing a gradient term. Subsequently, weak formulations are applied to these equations. Scheme continuity is enforced using the Lagrange multiplier technique. Finally, we derive an explicit formula for the entries of the matrix equation representing Lagrange multiplier unknowns resulting from hybridization. The method yields approximations of all variables, including the solution and gradient, with optimal order. Furthermore, the matrix representing the final linear algebra systems is not only symmetric but also positive definite. Numerical examples convincingly demonstrate the effectiveness of the hybrid mixed finite element method in addressing elliptic interface problems.

A lattice Boltzmann approach to mathematical modeling of myocardial perfusion.

A mathematical model of myocardial perfusion based on the lattice Boltzmann method (LBM) is proposed and its applicability is investigated in both healthy and diseased cases. The myocardium is conceptualized as a porous material in which the transport and mass transfer of a contrast agent in blood flow is studied. The results of myocardial perfusion obtained using LBM in 1D and 2D are confronted with previously reported results in the literature and the results obtained using the mixed-hybrid finite element method. Since LBM is not suitable for simulating flow in heterogeneous porous media, a simplified and computationally efficient 1D-analog approach to 2D diseased case is proposed and its applicability discussed.

Numerical simulation of foam displacement impacted by kinetic and equilibrium surfactant adsorption

This study explores the effectiveness of foam in reducing gas mobility in porous media impacted by surfactant adsorption in highly heterogeneous porous media. The research focuses on assessing the influence of kinetic and equilibrium adsorption during foam displacement on reservoir sweep efficiency, considering co-injection and SAG (Surfactant Alternating Gas) injection strategies. Including the surfactant on the liquid phase and considering the surfactant adsorption effects fitted by experimental data, the mathematical modeling combines Darcy’s law, fluid phase conservation, surfactant transport equations, and a non-Newtonian foam model in local equilibrium. The simulations utilize FOSSIL (FOam diSplacement SImuLator), an in-house software composed of a stable and conservative numerical algorithm with reduced numerical diffusion. These properties are achieved by combining a hybrid mixed finite element method to solve the Darcy system with a central-upwind finite volume scheme to approximate the transport equations and adopting an implicit adaptive method in time. The results reveal that while omitting adsorption phenomena enhances sweep efficiency due to reduced surfactant loss, differences between equilibrium and kinetic adsorption models are minor (up to 0.3% in production). Consequently, these results suggest using simple adsorption models, such as equilibrium isotherms, rather than more complex models, such as kinetics. Additionally, the SAG approach outperforms co-injection (up to 19% in production), despite the fact that the adsorption is detrimental to both of them, aligning with literature results and previous findings.

State of art for hybrid mixed finite element formulation in non-linear analysis of structures

Since the late 80’s the Structural Analysis Research Group of the Instituto Superior Técnico (IST) has been involved in the development of non-conventional finite element formulations in order to overcome some of the limitations associated with the use of the CFE method and to develop high performance numerical tools for the analysis of structural engineering problems. Several alternative models for the linear and non-linear structural analysis have been developed using hybrid and mixed models techniques. These works are summarized in this paper, in which their past and future applications of this formulation in non-linear analysis of structures are fully detailed.

Hybrid mixed discontinuous Galerkin finite element method for incompressible miscible displacement problem

A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve the pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin finite element procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach a global system with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the stability and optimal error estimates are also derived by the new technique. Meanwhile, numerical examples are provided to confirm the corresponding theoretical analysis.

Mathematical modeling of the multicomponent flow in porous media using higher-order methods

In this paper, we present a detailed numerical scheme for a single-phase compressible flow without diffusion of a multi-component mixture in porous media with the higher-order approximation in both space and time. The mathematical model consists of Darcy velocity, transport equations for each component of a mixture, pressure equation and associated relations for physical quantities such as viscosity or equation of state. The discrete problem is obtained using a combination of the discontinuous Galerkin method for the transport equations and the mixed-hybrid finite element method for the Darcy velocity and the pressure equation. In both methods the higher-order approximation is used. The resulting nonlinear problem for concentrations is solved with the fully mass-conservative iterative IMPEC method. Experimental order of convergence analysis (EOC) and some numerical experiments of a 2D flow are carried out.

Analyzing the relationship between doublet lifetime and thermal breakthrough at the Dogger geothermal exploitation site in the Paris basin using a coupled mixed-hybrid finite element model

Reinjection of produced water is a critical aspect of managing geothermal reservoirs. The sustainability of extracted thermal energy relies on the performance of well doublets, which in turn is linked to the hydrothermal properties of the reservoir, the thermal characteristics of confining layers, and operating conditions. This study presents an advanced mixed-hybrid finite element simulator that solves coupled hydrothermal equations on unstructured grids. The model is favorably compared with the conforming finite element method at a geothermal power plant site that exploits the Dogger reservoir in the Paris basin. The study also includes a parametric sensitivity analysis, using an ensemble simulation to assess doublet lifetime, thermal breakthrough, and their ratio as performance metrics. Our findings emphasize the significant influence of horizontal permeability anisotropy and production flow rate on doublet performance, with doublet orientation relative to the principal axes of the areal permeability tensor as a key control parameter. Other influential factors include the thermal conductivity of the confining beds, while longitudinal thermal dispersivity and the total productive thickness of the reservoir are less critical. We establish, for the first time, a simple relationship between doublet lifetime and thermal breakthrough, indicating that the former is twice as long as the latter. This underscores the importance of long-term monitoring to sustain geothermal exploitation. This relationship can inform enhanced management of the Dogger reservoir at the basin scale.

Block constrained pressure residual preconditioning for two-phase flow in porous media by mixed hybrid finite elements

This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set of system unknowns comprises both element and face pressures, in addition to the cell saturations, resulting in a 3×3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3\ imes 3$$\\end{document} block-structured Jacobian matrix with a 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} inner pressure problem. The CPR method is one of the most established techniques for reservoir simulations, but most research focused on solutions for Two-Point Flux Approximation (TPFA)-based discretizations that do not readily extend to our problem formulation. Therefore, we designed a dedicated two-stage strategy, inspired by the CPR algorithm, where a block preconditioner is used for the pressure part with the aim at exploiting the inner 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} structure. The proposed preconditioning framework is tested by an extensive experimentation, comprising both synthetic and realistic applications in Cartesian and non-Cartesian domains.

Coupling mixed hybrid and extended finite element methods for the simulation of hydro-mechanical processes in fractured porous media

Simulation of coupled hydro-mechanical (HM) processes in fractured porous media requires specific numerical schemes to deal with nonlinearity, computational burden, and heterogeneity. The mixed hybrid finite element method (MHFEM) is superior to the standard finite element method in simulating flow in fractured domains. However, MHFEM formulation cannot be efficiently used for the discretization of solid mechanics equations. The extended finite element method (XFEM) has significant advantages in modeling mechanical processes in cracked domains. The main goal of this paper is to extend the application of the MHFEM to HM processes in fractured domains by combining it with XFEM. MHFEM and XFEM are applied to flow and mechanical equations, respectively. This coupling allows for an efficient extending of the hybrid dimensional approach to deal with coupled HM processes in fractured domains. The mass lumping technique is generalized to fractured domains and mechanical processes. We show how this technique can be implemented with the fixed-stress split scheme in order to improve the performance and stability of the numerical scheme. The new scheme (MHFEM-XFEM) is validated against analytical and numerical solutions. Comparison against the standard finite element method shows that the new scheme significantly reduces the computational overhead while providing a high accuracy.

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

A hybrid mixed finite element method for convection-diffusion-reaction equation with local exponential fitting technique

A new hybrid mixed finite element method is proposed for solving the convection-diffusion-reaction equation with local exponential fitting technique. In this method, the exponential fitting technique in [7] is used to discretize the convection-diffusion-reaction equation by introducing a new variable at element level; then the hybrid mixed discontinuous Galerkin finite element method is used to approximate the discretized problem. The convergence of the proposed method is analyzed, and the corresponding a priori error estimate is derived. The numerical results are presented to confirm our theoretical analysis.

A Deforming Mixed-Hybrid Finite Element Model for Robust Groundwater Flow Simulation in 3D Unconfined Aquifers with Unstructured Layered Grids

Determining the water table shape and position in unconfined aquifers is fundamental to many groundwater flow assessment studies. The commonly used industry-standard fixed mesh models, contrary to popular belief, do not provide an accurate description of the phreatic surface. When using such models, the water table position is post-processed from the simulated groundwater heads, leading to an approximation error. This error becomes larger for coarse vertical grids. This paper introduces a novel moving mesh technique to simulate the groundwater table in three-dimensional unconfined aquifers under steady-state or transient conditions. We adopt the face-based mixed-hybrid finite element discretization approach in space, leading to a more accurate approximation of the specific discharge field. The model uses an adaptive unstructured but layered mesh which is iteratively adjusted until its top fits the phreatic surface. The developed algorithm accounts for a linearized form of the kinematic boundary condition prescribed on the moving boundary and also supports usual boundary conditions as well. The model was compared to the existing analytical, fixed mesh, and previously published solutions. The obtained results show that the developed model is superior in terms of its numerical stability, convergence behavior, and accuracy. Furthermore, the simulated phreatic surface is free from a cellwise interpolation error and independent of the vertical grid size as used in fixed mesh methods. We also found that the robustness of the moving mesh method cannot be surpassed by a fixed mesh alternative. The model’s efficiency is supported by an almost quadratic rate of convergence of the outer iteration loop.

A robust upwind mixed hybrid finite element method for transport in variably saturated porous media

Abstract. The mixed finite element (MFE) method is well adapted for the simulation of fluid flow in heterogeneous porous media. However, when employed for the transport equation, it can generate solutions with strong unphysical oscillations because of the hyperbolic nature of advection. In this work, a robust upwind MFE scheme is proposed to avoid such unphysical oscillations. The new scheme is a combination of the upwind edge/face centered finite volume method with the hybrid formulation of the MFE method. The scheme ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which permits employment of high-order time integration methods via the method of lines (MOL). Numerical simulations are performed in both saturated and unsaturated porous media to investigate the robustness of the new upwind MFE scheme. Results show that, contrarily to the standard scheme, the upwind MFE method generates stable solutions without under and overshoots. The simulation of contaminant transport into a variably saturated porous medium highlights the robustness of the proposed upwind scheme when combined with the MOL for solving nonlinear problems.

Configurable Open-source Data Structure for Distributed Conforming Unstructured Homogeneous Meshes with GPU Support

A general multi-purpose data structure for an efficient representation of conforming unstructured homogeneous meshes for scientific computations on CPU and GPU-based systems is presented. The data structure is provided as open-source software as part of the TNL library (https://tnl-project.org/). The abstract representation supports almost any cell shape and common 2D quadrilateral, 3D hexahedron and arbitrarily dimensional simplex shapes are currently built into the library. The implementation is highly configurable via templates of the C++ language, which allows avoiding the storage of unnecessary dynamic data. The internal memory layout is based on state-of-the-art sparse matrix storage formats, which are optimized for different hardware architectures in order to provide high-performance computations. The proposed data structure is also suitable for meshes decomposed into several subdomains and distributed computing using the Message Passing Interface (MPI). The efficiency of the implemented data structure on CPU and GPU hardware architectures is demonstrated on several benchmark problems and a comparison with another library. Its applicability to advanced numerical methods is demonstrated with an example problem of two-phase flow in porous media using a numerical scheme based on the mixed-hybrid finite element method (MHFEM). We show GPU speed-ups that rise above 20 in 2D and 50 in 3D when compared to sequential CPU computations, and above 2 in 2D and 9 in 3D when compared to 12-threaded CPU computations.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

Pressure robust hybrid mixed finite element method for Darcy–Forchheimer model

In this paper, we propose a pressure robust hybrid mixed finite element method for Darcy–Forchheimer model. The method uses piecewise polynomials of degrees for the velocity and k−1 for pressure in the interior of elements inside the domain, and piecewise polynomials of degree k for the numerical traces of the pressure on the inter-element boundaries. Our method can get the globally divergence-free approximation solution of velocity. Optimal error estimates are derived and the error estimates of velocity are pressure robust. Finally, numerical experiments confirm the theoretical results.

Conservative Finite Element Modeling of EEG and MEG on Unstructured Grids.

For interpretation of electroencephalography (EEG) and magnetoencephalography (MEG) data, multiple solutions of the respective forward problems are needed. In this paper, we assess performance of the mixed-hybrid finite element method (MHFEM) applied to EEG and MEG modeling. The method provides an approximate potential and induced currents and results in a system with a positive semi-definite matrix. The system thus can be solved with a variety of standard methods (e.g. the preconditioned conjugate gradient method). The induced currents satisfy discrete charge conservation law making the method conservative. We studied its performance on unstructured tetrahedral grids for a layered spherical head model as well as a realistic head model. We also compared its accuracy versus the conventional nodal finite element method ( P1 FEM). To avoid modeling singular sources, we completed our computations with a subtraction approach; the derived expression for the MEG response different from earlier published and involves integration of finite quantities only. We conclude that although the MHFEM is more computationally demanding than the P1 FEM, its use is justified for EEG and MEG modeling on low-resolution head models where P1 FEM loses accuracy.

A hybrid-mixed finite element method for single-phase Darcy flow in fractured porous media

We present a hybrid-mixed finite element method for a novel hybrid-dimensional model of single-phase Darcy flow in a fractured porous media. In this model, the fracture is treated as a (d−1)-dimensional interface within the d-dimensional fractured porous domain, for d=2,3. Two classes of fracture are distinguished based on the permeability magnitude ratio between the fracture and its surrounding medium: when the permeability in the fracture is (significantly) larger than in its surrounding medium, it is considered as a conductive fracture; when the permeability in the fracture is (significantly) smaller than in its surrounding medium, it is considered as a blocking fracture. The conductive fractures are treated using the classical hybrid-dimensional approach of the interface model where pressure is assumed to be continuous across the fracture interfaces, while the blocking fractures are treated using the recent Dirac-δ function approach where normal component of Darcy velocity is assumed to be continuous across the interface. Due to the use of Dirac-δ function approach for the blocking fractures, our numerical scheme allows for nonconforming meshes with respect to the blocking fractures. This is the major novelty of our model and numerical discretization. Moreover, our numerical scheme produces locally conservative velocity approximations and leads to a symmetric positive definite linear system involving pressure degrees of freedom on the mesh skeleton only. As an application, we extend the idea to a simple transport model. The performance of the proposed method is demonstrated by various benchmark test cases in both two- and three-dimensions. Numerical results indicate that the proposed scheme is highly competitive with existing methods in the literature

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

<p style='text-indent:20px;'>This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.</p>

Multi-phase compositional modeling in porous media using iterative IMPEC scheme and constant volume–temperature flash

In this paper, we present a numerical solution of a multi-phase compressible Darcy’s flow of a multi-component mixture in a porous medium. The mathematical model consists of mass conservation equation for each component, extended Darcy’s law for each phase, and an appropriate set of initial and boundary conditions. The phase split is computed using the phase equilibrium computation in the VTN-specification (known as VTN-flash). The transport equations are solved numerically using the mixed-hybrid finite element method and a novel iterative IMPEC scheme [1]. We provide examples showing the performance of the numerical scheme. The convergence of the numerical scheme is verified using the experimental orders of convergence (EOC). This is an extended version of conference paper [2].