Continuous Galerkin Finite Element Research Articles

Impact of micro-rotation on a double-diffusive radiative flow within a lid-driven enclosure fearuring Joule heating, porosity and Lorentz forces

The present paper explores the fluid dynamics, heat and mass transmission characteristics within hexagonal enclosure with square object. The phenomenon encompasses the interplay of various physical mechanism, including micro-rotation of fluid particles, Joule heating, thermal radiation, porous materials, and magnetic field. The governing nonlinear partial differential equations are transformed into non-dimensional form and then investigated numerically using continuous Galerkin finite element method. The pressure component of the momentum equation is removed by employing the Penalty parameter. In order to achieve the consistent solution the value of Penalty parameter is set to 107. The analysis includes several physical parameters, such as Buoyancy ratio (−1–1), micropolar parameter (2–6), Joule heating parameter (0–5), Richardson number (0.5–5), thermal radiation parameter (0.1 – 0.5), Lewis number (0.1–5), Darcy number (0.001–0.1) and Hartmann number (0–50). Results reveal notable trends regarding the impact of key parameters on flow behavior and thermal-solutal transfer within cavity. Increasing Darcy numbers intensify fluid motion and subsequently augments thermal and solutal transfer. The flow regime shifts from shear fricition-dominated at low Richardson number to buoyancy-induced at high Richardson numbers. For concentration-dominant counter flow, a uniform distribution pattern emerges, while aiding flow leads to increased fluid velocity, temperature, and concentration. Additionally, the micropolar parameter influences flow velocity, with varying effects on heat and mass transfer rates depending on the Hartmann number. Mass distribution expands with increasing Lewis number, while temperature rises significantly due to variations in radiation and Joule heating parameters. These findings contribute to a deeper understanding of enclosed flow dynamics and have implications for the optimization of diverse engineering systems.

Numerical simulation of CO2-enhanced oil recovery in fractured shale reservoirs using discontinuous and continuous Galerkin finite element methods

Introduction: This study explores the potential of enhancing shale oil recovery and reducing CO2 emissions through CO2 injection in fractured shale reservoirs. The importance of this approach lies in its dual benefit: improving oil extraction efficiency and addressing environmental concerns associated with CO2 emissions.Method: We employed a discrete fracture-matrix model to simulate CO2 flooding in fractured shale reservoirs, utilizing both discontinuous Galerkin (DG) and continuous Galerkin (CG) finite element methods. The DG-CG FEM’s accuracy was validated against the McWhorter problem, ensuring the reliability of the simulation results. Our model also considered various factors, including reservoir heterogeneity, fracture permeability, CO2 injection volume, and gas injection patterns, to analyze their impact on shale oil recovery.Result: Our simulations revealed that fractured reservoirs significantly enhance shale oil production efficiency compared to homogeneous reservoirs, with an approximate 48.9% increase in production. A notable increase in shale oil production, by 15.8%, was observed when fracture permeability was increased by two orders of magnitude. Additionally, a fourfold increase in CO2 injection rate resulted in a 31.5% rise in shale oil production. Implementing a step-by-step reduction in injection volume while maintaining the total CO2 injection constant proved to be more effective than constant-rate injections.Discussion: The study demonstrates the effectiveness of CO2 flooding in fractured shale reservoirs for enhancing shale oil recovery.

FEMFFUSION and its verification using the C5G7 benchmark

FEMFFUSION is an open source code that solves the multigroup neutron transport equation using the diffusion and the SPN approximations. This code uses the continuous Galerkin finite element method to be able to deal with any type of geometry, structured or unstructured, and problem dimension. To verify the program, a research study of the two-dimensional C5G7 benchmark has been performed, including the static, time-dependent and neutron noise computations. Numerical results include the effective multiplication factor, keff, the spatial neutron fluxes, and neutron power distributions for the static computations. The total neutron power evolution of a given transient and the neutron flux distribution evolution for the different energy groups. In addition, neutron noise frequency domain calculation were completed for perturbations of some cross-section in the reactor core. All computation have been performed using different SPN approximations. FEMFFUSION results are verified against the official OECD/NEA benchmark results.

Fully coupled hydro-mechanical modeling of two-phase flow in deformable fractured porous media with discontinuous and continuous Galerkin method

Discrete fracture and matrix modeling of coupled flow and geomechanics is instrumental for understanding flow in fractured media for various geoengineering applications such as enhanced geothermal energy systems and groundwater remediation. We employ the governing equations for two-phase flow in deformable fractured porous media with a stress-dependent porosity and permeability model of matrix, and variable fracture aperture and the corresponding permeability. A finite element framework is presented, in which fractures are regarded as low-dimensional objects, to discretize the coupled two-phase flow and geomechanics in fractured porous media. A hybrid method, combining discontinuous Galerkin (DG) and continuous Galerkin (CG) finite element methods (FEM), is utilized to solve for the two-phase flow, while the solid deformation is approximated using a discontinuous Galerkin FEM approach. Several benchmark cases are utilized to examine the accuracy of the proposed hybrid DG-CG FEM method. Further validation is performed using more complex, realistic fracture configurations. As the mechanical characteristics of the fracture and the surrounding matrix differ, the simulation results demonstrate that displacement and stress are discontinuous on both sides of the fracture. While the case with low permeability fractures exhibits pressure jump across the fractures, the pressure changes are reasonably smooth for the conduit fractures.

Numerical simulation and intelligent prediction of thermal transport of a water-based copper oxide nanofluid in a lid-driven trapezoidal cavity

This article investigates the fluid dynamics and heat transfer properties in a trapezoidal enclosure containing a heated cylindrical object. It involves the interaction of multiple physical processes such as the magnetic field, thermal radiation, porous materials, and aqueous copper oxide nanoparticles. The governing partial differential equations are analyzed numerically through the continuous Galerkin finite element algorithm. The analysis takes into account various physical parameter factors, including the Richardson number (0–5), the Hartmann number (5−40), the Darcy number (0.001−0.1), thermal radiation parameter (0.5−2), and nanoparticle volume concentration (0.01−0.1). The physical mechanism of thermal and mass transfer in the enclosure caused by various factors is fully explored. In addition, the multiple expression programming (MEP) technique is implemented to report a comparative analysis of flow profiles and thermal distribution. The findings demonstrated that at low Ri, the primary flow within the cavity is driven by the shear friction generated by the moving walls. The growing importance of radiative heat transfer reduces the effectiveness of convective heat transfer, resulting in a decline in the average Nusselt number with R. The heat transfer rate rises up to 27.7% as ϕ augments; however, its value declines by 9.37% against Ha. The expected results obtained by the MEP approach are very consistent with the numerical ones. There is no doubt that the new MEP concept provides a valuable tool for researchers to predict the heat transfer behavior of any data set in cavities of different shapes. It is expected to provide new idea for the development of efficient cooling systems and the improvement of energy efficiency in various engineering applications.

Linearization errors in discrete goal-oriented error estimation

This paper is concerned with goal-oriented a posteriori error estimation for nonlinear functionals in the context of nonlinear variational problems solved with continuous Galerkin finite element discretizations. A two-level, or discrete, adjoint-based approach for error estimation is considered. The traditional method to derive an error estimate in this context requires linearizing both the nonlinear variational form and the nonlinear functional of interest which introduces linearization errors into the error estimate. In this paper, we investigate these linearization errors. In particular, we develop a novel discrete goal-oriented error estimate that accounts for traditionally neglected nonlinear terms at the expense of greater computational cost. We demonstrate how this error estimate can be used to drive mesh adaptivity. We show that accounting for linearization errors in the error estimate can improve its effectivity for several nonlinear model problems and quantities of interest. We also demonstrate that an adaptive strategy based on the newly proposed estimate can lead to more accurate approximations of the nonlinear functional with fewer degrees of freedom when compared to uniform refinement and traditional adjoint-based approaches.

A Review of Continuous and Discontinuous Galerkin Finite Element Methodfor Differential Equations

A Review of Continuous and Discontinuous Galerkin Finite Element Methodfor Differential Equations

Solution of Neutron Diffusion Problems by Discontinuous Galerkin Finite Element Method With Consideration of Discontinuity Factors

Abstract Neutronics calculations are the basis of reactor analysis and design. Finite element methods (FEM) have gained increasing attention in solving neutron transport problems for the rigorous mathematical formulation and the flexibility of handling complex geometric domains and boundary conditions. In order to reduce the computational errors caused by the homogenization of cross sections, this paper adopts the discontinuous Galerkin finite element method (DG-FEM) to solve the generalized eigenvalue problem formulated by the neutron diffusion theory and compensates the homogenization error by incorporating discontinuity factors. The results show that the discontinuous Galerkin finite element method can introduce the discontinuity factors with clear mathematical and physical meanings. The computational results of the discontinuous Galerkin finite element method are slightly better than those of the continuous Galerkin finite element method. However, the computation cost of the former is higher than that of the latter. Although good parallel efficiency can be achieved, the discontinuous Galerkin finite element method is not preferable for large-scale problems unless the effect of the discontinuity factors is significant.

A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes

The Cahn-Hilliard Navier-Stokes (CHNS) system provides a computationally tractable model that can be used to effectively capture interfacial dynamics in two-phase fluid flows. In this work we present a semi-implicit, projection-based finite element framework for solving the CHNS system. We use a projection-based semi-implicit time discretization for the Navier-Stokes equation and a fully-implicit time discretization for the Cahn-Hilliard equation. We use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) formulation. Pressure is decoupled using a projection step, which results in two linear positive semi-definite systems for velocity and pressure, instead of the saddle point system of a pressure-stabilized method. All the linear systems are solved using an efficient and scalable algebraic multigrid (AMG) method. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. The overall approach allows the use of relatively large time steps with much faster time-to-solve than similar fully-implicit methods. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise and Rayleigh-Taylor instability.

A Priori and a Posteriori Error Analysis for Generic Linear Elliptic Problems

In this paper, a priori error analysis has been examined for the continuous Galerkin finite element method which is used for solving a generic scalar and a generic system of linear elliptic equations. We derived optimal order a priori error bounds in (energy) norm utilising standard a priori error analysis techniques and tools. Also, a posteriori error analysis is investigated for a generic scalar linear elliptic equation and for a generic system of linear elliptic equations. We derived optimal residual-based a posteriori error estimates energy technique in norm

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A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.

Finite element approximation and analysis of a viscoelastic scalar wave equation with internal variable formulations

We consider linear scalar wave equations with a hereditary integral term of the kind used to model viscoelastic solids. The kernel in this Volterra integral is a sum of decaying exponentials (The so-called Maxwell, or Zener model) and this allows the introduction of one of two types of families of internal variables, each of which evolve according to an ordinary differential equation (ODE). There is one such ODE for each decaying exponential, and the introduction of these ODEs means that the Volterra integral can be removed from the governing equation. The two types of internal variable are distinguished by whether the unknown appears in the Volterra integral, or whether its time derivative appears; we call the resulting problems the displacement and velocity forms. We define fully discrete formulations for each of these forms by using continuous Galerkin finite element approximations in space and an implicit ‘Crank–Nicolson’ type of finite difference method in time. We prove stability and a priori bounds, and using the FEniCS environment, https://fenicsproject.org/ (The FEniCS project version 1.5, Archive of Numerical Software, 3 (100), 9–23, 2015.) give some numerical results. These bounds do not require Grönwall’s inequality and so can be regarded to be of high quality, allowing confidence in long time integration without an a priori exponential build up of error. As far as we are aware this is the first time that these two formulations have been described together with accompanying proofs of such high quality stability and error bounds. The extension of the results to vector-valued viscoelasticity problems is straightforward and summarised at the end. The numerical results are reproducible by acquiring the python sources from https://github.com/Yongseok7717, or by running a custom built docker container (instructions are given).

Evaluation of the impact of strain-dependent permeability on reservoir productivity using iterative coupled reservoir geomechanical modeling

Permeability an important property plays a key role in reservoir performance, numerical reservoir simulation, drilling and production planning. In such reservoirs, stress and strain alters induced by extraction and injection of fluid may substantially change permeability in an irreversible manner. With regard to this phenomenon, several reservoirs may require to consider straindependent permeability, in order to have an accurate performance. In such reservoirs, especially carbonate reservoirs which are a significant proportion of the world’s hydrocarbon reserves, to present the mechanical behavior of rocks, the cap elastoplastic model is may be required. Therefore, the goal of this paper is to evaluate the strain-dependent permeability using coupled reservoir geomechanical modeling with considering the cap elastoplastic model. This coupling is implemented using a fixed stress iterative coupled scheme. In this coupling, the governing equation for fluid is presented by Darcy’s law in which the permeability is nonlinear. The geomechanical module is analyzed using Biot’s theory and cap elastoplastic model. To indicate the mechanical behavior of reservoir rock, the DiMaggio-Sandler elastoplastic model which is a cap envelope is selected. The numerical approximation is done by employing mixed finite element for reservoir module and a continuous Galerkin finite element for geomechanics. Several numerical simulations are performed to verify the models and analyze the effect of strain-dependent permeability on reservoir productivity.

Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach

A time-explicit Runge–Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection–reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ⩾ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results indicate that the proposed scheme is more robust and accurate in comparison to existing approaches based on the continuous Galerkin finite element method or the fast Fourier transform method.

A method of FE modeling multiphase compressible flow in hydrocarbon reservoirs

We propose a model and numerical method for multiphase multicomponent flow simulation in porous media for solving problems of hydrocarbon reservoir engineering. The method takes into account the complex structure of reservoirs, a large number of operating wells, various methods of specifying the dependences of the properties of phases on pressure and mass fractions of their components, the transition of components from one phase to another, as well as compressibility and gravity. The method provides a sufficiently high computational efficiency, local conservation of mass for each component and protects the requirement of constraint of phases saturations from violation due to the following aspects. To calculate the phase flows we use a combination of continuous Galerkin finite element method (CG FEM) with a new balancing method. This combination provides high accuracy of flows calculated from the pressure field on rather coarse meshes. We also propose to use the grouping of finite elements, which allows us to remove strict restrictions on the time step when calculating the phase flows between cells. Verification of the proposed method is performed by comparing with the results of SPE tests. The effectiveness of the method is demonstrated both on simple model problems and on problems of modeling a complex reservoir. For a real field of high-viscosity oil in Tatarstan (Russia), a good agreement between the computed and observed data is obtained for all high-rate wells over a fairly long period of field development.

Investigation of micro-shock waves in a planar magnetogasdynamic flow using the discontinuous Galerkin finite element method

The present work focuses on the numerical investigation of micro-shock wave propagation in a two-dimensional magnetogasdynamic flow in the framework of the Dis- continuous Galerkin-Finite Element Method (DG-FEM). The Lorentz force has been im- plemented in the compressible, viscous Navier–Stokes equations as a source term using first-order spatial and fourth-order temporal Runge–Kutta discretization schemes. To in- vestigate the effect of the electrical conductivity on the micro-shock wave propagation, a two-dimensional micro-shock channel problem with hydraulic diameter of 2.5 mm, length of 82 mm, and no-slip boundary conditions at the left and at the right wall is considered as a benchmark problem. In this case, acoustic waves are generated behind after the rupture of the membrane that separates two states of the same gas originally at different pressure and density and both initially at rest. The magnetic field is taken into account as uniform and stationary throughout the microchannel, and the numerical simulations are performed in a short physical time, before the reflection of the waves on the lateral wall. A detailed parametric study of the temperature, density, pressure, and u-velocity is carried out by a variation of the electrical conductivity of the magnetogasdynamic flow, under the assumption of low magnetic Reynolds numbers. It has been found that the jumps of the acoustic waves become significantly intensified when the electrical conductivity of the gas is increased. It has also been observed that the presence of the Lorentz force causes an acceleration in the gasflow towards the outlet section of the microchannel at the low Knudsen number of 0.05. The outcome of this research work could be relevant to biomedical applications where the ability to control the flow in a microchannel has a significant impact on the development of small devices aimed to deliver pharmaceutical drugs in specific locations.

A Phase-Field-Based Approach for Modeling Flow and Geomechanics in Fractured Reservoirs

Summary Modeling the geomechanical deformations of fracture networks has become an integral part of designing enhanced geothermal systems and recovery mechanisms for unconventional reservoirs. Stress changes in the reservoir can cause variations in the apertures of fractures resulting in large changes in their transmissivities. At the same time, sustained high-injection pressures can induce shear slipping along existing fractures and faults and trigger seismic activity. In this work, we extend the phase-field method to solve for flow and geomechanical deformations in naturally fractured reservoirs. The framework can predict the opening/closure of fractures as well as their shear slipping because of induced stresses and poromechanical effects. The flow through fractures is modeled on spatially nonconforming grids using the enhanced velocity mixed finite element method. The geomechanics equations are discretized using the continuous Galerkin (CG) finite element method. The flow and mechanics equations are decoupled using the fixed stress iterative scheme. The implementation is validated against the analytical solutions of Mandel’s problem and Sneddon’s benchmark test. Two synthetic examples are presented to illustrate the impact of poroelastic deformations and the accompanying dynamic behavior of fractures on the safety and productivity of subsurface projects.

Discontinuous Galerkin for the wave equation: a simplified a priori error analysis

Standard discontinuous Galerkin methods, based on piecewise polynomials of degree , are considered for temporal semi-discretization for second order hyperbolic equations. The main goal of this paper is to present a simple and straightforward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved. To this end, energy identities and stability estimates of the discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used, to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical results.

Energy stable and accurate coupling of finite element methods and finite difference methods

We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are present. The proposed coupling technique requires minimal changes in the existing schemes while maintaining strict stability, accuracy, and energy conservation. Results are demonstrated on linear and nonlinear scalar conservation laws in two spatial dimensions.