Galerkin Finite Element Method Research Articles

A robust weak Galerkin finite element method for two parameter singularly perturbed parabolic problems on nonuniform meshes

In this study, we proposed a weak Galerkin finite element method (WG-FEM) for solving two-parameter singularly perturbed parabolic problems (TP-SPPPs) of convection–diffusion–reaction type on nonuniform mesh. The WG-FEM approach incorporates an interpolation operator IN and achieves optimal order convergence in an energy-like norm for continuous and fully discrete schemes. In the fully-discrete analysis, we combined WG-FEM for spatial space on a Shishkin-type mesh with the Crank–Nicolson scheme for temporal discretization on an equidistant mesh. Since the energy norm is insufficient to capture the behavior of the boundary layer functions accurately, we have derived the optimal order of convergence in a strong balanced norm. This norm exhibits an order of O(N−1lnN)k in spatial convergence and second-order convergence in time when using an L2-projection QN as an intermediate operator. We conducted numerical examples to validate the proposed method’s theoretical findings. The results of these experiments confirmed the theoretical conclusions and demonstrated the robustness of the proposed method.

A comparative analysis of the vibrational behavior of various beam models with different foundation designs

This article discusses the modal behavior of elastically constrained beams under various types of foundations and provides insights into the effects of different factors on the eigenfrequencies of beams. Numerical and analytical techniques, specifically the Galerkin finite element method (GFM) and the separation of variables, are utilized to determine the eigenfrequencies and mode shapes of beams. Modal analysis of Timoshenko, shear, Rayleigh, and Euler-Bernoulli beams that are elastically constrained and resting on Winkler, Pasternak, and Hetényi foundations, considering non-classical boundary conditions, is included in the study. The effects of factors such as flexural rigidity, transverse modulus, and Winkler foundation constant on natural frequencies of different beam models are investigated. The proposed method efficiently converges to the exact solution without shear locking in the stiffness element. The results demonstrate that the natural frequencies of the beam rise because of the shear layer, flexural rigidity, and foundation constant. Furthermore, the Hetényi elastic foundation affects the natural frequency of the beam, depending on the relative values of beam stiffness and foundation stiffness. Additionally, incorporating both shear deformation and rotary inertia has a greater impact on the eigenfrequencies of Euler-Bernoulli beams compared to incorporating only one of these effects. The findings of this work provide valuable insights into the behavior of beams under different foundation conditions and have potential applications in the design and optimization of structures incorporating beams, thereby enhancing the understanding of beam analysis.

A discontinuous Galerkin finite element method for the Oldroyd model of order one

In this work, we analyze a discontinuous Galerkin finite element method for the equations of motion that arise in the 2D Oldroyd model of order one. We investigate the existence and uniqueness of semidiscrete discontinuous solutions, as well as the consistency of the scheme. We derive new a priori and regularity results for the discrete solution and establish optimal error estimates in ‐norm in time and energy norm in space for the velocity and ‐norm in both time and space for the pressure. Uniform estimates are derived for sufficiently small data. We next apply the backward Euler method to the semidiscrete formulation and establish optimal fully discrete error estimates. At the end, we conduct numerical experiments to support our theoretical results and analyze the findings.

Thermal storage study and enhancement of heat transfer through hybrid Jeffrey nanofluid flow in ducts under peristaltic motion with entropy generation

This article examines the analysis of peristaltic transport within a vertical sinusoidal wavy duct using a hybrid nanofluid composed of deionized blood and nanoparticles such as Graphene Oxide (Go) and Molybdenum disulfide (MoS2). The study also investigates volumetric entropy generation, employing the Jeffrey fluid model known for its representation of both viscous and elastic fluid behaviors. The mathematical framework integrates Navier–Stokes equations and the energy equation. Simplifying the analysis, non-dimensional equations are derived for the hybrid nanofluid in a two-dimensional coordinate system, assuming a long wavelength (δ < 1) and low Reynolds number (Re). The Galerkin finite element method via MATLAB is employed for numerical solutions. The results, depicted graphically using MATLAB, offer insights into temperature variations and the two-dimensional peristaltic flow concerning different parameters. To the best of authors knowledge entropy generation analysis is still not carried out for blood base Go-MoS2 hybrid nanofluid in a vertical wavy channel. The aim of the present study is to explore the contribution of significant parameters on entropy generation. One of the key findings includes that flow rate and Brinkmann number contribute to enhance entropy generation in presence of hybrid nanofluid.

A discrete-ordinate weak Galerkin method for radiative transfer equation

This research article discusses a numerical solution of the radiative transfer equation based on the weak Galerkin finite element method. We discretize the angular variable by means of the discrete-ordinate method. Then the resulting semi-discrete hyperbolic system is approximated using the weak Galerkin method. The stability result for the proposed numerical method is devised. A priori error analysis is established under the suitable norm. In order to examine the theoretical results, numerical experiments are carried out.

A posteriori error estimates of the weak Galerkin finite element methods for parabolic problems

In this paper, We propose the residual-based a posteriori error estimator of the weak Galerkin finite element method with the backward Euler time discretization for the linear parabolic partial differential equation. For the a posteriori error estimator, we introduce the Helmholtz decomposition technique to prove its reliability. We mainly study WG element (Pj(K),Pℓ(∂K),V(K,r)=RTj(K)). Numerical experiments based on the lowest order case, i.e. (P0(K),P0(∂K),RT0(K)), are provided to verify the theoretical research.

Generalized weak Galerkin finite element methods for second order elliptic problems

This article proposes and analyzes the generalized weak Galerkin (gWG) finite element method for the second order elliptic problem. A generalized discrete weak gradient operator is introduced in the weak Galerkin framework so that the gWG methods would not only allow arbitrary combinations of piecewise polynomials defined in the interior and on the boundary of each local finite element, but also work on general polytopal partitions. Error estimates are established for the corresponding numerical functions in the energy norm and the usual L2 norm. A series of numerical experiments are presented to demonstrate the performance of the newly proposed gWG method.

Heat transfer and fluid flow in nano-encapsulated PCM-filled undulated cavity

This article discusses the heat transfer characteristics of the mixed convection of NEPCM-fluid (Polyurethane/ Nonadecane dispersed in water) contained within a wavy-sided walls cavity. Mixed convection is caused by an adiabatic rotating cylinder located in the cavity center, while natural convection results from deferential heating of side walls. The upper and lower walls of the cavity are insulated. The Galerkin finite element method (GFEM) implemented in COMSOL CFD package is employed to solve the governing equations. The influence of the nanoparticles volume fraction (Ø = 0.02, and 0.05), cylinder radius (R = 0.05, 0.15, and 0.25), cylinder rotating speed (Ω = 100, −500, 500, and 1000), the dimensionless fusion temperature (θfu = 0.1, 0.25, and 0.5), wavy side walls undulation peaks number (N = 1 to 4) on the flow pattern and temperature field are presented. The results revealed that the heat transfer enhances by increasing the angular speed of the cylinder and its radius, also an enhancement in heat transfer is obtained by upgrading the volume fraction of NEPCM, however, it was noticed that for small cylinders, the counterclockwise rotation of the cylinder reduced the heat transfer rate as the rotation opposed the direction of natural heat convection. In addition, the waviness pattern has a specific impact on each case, for example, N = 4 denotes the best result for only the cases Ω = -500 or 500 when R = 0.25 and Ø = 0.05, while the heat transfer is negatively affected by increasing N for the other cases. On another hand, the impact of θfu on the heat transfer rate was found to be insignificant. Globally, at the biggest cylinder radius, higher Ø and highest angular speed an enhancement by more than 900 % can be achieved.

The uniform convergence of a weak Galerkin finite element method in the balanced norm for reaction–diffusion equation

In this paper, a weak Galerkin finite element method is implemented to solve the one-dimensional singularly perturbed reaction–diffusion equation. This weak Galerkin finite element scheme uses piecewise polynomial with degree k≥1 in the interior part of each element and piecewise constant function at the nodes of each element. The existence and uniqueness of the weak Galerkin finite element solution are proved. Based on the interpolation operator and the corresponding approximation properties, an ɛ−uniform error bound of O(N−(k+1)+ɛ(N−1lnN)k) in the energy-like norm is investigated rigorously. Furthermore, an ɛ−uniform error bound of O(N−(k+1/2)+(N−1lnN)k) in the balanced norm is established by the weighted local L2 projection and its corresponding approximation properties. Finally, numerical experiments validate the theoretical results. Moreover, numerical results show that this weak Galerkin finite element solution achieves the convergence rate of O((N−1lnN)k+1) in the L2 norm and the discrete L∞ norm uniformly with respect to the singular perturbation parameter ɛ.

Morphological applications of triadic hybridized nanoparticles in complex fluid with variable thermal radiations and bouncy force employing the Galerkin finite element method

The current development reveals an increment in thermophysical properties of viscous hybrid nanomaterial thought distribution in triadic hybrid nanoparticles involving quadratic and linear convection in Ellis’s fluid across a 3-D rotating sheet. Heat transfer takes place utilizing variable thermal radiation and heat sink/source. Four kinds of metallic oxide nanomaterials based on copper, aluminum, and titanium in motion of ethylene glycol are inserted. The phase of single-type triadic hybrid nanostructures is developed. Heat transfer and momenta are considered in the form of partial differential equations assuming multiple aspects. Similarity vectors are utilized to find a system of ODEs through PDEs. Finite element methodology is executed to achieve numerical consequences. The findings related to physical variables based on related distributions and associated quantities regarding physical consequences are captured in detail. The efficiency of thermal energy for the case of copper–aluminum/EG is higher than thermal energy efficiency for the case of copper–titanium oxide/EG and working fluid and nanofluid. Copper–aluminum/EG brings enhancement in the efficiency of temperature gradient for the case of copper–titanium oxide/EG. By higher values of solar thermal radiation number, the temperature field is magnified.

The baffle shape effects on natural convection flow and entropy generation in a nanofluid-filled permeable container with a magnetic field

Enhancing heat transfer rates within enclosures is a topic of considerable interest since it has several technical applications. Most heat transfer research projects focus on increasing the heat transfer rates of thermal systems since this will raise the systems' total efficiency. The geometry of the enclosure might have a substantial impact on heat transfer rates. This research studies quantitatively the natural convection of a nanofluid in a complicated form geometry with many baffle configurations. The system's governing equations were addressed by the Galerkin Finite Element Method (GFEM). The main consideration was given to the effects of the following factors: The Darcy number (Da), which ranges from 10–2 to 10–5; the Hartmann number (Ha), which ranges from 0 to 100; the volumetric fraction (ϕ), which ranges from 0 to 0.08, and the Rayleigh number (Ra) (102 to 106). The results suggested that raising Ra increases heat transfer discharge, whereas raising Ha and Da decreases it. In terms of heat transmission, case 1 (the case with a wavenumber of 1 and the zigzag pointing outward) is determined to be the optimum cavity structure, as it obtained the highest mean Nusselt (Nuavg) number when compared to other cases. At the highest studied Ra number, growing (ϕ) from 0 to 0.8 improved Nuavg by 25%, while growing Da from 10–2 to 10–5 and Ha from 0 to 100 declined Nuavg by 57% and 48%, respectively. The reason for the improvement in the values of the (Nu) is due to the speed of fluid movement within the compartment. Also, the shape of fins plays a major role in strengthening and weakening thermal activity.

Investigation of MHD oxytactic microorganisms with NEPCMs in rectotrapezoidal enclosure with FEM: Applications to energy storage technologies

Nanoparticle-enhanced phase change materials (NEPCMs) have gained significant attention in recent years due to their unique ability to enhance the thermal properties and overall performance of phase change materials (PCMs). The present work investigates the bioconvection flow of nano-encapsulated phase change material within a rectotrapezoidal cavity. The left wall of the is fixed at the constant high temperature whereas the right vertical wall is kept at cold temperature. The higher order Galerkin finite element method (GFEM) has been employed to simulate the proposed problem. The discrete systems of nonlinear equations are linearized by implementing the adaptive Newtons method. Influence of various pertinent parameters such as inclined magnetic field, oxytactic bacteria, radiation and Lewis number has been analyzed. It is observed that the average Sherwood and Nusselt number exhibit a positive correlation with the Hartmann and radiation parameter. Moreover, the average heat transfer exhibits a decreasing trend as the Hartmann, bioconvection Rayleigh and Peclet increase.

Comparison Study Between Galerkin Finite Element Method and Finite Volume Method for Diffusion Problem

The finite element method (FEM) is a robust and widely applied numerical scheme in the simulation of engineering problems, especially in structural mechanics. However, FEM is not as popular as the finite volume method (FVM) in Computational Fluid Dynamics (CFD), possibly due to its complicated numerical procedures. Indeed, FEM possesses tremendous advantages compared with FVM, particularly in dealing with complex geometry and rendering attractive flexibility to modify the interpolation functions. It is well-known that FEM and FVM differ in mathematical formulation, yet there is a lack of practical comparison between them. Therefore, the paper aims to develop a Galerkin FEM (GFEM) model, investigate its strengths and weaknesses compared with FVM, and discuss the conciliation between FEM and FVM. Our case study focuses on a two-dimensional diffusion problem comprising steady and transient cases, with and without heat generation. Our investigation revealed that GFEM does not possess conservative properties, which might yield spurious heat flux, leading to a 2 – 4% overestimation of the temperature field, depending on the amount of heat generation. Moreover, GFEM incurs approximately 34% higher computational time than FVM. However, FVM can be perceived as a special form of GFEM, and their relations were discussed.

A Discontinuous Galerkin–Finite Element Method for the Nonlinear Unsteady Burning Rate Responses of Solid Propellants

The unsteady combustion of solid propellants under oscillating environments is the key to understanding the combustion instability inside solid rocket motors. The discontinuous Galerkin–finite element method (DG-FEM) is introduced to provide an efficient yet flexible numerical platform to investigate the combustion dynamics of solid propellants. The algorithm is developed for the classical unsteady model, the Zel’dovich–Novozhilov model. It is then validated based on a special analytical solution. The DG-FEM algorithm is then compared with the classical spectral method based on Laguerre polynomials. It is shown that the DG-FEM works more efficiently than the traditional spectral method, providing a more accurate solution with a lower computational cost.

Analytical and computational analysis of vibrational behavior of axially loaded beams placed on elastic foundations

AbstractIn this study, the vibrational behavior of Rayleigh and shear beams placed over the Hetényi and Pasternak foundations is investigated, with special attention paid to the effects of axial stresses (both tensile and compressive). The uniqueness of the work comes from the thorough consideration of how axial forces, shear deformation, rotational inertia, and foundation factors affect the eigenfrequencies and eigenmodes that control beam vibrations. Eigenfrequencies and eigenmodes are determined through computational analyses with the Galerkin finite element method (GFEM) employing quadratic and cubic polynomials. Comparisons between finite element results and analytical outcomes, both in generalized and specialized contexts, serve to demonstrate the correctness and effectiveness of the approximate solution. The study shows that the Hetényi foundation's influence on eigenfrequency depends on foundation stiffness and relative beam magnitudes, and that shear deformation affects eigenfrequencies more than rotary inertia does. Higher frequencies are largely unaffected by compressive and tensile stresses, but the fundamental frequency is rigorously influenced. Further demonstrating its accuracy, the finite element approach shows great agreement in forecasting eigenmodes and eigenfrequencies. A variety of real‐world situations involving elastically confined beams resting on elastic foundations and being subjected to axial forces are the focus of the study. Axial forces commonly result from external loads, thermal expansion, or structural interactions in real‐world settings. Therefore, it is crucial to understand how they affect beam vibration when designing structures that can endure expected stresses and vibrations while maintaining safety and performance criteria.

Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity

Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity

Magnetic field effect on mixed convection flow inside an oval-shaped annulus enclosure filled by a non-Newtonian nanofluid

The phenomenon of flow around a static or rotating circular cylinder has significant applications in various engineering fields, such as building, bridge, and structure engineering. It can be used to control flow-induced vibration, blowing or suction, moving surfaces, acoustic thriller, and synthetic jet. Additionally, there is a need to enhance heat transfer rates in many life and industrial applications, including cavity shape enhancement, heating or cooling spread inside a cavity, heat dissipation in electronic devices, thermal mixing in storage units, solar collectors, desalination, and journal bearing lubrication. This study explores the effect of a horizontal magnetic field and a rotating inner circular cylinder on mixed convection within a two-dimensional oval-shaped enclosure filled with a non-Newtonian nanofluid. The Galerkin Finite Element Method (GFEM) is utilized for analysis, with the enclosure undergoing differential heating and cooling on the inner cylinder wall and the enclosure wall, respectively. The nanofluid consists of water infused with copper nanoparticles. The study explores various simulation parameters, including the Richardson number (Ri), Hartmann number (Ha), the inclination angle of the enclosure (β), power-law index (n), circle center eccentricity (e), and nanoparticles volume fraction (φ). It has been observed that as the Ha and n index increase for different Ri values, a wavy trend in local Nusselt number (Nu) profiles along the heat source surface is observed. Additionally, the heat transfer rate decreases with increasing Ri for various combinations of Ha and n index. Under specific conditions, such as Ri = 0.001, Ha = 60, e = 0, φ = 0.09, and β = 90, the results show that, compared to a Newtonian fluid (n = 1), the heat transfer rate exhibits percentage gains of approximately 6.31% for a pseudo-plastic fluid (n = 0.6) and 8.47% for a Dilatant fluid (n = 1.4). The convective heat transfer was significantly enhanced when the eccentricity of the cylinder was positioned in the lower or upper regions of the oval-shaped enclosure, particularly at e = - 0.3. The highest heat transfer rates were observed at β = 60 and 240.

Role of metal inserts for ‘targeted’, ‘uniform’ or ‘controlled’ microwave heating objectives involving generalized sets of dielectrics (Group 1 – 4)

Microwave heating occurs volumetrically leading to faster processing than conventional heating processes. Over the last few decades, the potential of microwave heating has been investigated for various thermal processing applications such as food processing, material synthesis, fuel production and waste treatment/remediation. Current work investigates the role of the metal inserts towards targeted, uniform and/or controlled microwave heating for the four groups of dielectrics (Group 1 – 4) involving lateral and isotropic microwave incidences. The microwave induced heat generation and evolving temperature distributions within the materials in the absence and presence of the metal insert are numerically determined via the mathematical models comprising of Helmholtz equation, energy balance equation and an integro-differential boundary condition representing the scattering of the microwave at the air-material interface. Galerkin finite element method with biquadratic elements along with Crank-Nicolson time integration scheme has been employed to solve the governing equations. It has been found that the metal insert can suitably alter the heating characteristics leading to either faster heating, better heating uniformity or more intense targeted heating of the materials. Uniform microwave heating is generally difficult to attain except for very small samples. This work shows that the metal inserts can significantly reduce the heating inhomogeneity of the moderately large samples involving Group 1 to Group 3 materials. In addition, the targeted heating effect of the materials (Groups 1 to 4) can also be intensified by the use of the metal insert, as has been illustrated for various test cases. Especially, the metal insert can lead to targeted heating at the incident surface irrespective of the material or sample dimension, and that is difficult to attain except for the large samples. The metal inserts are also found to be efficient to control the rapid microwave heating experienced by the small samples of lossy materials (Group 1 and 3 materials) or accelerate the slower heating of the large samples resulting in significant energy savings, especially for Group 1 to Group 3 materials.

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.

A fast linearized Galerkin finite element method for the nonlinear multi-term time fractional wave equation

This paper is concerned with a class of nonlinear multi-term time fractional wave equations. Based on the Galerkin finite element method (FEM) in space and the L2-1σ formula in time, combined with the corresponding fast algorithm, an efficient fully discrete scheme is constructed. The unconditional optimal convergence property in H1-norm of the proposed scheme is discussed by utilizing the time-space splitting technique. Furthermore, the optimal L2-norm convergence and the H1-norm superconvergence results of the method are obtained. Finally, several illustrative numerical experiments are performed to verify the theoretical results. Compared with the existing works, studying the convergence of FEMs for the nonlinear fractional wave equation is a great challenging task, and there are significant differences in research approaches.