Galerkin Finite Element Method Research Articles

On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations

Abstract In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

A Galerkin Finite Element Method for the Reconstruction of a Time-Dependent Convection Coefficient and Source in a 1D Model of Magnetohydrodynamics

The mathematical analysis of viscous magnetohydrodynamics (MHD) models is of great interest in recent years. In this paper, a finite element Galerkin method is employed for the estimation of an unknown time-dependent convection coefficient and source in a 1D magnetohydrodynamics flow system. In this inverse problem, two integral observations are posed and used to transform the inverse problem to a non-classical direct problem with a non-local parabolic operator. Then, the non-classical strongly coupled parabolic system is studied in various settings. The equivalence of the inverse problem (IP) and the direct one are proven. The Galerkin procedure is analyzed to proove the existence and uniqueness of the solution. The finite element method (FEM) has been developed for the solution of the variational problem. Test examples are discussed.

Impact of inclined magnetic field on natural convection in ferrofluid within a semi-circular enclosure with a heated wavy surface

Ferrofluids are liquids that exhibit distinctive magnetic properties when subjected to a magnetic field. Due to their magnetic susceptibility, ferrofluids have recently gained considerable scientific attention. They find tools in different fields, such as electronics, mechanical engineering, and materials science. The inclusion of the semi-circular enclosure inscribed around a heated wavy cylinder provides innovative perspectives essential for engineering applications involving induction furnaces and cancer therapy devices. This paper aims to conduct a computational investigation into the influence of an inclined magnetic field on the free convection within a semi-circular cage stuffed with a ferrofluid featuring an internally undulating surface. It is considered that the interior wavy cylinder was warmed ( T h ) , and the outer semi-circular enclosure is cold ( T c ) . The analyzed system’s governing equations are computed using the Galerkin finite element method (GFEM). The ferrofluid’s flow pattern and thermal transport rate are investigated about relevant non-dimensional values, such as the wave numbers, the number of Hartmann ( 0 ≤ Ha ≤ 20 ) , the number of Rayleigh ( 10 3 ≤ Ra ≤ 10 6 ) , and the volume fraction ( 0.00 ≤ ϕ ≤ 0.09 ) . Evaluating the results involves examining isotherms, analyzing the distribution of flow velocities, and calculating the Nu avg , all while considering variations in fundamental physical parameters. As the Hartmann number ( Ha ) decreases and both the number of Rayleigh ( Ra ) and volume fraction ( ϕ ) rises, the Nu avg of the ferrofluid exhibits an increased. As a result of incorporating ferroparticles, the simulation identifies a maximum value of 35.86 for the Nu avg . It is worth noting that an enhance in Hartmann numbers is associated with an 80% decrease in the velocity stream function. On the other hand, an escalation in ferroparticle values leads to a 9.7% reduction, while elevated Rayleigh numbers result in a 60.8% surge in the stream function. Moreover, the irreversibility stemming from fluid friction, transfer of heat, and the magnetic field can be minimized for ferrofluid fluids ( n = 1 ), by employing the optimal parametric combination.

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

Error Analysis of the Classical Artificial Diffusion Weak Galerkin Finite Element Method for the steady state- convection diffusion-reaction Equation in 2-D

In this study, we modified the error in the weak Galerkin method when solving problems in which diffusion is the dominant convection( h) in two dimensions. This is done by adding the artificial diffusion term (-δΔw, where δ=h-ϵ). The finite element method for discrete functions using a weakly defined gradient operator is presented in this study. The concept of the weak discrete gradient is introduced, which plays an important role when using numerical methods to solve partial differential equations. The goal of this study is to enhance the accuracy and stability of the solutions by studying the ellipticity and stability properties of the method, which works to ensure that the numerical method retains the properties of the original equation while reducing the fluctuations occurring with the weak galerkin finite element method. Specific theories have been used to estimate the error in parameters -norm, and . Practical examples demonstrate how this method improves the handling of partial equations characterized by convection-dominated diffusion, enhancing its potential for advancing numerical simulations in engineering and physics.

Optimization and heat transfer analysis for MHD radiative natural convective flow of hybrid nanofluid within a partially heated cavity

The current study consists of optimization and heat transfer analysis for magnetohydrodynamic (MHD) radiative-natural convective flow of hybrid nanofluid (mixture of silver and copper nanoparticles with ethylene glycol (base fluid)) saturated inside the partially heated isosceles triangular cavity. Additionally, the current analysis investigates the distraction of available energy during thermal mixing and flow friction through local entropy generation. The conservation laws of total mass, momentum, and energy balance equations have been utilized to govern the physical flow problem. After eradicating the pressure terms in the governing partial differential equations (PDEs) via the penalty technique, the Galerkin finite element method (FEM) with the Newton-Raphson technique has been utilized to solve it. The outcomes established that the concentration of silver nanoparticles, as compared to copper nanoparticles, gives better energy transport. An increase in the concentration of nano-particles from ( ϕ 1 = ϕ 2 = 0 ) to ( ϕ 1 = ϕ 2 = 0.1 ) increased the intensity of streamlines from ψ max = 1.5 to ψ max = 4 and energy transmit rate increases by 57.4 % . The influence of thermal radiation decreased the fluid flow friction or local entropy via fluid flow.

A discontinuous Galerkin method for a coupled Stokes–Biot problem

The interaction between fluid and poroelastic structure is a complex problem that couples the Stokes equations with the Biot system. In this work, we rewrite the poroelastic equations using three fields (displacement, fluid pressure, and total pressure) to account for locking phenomenon in fluid–structure interaction and the oscillation of numerical solutions. Then, in order to solve the coupled system, we use the weighted discontinuous Galerkin finite element method for spatial discretization and propose a semi-discrete scheme. In addition, we propose a fully discrete scheme using the backward Euler’s method for temporal discretization. In the framework of the Galerkin approximation, the stability and errors of semi- and fully discrete schemes are analyzed using the theory of differential algebraic equations and weak compactness arguments. Through numerical experiments, we validate the theoretical rate of convergence, explain the behavior of the method in modeling the interaction between surface and groundwater hydrological systems, investigate the robustness of the method with respect to physical parameters, and simulate channel filtering. In particular, the numerical results are free of locking features, reducing non-physical numerical oscillations.

A defect correction weak Galerkin finite element method for the Kelvin–Voigt viscoelastic fluid flow model

In this paper, we present a defect correction weak Galerkin finite element method for solving the Kelvin–Voigt viscoelastic fluid flow model, which can guarantee that the discrete velocity is globally divergence-free. The second-order Crank–Nicolson difference scheme is considered for temporal discretization. In particular, combined with the defect correction method, the numerical oscillation effect is reduced at high Reynolds number. Our algorithm not only gives the stability and convergence of the numerical solutions of velocity and pressure but also is uniformly valid at the retardation time κ→0. Several numerical experiments are performed to verify the method has good behaviors.

Optimisation of MHD flow within trapezoidal cavity containing hybrid nanofluid by artificial neural network

PurposeThis study aims to numerically investigate heat transport in a trapezoidal cavity using hybrid nanoparticles (Ag-$Al_2O_3$). Unlike previous studies, this one covers magnetohydrodynamics, joule heating with viscous dissipation, heat absorption and generation. The left and right sides of the chasm are frigid. The upper wall heats, whereas the bottom wall remains adiabatic.Design/methodology/approachAfter reducing the system of dimensional equations to dimensionless equations, the authors use the Galerkin finite element method to solve them numerically. Geometric parameters affect heating efficiency; thus, the authors use flow metrics such as the Reynold number Re, magnetic parameter M, volume fraction coefficient, heat absorption and Eckert number Ec. The authors use the finite volume method to solve the governing equations after converting them to dimensionless form. The authors also try the artificial neural network method to predict the innovative cavity’s heat response in future scenarios. Transition state charts, regression analysis, MSE and error histograms accelerate, smooth and accurately converge solutions.FindingsAs the magnetic parameter and Eckert number increase, the enclosure emits more heat. As Reynold and volume fraction coefficients rise, the Nusselt number falls. It rose as magnetic, Eckert and heat absorption characteristics increased. The average Nusselt number rises with Reynolds and volume fraction coefficients. The magnetic, Eckert and heat absorption characteristics have inverse values.Originality/valueThis study numerically investigates heat transport in a trapezoidal cavity using hybrid nanoparticles (Ag-$Al_2O_3$). Unlike previous studies, this one covers MHD, joule heating with viscous dissipation, heat absorption and generation. The left and right sides of the chasm are frigid. The upper wall heats, whereas the bottom wall remains adiabatic.

An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O(N−(k−1/2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O((N−1ln⁡N)(k−1/2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

Analysis of two discontinuous Galerkin finite element methods for the total pressure formulation of linear poroelasticity model

In this paper, we develop two discontinuous Galerkin (DG) finite element methods to solve the linear poroelasticity in the total pressure formulation, where displacement, fluid pressure, and total pressure are unknowns. The fully-discrete standard DG and conforming DG methods are presented based on the discontinuous approximations in space and the implicit Euler discretization in time. Compared to the standard DG method with penalty terms, the conforming DG method removes all stabilizers and maintains conforming finite element formulation by utilizing weak operators defined over discontinuous functions. The two methods provide locally conservative solutions and achieve locking-free properties in poroelasticity. We also derive the well-posedness and optimal a priori error estimates, which show that our methods satisfy parameter-robustness with respect to the infinitely large Lamé constant and the null-constrained specific storage coefficient. Several numerical experiments are performed to verify these theoretical results, even in heterogeneous porous media.

Effect of Wavy Interface on Natural Convection in Square Cavity Partially Filled with Nanofluid and Porous Medium using Buongiorno Model

nvective heat transfer improvement from wavy surfaces presents a new solution in industrial engineering for composite materials, including porous medium, and nanofluids to address the wavy irregular surfaces in heat transfer devices such as a wavy solar collector, energy absorption and filtration, thermal insulation, and geothermal power plants. This technique enables the performance of engineering applications. The numerical study is performed to examine the effects of a wavy interface separating two layers in the enclosure on heat exchange rates. This paper investigates numerically the natural convection flow in a square cavity partially filled with nanofluid-porous layers separated by a wavy horizontal interface. The left and right walls of the cavity are maintained at constant hot and cold temperatures, whereas the other walls are adiabatic. The Buongiorno model is used to describe nanofluid motion, taking into account the brownian and thermophoresis effects in the cavity. The Galerkin finite element method was applied to solve the differential governing equations. The dynamic, thermal field and heat transfer have been analyzed for various parameters such as Rayleigh number (10^3 ≤ Ra ≤ 10^6), the amplitude of interface (0 ≤ A ≤ 0.1), and undulation number (0 ≤ n ≤ 9). The results reveal that the flow intensity induced by buoyancy forces is more significant in the nanofluid layer than in the porous layer, since the heat transfer is enhanced while the flow is not sensitive to variations in amplitude and number undulation, and accordingly, the decline of average Nusselt and Sherwood numbers is insignificant. The effects of controlled parameters on the structure of nanofluid flow, heat, and mass transfer rate are insignificant.

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

Thermal analysis of hybrid nanoliquid contains iron-oxide (Fe3O4) and copper (Cu) nanoparticles in an enclosure

The major aim of this work is to analyze the thermophysical properties of hybrid nanoliquid inside a square lid driven enclosure under the effect of applied magnetic field and mixed convection. The Hybrid nanofluid is considered a suspension of Iron-oxide (Fe3O4) and Copper (Cu) nanoparticles into water. The cavity walls have varying properties such as top adiabatic wall is moving continuously while the lower wall is maintained at a high constant temperature, in addition, the vertical walls are kept at a low fixed temperature. The magnetic field is applied to the right vertical wall in a horizontal direction. Additionally, the effects of buoyancy forces caused by temperature gradients and shear forces caused by continuous lid motion are considered. The mathematical modelling of problem with all assumptions yields the nonlinear partial differential system. First this system is transferred into non-dimensional form and then transformed system is solved by using Galerkin finite element method (GFEM) along with Gaussian elimination method. Results for isotherms, streamlines and average Nusselt number are obtained versus Richardson number (0.1 ≤ Ri ≤ 15), Grashof number (10 ≤ Gr ≤ 10000), Hartmann number (0 ≤ Ha ≤ 100), volume fractions of solid nanoparticles (0.005 ≤ ϕ1 ≤ 0.02) and (0.005 ≤ ϕ2 ≤ 0.02). It is observed that the Richardson number and Grashof number have significant impacts on both flow velocity and temperature. In addition, concentration of hybrid nanoparticles improves the heat transfer performance. The obtained results may be crucial for thermal management in the cooling of electronics, heat exchangers, solar thermal collectors, industrial operations, medical equipment, HVAC systems, food processing, and renewable energy systems.

Capturing Local Temperature Evolution During Additive Manufacturing Through Fourier Neural Operators

Abstract High-fidelity, data-driven models that can quickly simulate thermal behavior during additive manufacturing (AM) are crucial for improving the performance of AM technologies in multiple areas, such as part design, process planning, monitoring, and control. However, complexities of part geometries make it challenging for current models to maintain high accuracy across a wide range of geometries. In addition, many models report a low mean-square error (MSE) across the entire domain of a part. However, in each time-step, most areas of the domain do not experience significant changes in temperature, except for the regions near recent depositions. Therefore, the MSE-based fidelity measurement of the models may be overestimated. This article presents a data-driven model that uses the Fourier neural operator to capture the local temperature evolution during the AM process. Besides MSE, the model is also evaluated using the R2 metric, which places great weight on the regions where the temperature changes significantly than MSE. The model was trained and tested on numerical simulations based on the discontinuous Galerkin finite element method for the direct energy deposition AM process. The results shows that the model maintains 0.983−0.999 R2 over geometries not included in the training data, which is higher than convolutional neural networks and graph convolutional neural networks we implemented, the two widely used architectures in data-driven predictive modeling.

Operator-splitting finite element method for solving the radiative transfer equation

AbstractAn operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

SUPG-based stabilized finite element computations of convection-dominated 3D elliptic PDEs using shock-capturing

This study is interested in stabilized finite element computations of advection-dominated convection–diffusion-type partial differential equations. The governing equations are assumed to be defined on the 3D unit cube and stationary. Towards that end, we stabilize the standard (Bubnov–) Galerkin finite element method (GFEM), which typically suffers from yielding numerical approximations polluted with node-to-node nonphysical oscillations in convection dominance, by employing the streamline-upwind/Petrov–Galerkin (SUPG) formulation. In order to achieve better shock representations near sharp layers, the SUPG-stabilized formulation is also enhanced with a simple and residual-based shock-capturing mechanism, the so-called YZβ technique. Several test computations are performed to assess the efficiency and robustness of the proposed formulation. The test problems are examined under more challenging conditions than those previously studied in the literature, i.e., for flows substantially dominated by convection phenomena. The numerical results and comparisons demonstrate that the SUPG-YZβ combination significantly eliminates both globally spread and localized numerical instabilities. Beyond that, the proposed formulation accomplishes that by using only linear elements on relatively coarser meshes and without making use of any adaptive mesh strategies.

Discontinuous Galerkin finite element method for solving non-linear model of gradient elution chromatography

Discontinuous Galerkin finite element method for solving non-linear model of gradient elution chromatography

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.

Coupled modal characteristics of a bladed rotor bearing system under aerodynamic and rotor excitations: Modelling and numerical study

Bladed Rotor Bearing System is considered in the coupled vibration analysis among blades and the rotor system. The governing equations of motion are derived based on Lagrange’s equation, which accounts for the coupled vibrational modes. The model is discretised based on the Galerkin finite element method. A comparative analysis on the influence of uniform and pre-twisted blade configurations on the BRBS and on individual blades is carried out. Free vibration analysis for various blade geometry configurations is studied and results show that natural frequencies of the blades are reduced in the presence of pre-twist whereas the same is increased with double-taper. The Campbell diagram revealed the presence of frequency veering among the blades and rotor loci of natural frequencies for a specific blade configuration. The influence of aerodynamic and external excitations on the blades and the rotor, respectively, are considered in the forced response analysis, where the resulting effects of both the excitations on the individual blades and the BRBS for different blade configurations are studied.