Galerkin-based Finite Element Method Research Articles

MHD mixed convection of nanofluid flow Ag- Mgo/water in a channel contain a rotational cylinder

This investigation explores the mixed convection phenomenon of a hybrid nanofluid flowing within a Channel oriented horizontally featuring a triangular cavity attached to the lower channel wall. The cavity houses a rotating circular cylinder with a radius of R = 0.12. The fluid within the enclosure is a water-based nanofluid containing Ag-MgO nanoparticles. Two scenarios of combined natural and forced convection are examined: in Case 1, all channel walls, the left wall of the cavity, and the surface of the rotating cylinder are adiabatic, while the inclined wall of the cavity is isothermal. In Case 2, all channel walls, the ready wall of the cavity, and the surface of the rotating cylinder are adiabatic, except for the left wall of the cavity, which is isothermal. The dimensionless governing equations are solved under steady flow conditions for various physical parameters using a higher-order and stable Galerkin-based finite element method implemented through the software package FlexPDE. Key governing parameters such as Re number (falling within the amplitude of 10 to 100), Ri number (ranging from Ri = 0.1 to 30), and the location of cylinder rotation (Xo = 0.65 to 0.9) are simulated. The study presents results in terms of average Nusselt numbers and contour maps of streamlines and isotherms. It is observed that the average Nusselt number increases with higher Reynolds number, angular rotation speed, cylinder location, and Richardson number. Specifically, heating the inclined surface suggests that the optimal location for the rotating cylinder is Xo = 0.9, whereas if the left wall of the cavity is heated, the preferred location for the cylinder is Xo = 0.75. These findings are corroborated by comparing them with previous research, demonstrating significant agreement.

Enhancing thermal performance in a magnetized square cavity: Novel insights from mixed convection of Ag-MgO nanofluid around a rotating cylinder

This numerical investigation considers the mixed convection of a magnetically influenced hybrid nanofluid in a two-dimensional square cavity containing a centrally positioned rotating circle cylinder with a radius of R = 0.12. The nanofluid within the enclosure is water-based and incorporates Ag-MgO nanoparticles, the chemical and physical interactions between MgO nanoparticles and Ag molecules play an important function in determining the thermal conductivity and dynamic viscosity of MgO-Ag nanofluid. These findings exhibit that MgO-Ag nanofluid has the potential to be used as an advanced heat transfer fluid in cooling systems and heat exchangers. Two distinct cases are examined: firstly, with all walls of the square cavity and the rotating circle cylinder being adiabatic, except for the bottom side, and secondly, with the right side of the cavity being maintained at an isothermal condition. A finite element software package (FLEXPDE) is used in the present study. The governing partial equations, rendered dimensionless and encompassing various physical parameters, are solved utilising the Galerkin-based finite element method. Novelty is injected into the exploration through a meticulous parametric study, delving into the influence of key parameters, including the angular velocity spanning from -25 to 25, Reynolds number (10 ≤ Re ≤ 100), and Richardson number (0 ≤ Ri ≤ 35). The outcomes are vividly portrayed through visualisation of the flow structure via streamlines, the depiction of the heat transfer process through isotherms, and the computation of average Nusselt numbers. A discernible trend emerges, demonstrating that an increase in the Reynolds number, angular rotation speed, and Richardson number corresponds to an elevation in the average Nusselt number. Increasing Reynolds number (Re) from 10 to 50 intensifies streamlines, indicating fivefold higher flow resistance, with isotherms at Re = 50 showing 80 % concentration near the hot bottom wall compared to Re = 10. Counterclockwise rotation induces 90 % curvature near the bottom, clockwise rotation near the top, highlighting directional effects on flow resistance and streamlines. Remarkably, an intriguing observation surfaces as the angular velocity intensifies counterclockwise, revealing a 10 % augmentation in the average Nusselt values compared to a clockwise direction. Furthermore, the adoption of nanofluids as cooling agents, in lieu of pure water, is shown to enhance the cavity's thermal and overall performance, surpassing previous results. This underscores the novel insight into the advantageous utilisation of nanofluids in improving the efficacy of thermal systems.

An ANN-assisted efficient enriched finite element method via the selective enrichment of moment fitting

Enrichment techniques that employ nonconforming mesh are effective in modeling structures with discontinuities because numerical issues regarding mesh quality are avoided. However, the accurate integration of the bilinear and linear forms on the discretized domain, which is required in the standard Galerkin-based finite element method, is computationally expensive due to the complexity of the enriched basis function. In this paper, we present a fast and accurate alternative method of numerical integration using nonlinear regression enabled by a multi-perceptron feedforward neural network. The relationship between an implicitly represented geometry and the quadrature rule derived from the moment fitting method is predicted by the neural network; the neural network-based regression model circumvents complex computation and significantly reduces the overall online time by avoiding expensive function evaluations. Through the selected numerical examples, we demonstrate the efficiency and accuracy of the current method, as well as the flexibility of the trained network to be used in different contexts.

Finite Element Model to Simulate Two-Phase Fluid Flow in Naturally Fractured Oil Reservoirs: Part I.

Naturally fractured reservoirs host more than 20% of the world’s total oil and gas reserves. To produce from such reservoirs efficiently, a good understanding of the reservoir behavior at various conditions is essential. This allows us to predict the reservoir performance in advance and assess its economic feasibility. However, the production from such reservoirs is challenging due to (a) uncertainty associated with the fracture map, (b) complex physics phenomena of fluid and rock interaction, and (c) lack of comprehensive knowledge of the extent, orientation, and permeability sensitivity of the fracture network. This paper addresses the abovementioned challenges by presenting a three-dimenisonal (3-D) two-phase fluid flow model in a poroelastic environment. The model is based on a hybrid methodology by combining single continuum and discrete fracture network approaches. Also, the capillary pressure effect, saturation, and relative permeability variations are considered. A mathematical formulation for three-dimensional, two immiscible fluid flows including rock deformation for the fracture network and the rock matrix is presented. A standard Galerkin-based finite element method is applied to discretize the poroelastic governing equations in space and time. The characteristic Galerkin discretization method is used to stabilize the solution of the convection equation in a finite element approach. The 3-D model is validated against IMEX commercial software and finite element package to test its robustness. The results show that the developed model has the ability to predict the two-phase flow behavior precisely, which can be used to assess the production performance of naturally fractured reservoirs. Numerical results of fluid flow profiles for single and multiple fractures with different orientations that match experimental oil drainage tests and field case studies are presented that proves the reliability of the developed multiphase flow model. Results of simulated well production data show an excellent match with the field production data.

Kalman filter based estimation algorithm for the characterization of the spatiotemporal hemodynamic response in the brain

The characterization of the spatiotemporal hemodynamic response (stHR) in the brain is important for understanding the interaction between neighboring brain voxels and regions. In this paper, we design an identification algorithm for the characterization of the cerebral stHR which is modeled by a system of coupled hyperbolic partial differential equation (PDE) and infinite-dimensional ordinary differential equation (ODE). The proposed algorithm provides estimates of the hemodynamic variables (cerebral blood flow and mass density contributed by blood) and physiological parameters using non-invasive Blood Oxygenation Level Dependent (BOLD) data measured with functional Magnetic Resonance Imaging (fMRI) modality. The proposed solution concept follows three main steps: (i) discretization of the stHR model using Galerkin-based finite element method; (ii) estimation of the output derivative using high-order sliding mode differentiator; and (iii) estimation of the state, input, and parameters from sampled-in-space measurements using the reduced-order approximation model and a constrained extended Kalman filter with unknown input algorithm. In addition, sufficient conditions that depend on the chosen discretization scheme, and which guarantee the structural identifiability of the input and parameters, and also the observability of the system are provided. The performance of the proposed algorithm is assessed using both synthetic and real data. The set of the used real data represents the 1-D BOLD signal collected from the visual cortex and acquired in 3 Tesla fMRI scanner.

Mixed convective magnetonanofluid flow over a backward facing step and entropy generation using extended Darcy–Brinkman–Forchheimer model

This article focuses on a numerical study for the effect of porosity and internal heat generation/absorption parameters on the mixed convective flow of nanofluid over a backward facing step along with the entropy generation. The channel downstream bottom wall is isothermally heated while the remaining walls of the channel are thermally insulated. The dimensionless governing equations are discretized utilizing the Galerkin-based finite element method. The discrete systems of nonlinear algebraic equations are computed using the Newton method and the corresponding linearized systems are treated using the monolithic geometric multigrid solver. Effect of different physical parameters in the specified ranges such as \((10 \le Re \le 200)\), \((0.01 \le Ri \le 20)\), \((0\le Ha \le 100)\), \((-10 \le q \le 10)\), \((10^{-6} \le Da \le 10^{-3})\), \((0.2 \le \epsilon \le 0.8)\) and \((0^\circ \le \gamma \le 180^\circ )\) on the fluid flow is analyzed with the help of the streamlines, isotherm patterns, and various graphs. It is noticed that the average heat transfer increases with the porosity parameter and reduces with the internal heat generation parameter. Furthermore, the entropy generation produced by the heat transfer and fluid friction is found to amplify with the porosity parameter, whereas a decline is recorded in it due to the magnetic field.

Water functionalized CuO nanoparticles filled in a partially heated trapezoidal cavity with inner heated obstacle: FEM approach

This frame work is established to investigate the thermal management of free convection enclosed in trapezoidal cavity filled with the water based copper oxide (CuO) nanofluid. As nanoparticles volume fraction play a significant role to handle the thermal conductivity of any working fluid, so we have addressed the complex nature real world model that widely used at the industrial level and many other mechanisms. An identical trapezoidal shape cavity is placed inside the big trapezoidal cavity that have three various constraints at the surface (cold, insulated and heated). Since bottom wall of the outer cavity is partially heated so various heated portion tests are applied to analyze the influence of heat transfer within the entire cavity. Aspect ratio that depends upon the size of the inner cavity is also determine. Complete and compatible mathematical model is constructed in the form of nonlinear coupled partial differential equation. These set of equations are characterized under the law of conservation of mass, momentum and energy equation along with the restricted domain of the cavity. Koo and Kleinstreuer-Li (KKL) model is used for effective thermal conductivity and viscosity of the nanofluid. A Galerkin based Finite Element method (FEM) is implemented to attain the suitable results in term of stream function and isotherms within the restricted domain of the cavity. Results are also obtained for velocity and temperature of the nanofluid at vertically mean position of the cavity. The simulations are performed for nanoparticles volume fraction 0⩽ϕ⩽0.2 heated portion length 0⩽LT⩽1 aspect ratio 0.5⩽AR⩽3.0, Rayleigh number 104⩽Ra⩽105.7, and three heated conditions (cold, adiabatic and hot) for inner trapezium. It is found that flow and thermal field are getting stronger due to increase in Rayleigh number. However, fluid velocity is decreasing with increasing nanoparticles volume fraction ϕ as the fluid is getting dens. Heat transfer rate is decreasing with the increase in ϕ and LT due to dominant convection.

Numerical Investigation of Forced Convective Heat Transfer Characteristics of a Porous Channel Filled With -Water Nanofluid in the Presence of Heaters and Coolers

ABSTRACTIn this analysis, forced convective heat transfer characteristics of -water nanofluid through a porous channel with several combinations of heaters and coolers is investigated numerically. The two-dimensional equations governing nanofluid flow and heat transfer through porous media are discretized using in-house code with Streamline Upwind Petrov Galerkin-based Finite Element Method. Darcy–Brinkman–Forchheimer’s generalized porous media model is used in this study. The average Nusselt number of basefluid without porous media, nanofluid with and without porous media cases are compared for different Peclet numbers and the effect of Peclet number on stream lines and isotherms are studied for nanofluid with and without porous media cases. In addition to this the effect of Darcy number, porosity, and nanoparticle volume fraction on the performance of average Nusselt number is investigated. From these results, it is observed that the average Nusselt number increases with decrease in Darcy number. From this analysis, it can be concluded that addition of porous media results in enhancement of heat transfer and can be used as a potential technique for electronic cooling applications.

Impact analysis of strain-rate-dependent composite plates with SMA wires in thermal environments: Proposing refined coupled thermoelasticity, constitutive, and contact models

Effects of the strain-rate dependency of the elastic moduli of the composite and shape memory alloy (SMA) materials on responses of rectangular composite plates with embedded SMA wires subjected to impacts with spherical indenters is considered in the present research, for the first time. The plates are assumed to be placed in thermal environments; so that in contrast to all the available researches, the shape memory and ferroelasticity effects are also taken into account in addition to the superelasticity. The governing equations are derived based on a proposed enhanced phase version free energy function and refined constitutive equations with coupled expressions and solved by a virtual-work-type Galerkin-based finite element method. The resulting highly non-linear and piecewise-defined governing equations are solved through using biquadratic rectangular elements, an iterative solution scheme, and a special phase transformation tracing algorithm. Hertz contact law is also revised accordingly based on a proper micromechanical model. Results are obtained for a range of ambient temperatures that are either lower or higher than the austenite start temperature. Results show that complex combinations of the strain-rate dependency of the material properties, impact-induced temperature rise, and ambient temperature may significantly affect the phase transformation phenomenon and the non-linear impact responses.

The modelling of instabilities in liquid-liquid miscible displacement processes by the Galerkin-based finite element method

The Galerkin-based finite element method has been used to successfully simulate the propagation of instabilities in the liquid-liquid miscible displacement process. The convective-dispersion equation is sufficient to model the liquid-liquid miscible displacement process under unstable conditions. The investigations reveal: i. An unconditional instability which will not disappear with time, ii. The rate of growth of the instabilities, tended to increased proportionally to a number to the M power, where M is the mobility ratio, iii. The propagation of instabilities can be achieved through the actions of viscous, gravity and heterogeneous forces, and iv. Heterogeneity in permeability introduces a macroscopic dispersion effect that attempts to stabilize the instabilities. Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 173-176

Numerical Simulation of Three-Phase Flow in Deforming Fractured Reservoirs

The mathematical formulation of a three-phase, three-dimensional fluid flow and rock deformation in fractured reservoirs is presented in this paper. The present formulation accounts for the significant influence of coupling between the fluid flow and solid deformation, an aspect usually ignored in the reservoir simulation literature. A Galerkin-based finite element method is applied to discretise the governing equations in space and a finite difference scheme is used to march the solution in time. The final set of equations, which contain the additional cross coupling terms as compared to similar existing models, are highly nonlinear and the elements of the coefficient matrices are updated implicitly during each iteration in terms of the independent variables. A field scale example is employed as an alpha case to test the validity and robustness of the currently formulation and numerical scheme. The results illustrate a significantly different behavior for the case of a reservoir where the impact of coupling is also considered.

Three-dimensional finite element simulation of three-phase flow in a deforming fissured reservoir

The development of a capacity to predict the exploitation of structurally complicated and fractured oil reservoirs is essential for the rational use of investment capital. A poor understanding of how the reservoir behaves during production may lead to inept, costly and inefficient development schemes. The mathematical formulation of a three-phase, three-dimensional fluid flow and rock deformation in fractured reservoirs is hence presented. The present formulation, consisting of both the equilibrium and multiphase mass conservation equations, accounts for the significant influence of coupling between the fluid flow and solid deformation, an aspect usually ignored in the reservoir simulation literature. A Galerkin-based finite element method is applied to discretise the governing equations in space and a finite difference scheme is used to march the solution in time. The final set of equations, which contain the additional cross coupling terms as compared to similar existing models, are highly non-linear and the elements of the coefficient matrices are updated implicitly during each iteration in terms of the independent variables. A field scale example is employed as an alpha case to test the validity and robustness of the currently formulation and numerical scheme. The results illustrate a significantly different behaviour for the case of a reservoir where the impact of coupling is also considered.

A novel finite element double porosity model for multiphase flow through deformable fractured porous media

Based on the theory of double-porosity, a novel mathematical model for multiphase fluid flow in a deforming fractured reservoir is developed. The present formulation, consisting of both the equilibrium and continuity equations, accounts for the significant influence of coupling between fluid flow and solid deformation, usually ignored in the reservoir simulation literature. A Galerkin-based finite element method is applied to discretize the governing equations both in the space and time domain. Throughout the derived set of equations the solid displacements as well as the fluid pressure values are considered as the primary unknowns and may be used to determine other reservoir parameters such as stresses, saturations, etc. The final set of equations represents a highly non-linear system as the elements of the coefficient matrices are updated during each iteration in terms of the independent variables. The model is employed to solve a field scale example where the results are compared to those of ten other uncoupled models. The results illustrate a significantly different behaviour for the case of a reservoir where the impact of coupling is also considered. © 1997 by John Wiley & Sons, Ltd.

A Taylor–Galerkin-based finite element method for turbulent flows

A finite element method is applied to solve the two-dimensional turbulent channel flows. Based on the fractional step techniques, the momentum and the k-ϵ equations are split into convection and diffusion equations. The convection equations are solved by the second-order Taylor-Galerkin finite element method, which can overcome the spurious oscillations with minimal artificial diffusion, and the diffusion equations are solved by the fully explicit Galerkin method. Since the same order interpolation is used for the velocity, pressure and turbulent quantities, the present method is computationally efficient. The sudden expansion flow and the obstructed turbulent channel flow are studied. The results are in good agreement with experimental observations.