Variational Finite Element Method Research Articles

Porosity-dependent stability analysis of bio-inspired cellular nanocomposite shells

This paper presents an innovative numerical model to investigate buckling behaviour of bio-inspired continuously graded porous (CGP) nanocomposite cylindrical shells. It is postulated that the shell subjected to combined lateral pressure and axial compressive load is constructed from metal foams with closed-cell structures that possess graded internal pores, which exhibit three types of continuously graded porosity profiles based on a power-law distribution. A scaling relation for the effective Young’s modulus of the cellular structure determined by a variational finite element method (FEM) is used. The effective constitutive law of an elastic isotropic metal matrix containing distributed elastic carbon-nanotubes (CNTs) is estimated in consideration of the impact of CNTs agglomeration using a continuum model based on the Eshelby–Mori–Tanaka (EMT) approach. In contrast to conventional approaches, the study employs Euler–Bernoulli beams to model stiffeners within the CGP shells. This choice allows for a more realistic representation of stiffener effects, as opposed to the prevalent approach of uniform smearing across the shell’s surface. The equilibrium equations of the CGP shell, based on the Reddy higher-order shell theory (RHST), is obtained through the application of the Euler equation. Subsequently, the equations for stability are obtained through the utilization of the variational method. This study emphasizes the effects of geometrical parameters, porosity variability, and distribution of CNTs on the buckling performance of the CGP shells. The intricate interplay between CNTs and porosity distributions critically influences the stability behaviour of CGP shells. CNTs arrangement remarkably impacts buckling behaviour at higher length-to-mean radius ratios, while symmetric porosity near the mid-surface significantly enhances stiffness. These findings provide valuable insights for designing closed-cell cellular stiffened shells with optimal porosity to enhance stability.

Моделювання деформування соляних структур внаслідок гравітаційної плавучості

We consider modeling and geophysical interpretation of the obtained results in the deforming process of the salt structures due to gravitational buoyancy (halokinesis). For solving this geophysical problem, we use variation finite element method of elastic problem resolving with calculation of heterogeneous rocks distribution into considering salt structures. We have defined that salt structures deforming amplitudes mainly depend on linear sizes (length and thickness) of the bottom parts of these structures. Decreasing of these parameters lead to noticeable drop of the press-strain state near the whole region of the salt structure (diapirs). Another hand forms and linear sizes of the top parts of the salt stocks influence only on the deforming of the local regions near these structure elements and don’t essentially influence on the whole region deforming around the stock. Quantity characteristics of linear sizes of the salt diapirs structural elements define the whole picture of the stress-strain state around these objects.

Heat Transfer Analysis using Finite Element Method under Convective Boundary Condition

The model in this study scrutinizes the effect of convective boundary conditions on the flow of a nanofluid across permeable flat plate. The fundamental equations get altered into a nonlinear form through choosing appropriate similarity transformations. In the process, they are solved mathematically by substantiated FEM code through use of variational finite element method. The outcomes clearly show the characteristics of relevant parameters such as temperature and velocity profiles. When the numerical analysis is evaluated in context of formerly published information, the reliability of the numerical code is conformed. Its found that there is a surge in thermal conductivity when proportion of nanoparticles rises in the fluid. Permeability of plate has a significant influence on the heat transfer and skin friction. The investigation supports the possibility of extending the work to flows of non-Newtonian fluid, three dimensional and for consideration of pressure gradients on arbitrary surfaces. The results practically aid the design of heat transfer systems for futuristic technology involving heat enhancement.

МОДЕЛЮВАННЯ ЗСУВНИХ ДЕФОРМАЦІЙ ПІД ДІЄЮ СИЛИ ТЯЖІННЯ

Nowadays, there are actual problems associated with destructive slope processes under the influence of gravity load. Indeed, gravitational slope processes, together with other erosion and tectonic processes, contribute a lot to the formation of the modern relief and at the same time very often complicate the rational using of the corresponding territory. Landslide processes can be distinguished among the most dangerous gravitational slope processes. These processes are characterized by widespread distribution and significant material losses and human casualties to which they lead. Due to its social importance and practical engineering significance, the problems of studying gravitational shear processes have a long history. Therefore, many works are devoted to these problems, but from another hand, the cases of strict mathematical and mechanical description and determination of certain quantitative mechanisms and criteria for the development of sliding gravitational processes have been considered in a rather limited way. In presented work on the base of variational finite-element method, we calculate the rotational deformation and failure criteria of a wide class of three-dimensional heterogeneous anticlinal geostructures under the gravity load conditions. The simulation results show that the shear deformation of anticlinal geostructures under the influence of gravity depends on the shape, size of the structure, and mechanical properties of the rocks that make up these geostructures. We have established that more compact geostructures are subjected to the smallest deformation. In the solid geostructures that retain elastic properties, the deformations are inversely proportional to the degree of rock stiffness, and a decreasing in the curvature radius of the geostructure leads to an inversely proportional increase in the deformation of the corresponding geostructure. We have shown that in order to maintain resistance to shear soil gravity failure, anticlinal geostructures cannot be completely composed of rocks softer than semi-solid dispersed soils. It was established that the resistance to shear gravity soil failure of heterogeneous anticlinal geostructures is mainly determined by the rigidity of the internal bearing rocks, while the influence of the stiffness of the external soft rocks is relatively insignificant and has a non-linear character.

Impact of gyrotactic microorganisms in nanofluid via porous media along an inclined stretching plate: Finite element analysis

In the current work, we examine the two-dimensional steady, laminar flow of boundary layers for a water-based nanofluid comprising gyrotactic microorganisms on a stretched surface that is inclined and embedded in a porous medium. Appropriate similarity transformations are used to transform the governing partial differential equations into nonlinear coupled ordinary differential equations with boundary conditions, and then, solved using extensively validated, highly efficient, variational finite element method. For dimensionless velocity, temperature, nanoparticle concentration, microbe conservation, local Nusselt number, local Sherwood number, Skin friction, and microbe’s density number with regard to modification in controlling parameters, the results are presented through tables and graphs and these all are seen to be slightly impacted by variations in the inclination angle, according to numerical results. The Prandtl number is treated as a variable to obtain realistic results. It has been determined that the provided numerical method perfectly aligns with the said comparable methods.

A mathematical model to study thermoregulation and heat-transfer processes in hypothermic neonates under variable physiological parameters

All neonates, whether term or preterm, are prone to hypothermia, which is a major health issue causing many health problems to infants and sometimes even death. Thus, such subjects are imperative to study to help researchers and biologists in neonatology, for developing certain methods, procedures and devices to prevent these abnormal temperature fluctuations to save the neonates from this health threat. To this purpose, a multi-node mathematical model is developed, to provide detailed insights and its applications to study the temperature profiles, thermoregulatory and heat-transfer mechanisms in hypothermic neonates. The model is constructed using the radial form of heat equation along with appropriate boundary and initial conditions. The model solution is obtained with the aid of the variational finite element method followed by the fundamental matrix method. The model outcomes obtained show the temperature fluctuations and tissue responses in hypothermic neonates. Finally, the model outcomes are compared with the published/experimental work to prove the feasibility and validity of the proposed work. Moreover, this work generalizes the several previously published works in the related field.

Heat distribution and the condition of hypothermia in the multi-layered human head: A mathematical model

The conduction, perfusion and metabolic heat generation based partial differential equation has been used to study the heat transfer in human head. The main objective of this study is to predict the temperature distribution at the multi-layered human head that results in hypothermic condition. The temperature profiles have been estimated at the interface points of brain, skull and scalp with respect to various parameters including atmospheric temperature, arterial temperature and metabolic heat generation. The variational finite element method and analytical method based on Laplace transform has been employed to establish the solution of the formulated model, and the resulting outcomes are illustrated graphically. Under cold exposure, the blood capillaries around scalp exchange core heat with the external cold environment and experience lowering in the tissue temperature of the blood in the scalp. It is reflected in the graphical view of the model that the prolonged exposure to cold transmits its effect into the deep brain capillaries, wherein the temperature gradually lowers down below the normal body temperature that results hypothermia and hence abnormal body homoeostasis.

Hellinger–Reissner variational principle for a class of specified stress problems

Abstract Aiming at the problem of the specified stress condition in a partial region of the structure, the Hellinger–Reissner (H–R) variational principle is studied to provide a theoretical basis for the finite element numerical analysis. By introducing the unknown non-elastic strain as an additional unknown quantity to fulfill the specified stress condition, the elastic mechanics governing equations for the specified stress problem are given. Both stress and unknown non-elastic strain are taken as independent variables to establish the complementary energy principle and virtual work equation which are equivalent to the elastic mechanical control equation of the specified stress problem. Based on the conventional H–R variational principle, using displacement, stress and unknown non-elastic strain as independent variables, a H–R variational functional that satisfies the specified stress conditions is established by using Lagrange multiplier method. Also the variational functional with displacement, elastic strain and unknown non-elastic strain as independent variables is deduced by transforming the stress into the elastic strain. The corresponding finite formulae are derived based on an intra-element stress hybridization method. The H–R variational principle for the specified stress problem takes non-elastic strain as an independent variable, so that the stress explicitly appears in the equilibrium equation of the element and structure, which expands the application range and capabilities of the existing variational principle and finite element method. The correctness and accuracy of the theory and algorithm are verified by numerical examples.

Impact of convective heat transfer and buoyancy on micropolar fluid flow through a porous shrinking sheet: An FEM approach

This paper represents steady two-dimensional boundary layer flow of micropolar fluid flow with impact of convective heat transfer and buoyancy force investigated numerically. The shrinking velocity has been expected to fluctuate linearly with the existence of a fixed point on the sheet. With the assistance of similarity transformations, the governing partial differential equations are transformed into a set of nonlinear ordinary differential equations; these nonlinear ODEs are solved numerically by using the variational finite element method. The current numerical results are obtained from the variational finite element method and compared with the previously published literature work, with which it exists in good agreement. The impact of the flow monitoring parameters on velocity, microrotation and temperature profiles is examined graphically and discussed. The skin friction coefficient and Nusselt numbers are impacts from adjusting various values of the physical parameters and relevant features which are studied.

Mathematical modelling of drug-diffusion from multi-layered capsules/tablets and other drug delivery devices

In this paper, two mathematical models have been formulated by extending the basic reaction–diffusion model, along with suitable initial and boundary conditions to study the drug delivery and its diffusion in biological tissues from multi-layered capsules/tablets and other drug delivery devices (DDDs), respectively. These devices are either taken orally or through other drug-administration routes. The formulated models are solved using the variational finite element method followed by the fundamental matrix method, to study the drug delivery and its diffusion more efficiently. The main aim of this work is to provide an effective model, using optimal mathematical techniques to help researchers and biologists in medicine in decreasing the endeavours and expenses in designing DDDs. The outcomes obtained are compared with the experimental data to demonstrate the validity and the feasibility of the proposed work.

Influence of magnetic field and radiation on heat and mass transfer flow of nano fluid over an inclined vertical plate embedded in porous medium

We have analyzed MHDboundary layer flow, heat and mass transfer analysis of nanofluid over an inclined vertical plate saturated by porous medium with thermal radiation, and heat generation/absorption. Suitable similarity variables are introduced to convert non-linear partial differential equations into ordinary differential equations and these equations together with associated boundary conditions are solved numerically by using versatile, extensively validated, variational Finite element method. The impact of various pertinent parameters on hydrodynamic, thermal and concentration boundary layers are examined in detail and the results are shown graphically. Furthermore, the impact of these parameters on local skin friction coefficient, rate of heat transfer andrate of mass transferis also investigated. The results are compared with the works published previously and found to be excellent agreement.

Non-Darcian Effects on Nanoliquid Flow Past a Stretching Sheet with Temperature Jump Condition and Thermal Radiation

This article explores the influence of non-Darcy porous medium on nanoliquid flow past a stretching plane with thermal radiation and thermal slip on surface. The pedesis and thermophoresis effects have been included into the nanofluid model. The equations governing the momentum and energy are initially cast into dimensionless form using the suitable non-dimensional variables. The resultant system of equations is then solved employing variational Finite Element Method (FEM). The impact of pertinent parameters such as thermal radiation, temperature jump, non-Darcy on the flow, temperature and local skin friction coefficient, local Nusselt number are investigated and presented graphically. The Forchheimer number is found crucial to increase temperature and lower the local heat transfer rate in the boundary layer.

A Comparative Analysis of Neutron Transport Calculations Based on Variational Formulation and Finite Element Approaches

The application of continuous and discontinuous approaches of the finite element method (FEM) to the neutron transport equation (NTE) has been investigated. A comparative algorithm for analyzing the capability of various types of numerical solutions to the NTE based on variational formulation and discontinuous finite element method (DFEM) has been developed. The developed module is coupled to the program discontinuous finite element method for neutron (DISFENT). Each variational principle (VP) is applied to an example with drastic changes in the distribution of neutron flux density, and the obtained results of the continuous and discontinuous finite element (DFE) have been compared. The comparison between the level of accuracy of each approach using new module of DISFENT program has been performed based on the fine mesh solutions of the multi-PN (MPN) approximation. The obtained results of conjoint principles (CPs) have been demonstrated to be very accurate in comparison to other VPs. The reduction in the number of required meshes for solving the problem is considered as the main advantage of this principle. Finally, the spatial additivity to the context of the spherical harmonics has been implemented to the CP, to avoid from computational error accumulation.

Finite element analysis of micropolar nanofluid flow through an inclined microchannel with thermal radiation

PurposeIn recent years, microfluidics has turned into a very important region of research because of its wide range of applications such as microheat exchanger, micromixers fuel cells, cooling systems for microelectronic devices, micropumps and microturbines. Therefore, in this paper, micropolar nanofluid flow through an inclined microchannel is numerically investigated in the presence of convective boundary conditions. Heat transport of fluid includes radiative heat, viscous and Joule heating phenomena.Design/methodology/approachGoverning equations are nondimensionalized by using suitable dimensionless variables. The relevant dimensionless ordinary differential systems are solved by using variational finite element method. Detailed computations are done for velocity, microrotation and temperature functions. The influence of various parameters on entropy generation and the Bejan number is displayed and discussed.FindingsIt is established that the entropy generation rate increased with both Grashof number and Eckert number, while it decreased with nanoparticle volume fraction and material parameter. Temperature is decreased by increasing the volume fraction of Ag nanoparticle dispersed in water.Originality/valueAccording to the literature survey and the best of the author’s knowledge, no similar studies have been executed on micropolar nanofluid flow through an inclined microchannel with effect of viscous dissipation, Joule heating and thermal radiation.

Variational finite element approach to study heat transfer in the biological tissues of premature infants.

The body temperature of newborn preterm infants depends on the heat transfer between the infant and the external environment. Factors that influence the heat exchange include the temperature and humidity of the air and the temperature of surfaces in contact with and around the infant. Neonatal thermoregulation has a different pattern as they have an immature thermoregulatory system. For this purpose, mathematical models can provide detailed insights for the heat transfer processes and its applications for clinical purposes. A new multi-compartment mathematical model of the neonatal thermoregulatory system is presented. The formulation of the model is based on the Pennes' bio-heat equation with suitable boundary and initial conditions. The variational finite element method has been employed to determine heat transfer and exchange in the biological tissues of premature infants. The results obtained in this paper have shown that premature infants are unable to maintain a constant core temperature and resemble the empirically obtained results, proving the validity and feasibility of our model. AMS (2010): SUBJECT CLASSIFICATION: 92BXX, 92CXX, 92C35, 92C50, 46N60.

Mathematical estimation of fluid concentration in human skin during water immersion

IntroductionThe concentration of fluid and its analysis in human skin is innately a challenge due to its continuous movement and involvement in maximum life processes. The concentration of the fluid gets affected by the diffusion of fluids through the skin, which acts as the main barrier between the human body and the external environment. Therefore, it becomes imperative to study the process and impact of the diffusion of fluids through the skin. The problem becomes more interesting when the human body is immersed in water. ObjectivesThe present paper studies the change in the fluid distribution of human skin during its immersion in water of different temperatures. The application part of the paper visualizes various impaired vascular function and muscle soreness by water immersion during the physiotherapy treatment. MethodsA mathematical model based on the two-dimensional diffusion equation, along with appropriate boundary conditions, has been formulated. The maximum of the relevant parameters, such as fluid regulation, transfer coefficient, evaporation rate, etc., influencing the fluid distribution, have been incorporated. The model has been solved by variational finite element method, and numerical results have been obtained by the Crank-Nicholson scheme. ResultsThe increase in fluid concentration due to treatment with cold and acute hot water immersion has been noted, and the role of water immersion in enhancing the recovery in exercise-induced muscular damage has been analyzed. ConclusionsThe paper addressed the issue of rate of water diffusion through human skin, which otherwise couldn't be drawn from the analogy of gas diffusion through the membrane due to the variation in permeabilities of the two processes. The paper has applications in water immersion therapies and other activities like monitoring swimming induced pulmonary edema, etc.

Finite Element Analysis of Variable Viscosity Impact on MHD Flow and Heat Transfer of Nanofluid Using the Cattaneo–Christov Model

In this mathematical study, magnetohydrodynamic, time-independent nanofluid flow over a stretching sheet by using the Cattaneo–Christov heat flux model is inspected. The impact of the thermal, solutal boundary and gravitational body forces with the effect of double stratification on the mass flow and heat transfer phenomena is also observed. The temperature-dependent viscosity impact on heat transfer through a moving sheet with capricious heat generation in nanofluids have studied, and the viscosity of the fluid is presumed to deviate as the inverse function of temperature. With the appropriate transformations, the system of partial differential equations is transformed into a system of nonlinear ordinary differential equations. By applying the variational finite element method, the transformed system of equations is solved. The properties of the several parameters for buoyancy, velocity, temperature, stratification, and Brownian motion parameters have examined. The enhancement in the concentration and thermal boundary layer thickness of the nanofluid sheet due to the increment in the viscosity parameter, also increased the temperature and concentration of nanoparticles. Moreover, the fluid temperature declined with the increasing values of thermal relaxation parameter. This displays that the Cattaneo–Christov heat flux model provides a better assessment of temperature distribution. Moreover, confirmation of the code and precision of the numerical method has inveterate with the valuation of the presented results with previous studies.

The Impact of Nanoparticles Due to Applied Magnetic Dipole in Micropolar Fluid Flow Using the Finite Element Method

The present work examines the effect of different magnetic nanoparticles and the heat transfer phenomena over the stretching sheet with thermal stratification and slips effect. The mixture of water (H 2 O) and ethylene glycol (C 2 H 6 O 2 ) is used as base fluid whereas the paramagnetic, diamagnetic, and ferromagnetic ferrites are taken as nanoparticles. In the presence of ferrite nanoparticles, the magnetic dipole has a significant effect in controlling the rate of heat transfer and the thermal boundary layers. By using suitable similarity transformations, the system of partial differential equations is transformed into nonlinear ordinary differential equations. The numerical solution of resulting equations is found out by using the variational finite element method. The effect of numerous emerging parameters on velocity, temperature, and micro-rotation velocity are represented graphically and analyzed numerically. It has been noticed that comparatively the diamagnetic ferrites have gained maximum thermal conductivity relative to the other nanoparticles. It was also observed that the thermal conduction of nanoparticles increases with the variation of volume fraction. Moreover, with increasing values of thermal stratification the thermal boundary layer thickness decreases and the heat transfer rate increases at the surface. Furthermore, the validation of code and the accuracy of the numerical technique has been confirmed by the assessment of current results with earlier studies.

Analysis of Magnetic Properties of Nano-Particles Due to a Magnetic Dipole in Micropolar Fluid Flow over a Stretching Sheet

This article explores the impact of a magnetic dipole on the heat transfer phenomena of different nano-particles Fe (ferromagnetic) and Fe3O4 (Ferrimagnetic) dispersed in a base fluid ( 60 % water + 40 % ethylene glycol) on micro-polar fluid flow over a stretching sheet. A magnetic dipole in the presence of the ferrities of nano-particles plays an important role in controlling the thermal and momentum boundary layers. The use of magnetic nano-particles is to control the flow and heat transfer process through an external magnetic field. The governing system of partial differential equations is transformed into a system of coupled nonlinear ordinary differential equations by using appropriate similarity variables, and the transformed equations are then solved numerically by using a variational finite element method. The impact of different physical parameters on the velocity, the temperature, the Nusselt number, and the skin friction coefficient is shown. The velocity profile decreases in the order Fe (ferromagnetic fluid) and Fe3O4 (ferrimagnetic fluid). Furthermore, it was observed that the Nusselt number is decreasing with the increasing values of boundary parameter ( δ ) , while there is controversy with respect to the increasing values of radiation parameter ( N ) . Additionally, it was observed that the ferromagnetic case gained maximum thermal conductivity, as compared to ferrimagnetic case. In the end, the convergence of the finite element solution was observed; the calculations were found by reducing the mesh size.

A Finite Element Simulation of the Active and Passive Controls of the MHD Effect on an Axisymmetric Nanofluid Flow with Thermo-Diffusion over a Radially Stretched Sheet

The present study investigated the steady magnetohydrodynamics of the axisymmetric flow of a incompressible, viscous, electricity-conducting nanofluid with convective boundary conditions and thermo-diffusion over a radially stretched surface. The nanoparticles’ volume fraction was passively controlled on the boundary, rather than actively controlled. The governing non-linear partial differential equations were transformed into a system of nonlinear, ordinary differential equations with the aid of similarity transformations which were solved numerically, using the very efficient variational finite element method. The coefficient of skin friction and rate of heat transfer, and an exact solution of fluid flow velocity, were contrasted with the numerical solution gotten by FEM. Excellent agreement between the numerical and exact solutions was observed. The influences of various physical parameters on the velocity, temperature, and solutal and nanoparticle concentration profiles are discussed by the aid of graphs and tables. Additionally, authentication of the convergence of the numerical consequences acquired by the finite element method and the computations was acquired by decreasing the mesh level. This exploration is significant for the higher temperature of nanomaterial privileging technology.