This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

This work assumes a basic familiarity with the finite element method. For those readers desiring a better understanding of the method, there are a number of excellent books varying from an introductory treatment to advanced topics listed in the bibliography [1] [2] [3] [4] [5]. In this chapter, we will review the fundamentals of the finite element method and introduce the notation which will be used throughout the book. For the examples shown in this book, either first or second order elements were used. For the worked examples first order triangles are used. The use of first order triangular elements simplifies the arguments and in no way reduces generality. In all cases the procedures given in this book can be used with high order and/or curved elements.

Introduction to Finite Elements

In this work, a new model of magneto-thermoelasticity theory has been constructed in the context of a new consideration of heat conduction with memory-dependent derivative. One-dimensional application for a perfect electrically conducting half-space of elastic material, which is thermally shocked, in the presence of a constant magnetic field has been solved by using Laplace transform technique. According to the numerical results and its graphs, conclusion about the new theory of magneto-thermoelasticity has been constructed and compared with dynamic classical coupled theory.

A novel magneto-thermoelasticity theory with memory-dependent derivative

The problem of two-temperature generalized thermoelastic thin slim strip is investigated in the context of Green and Lindsay theory. As an application of the problem, a particular type of moving he...

The effects of relaxation times and a moving heat source on a two-temperature generalized thermoelastic thin slim strip

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of magneto-thermoelasticity to investigate the transient phenomena for a fibre-reinforced thick plate having a heat source in the context of three-phase-lag model of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. The upper surface of the plate is free of traction having a prescribed surface temperature while the lower surface rests in a rigid foundation and is thermally insulated. Employing Laplace and Fourier transforms as tools, the problem has been solved analytically in the transformed domain. The inversion of the Fourier transform is carried out using suitable numerical techniques while the numerical inversion of Laplace transform is done incorporating a method on Fourier series expansion technique. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the presence of memory dependent derivative, magnetic field and reinforcement also.

Modeling of memory-dependent derivative in a fibre-reinforced plate

This paper is concerned with the determination of the thermoelastic stress, strain and conductive temperature in a piezoelastic half-space body in which the boundary is stress free and subjected to thermal loading in the context of the fractional order two-temperature generalized thermoelasticity theory (2TT). The two-temperature three-phase-lag (2T3P) model, two-temperature Green–Naghdi model III (2TGNIII) and two-temperature Lord–Shulman (2TLS) model of thermoelasticity are combined into a unified formulation introducing unified parameters. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain that is then solved by the state-space approach. The numerical inversion of the transform is carried out by a method based on Fourier series expansion techniques. The numerical estimates of the quantities of physical interest are obtained and depicted graphically. The effect of the fractional order parameter, two-temperature and electric field on...

One-dimensional problem of a fractional order two-temperature generalized thermo-piezoelasticity

This paper deals with the thermoelastic interactions in a transversely isotropic, infinite hollow cylinder in which the boundaries are stress-free. There is no temperature in the inner boundary and heat flux is applied on the outer boundary. In the context of two-temperature generalized thermoelasticity theory, the three-phase-lag thermoelastic model and Green Naghdi model III (GN-III) are employed to study the thermophysical quantities. The Laplace transform is used to transform the coupled equations into the Laplace transformed domain. Then two different methods, the Galerkin finite element technique and eigen-value approach, are employed to solve the resulting equations in the transformed domain. The numerical inversion of the transform is carried out using Fourier-series expansion techniques. The physical quantities have been computed numerically and presented graphically in a number of figures. A comparison of the results for different theories (GN-III and three-phase-lag model) and for two different...

Two-Temperature Generalized Thermoelasticity in a Fiber-Reinforced Hollow Cylinder Under Thermal Shock

This paper deals with a two-dimensional (2D) problem for a transverselyisotropic thick plate having heat sources and body forces. The upper surface of the plate is stress free with the prescribed surface temperature, while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of the generalized thermoelasticity proposed by Green and Naghdi. The governing equations for displacement and temperature fields are obtained in the Laplace-Fourier transform domain by applying the Laplace and Fourier transforms. The inversion of the double transform is done numerically. Numerical inversion of the Laplace transform is done based on the Fourier series expansion. Numerical computations are carried out for magnesium (Mg), and the results are presented graphically. The results for an isotropic material (Cu) are obtained numerically and presented graphically to be compared with those of a transversely isotropic material (Mg). The effect of the body forces is also studied.

Dynamic response in two-dimensional transversely isotropic thick plate with spatially varying heat sources and body forces

This paper deals with a two dimensional problem for a transversely isotropic thick plate having heat source The upper surface of the plate is stress free with prescribed surface temperature while the lower surface of the plate rests on a rigid foundation and is thermally insulated The study is carried out in the context of three-phase-lag thermoelastic model, GN model II (TEWOED) and GN model III (TEWED) The governing equations for displacement and temperature fields are obtained in Laplace–Fourier transform domain by applying Laplace and Fourier transform techniques The inversion of double transform has been done numerically The numerical inversion of Laplace transform is done by using a method based on Fourier Series expansion technique Numerical computations have been done for magnesium (Mg) and a comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) are presented graphically The results for an isotropic material (Cu) have been deduced numerically an

Study of Dynamical Response in a Two-Dimensional Transversely Isotropic Thick Plate Due to Heat Source

This paper deals with the problem of magnetothermoelastic interactions in a perfectly conducting elastic medium in which the boundary is stress free and subjected to thermal loading in the context of the fractional-order, two-temperature generalized thermoelasticity theory (2TT). The two-temperature, three-phase-lag (2T3P) model and two-temperature Lord–Shulman (2TLS) model of thermoelasticity are combined into a unified formulation introducing unified parameters. The governing equations of generalized thermoelasticity of these models under the influence of a magnetic field are established. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace-transform domain, which is then solved by the state-space approach. The numerical inversion of the transform is carried out by a method based on Fourier-series expansion techniques. Because of the short duration of the second sound effects, small time approximations of the solutions are studied. The numerical estim...

Mohsin Islam

Papers

One-dimensional problem of a fractional order two-temperature generalized thermo-piezoelasticity

Two-Temperature Generalized Thermoelasticity in a Fiber-Reinforced Hollow Cylinder Under Thermal Shock

Dynamic response in two-dimensional transversely isotropic thick plate with spatially varying heat sources and body forces

Study of Dynamical Response in a Two-Dimensional Transversely Isotropic Thick Plate Due to Heat Source

Short-time Analysis of Magnetothermoelastic Wave Under Fractional Order Heat Conduction Law