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

Free vibration analysis of rotating functionally graded CNT reinforced composite cylindrical shells with arbitrary boundary conditions

Nonlinear forced vibrations of functionally graded piezoelectric cylindrical shells under electric-thermo-mechanical loads

In this paper, the effect of loading frequency on the dynamic behavior of nanocomposite sandwich plates under periodic thermo-mechanical loadings has been investigated. The utilized sandwich plates are made of an isotropic polymer material and two symmetric face sheets reinforced by functionally graded (FG) distributions of carbon nanotube (CNT) agglomerations. In addition to periodic mechanical loads, these structures are also subjected to thermal gradient loads. Steady state response of the plates under thermal gradient load was assumed like a pre-stress for dynamic equations in conducting timeline vibrations of the structure. The material properties of polymeric matrix and CNTs were assumed to be temperature-dependent and the overall material properties of nanocomposites were estimated using Eshelby-Mori-Tanaka's approach. In order to achieve accurate results, a mesh-free method based on higher order shear deformation theory (HSDT) was utilized. The effects of mechanical loading frequency and thermal gradient load as well as CNTs cluster characterizations, essential boundary conditions and elastic foundation on forced vibration, resonance and phenomenon of beats behaviors were investigated. It was observed that thermal gradient loads and the formation of CNT agglomerations have significant effects on the amplitudes of vibrations in nanocomposite sandwich plates.

Frequency-dependent forced vibration analysis of nanocomposite sandwich plate under thermo-mechanical loads

As a first endeavor, the free flexural vibration behavior of doubly curved complete and incomplete sandwich shells with functionally graded (FG) porous core, FG carbon nanotube reinforced composite (FG-CNTRC) face sheets and integrated piezoelectric layers is investigated. The variable radii shells with the three most common types of geometries, i.e., elliptical, cycloid and parabolic, are considered. The system equations are derived based on the general higher-order shear deformation theory and Maxwell's equation. The generalized differential quadrature (GDQ) method is employed to discretize the governing partial differential equations subjected to different boundary conditions. The accuracy and reliability of the approach are verified by comparing the results with the existing solutions in open literature. The effects of porosity parameter and porosity distribution through the thickness direction, carbon nanotube (CNT) volume fraction, different boundary conditions and various shell geometrical parameters on the flexural vibrational behavior of the smart sandwich shell structures are investigated and useful results are presented.

Vibrational behavior of doubly curved smart sandwich shells with FG-CNTRC face sheets and FG porous core

In this paper, free and forced vibrations, and also resonance and pulse phenomena in sandwich plates with an isotropic core and composite reinforced by wavy carbon nanotube (CNT) face sheets are st...

Vibrational behavior of sandwich plates with functionally graded wavy carbon nanotube-reinforced face sheets resting on Pasternak elastic foundation:

In this article, free vibration analysis of simply supported sandwich plates with a flexible core and two functionally graded nanocomposite face sheets resting on two-parameter elastic foundation is carried out using Navier's solution. The nanocomposite face sheets are reinforced by randomly oriented and aggregated carbon nanotube (CNT). The material properties of the nanocomposites plates are graded along the thickness and are estimated though the Eshelby–Mori–Tanaka approach. In the proposed theory, 3-D elasticity theory and first order shear deformation theory are used for core and face sheets, respectively. Also, Hamilton's variational principle is used to derive the equations of motion.

Free vibration analysis of nanocomposite sandwich plates reinforced with CNT aggregates

In this paper, free vibration, forced vibration, resonance and stress wave propagation behavior in nanocomposite plates reinforced by wavy carbon nanotube (CNT) are studied by a mesh-free method based on first order shear deformation theory (FSDT). The plates are resting on Winkler-Pasternak elastic foundation and subjected to periodic or impact loading. The distributions of CNTs are considered functionally graded (FG) or uniform along the thickness and their mechanical properties are estimated by an extended rule of mixture. In the mesh-free analysis, moving least squares (MLS) shape functions are used for approximation of displacement field in the weak form of motion equation and the transformation method is used for imposition of essential boundary conditions. Effects of CNT distribution, volume fraction, aspect ratio and waviness, and also effects of elastic foundation coefficients, plate thickness and time depended loading are examined on the vibrational and stresses wave propagation responses of the nanocomposite plates reinforced by wavy CNT.

Dynamic analysis of functionally graded nanocomposite plates reinforced by wavy carbon nanotube

This study analyzes the free vibration, forced vibration, resonance, and stress wave propagation of orthotropic sandwich plates made of functionally graded materials (FGMs). Dynamic analyses are conducted using a mesh-free method based on first-order shear deformation theory and the shape functions constructed using moving least squares approximation. The sandwich plates are rested on a Pasternak elastic foundation and subjected to periodic or impact loading and essential boundary conditions, which are imposed through a transfer function method. The sandwich plates are assumed to be composed of a homogeneous orthotropic core and two orthotropic FGM face sheets made of two orthotropic materials. The volume fractions of the materials are varied smoothly along the thickness of the face sheets. The convergence and accuracy of the applied method are demonstrated, after which numerical analyses are conducted to investigate the effects of elastic foundation coefficients, material distributions, geometrical dimensions, time-dependent loading, and boundary conditions on the vibrational and dynamic characteristics of the orthotropic FGM sandwich plates.

http://macs.journals.semnan.ac.ir/article_2624_b408acaab5d0b87c655139a3e15abcd0.pdf

Mesh-free Dynamic Analyses of FGM Sandwich Plates Resting on A Pasternak Elastic Foundation

In this paper, static and free vibrations behaviors of the orthotropic functionally graded material (FGM) plates resting on the two-parameter elastic foundation are analyzed by the a mesh-free method based on the first order shear deformation plate theory (FSDT). The mesh-free method is based on moving least squares (MLS) shape functions and essential boundary conditions are imposed by transfer function method. The orthotropic FGM plates are made of two orthotropic materials and their volume fractions are varied smoothly along the plate thickness. The convergence of the method is demonstrated and to validate the results, comparisons are made with finite element method (FEM) and the others available solutions for both homogeneous and FGM plates then numerical examples are provided to investigate the effects of material distributions, elastic foundation coefficients, geometrical dimensions, applied force and boundary conditions on the static and vibrational characteristics of the orthotropic FGM plates.

http://jsm.iau-arak.ac.ir/article_531829_eb9114628e2e853c9b9b60ae752c649f.pdf

Hamed Momeni-Khabisi

Papers

Vibrational behavior of sandwich plates with functionally graded wavy carbon nanotube-reinforced face sheets resting on Pasternak elastic foundation:

Free vibration analysis of nanocomposite sandwich plates reinforced with CNT aggregates

Dynamic analysis of functionally graded nanocomposite plates reinforced by wavy carbon nanotube

Mesh-free Dynamic Analyses of FGM Sandwich Plates Resting on A Pasternak Elastic Foundation

Static and free vibration analyses of orthotropic fgm plates resting on two-parameter elastic foundation by a mesh-free method