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

THE appearance of a treatise in English upon the mathematical theory of elasticity is an event the potential importance of which may be judged by the that the author, in his frequent suggestions for collateral reading, refers to only three such, those of Southwell, Timoshenko, and Love. In spirit and content Sokolnikoff}s book differs greatly from each and all of these. It may be described by a possible sub-title: “A pure mathematician surveys topics related to certain problems in the mathematical theory of elasticity”. It is symptomatic of the change in outlook of American mathematics over the past few decades. Mathematical Theory Of Elasticity Prof. I. S. Sokolnikoff with the collaboration of Asst. Prof. R. D. Speche. Pp. xi + 373. (New York and London: McGraw-Hill Book Co., Inc., 1946.) 22s. 6d.

Mathematical Theory Of Elasticity

A review of theories for the modeling and analysis of functionally graded plates and shells

Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature

Variational methods in elasticity and plasticity

This book deals with the Finite Element Method for the analysis of elastic structures such as beams, plates, shells and solids. The modern approach of Unified Formulation (UF), as proposed by the lead author, deals with the consideration of one-dimensional (beams), two-dimensional (plates and shells) and three-dimensional (solids) elements. Applications are given for structures which are typically employed in civil, mechanical, and aerospace engineering fields. Additional topics include mixed order elements, extension to layered composite structures, and the analysis of multifield problems involving mechanical, electrical and thermal loadings.

Finite Element Analysis of Structures through Unified Formulation

A number of refined beam theories are discussed in this paper. These theories were obtained by expanding the unknown displacement variables over the beam section axes by adopting Taylor's polynomials, trigonometric series, exponential, hyperbolic and zig-zag functions. The Finite Element method is used to derive governing equations in weak form. By using the Unified Formulation introduced by the first author, these equations are written in terms of a small number of fundamental nuclei, whose forms do not depend on the expansions used. The results from the different models considered are compared in terms of displacements, stress and degrees of freedom (DOFs). Mechanical tests for thick laminated beams are presented in order to evaluate the capability of the finite elements. They show that the use of various different functions can improve the performance of the higher-order theories by yielding satisfactory results with a low computational cost.

Laminated beam analysis by polynomial, trigonometric, exponential and zig-zag theories

Free vibration analysis of rotating composite blades via Carrera Unified Formulation

This paper deals with the accurate evaluation of complete three-dimensional (3D) stress fields in beam structures with compact and bridge-like sections. A refined beam finite-element (FE) formulation is employed, which permits any-order expansions for the three displacement components over the section domain by means of the Carrera Unified Formulation (CUF). Classical (Euler-Bernoulli and Timoshenko) beam theories are considered as particular cases. Comparisons with 3D solid FE analyses are provided. End effects caused by the boundary conditions are investigated. Bending and torsional loadings are considered. The proposed formulation has shown its capability of leading to quasi-3D stress fields over the beam domain. Higher-order beam theories are necessary for the case of bridge-like sections. Various theories are also compared in terms of shear correction factors on the basis of definitions found in the open literature. It has been confirmed that different theories could lead to very different values of shear correction factors, the accuracy of which is subordinate to a great extent to the section geometries and loading conditions. However, an accurate evaluation of shear correction factors is obtained by means of the present higher-order theories.

Performance of CUF Approach to Analyze the Structural Behavior of Slender Bodies

This paper is a review of the most influential approaches to developing beam models that have been proposed over the last few decades. Essentially, primary attention has been paid to isotropic structures while a few extensions to composites have been given for the sake of completeness. Classical models - Da Vinci, Euler-Bernoulli, and Timoshenko - are described first. All those approaches that are aimed at the improvement of classical theories are then presented by considering the following main techniques: shear correction factors, warping functions, Saint-Venant based solutions and decomposition methods, variational asymptotic methods, the Generalized Beam Theory and the Carrera Unified Formulation (CUF). Special attention has been paid to the latter by carrying out a detailed review of the applications of 1D CUF and by giving numerical examples of static, dynamic and aeroelastic problems. Deep and thin-walled structures have been considered for aerospace, mechanical and civil engineering applications. Furthermore, a brief overview of two recently introduced methods, namely the mixed axiomatic/asymptotic approach and the component-wise approach, has been provided together with numerical assessments. The review presented in this paper shows that the development of advanced beam models is still extremely appealing, due to the computational efficiency of beams compared to 2D and 3D structural models. Although most of the techniques that have recently been developed are focused on a given number of applications, 1D CUF offers the breakthrough advantage of being able to deal with a vast variety of structural problems with no need for ad hoc formulations, including problems that can notoriously be dealt with exclusively by means of 2D or 3D models, such as complete aircraft wings, civil engineering constructions, as well as multiscale and wave propagation analyses. Moreover, 1D CUF leads to a complete 3D geometrical and material modeling with no need of artificial reference axes/surfaces, reduced constitutive equations or homogenization techniques

Enrico Zappino

Papers

Finite Element Analysis of Structures through Unified Formulation

Laminated beam analysis by polynomial, trigonometric, exponential and zig-zag theories

Free vibration analysis of rotating composite blades via Carrera Unified Formulation

Performance of CUF Approach to Analyze the Structural Behavior of Slender Bodies

Recent developments on refined theories for beams with applications