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

http://ieeexplore.ieee.org/iel5/5/31047/01444179.pdf

Fundamentals of acoustics

Damage identification in beam-type structures: frequency-based method vs mode-shape-based method

This paper is concerned with the free vibration problem for micro/nanobeams modelled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The small scale effect is taken into consideration in the former theory while the effects of transverse shear deformation and rotary inertia are accounted for in the latter theory. The governing equations and the boundary conditions are derived using Hamilton's principle. These equations are solved analytically for the vibration frequencies of beams with various end conditions. The vibration solutions obtained provide a better representation of the vibration behaviour of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant. The exact vibration solutions should serve as benchmark results for verifying numerically obtained solutions based on other beam models and solution techniques.

http://iopscience.iop.org/article/10.1088/0957-4484/18/10/105401/pdf

Vibration of nonlocal Timoshenko beams

In this paper, an efficient and simple higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with 3-dimensional and quasi-3-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.

An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates

Free vibration analysis of functionally graded plates using the element-free kp-Ritz method

Mechanical and thermal buckling analysis of functionally graded plates

Sound absorption of a finite flexible micro-perforated panel backed by an air cavity

This paper presents an improved moving least-squares (IMLS) approximation in which the orthogonal function system with a weight function is used as the basis function. The IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation, and does not lead to an ill-conditioned system of equations. By combining the element-free Galerkin (EFG) method and the IMLS approximation, an improved element-free Galerkin (IEFG) method for two-dimensional elasticity is derived. There are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method that is formed with the IMLS approximation fewer nodes are selected in the entire domain than are selected in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed. For two-dimensional fracture problems, the enriched basis function is used at the tip of the crack to give an enriched IEFG method. When the enriched IEFG method is used, the singularity of the stresses at the tip of the crack can be shown better than that in the IEFG method. To provide a demonstration, numerical examples are solved using the IEFG method and the enriched IEFG method.

Analyzing 2D fracture problems with the improved element-free Galerkin method

This paper presents a postbuckling analysis of functionally graded cylindrical shells under axial compression and thermal loads using the element-free kp-Ritz method. The formulation is developed to handle problems of small strains and moderate rotations, based on the first-order shear deformation shell theory and von Karman strains. The effective material properties of the shells are assumed to be continuous along their thickness direction, and are obtained using a power-law distribution of the volume fractions of the constituents. The approximations of the two-dimensional displacement fields are expressed in terms of a set of mesh-free kernel particle functions. The system bending stiffness is evaluated using a stabilized conforming nodal integration method and the membrane and shear terms are estimated using direct nodal integration to eliminate shear locking. The postbuckling path is traced using a combination of the arc-length and mesh-free kp-Ritz methods. The proposed formulation is validated by comparing the results of the proposed method with those in the literature. The postbuckling responses of two types of functionally graded conical shells, one composed of Al/ZrO 2 and the other of SUS304/Si 3 N 4 , are investigated and the effects of volume fraction, boundary condition, and length-to-thickness ratio on postbuckling behavior are discussed in detail.

Yiu-Yin Lee

Papers

Free vibration analysis of functionally graded plates using the element-free kp-Ritz method

Mechanical and thermal buckling analysis of functionally graded plates

Sound absorption of a finite flexible micro-perforated panel backed by an air cavity

Analyzing 2D fracture problems with the improved element-free Galerkin method

Postbuckling responses of functionally graded cylindrical shells under axial compression and thermal loads