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

Lagrangian dynamics of mechanical systems 1.1 Introduction 1.2 Kinetic state functions 1.3 Generalized coordinates, kinematic constraints 1.4 The principle of virtual work 1.5 D'Alembert's principle 1.6 Hamilton's principle 1.7 Lagrange's equations 1.8 Lagrange's equations with constraints 1.9 Conservation laws 1.10 More on continuous systems 1.11 References 2 Dynamics of electrical networks 2.1 Introduction 2.2 Constitutive equations for circuit elements 2.3 Kirchhoff's laws 2.4 Hamilton's principle for electrical networks 2.5 Lagrange's equations 2.6 References 3 Electromechanical Systems 3.1 Introduction 3.2 Constitutive relations for transducers 3.3 Hamilton's Principle 3.4 Lagrange's equations 3.5 Examples 3.6 General electromechanical transducer 3.7 References 4 Piezoelectric Systems 4.1 Introduction 4.2 Piezoelectric transducer 4.3 Constitutive relations of a discrete transducer 4.4 Structure with a discrete piezoelectric transducer 4.5 Multiple transducer systems 4.6 General piezoelectric structure 4.7 Piezoelectric material 4.8 Hamilton's principle 4.9 Rosen's piezoelectric transformer 4. 10 References 5 Piezoelectric laminates 5.1 Piezoelectric beam actuator 5.2 Laminar sensor 5.3 Spatial modal filters 5.4 Active beam with collocated actuator-sensor 5.5 Piezoelectric laminates 5.6 References 6 Active and Passive Damping with Piezoelectric Transducers 6.1 Introduction 6.2 Active strut, open-loop FRF 6.3 Active damping via 1FF 6.4 Admittance of the piezoelectric transducer 6.5 Damping via resistive shunting 6.6 Inductive shunting 6.7 Decentralized control 6.8 General piezoelectric structure 6.9 Self-sensing 6.10 Other active damping strategies 6.11 Remark 6.12 References Bibliography Index

Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems

http://wwwprof.uniandes.edu.co/~framirez/PUBLICATIONS/JSV2005.pdf

Free vibration response of two-dimensional magneto-electro-elastic laminated plates

Applicability of the crack-face electromagnetic boundary conditions for fracture of magnetoelectroelastic materials

An overview of layerwise theories for composite laminates and structures: Development, numerical implementation and application

Layerwise partial mixed finite element analysis of magneto-electro-elastic plates

Modelling of piezolaminated plates using layerwise mixed finite elements

This paper deals with the development of a layerwise finite element model for piezo-laminated plate structures. A Reissner mixed variational equation is used to derive the governing equations, using transverse stresses, transverse electrical displacement, displacement components and electrical potential as primary variables. The present model in contrast with the standard layerwise displacement finite elements fulfils the continuity of all primary variables at the interface between adjacent layers. Only the layers in-plane stress components and corresponding electric displacements are evaluated by post-processing through the piezolaminated constitutive equations. The applicability and performance of the proposed model is illustrated with an example. The predictions of the primary variables and the efficiency of the model mainly related with the interlaminar stresses are discussed and compared with alternative three-dimensional solutions.

Analysis of adaptive plate structures by mixed layerwise finite elements

Walsh series are used to develop a finite-element model based on first-order shear deformation theory that enables accurate prediction of static and dynamic response of laminated composite plates. The model is based on a single-layer quadrilateral plate element with mixed formulation, which can be applied to any arbitrary layup. The model has as degrees of freedom stress resultants and generalized displacements. The accuracy and efficiency of the model developed is discussed with four illustrative cases, for static and natural frequencies results, which are compared with alternative solutions.

R. Garcia Lage

Papers

Layerwise partial mixed finite element analysis of magneto-electro-elastic plates

Modelling of piezolaminated plates using layerwise mixed finite elements

Modelling of piezolaminated plates using layerwise mixed finite elements

Analysis of adaptive plate structures by mixed layerwise finite elements

Development of a Single-Layer Laminated Plate Finite-Element Model Based on Walsh Series