E-Book, Englisch, 400 Seiten
Reihe: Woodhead Publishing Series in Electronic and Optical Materials
Case Studies in Designing Devices
E-Book, Englisch, 400 Seiten
Reihe: Woodhead Publishing Series in Electronic and Optical Materials
ISBN: 978-0-85709-631-9
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark
Autoren/Hrsg.
Weitere Infos & Material
1 Overview of the ATILA finite element method (FEM) software code
K. Uchino, The Pennsylvania State University, USA Abstract:
The finite element method and its application to smart transducer systems are introduced in this chapter. The fundamentals of finite element analysis are introduced first. Then, the section ‘Defining the equations for the problem’ treats how to integrate the piezoelectric constitutive equations, and ‘Application of the finite element method’ describes the meshing. The last section ‘FEM simulation examples’ introduces six cases; multilayer actuator, ?-type linear ultrasonic motor, windmill ultrasonic motor, metal tube ultrasonic motor, piezoelectric transformer, and ‘cymbal’ underwater transducer, which includes most of the basic capabilities of ATILA FEM code. Key words finite element method piezoelectric constitutive equation node shape function variational principle discretization parent element assembly 1.1 An introduction to finite element analysis
Consider the piezoelectric domain O pictured in Fig. 1.1, within which the displacement field, u, and electric potential field, ?, are to be determined. The u and ? fields satisfy a set of differential equations that represent the physics of the continuum problem considered. Boundary conditions are usually imposed on the domain’s boundary, G, to complete the definition of the problem. 1.1 Schematic representation of the problem domain O with boundary G. The finite element method is an approximation technique for finding solution functions.1 The method consists of subdividing the domain O into sub-domains, or finite elements, as illustrated in Fig. 1.2. These finite elements are interconnected at a finite number of points, or nodes, along their peripheries. The ensemble of finite elements defines the problem mesh. Note that because the subdivision of O into finite elements is arbitrary, there is not a unique mesh for a given problem. 1.2 Discretization of the domain O Within each finite element, the displacement and electric potential fields are uniquely defined by the values they assume at the element nodes. This is achieved by a process of interpolation or weighing in which shape functions are associated with the element. By combining, or assembling, these local definitions throughout the whole mesh, we obtain a trial function for O that depends only on the nodal values of u and ? and that is ‘piecewise’ defined over all the interconnected elementary domains. Unlike the domain O, these elementary domains may have a simple geometric shape and homogeneous composition. We will show in the following sections how this trial function is evaluated in terms of the variation principle to produce a system of linear equations whose unknowns are the nodal values of u and ?.2 1.2 Defining the equations for the problem
1.2.1 The constitutive and equilibrium equations
The constitutive relations for piezoelectric media may be derived in terms of their associated thermodynamic potentials.3,4 Assuming the strain, x, and electric field, E, are independent variables, the basic equations of state for the converse and direct piezoelectric effects are written: [1.1] The quantities cE (elastic stiffness at constant electric field), e (piezoelectric stress coefficients), and ?x (dielectric susceptibility at constant strain) are assumed to be constant, which is reasonable for piezoelectric materials subjected to small deformations and moderate electric fields. Furthermore, no distinction is made between isothermal and adiabatic constants. On the domain, O, and its boundary, G, (where the normal is directed outward from the domain), the fundamental dynamic relation must be verified: [1.2] where u is the displacement vector, ? the mass density of the material, t the time, X the stress tensor, and r = < r1 r2 r3>, a unit vector in the Cartesian coordinate system. When no macroscopic charges are present in the medium, Gauss’ theorem imposes for the electric displacement vector, D: [1.3] Considering small deformations, the strain tensor, x, is written as: [1.4] Assuming electrostatic conditions, the electrostatic potential, ?, is related to the electric field E by [1.5] or, equivalently [1.6] Using Equations [1.2], [1.3] and [1.6] in combination with Equation [1.1] yields: [1.7] 1.2.2 Boundary conditions
Mechanical and electrical boundary conditions complete the definition of the problem. The mechanical conditions are as follows: • The Dirichlet condition on the displacement field, u, is given by: [1.8] where uo is a known vector. For convenience, we name the ensemble of surface elements subjected to this condition Su. • The Neumann condition on the stress field, X, is given by: [1.9] where n is the vector normal to G, directed outward, and fo is a known vector. For convenience, we name the ensemble of surface elements subjected to this condition SX. The electrical conditions are as follows: • The conditions for the excitation of the electric field between those surfaces of the piezoelectric material that are not covered with an electrode and are, therefore, free of surface charges is given by: [1.10] where n is the vector normal to the surface. For convenience, we name the ensemble of surface elements subjected to this condition Ss. Note that with the condition in Equation [1.9], we assume that the electric field outside O is negligible, which is easily verified for piezoelectric ceramics. • When considering the conditions for the potential and excitation of the electric field between those surfaces of the piezoelectric material that are covered with electrodes, we assume that there are p electrodes in the system. The potential on the whole surface of the pth electrode is: [1.11] The charge on that electrode is: [1.12] In some cases, the potential is used, and in others it is the charge. In the former case, ?p is known and Equation [1.11] is used to determine Qp. in the latter case, Qp is known and Equation [1.10] is used to determine ?. Finally, in order to define the origin of the potentials, it is necessary to impose the condition that the potential at one of the electrodes be zero (?o = 0). 1.2.3 The variational principle
The variational principle identifies a scalar quantity ?, typically named the functional, which is defined by an integral expression involving the unknown function, w, and its derivatives over the domain O and its boundary G. The solution to the continuum problem is a function w such that [1.13] ? is said to be stationary with respect to small changes in w, dw. When the variational principle is applied, the solution can be approximated in an integral form that is suitable for finite element analysis. In general, the matrices derived from the variational principle are always symmetric. Equation [1.7] and the boundary conditions expressed by Equations [1.8] to [1.12] allow us to define the so-called Euler equations to which the variational principle is applied such that a functional of the following form is defined that is stationary with respect to small variations in w. [1.14] Note that the first term of this expression for ? represents the...
