2000 IEEBEIA International Frequency Control Symposium and Exhibition TWO-DIMENSIONAL BOUNDARY ELEMENT ANALYSIS OF QUARTZ SURFACE WAVE RESONATORS Mitsunori Denda and Yook-Kong Yon& TMech. & Aero. Engineering Dept., Rutgers University, NJ, U.S.A. denda@ove.rutgers.edu $Civil & Environmental Engineering Dept., Rutgers University, NJ, U.S.A. yong@ove.rutgers.edu Abstract In this paper we will present the 2-D fundamental solutions for the time-harmonic dynamic problems of piezoelectric materials. Given a time-harmonic line force or charge at the origin of an infinite piezoelectric solid, we derive the displacement and electric potential at an arbi- trary point. The solution, obtained using the Radon trans- form, has an interesting feature that it can be split into two parts: singular static and regular dynamic parts. The singu- lar static part corresponds to the static fundamental solu- tion and the regular dynamic part provides frequency dependency. The regular part is presented in a form suit- able for numerical evaluation. The implementation of the fundamental solutions into the boundary element method (BEM) for the eigen frequency problems of the surface wave resonators will be outlined along with key technical features required for the successful BEManalysis. I. Introduction A properandrigorous analysis of a quartz surface wave resonator requires a numerical technique which accu- rately simulates not only the surface waves near the crystal surface, but also captures all the incipient bulk waves in the crystal substrate. In resonators which use leaky surface waves, the accurate simulation of the reflection of wavesat the bottom of the resonator is essential. The finite element method (FEM) can provide such solutions [l] but its com- puter memory and computing time requirements can be overwhelming evenfor today’s computers. We propose an alternative approach, based on the BEM. Wlule the domain consisting of the crystal surface, substrate and the electrodes is discretized in the FEM, only the boundary of the domain need to be discretized in the BEM. This reduction of the dimension bythe BEM makes it an ideal tool for the problem considered. However, the BEM requires the fundamental solutions, which have not been available for the time-harmonic dynamic problems of piezoelectric materials in a form convenient for numerical implementation. The primarily goal of this paper is the pre- sentation of these fundamental solutions and their imple- mentation into the BEM eigenvalue problems for the surface wave resonators. The derivation of the piezoelec- tric fundamental solutions here follows the framework set by Wang [2,3] for general anisotropic elastic materials. The general Green’s functions obtained for the SAW problems [4,5] provide the displacement’electric potential solution on the free surface due to a line force/charge placedon the same surface. Theadvantage is that they automatically satisfy the tractiodsurface charge free boundary condition on the surface and suited for the waves confined in the vicinity of the free surface. Since the source and the observation points are confined on the free surface, they cannot deal with bulk waves away from the surface. No such restriction is placed for the fundamentals solutions obtained in this paper. They provide influence functions for arbitrary selection of the source and observa- tion points. These fundamentalsolutions are used to satisfy the particular boundary conditions for the surface wave resonators under the guidance of the BEM. II. Piezoelectric Governing Equations We consider a two-dimensional elastodynamic prob- lem for an infinite piezoelectric solid defined in the (xl, x2 )-plane. Thebodyforceicharge densities andthe boundaryconditions are time-harmonicwith an angular frequency o . We adopt the convention that the real parts j.(k, t) = Rex(), o)e-iaf}, p’,(), t) = Re{ p$, represent the actual body forcelcharge densities, where i is an imaginary number. The displacement components and electric potential also have the form ui(k, t) = Re{ui(k, o)e-iOf} , +(k, t) = Re{cp(R, o)e-iaf} . In the following we suppress the symbol Re and the time factor cia‘ with a reminder that functions U,(?, a) , q(f, o) ,..., depend on o and are complex valued. Under the electrostatic approximation, the equations of motion andthe Gauss’s law are given by Ciajpuj, pa + epiaq , pa + po2ui = -4, (1) ea;pu;, Pa-Kapq , pa - -Pe? (2) - where p is the mass density. Note that the charge density -pe is defined with a minus sign. The coefficients cQpq , ejjk and ‘cij are the elastic moduli, piezoelectric moduli and dielectric permittivity constants, respectively.Inthis paper we employ three different suffix notations; a lower- case and an uppercase Roman suffixes range from 1 to 3 and 1 to 4, respectively, whilea Greek suffix takes the value of 1 and 2. The summation convention is applied to a repeated suffix over its range. The derivatives with respect to x, is written as ( ) .a or 5,. If we introduce 0-7803-5838-4/00/$10.00 0 2000 IEEE 290