Dual reciprocity BEM for dynamic piezoelectricity

Abstract

The piezoelectric phenomenon is widely utilized in many devices. An analysis of piezoelectric devices requires a solution of coupled mechanical and electrical partial differential equations. In this paper the boundary element method (BEM) is implemented to solve the eigenvalue two-dimensional problem of linear piezoelectricity. The piezoelectric material is modeled as: homogenous, transversal isotropic, linear elastic and dielectric. The numerical solution by the BEM requires fundamental solutions,which are derived using the Stroh formalism. The mass matrix of piezoelectric is computed by the dual reciprocity BEM. A numerical example shows that the dual reciprocity BEM formulation allows to analyze accurately dynamic problems of linear piezoelectricity. The dual reciprocity BEM formulation of dynamic piezoelectricityThe coupled field equations of piezoelectricity are given by the following system of partial differential equations [1, 2]:The tensors C ijkl , e lij , ε il denote elastic moduli, measured in a constant electric field, piezoelectric constants and dielectric constants, measured at constant strains, respectively; u k is the displacement vector, φ denotes the electric potential; b j is the body force vector per unit volume and ρ denotes the density. In equations (1) the intrinsic electric charge is neglected. To get the classical boundary-initial value problem formulation, equations (1) must be completed with the boundary and initial conditions.The boundary-initial value problem of linear piezoelectricity can be formulated in a much more convenient form using generalized quantities. The following vectors are introduced:where U K , T J and B J are the generalized displacement, traction and body force vector, respectively. The next step in the BEM formulation is application of well known reciprocity relation [2,3]. The system of equations is weighted with a test function and integrated by parts. When the test function is chosen as a fundamental solution of the static piezoelectric operator L JK and a limiting process to the boundary is done, the boundary integral formula is given by:where c KJ denotes a free term coefficient at the source point, U KJ is the test function -the fundamental solution, and T KJ depends on the derivative of the fundamental solution, Γ denotes the boundary and Ω denotes the region which is occupied by the piezoelectric body. In this equation the generalized body force vector is neglected. To obtain the fundamental solutions, the Stroh formalism is used [4,5]. The domain integral in equation (3), which describes the inertia effect, will be transformed into the boundary integral using the reciprocal theorem between two static states [2,3]. Let the generalized displacements be approximated using a sum of functions multiplied by unknown coefficients: