Three-dimensional piezoelectric boundary element method

The boundary element method is applied to problems of three-dimensional linear piezoelectricity. The continuum equations for conservation of linear momentum and charge are combined into one governing equation for piezoelectricity. A single boundary integral equation is developed from this combined field equation and Green's solution for a piezoelectric medium. Green's function and its derivatives are derived using the radon transform, and the resulting solution is represented by a line integral that is evaluated numerically using standard Gaussian quadrature. The boundary integral equation is discretized using eight-node isoparametric quadratic elements, resulting in a matrix system of equations. The solution of the boundary problem for piezoelectric materials consists of elastic displacements, tractions, electric potentials, and normal charge flux densities. The complete field solutions can be obtained once all boundary values have been determined, The accuracy of this linear piezoelectric boundary element method is illustrated with two numerical examples. The first involves a unit cube of material with an applied mechanical load. The second example consists of a spherical hole in an infinite piezoelectric body loaded by a unit traction on its boundary. Comparisons are made to the analytical solution for the cube and an axisymmetric finite element solution for the spherical hole. The boundary element method is shown to be an accurate solution procedure for general three-dimensional linear piezoelectric material problems.