Development of the boundary element method for 2D piezoelectricity

The analytical and numerical foundations are laid out for the formulation of the boundary element method (BEM) for plane piezoelectric solids. We extend a physical interpretation of Somigliana's identity to piezoelectricity and give a direct formulation of the BEM in terms of the continuous distributions of point forces/charges and displacement/electric potential discontinuities in the infinite piezoelectric domain. We adopt Stroh's complex variable formalism for piezoelectricity to derive the point force/charge and the displacement/electric potential discontinuity, their dipoles and continuous distributions systematically. The duality relations between the force/charge and the displacement/electric potential solutions, embedded in the Stroh formalism, are exploited as the foundations for the analytic and the numerical approaches to the piezoelectric boundary value problems in two dimensions. These approaches enable us to solve important problems of piezoelectricity with arbitrary geometry and composition.