BEM-based analysis of elastic banded material by using a contour integral method

Abstract

The combination of boundary element method (BEM) and a contour integral method is proposed for the first time to determine the band structures for banded elastic materials named as phononic/sonic crystals. Applying the Bloch theorem to a unit cell, the Bloch eigenvalue problem is derived to determine the dispersion relation which presents the band structures of the phononic crystals. Owing to the nonlinear form of the fundamental solution, the circular frequency, however, is involved in the coefficient matrix as the eigenvalue parameter implicitly, and the transcendental eigenequation makes it a nonlinear eigenvalue problem. To solve the nonlinear eigenvalue problem, the block Sakurai–Sugiura (SS) method is employed as an eigensolver, which extracts the eigenfrequencies within a closed contour integration path by evaluating the contour integral numerically. To avoid the complex fictitious eigenvalues, a new fusiform integration path is proposed to confine the solving domain to the neighborhood of the real axis. Numerical examples are presented to demonstrate the effectiveness of the proposed methodology for band structure calculation in 2D elastic problems.