Plane wave finite element model for the 2-D phononic crystal under force loadings

Abstract

Direct numerical simulations of the dynamic response of the phononic crystal to point or distributed forces via conventional numerical methods are still formidable currently due to large number of DOFs involved in the simulations. In this study, based on the plane wave expansion (PWE) method, a plane wave finite element (PWFE) model for the 2-D phononic crystal which is infinite in the horizontal direction and finite in the vertical direction and subjected to a point harmonic force is proposed. To this end, the point harmonic force is decomposed into the wavenumber domain (WND) components by the Fourier transform first. The variables of the 2-D phononic crystal under the WND force component are expanded into a series of plane waves along the horizontal direction, and the variables associated with each plane wave are discretized by the FE method in the vertical direction. By using the virtual work principle, FE equations for the plane waves of the phononic crystal are established. With the FE equations for the plane waves, the eigenvalue equation for the displacement of the plane waves and wavenumber is formulated. Superposing the eigenvectors of the above eigenvalue equation yields the displacement for the plane waves due to the WND force component. Synthetization of the contributions from all the plane waves and inversion of the Fourier transform with respect to the wavenumber via the contour integration method generate the response of the phononic crystal to the point force. Comparison of present results with those due to the transfer matrix method validates the proposed model. Also, presented numerical results suggest that the dynamic response of the phononic crystal to a point harmonic force exhibits remarkable resonance phenomenon.