A time domain approach for the analysis of periodic structures using finite element analysis

Abstract

The time domain analysis is an interesting alternative to spectral domain computations for vibrating structures or wave-guides exhibiting short impulse responses. we propose a development allowing for solving periodic transient problems. As for harmonic computations, we mesh only one period of the array and we then apply boundary conditions relating its edges one another. A periodic excitation coefficient similar to the one used in the spectral domain is defined and use to scan all the possible excitation figures. It is then shown how to derive mutual time domain coefficients that describe the way the different cells of the array are coupled together. It is remarkable that in this time domain representation, no singularity arises on the computed signals, yielding very favourable conditions for the derivation of mutual coefficients. but none take into account comprehensive periodic boundary conditions. In this paper, we propose a method to solve periodic transient problems. As in the case of harmonic computations, we mesh only one period of the array and we then apply periodic boundary conditions. A periodic excitation coefficient similar to the one employed in the spectral domain is defined and is used to scan all possible excitation situations. The time excitation is represented by Dirac (or Heaviside) impulses. It is then shown how mutual time domain coefficients can be derived that describe the way the different cells of the array are coupled together. It is remarkable that in this time domain representation, no singularities arise in the computed signals, resulting in very favourable conditions for the derivation of mutual coefficients. This time domain representation may be more accessible for most readers and should provide an efficient approach for the characterization of massively periodic devices with low quality factors. The first section is devoted to the fundamentals of the adopted integration scheme, i.e. the Newmark approach. Results are then reported in the case of a 2-2 piezocomposite structure and also in the case of a 2D micro-machined ultrasonic transducer (MUT) operating in a vacuum. Cross-talk phenomena due to acoustic propagation in these structures are identified thanks to the derivation of their mutual parameters (admittance, front velocity).