On spectral element methods for the analysis of acoustic and mechanical metamaterials

Chuanzeng Zhang

Ningbo University

Abstract: Acoustic and mechanical metamaterials are novel and artificially designed materials or structures exhibiting extraordinary wave propagation properties that cannot be found in natural or conventional materials. They have become a research hotspot and highlight of great interest in recent years. The analysis of the wave propagation problems in acoustic and mechanical metamaterials typically relies on the numerical methods to determine their band structures (dispersion relations) or transmission spectra. Among many numerical methods, the finite element method (FEM), time-domain finite difference method (TDFDM), boundary element method (BEM), plane wave expansion method (PWEM), and meshless method (MLM) are particularly common, while the spectral element method (SEM) is rarely used, despite its numerous advantages and superior features. In this presentation, two different SEMs extremely suitable for the numerical analysis of acoustic and mechanical metamaterials will be reported, namely, the Doyle-type SEM and the Patera-type SEM. The Doyle-type SEM uses the exact solutions of the equations of wave motion to construct the shape or interpolation functions and to derive the corresponding spectral element matrix. Its semi-analytical nature results in an extremely high computational accuracy and efficiency, but its applicability is primarily limited to one-dimensional (1D) wave propagation problems. On the other hand, the Patera-type SEM, which is in fact a higher-order FEM, is suited not only for 1D but also for two-dimensional (2D) and three-dimensional (3D) wave propagation problems. The conventional FEM typically uses low-order shape functions and uniform node distributions, whereas the Patera-type SEM utilizes higher-order Lagrangian shape functions and a Gauss-Legendre-Lobatto (GLL) node distribution. Several numerical examples will be shown to demonstrate the advantages and disadvantages, accuracy and efficiency of the two different SEMs for computing the band structures (dispersion relations) and transmission spectra of acoustic and mechanical metamaterials.

Bio: Professor Chuanzeng Zhang has received his PhD in 1986 at the TU Darmstadt, Germany. From 1986 to 1988, he was a postdoc with Professor Jan D. Achenbach at the Department of Civil Engineering, Northwestern University, USA. Before his appointment as Professor at the Ningbo University (China) in 2024, he was Associate Professor and Professor at the Department of Engineering Mechanics of the Tongji University (China), Professor at the Department of Civil Engineering, University of Applied Sciences Zittau/Görlitz (Germany), and Professor at the Department of Civil Engineering, University of Siegen (Germany). His research interests include computational mechanics, structural mechanics, acoustic and mechanical metamaterials, and mechanics of smart materials and structures. He has published 2 monographs and over 650 papers in peer-reviewed scientific journals with 25,000 citations and h-index 78 (Google Scholar). He is Co-Chief Editor of a book series, Associate Editor of 3 international journals, Advisory Editor, Guest Editor and Editorial Member of over 10 other scientific journals. He is Adjunct Professor, Guest Professor, Consulting Professor and Honorary Professor of over 10 universities. He was awarded as honorary doctorate (Dr. honoris causa) of Slovak University of Technology in Bratislava, honorary doctorate (Dr. honoris causa) of Aristotle University of Thessaloniki, member of European Academy of Sciences, member of European Academy of Sciences and Arts, and member of Academia Europaea.