Comparison of different higher order finite element schemes for the simulation of Lamb waves

Abstract

Structural Health Monitoring (SHM) applications call for both efficient and powerful numerical tools to predict the behavior of ultrasonic guided waves. When considering waves in thin-walled structures, so called Lamb waves, conventional linear or quadratic pure displacement finite elements soon reach their limits. The spatial as well as temporal discretisation, required to obtain good quality results has to be very fine. This results in enormous computational costs (computational time and memory storage requirements) when ultrasonic wave propagation problems are solved in the time domain. To resolve this issue several higher order finite element methods with polynomial degrees p>2 are proposed. The objective of the current article is to develop such higher order schemes and to verify their capabilities with respect to accuracy and numerical performance. To the best of the authors’ knowledge such comparison has not been reported in literature, yet. Specifically, spectral elements based on Lagarange polynomials (SEM), p-elements using the normalized integrals of the Legendre polynomials (p-FEM) and isogeometric elements utilizing non-uniform rational B-splines (NURBS, N-FEM) are discussed in this paper. By solving a two-dimensional benchmark problem, their advantages and drawbacks with respect to Lamb wave propagation are highlighted. The results of the convergence studies are then used to derive guidelines for estimating the optimal element size for a given finite element type and polynomial degree template. These findings serve the purpose to determine the optimal mesh configuration a priori and thus, save a considerable amount of computational effort. The proposed guideline is then tested on a three-dimensional structure with a conical hole showing an excellent agreement with the predicted behaviour.