Development, Validation and Comparison of Higher Order Finite Element Approaches to Compute the Propagation of Lamb Waves Efficiently

When considering structural health monitoring (SHM) applications efficient and powerful numerical methods are required to predict the behavior of ultrasonic guided waves and to design SHM systems. The existing commercial explicit finite element analysis tools based on standard linear displacement elements quickly reach their limits when applied to ultrasonic waves in thin plates, so called Lamb waves. It is known that the required temporal and spatial resolution causes enormous computational costs. One resort to overcome this problem is the application of special finite elements utilizing higher order polynomial shape functions (p > 2). The current paper is focused on the development of such higher order finite elements and the verification of their accuracy and performance. In the paper we compare and evaluate the capabilities of spectral finite elements, p-version finite elements and isogeometric finite elements. Their advantages and disadvantages with respect to ultrasonic wave propagation problems are discussed and their properties are demonstrated by solving a benchmark problem. Higher order finite elements with varying polynomial degrees in longitudinal and transversal direction (anisotropic ansatz space) are investigated and convergence studies are performed. The results of the convergence studies are summarized and a guideline to estimate the optimal discretization is prepared. If the required accuracy is known, the proposed guideline provides a helpful means to determine both the element size and the polynomial degree template for a given model.