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Simulation of Lamb waves using the spectral cell method

Today a steadily growing interest in on-line monitoring of structures is seen. Commonly referred to as structural health monitoring (SHM), the basic idea of this technique is to decrease the maintenance costs based on a continuous flow of information concerning the state of the structure. With respect to the aeronautic industry increasing the service time of airplanes is another important goal. A popular approach to SHM is to be seen in ultrasonic guided wave based monitoring systems. Since one focus is on typical lightweight materials elastic waves seem to be a viable means to detect delimitations, cracks and material degradation. Due to the complexity of such structures efficient numerical tools are called for. Several studies have shown that linear or quadratic pure displacement finite elements are not appropriate to resolve wave propagation problems. Both the mesh density and the spatial resolution needed to control the numerical dispersion are prohibitively large. Therefore, higher order finite element methods (p-FEM, SEM) are considered by the authors. One important goal is to simulate the propagation of guided ultrasonic waves in carbon/glass fiber reinforced plastics (CFRP, GFRP) or sandwich materials. These materials are typically deployed in aeronautical and aerospace application and feature a complex micro-structure. This micro-structure, however, needs to be resolved in order to capture effects like transmission, reflection and conversion of the different wave modes. It is known that using standard discretization techniques it is almost impossible to mesh the aforementioned heterogeneous materials without accepting enormous computational costs. Therefore, the authors propose to apply the finite cell method (FCM) and extend this approach by using Lagrange shape functions evaluated at a Gauss-Lobatto-Legendre grid. The latter scheme leads to the so called spectral cell method (SCM). Here, the meshing effort is shifted towards an adaptive integration technique used to determine the cell matrices and load vectors. Hence, a rectangular Cartesian grid can be used, even for the most complex structures. The functionality of the proposed approach will be demonstrated by studying the Lamb wave propagation in a two-dimensional plate with a circular hole. The perturbation is not symmetric with respect of the middle plane in order to introduce mode conversion. In the paper, an efficient method to simulate the elastic wave propagation in heterogeneous media utilizing the finite or spectral cell method is presented in detail.