The Finite Cell Method: A Higher Order Fictitious Domain Approach for Wave Propagation Analysis in Heterogeneous Structures

In this chapter a recently developed novel approach to simulate the propagation of ultrasonic guided waves in heterogeneous, especially cellular lightweight structures is presented. One of the most important drawbacks of traditional finite element-based approaches is the need for a geometry-conforming discretization. It is generally acknowledged that the mesh generation process constitutes the bottleneck in the current simulation pipeline. Therefore, different measures have been taken to at least alleviate the meshing bruden. One of these attempts is the finite cell method (FCM). It combines the advantages known from higher order finite element methods (FEM; exponential convergence rates) with those of fictitious domain methods (FDM; automatic mesh generation using Cartesian grids). In the framework of the FCM we do not rely on body-fitted discretizations and therefore shift the effort typically required for the mesh generation to the numerical integration of the system matrices which is performed by means of an adaptive Gaussian quadrature. The advantage of such a procedure in the context of wave propagation analysis is seen in the fully automated analysis process. Consequently, hardly any user input is required making the simulation very robust. Notwithstanding, it can be mathematically proven that the optimal rates of convergence are retained.