The finite and spectral cell methods for smart structure applications: transient analysis

This article introduces a robust and efficient numerical tool that is well suited for the simulation of ultrasonic guided waves and can be especially helpful when dealing with heterogeneous materials. The proposed method is based on a combination of high-order finite element methods (FEM) and the fictitious domain concept. If hierarchic shape functions, which are familiar from the p-version of the finite element method (p-FEM), are deployed the method is referred to as the finite cell method (FCM). Where Lagrange polynomials through Gau-Lobatto-Legendre points are used, we refer to it as the spectral cell method (SCM). The name, SCM, derives from the fact that the deployed shape functions are commonly utilized in the spectral element method. To model smart structure applications such as shape control problems, noise cancelation devices and the excitation/sensing of ultrasonic guided waves, a coupling between electrical and mechanical variables is also taken into account. In this context, we focus on including the linear theory of piezoelectricity in the variational formulation of the FCM and the SCM, respectively. Several numerical benchmark problems are then used to validate the proposed approach. The simulations show promising results with respect to the accuracy of the method and the computational effort. We observe similar convergence properties for the proposed high-order fictitious domain methods as with "conventional" high-order finite element approaches. Implementing the proposed method in existing finite element software is, moreover, a straightforward process. These properties make the method an efficient tool for practical applications in structural health monitoring problems and smart structure applications in general.