Moment Fitted Cut Spectral Elements for Explicit Simulation of Guided Waves in Damaged Media

Abstract

In nondestructive evaluation with guided waves, mechanical models of a damaged structure or component can be exploited in conjunction with an inverse solver to effectively localize and identify faults. The inherent physical complexity of a wave interacting with damage, along with the algorithmic complexity of an inverse problem, call for the accurate modeling of this phenomena, albeit with the necessity of reduced computational cost. The Spectral Cell Method has been shown to provide a set of qualities that are well suited in this context. Those lie in exploiting the properties of time-domain spectral elements, such as the availability of high order polynomials, and an optimally lumped mass matrix; while introducing additional domain features in the form of voids by means of hierarchical meshes and/or the Level Set method. While elements contained by the voids are penalized or discarded, the ones that are ``cut'' by the boundaries of such features require special integration rules, which often lead to the mass matrix not being diagonal. In this contribution, we propose an improved method aimed at preserving a diagonal mass matrix in the presence of voids. It consists in integrating monomials of the basis functions over the remaining portion of the element, and fitting the integration weights at the element's Gauss-Lobatto nodes. For this procedure, the moment fitting equations are transformed into a quadratic programming problem, enabling the application physically sound boundary conditions, thus avoiding the emergence of zero or negative diagonal coefficients in the resulting mass matrix. With this strategy, the decrease in the critical time step experienced by cut elements can be addressed with the use of a local time stepping solver, efficiently restricting the use of a finer time step to selected regions of the domain. The novel approach is firstly benchmarked against the analytical solution of a 2D plane beam with a straight cut, followed by a demonstration on the 3D modeling of an aluminium component with rivet holes. By overcoming the burden of explicitly meshing minor details, the choice of the element size (and thus the critical time step) can be tailored to the relevant frequency of the excitation, resulting in considerable speedup.