A Hybrid Spectral/Finite Element Method for Accurate and Efficient Modelling of Crack-Induced Contact Acoustic Nonlinearity

We propose a 2D hybrid spectral/finite element scheme for numerically resolving crack-induced contact acoustic nonlinearity in solid structures. While the high-order spectral element method (SEM) is much more efficient than the classical low-order finite element method (FEM), the fact that spectral elements are relatively large renders the SEM ineffective in discretizing microscopic cracks. The classical FEM, on the other hand, is good at modelling complex geometries. However, the computation of large-scale structures by the classical FEM can be extremely expensive. In this work, the coupling between high-order spectral elements and low-order finite elements is formulated by the Lagrange multipliers method. The nonlinear contact between crack surfaces is modelled using a penalty method. The aim of the proposed hybrid method is to simultaneously reduce the degrees of freedom (DOFs) in systems and ensure high numerical accuracy. A comprehensive mesh convergence study has been conducted to demonstrate the high convergence rate of the proposed method in modelling ultrasonic wave propagation, and its capability to converge for weak crack-induced contact acoustic nonlinearity. Through a series of numerical experiments on generation and propagation of nonlinear ultrasonic wave modes, the proposed method has been shown to possess a high accuracy and geometric flexibility, and require significantly less computational resources than the classical FEM. The proposed hybrid method will provide an efficient, accurate and robust numerical approach for studying crack-induced contact acoustic nonlinearity which nowadays is being widely exploited by non-destructive testing applications.