Elastic wave scattering from randomly rough surfaces

Abstract

Elastic wave scattering from randomly rough surfaces and a smooth surface are essentially different. For ultrasonic nondestructive evaluation (NDE) the scattering from defects with smooth surfaces has been extensively studied, providing fundamental building blocks for the current inspection techniques. However, all realistic surfaces are rough and the roughness exists in two dimensions. It is thus very important to understand the rough surface scattering mechanism, which would give insight for practical inspections. Knowledge of the stochastics of scattering for different rough surfaces would also allow the detectability of candidate rough defects to be anticipated. Hence the main motivation of this thesis is to model and study the effect of surface roughness on the scattering field, with focus on elastic waves. The main content of this thesis can be divided into three contributions.

First of all, an accurate numerical method with high efficiency is developed in the time domain, for computing the scattered waves from obstacles with arbitrary shapes. It offers an exact solution which covers scenarios where approximation-based algorithms fail. The method is based on the hybrid idea to combine the finite element (FE) and boundary integral (BI) methods. The new method efficiently couples the FE equations and the boundary integral formulae for solving the transient scattering problems in both near and far fields, which is implemented completely in the time domain. Several numerical examples are demonstrated and sufficiently high accuracy is achieved with different defects. It enables the possibility for Monte Carlo simulations of the elastic wave scattering from randomly rough surfaces in both 2D and 3D.

The second contribution relates to applying the developed numerical method to evaluate the widely used Kirchhoff approximation (KA) for rough surface scattering. KA is a high-frequency approximation which limits the use of the theory for certain ranges of roughness and incidence/scattering angles. The region of validity for elastic KA is carefully examined for both 1D and 2D random surfaces with Gaussian spectra. Monte Carlo simulations are run and the expected scattering intensity is compared with that calculated by the accurate numerical method. An empirical rule regarding surface parameters and angles is summarized to establish the valid region of both 2D and 3D KA. In addition, it is found that for 3D scattering problems, the rule of validity becomes stricter than that in 2D.

After knowing the region of validity, KA is applied to investigate how the surface roughness affects the statistical properties of scattered waves. An elastodynamic Kirchhoff theory particularly for the statistics of the diffused field is developed with slope approximations for the first time. It provides an analytical expression to rapidly predict the expected angular distribution of the scattering intensity, or the scattering pattern, for different combinations of the incidence/scattering wave modes. The developed theory is verified by comparison with numerical Monte Carlo simulations, and further validated by the experiment with phased arrays. In particular the derived formulae are utilized to study the effects of the surface roughness on the mode conversion and the 2D roughness caused depolarization, which lead to unique scattering patterns for different wave modes.