Session: 01-05-01: Congress-Wide Symposium on NDE & SHM: Computational Nondestructive Evaluation and Structural Health Monitoring

Paper Number: 141034

141034 - Modal Excitation Analysis for Interior and Surface-Breaking Cracks in Two-Dimensional Half Plane Using Boundary Element and Perturbation Methods 

Ultrasonic measurement for non-destructive testing is often modeled as wave scattering in a (semi-)infinite domain containing defects. This is because defect estimation is typically based on received waves over a finite time duration, excluding reflected waves from distant boundaries such as material surfaces. In this (semi-)infinite domain, wave scattering by defects is treated by considering the radiation conditions of elastic waves. Due to energy dissipation resulting from these radiation conditions, real-valued eigenfrequencies generally do not exist, except in cases where energy is perfectly trapped. These complex-valued eigenfrequencies can be physically interpreted as resonances with radiation attenuation. Understanding these resonances induced by complex-valued eigenfrequencies is crucial for ultrasonic imaging using local defect resonance and improving the signal-to-noise ratio in ultrasonic measurements. From this perspective, the present study conducts an excitation analysis of resonances with complex-valued eigenfrequencies induced by real-valued excitation frequencies, as these frequencies are realistic in measurement scenarios.

The orthogonality, normalization, and completeness of eigenfunctions (eigenvectors) for closed domains are well-established in vibration theory. Consequently, the normal mode expansion is often utilized for excitation analysis in closed domains to examine the relation between excitation and modal amplitude. However, research on the orthogonality, normalization, and completeness of eigenfunctions for open domains is ongoing. Although normalization and orthogonality have been achieved using a Perfectly Matched Layer (PML) in conventional research, unphysical eigenmodes arise due to the PML. Complete sets of physically meaningful eigenfunctions (i.e., without PML modes) have been constructed only for certain cases in conventional research. On the other hand, excitation analysis for eigenfrequencies using perturbation methods has been conducted for closed domains employing the Taylor expansion. Although orthogonality relations are necessary for excitation analysis involving multiple eigenmodes, it appears feasible to analyze excitation for isolated single eigenfrequencies without orthogonality. Hence, the present study proposes a simple excitation analysis method for open domains based on the perturbation method.

The eigenfrequency problem for an open domain must be solved for excitation analysis. In this problem, the amplitude of eigenfunctions tends to infinity as distance increases from scatterers in the spatial domain. Consequently, domain-type numerical methods such as the finite element method or finite difference method require absorbing boundaries (e.g., PML). Since the PML introduces non-physical eigenmodes, as mentioned earlier, identifying physically meaningful eigenmodes is necessary for practical application. Conversely, integral equation methods such as the boundary element method do not introduce such non-physical eigenmodes because the fundamental solution (or Green's function) exactly satisfies the radiation condition. Therefore, the boundary element method and the Sakurai-Sugiura method, one of the contour integral methods for nonlinear eigenvalue problems, are used for present eigenfrequency analysis. As a fundamental examination, this paper addresses wave scattering by interior and surface-breaking cracks in a two-dimensional half plane, as these problems do not involve fictitious eigenfrequency issues, which are well-known in boundary integral equations for crack problems. Several numerical results demonstrate that the present perturbation solutions well approximate modal amplitudes with real-valued frequency excitation.

Presenting Author: Taizo Maruyama Tokyo Institute of Technology

Presenting Author Biography: Taizo Maruyama is an associate professor in the Department of Civil and Environmental Engineering, School of Environment and Society, Tokyo Institute of Technology, Japan. He received his doctorate from the Graduate School of Information Science and Engineering, Tokyo Institute of Technology in 2016. His major research interests are in computational mechanics and nondestructive evaluation. He is mainly engaged in numerical modeling of wave scattering for ultrasonic nondestructive testing.

Authors:

Taizo Maruyama Tokyo Institute of Technology
Naohiro Sugita Tokyo Institute of Technology