Introduction

UNDE tests involve ultrasonic phenomena within regions spanning several hundreds of wave- lengths. At the same time, numerical methods for the exact modeling of ultrasound (as opposed to approximation methods) require discretization of the computational region on length-scales of the order of one-tenth of a wavelength for proper resolution of the ultrasonic fields. When such numerical methods are applied to model a UNDE test, a large number of discretization variables are required due to the relatively large size of the computational domain. Most UNDE models, therefore, treat different physical processes such as wave propagation, scattering, etc. separately, which allows the modeling of some or all of these processes via exact analytical or approximation methods, making the simulation effort tractable. Such composite models typically comprise three sub-models, including (a) an ultrasonic beam model to describe the propagation of ultrasonic waves,

(b) a scattering model to describe the scattering of ultrasonic fields by the defects, and (c) a model for electro-mechanical transduction.

The Kirchhoff approximation (KA) is a scattering model based on a ray-like approximation of waves. Similar to the geometrical optics and geometrical elastodynamics models, it works best in the high-frequency limit. Being an approximation method, it is computationally very efficient. Specifically, in terms of the computational time and memory usage, it outperforms full-wave nu- merical methods such as the finite element method (FEM) and the BEM by several orders of magnitude. Consequently, it is one of the widely used scattering models. Also, as seen from the survey of simulation softwares presented in Section 1.4, it is currently the only method that can handle arbitrary shaped 3D defects within UNDE simulation packages.

Due to the ray-like approximation, the KA treats specular reflections from flat surfaces accu- rately. Its accuracy decreases as the radius of curvature of the scattering surface becomes smaller and comparable to the wavelength. Notably, it does not model edge diffractions accurately. Further, secondary reflections and surface waves on defects are not considered in this method [22, Chap. 10]. Although the foregoing limitations of the KA are well-studied and understood, the application of the KA in UNDE measurement models raises a new question: as the scattering model is only a part of the UNDE measurement model, one might ask how the limitations of the KA affect the overall accuracy of the measurement model under various test scenarios. Clearly, the answer would depend on how sensitive the output of the measurement is to the process of scattering.

In this chapter, we look at three different measurement outputs. The accuracy of the KA in predicting these three quantities is studied by comparison with experiments under different testing scenarios. First output is the time-domain waveform of the receiver voltage. As mentioned in the previous chapter, this waveform is indeed the quantity that is represented in both A- and B-scans. However, in the C-scan and many other imaging methods, the output of interest is the maximum value of the amplitude of the receiver voltage pulse. This is the second quantity we consider. The third quantity is the far-field scattering spectra of the defect, which is obtained by deconvolving the effects of the transducers and electronics in the system. This parameter is commonly known as the

scattering amplitude (SA). For defects that are not relatively small, it is not possible to separate the effects of beam spreading and scattering. Therefore, a quantity analogous to the SA, but which includes the effects of beam spreading, is considered for relatively large defects.

One may expect to find the KA inadequate in predicting the measurements that are sensi- tive to the physical processes not fully represented in it. Diffraction from cracks, surface waves, secondary reflections, etc. are some examples of such processes. We try to confirm these expec- tations quantitatively by comparing predictions from KA-based models with measurements. Since any discrepancies observed in the model predictions may include the contributions of experimen- tal uncertainties, such a comparison will not indicate how well exact scattering models perform over the KA. Hence, a different set of results are obtained by replacing the KA with a full-wave scattering model while keeping all other elements in the composite model the same. A full-wave model is defined as any model that solves the complete wave equations. The particular full-wave scattering model considered here is based on the solution of a boundary integral equation (BIE) via the Nystr¨om method (NM), which is a boundary-element-type numerical method [114,115]. Com- parison of the NM-based model predictions with the measurement data will indicate the potential performance-gains of full-wave models over the KA.

Some studies have been conducted previously for the experimental verification of UNDE mea- surement models [27,47,63,116–120]. In [27,63,116–119], UNDE models were applied to simulate some benchmark problems proposed by the World Federation of NDE Centers (WFNDEC). Experi- mental data for these problems were also provided by the WFNDEC. The scattering models applied in these studies include the KA and full-wave analytical methods. As only few scattering problems admit exact analytical solutions, complex defects were treated only using the KA in these studies. Predictions of full-wave scattering models known as MOOT and T-matrix method were compared with measured data in [47]. A scattering model based on the BEM was used for the prediction of experimental pulse-echo signals from spherical voids or inclusions in [120]. Both [47, 120] were restricted to defects of simple shapes only.

In this chapter, composite UNDE models based on the KA and the NM are applied to the WFNDEC benchmark problems from the years 2004 and 2005. These problems include various pulse-echo immersion tests with standard scatterers such as flat bottomed holes, spherical voids, etc. in solids. Only echos resulting from longitudinal waves were considered in the simulations, although the simulation model has no restrictions on the type of wave modes. Experimental data for the benchmark problems were obtained from the WFNDEC archives [121]. The UNDE mea- surement model used in the simulations follows the development in [22]. This model has three quantities that are derived independently of the scattering problem: the system efficiency factor, which characterizes the electro-acoustic transduction process, and the velocity diffraction coeffi- cients of the transmitting and receiving transducers. The system efficiency factor corresponding to every measurement was extracted from a reference measurement performed with the same pulser and receiver settings as used in the original experiment. The velocity diffraction coefficients were calculated either using the multi-Gaussian beam model [22, 122] or by evaluating the Rayleigh- Sommerfeld integral for transducer radiation under the paraxial approximation as described in [13, Chap. 8]. In the following sections of this chapter, a description of the composite UNDE model is given first. Then, the methods employed to compute the elements of the composite model are described, which is followed by a presentation of the simulation results.