Dmitri Gridin

The high-frequency asymptotic description of pulses radiated by a circular normal transducer into an elastic half-space

A new method for simulating the propagation of pulses radiated by a circular normal ultrasonic transducer which is directly coupled to a homogeneous and isotropic elastic half-space is proposed. Both nonuniform and uniform high-frequency asymptotics inside geometrical regions as well as boundary layers (penumbra, an axial region, and a vicinity of the critical rays) have been used to describe the transient field by means of harmonic synthesis. The nonuniform asymptotic formulas involving elementary or well-known special functions elucidate the physics of the problem and give explicit dependence of the radiated waves upon the model parameters. The formulas are applicable in the radiating near field which is the near-field with the evanescent wave zone excluded. The code based on the uniform asymptotics has been tested in all regions against an exact numerical solution. It is orders of magnitude faster, but in many realistic cases the accuracy does not suffer. The limits of applicability of the model have b...

On the radiation of ultrasound into an isotropic elastic half-space via wavefront expansions of the impulse response

The problem of propagation of pulses in the radiating near zone of a large circular normal transducer directly coupled to a homogeneous and isotropic elastic half-space is re-visited. It is shown that for certain observation angles the impulse response approach is computationally inefficient. A new method based on the so-called wavefront expansions of the impulse response is developed instead. The expansions are obtained by the analytical harmonic synthesis of the high-frequency asymptotics of the transducer field. Unlike these asymptotics the wavefront expansions are expressed in terms of elementary functions only. The direct P, edge P and S waves as well as the transition regions (penumbra and axial region) are described. The uniform asymptotic expansions applicable throughout the radiating near zone are derived as well. The code based on the time convolution of the pressure input function with the wavefront expansions is compared to a direct numerical code. It is thousands of times faster but practical...

The radiating near field of a circular normal transducer of arbitrary apodization on an elastic half-space

A new fast method is developed for simulating the propagation of pulses radiated by a circular normal transducer of arbitrary apodization into an isotropic and homogeneous elastic half-space. First, the model proposed in Fradkin, Kiselev and Krylova for a time-harmonic uniform transducer [J. Acoust. Soc. Am. 104, 1178–1187 (1998)] is extended to the case of nonuniform load to obtain high-frequency asymptotics of the time-harmonic field. Then, the transient field is described by means of harmonic synthesis. The asymptotics elucidate the physics of the problem and give explicit dependence of the radiated waves on model parameters. The formulas are applicable in the radiating near field, that is the near field with the evanescent wave zone excluded. They involve in geometrical regions elementary and inside boundary layers, well-known special functions (Fresnel integral and Bessel functions). Unlike the uniform load case, the direct shear wave is present and the edge waves may be practically eliminated. The asymptotic code has been tested against an exact numerical solution when apodization is parabolic. It has proved to be hundreds of times faster but in many realistic cases the accuracy does not suffer much.

Diffraction coefficients for tilted surface-breaking cracks

Many cracks of practical interest in ultrasonic non-destructive testing (NDT) are surface-breaking (Fig. 1). In order to model inspection of such cracks with the geometrical theory of diffraction (GTD), the diffraction coefficients for surface corners need to be calculated. The corresponding canonical problem is that of the diffraction of a plane wave by the vertex of an elastic wedge of less than 180°.

The complete far–field asymptotic description of a point source acting on a transversely isotropic half–space

The elastic wavefield generated by a point source of tractions acting on the surface of a transversely isotropic half–space is studied. The symmetry axis of the solid is oriented arbitrarily with respect to the surface of the half–space. First, the integral representation of the time–harmonic Greens tensor is given. Then the complete far–field asymptotic approximation of a quasi–longitudinal (qP) and two quasi–shear (qSH and qSV) waves is derived. The qP wave is described by the leading term of the ray series, since there is only one arrival of this wave. The qSH wave is treated similarly everywhere apart from the so–called kissing–point boundary layer, where the qSH and qSV wavefronts are tangentially close to each other. A special asymptotic formula is obtained for this case. The qSV sheet of the wave surface is allowed to have conical points and cuspidal edges. Thus, the far–field approximation of the qSV wave involves ray–asymptotic expressions while inside the geometrical regions (where either one or three qSV arrivals exist), or else boundary–layer asymptotics inside conical–point, cuspidal–edge and kissing–point boundary layers. At the end of the paper we present numerical results of the simulation of pulse propagation. A good agreement between the asymptotic and direct numerical codes is achieved but the former is orders of magnitude faster.

Modeling the quasicompressional wave field of a rectangular transducer in a transversely isotropic solid

A fast model for simulating the transient quasicompressional wave field of a rectangular ultrasonic transducer directly coupled to a transversely isotropic elastic half-space of general orientation is developed. The so-called two-tier asymptotic approach and the uniform stationary phase method are used to derive the high-frequency asymptotics of time-harmonic displacements. Then, transient fields are modeled by means of harmonic synthesis. In geometrical regions, the formulas involve elementary and inside boundary layers, well-known special functions (Fresnel integral and generalized Fresnel integral), and are applicable in the radiating near field. The asymptotics elucidate the physics in terms of various arrivals and give explicit dependence of the radiated waves upon model parameters. The asymptotic code is tested against a direct numerical solution. It is at least a thousand times faster but describes accurately both arrival times and amplitudes of various pulses radiated by the transducer.

Mathematical modelling of ultrasonic non-destructive evaluation

High-frequency asymptotics have been used at our Centre to develop codes for modelling pulse propagation and scattering in the near-field of the ultrasonic transducers used in NDE (Non-Destructive Evaluation), particularly of walls of nuclear reactors. The codes are hundreds of times faster than the direct numerical codes but no less accurate.

Modeling point-source and transducer wavefields in transversely isotropic half-spaces

The elastic wavefields generated by surface loads in a transversely isotropic (TI) half-space of general orientation are studied. First, we consider the field of a point source, that is, Green’s tensor of Lamb’s problem for a TI solid. The complete far-field asymptotics of a quasi-longitudinal (qP) and two quasi-shear (qSH and qSV) waves are derived, including irregular directions. Then we present new computationally efficient high-frequency asymptotic models of the wavefields radiated by rectangular and circular transducers into a TI half-space.

The far-field of a point source in a transversely isotropic elastic solid

The elastic field of a time-harmonic point source acting in a transversely isotropic, homogeneous, linearly elastic solid is studied. First, the representation of the Green’s tensor as an integral over the unit sphere is obtained. It consists of three waves: quasi-longitudinal (P), shear-horizontal (SH) and quasi-shear (SV). Then, an original exact analytical solution for the SH wave in terms of elementary functions is derived. The complete far-field asymptotic approximation of P and SV waves is obtained next, using the uniform stationary phase method. For the P wave it involves the leading term of the ray series since there is only one arrival of this wave. The wave surface for the SV wave contains conical points and cuspidal edges. The asymptotic description applicable near these singular directions is derived involving the Airy and Bessel functions. The directions close to the points of tangential contact of the SH and SV sheets of the wave surface are also treated. Numerical results in both frequency ...

The Radiating Near Field of an Ultrasonic Transducer Directly Coupled to an Isotropic Solid Half-Space

Simulating the propagation of pulses radiated by ultrasonic transducers into an elastic half-space is a first step toward the system model of ultrasonic non-destructive evaluation (NDE) of industrial materials. The case of a time-harmonic transducer acting on the surface of a homogeneous and isotropic elastic half-space has been considered by Miller and Pursey [1]. The solutions were obtained in the integral form using the Hankel transform, and then the far-field asymptotics of the compressional and shear waves were found for the case of a point-like transducer. However, in practice large transducers are used and for the directly coupled large transducers mostly the near field is of interest. Its structure is rather complicated. For this reason, for the past two decades the problem has been intensively studied by various full numerical schemes (e.g. [2,3]). The schemes are time-consuming and do not produce explicit dependence of the radiated waves upon the model parameters. Therefore, a considerable effort has been put into developing analytical and approximate methods instead.