Direct Numerical Code Research Articles

A New Approach for a Class of the Blasius Problem via a Transformation and Adomian’s Method

The main feature of the boundary layer flow problems is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the series methods when applied to solve such problems. To the best of the authors’ knowledge, two procedures were used extensively in the past two decades to deal with the boundary conditions at infinity, either the Padé approximation or the direct numerical codes. However, an intensive work is needed to perform the calculations using the Padé technique. Regarding this point, a new idea is proposed in this paper. The idea is based on transforming the unbounded domain into a bounded one by the help of a transformation. Accordingly, the original differential equation is transformed into a singular differential equation with classical boundary conditions. The current approach is applied to solve a class of the Blasius problem and a special class of the Falkner-Skan problem via an improved version of Adomian’s method (Ebaid, 2011). In addition, the numerical results obtained by using the proposed technique are compared with the other published solutions, where good agreement has been achieved. The main characteristic of the present approach is the avoidance of the Padé approximation to deal with the infinity boundary conditions.

The high-frequency description of scatter of a plane compressional wave by an elliptic crack

High-frequency approximations that can be interpreted in terms of the Uniform Geometrical Theory of Diffraction (UGTD) and Uniform Kirchhoff Approximation (UKA) are used to develop a code for modeling ultrasonic scatter of a plane compressional wave by an elliptic crack in the radiating near field. The approximations are intercompared and partially validated against a direct numerical code based on an FD (Finite-Difference) scheme. At present, in many realistic situations the approximate codes of the type described here offer the only viable simulation tool; purely numeric codes are not only much slower, they still require too much computer memory to simulate the complex structure of the radiating near fields.

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 Green9s 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.

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.

Far–field asymptotics of the Green's tensor for a transversely isotropic solid

The dynamic time–harmonic Green's tensor for 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 parts: quasi–longitudinal (P), shear–horizontal (SH) and quasi–shear (SV). Then, an exact analytical solution for the SH part in terms of elementary functions is derived. The complete far–field asymptotic approximation of P and SV parts 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. At the end of the paper numerical results in both frequency and time domain are presented. They show that the agreement between the outputs of the asymptotic and direct numerical codes is very good throughout all regions but the former can be orders of magnitude faster.

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 practically just as accurate except that the phenomena related to the head waves are not described. Formulas pertaining to the far field are also offered.