NDT.net - March 2003, Vol. 8 No.3 |

(Elec. Eng. Dept., The University of Akron, OH 44325-3904, USA)

(rc17@uakron.edu)

Nathan Ida

(Elec. Eng. Dept., The University of Akron, OH 44325-3904, USA)

The Transmission Line Matrix ( TLM ) is a physical discretization approach used to solve the wave equation numerically. The method replaces a continuous system by a network or an array of lumped elements. The TLM method involves dividing the solution region into a rectangular mesh of the transmission line segments. The nodes of the mesh are the points of discontinuity for acoustic impedances.

A computer program based on transmission-line matrix (TLM) model was developed to simulate the ultrasound propagation media with different acoustic impedances. The numerical model provides both frequency and time domain responses. The influence of variations in the shape of the incident pulse is discussed. The numerical results are compared with those obtained from experiments.

To improve the results obtained in ultrasonic NDT , considerable theoretical effort is involved in developing reliable mathematical models of wave propagation in different media. Due to the complexity of the problems, numerical methods have proven to be an appropriate approach. Among other numerical methods applied in the time domain, the TLM was found to be suitable for complex geometries and various types of input signals. These advantages make this method a suitable method for ultrasonic NDT.

The transmission line method was proposed by Johns (1). The method is a direct numerical implementation of the Huygens principle (2). The wave front at each iteration (instant in time) for a certain point is a result made of the waveforms generated at a neighbouring point in the previous iteration. The TLM is a physical discretization approach and this method does not require the solution of the differential equation. A continuous system is replaced by a network or an array of lumped elements. The TLM requires division of the solution region into a rectangular mesh of the transmission lines. The nodes of the mesh are points of discontinuity for impedances.

To solve a problem using the TLM, a set of boundary conditions and material constituents must be provided. An initial excitation must also be given. Then the impulses are propagated throughout the mesh using scattering theory on transmission lines. There is no limitation regarding the frequency of interest, but the size of the mesh imposes an upper limit on the frequency response analysis.

**TLM algorithm**

Considering Kirchhoff’s laws for a shunt transmission the wave equation for the voltage
can be written (3):

(1) |

Assuming that the transmission line is lossless and nondispersive. The elements of the
circuit (L and C) are chosen to model the propagation in a homogenous infinite space with
acoustic impedance *Z*_{m} :By combining the continuity equation and the equation of motion
for a medium with a uniform density
**r** and compressibility **k**
, the acoustic wave equation
written for pressure *p* has the form (4):

(2) |

Equations (1) and (2) show that the following equivalencies can be enforced:

(3) |

The equivalencies between the current components (*I _{x}*,

(4) |

where [*p*]* ^{r}* and [

(5) |

The positive integer variable k is the number of iterations and represents the number of time steps Dt that have passed since the beginning of the computation.

The presence of a medium *l* is modelled in two ways. The first method is modelled by
modifying the reflection coefficients at the boundary between the two media. The second
way to model the presence of different media in a TLM mesh is to modify the scattering
matrix. The discontinuity in impedances that exists at the interface between two media is
modelled by incorporating an open circuit stub of length
*Dl*/2 (5). This implies introducing
a supplementary leg that will match impedances of two different layers. For this case
equation (5) becomes:

(6) |

The pressure at node (i,j) at iteration k is found as:

(7) |

Equation (7) shows that the method discussed above is equivalent to the finite difference
time domain (FD-TD) method with respect to the final result, but the TLM does not require
explicitly finding the solution for the wave equation (2). The equivalence between
admittances *Y _{l}* and

(8) |

Attenuation of the ultrasonic pulse in medium *l* is modelled by assuming that the losses are
distributed continuously. Therefore the amplitude of the reflected wave is attenuated by the
attenuation factor
a along the mesh lines. This was implemented in the TLM code using
the relation:

(9) |

Wave propagation is modelled by the so called connection process. In this process, the
reflected pulse for a certain node at time *(k+1)Dt* becomes the incident pulse for
neighbouring nodes at the same time *(k+1)Dt*.

Equations (6) - (9) describe the wave propagation at any coordinate (x,y) at any instant *kDt*.
To initiate the process an input energy needs to be provided. This energy is called
excitation. Due to the equivalence with a microwave network the initial conditions of the
problem are modelled by voltage sources that can be placed at any node. A TLM structure
can be excited at any location with practically any kind of excitation. The physical problem
determines the locations and the type of excitation. It is possible for a continuous
waveform to be excited at appropriate input nodes. This excitation is implemented by
keeping, for all iterations, the voltages for input points at pre-set values. When the
characteristics of a structure have to be investigated over a wide frequency range a single
localised pulse is used. Initially, the voltage amplitudes at all nodes are set to zero except at
the selected input point. An impulse is then applied there. The minimum time interval for a
pulse is where *v* is the phase velocity in the mesh. To minimise the effect of dispersion
the minimum distance between the nodes,
*Dl*, is related to the smallest wavelength
l_{min} of
interest by the following relationship (7):

(10) |

For the purpose of the present discussion a sinusoidal Gauss-modulated pulse is considered:

(11) |

The output impulse function at a particular point is obtained by summing the total node
voltage at *k* for all iteration. Mathematically this is written as a sum over the total number
of iterations *Nit*:

(12) |

Similarly, the excitation function can be written as:

(13) |

For example, for a sinusoidal excitation at frequency *f*, such as that given in equation (11),
the coefficients *a(i,j)* at iteration *k* are:

(14) |

The frequency response, *H ( jw)* = *H ( j2pf)*, is obtained as the Fourier Transform of the time response function given by equation. (14):

(15) |

The concepts of the TLM algorithm presented in previous section are demonstrated in figure 1. A Gaussian pulse was launched in a 101 by 101 mesh modelling free space. The interaction of this wave with a perfect reflecting square is shown for two iterations. The first iteration is when the wave front touches the square ( iteration 39). The second iteration was chosen when the wave front reaches the boundary of mesh ( iteration 60).

Fig 1: The simulated waveforms for a Gaussian pulse propagating in 101x 101 mesh, containning a perfect reflecting square. Two instances are shown: a) iteration 39 ( when the waveform touches the corner placed in position (70,70); b) iteration 60. |

The ultrasonic images for this work were obtained in two different experimental arrangements in the same structure (8) : a data acquisition board with a high sampling rate : 80 MHz (128 MHz ) maximum amplitude of 400V and 9ns rise time; a mechanical system based on a computer controlled stepper motor allowing achievement of ± 0.05mm resolution.

A multilayered structure made of two plastic sheets ( 0.5mm thickness) with a water gap
between them was investigated. The ultrasonic image of the structure is shown in figure 2.
The same experiment was reproduced numerically. A sinusoidal Gaussian pulse was
injected in the TLM mesh according to equation (11). The parameters that were used to
generate this pulse are: *A*= 400V;
s = 6.7; *t _{o}* = 0.3ms;

Fig 2: The ultrasonic image of the water-plastic-water-plastic-water structure investigated. |

The comparaison between the numerically generated signal ( continue line) and the reflected pulse (dashed line) obtained from structure under investigation is shown in figure 3. As shown in figure 3 TLM model reproduced the real pulse all boundaries have been solved corectly.

Fig 3: TLM reflected pulse from a multilayer structure: water-plastic-water- plastic-water. |

A metallic multilayer bearing containing a disbound was considered next. This structure is
made of a steel base and an antifriction layer ( 6 mm thickness) . The ultrasonic image was
obtained in immersion using and emitter-receiver transducer ( 5MHz). The TLM model
consisted of a 1000 by 1000 mesh in which a Gaussian pulse with the following
parameters was used for excitation: *A*= 400V;
s = 7.7; *t _{o}* = 0.2ms;

Fig 4: Comaparison between frequency response obtained from experiment and the numerically generated response. The structure under investigation was a metallic multilayer bearing containing a disbound. |

A numerical model for ultrasonic wave propagation in layered media is proposed. The model is based on the TLM algorithm. The model proposed was implemented in a FORTRAN program. The results shown in this paper demonstrate that the model can be applied to characterisation of the flaw in multilayered structure. Samples with different acoustic impedance profiles have been investigated. Comparison between numerically generated signals and real ultrasonic signals validate the proposed model.

- P. B. Johns, R. L. Beurle, “ Numerical solution of Two-Dimensional Scattering
Problems Using a Transmission-Line Matrix”,
*Proc. IEEE*, vol**118**, pp. 1203-1209, 1971. - Y. Kagawa, T. Tsuchiya, B. Fujii, and K. Fujioka, “ Discrete Huygens ‘ model
approach to sound wave propagation”,
*Journal of sound and vibration***218**(3), pp. 419-444, 1998. - N. Ida,
*Engineering Electromagnetics*, Springer , pp.890-892, 2000. - P.M. Morse, K.U. Ingard,
*Theoretical acoustics*, McGraw-Hill, pp. 242-243, 1968. - C. Christopoulos,
*The Transmission-Line Modeling Method*, IEEE Press, 1995. - A.H.M. Saleh, P. Blanchfield, “Analysis of acoustic radiation patterns of array
transducers using the TLM Method”
*Int. Journ.Numer. Model.*, .**3**,pp. 39-56 1990. - W.J.R. Hoefer, P.M. So,
*The electromagnetic wave simulator*, John Wiley & Sons, pp. 87-99 1993. - R. Ciocan, M. Soare, V. Revenco, “The quality evaluation of the end-plate welds and
brazed joints for CANDU nuclear fuel by an ultrasonic imaging method”,
Insight-Non-Destructive Testing and Condition Monitoring, vol.
**39**. vol.39. no.9 pp. 622-625, 1997

© NDT.net - info@ndt.net | |Top| |