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In this paper, a receiver model for ultrasonic ray tracing simulation is described. This is a complementary part of an existing simulation model and is the next step towards a numerical solution to the inverse problem and thus a NDT methodology for characterization of the dendrite orientation in a weld. The establishment of the receiver model is based on the electromechanical reciprocity principle. A concise retrospect of the weld model and the 2D model is made. The reciprocity principle is applied in an original way to handle the model problem including the back wall. Experimental qualitative validations for both P and SV waves on a specific weld are also made for C-scans included in this paper. Two different cases are studied. The first is a direct incidence of an ultrasonic ray towards the weld, and the second is a reflection from the back surface in the base material followed by an incidence to the weld. Even though mode-converted rays are excluded in the simulations, both the P and SV probe-models show the same behavior as the experimental results. The qualitative validation though reveals that it even if a thorough time-gating of received information would enable exclusion of mode-conversion in the model, inaccuracy of experimental results is affecting the evaluation of the weld model.

In-service inspection of components that includes welds in austenitic stainless steel and Inconel metal has revealed systematic faults that are due to unpredictable paths of the ultrasound in the welded material. These welds exhibit not only highly anisotropic behavior but also involve inhomogeneous ultrasonic properties. This is caused by the solidification process and the orientation of the dendrites (i.e. large grain structures) is governed by the temperature gradient in the cooling process.

Modeling of ultrasonic non-destructive testing (NDT) is important and helpful e.g. in predicting the response of an NDT inspection, correctly analyzing output data or acquiring a medium’s mechanical properties. Many endeavors have been made to simulate this process. Among them, some special efforts are made to ultrasound propagation through an anisotropic weld [

This paper is a part of an initiative to develop a methodology that estimates grains’ orientations in an anisotropic weld by using ultrasonic information in an inverse scheme. Well defined anisotropy in the simulated volume is prerequisite in order to make simulations of the forward problem, i.e. ultrasonic inspection of an anisotropic weld. The framework of such an initiative consists of a weld model, a forward 2D ray tracing model, experiments and an inverse problem solver. The latter justifies the limitations in the simplified model presented in this paper (e.g. 2D model, no mode conversion and ray tracing). The assumption of the weld being two dimensional and transversely isotropic is often used [

Another 2D ray tracing model was recently validated by using EFIT calculations [

In the present paper, a receiver model is presented, which is a continuation of a previously developed forward ray tracing model [

The forward model is composed of four main elements: a weld model, a transmitter model, a 2D ray tracing algorithm and a receiver model. In a previous paper [

The weld model is established by studying the crystalline structure of a V-butt weld. The prototype is a weld specimen provided by Swedish Qualification Center (SQC) and defined as weld B27 in [

The model of the transmitter is created from a truncated traction distribution representing the pressure produced by the probe on the surface of the component [

where

A truncation of the corresponding traction distribution along the upper surface of the half-space is performed, which provides

A weld model is generated by studying the macrograph of a typical weld

The transmitter model is generated from a presumed plane wave propagating in an isotropic half-space

where

If Fourier transform in

In this expression,

Hence, Equation (3) is taken as the model of the ultrasonic transmitter.

A simple 2D ray tracing algorithm has also been developed. The ray direction, which is also the ultrasonic energy direction, is derived from the relationship between the phase velocity and the energy velocity (the energy velocity must always be normal to the slowness surface). Since the group velocity and the energy velocity are identical for acoustic waves in a lossless medium, an expression can be utilized to calculate the energy velocity [

with

In the modeling process, the ultrasonic testing is assumed to be performed with a transmitter-receiver structure of pitch-catch configuration. According to Auld’s reciprocity principle, the response of the receiver model can be expressed by the change of the electrical transmission coefficient between two states that correspond to the situations with or without a scatterer. Hence, the response of the ultrasonic receiver model can be calculated as

here,

For the receiver model in this paper, state 1 is chosen as the actual testing situation. As shown in the left part of

Illustrations of state 1 and 2 in the reciprocity model

received signal is to determine the displacement

Since the weld’s upper border

On the side of incidence, two different cases are considered in state 1. The first case is that ultrasound impinges directly on the boundary

Let us assume a

As in Equation (1), A is the displacement amplitude of the

where

Reflected fields on the boundary

Since neither mode conversion, viscous damping or any coarseness of the weld are included in the model at this stage, the intention with the validation is to identify necessary modifications in the experimental procedure or essential limitations in selecting weld, when the inverse problem in a later stage is to be addressed. The validation only intends to qualitatively validate the variation in anisotropic properties in the welding direction, in an ultrasonic pitch-catch perspective.

The experimental part of this project was performed by the Swedish Qualification Center (SQC) according to instruction (i.e. procedure) developed in collaboration with Chalmers. The purpose to collect data from real inspection objects with material structure defined by the welding specifications is to compare experimental data with theoretically calculated values. The three-dimensional welded volume is piecewise modeled by two dimensional line scans with variation of the predefined anisotropic orientation in each individual scan.

Collection of data has been performed by keeping one probe on a fixed distance from the weld centre line and moving the receiving probe perpendicular to the weld in line with the other probe (see

The output of the ultrasonic NDT system, expressed by the change in the electrical transmission coefficient, is calculated according to Equation (5), but in a discrete form on each boundary. The expression in Equation (5) is approximated by

In the calculations, 100 discrete points are preset evenly along the fusion lines

where

A picture of the experimental setup with the transmitter and re- ceiver in a tandem configuration

is referred to. It says that, for a circular piston oscillator, the divergence of the beam

In this expression,

In this paper’s calculation,

Two different groups of simulations are performed for the same weld model, one for the _{1} axis), respectively. For the SV wave,

Experimental results of ultrasonic C-scan are displayed in

. Experimental parameters on a specific weld

Wave type | Probe angle (˚) | Probe frequency (MHz) | Transmitter position 1 (mm) | Transmitter position 2 (mm) | Transmitter position 3 (mm) |
---|---|---|---|---|---|

P | 60 | 2.25 | −18 | −33 | −67 |

SV | 45 | 1 | −18 | −24 | −36 |

The relationship of the coordinate systems used in the simulation and the experiment

Simulation result of the P wave (probe position is −18 mm). (a) Ray tracing plot; (b) Receiver model output 1; (c) Receiver model output 2

Experimental result of the P wave (probe position is −18 mm)

Experimental result of the P wave (probe position is −33 mm). (a) Ray tracing plot; (b) Receiver model output 1; (c) Receiver model output 2

Experimental result of the P wave (probe position is −33 mm)

Simulation result of the P wave (probe position is −67 mm). (a) Ray tracing plot; (b) Receiver model output 1; (c) Receiver model output 2

Experimental result of the P wave (probe position is −67 mm)

Simulation result of the SV wave (probe position is −18 mm). (a) Ray tracing plot; (b) Receiver model output 1; (c) Receiver model output 2

Experimental result of the SV wave (probe position is −18 mm)

Simulation result of the SV wave (probe position is −24 mm). (a) Ray tracing plot; (b) Receiver model output 1; (c) Receiver model output 2

Experimental result of the SV wave (probe position is −24 mm)

Simulation result of the SV wave (probe position is −36 mm). (a) Ray tracing plot; (b) Receiver model output 1; (c) Receiver model output 2

Experimental result of the SV wave (probe position is −36 mm)

each figure of the experimental result. The scales of the abscissa on the first line are all negative, from about −30 mm to about −80 mm. For the sake of making a better comparison between the simulation results and the experimental outputs, a coordinate transformation is executed as a post processing. With this coordinate transformation, the coordinate of the receiver is switched to be associated with the weld center, rather than the transmitter. This is realized by

When considering the simulation results, it is observed from the subplots (a) that the P wave is less affected by the weld than the SV wave is. For the P wave, all rays can reach the upper surface of the weld model, while some of the SV rays terminate in the weld without further transmission. In addition, some of the SV rays behave irregularly when propagating through the weld model. This phenomenon implies that SV waves are more sensitive to the grains’ crystal orientations, which coincides with common knowledge of the SV wave propagation in an anisotropic medium. On the other hand, this indicates that the weld model has more influence on the SV wave propagation than on corresponding P wave. Therefore, the partition of the weld, as well as the setup of the boundaries between sub-regions is essential for a successful simulation, especially for the SV wave. In addition, mode conversion is not considered in the modeling, which also possibly makes the number of transmitted SV waves too few.

The receiver model presents the distribution of the signal in subplots (b) and (c). A simulated C-scan plot is implemented by the adoption of randomness in the modeled grain orientations. It is noticed that the distribution of the receiver model’s output does not agree with that of the ray tracing plot perfectly. In

When considering the experimental outputs, it is found that since the scans only cover an interval of 20 mm (from 30 mm to 50 mm) along the weld, marked difference among the sections of a C-scan cannot be noticed for the P wave results. But for the SV wave results, difference among the sections of a C-scan can be observed, e.g. in

If the receiver model’s output 2 and the C-scan plot of the experimental result are compared, the result is not very satisfying. Even if only the maximum response position is considered, there is a difference between the simulation result and the experimental result. For the P wave experiments, in

A receiver model for a 2D ultrasonic ray tracing program is proposed in this paper. It is based on Auld’s electromechanical reciprocity principle. Two different states are employed in the calculation. The first state represents the actual testing situation. The “scatterer” is then present and the transmitter works at the same position as in the actual measurement. For the second state, no “scatterer” is present and the transmitting probe is positioned on the other side of the weld (receiving side). Numerical calculations are performed for both P and SV waves. In each case, three different transmitter positions are used. The distribution of the detected signal then is presented by the receiver model. Simulation results are compared with the C-scan plots from the experiments. In some of these cases there are obvious differentiations. This involves a number of possibilities due to deviations between the idealized mathematical model of the NDT inspection situation and the actual experimental data collected by conventional equipment in an industrial environment. Beside the previous mentioned exclusion of viscous damping in the model also the simplification of the texture in the weld could be part of the explanation. Another plausible cause could be inaccuracies in the collection procedure of experimental data (described in [

A point achieved from the validating process and the comparison is that the simulation presents the maximum received signal at a point, while the experiments present the maximum output over an area. Hence, the resolution of the experimental results is vital to the evaluation of the later inversion calculation. How to obtain a reliable result from the experiments and then apply it to the comparison with the forward simulation result is a new challenge.

This project is financed by the Swedish Qualification Center (SQC). Kjell Högberg, Gunnar Werner and Jeanette Gustafsson from SQC provided great help in the experiments. This is gratefully acknowledged.