
Ultrasonic Models 





Simultsonic 

An effective twodimensional model based on the ray tracing methods was used to simulate the ultrasonic wave propagation in structures with crack like defects. The package was developed in VC++ operating in Microsoft Windows environment, which can take input parameters defining the experimental conditions for the phased array experimental setup on a pipe model and generates Ascan data. The Ascans can then be exported and stacked to generate the Bscan image. A three cycle Hanning window pulse was chosen for the simulations. The model employs a timestepping scheme to launch rays from the transducer, traces its path to the nearest interface, and generates new rays at the interface satisfying Snell’s law, tracks mode conversion all in a recursive manner.
Each ray is characterized by the position of its head, which is given by the global X and Y. The ray also has direction cosines that describe the direction in which the ray propagates. Each ray is also characterized by its type (whether shear or longitudinal) and its energy. The ray also has information about the amount of time lapsed from the time it was created at the transducer. Ray tracing is carried out in specific time steps and checks are made at every stage to ascertain the ray position in the domain as a function of the time elapsed. Each ray is propagated until it meets an interface or has reached the end of the domain or until the given time. Once a ray meets an interface, new reflected and refracted rays are created based on the angle of incidence of the rays taking into account mode conversion. Generation of the reflected ray or the refracted ray is also governed by the critical angles, evaluated using the sound speeds in the media.






(a) Bscan showing the defect 

Fig. 3: Comparison of Simulated and Experimental Bscans of 7.5 mm crack
in a 14.33mm thick pipe specimen 






References : 1. Krishnamurthy, A., Karthikeyan, S., Krishnamurthy, C.V., and Balasubramaniam, K., “Simultsonic: A Simulation Tool for Ultrasonic Inspection”,
AIP Conf. Proc. 820, 1894 (2006).
2. L. Satyarnarayan, DM. Pukazhendhi, Krishnan Balasubramaniam, C.V.Krishnamurthy, D.S. Ramachandra Murthy, “Ultrasonic Phased
Array
Measurement of Fatigue Crack Growth Profiles in SS Pipes”, ASME International Journal of pressure vessel technology (In Press). 









Point Spread Function for Flaw Sizing 

An explicit Point Spread Function (PSF) evaluator in the frequency domain has been developed for an ultrasonic transducer operating in the pulseecho mode. The PSF evaluator employs the patch element model (PEM) for transducer field determination and scattered field assessment from a small but finite “point” reflector. The PSF for a planar transducer in a medium has been evaluated in the near and the farfield. The computed PSFs’ have been used to deconvolve and restore surface images, obtained experimentally, of a single hole and a five hole cluster in an Al calibration block.
A calibration plot is arrived at for estimating, without the need for deconvolution, the actual diameters of circular reflectors from apparent diameters obtained experimentally for a single medium imaging configuration. The PSF, when the transducer and the point reflector are in two media separated by a planar interface, has been evaluated in the near and farfield. The computed PSFs’ have been used to deconvolve and restore subsurface images, obtained experimentally, of Flat Bottom Holes (FBHs) in an Al calibration block. We show that the PSF, in the presence of a planar interface, can be obtained from a single medium PSF model using an effective single medium path length concept. The PSFs’ and MTFs’ are evaluated for spherical focused and annular transducers and compared with that for the planar transducer. It was also attempted to identify imaging distances to get betterresolved images when using planar, spherical focused and annular transducers.
The PSF computation, like transducer characterization, is a
one time procedure to ascertain the resolving capabilities of a
transducer. Calculation of the PSF requires characterizing the
transducer. Given the transducer parameters, the PSF is evaluated for any specific imaging configuration. Examples of how ultrasound images can be restores and defect sizing can be carried out through deconvolution using the computed PSF with minimal noise have been demonstrated. It is significant to note that the capability of evaluating the PSF helps imaging parameters such as imaging distance, scan step and frequency to be optimized for a variety of apertures and apodizations.
The medium in which the beam is launched and collected is found to determine the PSF even in cases where the flaws need to be imaged across planar interfaces separating two media. In addition to the commonly encountered immersionbased imaging, contact mode imaging with wedge or roller based transducers would also benefit from this feature as it simplifies the PSF computation. The PSF evaluator developed at CNDE is capable of handling array configurations and/or newer imaging configurations that may lead to possibly higher lateral resolutions in the nearfield as will as in the farfield.





Test block with holes. 


Experimental Image obtained 



 K. Balasubramaniam, C.V. Krishnamurthy, Ramsharan R., Ultrasonic Imaging using Computed PSF IEEE trans. Ult. Ferro. Freq. in press (2007)







Finite Difference Time Domain Models 

In FDTD, differential form of equations are modified to centraldifference equations, discritized. The equations are solved in a leapfrog manner, i.e., in the case of wave equation, displacement field at a given instant in time uses the displacement field at the previous time, and the process is repeated over and over again. This equations are coupled first order homogeneous equation and hence it are solved by staggered grid formulation.
The idea behind the PML is to define a set of equations in the PML region that have large attenuation and at the same time, do not produce any reflections from the PML region. The PML region has the same density as the acoustic domain and hence no impedance mismatch occurs between the two layers.
Wave propagation in acoustic medium
The propagation speeds of traveling waves are characteristic of the media in which they travel and are generally not dependent upon the other wave characteristics such as frequency, period, and amplitude. The speed of sound in air and other gases, liquids, and solids is predictable from their density and elastic properties of the media (bulk modulus).
Wave propagation in elastic medium
The FDTD model for the simulation and visualization of the elastic wave propagation is a secondorder accurate algorithm in space and time . It is based on a velocitystress finitedifference method for homogeneous isotropic material. The equation of motion, stress strain relation together with constitutive equations completely describes the elastic wave motion in a homogenous material.















WFNDEC Ultrasonics Bench Mark Problem Solutions 

At CNDE, as part of a large research program, the need to develop a raybased assessment code for simulating UT inspection is being addressed through a algorithm SIMULTSONIC. The code is designed to describe pulse propagation characteristics directly in time domain through a ray picture. It is written in Visual C++ operating in Microsoft Windows environment and provides an interactive user interface.
A raybased simulation tool has been developed in LABVIEW environment to facilitate the simulation of the timeofflightdiffraction (TOFD) based inspection. The tool has been used to optimize design parameters such as the probe separation and the probe angle.
Concurrent to this effort in developing raybased models, CNDE is involved in developing fieldbased models that provide more accurate amplitude and phase information of the ultrasonic signals. A frequencydomain patchelement model (PEM) has been successfully developed to evaluate beam profiles within a single medium or in solids insonified by either an immersion transducer or a contact transducer mounted on a shoe. In addition to handling the beam profiles of focused transducers, the PEM has also been used to model the beam profiles of phased arrays in steered and steered focused modes of operation. The PEM has been used to characterize transducers and their imaging capabilities.
In the timedomain, fieldbased models have been developed using the FDTD and FEM schemes in 1D and 2D. Algorithms that provide efficient absorbing boundary conditions that reduce the computational burden have been developed. Models developed at CNDE are being constantly verified and validated with published results and inhouse experiments. In addition, the benchmark problems defined every year by the WFNDE, of which CNDE is a member, has been providing a forum for verification and validation.





Fig 1. Examples of PEM Calculations at 5 MHz of Radiated Pressure Fields in Water for Various Transducer Shapes




References :1. E. Kannan, B. W. Maxfield and Krishnan Balasubramaniam, “E. Kannan, B. W. Maxfield and Krishnan Balasubramaniam”,
Review of Progress in Quantitative NDE, July 22 – July 27, 2007 Colorado School of Mines, Golden, Colorado, USA. 





Guided Wave Models in multilayered cylinders and rods using Taylor Series Expansion Approximations 

The problem of elastic wave motion in media with coatings of constant thicknesses comparable to (but less than) wavelength was considered. A general algorithm for evaluation of higher order effective boundary conditions (O(hn) model) for problems of elastic wave motion in media with coatings that are homogeneous isotropic and elastic was developed and implemented. Analytical formulation for evaluation of effective boundary conditions using an arbitrary order Taylor power series expansion of field quantities in terms of thickness, h, of the coating layer is presented. The effective boundary conditions appear as preservative terms added to the exact boundary conditions of the uncoated substrate medium.
The effective boundary conditions are approximate, reproducing the effect of the thickness up to any order, depending on the order of the Taylor series used. The formulation assumes a general curvilinear shape of the coatingsubstrate medium interface. An algorithm is designed for easy implementation of the model on a symbolic software. The model was numerically verified against global matrix method for the case of longitudinal guided waves in rods with thin alternative coatings and thicker nonalternative coatings using different orders of the Taylor series.
This technique is an approximate approach that can be used to study the propagation of elastic waves in a substrate medium coated with a layer of constant thickness. The method solves the elastic wave problem for the substrate medium by replacing the boundary condition at the interface substratelayer with a suitable equation which relates stresses and displacement within the substrate only. The equation is obtained by combining the boundary conditions at the interface, the traction free condition at the top of the layer and a generalized Taylor expansion of the displacement and stress fields within the layer.
The model was evaluated for case studies involving coatings (viscous fluids and solids) on isotropic solid rod. For coating thicknesses which are comparable to the guided wave wavelength, the phase velocity and attenuation computed using the O(hn) model, are in good agreement with the exact solution over the range of viscosities and thicknesses considered. The method was found to be numerically stable for all the cases considered. The O(hn) model therefore can model coatings of microscopic(<100μm) and microscopic (<1mm) range of thicknesses depending on the number of orders that can be computed.












Ultrasonic Wave Response from Stacks 

The time domain and frequency domain response of finite mirror periodic and periodic stacking of layered materials are presented. The pulse propagation characteristics are simulated using transient finite element analysis and the frequency domain characteristics are simulated using the stiffness matrix based calculations for an AluminumPlexiglas multilayered material. Simulation of the band gaps in a periodic stack and the occurrence of local resonances within the band gaps in a mirror periodic stack obtained using these numerical methods agree well with each other. Experiments carried out on the AluminumPlexiglas stack at ultrasonic frequencies for periodic, mirror periodic and perturbed (randomly flipped) layering, validate the simulations very well.
The response of such layered media to oblique incidence is also studied in the frequency domain using the stiffness matrix method. It is found that such dispersion curves generated for layered media reveal interesting features that can be exploited to provide characteristic signatures for mirror periodic stack and perturbed periodic stack. Such an approach, it is believed, can facilitate nondestructive detection and assessment of anomalies in the layering of platelike structures. Several significant results have been obtained:


For a finite number of unit cells, mirror periodic stack response can be distinguished from that of a periodic stack even under normal incidence ultrasonic wave propagation conditions. Such a distinct response may be exploited in the nondestructive assessment of periodic and mirror periodic layered media.



For a finite number of unit cells, randomly flipped sequences in a periodic stack have characteristic features under normal incidence as well as oblique incidence conditions. In particular, interesting telltale signatures appear in the dispersion plots or “images” that may be utilized to assess the layup of periodic layered media.



It is possible to define an effective medium empirically such that its low frequency response matches with that of the corresponding finite periodic stack. The low frequency response of a mirror periodic stack is no different as is to be expected.



While the frequency domain approach continues to be useful to extract material characteristics, the success of the time domain approach is of considerable significance as it permits inversion schemes to be developed in time domain for the determination of the actual layup of the stack.







References : B. C. Ram, Kumar P. Padma, C. V. Krishnamurthy, and Krishnan Balasubramaniam, Ultrasonic Band Gaps in Periodic Stack of
Plates — Simulation and Experiments, AIP Conf. Proc. 820, 1043 (2006) 





Phased Array Models using FDTD 

The 2D FDTD model for the simulation and visualization of the elastic wave propagation is based on a first order velocitystress finitedifference method for a homogeneous isotropic material. Absorbing Boundary Conditions (ABC) were applied on the appropriate domain boundaries.
A two dimensional model was developed in MATLAB® using the FDTD technique to simulate the ultrasonic phased array wave propagation in pipe, elbow and tee sections. The code takes input parameters that define a phased array experimental setup and generates Ascans and Bscans that can be compared with the corresponding experimental Ascans and Bscans.
The time delay/ focal laws have been used to simulate phased array transmission and reception of unfocused phase steered beams. Simulations of the linear scan (for the generation of the Bscan images) and the sector scan features of the phased array were successfully implemented using this scheme. The simulation results are validated by comparing with the experimental results.


Fig 3. Comparison of Simulated and Experimental
sector scan image of
a 7mm notch using phased array 



Fig 1. Interaction of a phase steered
wave with the notch tip and corner 


Fig 2. Comparison of Simulated and Experimental Bscan image of a 7mm
notch using phased array 






Reference : 1. L. Satyanarayan, C. Sridhar, C.V. Krishnamurthy, Krishnan Balasubramaniam, “FDTD Simulation of Ultrasonic Phased Array Technique for Imaging and Sizing of Defects using Longitudinal Waves”, Journal of Pressure Vessels and Piping (In Press, Available online).
2. L. Satyanarayan, Mohan K .V., Krishnan Balasubramaniam, C.V.Krishnamurthy, “Finite Difference Time Domain Simulation of Ultrasonic
Phased Array Sector Scan for Imaging Surface Cracks in Pipes, Elbows and Tee Sections” , Research in Non Destructive Evaluation(In Press).






Rheological Models for Ultrasonic Attenuation in Fluids 

Monitoring the changes in density and viscosity has been an important process monitoring parameter in industries. Currently, these are measured by methods such as rectilinear flow measurements, rotational flow measurements at low frequencies, flow measurements at zero frequency such as falling ball technique, oscillating plate viscometer and longitudinally vibrating piezoelectric transducer.
Here, the attenuation measurements of the fundamental longitudinal and torsional wave modes, propagating in a circular wire like wave guide, are used to decouple the densityviscosity product using an inverse modeling approach. In the method proposed, it is enough to dip one end of the thin wire waveguide into the fluid and the piezoelectric elements can be kept far away from the point of actual measurement.
It was observed that the accuracy of the results obtained greatly depends on accuracy in the measurement of immersion depth. Hence, a more accurate and reliable method of measuring the immersion depth needs to be devised in order to realize the full usefulness of the method. Further, it was also shown that the accuracy of the measurement procedure tends to increase with an increase in the viscosity or decrease in the density of the fluid. With the current measuring system with a least count of 1 mm in immersion depth measurement, precision in the measurement of density and viscosity was around 710% when signals are recorded at immersion depth of 5060 mm.








ElectroMagnetic Models 




Obscuration Effects on Radar Antenna Performance 

The performance of a receiving antenna tends to degrade significantly from its free space pattern in certain angular sectors in the azimuthal plane and/or elevation plane when mounted on or in proximity of electrically large structures constituting the platform that carries the receiver antenna along with a host of electrically passive and active elements. The degradation depends on the operating frequency and on the spatial location of the receiver with respect to the obscuring structures making up the platform.
Codes have been developed to assess the degradation on the receive pattern when perfectly conducting objects such as cylinders and plates are in the vicinity of the antenna. The calculations are based on exact schemes such as the separation of variables (SOV) for cylinder scattering, and high frequency approximation schemes such as the uniform theory of diffraction (UTD) and the modified physical optics method (MPO) for plates of arbitrary shapes and orientations.




Screen shot of the GUI for assessment of obscuration effects due to a cylinder. 



Elevation pattern at 0.5 GHz for a receiver located 0.5 m below a 6 m long plate
with a tilted segment. 


Azimuthal pattern (in dB scale) for a receiver 2l away from a cylinder of radius l. 






Veeraraghavan Sundararaghavan1, and Krishnan Balasubramaniam,”Eddy Current Benchmark Problem 3  Optimisation of Pipe Inspection probe Design,
Review of Quantitative Nondestructive Evaluation, American Institute of Physics, Vol. 23, ed. by D. O. Thompson and D. E. Chimenti, pp. 157480,(2004). 







Microwave Models 

Two different probes were modeled to be used in microwave nondestructive evaluation using the finite difference time domain method. The probes were analyzed for their different characteristics and usage. Theoretical results were matched with the experimental data available with the literature and once the verification of the model was done, it was used to generate results for various samples having surface and intrinsic defects. It was realized that the two probes having different geometrical parameters can be used for different types of non destructive evaluation tests
The coaxial tip is a noncontact type probe and has a strong and directional field, whereas, the coaxial line is to be used as a contact type probe used for material detection. The major results from the models were :


For a finite number of unit cells, randomly flipped sequences in a periodic stack have characteristic features under normal incidence as well as oblique incidence conditions. In particular, interesting telltale signatures appear in the dispersion plots or “images” that may be utilized to assess the layup of periodic layered media.



An optimization was done for the tip length of the coaxial tip probe so that the directivity can be improved.



Effect of the presence of a dielectric in the space between two conductors of the coaxial tip was understood.



Physical limitations on the flange size of the tip were analyzed and a correlation was developed to realize the theoretical formulation under manufacturing constraints.



Results were developed for material identification by the coaxial line probe with and without the standoff distance.






Geometry for Coaxial Tip fed through an image plane by a coaxial line 


Bounce Diagram for a Gaussian pulse
traveling through the coaxial tip 


Comparison of radiation from a
Teflon filled coaxial tip 








Eddy Current Models 

ANSYS MODELS: The axisymmetric finite element modelling of eddy current test set up used for crack detection. It is used to predict theoretically the probe output signal for a given test configuration. Eddy current test set ups for crack detection using absolute and differential eddy current probes are modeled using ANSYS. Axisymmetric finite element analysis technique was used to predict absolute eddy current probe signals when it is passed through a defect tube.. Impedance plane plots for various defect sizes and at different frequencies were also obtained. Signals for inside, subsurface and outside defects are obtained and parameters that help in distinguishing them are discussed.
Similarly finite element analysis technique is also used to predict differential eddy current probe signals for an inner diameter and outer diameter axisymmetric slots. Coil spacing and size is varied and its effect on impedance plane trajectory is studied. Eddy current test set up using differential eddy current probe in steam generator tube testing is also modeled and results are compared with standard experimental results.
EDDYSIMULATE:











WFNDEC Benchmark EC Models 

The objective of the third eddy current benchmark problem proposed by the World federation of NDE centers (www.wfndec.org) was to determine optimal operating frequency and size of the pancake coil designated for Inconel tubes testing . The optimal design of the sensor should provide the highest possible sensitivity (the maximum change in inductance as it moves past the defect). A resolution of the sensor (ability to discriminate two parallel flaws located very closely) should also be determined.
The tube has an inside diameter Di =19.69 mm and an outside diameter D0 = 22.23 mm. The tube contains an outside flaw whose depth h is 20% of the tube wall thickness. Semi angle, coil height and coil width were assumed as an unknown parameters.
The coil geometry at which a maximum inductance change of the coil is achieved in the presence of a defect was studied as a part of the WFNDEC benchmark problem –3. A change of 9.12% in the inductance of the coil was obtained at the coil parameters listed below,



Outer Radius = 2.548 mm
Inner Radius = 2.148 mm
Coil thickness = 2 mm
Frequency = 400 kHz
Coil Turns = 400 

Further the resolution of the sensor was calculated. In order to evaluate the resolution, an ability to discriminate two parallel flaws located very closely (distance between flaws is of 1 mm) was studied. 




The schematic representation of the
WFNDEC EC Benchmark Model version 3. 

Typical results obtained from
the
ANSYS Model 






Veeraraghavan Sundararaghavan1, and Krishnan Balasubramaniam,”Eddy Current Benchmark Problem 3  Optimisation of Pipe Inspection probe Design,
Review of Quantitative Nondestructive Evaluation, American Institute of Physics, Vol. 23, ed. by D. O. Thompson and D. E. Chimenti, pp. 157480,(2004). 






Pulsed Eddy Current Models 

The theoretical model is derived from the analytical solutions of Cheng, Dodd, and Deeds for a coil above a multilayered halfspace. The model is slightly modified by replacing the reflection coefficients in the Cheng, Dodd and Deed model by transmission line equation which calculates them. The basic equations are formulated in the frequency domain and then the inverse Fourier transform is performed to obtain the predictions of timedomain PEC response.
Fig.1 shows the schematic diagram where a single layer of either conductive or non conductive material is coated on the substrate material. Fig.2 plots the model results for different thickness [75μm to 375 μm] of paint (nonconductive) coating on a metallic substrate. As the paint thickness increases the peak voltage amplitude decreases since only a partial amount of coil magnetic field will interact with the metallic substrate due to lift off caused by the paint thickness. Fig.3 plots the model results for 50 μm and 100 μm stainless steel coating on a titanium halfspace. As the coating thickness of the metallic layer increases there is a more amount of drop in the amplitude of the voltage.


Fig. 3 Model results for SS coating on Titanium 



Schematic diagram 


Fig. 2 Models results for various thickness of paint coating on metal 






 Cheng CC, Dodd CV, Deeds WE, ‘General analysis of probe coils near stratified conductors’, Intl. Journal of NDT, 1971, Vol. 3, 109 – 130.
2.
Tai CC, Rose JH, Moulder JC, ‘Thickness and conductivity of metallic layers from pulsed eddycurrent measurements’, Rev. Sci. Instrum. Vol. 67, No. 11, November 1996, 3965 – 3972.






Thermal Imaging Models 





Finite Element Modeling of Thermal Imaging 

Transient pulse propagation in various materials such as Aluminum, Steel, and Zirconium 2 alloy has been modeled using FEM in 1D and 2D. Simulation of flaws such as air voids and air gaps at interfaces have been carried out. Finite heat pulse duration effects on the transient response of defective and nondefective materials has been studied using commercial flash lamps. Facility to determine thermal diffusivity of materials and flaw depths using the flash method has been developed for planar and moderately curved geometries. Transient response of complex structures such as Honeycomb panels have been modeled and experimentally validated.




Simulated transient response of a 4.5 mm thick tube wall to heat pulse. The hot spot is the region of 10% wall thinning. 


















