Theoretical and numerical study of the reflection of an ultrasonic pulse radiated by a linear phased
array transducer at a fluid-fluid interface
OUDINA Assia Materials Physics Laboratory U.S.T.H.B, Faculty of Physics
Algiers, Algeria hkourgli@yahoo.fr
DJELOUAH Hakim Materials Physics Laboratory U.S.T.H.B, Faculty of Physics
Algiers, Algeria djelouah_hakim@yahoo.fr
Abstract—This study is devoted to the calculation, in transient mode, of the ultrasonic field emitted by a linear array and reflected from a fluid-fluid interface thanks to a finite element package widely used in computer simulations for solving partial differential equations describing such physical phenomena. The results obtained show the various waves emerging at the interface: direct and edge waves, specular reflection and the appearance of radiating surface waves at critical angle. The various waves are identified by their arrival times calculated by using the ray method.
Keywords : Transient ultrasonic;liquid-liquid interface;reflection;phased array.
I. INTRODUCTION
Ultrasonictechnics are widelyused in medicine, both for diagnostics and therapy. The ultrasound probe is the heart of any medical ultrasound imaging system because it determines the image quality. Most ultrasound systems use ultrasonic phased array probes composed of an arranged network of transducers [1][2][3]. There are several methods for calculating the ultrasound field radiated by a linear array and reflected at an interface between two media.Among these methods we can note the semi-empirical methods [4], the full representationmethod [5][6] and the exact methods [7]. Other purely numerical methods such as finite difference method and finite element method are also used [8][9]. In our case we have used the finite element method.This work is devoted to the study of the pulsed ultrasonic field emitted by a linear array phased array and reflected by a plane interface separating two homogeneous lossless fluid media. The modeling is based on the finite element method (FEM) widely used in numerical simulations, particularly for solving partial differential equations describing physical phenomena. For this study, we used the software package Comsol. The results of the numerical simulation show the different waves generated.
These results will be validated and interpreted through the ray model.
Fig. 1. Geometry of the studied problem.
II. NUMERICAL SIMULATION
For the numerical simulation, the phased array is excited by a four periods long sinusoidmodulated by a Gaussian of the form:
f(t)= A0sin (2πft) exp[-(t-T0)/(0.4T0)]2 (1) The phased arrayconsists of 15 elements length L=16,6mm and center frequency F=1MHz, each element of the array has a width of 1mm and inter element space measuring 0.1mm (Fig.
1). The medium 1 is characterized by the velocity c1=1500m/s and density ρ1=1000kg/m3, the medium 2 is characterized by the velocity c2=1904m/s and densityρ2=1260kg/m3. For modeling the propagation in the fluid we chose Acoustics
module in COMSOL Multiphysics.
To have a good result, it is necessary to ensure the numerical stability of the model with adequate sampling both in space and time domains. In particular time sampling must respect Shannon theorem. In our case, the media are sampled with a spatial stepΔx=0.3mm and a temporal stepΔt=0.1µs. The
EW
RDW
REW EW solver used is MUMPS type and mesh type is free triangular
type.
III. RESULTS AND DISCUSSION
The results of the numerical simulation are shownbelow in the case where c2>c1and where c1>c2.
A. Case when c2>c1
Fig. 2shows the reflected pressure at a fixedheight h=2mm from the interface and at two different axis positions x=0mm and x=8mm.Differentpulses results, they corresponds to the direct wave, theedge wave, the directreflectedwave, and the edge reflected waves, this two last ones corresponds to the specularreflection. These pulses are identified by the ray method. The time tDW=1,33µs corresponds to the arrival of the direct wave with a positive polarity followed by the edge wave witha negative polarity arriving at tEW=5,69µs. These arrival times have been calculated from the following expressions:
tEW=[h2+ (L/2)2]1/2/c1 (3)
Due to specular reflection, the direct wave is reflected at tDRW=12µswich generates direct reflected wave. In the same way a reflected edge wave is generated at tREW=13,20µs .
Whenc2> c1there is a critical angleθl =asin(c1/c2). In this case the angle of refraction is π/2 and we observe the appearance of longitudinal head wave which propagates with avelocityc2. As c2> c1 she will be re-emitted in the first medium at an angleθlcorresponding to a conical wave arrivingattLW=12,02µs. The amplitude of this wave is very low.
tLW= (L/2-(2z-h) tanθl)/c2+ (2z-h)/( c1cosθl) (4)
We also notice the appearance of other direct reflected waves through the pathdRDW. For the position x=8mm we find the same waves as previously but with a cleardecrease of the amplitude of the edge waves. We also have represented the instantaneous pressure field p(x,z) att=1,4e-5s(Fig. 3).The figure highlights the various wave types: reflected direct wave (RDW), edge waves (EW) and reflected edge waves (REW).
The first wave front detected is thedirect wave front which propagates according to the z direction, followed by the edge wave front.
B. Case when c1>c2
Fig. 4shows the reflected pressure at a fixedheight h=2mm from the interface and at two different axis positions x=0mm and x=8mm. We observe the same waves as previously: direct wave which arrives firstlyattDW=1,05µs followed by the edge wave attEW=4,48µs, the reflecteddirect wavecorresponds to the plane wave front propagating in the zdirection arriving at instanttRDW=9,45µs and followed by the reflected edge wave arrival at tREW=13,02µs.
Fig. 2. Calculated pressure at height h=2mm and -axis positions x=0mm and x=8mm.
Fig. 3. Instantaneous pressure representation atinstantt=1.4e-5sfor c2> c1
Fig. 4. Calculated pressure at height h=2mm and -axis positions x=0mm and x=8mm.
tDW=h/c1 (2)
dEW
dDW
dRDW
dREW dRDW1
dθl
.
Fig. 5. Instantaneous pressure representation at instant t=1,23e-5sfor c1>c2.
Fig.6.Representation of the different paths traveled by the various waves.
As the velocity of the medium1 is greater than the velocity ofthe medium 2, there will be no limit angle or conversion wave. For x=8mm, we notice an apparent decrease of the amplitudes of waves especially for edges waves.The instantaneous pressure field p(x,z) at t=1.23e-5sis represented in (Fig. 5).The first wave which arrives is the plane wave followed by the edge wave.The paths followed by the various waves are shown inFig. 6.
IV. CONCLUSION
The numerical finite element simulation allowed us to study the fluid-fluid interface problems relating to the propagation and reflection of short ultrasonic pulse waves.
References
[1] E.Lacaze, S. Michau, R. Dufait, and A. Flesch, “A l’intérieur d’une barrette échographique”,Echographie technologievol 20, pp. 233-242 1998.
[2] R.Huang, and LW.Schmerr Jr, “Characterization of the system functions of ultrasounds linear phased array inspection systems”, Ultrasonics vol 49, pp 219-2252009.
[3] N.Samet, P. Maréchal, and H. Duflo, “Ultrasound monitoring of bubble size and velocity in a fluid model using phased array”,NDT&E international vol 44, pp 621-627 2011.
[4] J. PWeight, “A model for the propagation of short pulses of ultrasound in a solid”,J.Acoust.Soc.Am vol81(4), pp 815-826 1987.
[5] R.P Stephanishen, “Transient radiated from pistons in an infinite planar baffle”,J.Acoust.Soc.Am vol49, pp 1629-1638, 1971.
[6] A.T. de Hoop,” A modification of Cagniard’s method for solving seismic pulse problems”, Applied Scientific Research vol 8 pp 349-356, 1960.
[7] H.Djelouah, and JC.Baboux, “Transient ultrasonicfieldradiatedby a circulartransducer in a solid medium”,J.Acoust.Soc.Am vol 92(5),pp 2932-2941, 1992.
[8] A.Ilan, and J.Weight, “The propagation of short pulses of ultrasound from a circular source coupled to an isotropic solid”,J.Acoust.Soc.Amvol 88(2), pp1142-1151 1990.
[9] A.Alia, H.Djelouah,and N. Bouaoua, “Finitedifference modelingof the
ultrasonic fieldradiated by circular
transducers”,J.Comput.Acoustvol49(4),pp 475-499,2004.
