Extension of the distributed point source method for ultrasonic field modeling

The distributed point source method (DPSM) was recently proposed for ultrasonic field modeling and other applications. This method uses distributed point sources, placed slightly behind transducer surface, to model the ultrasound field. The acoustic strength of each point source is obtained through matrix inversion that requires the number of target points on the transducer surface to be equal to the number of point sources. In this work, DPSM was extended and further developed to overcome the limitations of the original method and provide a solid mathematical explanation of the physical principle behind the method. With the extension, the acoustic strength of the point sources was calculated as the solution to the least squares minimization problem instead of using direct matrix inversion. As numerical examples, the ultrasound fields of circular and rectangular transducers were calculated using the extended and original DPSMs which were then systematically compared with the results calculated using the theoretical solution and the exact spatial impulse response method. The numerical results showed the extended method can model ultrasonic fields accurately without the scaling step required by the original method. The extended method has potential applications in ultrasonic field modeling, tissue characterization, nondestructive testing, and ultrasound system optimization.     

Pencil method in elastodynamics: application to ultrasonic field computation

The principles of pencil elastodynamics and, in more detail, some selected applications of pencil techniques to elastodynamics are described. It is shown how a systematic use of a matrix representation for the wave front curvature and for its transformations simplifies the handling of arbitrary pencils and, consequently, the field computations. Pencil matrix representations for the propagation into homogeneous solids made of isotropic or anisotropic media are derived. The use of matrix representations for pencil reflections on, or refractions through, arbitrarily curved interfaces, together with matrix representations for propagation into homogeneous media, allow us to derive an overall matrix formulation for elastodynamic propagation into complex heterogeneous structures. Combined with the classical Rayleigh integral to account for transducer diffraction effects, the proposed theory is applied to the prediction of ultrasonic fields radiated into complex structures by arbitrary transducers. Examples of interest for application to ultrasonic non-destructive testing are given.     

Controlled Space Radiation concept for mesh-free semi-analytical technique to model wave fields in complex geometries

Numerical modelling of the ultrasonic wave propagation is important for Structural Heath Monitoring and System Prognosis problems. In order to develop intelligent and adaptive structures with embedded damage detector and classifier mechanisms, detailed understanding of scattered wave fields due to anomaly in the structure is inevitably required. A detailed understanding of the problem demands a good modelling of the wave propagation in the problem geometry in virtual form. Therefore, efficient analytical, semi-analytical or numerical modelling techniques are required. In recent years a semi-analytical mesh-free technique called Distributed Point Source Method (DPSM) is being used for modelling various ultrasonic, electrostatic and electromagnetic wave field problems. In the conventional DPSM approach point sources are placed along the transducer faces, problem boundaries and interfaces to model incident and scattered fields. Every point source emits energy in all directions uniformly. Source strengths of these 360° radiation sources are obtained by satisfying interface and boundary conditions of the problem. In conventional DPSM modelling approach it is assumed that the shadow zone does not require any special consideration. 360° Radiation point sources should be capable of properly modelling shadow zones because all boundary and interface conditions are satisfied. In this paper it is investigated how good this assumption is by introducing the ‘shadow zone’ concept at the point source level and comparing the results generated by the conventional DPSM and by this modified approach where the conventional 360° radiation point sources are replaced by the Controlled Space Radiation (CSR) sources.     

Minimum radiation force target size for power measurements in focused ultrasonic fields with circular symmetry

The time-averaged ultrasonic power emitted by medical ultrasonic equipment is mostly measured using a radiation force balance, and the question of the necessary target size is of practical importance. The question is answered here by calculations based on a Rayleigh integral algorithm for fields from circular, focusing transducers. This case occurs particularly in the field of high-intensity therapeutic ultrasound. The calculation yields the necessary size of an absorbing target so that the radiation force is 98% of that exerted on an absorber of infinite lateral size, and this as a function of the transducer-to-target distance, of the transducer radius in comparison with the wavelength and of the focus (half-)angle. Several distributions of the transducer vibration amplitude are considered. The Rayleigh integral strictly applies only to planar transducers, but among the amplitude distributions there is also one that allows the simulation of the spherically curved transducer type often found in practice.     

Scattering of focused ultrasonic beams by cavities in a solid half-space

The ultrasonic field generated by a point focused acoustic lens placed in a fluid medium adjacent to a solid half-space, containing one or more spherical cavities, is modeled. The semi-analytical distributed point source method (DPSM) is followed for the modeling. This technique properly takes into account the interaction effect between the cavities placed in the focused ultrasonic field, fluid-solid interface and the lens surface. The approximate analytical solution that is available in the literature for the single cavity geometry is very restrictive and cannot handle multiple cavity problems. Finite element solutions for such problems are also prohibitively time consuming at high frequencies. Solution of this problem is necessary to predict when two cavities placed in close proximity inside a solid can be distinguished by an acoustic lens placed outside the solid medium and when such distinction is not possible.     

Optimal focusing by spatio-temporal inverse filter. I. Basic principles

A focusing technique based on the inversion of the propagation operator relating an array of transducers to a set of control points inside a medium was proposed in previous work [Tanter et al., J. Acoust. Soc. Am. 108, 223-234 (2000)] and is extended here to the time domain. As the inversion of the propagation operator is achieved both in space and time, this technique allows calculation of the set of temporal signals to be emitted by each element of the array in order to optimally focus on a chosen control point. This broadband inversion process takes advantage of the singular-value decomposition of the propagation operator in the Fourier domain. The physical meaning of this decomposition is explained in a homogeneous medium. In particular, a definition of the number of degrees of freedom necessary to define the acoustic field generated by an array of limited aperture in a focal plane of limited extent is given. This number corresponds to the number of independent signals that can be created in the focal area both in space and time. In this paper, this broadband inverse-focusing technique is compared in homogeneous media with the classical focusing achieved by simple geometrical considerations but also with time-reversal focusing. It is shown that, even in a simple medium, slight differences appear between these three focusing strategies. In the companion paper [Aubry et al., J. Acoust. Soc. Am. 110, 48-58 (2001)] the three focusing techniques are compared in heterogeneous, absorbing, or complex media where classical focusing is strongly degraded. The strong improvement achieved by the spatio-temporal inverse-filter technique emphasizes the great potential of multiple-channel systems having the ability to apply completely different signal waveforms on each transducer of the array. The application of this focusing technique could be of great interest in various ultrasonic fields such as medical imaging, nondestructive testing, and underwater acoustics.     

多层介质中超声相控阵声场仿真*

由于良好的声束偏转与聚焦特性,超声相控阵已经广泛应用于多层固体介质的缺陷检测。当超声束经过多层介质时,由于反射、透射以及模式转换的存在,多种声束存在于这种结构中,使得声场分析变得复杂。为了提高多层介质检测的准确性,有必要对超声声场的分布规律进行深入地了解。该文结合高斯声束等效点源模型以及射线追踪法,给出了相控阵声源在多层固体介质中激发声场的仿真方法,并且模拟计算了一维线型相控阵在楔块-铝-黄铜-钢四层固体介质中的辐射声场。通过对不同延时法则的计算,实现了声波在这种复杂介质中的偏转与聚焦,进而研究了不同焦点处聚焦声场的分布。结果表明:相控阵方法能使聚焦点处的声场幅值增大,能量集中,提高了检测分辨率;不同聚焦点处声场聚焦效果不同,实际检测时应根据检测区域结构及位置特点,合理放置相控阵换能器。与瑞利积分法的比较表明,该文的仿真方法适用于多层介质相控阵声场的计算。     

Nonparaxial multi-Gaussian beam models and measurement models for phased array transducers

A nonparaxial multi-Gaussian beam model is proposed in order to overcome the limitation that paraxial Gaussian beam models lose accuracy in simulating the beam steering behavior of phased array transducers. Using this nonparaxial multi-Gaussian beam model, the focusing and steering sound fields generated by an ultrasonic linear phased array transducer are calculated and compared with the corresponding results obtained by paraxial multi-Gaussian beam model and more exact Rayleigh-Sommerfeld integral model. In addition, with help of this novel nonparaxial method, an ultrasonic measurement model is provided to investigate the sensitivity of linear phased array transducers versus steering angles. Also the comparisons of model predictions with experimental results are presented to certify the accuracy of this provided measurement model.     

Acoustic power calibration of high-intensity focused ultrasound transducers using a radiation force technique

To address the challenges associated with measuring the ultrasonic power from high-intensity focused ultrasound transducers via radiation force, a technique based on pulsed measurements was developed and analyzed. Two focused ultrasound transducers were characterized in terms of an effective duty factor, which was then used to calculate the power during the pulse at high applied power levels. Two absorbing target designs were used, and both gave comparable results and displayed no damage and minimal temperature rise if placed near the transducer and away from the focus. The method yielded reproducible results up to the maximum pulse power generated of approximately 230 W, thus allowing the radiated power to be calibrated in terms of the peak-to-peak voltage applied to the transducer.     

Gaussian-DPSM (G-DPSM) and Element Source Method (ESM) modifications to DPSM for ultrasonic field modeling

In the last few years, Distributed Point Source Method (DPSM) a mesh-free semi-analytical technique has been developed. In spite of its many advantages, one shortcoming of the conventional DPSM method is that the field obtained by conventional DPSM method needs to be scaled to match the theoretical solutions. Two modification techniques called Gaussian-DPSM (G-DPSM) and Element Source Method (ESM) are developed here to avoid the scaling need. G-DPSM technique introduces additional fictitious point sources around every parent point source. Gaussian weight functions determine the strength of these additional fictitious point sources that are denoted as child point sources. ESM replaces discrete point sources used in the conventional DPSM by continuous sources. In the ESM formulation individual point sources are denoted as nodes. Special elements are formed on the boundary by connecting these nodes. The source strength inside the element can vary linearly or non-linearly depending on the order of the interpolation function used inside the element. Results generated by both these methods are compared with the conventional DPSM solution and analytical solution. It is shown that the ultrasonic field in front of the transducer computed by G-DPSM and ESM matches very well with the theory without using any scaling factor.     

Elastic wave propagation in sinusoidally corrugated waveguides

The ultrasonic wave propagation in sinusoidally corrugated waveguides is studied in this paper. Periodically corrugated waveguides are gaining popularity in the field of vibration control and for designing structures with desired acoustic band gaps. Currently only numerical method (Boundary Element Method or Finite Element Method) based packages (e.g., PZFlex) are in principle capable of modeling ultrasonic fields in complex structures with rapid change of curvatures at the interfaces and boundaries but no analyses have been reported. However, the packages are very CPU intensive; it requires a huge amount of computation memory and time for its execution. In this paper a new semi-analytical technique called Distributed Point Source Method (DPSM) is used to model the ultrasonic field in sinusoidally corrugated waveguides immersed in water where the interface curvature changes rapidly. DPSM results are compared with analytical solutions. It is found that when a narrow ultrasonic beam hits the corrugation peaks at an angle, the wave propagates in the backward direction in waveguides with high corrugation depth. However, in waveguides with small corrugation the wave propagates in the forward direction. The forward and backward propagation phenomenon is found to be independent of the signal frequency and depends on the degree of corrugation.     

Ultrasound field estimation method using a secondary source-array numerically constructed from a limited number of pressure measurements

A new and faster method for the accurate estimation of acoustic fields of underwater ultrasonic transducers was developed, tested experimentally, and compared to previously reported methods. Using a limited number of pressure measurements close to the transducer's face, the method numerically constructs a virtual secondary source-array whose acoustic field is similar to the field generated by the actual transducer (primary source). The measured data are used to obtain the normal particle velocity on the surface of the virtual secondary source-array, which in turn permits the calculation of the forward propagating field using the Rayleigh-Sommerfeld diffraction integral. The method is novel in that it constructs a virtual secondary source-array, thus eliminating the problems associated with obtaining the excitation source of a real transducer; and it is faster because it uses finite differences instead of a matrix inversion to obtain the excitation source. Results showed that predicted ultrasound fields agreed quantitatively and qualitatively with measured fields for three commonly used transducer types: two planar radiators (one circular, 0.5 MHz, 1.9-cm diam.; and one square, 1 MHz, 1.2 cm on a side), and a sharply focused radiator (1.5 MHz, 10-cm diam., 10-cm radius of curvature). The agreements suggest that the secondary source-array method (SSAM) is applicable to a wide range of radiator sizes, shapes, and operating frequencies. The SSAM was also compared to these authors' previous equivalent phased array methods (EPAM) [J. Acoust. Soc. Am. 102, 2734-2741 (1997); and Concentric ring equivalent phased array method (CREPAM), UFFC 46, 830-841 (1999)] which require matrix inversions. The SSAM proved to be much faster and equally or more nearly accurate than the previous methods.     

聚焦换能器自由场互易校准的冲击响应积分法衍射系数计算

在聚焦换能器的会聚球面波自由场互易校准法中，衍射损失修正系数是获得互易常数的一个关键参数。本文利用冲击响应函数积分法系统地计算了超声治疗头、机械扫描B超探头以及多普勒诊断探头的衍射损失系数，得出了一些典型参数下的计算曲线，可供不同类型的聚焦换能器会聚球面波互易校准时引用。与传统的瑞利积分法分析比较显示，本文中所采用的冲击响应函数积分法显著地提高了运算速度，同时基本保持了与瑞利积分算法计算结果的一致性。     

光弹法测量超声换能器声场

超声换能器声场的测试对于超声检测具有基础性作用,而传统的超声换能器声场测试方法具有一些局限性。本文介绍了搭建的动态激光光弹实验平台,并利用动态光弹法测量了纵波换能器和横波换能器辐射声场的特征,由瞬态声场图像获得了传声介质的声波速度及超声换能器的中心频率;由稳态声场图像获得了声场的近场长度、指向性和扩散角等参数;分析了光弹实验系统和测量方法可能引起测量误差。本文结果表明动态光弹法是一种有效的定量测量超声换能器声场的方法。     

Some basic relations for ultrasonic fields from circular transducers with a central hole

Ultrasonic transducers with a central hole are increasingly in use in high-intensity therapeutic ultrasound and similar medical applications. The hole is intended for the addition of a diagnostic device. Some fundamental properties of the fields produced by such transducers are investigated here theoretically, based on a Rayleigh integral algorithm. The approach is restricted to those cases where the Rayleigh integral can be fully solved to yield closed-formula results that can be easily used by the reader. This means a concentration on points on the field axis which, on the other hand, is most important under the aspect of the safety of the patient (maximum amplitudes). Closed-form expressions describing the influence of the central hole on the acoustic pressure, on the particle velocity and on the time-averaged intensity are presented. The relation between the true intensity and the intensity expression derived from the square of the acoustic pressure is discussed in some detail, an aspect which is important in ultrasonic measurement practice where the local intensity value is mostly assessed based on the square of a hydrophone signal.     

带楔块二维面阵列超声相控阵声场特性分析

开展了带楔块二维面阵脉冲超声相控阵辐射声场特性的研究。将安装在一定角度斜楔块上二维面阵相控阵换能器声场问题简化为液固界面情况进行讨论。以单阵元在液固平界面条件下的辐射声场为基础,推导了聚焦偏转法则,给出了带楔块二维面阵超声相控阵声场计算方法。以检测材料为钢板,安装在倾斜角为36°有机玻璃楔块上的频率为5 MHz、8×8二维面阵相控阵的辐射声场计算为例,分析了在不同偏转角和不同聚焦深度下检测材料中的辐射声场特性。计算结果表明该方法可有效的分析带楔块二维面阵超声相控阵声场特性并用于指导二维超声面阵角束探头的设计。      

An efficient calculation algorithm for focusing by plane ultrasonic transducers

In the present paper, we simulate focused wave fields generated by a system of plane transducers used in an immersion ultrasonic technique. The ultrasonic beam penetrates into the elastic medium through the fluid-solid boundary. A real-time computer algorithm is proposed to calculate stress components in the solid medium. It is shown that a good focusing wave structure can be provided by a pair of plane rectangular transducers, inclined with respect to each other and to the beam acoustic axis.     

Study on the compound multifrequency ultrasonic transducer in flexural vibration

A new type of compound multifrequency ultrasonic transducer is analyzed in this paper. The compound multifrequency ultrasonic transducer consists of two sandwiched ultrasonic transducers and a rectangular radiator. In virtue of the coupling between longitudinal vibration of the sandwiched ultrasonic transducers and flexural vibration of the rectangular radiator, the compound multifrequency ultrasonic transducer can produce several resonance frequencies. Some compound multifrequency ultrasonic transducers are designed and simulated by finite element method (ANSYS), and modal shapes and harmonic response are analyzed. The compound multifrequency ultrasonic transducers are designed and manufactured. The resonance frequencies are measured and compared with the numerical results. The effect of the geometrical dimensions of the compound multifrequency ultrasonic transducer and the location of two sandwiched ultrasonic transducers on the compound multifrequency ultrasonic transducer is discussed. It is shown experimentally and numerically that the compound multifrequency ultrasonic transducer has several resonance frequencies.     

The ultrasonic probe for the investigating of internal object structure by ultrasound transmission tomography

The main component of every device used for investigating internal object structure by Ultrasound Transmission Tomography (UTT) is a special ultrasonic probe. This paper presents the structure of model multielement ring probe for examining objects using this method in divergent ray projection geometry. The probe is made up of 1024 rectangular separate piezoelectric transducers working at frequency f=1.7 MHz, placed inside a ring with diameter D=30 cm and height h=9 cm. Each element of the probe is equipped with a quarter-wave matching layer. All the transducers function as transmitters and receivers, and can be joined in groups both during transmitting and during receiving. Some examples of admittance characteristics of a single piezoelectric transducer and examples of shapes of pulses generated and received by particular transducers were presented. Important factors affecting the measurement resolution are the sizes of the active surface of the transducers.