Transducer design simulation using finite element method

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1 Transducer design simulation using finite element method Wenwu Cao Whitaker Center for Medical Ultrasonic Transducer Engineering Department of Mathematics and Materials Research Laboratory The Pennsylvania State University University Park, Pennsylvania 1682 Keywords: ultrasonic, medical imaging, transducers, arrays, simulation, design optimization, beam pattern, finite element method, composite transducers, spurious resonance ABSTRACT The fast development of computer power and computational algorithms has made it possible to design complex transducers and arrays using computer simulation. Practical examples are given here for transducer modeling using 3-D finite element method (FEM) instead of the traditional 1-D equivalent circuit models. The merits and deficiencies of the frequency domain and time domain FEM formulations will also be analyzed. 1. INTRODUCTION Many factors need to be considered in medical transducer design, for example, sensitivity, ringdown, axial resolution, lateral resolution, acoustic impedance matching, electrical impedance matching, beam size and focus ability, physical geometry as well as electrical and mechanical interconnects. More and more complex transducers are designed to meet the demand of medical ultrasonic imaging. Unfortunately, the current resolution limitation in medical ultrasound is still mainly caused by inadequate transducers. Traditional transducer design modeling based on equivalent circuit1 becomes more and more inadequate due to the complexity of the transducers and the one dimensional nature of the models. Problems such as cross-talk, beam focusing, nonuniformity of transducer surface velocity all require the knowledge of a 3- dimensional data base. There is an urgent need for better design tools which can give more accurate and direct guidance to transducer manufactures. In this paper, the current status of transducer design using 3-D computer simulation will be reviewed and some new results will be reported. The design simulation discussed here is based on finite element method (FEM) for the spatial variable. The time variation can either be integrated step-by-step in time domain or be solved by eigen function expansion in frequency domain. The time domain algorithm is efficient for transient analysis of short time or high frequency. On the other hand, frequency domain method works better for modal analysis and low frequency situation, including static deformation. Generally speaking, frequency domain algorithm is more computational expensive but more stable. Here we show 2 examples of using time domain and frequency domain FEM to analyze 2-2 and 1-3 composite transducers, respectively, particularly on the resonance behavior and the nonuniform surface vibration profile. The FEM can also be used to simulate cross-talk in an array transducer and the propagation of waves inside the transducer and in the medium. In addition, the beam pattern of a given transducer can also be calculated using FEM combined with the Helmholtz integral. 2. FINITE ELEMENT METHODS The fundamental equations for the structural analysis of transducers are the equations of motion in solids and the acoustic wave equations in liquid. One must pay special attention to the piezoelectric coupling while SPIE Vol X/971$1. 13 Downloaded from SPIE Digital Library on 18 Feb 21 to Terms of Use:

2 describing the active elements of a transducer, which reflects the electromechanical interaction. As a general practice, quasi-static approximation is used for the electromagnetic equations considering the fact that the acoustic velocity is much slower than that of the electromagnetic waves. In finite element computations, the continuous medium is meshed into a set of discrete nodes and the governing equations can be written in terms of the nodal displacement {u } and nodal potential {v } in the following form: [ [lvi] [] { U} [a] [] { i [K] [KZ] {u }1 I { f} I.. ;+ +?='? (1) L [] [] J {v}j [] [] [KT] [Kd] {v}j UQ}J where [M] = S [NU] [NUJT dv (2) element is the mass matrix, [a] is the structural damping, [K] =.1 [BuIT [c] [Bu] dv (3) element is the elastic stiffness matrix, [Kd] - J [Bv]T [] [BvJ dv (4) element is the dielectric constant matrix, and [KZ} = J [BuIT [e] {BvJ dv (5) element is the piezoelectric coupling matrix. The two vectors on the r.h.s. of Eq. (1), {f},and {Q}, are the generalized nodal force and nodal charge, respectively. Equation (1) can be solved either in frequency domain using harmonic function expansion or in time domain to integrate step-by-step. There are advantages and disadvantages in both methods, one needs to first identify the objective of the analysis before choosing a methodology. 3. RESONANCE BEHAVIOR OF A 1-3 COMPOSITE TRANSDUCER The 1-3 composite2 structure is made of ceramic rods embedded in a polymer matrix. Because of its low acoustic impedance and high electromechanical coupling coefficient, many single element transducers for medical and under water acoustic applications are made of 1-3 composites.3 One of the fundamental questions in the design of composite transducers is how to accurately calculate the resonance frequency. The isostrain approximation4 does not hold because of the large difference in elastic stiffness between the two constituents. One would expect different vibration amplitude in polymer and in ceramic, which has been confirmed both through optical interferometer measurements and theoretical analysis of a simplified system.5'6 Here we show that such nonuniform vibration profile can be easily obtained using frequency domain FEM. The surface vibration profile shown in Fig. 1 (a) was obtained from interferometer measurements by a group at the Navy Air Warfare Center (NAWC) at Warmingster, Pennsylvania. Because the measurement was performed using a continuous sine wave drive, the profile is for a steady state. 14 Downloaded from SPIE Digital Library on 18 Feb 21 to Terms of Use:

3 Obviously, for a single frequency drive the finite element analysis (FEA) is simpler to perform in frequency domain. Fig. 1(b) shows the vibration amplitude distribution at the thickness resonance calculated by ANSYS. One can see from Fig. 4 that the FEA accurately simulated the nonuniform vibration profile at resonance. In addition it also gives information on the interior displacement distribution which can not be measured experimentally. We also obtained, at the same time, internal stress concentration, electric field concentration and surface velocity map. (a) (b) Fig. 1. Comparison between (a) the experimentally measured surface vibration profile and (b) the simulation results (b) of a steady state vibration profile in a 1-3 composite. 4. BEAM PATTERN OF A 1-3 COMPOSITE TRANSDUCER The radiation beam pattern of a transducer depends on the surface displacement and velocity distributions. Because FEA can give us the needed information, it is natural to extend the computation to the transducer radiation beam pattern. However, we found that the FEA results diverge when the system becomes very large due to the inclusion of water medium. A compromise solution is to add an adequate sized piece of liquid to the transducer and specify the absorbing boundary condition at the outer boundaries of the liquid. Such a model can give us the surface velocity and amplitude distribution under water loading. With these velocity and amplitude information, the radiation pattern can be calculated by the Helmholtz integral using boundary element method. 15 Downloaded from SPIE Digital Library on 18 Feb 21 to Terms of Use:

4 Shown in Fig. 2(a) is the surface velocity distribution of a 1-3 composite transducer with square cross section and 4 X 4 =16 PZT rods. We found that the vibration amplitude is larger in the center region and is much smaller in the polymer phase than in the ceramic phase under water loading. Using the FEA results, the beam pattern can be calculated as shown in Fig. 2(b). The nearfield pattern, which is difficult to measure experimentally, reflects the nature of the composite structure. One can also see that the polymer phase does contribute to the radiation pattern but the ceramic is the dominant radiator. More information can be obtained from computation, such as the focal point, focal size, side lobs and total radiation energy. In a sense, the 3- D simulation modeling can provide more accurate information than direct measurements since the nearfield measurements always disturb the field to be measured while the simulation does not. (a) Fig. 2. The radiation beam pattern of a 1-3 composite transducer in water. (a) Surface displacement distribution; (b) beam pattern. 5. CROSS-ELEMENT COUPLING IN ARRAY TRANSDUCER One of the important issues in array design is the cross-element coupling. It can affect the sensitivity and the acceptance angle. Since arrays in medical imaging are generally broadband and operated in pulse mode, the time duration for the dynamical process is relatively short. As a consequence, time domain FEA shows some advantages for such studies. This is because a short time duration corresponds to a broad frequency spectrum, it requires much larger frequency domain computation to achieve desired resolution in time domain. In addition, one can effectively treat a smaller spatial system to reduce the computation task since the local disturbance can only travel to a limited distance in given time. An example using the time domain package PZ-Flex developed by Weidlinger Associates is given here to demonstrate the procedure. Shown in Fig. 3 is a cross section of a typical linear array transducer. We can use a 2-D model to describe the linear anay. The aim of the FEA is to look at the problem of mechanical isolation through different dicing strategy. The element at the center of the transducer labeled by an "F" is fired using a half cycle sine-pulse, the mechanical excitation will propagate through the filler, backing and matching layer and excite the neighboring element via piezoelectric effect. Here we will sample the electric response at the center of each neighboring elements labeled "1", "2", "3", and "4" as shown in Fig.3 and study the influence of subdicing of the matching layer. (b) 16 Downloaded from SPIE Digital Library on 18 Feb 21 to Terms of Use:

5 Fig. 3. The 2-D FEA model for a 1 -D array transducer. The backing and the filler are the same material for simplicity. Showing in Fig. 4 is the electrical response detected at the center of each neighboring elements. We can see that with the configuration shown in Fig.3, the coupling is very strong and the reduction rate away from the element is very slow. In fact, the coupling is very strong even at the 1th neighbor. This is the reason why arrays are subdiced in practice. The backing is generally made of more attenuating materials compared to the matching layer. Therefore, it has the least effect in mechanical isolation. But it is very important for the ringdown and mechanical stability. The subdicing of the matching layer should be very effective for the mechanical isolation because it can not be made of attenuating material. The filler, although can be made attenuative, is in direct contact with the whole elements, therefore it also contributes a lot to the cross-talk. With this consideration, we dice through the matching layer and the filler to reduce the cross-element coupling. Fig. 5 is the neighboring element response after the subdicing, we can see substantial improvement compared with the non-diced design. At the third neighbor, the response reduction is 16 db. Another advantage of the FEM is its ability to simulate the pulse propagation inside a complex structure. Fig. 6. is a 'snapshot of the pressure propagation from a pulse excitation of a single element in the subdiced linear array described above. One can clearly see the nature of the cross-element coupling through the backing in this snapshot. It also shows the interference nearfield pattern due to the electromechanical coupling of the neighboring elements. Such a complicated pressure field can only be obtained through numerical simulation. 6. SUMMARY AND CONCLUSIONS While the development of FEM and computational power makes many simulations possible using real dimensions and real time, it is important to realize some of the limitations in time domain and frequency domain formulations and also take advantage of their merits. Generally speaking, frequency domain formulation is most appropriate to study static deformation, modal shapes, resonance frequencies and steady state; while time domain formulation is ideal for transient analysis and local disturbance of short time 17 Downloaded from SPIE Digital Library on 18 Feb 21 to Terms of Use:

6 duration. Although time domain and frequency domain results can be connected through Fourier Transform, it is time consuming if one does not use the right formulation. All numerical techniques involve truncation errors and rounding errors. The grid size and shape function also limit the accuracy of the discrete representation of the continuous solution. These errors will propagate with computational steps either in time or in space. In some cases, there will be a catastrophic failure if a system is being integrated over a long distance in space or over a long period of time. Other times, the deviation may not be catastrophic and hard to be detected, which is the most damaging situation since there is no easy way i \. First Neighbor Response,L1,i, / Second Neighbor Response,fI;. (f\!\r, _._ L/"' I,1!,. ILl f Third Neighbor Response L I t V H - i.1 ) / 1/ I Fourth Neighbor Response \ /fl\/--\/_..._._..._._ Time ( tsec.) I 4 I. Fig. 4 The electrical response detected at different neighboring elements. The first plot is the half sine drive signal. 18 Downloaded from SPIE Digital Library on 18 Feb 21 to Terms of Use:

7 to tell if the solutions are correct. Therefore, one must design the FEM problem with these errors in mind and treat the results with caution. It is always a good practice to perform some experimental verification as a cross check, at least on the validity of the fundamental model with some easily detectable quantities, such as the resonance frequencies and the impedance curves. Once the fundamental model is verified, one can have enough confidence on other analysis results which are obtained with minor modification of the fundamental model. FEM design simulation gives more flexibility to transducer designers to venture new designs without high cost. All experimental measurable quantities, including the electrical impedance, beam pattern and impulse response can all be calculated using FEM. Proper use of the simulation technique can promote more innovative transducer designs and also greatly reduce the cost of time and money for design optimization r _L F p- \ - Voltage on the Active Element First Neighbor Response I I Response / 7 \\ I/ / \ /,// \ 7 " Time ( Sec.) d p Fig. 5 The electrical response detected at different neighboring elements after the subdicing. 19 Downloaded from SPIE Digital Library on 18 Feb 21 to Terms of Use:

8 Fig. 6 A 'snap-shot' of the pressure distribution generated by one element in the subdiced linear array transducer. ACKNOWLEDGMENTS This research is sponsored by the Office of Naval Research and the Whitaker Foundation under special opportunity award. The author is indebted to the Weidlinger Associates for their generosity in providing the PZ-Flex code. REFERENCES 1. R. Krimholtz, D. A. Leedom and G. L. Matthaei, 'New Equivalent Circuits for Elementary Piezoelectric Transducers", Electronic Lett., Vol. 6, , R. E. Newnham, D. P. Skinner, and L. E. Cross, "Connectivity and Piezoelectric-Pyroelectric composites", Mater. Res. Bull. Vol. 13, pp , W. A. Smith, "The Role of Piezocomposites in Ultrasonic Transducers," Proc IEEE Ultrason. Symp., pp , W. A. Smith and B. Auld, "Modeling 1-3 Composite Piezoelectrics: Thickness-Mode Oscillations', IEEE Trans. Ultrasonic, Ferroelectrics and Frequency Control, Vol. 38, pp 4-47 (199). 5. W. Cao, Zhang Q. and Cross L. E., "Theoretical Study on the Static Performance of Piezoelectric Ceramic-polymer Composites with 1-3 Connectivity," J. App!. Phys., Vol. 72, pp , Q. M. Zhang, Wenwu Cao, Hong Wang and L. Eric Cross, "Characterization of the Performance of 1-3 Type Piezocomposites for Low-Frequency Applications" J. Appi. Phys.,vol. 73, (1993). 11 Downloaded from SPIE Digital Library on 18 Feb 21 to Terms of Use: