SIGNAL MODELLING USING THE FLOSIM SYSTEM MODEL IN ULTRASONIC INSTRUMENTATION FOR INDUSTRIAL APPLICATIONS

Per Lunde1, Remi André Kippersund1 and Magne Vestrheim2

1 Christian Michelsen Research AS (CMR), P.O. Box 6031, N-5892 Bergen, Norway, per.lunde@cmr.no 2 University of Bergen (UoB), Department of Physics, Allégaten 55, N-5007 Bergen, Norway

ABSTRACT

A development of the FLOSIM system model for de-scription of signal transmission in ultrasonic transmit-receive measurement systems incorporating finite-element modelling (FEM) of the piezoelectric transduc-ers and the fluid medium, is described. An example from high-precision custody transfer flow metering of natural gas is used to illustrate the capabilities and pos-sibilities of a system model of this type, in relation to system design, signal properties and signal processing. Challenges related to use of FE modelling in FLOSIM type of descriptions are addressed and discussed.

1. INTRODUCTION

Ultrasonics is today used within a broad spectrum of in-dustrial applications, such as medical ultrasound, mili-tary and civil marine acoustics, fishery acoustics, oil and gas flow metering, process metering, subsea instrumen-tation, downhole instrumentation, non-destructive test-ing, meteorology and atmospheric research, etc. The role of ultrasound technology is becoming increasingly important. Although the various ultrasonic instruments and applications are subject to different challenges, and different signal processing strategies may be used, there are common factors for many of these instruments which need to be taken into account in the signal processing strategy.

An ultrasonic measurement system is normally based on one (or several) transmitting transducer(s) with dedi-cated electronics, a medium in which the ultrasound propagates (and which normally is to be measured or characterised in some way), one (or several) receiving transducer(s) with its dedicated electronics, signal filter-

ing and signal detection / processing. Such components of the ultrasonic measurement system (transducers, elec-tronics, filters, etc.) have their specific frequency charac-teristics, which in practice strongly influences on the ul-trasound signal, and thus on the signal processing strate-gies and solutions being chosen. For several reasons, such as e.g. to optimise the physical hardware compo-nents of the system and the signal processing methods, there may be significant advantages in being able to de-scribe the ultrasonic transmit-receive system as a unit, with such components working together.

An extensive effort has been done internationally particularly during the recent decennia on developing powerful signal simulation models for such ultrasonic transmit-receive measurement systems. Some of this work is discussed in the references cited here [1-23] and in the further works referred to there.

The present paper describes a development of the FLOSIM numerical simulation model to improve the de-scription of such measurement systems.

2. THE FLOSIM SYSTEM MODEL

2.1. Simplified approach using 1D acoustic modelling

A first version of the FLOSIM system model for descrip-tion of an ultrasonic transmit-receive measurement sys-tem was described in [1,2]. FLOSIM consists of models for the signal generator, the transmitting electronics, the transmitting piezoelectric transducer, the propagation medium, the receiving piezoelectric transducer, the re-ceiving electronics, filters, and the electrical termination (e.g. analog-to-digital conversion, ADC), cf. Fig. 1.

V0 V1 V2 V3 u4 P5P6

V7 V8

Bandpassfilter

Electronics

Sendertransducer

MEDIUM

Receivertransducer

Electronics

Electricterminator

Signalgenerator

Fig. 1. Schematic overview of the module-based FLOSIM numerical simulation model for time and frequency domain modelling of an acoustic measurement system.

Interactive modelling in time and frequency domain is possible, including modelling and analysis of the sig-nal and its frequency spectrum as it propagates from one node to another in the component network, as well as the transfer function and its impulse response between any two nodes in the network.

For example, in the configuration shown in Fig. 1, the output voltage frequency spectrum at node no. 8 is calculated as

7

8

6

7

5

6

4

5

3

4

2

3

1

2

0

108 V

VpV

pp

up

Vu

VV

VV

VV

VV = , (1)

where V denotes voltage, p sound pressure in the fluid propagation medium, and u volume velocity. Subscripts represent node numbers in the model, at which the time signals, frequency spectra, transfer functions and im-pulse responses can be calculated. Note that node nos. 5 and 6 represent the sound pressure in the fluid at an axial distances of 1 m and at the receiver, respectively, the latter averaged over the receiver front (denoted by ⋅ ). The main calculations are performed in the frequency domain. Time domain responses (signal traces and im-pulse responses) are obtained through inverse Fast Fou-rier Transform (FFT) techniques [1].

The FLOSIM model has been further developed and used in industrial development work at CMR for more than a decade, for applications within e.g. custody trans-fer metering of natural gas and oil, wet gas metering, separator monitoring, high-temperature ultrasonics, flare gas metering, etc. (cf. e.g. [3-9]).

Signalgenerator

Receivingcables &electronics

Transmittingcables &electronics

Pulsedetection

Transmittingtransducer

TOP VIEW FRONT VIEW

Receivingtransducer

2Ryi

z

xL pi

φi

Fig. 2. (a) Schematic illustration of a single path in a multi-

path ultrasonic transit time flow meter (USM), and (b) photograph of FMC Kongsberg Metering’s MPU 1200 ultrasonic fiscal gas flow meter, originally developed by CMR [11,13,14].

The use of the FLOSIM model has been based on

utilizing simplified one-dimensional (1D) Mason type of

models for describing the effects of the transmitting and receiving transducers [10, 2], and using the plane baffled piston model for describing the sound field and diffrac-tion effects in the propagation medium. As is well known, use of such models can impose severe limita-tions on the transducer and sound field simulations. Only some vibration modes of the piezoelectric elements can be modelled in this approach, in an idealized (one-dimensional (1D)) way, where the complicated vibration patterns experienced in practice are not properly repre-sented. The propagation of more complicated vibration patterns into the near field and far field is also not prop-erly described. The description of effects of other con-structional components, for a full transducer structure, such as matching layers, backing layers, transducer cas-ing, etc., is also usually insufficiently described. There is thus a significant need to implement far more powerful modelling tools both for the piezoelectric transducers and their interaction with sound fields, in order to obtain more accurate simulation of ultrasonic measurement sys-tems in a system simulation model such as FLOSIM.

2.2. Incorporation of finite element acoustic model-ling (FEM) in FLOSIM

To overcome such simplifications and limitations, work has been started on using data resulting from finite ele-ment modelling (FEM) of piezoelectric transducers and sound radiation in the fluid medium, in the FLOSIM model. FEM is used for numerical calculation of the transfer functions related to the transmitting transducer, fluid medium and receiving transducer modules, cf. Fig. 1. That is, the transfer functions

3

4Vu

, 4

5up

, 5

6

pp

and 6

7pV

(2)

may be calculated using FEM, as an additional function-ality to the Mason and plane piston type of models which have been available earlier, cf. Section 2.1.

The FE model used is based on the FEMP 3.0 model previously developed in a cooperation between CMR and UoB [17-19]. This model can describe effects of dif-ferent parts in the transducer structure as well as the ef-fects of the resulting vibrations on the radiated sound field, under the assumption of axial symmetry. In addi-tion to being used and tested versus measurement results and numerical accuracy in connection with transducer design and construction, FEMP has been further devel-oped such as to describe also the receiving transducer (voltage and current reception), animation of transducer vibration and sound propagation, inclusion of fluid me-dium losses, adaptation to FLOSIM time domain type of modelling, etc. The present code is denoted FEMP 3.4.

(b)

(a)

3. RESULTS AND DISCUSSION

3.1. Application to custody transfer flow metering of natural gas and oil

To illustrate the possibilities, capabilities and potentials of a system model of this type, examples from high-precision custody transfer flow metering of natural gas and oil [11-16] are used, cf. Fig. 2.

0 0.01 0.02 0.03 0.04 0.05

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

r [m]

z [m

]

Fig. 3. (a) Illustration (largely simplified) of type of mesh used in FE modelling of transducers and sound fields

(finite and infinite elements), and (a) example of a metal encapsulated ultrasonic transducer used in high-precision ultrasonic flow metering of oil.

In Fig. 3b a metal encapsulated ultrasonic transducer is shown as an example of related applications in ultra-sonic precision flow meters (USM) for fiscal measure-ment of natural gas and oil (sales metering), which has been designed using FEMP as one disign tool. Fig. 3a illustrates the geometry and finite element mesh used for the modelling of a more simplified transducer structure example, consisting of a piezoelectric element with a front matching layer and a backing layer. A 12 x 3 mm Pz27 type of piezoelectric element [20]) was used here.

The 12 x 4 mm front layer (quarter wavelength thick at 150 kHz) was modelled using a density of 1100 kg/m3, compressional velocity of 2500 m/s, Poisson’s ra-tio of 0.31, and compressional and shear Q factors of 20. The 12 x 18 mm tungsten/araldite backing layer was modelled using a density of 10000 kg/m3, compressional velocity of 1600 m/s, Poisson’s ratio of 0.2, and com-pressional and shear Q factors of 5. The propagation medium (surrounding the transducer) was natural gas at 25 bar, with density and sound velocity taken as 20 kg/m3 and 400 m/s, respectively.

0 50 100 150 200 250 300 350

-120

-110

-100

-90

-80

-70

Electrical conductance

Frequency [kHz]

20lo

g 10 G

[dB

re 1

S]

FE modelling1D/Piston modelling

0 50 100 150 200 250 300 35060

70

80

90

100

Frequency [kHz]

20lo

g 10|Z

| [dB

re 1Ω

]

Electrical impedance

FE modelling1D/Piston modelling

0 50 100 150 200 250 300 350-60

-50

-40

-30

-20

-10

0

Frequency [kHz]Mag

nitu

de [d

b re

1P

a/V

at 1

m]

Source Sensitivity

FE modelling1D/Piston modelling

0 50 100 150 200 250 300 350

-100

0

100

200

300

Frequency [kHz]

Slo

w P

hase

[deg

rees

]

Phase of Source Sensitivity divided by phase factor e-ik

FE modelling1D/Piston modelling

5.0 dB/div

180°

170°

160°

150° 140°

130°

120°

110°

100°

90°

80°

70°

60°

50°

40°

30

° 20

°

10

°

0

°

-10

°

-20

°

-30

°

-40

°

-50°

-60°

-70°

-80°

-90°

-100°

-110°

-120° -130°

-140° -150° -160°

-170°

-180°

(a) (b)

(c) (d)

(a) (b)

Fig. 4. Examples of FE modelling results (thick lines) for the transducer shown in Fig. 3a, and the radiated sound field (in natural gas): (a) input electrical conductance and impedance, (b) source sensitivity (magnitude and slow phase), (c) transducer vi-bration (exaggerated) and radiated sound field (nearfield) at 150 kHz, and (d) directivity (farfield, at 1 m dist.) at 150 kHz. Corresponding Mason type 1D / plane piston modelling results are shown in (a) and (b) using thin lines, for comparison.

As an example, a single path in a USM, with source-receiver distance 25 cm is considered here. Infinite ele-ments are used for the outermost radiated field, beyond 2.5 cm, cf. Fig. 3a. For clarity, only a reduced mesh consisting of 292 elements is shown in Fig. 3a. (The re-sults given in Figs. 4 and 5 have been calculated using a mesh of 16736 8-node isoparametric finite elements.)

In Fig. 4a the input electrical conductance and the impedance magnitude of the transducer structure in Fig. 3a are shown over a frequency range up to 350 kHz, as examples of FE calculations using the FEMP model (thick lines). Fig. 4b shows the calculated magnitude and phase of the voltage source sensitivity (i.e. p5/V3, cf. Fig. 1) for the same transducer and frequency range (thick lines). The calculated transducer vibration and sound ra-diation from this transducer (only the nearfield shown here) and the corresponding directivity pattern (farfield) are shown in Figs. 4c and d, respectively, both at 150 kHz (i.e. close to the fundamental radial mode of the pie-zoelectric element).

Frequency response functions such as those demon-strated in Figs. 4a and b are needed in the FLOSIM model, cf. Section 2.1. In order to obtain accurate re-sults in the time domain, the frequency domain response functions (magnitude and phase) need to be calculated with sufficient accuracy and resolution, over a suffi-ciently wide frequency range.

For the present calculations, the excitation voltage signal V0(t) is given as a 10 cycle constant amplitude (2 Vp-p) tone burst, starting positive. The internal imped-ance of the signal generator is Zs = 50 Ω (real). The car-rier frequency of the excitation signal is 150 kHz, as a representative example of gas USMs. 6 MHz sampling frequency and 8192 sampling points were used. FE cal-culations were made to 1 MHz, above which the system response was set to zero (implying a rectangular lowpass filter). To avoid effects of (a) numerical inaccuracies of the FE calculations in the 400 - 1000 kHz band, and (b) effects of the truncation at 1 MHz, a Butterworth low-pass filter (LPF) was used, with -3dB cutoff at 300 kHz, and 40 dB/oct decay rate. To make the system as simple

as possible, and still with sufficient realism to illustrate the main items, no specific electronics was used in these simulations.

The numerical inaccuracies of the FE calculations in the 400 - 1000 kHz band (not shown in Fig. 4) are re-lated to insufficient number of finite elements per wave-length used in this high frequency band for the calcula-tions shown here. High accuracy in this band is not critical for the results shown in the present work.

Fig. 5a shows an example of a signal trace calculated at node 6 in the network of Fig. 1 (the sound pressure in the gas at the receiving transducer front, averaged over the front surface, <p6>).

3.2. Comparison with simplified Mason 1D and plane piston type modelling

For comparison, corresponding calculations have been made using the Mason type 1D model for the transduc-ers, combined with the plane-piston-in-infinitely-hard-baffle model for the sound propagation in the gas be-tween the transducers. Note that since today there is no useful analytical model available for the piezoelectric element’s radial modes describing the effects of front layer, backing and radiation load, a Mason type thick-ness-extensional (TE) mode model was used to model the transducer vibration in the frequency range of the fundamental radial mode (150 kHz). That is, in these 1D simulations the thickness of the piezoelectric element was set to 13 mm (instead of 3 mm). Otherwise the transducer and the medium were the same as in the FE simulations (except that shear waves are not accounted for in the Mason model). Use of the TE mode to “de-scribe” the radial mode is of course an incorrect simpli-fication, but still perhaps the only approach available to-day in terms of 1D modelling.

In Figs. 4a and b these simplified modelling results are shown using thin lines. Although many of the same response features are indeed present in these simplified simulation results as in the FE results, it appears that both the electrical and acoustical responses are signifi-

600 620 640 660 680 700 720 740 760 780 800

-1.5

-1

-0.5

0

0.5

1

1.5

Sound pressure in gas at receiving transducer (averaged over front surface)

Time [µs]

Pre

ssur

e am

plitu

de [P

a]

FE modelling

600 620 640 660 680 700 720 740 760 780 800

-1.5

-1

-0.5

0

0.5

1

1.5

Sound pressure in gas at receiving transducer (averaged over front surface)

Time [µs]

Pre

ssur

e am

plitu

de [P

a]

1D/Piston modelling

(a) (b)

Fig. 5. Examples of FLOSIM time domain modelling results corresponding to Figs. 3b and 4; the sound pressure in the gas at the receiving transducer front, averaged over the front surface, <p6> (node 6 in Fig. 1). Transducer response and sound propa-gation in the gas calculated using (a) finite-element modelling (FEMP) and (b) 1D Mason 1D type transducer model, com-bined with the plane-piston-in-infinitely-hard-baffle sound propagation model.

cantly different from the FE calculated responses. The electrical impedance is one order of magnitude higher, the source sensitivity is lower, the response has a differ-ent “shape”, with incorrect description of details, and the bandwidth is different.

All of these properties contribute to the shape and properties of the acoustic signal, e.g. the pressure at the receiver, shown in Fig. 5b.

By comparison with Fig. 5a, the difference in signal form and level is apparent. In USMs, the signal form (e.g. the signal rise time and possible “overshoot”) may be important in relation to achieving reliable and high precision transit time detection. Normally a rapid rise (wide bandwidth) is desired, and the 1D / plane piston modelling approach is obviously not capable of describ-ing the signal and its shape sufficiently accurate.

This new capability to use FE modelling as an inte-grated part of the system modelling opens up highly in-teresting perspectives for improved design of ultrasonic measurement systems, including signal generation, elec-tronics, transducers and signal processing. For optimum design, these parts need to be treated as a unit.

For example, in relation to USM flow metering of oil and gas, essential factors in relation to signal processing are e.g. requirements to achieve reciprocity, optimum signal shaping, high-precision transit time detection (or-der of ns for gas meters, and tens of ps for oil meters), correction of transit times (e.g. transducer delay, diffrac-tion effects, possible lack of reciprocity), etc. For all of these factors, the capability to accurately model the elec-troacoustic system for the transducer vibration mode and system frequency response in question provides perspec-tives for improved USM system design.

3.3. Challenges

There are definitely challenges related to use of models such as FLOSIM and FEMP for applications as dis-cussed above. Some of the ongoing work is concen-trated on: (a) ensuring sufficiently reliable and accurate results for the FE calculated functions to be used in FLOSIM, with respect to magnitude and phase re-sponses, (b) ensuring correct physical interpretations and uses of the FEM results, (c) obtaining FEM based inte-grated diffraction effects (diffraction corrections) on the receiving transducer, and (d) performing comparisons with simulations using the more traditional models for the transducers and sound field which are already avail-able in FLOSIM, and with experimental results.

In the comparisons with experimental results, the available data for the material constants become crucial. For piezoelectric materials, for example, only typical data with a low accuracy (5-20 %) may be available from the manufacturer, at best. Such data are obtained using standardized measurement methods (cf. e.g. [21]).

However, the standardized methods have been shown to be physically incorrect because only one-dimensional models are used in the analysis of the measurements, re-sulting in systematic errors which can be up to several percent [22]. Practical work shows that adjusted values of the material constants may have to be used in the simulations. That has been done in the simulations shown in Figs. 4 and 5.

Also for non-pezoelectric materials the availability of reliable material data may be a challenge. The impor-tance of this topic may be illustrated by an example: by changing the Poisson’s ratio of the transducer’s front layer to 0.4, the frequency response and signal form is considerably changed. This may also illustrate the im-portance of using FE modelling, - such change would not be described using the Mason type of model.

As FEM accuracy is closely connected to number of elements per wavelength, gas media are normally - for a given frequency, due to the longer wavelength - numeri-cally more demanding than liquid media, with respect to computer time, memory, accuracy etc.

4. CONCLUSIONS

Since in order to correctly process your signal you need to know your signal, better knowledge of the measure-ment signal will lead to improved processing and detec-tion capabilities, with a more accurate, powerful and economic measurement system as a result.

By implementing FE based modelling (of the trans-mitting and receiving transducers, and their interactions with and propagation in the sound field) in an ultrasonic system models such as FLOSIM, a significantly more powerful and versatile simulation tool for design of ul-trasonic instruments is obtained. A considerably more extensive and relevant range of problems can be run much more physically correctly and accurately than ear-lier. This results in the capability for use in more exten-sive simulations and studies of the signal propagation and signal processing in ultrasonic measurement systems than has been available before. Challenges are definitely faced in this approach, such as with respect to availabil-ity and correctness of material data, control of model ac-curacy and numerical accuracy (especially at high fre-quencies), etc. Developments are being addressed to overcome such challenges.

5. ACKNOWLEDGEMENTS

The present work has been supported by The Research Council of Norway (NFR), under a 4-year strategic insti-tute programme “Ultrasonic technology for improved exploitation of petroleum resources” (2003-06). In addi-tion, the work has evolved from project cooperation over time related to USM fiscal metering of gas and oil (cus-tody transfer, sales and allocation metering), involving

several partners: The Norwegian Society of Oil and Gas Measurement (NFOGM), the Norwegian Petroleum Di-rectorate (NPD), GERG (Groupe Européen de Reserches Gazières), FMC Kongsberg Metering, FMC Smith Meter (USA), Roxar Flow Measurement, Statoil, Norsk Hydro and ConocoPhillips.

6. REFERENCES

[1] A. Lygre, M. Vestrheim, P. Lunde, V. Berge, ”Nu-merical simulation of ultrasonic flowmeters”, Proc. Ultrason. Intern. 1987, London, UK, July 6-9, 1987, Butterw. Scient. Ltd., UK, 1987, p. 196-201.

[2] P. Lunde and M. Vestrheim, "Piezoelectric transducer modelling. Thickness mode vibration", CMI Report No. CMI-91-A10003, Chr. Michelsen Institute, Bergen, Norway, Dec. 1991.

[3] M. Vestrheim, R. Bø, P. Lunde and A. Lygre, "Aluminium elektrolysecelle. Ultralydmålinger av sjikttykkelser. Forprosjekt", CMI Rep. 89/10400-1, Chr. Michelsen Institute, Bergen, Norway, June 1989. (Confidential, in Norwegian.)

[4] M. Vestrheim, R. Bø, P. Lunde, A. Lygre, and K. S. Mylvaganam, "Aluminium elektrolysecelle. Ultralydmåling av sjikttykkelser. Akustisk transmisjon i flytende kryolitt", CMI Rep. CMI-90-A10005, Chr. Michelsen Institute, Bergen, Norway, April 1990. (In Norwegian.)

[5] P. Lunde, M. Vestrheim and R. Bø, "Aspects of transducer modelling", Proc. 13th Scand. Coop. Meeting in Acoustics/Hydrodyn., Ustaoset, 28-31 January 1990. Scient./Tech. Rep. 227, Dept. of Physics, Univ. of Bergen, Norway, 1990, pp. 59-72.

[6] Ø. Nesse and P. Lunde, "Separator instrumentation. Preliminary study of ultrasonic interface detection through a separator steel wall. TOP - Online monitoring of oil and gas processes", CMR Work Note 93-10-2, Christian Michelsen Research AS, Bergen, Norway, Nov. 1993. (Confidential.)

[7] P. Lunde and Ø. Nesse, "Wet gas measurement. Ultrasonic transducer for thickness measurement of liquid film. TOP - Online monitoring of oil and gas processes", CMR Rep. CMR-95-F10041, Christian Michelsen Research AS, Bergen, Norway, Dec. 1995. (Confidential.)

[8] P. Lunde and Frøysa, K.-E., "Influence of piezoelectric transducer response on transit time determination using cross-correlation of chirp signals", in Proc. 20th Scand. Meeting in Phys. Acoustics, Ustaoset, Norway, February 2-5, 1997. Scient./Tech. Rep. 1997-04, Univ. of Bergen, Dept. of Physics, April 1997, p. 12.

[9] P. Lunde, K.-E. Frøysa and M. Vestrheim, “Tran-sient diffraction effects in ultrasonic flow meters for gas and liquid”, in Proc. 26th Scand. Symp. on Phys. Acoustics, Ustaoset, Norway, 26-29 January 2003. Scient./Tech. Rep. 420304, Norw. Univ. of

Sc. and Tech., Dept. of Telecomm./Acoustics, Trondheim, June 2003. (CD issue only.)

[10] D.A. Berlincourt, D.R. Curran and H. Jaffe, “Pie-zoelectric and piezomagnetic materials and their function in transducers “, in: Physical Acoustics. Principles and Methods. Volume I - Part A, ed. W.P. Mason, Academic Press, New York, 1964, Chapter 3.

[11] A. Lygre, T. Folkestad, R. Sakariassen and D. Al-dal, “A new multi-path ultrasonic flow meter for gas”, in Proc. of 10th North Sea Flow Measurem. Worksh., Peebles, Scotland, 27-29 October, 1992.

[12] A. Lygre, P. Lunde and K.-E. Frøysa, Present status and future research on multi-path ultrasonic gas flow meters, GERG Tech. Monograph TM-8, Groupe Européen de Recherches Gazières (GERG), Groeningen, The Netherlands, 1995.

[13] K.-E. Frøysa, P. Lunde, R. Sakariassen, J. Grendstad and R. Norheim, “Operation of multi-path ultrasonic gas flow meters in noisy environ-ment”, in Proc. North Sea Flow Measurement Workshop, Scotland, 28-31 Oct. 1996.

[14] P. Lunde, K.-E. Frøysa, J. B. Fossdal and T. Heis-tad, “Functional enhancements within ultrasonic gas flow measurement”, in Proc. 17th North Sea Flow Measurem. Worksh., Oslo, Norw., 25-28 Oct. 1999.

[15] P. Lunde, P., K.-E. Frøysa and M. Vestrheim (eds.), GERG project on ultrasonic gas flow me-ters, Phase II, GERG Tech. Monograph TM 11, Groupe Européen de Recherches Gazières (GERG), VDI Verlag, Düsseldorf, 2000.

[16] P. Lunde and K.-E. Frøysa, Handbook of uncer-tainty calculations. Ultrasonic fiscal gas metering stations, prep. on behalf of the Norwegian Society for Oil and Gas Measurement (NFOGM) and the Norwegian Petroleum Directorate (NPD), ISBN 82-566-1009-3, Dec. 2001.

[17] J. M. Kocbach, P. Lunde and M. Vestrheim, “FEMP - Finite element modeling of piezoelectric structures. Theory and verification for piezoce-ramic disks”, Scient../Tech.Rep. 1999-07, Univ. of Bergen, Dept. of Physics, 1999.

[18] J. M. Kocbach, “Finite element modeling of ultra-sonic piezoelectric transducers. Influence of ge-ometry and material parameters on vibration, re-sponse functions and radiated field”, Dr. Scient. thesis, Univ. of Bergen, Dept. of Physics, Norway, Sept. 2000.

[19] J. M. Kocbach, P. Lunde and M. Vestrheim, “Reso-nance frequency spectra with convergence tests for piezoceramic disks using the Finite Element Method”, Acustica, 87, 271-285, 2001.

[20] “Piezoelectric ceramics”, Data brochure, Ferrop-erm, Denmark (2002).

[21] “IEEE standard on piezoelectricity”, ANSI/IEEE std. 176-1987, The Inst. of Electr. and Electron. Eng., Inc., New York, 1988.

[22] M. Vestrheim and R. Fardal, “Methods for deter-mining constants of piezoelectric ceramic materi-als”, in Proc. 25th Scand.. Symp. on Phys. Acous-tics, Ustaoset, Norway, 27-30 January, 2002. Sci-ent./Tech. Rep. 420103, Norw. Univ. of Sc. and Techn., June 2002. (CD issue only).

[23] M. Vestrheim, “Akustiske målesystemer”, Lecture notes for the course PHYS373 Akustiske målesys-temer, Dept. of Physics, Univ. of Bergen, Norway. Revised and extended version spring 2004 (in Norwegian).