Finite Element modeling of Ultrasonic Piezoelectric Transducers Influence

Finite Element modeling of Ultrasonic Piezoelectric Transducers Influence of geometry and material parameters on vibration, response functions and radiated field Jan Kocbach Dr. Scient. Thesis Department of Physics University of Bergen November 2000

Outline è Motivation è Piezoelectric materials è Piezoelectric transducers - noe om objectives med arbeidet må sies tidlig! NB! Fokuser mere på hva som er nytt i dette arbeidet, istedenfor å vise masse forskjellige enkeltting. Se på konklusjoner, objectives osv. for å se hva det er som er nytt!! è FE modeling of transducers and surrounding fluid è Results: – Systematic analysis for piezoelectric disks Egenmoder: – Systematic analysis for piezoelectric disks with a -front layer Nytt at è Conclusions ny mode klassifiserings system. Resp. fu: Nytt at systematisk vist. - Frontlag: Mye nytt og spennende

A typical example of a combined numerical/experi mental trnasducer design process. Motivation è Increased understanding and control in a transducer design process Choose a basic design based on needs and experience Repeat until prototype has desired properties Without appropriate simulation tools, all of the variation of design parameters must be made using Vary design parameters inmeasurements only. simulation tool until desired This is a time. Candidate consuming and properties expensive process. Build a prototype (expensive!) Accurate simulation tool few repetitions required

Piezoelectric materials è Piezoelectric effect: – Mechanical stress applied – Voltage applied surface electrode piezoelectric bar – AC field alternate in size voltage produced mechanical distortion New figure here? sound radiated

Piezoelectric materials è Equations governing a piezoelectric material: Sjekk disse ligningene p[ tensor form!!!

Piezoelectric transducers è Found in a wide range of different applications – e. g. medical ultrasound, ultrasonic measurement Need som more here. systems, sonars, ultrasonic distance measurement, non. Fra sluttrapport! destructive testing, process instrumentation. . . è Many different types of transducer structures – e. g. disk transducers, ring transducers, Needbar som transducers, more here transducer arrays, unimorph/bimorph transducers. . . è Present work restricted to axisymmetric piezoelectric disk transducers

The central part of a piezoelectric disk transducer is the piezoelectric disk. There is a large impedance mismatch between typical piezoelectric materials applied in a piezoelectric disk transducer and air or water, and therefore a front layer is applied for better matching between the piezoelectric disk and the fluid medium. A backing layer is applied at the back of the disk to damp down vibrations. A transducer housing is also often applied, to protect the transducer from the operating environment. Piezoelectric transducers è Axisymmetric piezoelectric disk transducers + Fluid medium Frontlayer Piezoelectric disk Backing layer + Transducer housing

The central part of a piezoelectric disk transducer is the piezoelectric disk. There is a large impedance mismatch between typical piezoelectric materials applied in a piezoelectric disk transducer and air or water, and therefore a front layer is applied for better matching between the piezoelectric disk and the fluid medium. A backing layer is applied at the back of the disk to damp down Present work: vibrations. A transducer Focus on the housing is also often applied, to parts protect basic of the transducer from the transducer: operating environment. Piezoelectric transducers è Axisymmetric piezoelectric disk transducers + Fluid medium Frontlayer Piezoelectric disk Backing layer disk + frontlayer + fluid medium + Transducer housing

Piezoelectric transducers è Transducer properties studied: –Transducer vibration –Transducer response functions - Impedance/Admittance - Source sensitivity response –Radiated sound field - Directivity pattern - Near and farfield pressure field Nevne andre ”transducer properties”, som for eksempel mottakerfølsomhet o. l.

Transducer modeling è 3 D approach required for accurate modeling è Possible approaches: Use commercial simulation tool Develop new simulation tool è New simulation tool developed: FEMP è Why develop new? ¬ Research tool: • Implement/evaluate different methods and identify best approach Systematic analysis • Commercial tools difficult to tailor for efficient systematic analyses

posding No need to mesh fluid region & negding Part of fluid region must be meshe posding Small BE matrices & negding Large FE matrices negding BE matrices are full & posding FE matrices are sparse negding Problems at characteristic frequencies & posding No problems at any freque negding Evaluation of singular integrals& posding All integrals may be evaluated ea negding Computationally intensive to calculate field & posding Field is calculated au negding BE matrices must be set up for every frequeny & posding FE matrices set u Astley-Leis inf. el: Applied of simulation tool in near field. Field easily calculated in far field. Say something about which properties are important for the method implemented here FEMP: Modeling approach è Different modeling approaches evaluated to find best approach – Desired properties • Model complete axisymmetric piezoelectric transducer + fluid medium • Efficient calculation of vibration, response functions and sound field • Efficient calculation in a large frequency band – Modeling approach for piezoelectric medium: • Finite element method / Finite difference method / Boundary element method – Modeling approach for infinite fluid medium : • Fluid finite elements + infinite elements / Fluid boundary elements / Fluid Si enfinite del om de elements + dampers / Fluid finite elements + analytical solution / etc. forskjellige metodene, • conjugated Astley-Leis inf. el. / unconjugated Astley-Leis inf. el. / conjugated or ulemper/fordeler unconjugated Burnett inf. el. / Olson-Bathe inf. el. / etc. osv. her! – Harmonic analysis method : • Mode superposition method / Direct harmonic solution / FFT from time domain – Loss model for piezoelectric medium : • Complex material constants / Structural friction force / Rayleigh damping

FEMP: Theory è FE formulation set up using Galerkin method Set up weak formulation variational formulation Divide region of analysis into elements and nodes Set up FE equations from variational formulation

FEMP: Theory Set up weak formulation (piezoelectric) ( ) (piezoelectric) (fluid) – apply boundary conditions – results in variational formulation ( )

FEMP: Theory Divide region of analysis into elements and nodes Piezoelectric finite elements Fluid infinite elements

FEMP: Theory Divide region of analysis into elements and nodes – Unknowns approximated by nodal values in each element using interpolation functions û 4 û 7 û 8 û 1 û 3 û 6 û 5 û 2 Interpolation functions (quadratic + variable order)

Calculation of one matrix as example? ? the region of analysis is divided into nodes FEMP: Theory When and elements, a set of matrix equations, also called th Size of system? FE equations, may be set up. Boundary Set up FE equations : set of matrix equations conditions? è FE matrices calculated for each local element è Assembled to global FE matrices è Solved for unknown quantities for each frequency

FEMP: Verification Comparison with other FE codes • ABAQUS, ANSYS, CAPA: < 5 ppm difference (resonance freq. ) Comparison with measurements • good qualitative agreement (resonance freq. , response funct. ) Input conductance [S] Input conductance of PZT-5 A disk with D/T=12 in water. f [k. Hz]

FEMP: Verification Comparison with other FE codes • ABAQUS, ANSYS, CAPA: < 5 ppm difference (resonance freq. ) Comparison with measurements • good qualitative agreement (resonance freq. , response funct. ) Comparison with analytical solution normalized pressure p • good quantitative agreement (plane piston pressure field) On-axis pressure field from plane-piston radiator normalized distance S

FEMP: Verification Comparison with other FE codes • ABAQUS, ANSYS, CAPA: < 5 ppm difference (resonance freq. ) Comparison with measurements • good qualitative agreement (resonance freq. , response funct. ) Comparison with analytical solution • good quantitative agreement (plane piston pressure field) Comparison with literature results • good qualitative agreement (resonance freq. , response funct. )

Analysis results è Present analysis made for piezoelectric disks and piezoelectric disks with a front layer • D/T ratio of disk • Thickness of front layer • Disk material • Frontlayer material • Fluid medium (air/water) Dette kan evt. gjøres annerledes ved at man Fluid medium viser hvordan design parametrene endres grafisk, kutter T ut Frontlayer menypunkter påfront at man kun ser på Piezoelectricdisk+frontlag disk T før). (sagt D è Focus of investigations: – How does systematic variation of geometry and material parameters influence on transducer properties?

Challenges related to analysis è All of the analysis: • Quantify the accuracy in the results NB! Ta med Forklare konkl. av FE for bakgrunnen program valg av o. l. tidligere et sted, kvartbølgelag, refine previous mode classification scheme eventuelt skrive og hva som det inn her! forventes av et relation vibrational modes peaks in response functions kvartbølgelag, relation disk vibration radiated sound field hva de 1 D modellene sier, og at det stort sett har blitt analysis for thick disks brukt 1 D modeller i analysis for radial mode transducers bestemmelse av differences in optimal front layer thickness and materials optimal frontlags tykkelse og how to change geometry and material parameters materialvalg. to avoid è Piezoelectric disks: • • • è Piezoelectric disks with a front layer: • • Komme med konklusjonene av analysen her, og så trekke ut eksempler etterpå! unwanted peaks and dips in response functions

Accuracy considerations è Literature: no clear answer for piezoelectric media – 2 -10 elements per needed for “sufficient accuracy” è What is “sufficient accuracy”? – Application dependent • 5 -10% error tolerated in some applications • Resonance freq. (e. g. for material constant evaluation): ppm level è Challenge: – How many elements to get a specified accuracy? • Answer through convergence tests • Accuracy found as function of elements per wavelength

Accuracy considerations è Example: Convergence test for resonance freq. (quadratic elements) 5 elements per wavelength max 0. 25% error 10 elements per wavelength max 100 ppm error è Corresponding convergence tests made for response functions and radiated field

Komme med konklusjonene av analysen her, og så trekke ut eksempler etterpå! Analysis results: Piezoelectric disks è Mode classification scheme NB! Ta med Forklare konkl. av FE for bakgrunnen program valg av o. l. tidligere et sted, kvartbølgelag, eventuelt skrive og hva som det inn her! forventes av et kvartbølgelag, hva de 1 D modellene sier, og at det stort sett har blitt brukt 1 D modeller i bestemmelse av optimal frontlags tykkelse og materialvalg. è Relation between eigenmodes and radiated field è Relation between eigenmodes and response functions

ode classification scheme in the present work s for the remainder of the work. Piezoelectric disks: mode classification è Mode classification scheme developed based on: – resonance frequency spectra – vibration of disks for varying geometry and materials è Vibrational modes classified into – R modes – E modes – A modes – L modes è Thickness extensional modes and thickness shear modes special types of A and L modes

ode classification scheme in the present work s for the remainder of the work. frequency-thickness product [k. Hz mm] Piezoelectric disks: mode classification Example: PZT-5 A disks – Resonance frequency spectrum of PZT-5 A disks Thick disks: – Resonance frequencies of disks shown as a function of D/T ratio D/T=1 Thin disks: D/T=20 Diameter/Thickness ratio

frequency-thickness product [k. Hz mm] Piezoelectric disks: R modes Mode classification difficult due to strong mode coupling to E, A and L modes R modes, associated with 1 st order symmetric Lamb wave in infinite plate Diameter/Thickness ratio

frequency-thickness product [k. Hz mm] Piezoelectric disks: R modes Fundamental radial mode: Disk expands in thickness direction when it contracts in radial direction. R 1: D/T=5 R 1: D/T=10 R 1: D/T=20 Diameter/Thickness ratio

frequency-thickness product [k. Hz mm] Piezoelectric disks: R modes R 4: D/T=10 R 3: D/T=10 R 2: D/T=10 Diameter/Thickness ratio Higher order radial modes: number of nodal circles with zero radial displacement increases with order of radial modes.

frequency-thickness product [k. Hz mm] Piezoelectric disks: R modes R 1: D/T=5 (water) Diameter/Thickness ratio

frequency-thickness product [k. Hz mm] Piezoelectric disks: E mode: D/T=10 E mode: D/T=5 Diameter/Thickness ratio E mode Large axial displacement at circular edge of disk.

frequency-thickness product [k. Hz mm] Piezoelectric disks: E mode E-mode: D/T=5 (water) Diameter/Thickness ratio

Piezoelectric disks: A modes frequency-thickness product [k. Hz mm] New numbering scheme for A modes suggested A 2 mode: D/T=10 A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 1 (TS 1) mode: D/T=8 A 1 (TS 1) mode: D/T=5 Diameter/Thickness ratio A modes, associated with 2 nd order symmetric Lamb mode in infinite plate PZT-5 A: A 1 = TS 1

frequency-thickness product [k. Hz mm] Piezoelectric disks: L modes L 2 mode: D/T=10 L 1(TE 1) mode: D/T=14 L 1(TE 1) mode: D/T=10 ideal: no R mode coupling Diameter/Thickness ratio L modes, associated with 3 rd order symmetric Lamb mode in infinite plate PZT-5 A: L 1 = TE 1

frequency-thickness product [k. Hz mm] Piezoelectric disks: L modes L 1 (TE 1) mode D/T=5 (water) Diameter/Thickness ratio

Piezoelectric disks: response functions è Relation between peaks in response functions and eigenmodes: – Found by comparing resonance frequency spectrum and response functions – Example for electrical input conductance of PZT-5 A disk with D/T=10

Piezoelectric disks: response functions R 12 L 1 (TE 1) A 4 R 9 A 3 A 2 A 1 R 8 R 7 R 6 E R 5 R 4 R 3 R 2 R 1 Diameter/Thickness ratio Conductance [m. S] frequency-thickness product [k. Hz mm] – Relation: Resonance frequency spectrum response functions

Komme med Analysis results: konklusjonene analysen her, Piezoelectric disk with a frontlayerav og så trekke ut è Why to use a frontlayer - general theory eksempler etterpå! è 1 D model results valid for thin disks NB! Ta med Forklare konkl. av FE for bakgrunnen program valg av o. l. tidligere et sted, kvartbølgelag, eventuelt skrive og hva som det inn her! forventes av et kvartbølgelag, hva de 1 D modellene sier, og at det stort sett har blitt brukt 1 D modeller i bestemmelse av optimal frontlags tykkelse og materialvalg. è 3 D model results for thick disks è 3 D model results for radial mode transducers

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Så sette frontlag imellom, og illustrere at Fluid mer medium: Z 400 rayl energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk Piezoelectric disk: Z 35 illustrasjon på dette? Skiven ”ser” større strålingsimpeda D ns! Piezoelectric disks with a front layer: General theory è Piezoelectric disk in water/air/gas: -1. 5 Mrayl quarterwave transformator? T – large acoustic impedance mismatch: • low bandwidth • low acoustic transmission coefficient -> of intermediate acoustic impedance

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Så sette frontlag imellom, og illustrere at Fluid mer medium: Z 400 rayl -1. 5 Mrayl energi kommer gjennom pga. transformator. Frontlayer: Z = ? ? ? Hvordan illustrere? Grafisk Piezoelectric disk: Z 35 Mrayl illustrasjon på dette? Skiven ”ser” større strålingsimpeda D ns! Piezoelectric disks with a front layer: General theory è Piezoelectric disk in water/air/gas: Tfront T Solution: layer -> of for better intermediate acoustic transmission impedance quarterwave transformator? matching – large acoustic impedance mismatch: Important parameters • low bandwidth • low acoustic transmission coefficient for bandwidth and transmission

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Så – Quarterwave thick frontlayer for optimal transmission sette frontlag – Optimal acoustic impedance of front layer given by: imellom, og illustrere at mer – Zfront = (Zpiezo Zfluid)1/2 quarterwave energi kommer 1/3 2/3 transformator? gjennom pga. – Zfront =2 (Zpiezo ) (Zfluid) transformator. – Z -> of 1/3 ( Z 2/3 front = (Zpiezo ) fluid) Hvordan intermediate illustrere? acoustic Grafisk impedance – Other optimal values for frontlayer thickness and acoustic impedance illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: Example è Most previous work: 1 D, thin disks/plates, TE 1 mode è Other transducer configurations and modes: è Example: – PZT-5 A (D/T=5) with epoxy frontlayer (Zfront=4. 17 Mrayl ) in water – Comparison 1 D model and 3 D model for TE 1 mode region

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Så Without sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? 1 D model Grafisk result illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: Thick disk, TE 1 mode region – Without frontlayer: One narrow peak – Two peaks in response functions with positions varying with Tquarterwave front. normalized frequency f/f. TE 1 transformator? -> of intermediate acoustic impedance normalized frontlayer thickness Tfront /T /4, TE 1 conductance [d. B re 1 m. S] frontlayer

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Tfront Så = 1. 0 sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: Thick disk, TE 1 mode region – Equal height at quarterwave thickness – Low Zfront one wide peak at quarterwave thickness normalized frequency f/f. TE 1 quarterwave transformator? -> of intermediate acoustic impedance normalized frontlayer thickness Tfront /T /4, TE 1 conductance [d. B re 1 m. S] T /4, TE 1 Zfront = 4. 17 Mrayl

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Så Without front sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan TE 1 illustrere? A 2 Grafisk illustrasjon på A 1 dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: Thick disk, TE 1 mode region normalized frequency f/f. TE 1 quarterwave transformator? -> of intermediate acoustic impedance normalized frontlayer thickness Tfront /T /4, TE 1 conductance [d. B re 1 m. S] layer

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Tfront Så = 0. 6 T /4, TE 1 Zfront sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? A 2 TE 1 Grafisk illustrasjon på dette? Skiven. A 1 ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: Thick disk, TE 1 mode region normalized frequency f/f. TE 1 quarterwave transformator? -> of intermediate acoustic impedance normalized frontlayer thickness Tfront /T /4, TE 1 conductance [d. B re 1 m. S] = 4. 17 Mrayl

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Tfront Så = 0. 8 T /4, TE 1 Zfront sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk A 2 TE 1 illustrasjon på dette? Skiven A 1 ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: Thick disk, TE 1 mode region normalized frequency f/f. TE 1 quarterwave transformator? -> of intermediate acoustic impedance normalized frontlayer thickness Tfront /T /4, TE 1 conductance [d. B re 1 m. S] = 4. 17 Mrayl

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Tfront Så = 1. 0 T /4, TE 1 sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på A 2 dette? Skiven ”ser” større strålingsimpeda ns! Zfront = 4. 17 Mrayl normalized frequency f/f. TE 1 problem quarterwave transformator? -> of intermediate acoustic impedance normalized frontlayer thickness Tfront /T /4, TE 1 conductance [d. B re 1 m. S] Piezoelectric disks with a front layer: Thick disk, TE 1 mode region

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Så sette frontlag – Several different front layer materials imellom, og illustrere at mer quarterwave – Avoid dips in otherwise flat response by minor change of energi kommer transformator? gjennom pga. D/T ratio transformator. -> of – Highest bandwidth found for frontlayer which is 10 -15% Hvordan intermediate illustrere? acoustic thinner than a quarterwave thick Grafisk impedance illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: Thick disk, TE 1 mode region è Corresponding systematic analysis made for the source sensitivity response (acoustic response):

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive eksplisitt? ). Så sette frontlag imellom, –og Similar behaviour as for TE 1 mode region? illustrere at mer quarterwave – Influence from flexural modes in the disk? transformator? energi kommer gjennom pga. – Influence of changing shear velocity of front layer? transformator. -> of Hvordan intermediate illustrere? acoustic Grafisk impedance illustrasjon på dette? Skiven – PZT-5 A (D/T=3 / D/T=10) with epoxy frontlayer (Zfront=2. 64 Mrayl ”ser” større strålingsimpeda – Source sensitivity response (Rayleigh integral) ns! Piezoelectric disks with a front layer: R 1 mode region è No previous systematic analysis for R 1 mode region è Questions: è Example: R 1 mode region (in-air analysis) )

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi gjennom (skrive D/T=3, eksplisitt? ). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: R 1 mode region normalized frequency f/f. R 1 quarterwave transformator? -> of intermediate acoustic impedance R 1 F normalized frontlayer thickness Tfront /T /4, R 1 sensitivity [d. B re 1 Pa/V] without frontlayer

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi Tfront (skrive = 1. 0 T /4, R 1 gjennom eksplisitt? ). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: R 1 mode region normalized frequency f/f. R 1 Similar to TE 1 mode region: Splitting into two symmetric quarterwave peaks with equal height. transformator? -> of intermediate acoustic impedance R 1 F normalized frontlayer thickness Tfront /T /4, R 1 sensitivity [d. B re 1 Pa/V] Zfront = 2. 64 Mrayl, D/T=3

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi Tfront =(skrive 1. 0 T /4, R 1 gjennom eksplisitt? ). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: R 1 mode region normalized frequency f/f. R 1 D/T ratio of piezoelectric disk changedquarterwave from 3 to 10 normalized frequency f/f. R 1 transformator? -> of intermediate acoustic impedance F R 1 normalized frontlayer thickness Tfront /T /4, R 1 sensitivity [d. B re 1 Pa/V] Zfront = 2. 64 Mrayl, D/T=10

Først disk+medium. Vise impedans. Stor impedansforskje ll -> lite energi Tfront =(skrive 1. 0 T /4, R 1 gjennom eksplisitt? ). Så sette frontlag imellom, og illustrere at mer energi kommer gjennom pga. transformator. Hvordan illustrere? Grafisk illustrasjon på dette? Skiven ”ser” større strålingsimpeda ns! Piezoelectric disks with a front layer: R 1 mode region normalized frequency f/f. R 1 Shear velocity of front layer lowered 30% quarterwave normalized frequency f/f. R 1 transformator? -> of intermediate acoustic impedance F R 1 normalized frontlayer thickness Tfront /T /4, R 1 sensitivity [d. B re 1 Pa/V] Zfront = 2. 64 Mrayl, D/T=10

Conclusions è FE code developed : – based on evaluation of different modeling approaches – accuracy in FE results quantified è Contribute to increased control and understanding in a transducer design process – based on varying design parameters for simple piezoelectr disk transducers è Suggestions for further work: – Corresponding analysis for more complex transducers – Include fluid flow and receiving transducer in analysis