Design optimization of embedded ultrasonic transducers for ......Design optimization of embedded ultrasonic transducers for concrete structures assessment Cedric Dumoulin´ a,, Arnaud

Design optimization of embedded ultrasonic transducers for concrete structuresassessment

Cédric Dumoulina,∗, Arnaud Deraemaeker a,∗∗

aUniversité libre de Bruxelles (ULB), École polytechnique de Bruxelles, Building, Architecture and Town Planning (BATir) department

Abstract

In the last decades, the field of structural health monitoring and damage detection has been intensively explored. Active vibrationtechniques allow to excite structures at high frequency vibrations which are sensitive to small damage. Piezoelectric PZT trans-ducers are perfect candidates for such testing due to their small size, low cost and large bandwidth. Current ultrasonic systems arebased on external piezoelectric transducers which need to be placed on two faces of the concrete specimen. The limited accessibilityof in-service structures makes such an arrangement often impractical. An alternative is to embed permanently low-cost transducersinside the structure. Such types of transducers have been applied successfully for the in-situ estimation of the P-wave velocity infresh concrete, and for crack monitoring. Up to now, the design of such transducers was essentially based on trial and error, or in afew cases, on the limitation of the acoustic impedance mismatch between the PZT and concrete. In the present study, we explore theworking principles of embedded piezoelectric transducers which are found to be significantly different from external transducers.One of the major challenges concerning embedded transducers is to produce very low cost transducers. We show that a practicalway to achieve this imperative is to consider the radial mode of actuation of bulk PZT elements. This is done by developing a simplefinite element model of a piezoelectric transducer embedded in a infinite medium. The model is coupled with a multi-objectivegenetic algorithm which is used to design specific ultrasonic embedded transducers both for hard and fresh concrete monitoring.The results show the efficiency of the approach and a few designs are proposed which are optimal for hard concrete, fresh concrete,or both, in a given frequency band of interest.

Keywords: Embedded Piezoelectric Transducer, Smart Aggregate, PZT Ultrasonic testing, Concrete Monitoring

1. Introduction

Assessing the state of health of concrete is a major issue foreveryone for whom the reliability of the structure is essentialboth for safety and economical reasons (operators of transportnetwork, nuclear power plants, dams, etc.). Visual inspectionsor destructive tests are the most widely used methods. Suchtechniques require specific equipment and are labor intensive.They are therefore costly and hardly efficient since they are nec-essarily sporadic. In the framework of civil engineering struc-tures, an alternative is to set up large sensors networks with thepurpose of measuring the dynamic signature of the structure[1]. Large scale effects can be monitored by analyzing the firstvibrations modes which are generally excited by the ambientvibrations (wind, traffic).

The detection of local defects requires however to study theinformation carried by higher frequency vibrations. Such wavescan be generated by the appearance of a crack. They can bemeasured with the help of a large network of sensors whichallows to localize the defect. This is the concept of AcousticEmission (AE) testing [2, 3].

∗Corresponding Author∗∗Principal Corresponding Author

Email addresses: [email protected] (Cédric Dumoulin ),[email protected] (Arnaud Deraemaeker )

URL: batir.ulb.ac.be (Arnaud Deraemaeker )

The wave can also be generated by the monitoring systemitself. Such active methods are called Ultrasonic (US) testing.Both AE and US methods require specific transducers whichallow to detect and generate waves in a given frequency band-width. Such transducers are generally made of Lead ZirconateTitanate (PZT) which is a piezoelectric material. Piezoelectrictransducers are currently widely used for nondestructive testing(NDT) due to their small size, low cost and their ability to workboth as actuator or sensor.

The large external probes which are generally used sufferfrom several drawbacks. AE and US methods rely on highfrequency waves (20 kHz to 500 kHz) which are strongly at-tenuated in concrete. Consequently, the measurement must beperformed near the source. The measurement should thereforebe done on small size specimens or in really restricted areas.Additionally, the use of such external transducers is restrictedby the need of flat surfaces and coupling agents which poten-tially reduce the efficiency of the transducers.

In order to overcome these drawbacks, several researchershave studied the possibility of embedding low-cost piezoelec-tric transducers in the concrete structure. These embeddedpiezoelectric transducers allow much more flexible configura-tions of measurement network and avoid the need of couplingagents. These transducers can be divided in two main designcategories. The first type of transducers is based on the design

Preprint submitted to Ultrasonics January 24, 2017

of classical piezoelectric transducers which consists in a piezo-electric patch surrounded by several matching or coating layers[4–9] while the second consists in cement-based piezoelectriccomposites [10–14].

At ULB-BATir, several designs of the first category havebeen manufactured and successfully used both for monitoringthe Young’s Modulus at very early age and damage detection[15–17]. These experiments have demonstrated the efficiencyof such transducers for structural health monitoring but havealso revealed the great importance of optimizing the design ofthe transducer for each specific application.

The main objective of the current study is to develop an ef-ficient method to characterize the performances of embeddedpiezoelectric transducers. In this study, it is specifically pointedout that the working principle of embedded transducers is dif-ferent from external transducers. More specifically, it is shownthat the methods classically used to optimize external transduc-ers cannot be used for embedded transducers.

One of the major issues for permanently embedding trans-ducers into the structure is to reduce their cost and their size asmuch as possible. It is shown that a pragmatic way to achievethis target is to benefit from the radial mode of actuation. Whilethe behavior of the transducers in the thickness mode can bestudied with simple analytic models such as the KLM model[18–20], this is not the case for the behavior in the radial modefor which a much more advanced finite element model is re-quired. To prevent the results from being impacted by theexternal boundaries, the transducer is embedded in an infinitemedium. This can be achieved through specific strategies suchas viscous damping boundaries [21, 22] and perfectly matchedlayers [23, 24]. Both methods are implemented and comparedin order to select the most promising technique. The first partof the present study deals with the development of a simple andreliable model for characterizing embedded transducers.

The second part of the study concerns the optimization ofthe transducers. For that purpose, the model is coupled with amulti-objective genetic optimization algorithm in order to de-termine new designs of transducers based on specific expectedproperties. More specifically, the mechanical properties of con-crete strongly evolve with the setting process. This requires thetransducer to work in a medium and a related frequency band-width of interest which are evolving with the maturity of con-crete and results in different optimal designs. This is here high-lighted through multiple optimization cases aiming at definingoptimal designs of transducers in fresh and hard concrete.

2. Modeling embedded transducers

The present section is aimed at developing a simple and ac-curate model of an embedded piezoelectric transducer. Thismodel should be sufficiently elaborate to properly represent thebehavior of the transducer in a given medium while being ef-fective in terms of computational costs. The first section of thispart is devoted to show how external transducers are generallyoptimized and why these methods cannot be used for embeddedtransducers. The second section deals with the developmentand the validation of an appropriate finite element model.

2.1. External and embedded transducersThe general design of external transducers consists in a

piezoelectric patch surrounded on one side by a series of match-ing layers which aim at transmitting the wave from the piezo-electric element to the tested material in a specified frequencybandwidth. On the other side, the piezoelectric element isbounded by a backing material which aims at both absorbingthe wave which is propagating in the opposite direction to thetested material and avoiding any reflection between the piezo-electric material and the backing material [25–28]. Such a de-sign allows to restrict the model to a one-dimensional problem.The well-known KLM model is the most widely used model toestimate the efficiency of piezoelectric transducers [18–20, 29–31]. It is briefly presented in Appendix A. Nevertheless, the de-sign of piezoelectric transducers is still often determined by op-timizing the acoustic impedance matching between the piezo-electric material and the tested material [32–35].

The difference of behavior between the external and embed-ded transducers is illustrated through a simple example (Fig. 1).It is suggested to compare the axial displacement uz at the ex-ternal boundary of the transducer due to an applied voltage tothe PZT, for several matching (transition) materials. The piezo-electric material is a piezoelectric patch of a thickness of 2 mm(Meggitt Pz26, see Table B.10) and the tested material corre-sponds to hard concrete. The different matching materials aregiven in Table 1 where Optim. material is the theoretical op-timal matching material between PZT (Zp) and concrete (Zc)which is given by

ZOptim =√

ZcZp (1)

where Z = ρVp [Rayls] is the acoustic impedance of the mate-rial (ρ and Vp are respectively the density and the P-wave ve-locity in the medium). This optimal value is the one whichmaximizes the transmission coefficient given by

T = 1 − R R =∣∣∣∣∣∣Zp − ZeqZp + Zeq

∣∣∣∣∣∣2 (2)where R is the reflection coefficient which is defined as thesquare of the ratio between the amplitude of the reflected waveand the amplitude of the incident wave at the interface of thePZT material and Zeq is the equivalent acoustic impedance asseen from the PZT patch. For a unique matching layer, Zeq issimply expressed by (see Eq. A.3)

Zeq = ZnZc cos(kntn) + jZn sin(kntn)Zn cos(kntn) + jZc sin(kntn)

(3)

where Zn, kn and tn are respectively the acoustic impedance,the wave-number and the thickness of the matching layer. Forthe present example, the thickness of the transition materialsis arbitrary chosen in order to correspond to the quarter of thewavelength in the material at 300 kHz (tλ/4 = λ/4).

The transducers are modeled with the KLM model. The tra-ditional (external) transducer (Figure 1a) is bounded on one sideby the matching material and the tested semi-infinite material,and on the other side, by a semi-infinite backing material of

2

2 mm

In�nite Material In�nite MaterialPZT

Absorbing In�nite Material PZT In�nite Material

2 mm

a) Traditionnal Transducers

b) Embedded Transducers

t

uz

uz

ConcreteMatching LayerPiezoelectric Material

Edge of the transducer


Figure 1: Comparison between tradition external transducer with backing ma-terial and embedded transducer.

Table 1: Transition materials properties

Materials E[GPa] ρ [kg/m3] Z [MRayls] tλ/4[mm]

Glue X60 6 900 2.44 2.27Hard Concrete 30 2200 8.57 3.25Optim. 54 4142 15.7 3.17Steel 210 7800 46.96 4.56

the same acoustic impedance as the PZT which aims at avoid-ing any wave reflection at the backside of the transducer. Thissemi-infinite non-reflecting backing material corresponds to aidealized case of a highly absorbing material actually used asbacking layer in real transducers. The embedded transducer(Fig. 1b) is symmetrically bounded by the matching materialand the tested semi-infinite material.

The transmission coefficients T between PZT and concretefor the different matching layers as computed with Eq. 2 areshown in Fig. 2 where it clearly appears that the optimal transi-tion material allows to increase the amplitude of the wave whichis transmitted to the tested material. It is important to note thatthe transmission coefficient T is maximum at odd multiples ofthe central frequency fc = Vp/4tn only if the impedance Zn ofthe matching material is in the interval Zc < Zn < Zp, otherwise,the maximum value of T will be reached at even multiples offc.

Figures 3a and b show the amplitude of the displacement atthe edge of the transducer (including the matching material)due to an applied unit voltage as a function of the frequencyfor both configurations. It can be observed that the behavior ofthe traditional transducers has a similar trend as the transmis-sion coefficient while it is different in the case of the embeddedtransducer. This difference is due to the appearance of strongresonances for the embedded transducer as shown in Fig. 3b.Such strong resonances are not present for the traditional trans-ducer because of the presence of the absorbing backing layer.They have been studied using a finite element model with vis-cous damping boundaries as explained in Section 2.2.

With such an approach, the complex mode shapes {ψ} are the

solution of the second order eigenvalue problem[[M] λ2j + [C] λ j +

[[K] + [D]

]]{ψ} = 0 (4)

while the normal mode shapes {ϕ} are the solution of the un-damped eigenvalue problem[

− [M]ω2j + [K]]{ϕ} = 0 (5)

[M] and [K] are respectively the mass and the stiffness matri-ces of the system while [D] is the material (hysteretic) damp-ing matrix and [C] is the viscous damping matrix which in thepresent model only includes the viscous damping boundariesand is therefore diagonal. ω j is the jth eigenfrequency of the un-damped system, λ j is the jth complex eigenvalue of the dampedmodel for which ω2j = |λ2j | [36–38]. In Fig. 4, we plot the realpart of the complex mode shapes for the different transducers ina infinite medium. They are normalized so that the value of thedisplacement is situated between −1 and 1. The correspondingeigenfrequencies are given in Table 2.

We have found that these resonant frequencies are in goodcorrespondence with the axial mode of the full transducer (thepiezoelectric patch and the matching layers) either with free(first axial mode, f1, f ) or clamped (second axial mode, f2,c)boundary conditions, depending on the relative stiffness of thematching layer in comparison to the infinite media (see thedashed and dotted lines in Fig. 4). The free boundary condi-tion corresponds to a transducer with top and bottom surfacesmechanically free to move, while in the clamped case, the dis-placements of the top and bottom surfaces are constrained to azero value.

Note that the axial mode refers to the resonance of the me-chanical system and must be distinguished from the thicknessmode of the piezoelectric element which is situated around1 MHz for the present geometry.

Concrete

Optim

Glue

Steel

0

0.68

1

T

0 100 200 300 400 500 600Frequency [kHz]

Figure 2: Transmission Coefficient between PZT and Concrete for differenttransition materials.

Table 2: Transition materials properties

Materials [kHz]

Glue X60 f2,c,glue 569Optim f1, f ,optim 245Steel f1, f ,S teel 195

3

0 100 200 300 400 500 6000

2

4 x 10-4

Frequency [kHz]

4

8

12

x 10-416

a) Traditional Transducer

b) Embedded Transducer

Concrete

Optim

Glue

Steel

ConcreteOptim Glue

Steel

f2,f,steel f2,f,optim f2,c,glue

|uz|

[µm

/V]

|uz|

[µm

/V]

0 100 200 300 400 500 600

Figure 3: Acoustic Response (displacement/Volt) as a function of frequencyfor a) traditional external transducers and b) embedded transducers for differenttransition materials.

0

1

-1

Free modesFixed modesComplex modes

0 tp/2-tp/2 z

uz / uz,max f1,f Optim

f2,c Gluef1,f Steel

Figure 4: Axial (z) component of the displacement of the mode shapes alongthe transducer (z-axis) for different transition materials (Table 1). The solidlines present the real part of the complex mode shapes, the dashed and dottedlines respectively present the normal mode shapes in fixed and free boundaryconditions. The corresponding natural frequencies are given in Table 2. tp isthe thickness of the piezoelectric element. The materials and the correspondingthickness of the surrounding layers are given in Table 1.

This leads to the first conclusion of the present study : acous-tic impedance matching theory can be used for the design ofexternal transducers but not for embedded transducers. Indeed,in the case of external transducers the incident wave is propa-gating from a media (PZT) to another (concrete), from left toright in Fig. 1, and the back propagating wave in the piezo-electric material resulting from the multiple reflections at thesuccessive interfaces is not in turn reflected due to the backingmaterial. As a consequence, the resonance of the piezoelectricelement is highly damped. This implies that the main mech-anism behind the acoustic response of an external transducer

is the resonant and anti-resonant vibration modes of each in-dividual layer, which can be either in-phase (constructive) orout-of-phase (destructive) with the incident wave, dependingon the surrounding materials. This is immediately related to thematching layer theory. For embedded transducers, as demon-strated here above, the acoustic response of the transducer isrelated to the overall dynamic behavior of the transducer in itsenvironment and in some cases high amplitude resonances arepresent, so that the matching layer theory is not sufficient topredict the behavior of the transducer.

The KLM model which is usually used to model piezoelec-tric transducers only considers the thickness modes of vibra-tion. Nevertheless, it can be shown for typical geometries oftransducers that the first vibration mode is the radial mode. It ispossible to find an analytic solution that combines these modesfor simplified 3D geometries [39–43]. But these analytic mod-els are difficult to couple with analytic wave transmission mod-els which makes them actually hardly usable. Furthermore,other modes of vibration which are not considered with theseanalytic models have to be considered.

The difference between the radial mode and thickness modeof vibration lies on the main direction in which a specimen isdeformed. Fig. 5 shows the displacement field of specific modeshapes for two geometries. The dimensions of the first sam-ple have been defined in order to ensure that the first vibrationmode is a pure radial mode (Fig. 5a), also referred as radial ex-tensional mode (R1). For the present geometry, the first radialmode is approximately at 210 kHz. The first thickness exten-sional mode (TE1) occurs at 990 kHz. This mode of vibra-tion is specific to thin piezoelectric disks and is described bya large displacement at the center and very low displacementat the disk edges (Fig. 5b). In practice, this mode of vibrationis actually hardly usable since it is most often strongly cou-pled to other vibration modes such as the overtones of the ra-dial mode or other modes of vibrations such as edges modes(E), thickness shear modes (TS). The description of these vi-bration modes are clearly beyond the scope of the present studyand are extensively described by Kocbach [44]. As a conse-quence, to observe the behavior of the transducer under a purethickness mode of vibration, the geometry of the piezoelectricelement has to be modified in order to decrease the frequencycorresponding to the thickness mode of vibration and increasethe frequency of the first radial mode (Fig. 5c). Since the di-ameter of the disk has the same dimension as the thickness,such a geometry cannot be strictly described as a disk. Wehave therefore considered more appropriate to call the vibra-tion mode displayed in Fig. 5c a longitudinal extension mode(LE1) which usually refers to long cylinders. However, for thepresent geometries, the boundary between both modes is notclearly defined so that in the rest of the present study one willonly refer to them as thickness modes.

It is important to note that today, many external transduc-ers are made of 1-3 piezoelectric composites, allowing to sub-stantially reduce the impact of the undesirable vibration modeswhich are coupled with the thickness mode, or to design phasedmatrix array transducers [30, 45]. Nevertheless, such compos-ite materials are much more expensive in comparison to bulk

4

a) Radial extensional (R1) mode

c) Longitudinal extensional (LE1) mode

Axis of symmetry

Axis of symmetry

R= 5mm

R= 2 mm

t= 5 mm

t= 2 mm

b) Thickness extensional (TE1)mode

3

1

0 MaxDisplacement

Figure 5: Comparison between radial resonant mode of vibration (R1) (a),thickness extension resonant mode of vibration (TE1) (b) and longitudinal ex-tension resonant mode of vibration (LE1). The colors corresponds to the normof the displacement in a radial section of the piezoelectric elements. The di-mensions of the piezoelectric patches have been chosen to ensure that the firstvibration modes corresponds to a) the radial mode (Radius > Thickness) and b)the thickness (longitudinal) mode (Radius < Thickness)

piezoceramic elements which makes their use beyond the scopeof the present study. Indeed, one of the major challenges con-cerning the embedded transducers is to obtain a sufficiently lowcost transducer which can be lost in concrete structures.

In order to prevent any local mechanical weakness in thestructure, the size of the transducer should at most be of thesame order of magnitude as the largest aggregates in the con-crete structures (around 10 mm diameter). The frequency rangeof interest for concrete applications (Section 3.3) requires touse thick (and consequently expensive) piezoelectric elementswhich have a thickness resonant mode at a sufficiently low fre-quency (around 20 mm for a thickness mode resonant frequencyof 100 kHz). A pragmatic solution is to benefit from the ra-dial mode of actuation which allows to reduce the resonant fre-quency of the transducer. This can be achieved by transformingthe radial displacement to thickness displacement with the helpof specific structures such as moonies [46], but their use wouldlead to high cost transducers. In the present study, it is sug-gested to directly benefit from the ability of affordable piezo-electric disc elements to generates axial displacements whiletheir main vibration mechanism in the working frequency rangeis the radial mode as illustrated in Fig. 5a. Characterizingthe performances of such transducers requires a finite elementmodel. The next section deals with the development of such amodel which is sufficiently accurate while limiting as much aspossible the required computational resources.

2.2. Finite element model of embedded piezoelectric transduc-ers

As illustrated in the previous section, designing embeddedpiezoelectric transducers requires much more advanced mod-els in comparison to those usually employed for external trans-ducers. In this section, a finite element model is suggested for

the purpose of being intensively used in a genetic optimiza-tion algorithm. This model should therefore be simultaneouslysufficiently accurate to properly estimate the performances ofthe embedded transducer while being sufficiently economicalin terms of computational costs.

In order to prevent the results to be impacted by the externalgeometry and boundary conditions, it is suggested to embed thetransducer in an infinite medium. This can be achieved with thehelp of specific elements such as boundaries elements, infiniteelements, viscous damping boundaries (VDB)[21, 22, 47–49]or perfectly matched layers (PML)[23, 24, 50]. The last two areby far the most widely used methods since they can be easilyimplemented in a finite element software. Both methods havebeen implemented with SDT, an open and extendible finite ele-ment modeling MATLAB based toolbox for vibration problems[51] and are briefly detailed here below.

Perfectly matched layers are unquestionably the most ac-curate elements since they are known to appropriately absorbcompression, shear and surface waves, evanescent and propa-gating waves, at any angle of incidence [23, 24, 50]. But theiruse can lead to heavy computational costs.

Ω = Physical Domain ΩPML

x0

LPML

Ω = Physical Domain

x

Ω∞

Wave Amplitude

xt

a) Unbounded Medium

b) Medium bounded by PML

b

a

Figure 6: Concept of perfectly matched layer. The wave is the same in anunbounded medium and in a medium bounded by perfectly matched layers.

Perfectly Matched Layers method consists in replacing asemi-infinite medium Ω∞ bounding a physical domain Ω bya finite absorbing bounding domain ΩPML so that the elastody-namic behavior in the physical domain Ω remains unchanged(Fig. 6). The PML domain ΩPML should absorb progressivelythe wave so that no reflection occurs both at the interface be-tween the physical domain and at the external boundaries of thePML domain.

The choice of the attenuation function is crucial to properlyattenuate both types of waves. An extensive discussion relativeto the choice of these parameters is given in François et al. [50].The values of the attenuation parameters in the direction i (i =x, y, z) f ei,0 and f

pi,0 which respectively control the damping of

evanescent and propagating waves used in the present study aregiven in Table 3.

5

Table 3: Attenuation functions parameters

Materials ≤ 150 kHz > 150 kHz

f ei,0 5 0f pi,0 20 20

Incident P-Wave

Reflected P-Wave

Reflected S-Wave

Incident S-Wave

Reflected S-Wave

ReflectedP-Wave

a) Incident P-Wave b) Incident S-Wave

13

τ13

σ1

τ13

σ1

Figure 7: Wave reflection at a boundary due to a incident P-Wave (a) and S-Wave (b). The viscous boundary reaction stresses absorb the incident waveaccording to Eq. 6.

Viscous damping boundary method is a cheaper option, theyare known to prevent the reflection of both compression andshear propagating waves but their efficiency is strongly im-pacted by the angle of incidence of the wave [21, 22, 47–49].The basic idea of viscous damping boundary method consistsin applying dynamic boundary stresses at the surfaces of thephysical domain in order to balance the stresses generated byincoming waves (see Fig. 7). The boundary stresses are definedby

σ1 = aρVp u̇1τ13 = bρVs u̇3

(6)

where Vp and Vs are respectively the P and S wave velocities,a and b are coefficients that depend on the angle of the incidentwave but are generally given as a = b = 1, u̇1 and u̇3 are the par-ticle velocities respectively normal and tangent to the externalsurfaces. Applying a VDB as expressed in Eq. 6 requires theuse of dashpots applied to the nodes of the external surface. InSDT, this can be achieved by spring-dashpot CBUSH elements.

ConcreteMatching LayerPiezoelectric Material

In�nite Domain

Transducer radially free

In�nite Domain

Figure 8: Transducer Embedded in an unbounded domain. The radial displace-ments of the transducer are left free.

The present section is aimed at selecting an accurate methodto model the behavior of embedded piezoelectric transducers.

For this purpose, a cylindrical transducer for which the radialdisplacements are kept free is embedded in an infinite elasticmaterial (Fig. 8).

The transducer is composed of a PZT disc which is boundedby a transition layer of the same material and thickness as pre-sented in Section 2.1 (see Table 1 and Fig. 1). The PZT elementis made of Meggitt-Pz26 hard piezoceramic with a thickness of2mm and 10mm of diameter. The material data for a finiteelement computation for this piezoceramic can be found in Ta-ble B.10 in Appendix B. The first free resonance frequency ofthe PZT disc is located around 210 kHz and corresponds to a ra-dial mode, the thickness mode is situated around 1 MHz. Twomethods to model the infinite part of the model are here con-sidered. The first consists in using viscous damping boundarieson the external surfaces of the physical domain (see Fig. 9a).Since the transducer is cylindrical, the computation of the trans-ducer can be reduced to the computation of a single slice withperiodic boundary conditions. The loading case consists in en-forced voltage on the electric DOFs corresponding to the elec-trodes which are respectively the upper and the bottom faces ofthe piezoelectric element [52].

This method presents the main advantage of involving a re-duced number of degrees of freedom (approx. 10 000 DOFs).But, as aforementioned, the efficiency of the method is stronglyimpacted by both the type and the angle of incidence of thewaves in the physical domain.

The second method consists in bounding the domain withperfectly matched layers (see Fig. 9b and c). Such a modelshould provide more accurate results since PML enable to ab-sorb the incident waves regardless of their type or the incidentangle. This method is usually used with rectangular boundarieswhich implies to compute the solution on the full 3D domain(Fig. 9c). However, such model leads to substantially highernumber of degrees of freedom (approx. 240 000 DOFs) andconsequently higher computational costs. Nevertheless, thesymmetry of the present case allows to use cyclic boundaryconditions so the number of DOFs in the model (Fig. 9b) isthen considerably reduced (approx. 30 000 DOFs) in compar-ison with the full model. Although the number of DOFs canbe significantly reduced by considering cyclic symmetry, thePML method leads to much higher computational costs whichis accentuated by the need of reassembling the system for eachcomputed frequency (the matrices are frequency dependent).

The 3D elements used in the different models in Fig. 9 arequadratic and the size of the elements are between 1/10th (c)and 1/15th (a and b) of the shortest wavelength in the medium,which are usual requirements to properly estimate the wavepropagation using a finite element model [50, 53, 54].

The constitutive equations for a piezoelectric material aregiven by Eq. 7{

TD

}=

[cE

][e]T

[e][εS

] {SE}

(7)

where T and S are the mechanical stress and the mechanicalstrain vectors while D and E are the electric displacement andthe electric field vectors,

[cE

]is the stiffness matrix. PZT is con-

6

8 mm

t2 mm

b) Cyclic PML± 40 000 DOFs

CBUSH Elements

8 mm

t

2 mm

a) Cyclic VDB ± 10 000 DOFs

ConcreteMatching Layer

Piezoelectric MaterialAborbing Material

Cyclic Symmetry

8 mm

t 2 mm

PML Cyclic Symmetry

c) Full PML± 240 000 DOFs


Electrodes

Figure 9: Finite element meshes when the domain is bounded with a) viscous damping boundaries and considering a cyclic symmetry, b) perfectly matched layersand considering a cyclic symmetry and c) perfectly matched layers. In each case, the piezoelectric element is a cylinder of a thickness of 2 mm and a diameter of10 mm. The properties and the thickness t of the matching materials are given in Table 1.

sidered as an elastic and transversely isotropic material.[εS

]is

the permittivity matrix at constant strain and [e] is the piezo-electric coupling constants matrix which relates the electricaland the mechanical variables of the equation. These differentmatrices and the related Meggitt Pz26 material properties aregiven in Appendix B. Considering the elastodynamic and theelectrostatic equilibrium equations and remembering that thestrain field and the electric field derive respectively from thedisplacement field and the electric potential, one can obtain thediscrete form of the variational piezoelectric equations used forfinite elements analysis [53, 55–57](−ω2

[Muu 0

0 0

]+ iω

[Cuu 00 0

]+

[Kuu KuφKϕu Kϕφ

]) {uϕ

}=

{FQ

}(8)

where the subscripts u and ϕ denote respectively the mechani-cal and the electrical part of the equation. u and ϕ are respec-tively the nodal displacement vector and nodal electrical poten-tial vector and by extension, F and Q are the nodal vectors ofmechanical forces and electrical charges. The damping consid-ered in the present model is a hysteretic damping so the stiffnessmatrix K is complex and the viscous damping matrix C onlyapplies for the viscous damping boundaries, which only havemechanical DOFs.

In order to properly compare the different models, it is sug-gested to compute both the electrical input impedance betweenthe electrodes of the piezoelectric element and the acoustic re-sponse of the system for each case. In the present case, theelectrodes are equipotentials which are respectively located atthe bottom and the upper surfaces of the piezoelectric element.In the finite element models, this is achieved by imposing thedegrees of freedom corresponding to the electric potential (ϕ)of each node located on these respective surfaces to be equal.

The actuation of the transducer is then performed by impos-ing for each computed frequency a unit voltage (ϕA = 1) onthe electric DOFs corresponding to one electrode (either theupper or the lower) and the electric DOFs of the other elec-trode are grounded (ϕG = 0). The electrical input impedance ofthe transducer is given by Zin(ω) = V(ω)/I(ω) where V(ω) =ϕA − ϕG = 1 is the imposed voltage and I(ω) is the resultingcurrent which is actually obtained by computing the resultingcharge Q(ω) on one electrode from which the current is simplygiven by I(ω) = iωQ(ω).

Fig. 10 shows the evolution of the electrical input impedanceat the terminals of the PZT disc. A really good match betweenthe results of the different models can be observed.

102

103

104

0 50 100 150 200 250 300Frequency [kHz]

|Zin| [Ω

]

105

ConcreteOptim

Glue

Steel

Full PML Model

Cyclic VDP ModelCyclic PML Model

Figure 10: Comparison of the electrical input impedance Zin as computed withthe full PML model (solid lines), the cyclic PML model (dotted lines) and thecyclic VDP model (dashed line) for different matching materials given in Ta-ble 1).

Estimating the acoustic response of the transducer consists

7

3Full PML Model

Cyclic VDP Model

Frequency [kHz]0 50 100 150 200 250 300

x 10-3

0

1.5

1

1.5

2

2.5 Cyclic PML Model

|uz|

[µm

/V]

Figure 11: Comparison of the acoustic response (|uz | [µm/Volt]) at the edge ofthe transducer (including the matching layer) as computed with the full PMLmodel (solid lines), the cyclic PML model (dotted lines) and the cyclic VDPmodel (dashed line) for different matching materials given in Table 1).

in computing the amplitude of the transmitted wave for a givendriving voltage. It has to be noted that the amplitude of the dis-placement may vary depending on the measured location. Thiscan be a significant issue for short wavelengths but not a majormatter for the frequency range of interest in the present study(< 300 kHz). It is then suggested to only consider the averageamplitude of the vertical displacement |uz| of the upper surfaceof the transducer (including the matching material). Fig. 11shows that the evolution of the displacement spectra for the dif-ferent models are really well matched. The resonance whichappears in Fig. 10 and Fig. 11 corresponds to the radial modeof vibration. As mentioned above, the thickness mode of vi-bration for the present geometry is situated around 1 MHz, thecoupling between these two modes is very low for the frequen-cies displayed on the present figures. Nevertheless, it has to bementioned that above 300 kHz, other modes of vibrations inter-act and are superimposed, which makes the acoustic responsedifficult to interpret for much higher frequencies.

This leads to the conclusion that the three models provideequivalent results. According to that observation, the viscousdamping boundary method seems more appropriate in an op-timization process since it requires significantly less computa-tional resources.

2.3. Effect of the radial modeFig. 11 displays the acoustic responses of a piezoelectric

transducer which is composed of a piezoceramic element fordifferent surrounding layers. In the frequency band for whichthe acoustic responses have been computed, only the radialmode of actuation of the piezoelectric element is excited. Forthat purpose, the radial displacement of the piezoelectric ele-ment has been kept free (see Figures 8 and 9). These choiceslead to two mains issues. The first concerns the performance ofthe transducer if it is directly radially surrounded by the testingmaterials. The second concerns the impact of using the radialmode of actuation instead of the thickness mode.

It is suggested to address these issues by comparing theacoustic response (|uz|) for three different cases. For each case,the acoustic response is evaluated from a finite element model

with cyclic symmetry and with viscous damping boundaries asinfinite material. The two first cases correspond to ultrasonictransducers for which the actuation mode is the radial mode.The radial displacement of the transducer is first kept free.This case is therefore fully identical as in the previous section(Fig. 9a). In a second step, the transducer is radially connectedto the tested material. This is simply achieved by adding VDBto the radial outline of the transducer, the transducer is then saidradially constrained (RC). For these two cases, the geometry ofthe piezoelectric element is kept the same as previously (thick-ness of 2 mm, diameter of 10 mm). The third case is aimed atcomparing the efficiency of ultrasonic transducers working inradial or thickness mode. As explained in 2.1, the thicknessmode of a piezoelectric disk of the same geometry is not us-able since it is strongly coupled with other vibration modes. Itis then suggested to consider a different geometry for whichthe frequency of the fundamental thickness mode roughly cor-responds to the frequency of the first radial mode of the initialgeometry. For that purpose, the thickness of the piezoelectricdisk is increased to 6 mm and the diameter of the transducersin reduced to 2 mm to sufficiently raise the resonant frequencyof the radial mode in order to avoid any coupling between thesetwo modes. For the latter case, the transducer is kept radiallyfree.

The acoustic responses for the different cases and for differ-ent surrounding layers (Table 1) are presented in Fig. 12. Asone might expect, bounding the radial edge of the transducerleads to severely damp the resonance of the transducer. Moresurprisingly, according to Fig. 12, using the thickness mode ofa piezoelectric element does not enhance the efficiency of thetransducer. This demonstrates that using the radial mode of vi-bration of piezoeceramic disk is not only a pragmatic choice re-garding the cost and the geometry but also an efficient solutionprovided that the radial displacement of the transducer is notconstrained. This result is one of the key points of the presentstudy. In practice, this could for instance be achieved with aproper housing (stainless steel or aluminum) joined to the trans-ducer with a very soft and light potting material such as specificfoams, cork or polyurethane for which both the stiffness anddensity are lower of several orders of magnitude in comparisonwith piezoceramic materials. The impact of the radial bondingis therefore drastically reduced and can be neglected in a firstestimate. Such a kind of design is very common in the industryof ultrasonic transducers.

3. Optimization of the transducer

Optimizing a piezoelectric transducer consists in looking forthe optimal design for a specific application. In the presentstudy, it is suggested to select the material and the thickness ofsuccessive transition materials with the purpose of both maxi-mizing the amplitude and the frequency bandwidth of the trans-mitted wave. These requirements lead to the definition of twoobjectives functions that will be presented hereafter. These ob-jectives are used in a multi-objective evolutionary algorithm(EA) called nondominated sorting genetic algorithm II (NSGA-II) [58]. This elitist algorithm consists in constructing each

8

0 50 100 150 200 250 3000

1

2

3 x 10-3

Frequency [kHz]

Am

plitu

de [µ

m/V

]

Axial mode

Radial mode ‘RC’ Radial mode

Concrete Optim

Glue

Steel

0

1

2

3

Am

plitu

de [µ

m/V

]

x 10-3

0 50 100 150 200 250 300

Frequency [kHz]

0

1

2

3

Am

plitu

de [µ

m/V

]

0

1

2

3

Am

plitu

de [µ

m/V

]

Figure 12: Comparison of the acoustic response (|uz | [µm/Volt]) at the edge of the transducer (including the matching layer) as computed with the VDB model(solid lines) and the VDP model (dashed line) for different matching materials

offspring population from the best ranked fronts of the parentpopulation, where the rank corresponds to the nondominationlevel of the solution. The parent population Pi+1 of each off-spring generation Qi+1 is composed of the most nondominatedmembers of the population Ri composed of both the currentgeneration Qi and their own parents Pi (Ri = Qi ∪ Pi). Theelitist aspects of the method is ensured since all previous andcurrent population members are included in Ri. This specificalgorithm is known to be fast and efficient for any shape ofPareto-Optimal front (convex, non-convex, disconnected, etc.).The choice of EA to optimize the transducers results from thedifficulty to compute the derivatives of the objective functionswith respect to the variables, in particular for discrete variables.

3.1. Objectives definition

The objective functions have to be adequately defined de-pending on the characteristics of the transducer that are ex-pected. One can define objectives function in order to optimizethe bandwidth, the transmitted energy, the maximum amplitudeof the transmitted acoustic wave, the input energy, the focus ofthe wave and so on [59–62].

In the present study, it is suggested to consider both the en-ergy and the bandwidth of the transmitted wave as criteria tooptimize the embedded transducers.

The first objective F1 (Equation 9) estimates the mean squareof the displacement at the edge of the transducer (including thematching layers). The displacement is the value of the ampli-tude of the vertical displacement |uz| as explained in Section 2.2.The first objective function is then expressed as:

F1 = −∫ f2

f1|uz( f )|2d f (9)

where f1 and f2 are respectively the lowest and the highest fre-quencies of interest.

|uz|

Frequencyf1 f2

-6dB∆f6dB

∆f12

∆f3dB-3dB

|uz,max|

Figure 13: Definition of the parameters used to compute the objective functionsF1 (Equation 9) and F2 (Equation 10).

The second objective function is related to the bandwidth ofthe transmitted wave. Figure 13 shows two spectra of displace-ment |uz|. The bandwidth is defined as the range of frequencies∆ fxdB for which the amplitude is −x dB above the peak of thespectrum. A low value of ∆ fxdB will characterize a narrow-band transducer and conversely, a high value will characterizea broadband transducer. In the present study, the bandwidth∆ fxdB is normalized by the required bandwidth ∆ f12 = f2 − f1for the application. It is here suggested to define an objectivefunction that both considers bandwidth for −3 dB (≈ 0.7|uz,max|)and −6 dB (≈ 0.5|u f ,max|). This criterion includes both the band-width and the sharpening of the spectrum. It is expressed by

F2 = −12

(∆ f3dB∆ f12

+∆ f6dB∆ f12

)(10)

This is illustrated in Fig. 13 where both spectra have similarvalues of F1 and ∆ f6dB. Nevertheless, the spectrum depicted bythe black line has clearly a flatter shape than the one of the gray

9

line, which is taken into account with the ∆ f3dB bandwidth.

3.2. Variables

Table 4: Variables constraints for the thickness of the layers

t1 [mm] t2 [mm] t3 [mm]

Lower Bounds 0.1 0 0Upper Bounds 4 4 4

The optimization process is performed on the VDB model(Fig. 9a) and considering three matching layers. The first threevariables are continuous and correspond to the thickness of eachlayer. They are constrained according to Table 4 where it canbe observed that the minimum thickness for two layers is zero,which enables to consider transducers with one to three tran-sition layers. To each layer corresponds a material which hasto be chosen in a restricted list of materials given in Table 5where E is the Young’s modulus, ν is the Poisson’s ratio, ρ isthe density, η is the mechanical loss factor. This list is chosenin order to cover a wide range of stiffnesses and densities whileconsidering only a reduced number of existing materials. Thelast parameter corresponds to the diameter Φ of the piezoelec-tric element. The diameters are restrained to the standard valuesof the Meggitt-PZ26 piezoelectric disc elements.

Table 5: Properties of the materials and geometry of the piezoelectric elements(standard geometry for Pz26 elements) considered in the optimization process

Materials E[GPa] ν ρ[ kgm3

] η Z[MRayls]

1 Glue (X60) 6 0.4 900 0.1 2.322 Mortar 30 0.2 2200 0.04 8.123 Marble 50 0.2 3000 0.01 12.254 Low Stiff. Glass 65 0.22 2500 0.03 12.755 Aluminum 70 0.35 2700 0.02 13.756 High Stiff. Glass 80 0.25 2500 0.03 14.147 Brass 100 0.31 8500 0.02 29.158 Titanium 116 0.34 4500 0.02 22.859 DC53 Steel 150 0.28 7800 0.03 34.1210 Steel 200 0.3 7800 0.03 39.5011 High Stiff. Steel 210 0.3 7800 0.03 40.47

Fresh Concrete 5 0.2 2200 0.04 3.32Hard Concrete 30 0.2 2200 0.04 8.12

Diameters Φ [mm] 10 12.7 16 20 25 30

3.3. Frequency range of interestThe transducers designed in the current study are dedicated

to US applications in concrete from early age to damage detec-tion in hardened concrete structures. One of the main interestsof using embedded transducers is their ability to catch local in-formation. The frequency domain of interest should thereforebe located above the stationary wave regime corresponding tothe first vibration modes of the structure.

The frequency band of interest is guided by the wavelengthcorresponding to the shortest characteristic length of the struc-ture (i.e from the average size of aggregates to several centime-ters). For concrete application, the domain evolves with the

setting process of the concrete. The evolution of the frequencyrange is directly related to the evolution of the wave velocityin the material relative to the evolution of the Young’s modulus[15, 63]. Fig. 14 shows the typical evolution of the frequenciescorresponding to wavelengths from 1 cm which roughly corre-sponds to the average size of the aggregates, to 10 cm which isa lower limit of the characteristic length of concrete specimens,relative to the wave propagation regimes as defined by Planèset al. [64]. Nevertheless, the limits between the propagationregimes in Fig. 14 should be viewed as a general trend ratherthan strict frontiers. Indeed, the transition between two propa-gation regimes is smooth and strongly depends on the concreteitself.

0 10 20 30 40 50 600

200

400

Age [h]

Freq

uenc

y [k

Hz]

Frequency Bandof interest

λ≈1 cm

λ≈10 cm

600

Modal Analysis

Simple scattering regime

Multiple scatteringregime

Strong absorptionRegime

800

FreshConcrete

HardenedConcrete

Figure 14: Evolution of the frequency for different wavelengths with setting ofconcrete in comparison to the corresponding approximated wave propagationregime.

For hardened concrete, according to Fig. 14 the frequencyband of interest is therefore located largely above the modalanalysis regime ( f � 10 kHz) and sufficiently below the at-tenuation regime ( f � 500 kHz) where the wave is both toostrongly scattered and absorbed. The working frequency bandis ranging from the simple wave scattering regime to the mul-tiple wave scattering regime. Nevertheless, the frequencyrange of interest depends on the targeted application. At ULB-BATir, we are mainly working in the simple scattering regimefor which the frequency range of interest can be restricted tothe domain defined by f1 = 20 kHz and f2 = 200 kHz. In thefirst few hours (fresh concrete), the frequency band of interestis evolving fast so that in our case, the frequency domain canbe kept the same as for hard concrete 20 kHz to 200 kHz.

3.4. Cases

In the present study, it is suggested to define designs of trans-ducers specifically optimized for hard concrete (Optim. CaseA) and early age applications (Optim. Case B). The mechani-cal properties of fresh and hard concrete are given in Table 5.The piezoelectric transducers are standard piezoceramic disc el-ements (Meggitt Pz26, see B.10).

10

Table 6: Summary of the frequency limits, the geometry of the piezoelectricelement and the state of the concrete considered in the different optimizationcases.

Case f1 [kHz] f2 [kHz] S tate

Optim. A 20 200 Fresh ConcreteOptim. B 20 200 Hard Concrete

Optim. C 20 200 Fresh and Hard

Besides these scenarios, a complementary cases (Optim. C)which combines the behavior in fresh and hard concrete is con-sidered. For that purpose, the objective functions are slightlymodified in order to take into account both cases. F1 and F2 aretherefore the average value of the respective objective functionsin both hard and fresh concrete as expressed in

Fi =12

(Fi, f resh + Fi,hard

)i = 1, 2 (11)

where Fi, f resh and Fi,hard are the objectives functions which areextracted from the acoustic responses respectively in fresh andhard concrete as given in Eq. 9 and Eq. 10.

The different scenarios are summarized in Table 6 where f1and f2 define respectively the lower and the upper bound of thefrequency domain.

4. Results

The result of the method has a strong dependence on the ini-tial population since the following generations directly descentfrom that latter. The first generation should therefore be suffi-ciently large to be representative of the possible solutions oth-erwise the algorithm runs the risk of converging on a reducedpart of the optimal Pareto front. The EA optimization process isperformed considering 40 generations with a population of 400individuals for each generation. Each optimization process isrepeated three times with a different (randomly chosen) startingpopulation. This allows to ensure that the process has actuallyconverged. For each optimization, the optimal Pareto front isthen composed of the first ranked members of the populationR40 which combines the members of Q40 and P40, respectivelythe last offspring generation and their parents as explained inSection 3. In order to benefit of the repeated processes, the finaloptimal population is generated from the top ranked individualsof a population which mixes up the results of each process.

This section is aimed at discussing the results for each casepresented in Table 6. For each case, the final optimal Paretofront is shown as well as the solutions at each process. It isthen possible to compare the Pareto-Optimal fronts for each ofthem. Several Pareto-Optimal solutions are selected in order tocover the entire front. For these solutions, the geometry and theacoustic response are presented.

4.1. Optim. A (Fresh Concrete 20-200 kHz)Fig. 15 shows the Pareto-Optimal front for the first optimiza-

tion case (Optim. A in Table 6). The individuals of the Pareto-Optimal front are numbered in the ascending order of F2. The

colors of the circles which shape the final Pareto front are aimedat highlighting the solutions which involve a identical PZT di-ameter. The grayed crosses are the respective Pareto frontsfor three different starting populations which appear to be wellmatched. The front is split in two main parts, each correspond-ing to a specific PZT geometry (see Table 7). A piezoelectric el-ement of 16 mm (blue circles) allows to obtain solutions whichprovide more energy to the transmitted wave while a diameterof 10mm (red circles) leads to more broadband solutions.

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

−1

−0.8

−0.6

−0.4

−0.2

0

340

8790108

220

F1

F2

Figure 15: Optim. A (fresh concrete, 20 to 200 kHz). Multi-Objectives Opti-mization Computation. The colored bullets correspond to the best ranked solu-tions of population mixing the optimal solutions of three optimization process(gray crosses). Red and blue filled circles correspond respectively to geometrieswith a piezoelectric element of 10 mm and 16 mm.

Table 7: Optim. A (fresh concrete, 20 to 200 kHz). Geometry of the selectedPareto-Optimal solutions. The dimensions (Φ, t1, t2, t3) are in mm.

PO Φ Lay. 1 t1 Layer 2 t2 Layer 3 t3

3 10 Mat 5 0.73 Mat 1 3.95 Mat 8 0.2740 10 Mat 1 1.95 Mat 1 0.40 Mat 8 1.0687 10 Mat 1 1.69 Mat 2 0.73 Mat 7 0.35


0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3x 10

−3

Frequency [kHz]

Am

plitu

de [µ

m/V

]

frequency band of interest

34087

90108220

Figure 16: Optim. A (fresh concrete, 20 to 200 kHz). Acoustic response forthe selected Pareto-Optimal solutions. The colors of the lines refer to a specificdiameter: red (10 mm), blue (16 mm)

11

Six Pareto-Optimal (PO) geometries are selected in order tocover all the Pareto front (see Fig. 15). The corresponding ge-ometries are shown in Table 7. The acoustic responses for thesesolutions are shown on Fig. 16 where the amplitude spectrumof the transmitted wave in the frequency band of interest pro-gressively evolves from a relatively flat shape (e.g. line 3) toa narrow-band response (e.g. line 220). Fig. 16 also illustratesthat the transducers which lead to flatter acoustic responses (redlines) are working below the radial resonance frequency of thePZT element while more energy can be transmit to the testedmaterial by benefiting of the resonance of the element. Never-theless, it can also be pointed out that it is possible to obtainbroadband transducers for which the resonant frequency of thepiezoelectric element is located in the frequency band of inter-est (see e.g. lines 90 and 108).

4.2. Optim. B (Hard Concrete 20-200 kHz)The final Pareto front for hard concrete (Optim. B) is dis-

played on Fig. 17 where it clearly appears that the different op-timization processes lead to really well matched solution do-mains (gray crosses). The colors of the circles correspond to aspecific diameter of the PZT patch while the geometries of sixof the PO solutions are given is Table 8.

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0

−1

−0.8

−0.6

−0.4

−0.2

0

1

302303

413454

581

F1

F2

−1.4

Figure 17: Optim. B (hard concrete, 20 to 200 kHz). Multi-Objectives Opti-mization Computation. The colored bullets correspond to the best ranked solu-tions of population mixing the optimal solutions three optimization processes(gray crosses). The colors of the filled circles refer to a specific diameter: red(10 mm), blue (16 mm) and orange (20 mm).

Table 8: Optim. B (hard concrete, 20 to 200 kHz). Geometry of the selectedPareto-Optimal solutions. The dimensions (Φ, t1, t2, t3) are in mm.


1 10 Mat 1 1.86 Mat 1 0.05 Mat 10 1.88302 10 Mat 1 1.04 Mat 5 0.01 Mat 7 1.87

454 16 Mat 1 0.49 Mat 2 1.28 Mat 9 1.91


As for fresh concrete (Optim. A, Fig. 15), the optimal frontis divided in two main groups each corresponding to a specificworking principle. Indeed, the solutions which lead to the most

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3 x 10−3

Frequency [kHz]

Am

plitu

de [µ

m/V

]


1302

303413

454

581

Figure 18: Optim. B (hard concrete, 20 to 200 kHz). Acoustic response forthe selected Pareto-Optimal solutions. The colors of the lines refer to a specificdiameter: red (10 mm), blue (16 mm) and orange (20 mm)

broad-band acoustic response (Fig. 18) are obtained by using apiezoelectric disk with a smaller diameter and whose frequencycorresponding to the radial mode of actuation is located abovethe frequency band of interest (see lines 1 and 102 in Fig. 18).Nevertheless, these solutions are not able to transmit a lot of en-ergy into the the system in comparison to solutions which takeadvantage of the radial resonance mode of the piezoelectric ele-ment. In particular, solutions 302 and 303 have almost identicalvalues of F2 (which indicates the bandwidth of the transducer)while the solution 303 has a value of F1 (which refers to thetransmitted energy) almost twice as large as the value of F1 forsolution 302.

4.3. Optim. C (Fresh and Hard Concrete 20-200 kHz)

Comparing the optimal solutions resulting from Optim. Aand Optim. B in order to determine similarities and deducinga design which would be optimal in both cases looks to be adifficult challenge. On the one hand, the diameter of the Pareto-Optimal solution differs on a large range of the Pareto front.On the other, among the geometries presented in Table 7 andTable 8 which are of the same diameter, it is difficult to drawgeneral conclusions.

Table 9: Optim. C (fresh and hard, 20 to 200 kHz). Geometry of the selectedPareto-Optimal solutions. The dimensions (Φ, t1, t2, t3) are in mm.




Fig. 19 shows the PO solutions for a mixed case where theefficiency of the transducer is balanced between optimal per-formance in fresh and hard concrete according to Eq. 11. ThePareto front has a similar trend as observed for Optim. A andOptim. B. Specifically, the results can be clearly separated intwo main groups each corresponding to a specific geometry of

12

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

−1

−0.8

−0.6

−0.4

−0.2

0

119

114240

261298

F1

F2

Figure 19: Optim. C (fresh and hard, 20 to 200 kHz). Multi-Objectives Opti-mization Computation. The colored bullets correspond to the best ranked solu-tions of population mixing the optimal solutions of three optimization process(gray crosses). Red and blue filled circles correspond respectively to geometrieswith a piezoelectric element of 10 mm and 16 mm.

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3x 10

−3

Frequency [kHz]

Am

plitu

de [µ

m/V

]


a) Fresh concrete

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3 x 10−3

Frequency [kHz]

Am

plitu

de [µ

m/V

]


119114

240261298

b) Hardened concrete

119114

240261298Optim. A (220)

Optim. B (581)

Figure 20: Optim. C (fresh and hard concrete, 20 to 200 kHz). Acoustic re-sponse in a) fresh concrete and b) hardened concrete for the selected Pareto-Optimal solutions. The colors of the lines refer to a specific diameter: red(10 mm), blue (16 mm)

the piezoelectric elements which can be identified on the Paretofront by the colored circles. As for the two previous optimiza-tion cases, each group corresponds to a specific working prin-ciple and the same remarks concerning the transmitted energy

and the bandwidth hold. On the contrary to Optim. B and as forOptim. A, the domain of solutions only contains two differentdiameters.

The geometry of six PO solutions is presented on Table 9 andtheir respective locations in the Pareto front can be observedon Fig. 19 while the acoustic responses in both fresh and hardconcrete are respectively displayed on Fig. 20a and b. The resulthas to be compared to Optim. A and Optim. B. This is firstachieved by comparing the acoustic response either in fresh andhard concrete with one solution obtained in Optim. A (PO 220)and Optim. B (PO 581), see dashed lines in Fig. 20. The resultsobviously differ but the actual gain of a proper optimizationprocess for each specific cases does not clearly appear.

The objectives functions F1 and F2 of the PO solutions forOptim. C are now evaluated separately in fresh and hard con-crete. The couples (F1, F2) for the different PO solutions in Ta-ble 9 are then displayed in Fig. 21a and b (black circles) wherethey are compared to the Pareto fronts for Optim. A and Optim.B (gray circles). Such a representation allows to clearly observewhich solutions are more optimized in one case than the other.Nevertheless, Fig. 21 also highlights that it is possible to obtainsolutions that are almost optimal in both cases as for PO 261and 298.

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

−1

−0.8

−0.6

−0.4

−0.2

0

1114

19

240261

298

F1

F2

Optim. BOptim. C

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0

−1

−0.8

−0.6

−0.4

−0.2

0

1

114

19

240261298

F1

F2

−1.4

Optim. AOptim. C

a) Fresh concrete

b) Hardened concrete

Figure 21: Pareto-Optimal solutions for the mixed cases (Optim. C, Table 9)compared to the Pareto-Optimal front in a) fresh concrete (Optim. A), and b)hard concrete (Optim. B).

4.4. Discussion of the resultsThree optimization cases have been considered with the pur-

pose of covering the frequency domain of interest for concrete

13

assessment at very early age (Optim. A), in hardened concrete(Optim. B) or in both cases (Optim. C).

The aim of this section is to draw general conclusions fromthe previous results. One of the main goals of using a meta-heuristic optimization algorithm is to obtain solutions to a sys-tem which is difficult to predict. In return, analyzing the resultsfrom such a process also leads to difficulties. Specifically, itappears difficult to draw general design rules from the resultsobtained with this method. And it is still worst with an in-creased number of parameters. Indeed, the algorithm only pro-vides a range of feasible optimal solutions according to presetconstraints. Furthermore, it has to be noted that only six out ofhundreds of PO solutions are presented for each optimizationprocess. They have been chosen in order to cover the entirePareto front and to provide a relatively representative picture ofthe geometry associated to a part of the front. In order to re-main focused on the main objective of the current research, theothers geometries are not presented. Nevertheless, many othergeometries are feasible and some points that are really close inthe Pareto chart can be associated to geometries that are quitedifferent both in terms of materials and thicknesses of the sur-rounding layers. However, it is comforting to observe that inmost cases, PO solutions which have similar geometries are lo-cated in the same part of the chart.

The quantification of the effect of perturbations in the geom-etry and in the material properties on the acoustic response isfundamental and should be the object of specific studies. De-signs could appear more robust than other to perturbations andshould therefore be preferred.

It is then to the user to select the geometry depending on boththe required acoustic response, the technical feasibility as wellas economic considerations. Specifically, gluing two succes-sive materials together is not a trivial task. In the present study,the link between two layers has been considered as ideal whichis never the case in practice. As a consequence, the impact ofa layer of glue has to be carefully studied. Such a study musthowever be performed taking into account the current techno-logical limits. However, the analysis of the dynamic behavior ofthin bounding layers is still on the spotlight of the research [65–67] and properly including such kind of material in the modelhas to be performed with the utmost care [68–70].

Although the number of materials has been restricted to ac-cessible and affordable materials, the practical way to manu-facture these different solutions is not considered in the presentstudy. It is however obvious that certain optimal designs areeasier and less expensive to manufacture and are therefore moreappropriate for the actual fabrication of the new transducers.More specifically, certain solutions involve a reduced numberof materials since two successive layers are made of the samematerial, or two successive materials can be more or less easyto bind together.

Before the earliest stages of setting of the concrete, the fre-quency range of interest is clearly situated below 100 kHz. Nev-ertheless, as observed in Fig. 14 this upper limit evolves rapidlyonce the setting process has started. Depending on the concreteand the conditions in which it is set up, its properties evolverapidly and at 6 to 10 hours this upper limit roughly reaches

200 kHz. For hardened concrete, the choice of the frequencyband of interest will strongly depend on the range of applica-tions for which the transducer is dedicated. For instance, ul-trasonic pulse velocity tests (UPV), AE or more advanced ul-trasonic testing such as nonlinear ultrasonic wave spectroscopyNRUS [71] or diffuse ultrasound [72–74] will require a differentbandwidth and consequently a different design.

5. Conclusions

In this study, the fundamental difference in terms of work-ing principle between external transducers and embedded trans-ducers is first shown through a simple example. The use ofthe radial mode of actuation of the piezoelectric transducer isexplored. Such a mode of actuation is generally seen as anundesirable mode leading to the use of expensive piezocom-posites which considerably reduce its effect. The resonant fre-quency corresponding to the radial mode for typical geometriesof transducers is generally much lower than the thickness mode.The frequency range of interest for concrete application can bereached with smaller piezoelectric elements. Using the radialmode can thus be viewed as a pragmatic choice to produceeconomical transducers with reduced dimensions. The perfor-mance of the transducer for such mode of actuation is difficultto estimate and necessitates a finite element model.

In order to prevent the impact of the external geometry suchas wave reflection or global modes of vibration, the trans-ducer should be embedded in an infinite medium. This canbe achieved by using non-reflecting boundary conditions suchas viscous damping boundaries or specific elements such asinfinite elements or perfectly matched layers. In the presentstudy, both VDB and PML are used and compared. It is shownthat both methods lead to similar results. Since the use ofVDB requires significantly less computational resources, it isselected for optimizing the design of the transducers with amulti-objective genetic algorithm. The objective functions usedin this study are aimed at characterizing both the bandwidth andthe transmitted energy in the tested medium.

Several optimization cases are considered in order to defineefficient designs of transducers either in fresh or hardened con-crete. It is shown that the method allows to design a transducerwhose performances match specific requirements. The methodis general and allows to either define additional objective func-tions or to modify the definition of the objectives depending onthe expected specification for the transducer.

Further research will be focused on the fabrication of the newtransducers as designed in the present study. These new trans-ducers will be experimentally characterized and then used forthe development of new efficient structural health monitoringtechniques in concrete structures.

Appendix A. KLM Model

The one dimensional piezoelectric KLM model [18–20] isschematically presented in Fig. A.22 where CS0 , X and Φ are

14

given by

CS0 =AεS33

tp

X =h233Aω2

sin kptp

Φ =2h33

AωZpsin kptp/2

(A.1)

where 3 = z is the poling axis in the IEEE standards [75],h33 = e33/εS33 is the piezoelectric constant in the poling direc-tion, e33 is the piezoelectric stress constant, εS33 is the dielectricconstant (permittivity) at constant strain, cD33 is the elastic stiff-ness at constant electric displacement field D, ω is the angularfrequency, A is the area of the transducers, Zp, kp and tp and arerespectively the acoustic impedance, the wave-number and thethickness of the piezoelectric element.

Backing Piezoelectric material (KLM) Front

ZF

tF

ZB

tB

Ff

Fb

FfFb

vb

vb

-vf

VΦ :

I

tp2

, Zptp2

, Zp

-vf

Zb

Zf

Figure A.22: KLM model: Piezoelectric transducer in an acoustic transmissionline.

The front and backing materials are linked to the KLM modelthrough the acoustic ports of the model. The successive trans-mission matrices are given by

[Tn] =

cos kntn jAZn sin kntn

jsin kntn

AZncos kntn

(A.2)which relates the force F and the particle velocity v at the leftside of each acoustic layer to the force and the particle veloc-ity at the right side of the layer. The backing and front semi-infinite materials are simply given by their respective acousticimpedance Zb and Z f . For an unique transmission layer, theequivalent acoustic impedance Zeq of the front side as viewedby the transducer is given by

Zeq =F′fAv′f

= ZnZ f cos(kntn) + jZn sin(kntn)Zn cos(kntn) + jZ f sin(kntn)

(A.3)

Appendix B. Piezoelectric properties for finite elementsand the KLM model

The piezoelectric elements used in the present study aremade of Meggitt Pz26 which is a Navy type I hard PZT. The

material data for finite element computations are given in Ta-ble B.10. These values can be retrieved from the material data-sheet using the relation given in [75–77].

Table B.10: Pz26 Properties to Introduce in FEM and Analytic Modeling

Material property Value Unit

Piezoelectric Constants

d33 300 10−12 C/Nd31 −130 10−12 C/Nd15 330 10−12 C/N

Permittivity

εT33 1300 ε0 F/mεT11 1335 ε0 F/m

Mechanical Data

Ep 74.17 GPaEz 59.14 GPa

Gzp 25.1 GPaGp 27.89 GPaνp 0.329νzp 0.3νpz 0.376ρ 7700 kg/m3

The different values required for the finite element model andthe KLM model can be retrieved from Table B.10 by the follow-ing set of equations:

[cE

]=

[sE

]−1[e] = [d]

[sE

][εS

]=

[εT

]− [d]T [e]

[h] =[εS

]−1[e][

cD]

=[sE

]−1+ [e]T [h]

(B.1)

where[sE

]is the compliance matrix at constant electric field

which is given by

[sE

]=

1Ex

−νyx

Ey−νzx

Ez0 0 0

−νxy

Ex

1Ey

−νzy

Ez0 0 0

−νxzEx

−νyz

Ey

1Ez

0 0 0

0 0 01

Gyz0 0

0 0 0 01

Gxz0

0 0 0 0 01

Gxy

(B.2)

for an orthotropic material,[cD

]is the stiffness matrix at con-

stant electric displacement field,[εS

]is the electric permittivity

matrix at constant strain (S ) and[εT

]is the electric permittivity

15

matrix at constant stress (T ) which is given by

[εT

]=

εT11 0 00 εT22 00 0 εT33

(B.3)where εT22 = ε

T11. [h], [e] and [d] are piezoelectric constants

matrices. For PZT materials, [d] is given by

[d] =[ 0 0 0 0 0 d15 0

0 0 0 0 d24 0 0d31 d32 d33 0 0 0

](B.4)

where d24 = d15 and d32 = d31.

Acknowledgments

Cédric Dumoulin is a Research Fellow of the Fonds de laRecherche Scientifique - FNRS. The authors would like tothank Mr Alexis Tugilimana (ULB) and Prof. Geert Lombaert(KULeuven) for their help.

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IntroductionModeling embedded transducersExternal and embedded transducersFinite element model of embedded piezoelectric transducersEffect of the radial mode

Optimization of the transducerObjectives definitionVariablesFrequency range of interestCases

ResultsOptim. A (Fresh Concrete 20-200 kHz)Optim. B (Hard Concrete 20-200 kHz)Optim. C (Fresh and Hard Concrete 20-200 kHz)Discussion of the results

ConclusionsKLM ModelPiezoelectric properties for finite elements and the KLM model

Documents

Design optimization of embedded ultrasonic transducers for ......Design optimization of embedded ultrasonic transducers for concrete structures assessment Cedric Dumoulin´ a,, Arnaud