-
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
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
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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
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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
Edge of the transducer
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
90 16 Mat 2 3.16 Mat 3 3.73 Mat 3 1.11108 16 Mat 1 0.33 Mat 2
2.70 Mat 3 3.84220 16 Mat 1 1.60 Mat 2 1.24 Mat 5 1.33
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.
PO Φ Lay. 1 t1 Layer 2 t2 Layer 3 t3
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
303 20 Mat 2 2.26 Mat 11 2.44 Mat 9 1.89413 20 Mat 2 2.08 Mat 5
2.42 Mat 9 1.93581 20 Mat 1 0.68 Mat 2 2.00 Mat 11 1.53
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
]
frequency band of interest
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.
PO Φ Lay. 1 t1 Layer 2 t2 Layer 3 t3
1 10 Mat 6 2.10 Mat 1 3.68 Mat 5 0.3719 10 Mat 5 0.78 Mat 1 1.84
Mat 6 2.43114 10 Mat 1 1.10 Mat 5 1.20 Mat 11 0.99
240 16 Mat 2 3.07 Mat 3 2.03 Mat 7 0.96261 16 Mat 1 0.69 Mat 2
1.82 Mat 7 1.13298 16 Mat 1 1.03 Mat 2 1.49 Mat 11 0.97
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
]
frequency band of interest
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
]
frequency band of interest
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