Design optimization of embedded ultrasonic transducers for concrete structures assessment

Design optimization of embedded ultrasonic transducers for concrete structures assessment

Ultrasonics 79 (2017) 18–33 Contents lists available at ScienceDirect Ultrasonics journal homepage: www.elsevier.com/locate/ultras Design optimizat...

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Ultrasonics 79 (2017) 18–33

Contents lists available at ScienceDirect

Ultrasonics journal homepage: www.elsevier.com/locate/ultras

Design optimization of embedded ultrasonic transducers for concrete structures assessment Cédric Dumoulin ⇑, Arnaud Deraemaeker 1 Université libre de Bruxelles (ULB), École polytechnique de Bruxelles, Building, Architecture and Town Planning (BATir) Department, Belgium

a r t i c l e

i n f o

Article history: Received 21 September 2016 Received in revised form 6 April 2017 Accepted 8 April 2017 Available online 09 April 2017 Keywords: Embedded piezoelectric transducer Smart aggregate PZT ultrasonic testing Concrete monitoring

a b s t r a c t In the last decades, the field of structural health monitoring and damage detection has been intensively explored. Active vibration techniques allow to excite structures at high frequency vibrations which are sensitive to small damage. Piezoelectric PZT transducers are perfect candidates for such testing due to their small size, low cost and large bandwidth. Current ultrasonic systems are based on external piezoelectric transducers which need to be placed on two faces of the concrete specimen. The limited accessibility of in-service structures makes such an arrangement often impractical. An alternative is to embed permanently low-cost transducers inside the structure. Such types of transducers have been applied successfully for the in-situ estimation of the P-wave velocity in fresh concrete, and for crack monitoring. Up to now, the design of such transducers was essentially based on trial and error, or in a few cases, on the limitation of the acoustic impedance mismatch between the PZT and concrete. In the present study, we explore the working 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 practical way to achieve this imperative is to consider the radial mode of actuation of bulk PZT elements. This is done by developing a simple finite element model of a piezoelectric transducer embedded in an infinite medium. The model is coupled with a multi-objective genetic 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. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Assessing the state of health of concrete is a major issue for everyone for whom the reliability of the structure is essential both for safety and economical reasons (operators of transport network, nuclear power plants, dams, etc.). Visual inspections or destructive tests are the most widely used methods. Such techniques require specific equipment and are labor intensive. They are therefore costly and hardly efficient since they are necessarily sporadic. In the framework of civil engineering structures, an alternative is to set up large sensors networks with the purpose of measuring the dynamic signature of the structure [1]. Large scale effects can be

⇑ Corresponding author. E-mail addresses: [email protected] (C. Dumoulin), [email protected] (A. Deraemaeker). URL: http://www.batir.ulb.ac.be (A. Deraemaeker). 1 Principal corresponding author. http://dx.doi.org/10.1016/j.ultras.2017.04.002 0041-624X/Ó 2017 Elsevier B.V. All rights reserved.

monitored by analyzing the first vibrations modes which are generally excited by the ambient vibrations (wind, traffic). The detection of local defects requires however to study the information carried by higher frequency vibrations. Such waves can be generated by the appearance of a crack. They can be measured with the help of a large network of sensors which allows to localize the defect. This is the concept of Acoustic Emission (AE) testing [2,3]. The wave can also be generated by the monitoring system itself. Such active methods are called Ultrasonic (US) testing. Both AE and US methods require specific transducers which allow to detect and generate waves in a given frequency bandwidth. Such transducers are generally made of Lead Zirconate Titanate (PZT) which is a piezoelectric material. Piezoelectric transducers are currently widely used for nondestructive testing (NDT) due to their small size, low cost and their ability to work both as actuator or sensor. The large external probes which are generally used suffer from several drawbacks. AE and US methods rely on high frequency waves (20–500 kHz) which are strongly attenuated in concrete.

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33

Consequently, the measurement must be performed near the source. The measurement should therefore be done on small size specimens or in really restricted areas. Additionally, the use of such external transducers is restricted by the need of flat surfaces and coupling agents which potentially reduce the efficiency of the transducers. In order to overcome these drawbacks, several researchers have studied the possibility of embedding low-cost piezoelectric transducers in the concrete structure. These embedded piezoelectric transducers allow much more flexible configurations of measurement network and avoid the need of coupling agents. These transducers can be divided in two main design categories. The first type of transducers is based on the design of classical piezoelectric transducers which consists in a piezoelectric patch surrounded by several matching or coating layers [4–9] while the second consists in cement-based piezoelectric composites [10–14]. At ULB-BATir, several designs of the first category have been manufactured and successfully used both for monitoring the Young’s Modulus at very early age and damage detection [15– 17]. These experiments have demonstrated the efficiency of such transducers for structural health monitoring but have also revealed the great importance of optimizing the design of the transducer for each specific application. The main objective of the current study is to develop an efficient method to characterize the performances of embedded piezoelectric transducers. In this study, it is specifically pointed out that the working principle of embedded transducers is different from external transducers. More specifically, it is shown that the methods classically used to optimize external transducers cannot be used for embedded transducers. One of the major issues for permanently embedding transducers into the structure is to reduce their cost and their size as much as possible. It is shown that a pragmatic way to achieve this target is to benefit from the radial mode of actuation. While the behavior of the transducers in the thickness mode can be studied with simple analytic models such as the Krimholtz-Leedom-Matthaei (KLM) model [18–20], this is not the case for the behavior in the radial mode for which a much more advanced finite element model is required. To prevent the results from being affected by the external boundaries, the transducer is embedded in an infinite medium. This can be achieved through specific strategies such as Absorbing Boundary Conditions [21,22] and Perfectly Matched Layers [23,24]. Both methods are implemented and compared in order to select the most promising technique. The first part of the present study deals with the development of a simple and reliable model for characterizing embedded transducers. The second part of the study concerns the optimization of the transducers. For that purpose, the model is coupled with a multiobjective genetic optimization algorithm in order to determine new designs of transducers based on specific expected properties. More specifically, the mechanical properties of concrete strongly evolve with the setting process. This requires the transducer to work in a medium and a related frequency bandwidth of interest which are evolving with the maturity of concrete and results in different optimal designs. This is here highlighted through multiple optimization cases aiming at defining optimal designs of transducers in fresh and hard concrete.

2. Modeling embedded transducers The present section is aimed at developing a simple and accurate model of an embedded piezoelectric transducer. This model should be sufficiently elaborate to properly represent the behavior of the transducer in a given medium while being effective in terms of computational costs. The first section of this part is devoted to

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show how external transducers are generally optimized and why these methods cannot be used for embedded transducers. The second section deals with the development and the validation of an appropriate finite element model. 2.1. External and embedded transducers The general design of external transducers consists in a piezoelectric patch surrounded on one side by a series of matching layers which aim at transmitting the wave from the piezoelectric element to the tested material in a specified frequency bandwidth. On the other side, the piezoelectric element is bounded by a backing material which aims at both absorbing the wave which is propagating in the opposite direction to the tested material and avoiding any reflection between the piezoelectric material and the backing material [25–28]. Such a design allows to restrict the model to a one-dimensional problem. The well-known KLM model is the most widely used model to estimate the efficiency of piezoelectric transducers [18–20,29–31]. It is briefly presented in Appendix A. Nevertheless, the design of piezoelectric transducers is still often determined by optimizing the acoustic impedance matching between the piezoelectric material and the tested material [32–35]. The difference of behavior between the external and embedded transducers is illustrated through a simple example (Fig. 1). It is suggested to compare the axial displacement uz at the external boundary of the transducer due to an applied voltage to the PZT, for several matching (transition) materials. The piezoelectric material is a piezoelectric patch of a thickness of 2 mm (Meggitt Pz26, see Table B.10) and the tested material corresponds to hard concrete. The different matching materials are given in Table 1 where Optim. material is the theoretical optimal matching material between PZT (Z p ) and concrete (Z c ) which is given by

Z Optim ¼

qffiffiffiffiffiffiffiffiffiffi Zc Zp

ð1Þ

where Z ¼ qV p ½Rayls is the acoustic impedance of the material (q and V p are respectively the density and the P-wave velocity in the medium). This optimal value is the one which maximizes the transmission coefficient given by

  Z p  Z eq 2  T ¼ 1  R R ¼  Z p þ Z eq 

ð2Þ

where R is the reflection coefficient which is defined as the square of the ratio between the amplitude of the reflected wave and the amplitude of the incident wave at the interface of the PZT material and Z eq is the equivalent acoustic impedance as seen from the PZT patch. For a unique matching layer, Z eq is simply expressed by (see Eq. (A.3))

Z eq ¼ Z n

Z c cosðkn t n Þ þ jZ n sinðkn tn Þ Z n cosðkn t n Þ þ jZ c sinðkn tn Þ

ð3Þ

where Z n ; kn and t n are respectively the acoustic impedance, the wave-number and the thickness of the matching layer. For the present example, the thickness of the transition materials is arbitrary chosen in order to correspond to the quarter of the wavelength in the material at 300 kHz (tk=4 ¼ k=4). The transducers are modeled with the KLM model. The traditional (external) transducer (Fig. 1a) is bounded on one side by the matching material and the tested semi-infinite material, and on the other side, by a semi-infinite backing material of the same acoustic impedance as the PZT which aims at avoiding any wave reflection at the backside of the transducer. This semi-infinite non-reflecting backing material corresponds to an idealized case of a highly absorbing material actually used as backing layer in real

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C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33

a) Traditionnal Transducers

Optim

1

Edge of the transducer

Concrete 0.68

uz

T Steel

Absorbing PZT

Glue

0 0

2 mm t

200

300

400

500

600

Frequency [kHz]

Concrete Matching Layer Piezoelectric Material

b) Embedded Transducers

100

Fig. 2. Transmission coefficient between PZT and concrete for different transition materials.

Edge of the transducer

a) Traditional Transducer 16

uz

x 10-4

|uz| [μm/V]

Optim

PZT 2 mm

12 Concrete

Glue

Fig. 1. Comparison between tradition external transducer with backing material and embedded transducer.

4 0

h

i ½Mk2j þ ½C kj þ ½½K  þ ½D fwg ¼ 0

ð4Þ

while the normal mode shapes fug are the solution of the undamped eigenvalue problem

100

200

300

400

500

600

b) Embedded Transducer -4 4 x 10

|uz| [μm/V]

transducers. The embedded transducer (Fig. 1b) is symmetrically bounded by the matching material and the tested semi-infinite material. The transmission coefficients T between PZT and concrete for the different matching layers as computed with Eq. (2) are shown in Fig. 2 where it clearly appears that the optimal transition material allows to increase the amplitude of the wave which is transmitted to the tested material. It is important to note that the transmission coefficient T is maximum at odd multiples of the central frequency f c ¼ V p =4tn only if the impedance Z n of the matching material is in the interval Z c < Z n < Z p , otherwise, the maximum value of T will be reached at even multiples of f c . Figs. 3a and b show the amplitude of the displacement at the edge of the transducer (including the matching material) due to an applied unit voltage as a function of the frequency for both configurations. It can be observed that the behavior of the traditional transducers has a similar trend as the transmission coefficient while it is different in the case of the embedded transducer. This difference is due to the appearance of strong resonances for the embedded transducer as shown in Fig. 3b. Such strong resonances are not present for the traditional transducer because of the presence of the absorbing backing layer. They have been studied using a finite element model with Absorbing Boundary Conditions as explained in Section 2.2. With such an approach, the complex mode shapes fwg are the solution of the second order eigenvalue problem

Steel

8

Steel

Optim

2

Concrete

Glue

0 0

100

200

300

400

500

600

Frequency [kHz] f2,f,steel f2,f,optim

f2,c,glue

Fig. 3. Acoustic response (displacement/Volt) as a function of frequency for (a) traditional external transducers and (b) embedded transducers for different transition materials.

h i ½Mx2j þ ½K  fug ¼ 0

ð5Þ

½M and ½K  are respectively the mass and the stiffness matrices of the system while ½D is the material (hysteretic) damping matrix and ½C  is the viscous damping matrix which in the present model only includes the Absorbing Boundary Conditions and is therefore diagonal. xj is the jth eigenfrequency of the undamped system, kj is the jth complex eigenvalue of the damped model for which

x2j ¼ jk2j j [36–38]. In Fig. 4, we plot the real part of the complex mode shapes for the different transducers in an infinite medium. They are normalized so that the value of the displacement is situated between 1 and 1. The corresponding eigenfrequencies are given in Table 2. We have found that these resonant frequencies are in good correspondence with the axial mode of the full transducer (the piezo-

Table 1 Transition materials properties. Materials

E [GPa]

q [kg/m3]

Z [MRayls]

t k=4 [mm]

Glue X60 Hard concrete Optim. Steel

6 30 54 210

900 2200 4142 7800

2.44 8.57 15.7 46.96

2.27 3.25 3.17 4.56

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33

f1,f Steel 1

uz / uz,max

f2,c Glue

Free modes Fixed modes Complex modes

0

f1,f Optim

-1 -t p/2

0

t p/2

z

Fig. 4. Axial (z) component of the displacement of the mode shapes along the transducer (z-axis) for different transition materials (Table 1). The solid lines present the real part of the complex mode shapes, the dashed and dotted lines respectively present the normal mode shapes in fixed and free boundary conditions. The corresponding natural frequencies are given in Table 2. t p is the thickness of the piezoelectric element. The materials and the corresponding thickness of the surrounding layers are given in Table 1.

Table 2 Resonant frequencies of the axial modes for different transition materials. Materials

[kHz]

Glue X60

f 2;c;glue

569

Optim

f 1;f ;optim

245

Steel

f 1;f ;Steel

195

electric patch and the matching layers) either with free (first axial mode, f 1;f ) or clamped (second axial mode, f 2;c ) boundary conditions, depending on the relative stiffness of the matching layer in comparison to the infinite medium (see the dashed and dotted lines in Fig. 4). The free boundary condition corresponds to a transducer with top and bottom surfaces mechanically free to move, while in the clamped case, the displacements of the top and bottom surfaces are constrained to a zero value. Note that the axial mode refers to the resonance of the mechanical system and must be distinguished from the thickness mode of the piezoelectric element which is situated around 1 MHz for the present geometry. This leads to the first conclusion of the present study: acoustic impedance matching theory can be used for the design of external transducers but not for embedded transducers. Indeed, in the case of external transducers the incident wave is propagating from a medium (PZT) to another (concrete), from left to right in Fig. 1, and the back propagating wave in the piezoelectric material resulting from the multiple reflections at the successive interfaces is not in turn reflected due to the backing material. As a consequence, the resonance of the piezoelectric element is highly damped. This implies that the acoustic response of an external transducer is mainly governed by the resonant and anti-resonant vibration modes of each individual layer, which can be either in-phase (constructive) or out-of-phase (destructive) with the incident wave, depending of the surrounding materials. This is immediately related to the matching layer theory. For embedded transducers, as demonstrated here above, the acoustic response of the transducer is related to the overall dynamic behavior of the transducer in its environmentand in some cases high amplitude resonances are present, so that the matching layer theory is not sufficient to predict the behavior of the transducer. The KLM model which is usually used to model piezoelectric transducers only considers the thickness modes of vibration. It can be shown for typical geometries of transducers that the first

21

vibration mode is the radial mode. It is possible to find an analytic solution that combines these modes for simplified 3D geometries [39–43]. These analytic models are difficult to couple with analytic wave transmission models which makes them actually hardly usable. Furthermore, other modes of vibration which are not considered with these analytic models have to be considered. The difference between the radial mode and thickness mode of vibration lies on the main direction in which a specimen is deformed. Fig. 5 shows the displacement field of specific mode shapes for two geometries. The dimensions of the first sample have been defined in order to ensure that the first vibration mode is a pure radial mode (Fig. 5a), also referred to as radial extensional mode (R1). For the present geometry, the first radial mode is approximately at 210 kHz. The first thickness extensional mode (TE1) occurs at 990 kHz. This mode of vibration is specific to thin piezoelectric disks and is described by a large displacement at the center and very low displacement at the disk edges (Fig. 5b). In practice, this mode of vibration is actually hardly usable since it is most often strongly coupled to other vibration modes such as the overtones of the radial mode or other modes of vibrations such as edge modes (E), thickness shear modes (TS). The description of these vibration modes is clearly beyond the scope of the present study and is extensively described by Kocbach [44]. As a consequence, to observe the behavior of the transducer under a pure thickness mode of vibration, the geometry of the piezoelectric element has to be modified in order to decrease the frequency corresponding to the thickness mode of vibration and increase the frequency of the first radial mode (Fig. 5c). Since the diameter of the disk has the same dimension as the thickness, such a geometry cannot be strictly described as a disk. We have therefore considered it more appropriate to call the vibration mode displayed in Fig. 5c a longitudinal extension mode (LE1) which usually refers to long cylinders. For the present geometries, the boundary between both modes is not clearly defined so that in the rest of the present study one will only refer to them as thickness modes. It is important to note that today, many external transducers are made of 1–3 piezoelectric composites, allowing to substantially reduce the effect of the undesirable vibration modes which are coupled with the thickness mode, or to design phased matrix array transducers [30,45]. Nevertheless, such composite materials are much more expensive in comparison to bulk piezoceramic elements which makes their use beyond the scope of the present study. Indeed, one of the major challenges concerning the embedded transducers is to obtain a sufficiently low cost transducer which can be lost in concrete structures. In order to prevent any local mechanical weakness in the structure, the size of the transducer should at most be of the same order of magnitude as the largest aggregates in the concrete structures (around 10 mm diameter). The frequency range of interest for concrete applications (Section 3.3) requires to use thick (and consequently expensive) piezoelectric elements which have a thickness resonant mode at a sufficiently low frequency (around 20 mm for a thickness mode resonant frequency of 100 kHz). A pragmatic solution is to benefit from the radial mode of actuation which allows to reduce the resonant frequency of the transducer. This can be achieved by transforming the radial displacement to thickness displacement with the help of specific structures such as moonies [46], but their use would lead to expensive transducers. In the present study, it is suggested to directly benefit from the ability of inexpensive piezoelectric disk elements to generate axial displacements while their main vibration mechanism in the working frequency range is the radial mode as illustrated in Fig. 5a. Characterizing the performances of such transducers requires a finite element model. The next section deals with the development of such a model which is sufficiently accurate while limiting as much as possible the required computational resources.

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C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33 Axis of symmetry

a) Unbounded Medium

3 1

Ω = Physical Domain

a) Radial extensional (R1) mode

Ω∞

t= 5 mm

b) Medium bounded by PML

Axis of symmetry

Ω = Physical Domain t= 2 mm

ΩPML

R= 2 mm R= 5mm

b) Thickness extensional (TE1) mode

0

Displacement

LPML Max

c) Longitudinal extensional (LE1) mode

Wave Amplitude

Fig. 5. Comparison between radial resonant mode of vibration (R1) (a), thickness extension resonant mode of vibration (TE1) (b) and longitudinal extension resonant mode of vibration (LE1). The colors corresponds to the norm of the displacement in a radial section of the piezoelectric elements. The dimensions of the piezoelectric patches have been chosen to ensure that the first vibration modes corresponds to (a) the radial mode (Radius > Thickness) and (b) the thickness (longitudinal) mode (Radius < Thickness). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

a b x0

xt x

Fig. 6. Concept of perfectly matched layer. The wave is the same in an unbounded medium and in a medium bounded by perfectly matched layers.

2.2. Finite element model of embedded piezoelectric transducers As illustrated in the previous section, designing embedded piezoelectric transducers requires much more advanced models in comparison to those usually employed for external transducers. In this section, a finite element model is suggested for the purpose of being intensively used in a genetic optimization algorithm. This model should therefore be simultaneously sufficiently accurate to properly estimate the performances of the embedded transducer while being sufficiently economical in terms of computational costs. In order to prevent the results to be affected by the external geometry and boundary conditions, it is suggested to embed the transducer in an infinite medium. This can be achieved with the help of specific elements such as boundary elements, infinite elements, Absorbing Boundary Conditions (ABC) [21,22,47–49] or Perfectly Matched Layers (PML) [23,24,50]. The last two are by far the most widely used methods since they can be easily implemented in a finite element software. Both methods have been implemented with SDT, an open and extendible finite element modeling MATLAB based toolbox for vibration problems [51] and are briefly detailed here below. Perfectly Matched Layers are unquestionably the most accurate elements since they are known to appropriately absorb compression, shear and surface waves, evanescent and propagating waves, at any angle of incidence [23,24,50]. Their use can lead to heavy computational costs. Perfectly Matched Layers method consists in replacing a semiinfinite medium X1 bounding a physical domain X by a finite absorbing bounding domain XPML so that the elastodynamic behavior in the physical domain X remains unchanged (Fig. 6). The PML domain XPML should absorb progressively the wave so that no reflection occurs both at the interface between the physical domain and at the external boundaries of the PML domain. The choice of the attenuation function is crucial to properly attenuate both types of waves. An extensive discussion relative to the choice of these parameters is given in François et al. [50]. The values of the attenuation parameters in the direction i e p (i ¼ x; y; z) f i;0 and f i;0 which respectively control the damping of

evanescent and propagating waves used in the present study are given in Table 3. Absorbing Boundary Conditions method is a cheaper option, they are known to prevent the reflection of both compression and shear propagating waves but their efficiency is strongly affected by the angle of incidence of the wave [21,22,47–49]. The basic idea of Absorbing Boundary Conditions method consists in applying dynamic boundary stresses at the surfaces of the physical domain in order to balance the stresses generated by incoming waves (see Fig. 7). The boundary stresses are defined by

r1 ¼ aqV p u_ 1 s13 ¼ bqV s u_ 3

ð6Þ

where V p and V s are respectively the P and S wave velocities, a and b are coefficients that depend on the angle of the incident wave but are generally given as a ¼ b ¼ 1; u_ 1 and u_ 3 are the particle velocities respectively normal and tangent to the external surfaces. Applying ABC as expressed in Eq. (6) requires the use of dashpots applied to the nodes of the external surface. In SDT, this can be achieved by spring-dashpot CBUSH elements. The present section is aimed at selecting an accurate method to model the behavior of embedded piezoelectric transducers. For this purpose, a cylindrical transducer for which the radial displacements are kept free is embedded in an infinite elastic material (Fig. 8). The transducer is composed of a PZT disk which is bounded by a transition layer of the same material and thickness as presented in Section 2.1 (see Table 1 and Fig. 1). The PZT element is made of Meggitt-Pz26 hard piezoceramic with a thickness of 2 mm and 10 mm of diameter. The material data for a finite element computation for this piezoceramic can be found in Table B.10 in Appendix B. The first free resonance frequency of the PZT disk is located around 210 kHz and corresponds to a radial mode, the thickness mode is situated around 1 MHz. Two methods to model the infinite part of the model are here considered. The first consists in using Absorbing Boundary Conditions on the external surfaces of the physical domain (see Fig. 9a). Since the transducer is cylindrical,

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C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33 Table 3 Attenuation functions parameters. Materials

6150 kHz

>150 kHz

e f i;0 p f i;0

5

0

20

20

3 Reflected P-Wave

Reflected P-Wave

1 Reflected S-Wave

Reflected S-Wave

σ1

a) Incident P-Wave

τ13

Incident S-Wave

τ13

Incident P-Wave

σ1

b) Incident S-Wave

Fig. 7. Wave reflection at a boundary due to an incident P-Wave (a) and S-Wave (b). The viscous boundary reaction stresses absorb the incident wave according to Eq. (6).

Concrete Matching Layer Piezoelectric Material

The second method consists in bounding the domain with Perfectly Matched Layers (see Fig. 9b and c). Such a model should provide more accurate results since PML enable to absorb the incident waves regardless of their type or the incident angle. This method is usually used with rectangular boundaries which implies to compute the solution on the full 3D domain (Fig. 9c). Such model leads to substantially higher number of degrees of freedom (approx. 240 000 DOFs) and consequently higher computational costs. The symmetry of the present case allows to use cyclic boundary conditions so the number of DOFs in the model (Fig. 9b) is then considerably reduced (approx. 30 000 DOFs) in comparison with the full model. Although the number of DOFs can be significantly reduced by considering cyclic symmetry, the PML method leads to much higher computational costs which is accentuated by the need of reassembling the system for each computed frequency (the matrices are frequency dependent). The 3D elements used in the different models in Fig. 9 are quadratic and the size of the elements is between 1=10th (c) and 1=15th (a and b) of the shortest wavelength in the medium, which is sufficiently fine to obtain an accurate finite element solution of the wave propagation [50,53,54]. The constitutive equations for a piezoelectric material are given by Eq. (7)



T D

Transducer radially free

Fig. 8. Transducer embedded in an unbounded domain. The radial displacements of the transducer are left free.

the computation of the transducer can be reduced to the computation of a single slice with periodic boundary conditions. The loading case consists in enforced voltage on the electric DOFs corresponding to the electrodes which are respectively the upper and the bottom faces of the piezoelectric element [52]. This method presents the main advantage of involving a reduced number of degrees of freedom (approx. 10 000 DOFs). As aforementioned, the efficiency of the method is strongly affected by both the type and the angle of incidence of the waves in the physical domain.



"  #  S cE ½eT ¼  S E ½e  e

where T and S are the mechanical stress and the mechanical strain vectors while D and E are the electric displacement and the electric   field vectors, cE is the stiffness matrix. PZT is considered as an   elastic and transversely isotropic material. eS is the permittivity matrix at constant strain and ½e is the piezoelectric coupling constants matrix which relates the electrical and the mechanical variables of the equation. These different matrices and the related Meggitt Pz26 material properties are given in Appendix B. Considering the elastodynamic and the electrostatic equilibrium equations and remembering that the strain field and the electric field derive respectively from the displacement field and the electric potential, one can obtain the discrete form of the variational piezoelectric equations used for finite elements analysis [53,55–57]

 M uu x2 0

Cyclic Symmetry

8 mm Edge of the transducer

t 2 mm

Electrodes

a) Cyclic ABC ± 10 000 DOFs

0



þ ix



C uu

0

0

0





þ

K uu

K u/

K uu

K u/



u

u





¼

F



Q

Aborbing Material Piezoelectric Material Concrete Matching Layer

PML

8 mm

8 mm

t 2 mm

t 2 mm

b) Cyclic PML ± 40 000 DOFs

0

ð8Þ

CBUSH Elements Cyclic Symmetry

ð7Þ

c) Full PML ± 240 000 DOFs

Fig. 9. Finite element meshes when the domain is bounded with (a) absorbing boundary conditions and considering a cyclic symmetry, (b) perfectly matched layers and 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 of 10 mm. The properties and the thickness t of the matching materials are given in Table 1.

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33

2.3. Effect of the radial mode Fig. 11 displays the acoustic responses of a piezoelectric transducer which is composed of a piezoceramic element for different surrounding layers. In the frequency band for which the acoustic responses have been computed, only the radial mode of actuation of the piezoelectric element is excited. For that purpose, the radial displacement of the piezoelectric element has been kept free (see Figs. 8 and 9). These choices lead to two mains issues. The first concerns the performance of the transducer if it is directly radially surrounded by the testing materials. The second concerns the impact of using the radial mode of actuation instead of the thickness mode.

105

|Zin| [Ω]

where the subscripts u and u denote respectively the mechanical and the electrical part of the equation. u and u are respectively the nodal displacement vector and nodal electrical potential vector and by extension, F and Q are the nodal vectors of mechanical forces and electrical charges. The damping considered in the present model is a hysteretic damping so the stiffness matrix K is complex and the viscous damping matrix C only applies for the Absorbing Boundary Conditions, which only have mechanical DOFs. In order to properly compare the different models, it is suggested to compute both the electrical input impedance between the electrodes of the piezoelectric element and the acoustic response of the system for each case. In the present case, the electrodes are equipotentials which are respectively located at the bottom and the upper surfaces of the piezoelectric element. In the finite element models, this is achieved by imposing the degrees of freedom corresponding to the electric potential (u) of each node located on these respective surfaces to be equal. The actuation of the transducer is then performed by imposing for each computed frequency a unit voltage (uA ¼ 1) on the electric DOFs corresponding to one electrode (either the upper or the lower) and the electric DOFs of the other electrode are grounded (uG ¼ 0). The electrical input impedance of the transducer is given by Z in ðxÞ ¼ VðxÞ=IðxÞ where VðxÞ ¼ uA  uG ¼ 1 is the imposed voltage and IðxÞ is the resulting current which is actually obtained by computing the resulting charge Q ðxÞ on one electrode from which the current is simply given by IðxÞ ¼ ixQ ðxÞ. Fig. 10 shows the electrical input impedance at the terminals of the PZT disk. A really good match between the results of the different models can be observed. Estimating the acoustic response of the transducer consists in computing the amplitude of the transmitted wave for a given driving voltage. It has to be noted that the amplitude of the displacement may vary depending on the measured location. This can be a significant issue for short wavelengths but not a major issue for the frequency range of interest in the present study (<300 kHz). It is then suggested to only consider the average amplitude of the vertical displacement juz j of the upper surface of the transducer (including the matching material). Fig. 11 shows that the acoustic responses for the different models are really well matched. The resonance which appears in Figs. 10 and 11 corresponds to the radial mode of vibration. As mentioned above, the thickness mode of vibration for the present geometry is situated around 1 MHz, the coupling between these two modes is very low for the frequencies displayed on the present figures. It has to be mentioned that above 300 kHz, other modes of vibrations interact and are superimposed, which makes the acoustic response difficult to interpret for much higher frequencies. This leads to the conclusion that the three models provide equivalent results. According to that observation, the Absorbing Boundary Conditions method seems more appropriate in an optimization process since it requires significantly less computational resources.

10

Full PML Model Cyclic PML Model Cyclic ABC Model

4

Glue Optim

Concrete

Steel 103

102 0

50

100

150

200

250

300

Frequency [kHz] Fig. 10. Comparison of the electrical input impedance Z in as computed with the full PML model (solid lines), the cyclic PML model (dotted lines) and the cyclic ABC model (dashed line) for different matching materials given in Table 1.

x 10 -3

3

Full PML Model Cyclic PML Model Cyclic ABC Model

2.5

|uz| [μm/V]

24

2 1.5 1 1.5 0 0

50

100

150

200

250

300

Frequency [kHz] Fig. 11. Comparison of the acoustic response (juz j [lm/Volt]) at the edge of the transducer (including the matching layer) as computed with the full PML model (solid lines), the cyclic PML model (dotted lines) and the cyclic ABC model (dashed line) for different matching materials given in Table 1.

It is suggested to address these issues by comparing the acoustic response (juz j) for three different cases. For each case, the acoustic response is evaluated from a finite element model with cyclic symmetry and with Absorbing Boundary Conditions as infinite material. The first two cases correspond to ultrasonic transducers 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 connected to the tested material. This is simply achieved by adding ABC to the radial outline of the transducer, the transducer is then said radially constrained (RC). For these two cases, the geometry of the piezoelectric element is kept the same as previously (thickness of 2 mm, diameter of 10 mm). The third case is aimed at comparing the efficiency of ultrasonic transducers working in radial or thickness mode. As explained in 2.1, the thickness mode of a piezoelectric disk of the same geometry is not usable since it is strongly coupled with other vibration modes. It is then suggested to consider a different geometry for which the frequency of the fundamental thickness mode roughly corresponds to the frequency of the first radial mode of the initial geometry. For that purpose, the thickness of the piezoelectric disk is increased to 6 mm and the diameter of the transducers is reduced to 2 mm to sufficiently raise the resonant frequency of the radial mode in order to avoid any coupling between these two modes. For the latter case, the transducer is kept radially free. The acoustic responses for the different cases and for different surrounding layers (Table 1) are presented in Fig. 12. As one might expect, bounding the radial edge of the transducer strongly damps the resonance of the transducer. More surprisingly, according to

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33

Fig. 12, using the thickness mode of a piezoelectric element does not enhance the efficiency of the transducer. This demonstrates that using the radial mode of vibration of piezoeceramic disk is not only a pragmatic choice regarding the cost and the geometry but also an efficient solution provided that the radial displacement of the transducer is not constrained. This result is one of the key points of the present study. In practice, this could for instance be achieved with a proper housing (stainless steel or aluminum) joined to the transducer with a very soft and light potting material such as specific foams, cork or polyurethane for which both the stiffness and density are several orders of magnitude lower in comparison with piezoceramic materials. The impact of the radial bonding is therefore drastically reduced and can be neglected in a first estimate. Such a kind of design is very common in the industry of ultrasonic transducers. 3. Optimization of the transducer Optimizing a piezoelectric transducer consists in looking for the optimal design for a specific application. In the present study, it is suggested to select the material and the thickness of successive transition materials with the purpose of both maximizing the amplitude and the frequency bandwidth of the transmitted wave. These requirements lead to the definition of two objective functions that will be presented hereafter. These objectives 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 offspring population from the best ranked fronts of the parent population, where the rank corresponds to the nondomination level of the solution. The parent population Piþ1 of each offspring generation Q iþ1 is composed of the most nondominated members of the population Ri composed of both the current generation Q i and their own parents Pi (Ri ¼ Q i [ Pi ). The elitist aspects of the method is ensured since all previous and current population members are included in Ri . This specific algorithm is known to be fast and efficient for any shape of Pareto-Optimal front (convex, non-convex, disconnected, etc.). The choice of EA to optimize the transducers results from the difficulty to compute the derivatives of the objective functions with respect to the variables, in particular for discrete variables. 3.1. Definition of objective functions The objective functions have to be adequately defined depending on the characteristics of the transducer that are expected. One can define objective functions in order to optimize the bandwidth, the transmitted energy, the maximum amplitude of the transmitted acoustic wave, the input energy, the focus of the wave and so on [59–62]. In the present study, it is suggested to consider both the energy and the bandwidth of the transmitted wave as criteria to optimize the embedded transducers. The first objective F 1 (Eq. (9)) estimates the mean square of the displacement at the edge of the transducer (including the matching layers). The displacement is the value of the amplitude of the vertical displacement juz j as explained in Section 2.2. The first objective function is then expressed as:

Z F1 ¼ 

f2

juz ðf Þj2 df

ð9Þ

f1

where f 1 and f 2 are respectively the lowest and the highest frequencies of interest. The second objective function is related to the bandwidth of the transmitted wave. Fig. 13 shows two spectra of displacement juz j.

25

The bandwidth is defined as the range of frequencies Df xdB for which the amplitude is xdB above the peak of the spectrum. A low value of Df xdB will characterize a narrowband transducer and conversely, a high value will characterize a broadband transducer. In the present study, the bandwidth Df xdB is normalized by the required bandwidth Df 12 ¼ f 2  f 1 for the application. It is here suggested to define an objective function that both considers bandwidth for 3dBð 0:7juz;max jÞ and 6dBð 0:5juf ;max jÞ. This criterion includes both the bandwidth and the sharpening of the spectrum. It is expressed by

F2 ¼ 

1 Df 3dB Df 6dB þ 2 Df 12 Df 12

ð10Þ

This is illustrated in Fig. 13 where both spectra have similar values of F 1 and Df 6dB . The spectrum depicted by the black line has clearly a flatter shape than the one of the gray line, which is taken into account with the Df 3dB bandwidth. 3.2. Variables The optimization process is performed on the ABC model (Fig. 9a) and considering three matching layers. The first three variables are continuous and correspond to the thickness of each layer. They are constrained according to Table 4 where it can be observed that the minimum thickness for two layers is zero, which enables to consider transducers with one to three transition layers. To each layer corresponds a material which has to be chosen in a restricted list of materials given in Table 5 where E is the Young’s modulus, m is the Poisson’s ratio, q is the density, g is the mechanical loss factor. This list is chosen in order to cover a wide range of stiffnesses and densities while considering only a reduced number of existing materials. The last parameter corresponds to the diameter U of the piezoelectric element. The diameters are restrained to the standard values of the Meggitt-PZ26 piezoelectric disk elements. 3.3. Frequency range of interest The transducers designed in the current study are dedicated to US applications in concrete from early age to damage detection in hardened concrete structures. One of the main interests of using embedded transducers is their ability to catch local information. The frequency domain of interest should therefore be located above the stationary wave regime corresponding to the first vibration modes of the structure. The frequency band of interest is guided by the wavelength corresponding to the shortest characteristic length of the structure (i.e. from the average size of aggregates to several centimeters). For concrete application, the domain evolves with the setting process of the concrete. The evolution of the frequency range is directly related to the evolution of the wave velocity in the material relative to the evolution of the Young’s modulus [15,63]. Fig. 14 shows the typical evolution of the frequencies corresponding to wavelengths from 1cm which roughly corresponds to the average size of the aggregates, to 10cm which is a lower limit of the characteristic length of concrete specimens, relative to the wave propagation regimes as defined by Planès et al. [64]. The limits between the propagation regimes in Fig. 14 should be viewed as a general trend rather than strict frontiers. Indeed, the transition between two propagation regimes is smooth and strongly depends on the concrete itself. For hardened concrete, according to Fig. 14 the frequency band of interest is therefore located largely above the modal analysis regime (f  10 kHz) and sufficiently below the attenuation regime (f  500 kHz) where the wave is both too strongly scattered and absorbed. The working frequency band is ranging from the simple

Amplitude [μm/V]

3

x 10-3

Concrete

2

1

0 3

Amplitude [μm/V]

Amplitude [μm/V]

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33

x 10-3

Glue 2

1

Amplitude [μm/V]

26

3

2

Optim

1

0 3 Axial mode Radial mode Radial mode ‘RC’

2

Steel 1

0

0 0

50

100

150

200

250

300

0

50

Frequency [kHz]

100

150

200

250

300

Frequency [kHz]

Fig. 12. Comparison of the acoustic response (juz j [lm/Volt]) at the edge of the transducer (including the matching layer) as computed with the ABC model for axial mode (solid lines), radial mode (dotted lines) and radial mode constrained (dashed line) for different matching materials.

tric transducers are standard piezoceramic disk elements (Meggitt Pz26, see B.10). Besides these scenarios, a complementary cases (Optim. C) which combines the behavior in fresh and hard concrete is considered. For that purpose, the objective functions are slightly modified in order to take into account both cases. F 1 and F 2 are therefore the average value of the respective objective functions in both hard and fresh concrete as expressed in

|uz,max|

-3dB

Δf3dB

|uz| -6dB

Δf6dB

Fi ¼

f1

f2

Frequency

Δf12

1 F i;fresh þ F i;hard 2

i ¼ 1; 2

ð11Þ

where F i;fresh and F i;hard are the objective functions which are extracted from the acoustic responses respectively in fresh and hard concrete as given in Eqs. (9) and (10). The different scenarios are summarized in Table 6 where f 1 and f 2 define respectively the lower and the upper bound of the frequency domain.

Fig. 13. Definition of the parameters used to compute the objective functions F 1 (Eq. (9)) and F 2 (Eq. (10)).

4. Results Table 4 Variables constraints for the thickness of the layers.

Lower bounds Upper bounds

t 1 [mm]

t2 [mm]

t 3 [mm]

0.1 4

0 4

0 4

wave scattering regime to the multiple wave scattering regime. The frequency range of interest depends on the targeted application. At ULB-BATir, we are mainly working in the simple scattering regime for which the frequency range of interest can be restricted to the domain defined by f 1 ¼ 20 kHz and f 2 ¼ 200 kHz. In the first few hours (fresh concrete), the frequency band of interest is evolving fast so that in our case, the frequency domain can be kept the same as for hard concrete 20–200 kHz. 3.4. Cases In the present study, it is suggested to define designs of transducers specifically optimized for hard concrete (Optim. Case A) and early age applications (Optim. Case B). The mechanical properties of fresh and hard concrete are given in Table 5. The piezoelec-

The result of the method strongly depends on the initial population since the following generations directly descent from that latter. The first generation should therefore be sufficiently large to be representative of the possible solutions otherwise the algorithm runs the risk of converging on a reduced part of the optimal Pareto front. The EA optimization process is performed considering 40 generations with a population of 400 individuals for each generation. Each optimization process is repeated three times with a different (randomly chosen) starting population. This allows to ensure that the process has actually converged. For each optimization, the optimal Pareto front is then composed of the first ranked members of the population R40 which combines the members of Q 40 and P40 , respectively the last offspring generation and their parents as explained in Section 3. In order to benefit of the repeated processes, the final optimal population is generated from the top ranked individuals of a population which mixes up the results of each process. This section is aimed at discussing the results for each case presented in Table 6. For each case, the final optimal Pareto front is shown as well as the solutions at each process. It is then possible to compare the Pareto-Optimal fronts for each of them. Several Pareto-Optimal solutions are selected in order to cover the entire

27

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33 Table 5 Properties of the materials and geometry of the piezoelectric elements (standard geometry for Pz26 elements) considered in the optimization process.

1 2 3 4 5 6 7 8 9 10 11

Materials

E [GPa]

m

q [kg m3]

g

Z [MRayls]

Glue (X60) Mortar Marble Low stiff. glass Aluminum High stiff. glass Brass Titanium DC53 steel Steel High stiff. steel

6 30 50 65 70 80 100 116 150 200 210

0.4 0.2 0.2 0.22 0.35 0.25 0.31 0.34 0.28 0.3 0.3

900 2200 3000 2500 2700 2500 8500 4500 7800 7800 7800

0.1 0.04 0.01 0.03 0.02 0.03 0.02 0.02 0.03 0.03 0.03

2.32 8.12 12.25 12.75 13.75 14.14 29.15 22.85 34.12 39.50 40.47

Fresh concrete Hard concrete

5 30

0.2 0.2

2200 2200

0.04 0.04

3.32 8.12

Diameters U [mm]

10

12:7

16

20

25

30

800 Strong absorption Regime

Frequency [kHz]

600 λ≈1 cm

Multiple scattering regime

400 Frequency Band of interest 200

Simple scattering regime λ≈10 cm

Modal Analysis

0 0

10

20

30

40

50

60

Age [h] Fresh Concrete

Hardened Concrete

allows to obtain solutions which provide more energy to the transmitted wave while a diameter of 10 mm (red circles) leads to more broadband solutions. Six Pareto-Optimal (PO) geometries are selected in order to cover all the Pareto front (see Fig. 15). The corresponding geometries are shown in Table 7. The acoustic responses for these solutions are shown on Fig. 16 where the amplitude spectrum of the transmitted wave in the frequency band of interest progressively evolves from a relatively flat shape (e.g. line 3) to a narrow-band response (e.g. line 220). Fig. 16 also illustrates that the transducers which lead to flatter acoustic responses (red lines) are working below the radial resonance frequency of the PZT element while more energy can be transmitted to the tested material by benefiting of the resonance of the element. It can also be pointed out that it is possible to obtain broadband transducers for which the resonant frequency of the piezoelectric element is located in the frequency band of interest (see e.g. lines 90 and 108). 4.2. Optim. B (hard concrete 20–200 kHz)

Fig. 14. Evolution of the frequency for different wavelengths with setting of concrete in comparison to the corresponding approximated wave propagation regime.

Table 6 Summary of the frequency limits, the geometry of the piezoelectric element and the state of the concrete considered in the different optimization cases. Case

f 1 [kHz]

f 2 [kHz]

State

Optim. A Optim. B

20 20

200 200

Fresh concrete Hard concrete

Optim. C

20

200

Fresh and hard

front. For these solutions, the geometry and the acoustic response are presented.

4.1. Optim. A (fresh concrete 20–200 kHz) Fig. 15 shows the Pareto-Optimal front for the first optimization case (Optim. A in Table 6). The individuals of the Pareto-Optimal front are numbered in the ascending order of F 2 . The colors of the circles which shape the final Pareto front are aimed at highlighting the solutions which involve an identical PZT diameter. The gray crosses are the respective Pareto fronts for three different starting populations which appear to be well matched. The front is split in two main parts, each corresponding to a specific PZT geometry (see Table 7). A piezoelectric element of 16 mm (blue circles)

The final Pareto front for hard concrete (Optim. B) is displayed on Fig. 17 where it clearly appears that the different optimization processes lead to really well matched solution domains (gray crosses). The colors of the circles correspond to a specific diameter of the PZT patch while the geometries of six of the PO solutions are given is Table 8. As for fresh concrete (Optim. A, Fig. 15), the optimal front is divided in two main groups each corresponding to a specific working principle. Indeed, the solutions which lead to the most broadband acoustic response (Fig. 18) are obtained by using a piezoelectric disk with a smaller diameter and whose frequency corresponding to the radial mode of actuation is located above the frequency band of interest (see lines 1 and 102 in Fig. 18). These solutions are not able to transmit a lot of energy into the system in comparison to solutions which take advantage of the radial resonance mode of the piezoelectric element. In particular, solutions 302 and 303 have almost identical values of F 2 (which indicates the bandwidth of the transducer) while the solution 303 has a value of F 1 (which refers to the transmitted energy) almost twice as large as the value of F 1 for solution 302. 4.3. Optim. C (fresh and hard concrete 20–200 kHz) Comparing the optimal solutions resulting from Optim. A and Optim. B in order to determine similarities and deducing a design which would be optimal in both cases looks to be a difficult challenge. On the one hand, the diameter of the Pareto-Optimal solution differs on a large range of the Pareto front. On the other,

28

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33

F1 −1.4

−1.2

−1

−0.8

F1 −0.6

−0.4

−0.2

0

−1.4

0

−1.2

−1

−0.8

−0.6

−0.4

−0.2

−0.2 −0.4 220

90

−0.4

581

F2

454 413

−0.6 303

87

0 −0.2

−0.6 108

0

F

2

302 −0.8

−0.8 40 3

−1

Fig. 15. Optim. A (fresh concrete, 20–200 kHz). Multi-objectives optimization computation. The colored bullets correspond to the best ranked solutions of population mixing the optimal solutions of three optimization process (gray crosses). Red and blue filled circles correspond respectively to geometries with a piezoelectric element of 10 mm and 16 mm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

1

−1

Fig. 17. Optim. B (hard concrete, 20–200 kHz). Multi-objectives optimization computation. The colored bullets correspond to the best ranked solutions 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). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 7 Optim. A (fresh concrete, 20–200 kHz). Geometry of the selected Pareto-Optimal solutions. The dimensions (U; t1 ; t2 ; t3 ) are in mm. PO

U

Lay. 1

t1

Layer 2

t2

Layer 3

t3

3 40 87

10 10 10

Mat 5 Mat 1 Mat 1

0.73 1.95 1.69

Mat 1 Mat 1 Mat 2

3.95 0.40 0.73

Mat 8 Mat 8 Mat 7

0.27 1.06 0.35

90 108 220

16 16 16

Mat 2 Mat 1 Mat 1

3.16 0.33 1.60

Mat 3 Mat 2 Mat 2

3.73 2.70 1.24

Mat 3 Mat 3 Mat 5

1.11 3.84 1.33

3

x 10

−3

3 40 87

Amplitude [µm/V]

2.5 2

90 108 220

frequency band of interest

1.5 1 0.5 0 0

50

100

150 200 Frequency [kHz]

250

300

Fig. 16. Optim. A (fresh concrete, 20–200 kHz). Acoustic response for the selected Pareto-Optimal solutions. The colors of the lines refer to a specific diameter: red (10 mm), blue (16 mm). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

among the geometries presented in Tables 7 and 8 which are of the same diameter, it is difficult to draw general conclusions. Fig. 19 shows the PO solutions for a mixed case where the efficiency of the transducer is balanced between optimal performance in fresh and hard concrete according to Eq. (11). The Pareto front has a similar trend as observed for Optim. A and Optim. B. Specifically, the results can be clearly separated in two main groups each corresponding to a specific geometry of the piezoelectric elements which can be identified on the Pareto front by the colored circles. As for the two previous optimization cases, each group corresponds to a specific working principle and the same remarks concerning the transmitted energy and the bandwidth hold. On the contrary to Optim. B and as for Optim. A, the domain of solutions only contains two different diameters.

The geometry of six PO solutions is presented on Table 9 and their respective locations in the Pareto front can be observed on Fig. 19 while the acoustic responses in both fresh and hard concrete are respectively displayed on Fig. 20a and b. The result has to be compared to Optim. A and Optim. B. This is first achieved by comparing the acoustic response either in fresh and hard concrete with one solution obtained in Optim. A (PO 220) and Optim. B (PO 581), see dashed lines in Fig. 20. The results obviously differ but the actual gain of a proper optimization process for each specific cases does not clearly appear. The objective functions F 1 and F 2 of the PO solutions for Optim. C are now evaluated separately in fresh and hard concrete. The couples (F 1 ; F 2 ) for the different PO solutions in Table 9 are then displayed in Fig. 21a and b (black circles) where they are compared to the Pareto fronts for Optim. A and Optim. B (gray circles). Such a representation allows to clearly observe which solutions are more optimized in one case than the other. Fig. 21 also highlights that it is possible to obtain solutions that are almost optimal in both cases as for PO 261 and 298. 4.4. Discussion of the results Three optimization cases have been considered with the purpose of covering the frequency domain of interest for concrete 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 from the previous results. One of the main goals of using a meta-heuristic optimization algorithm is to find solutions to multimodal, nonlinear and discontinuous problems for which the gradient of the objective functions cannot be estimated. In return, analyzing the results from such a process also leads to difficulties. Specifically, it appears difficult to draw general design rules from the results obtained with this method. And it is still worse with an increased

29

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33 Table 8 Optim. B (hard concrete, 20–200 kHz). Geometry of the selected Pareto-Optimal solutions. The dimensions (U; t1 ; t2 ; t3 ) are in mm. PO

U

Lay. 1

t1

Layer 2

t2

Layer 3

t3

1 302

10 10

Mat 1 Mat 1

1.86 1.04

Mat 1 Mat 5

0.05 0.01

Mat 10 Mat 7

1.88 1.87

454

16

Mat 1

0.49

Mat 2

1.28

Mat 9

1.91

303 413 581

20 20 20

Mat 2 Mat 2 Mat 1

2.26 2.08 0.68

Mat 11 Mat 5 Mat 2

2.44 2.42 2.00

Mat 9 Mat 9 Mat 11

1.89 1.93 1.53

a) Fresh concrete

3 x 10 −3

1 302 454 303 413 581

2 frequency band of interest

1.5

x 10

3

1 0.5

1 19 114

frequency band of interest

240 261 298

2 Optim. A (220)

1.5 1 0.5

0 0

50

100 150 200 Frequency [kHz]

250

300 0 0

Fig. 18. Optim. B (hard concrete, 20–200 kHz). Acoustic response for the selected Pareto-Optimal solutions. The colors of the lines refer to a specific diameter: red (10 mm), blue (16 mm) and orange (20 mm). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

50

−0.8

−0.6

−0.4

−0.2

0

Amplitude [µm/V]

−1

0 −0.2 −0.4

298 261 240

250

−0.6

F

2

114

300

−3

1 19 114

2.5

F1 −1.2

100 150 200 Frequency [kHz]

b) Hardened concrete 3 x 10

−1.4

−3

2.5

Amplitude [µm/V]

Amplitude [µm/V]

2.5

frequency band of interest 2 1.5

240 261 298

Optim. B (581)

1 0.5 0 0

−0.8

19

50

100 150 200 Frequency [kHz]

1 −1 Fig. 19. Optim. C (fresh and hard, 20–200 kHz). Multi-objectives optimization computation. The colored bullets correspond to the best ranked solutions of population mixing the optimal solutions of three optimization process (gray crosses). Red and blue filled circles correspond respectively to geometries with a piezoelectric element of 10 mm and 16 mm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

number of parameters. Indeed, the algorithm only provides a range of feasible optimal solutions according to preset constraints. Furthermore, it has to be noted that only six out of hundreds of PO

250

300

Fig. 20. Optim. C (fresh and hard concrete, 20–200 kHz). Acoustic response 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). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

solutions are presented for each optimization process. They have been chosen in order to cover the entire Pareto front and to provide a relatively representative picture of the geometry associated to a part of the front. In order to remain focused on the main objective of the current research, the other geometries are not presented.

Table 9 Optim. C (fresh and hard, 20–200 kHz). Geometry of the selected Pareto-Optimal solutions. The dimensions (U; t1 ; t2 ; t3 ) are in mm. PO

U

Lay. 1

t1

Layer 2

t2

Layer 3

t3

1 19 114

10 10 10

Mat 6 Mat 5 Mat 1

2.10 0.78 1.10

Mat 1 Mat 1 Mat 5

3.68 1.84 1.20

Mat 5 Mat 6 Mat 11

0.37 2.43 0.99

240 261 298

16 16 16

Mat 2 Mat 1 Mat 1

3.07 0.69 1.03

Mat 3 Mat 2 Mat 2

2.03 1.82 1.49

Mat 7 Mat 7 Mat 11

0.96 1.13 0.97

30

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33

a) Fresh concrete −1.4

−1.2

−1

F1 −0.8

−0.6

−0.4

−0.2

0

Optim. A Optim. C

0 −0.2 −0.4

298 261

114

240

−0.6

F

2

−0.8

19

5. Conclusions

−1

1

b) Hardened concrete F 1 −1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Optim. B Optim. C

0 −0.2

298 261

observed in Fig. 14 this upper limit evolves rapidly once the setting process has started. Depending on the concrete and the conditions in which it is set up, its properties evolve rapidly and at 6–10 h this upper limit roughly reaches 200 kHz. For hardened concrete, the choice of the frequency band of interest will strongly depend on the range of applications for which the transducer is dedicated. For instance, ultrasonic pulse velocity tests (UPV), AE or more advanced ultrasonic testing such as nonlinear ultrasonic wave spectroscopy NRUS [71] or diffuse ultrasound [72–74] will require a different bandwidth and consequently a different design.

−0.4 240 −0.6 114 19

F2

1 −0.8 −1

Fig. 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).

Many other geometries are feasible and some points that are really close in the Pareto chart can be associated to geometries that are quite different both in terms of materials and thicknesses of the surrounding layers. It is comforting to observe that in most cases, PO solutions which have similar geometries are located in the same part of the chart. The quantification of the effect of perturbations in the geometry and in the material properties on the acoustic response is fundamental and should be the object of specific studies. Designs could appear more robust than other to perturbations and should therefore be preferred. It is then to the user to select the geometry depending on both the required acoustic response, the technical feasibility as well as economic considerations. Specifically, gluing two successive materials together is not a trivial task. In the present study, the link between two layers has been considered as ideal which is never the case in practice. As a consequence, the impact of a layer of glue has to be carefully studied. Such a study must however be performed taking into account the current technological limits. The analysis of the dynamic behavior of thin bounding layers is still on the spotlight of the research [65–67] and properly including such kind of material in the model has to be performed with the utmost care [68–70]. Although the number of materials has been restricted to accessible and affordable materials, the practical way to manufacture these different solutions is not considered in the present study. It is however obvious that certain optimal designs are easier and less expensive to manufacture and are therefore more appropriate for the actual fabrication of the new transducers. More specifically, certain solutions involve a reduced number of materials since two successive layers are made of the same material, or two successive materials can be more or less easy to bind together. Before the earliest stages of setting of the concrete, the frequency range of interest is clearly situated below 100 kHz. As

In this study, the fundamental difference in terms of working principle between external transducers and embedded transducers is first shown through a simple example. The use of the radial mode of actuation of the piezoelectric transducer is explored. Such a mode of actuation is generally seen as an undesirable mode leading to the use of expensive piezocomposites which considerably reduce its effect. The resonant frequency corresponding to the radial mode for typical geometries of transducers is generally much lower than the thickness mode. The frequency range of interest for concrete application can be reached with smaller piezoelectric elements. Using the radial mode can thus be viewed as a pragmatic choice to produce economical transducers with reduced dimensions. The performance of the transducer for such mode of actuation is difficult to estimate and necessitates a finite element model. In order to prevent the impact of the external geometry such as wave reflection or global modes of vibration, the transducer should be embedded in an infinite medium. This can be achieved by using non-reflecting boundary conditions such as Absorbing Boundary Conditions or specific elements such as infinite elements or Perfectly Matched Layers. In the present study, both ABC and PML are used and compared. It is shown that both methods lead to similar results. Since the use of ABC requires significantly less computational resources, it is selected for optimizing the design of the transducers with a multi-objective genetic algorithm. The objective functions used in this study are aimed at characterizing both the bandwidth and the transmitted energy in the tested medium. Several optimization cases are considered in order to define efficient designs of transducers either in fresh or hardened concrete. It is shown that the method allows to design a transducer whose performances match specific requirements. The method is general and allows to either define additional objective functions or to modify the definition of the objectives depending on the expected specification for the transducer. Further research will be focused on the fabrication of the new transducers as designed in the present study. These new transducers will be experimentally characterized and then used for the development of new efficient structural health monitoring techniques in concrete structures. Acknowledgments Cédric Dumoulin is a Research Fellow of the Fonds de la Recherche Scientifique – FNRS. The authors would like to thank Mr. Alexis Tugilimana (ULB) and Prof. Geert Lombaert (KULeuven) for their help.

Appendix A. KLM model The one dimensional piezoelectric KLM model [18–20] is schematically presented in Fig. A.22 where C S0 ; X and U are given by

31

C. Dumoulin, A. Deraemaeker / Ultrasonics 79 (2017) 18–33 Table B.10 Pz26 properties to introduce in FEM and analytic modeling. Material property

Value

Unit

300

1012 C/N

Piezoelectric constants d33 d31

130

1012 C/N

d15

330

1012 C/N

Permittivity

eT33 eT11

Fig. A.22. KLM model: piezoelectric transducer in an acoustic transmission line.

C S0 ¼

AeS33 tp 2

h33 sin kp tp Ax 2 2h33 U¼ sinkp tp =2 AxZ p

where 3 ¼ z is the poling axis in the IEEE standards [75], h33 ¼ e33 =eS33 is the piezoelectric constant in the poling direction, e33 is the piezoelectric stress constant, eS33 is the dielectric constant (permittivity) at constant strain, cD33 is the elastic stiffness at constant electric displacement field D; x is the angular frequency, A is the area of the transducers, Z p ; kp and t p and are respectively the acoustic impedance, the wave-number and the thickness of the piezoelectric element. The front and backing materials are linked to the KLM model through the acoustic ports of the model. The successive transmission matrices are given by

"

½Tn  ¼

cos kn t n j sinAZknn tn

jAZ n sin kn tn cos kn tn

#

ðA:2Þ

which relates the force F and the particle velocity v at the left side of each acoustic layer to the force and the particle velocity at the right side of the layer. The backing and front semi-infinite materials are simply given by their respective acoustic impedance Z b and Z f . For a unique transmission layer, the equivalent acoustic impedance Z eq of the front side as viewed by the transducer is given by

Z eq ¼

F 0f

Av 0f

¼ Zn

Z f cosðkn tn Þ þ jZ n sinðkn tn Þ Z n cosðkn tn Þ þ jZ f sinðkn tn Þ

ðA:3Þ

Appendix B. Piezoelectric properties for finite elements and the KLM model The piezoelectric elements used in the present study are made of Meggitt Pz26 which is a Navy type I hard PZT. The material data for finite element computations are given in Table B.10. These values can be retrieved from the material data-sheet using the relation given in [75–77]. The different values required for the finite element model and the KLM model can be retrieved from Table B.10 by the following set of equations:

 E   E 1 c ¼ s   ½e ¼ ½d sE  S  T  e ¼ e  ½dT ½e  1 ½h ¼ eS ½e  D   E 1 þ ½eT ½h c ¼ s

ðB:1Þ

F/m F/m

74:17 59:14 25:1 27:89 0:329 0:3 0:376 7700

GPa GPa GPa GPa

Mechanical data Ep Ez Gzp Gp

mp mzp mpz q

ðA:1Þ



1300e0 1335e0

kg/m3

  where sE is the compliance matrix at constant electric field which is given by

2

1 Ex

6 mxy 6 6 Ex 6 mxz  E 6 6  Ex s ¼6 6 0 6 6 6 0 4

 Eyxy

m

 mEzxz

0

0

1 Ey

 Ezyz

0

0

 Eyzy

1 Ez

0

0

0

0

1 Gyz

0

0

0

0

1 Gxz

0

0

0

0

m

m

0

0

3

7 07 7 7 07 7 7 07 7 7 07 5

ðB:2Þ

1 Gxy

  for an orthotropic material, cD is the stiffness matrix at constant  S electric displacement field, e is the electric permittivity matrix   at constant strain ðSÞ and eT is the electric permittivity matrix at constant stress ðTÞ which is given by

2  T

eT11

0

e ¼6 4 0

eT22

0

0

3 0 7 0 5

ðB:3Þ

eT33

where eT22 ¼ eT11 . ½h; ½e and ½d are piezoelectric constants matrices. For PZT materials, ½d is given by

2

0 6 ½d ¼ 4 0 d31

0 0

0 0

d32

d33

0 0 0 d24 0

0

d15 0

3 0 7 05

ðB:4Þ

0

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