- Email: [email protected]
Topology optimized design of functionally graded piezoelectric ultrasonic transducers
Available online at www.sciencedirect.com
Physics Procedia 00 (2009) 000–000 Physics Procedia 3 (2010) 891–896 www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
International Congress on Ultrasonics, Universidad de Santiago de Chile, January 2009
Topology Optimized Design of Functionally Graded Piezoelectric Ultrasonic Transducers Wilfredo Montealegre Rubio*, Flávio Buiochi, Julio Cezar Adamowski, Emílio C. N. Silva Department of Mechatronics and Mechanical Systems Engineering, University of São Paulo, Av. Prof. Mello Moraes, 2231 - Cidade Universitária, São Paulo – SP, CEP: 05508-900, Brazil Elsevier use only: Received date here; revised date here; accepted date here
Abstract This work presents a new approach to systematically design piezoelectric ultrasonic transducers based on Topology Optimization Method (TOM) and Functionally Graded Material (FGM) concepts. The main goal is to find the optimal material distribution of Functionally Graded Piezoelectric Ultrasonic Transducers, to achieve the following requirements: (i) the transducer must be designed to have a multi-modal or uni-modal frequency response, which defines the kind of generated acoustic wave, either short pulse or continuous wave, respectively; (ii) the transducer is required to oscillate in a thickness extensional mode or piston-like mode, aiming at acoustic wave generation applications. Two kinds of piezoelectric materials are mixed for producing the FGM transducer. Material type 1 represents a PZT-5A piezoelectric ceramic and material type 2 represents a PZT-5H piezoelectric ceramic. To illustrate the proposed method, two Functionally Graded Piezoelectric Ultrasonic Transducers are designed. The TOM has shown to be a useful tool for designing Functionally Graded Piezoelectric Ultrasonic Transducers with uni-modal or multi-modal dynamic behavior.
PACS: 07.05.Tp; 43.38.FX Keywords: Piezoelectric transducers; topology optimization method; functionally graded materials; piston-like mode
1. Introduction A new concept has been applied to design piezoelectric transducers: Functionally Graded Material (FGM) concept. In this work, these piezoelectric transducers are named Functionally Graded Piezoelectric Transducers (FGPT). In general, FGMs possess continuous graded properties with gradual change in microstructure [1]. The FGM concept can be applied to piezoelectric ultrasonic transducers to reduce reflection wave inside them [2], and to obtain smaller time waveform (large bandwidth) [3, 4], which is desirable in medical imaging and non-destructive testing applications. Although there exist literature concerning FGPT manufacturing, research that focuses on the systematic design and analysis of FGPTs is rather scarce and deals with analytical techniques, which grade a single material property.
* Corresponding author. Tel.: +55-11-92920133; fax: +55-11-30915461. E-mail address: [email protected].
doi:10.1016/j.phpro.2010.01.114
892
W.M. Rubio et al. / Physics Procedia 3 (2010) 891–896 Rubio et al./ Physics Procedia 00 (2010) 000–000
On the contrary, by using finite element approaches, 2D and/or 3D FGPT can be designed, considering one or more gradation properties. Many of the existing numerical approaches are based on multilayer design, where the material properties vary from layer to layer and are uniform at each layer [5]. Additionally, several works have shown that the gradation function of an FGPUT can influence its performance [5], though the best property gradation function for a specific application has not been demonstrated. This research proposes a new, generic and systematic topology optimization formulation to find the optimum material gradation of FGPTs aiming ultrasonic applications. FGPTs are designed by using the Topology Optimization Method (TOM). The TOM is a generic and systematic optimization technique, which combines optimization algorithms with Finite Element Method to maximize a user-defined objective function. The main goal is to find the optimal material distribution of Functionally Graded Piezoelectric Ultrasonic Transducers (FGPUT) to achieve the following requirements: (i) designing uni-modal transducers; (ii) designing multi-modal transducers. The Modal Assurance Criterion [6] is applied to track the desirable piston-like mode. The applied optimization algorithm is based on Sequential Linear Programming and the concept of the Continuum Approximation of Material Distribution is considered for achieving continuous material gradation [7].
2. FGPUT Modeling The constitutive relations for piezoelectric media may be derived in terms of their associated thermodynamic potentials. Assuming the strain tensor S and the electrical field E as independent variables, the constitutive piezoelectric equations are written (with Einstein’s convention) [8]:
Tij
E x, y S kl ekij x, y Ek Cijkl
Di
eikl x, y S kl İikS x, y Ek
for i,j,k,l = 1,2,3
(1)
where T and D are respectively the stress tensor and the electric displacement vector. The elasticity tensor CE (elastic stiffness at constant electric field), the piezoelectric coefficient tensor e, and the dielectric constant tensor HS (dielectric susceptibility at constant strain) are assumed to vary along the Cartesian coordinates x and y (for a twodimensional model). Completing the piezoelectric model, the mechanical and electrical equilibrium equations are applied. The mechanical balance corresponds to the calculation of the forces expressed by the Newton’s equation and the electrical balance is based on the Gauss’s theorem. After a mathematical manipulation, the equilibrium equations for a piezoelectric medium (including equilibrium balance) are written as: - ȡ x, y Ȧ 2 u i
w E x, y S kl ekij x, y E k ; C ijkl wr j
w eikl x, y S kl İ ikS x, y E k wr j
(2)
0
where U is the density of the material, which varies along the Cartesian coordinates x and y; r is a unit vector in the Cartesian coordinate system; u is the displacement vector; and Z is the circular frequency.
3. Optimization Problem The topology optimization problem is formulated to find the optimal gradation law of a FGPUT. The transducer must be designed to have a multi-modal or uni-modal frequency response, which defines the kind of generated acoustic wave, either short pulse or continuous wave, respectively. Furthermore, the transducer is required to oscillate in a thickness mode (piston-like mode), aiming at acoustic wave generation applications. For uni-modal transducers, the electromechanical coupling of a desirable mode k must be maximized, and the electromechanical coupling of the adjacent modes (mode number k + a1 with a1 = 1, 2,., A1, and/or k – a2 with a2 =
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W.M. Rubio et al. / Physics Procedia 3 (2010) 891–896 Rubio et al./ Physics Procedia 00 (2010) 000–000
1, 2,…, A2) must be minimized. Additionally, the resonance frequencies of the modes k1 = k + a1 with a1 = 1, 2,…, A1 must be maximized, and the resonance frequencies of the modes k2 = k – a2 with a2 = 1, 2,…, A2 must be minimized. For multi-modal transducers, the electromechanical couplings of a mode set must be maximized, and their resonance frequencies must be approximated. For uni-modal (F1) and multi-modal (F2) transducers, the objective functions are given as:
ª 1 § A1 wk Ark « ¨ w1k1 Ark 1 «¬ Į1 ¨© k1 1
¦
F1
ª 1 § A1 « ¨ w3 k Ȝrk «¬ Į3 ¨© k1 1 1 1
¦
·º n3 ¸ » ¸» ¹¼
1
n3
n1
·º ¸» ¸» ¹¼
ª1 « «¬ Į4
1
n1
ª1 « «¬ Į2
§ A2 ¨ w A ¨ k 1 2 k2 rk2 ©2
¦
§ A2 ¨ w Ȝ ¨ k 1 4 k2 rk2 ©2
¦
·º n4 ¸ » ¸» ¹¼
1
n2
·º ¸» ¸» ¹¼
1
n2
... (3)
n4
with: A1
A2
¦
Į1
¦
w1k1 ; Į2
k1 1
F2
ª1 « «¬ Į 1
A1
¦
w2 k2 ; Į3
k2 1
§ ¨ ¨ ©
m
¦ w A k
n1
rk
k 1
A2
w3 k1 ;
¦w
Į4
4 k2
·º ¸» ¸ ¹ »¼
1
n1
ª1 « «¬ Į 2
m
º 1 2 Ȝr Ȝ02k » Ȝ02k k »¼
¦ k 1
; nm
1, 3, 5, 7... ; m = 1, 2, 3, 4
k2 1
k1 1
1
n2
(4)
with: m
Į1
¦
m
wk ;
Į2
k 1
1
¦Ȝ k 1
2 0k
;
Ȝrk
Ȧr2k ;
n1
1, 3, 5, 7... ; n 2
r 2 , r 4 , r 6 , r 8 ...
where, for uni-modal transducers (see (3)), the terms Ark , Ark1 and Ark2 represent the electromechanical coupling (measured by the Piezoelectric Modal Constant – PMC [8]) of the desirable mode, the left and the right adjacent modes, respectively. The terms w k , wik1 (i = 1, 3) and wjk2 (j = 2, 4) are the weight coefficients for each mode considered in the objective function F1. Finally, the terms Ȝrk1 and Ȝrk2 represent the eigenvalues of the left and the right modes with relation to the desirable one (mode number k), and the term n is a given power. For multimodal transducers (see (4)), the constant m is the number of modes considered, the terms Ȝ rk and Ȝ0 k are the current and desirable (or user-defined mode) eigenvalues for mode k (k = 1, 2,…, m), respectively; and Ȧ rk are the resonance frequencies for mode k (k = 1, 2,…, m). For uni-modal and multi-modal transducers, the proposed optimization problem is expressed as: maximize ȡ
TOM
x, y
subjected to :
Fi
i 1 or 2
³ȡ
TOM
x, y dȍ - ȍs d 0
;
(5)
ȍ
0 d ȡTOM x, y d 1 Equilibriu m and Constituti ve Equation
where U720(x, y) is the design variable (pseudo-density) at Cartesian coordinates. The term :s describes the amount of piezoelectric material at two-dimensional domain :. The second requirement (mode shape tracking) is achieved by using the Modal Assurance Criterion (MAC) [6], which is widely used to compare experimental and theoretical modal analyses. The optimization problem is formulated as finding the material gradation of an FGPUT,
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W.M. Rubio et al. / Physics Procedia 3 (2010) 891–896 Rubio et al./ Physics Procedia 00 (2010) 000–000
which maximizes the multi-objective function F subjected to a piezoelectric volume constraint. That constraint is implemented to control the amount of piezoelectric material into the design domain, :. Since FGPUT can be constructed by sintering a layer-structured ceramic green body without using adhesive material, the optimization problem is arranged as a layer-like optimization problem; in other words, the design variables are considered equals at each layer line. This makes the FGPUT manufacturing possible. However, the layer-like optimization implies that the finite element mesh must be homogeneous.
4. Numerical Implementation To treat the gradation in FGPUTs, the material property is continuously interpolated inside each finite element based on property values at each finite element node. This approach is named Graded Finite Element (GFE) formulation [9]. In this research, the piezoelectric GFE employs the same shape functions Nn to interpolate the unknown nodal displacements and electrical potentials, spatial coordinates, and the material constants. Thus, the elastic CE, piezoelectric e, and dielectric HS material properties in (2) are respectively given by: E x, y Cijkl
nd
¦ N x, y C
E ijkl n
n
,
n 1
eikl x, y
nd
¦ N x, y e n
ikl n
, and
(6)
n 1
İikS x, y
nd
¦ N x, y İ
S ik n
n
for i, j, k, l 1, 2, 3
n 1
where nd is the number of nodes per finite element. On the other hand, the Continuum Approximation of Material Distribution (CAMD) concept [7] is used to continuously represent the material distribution. The CAMD considers that the design variables inside the finite element are interpolated by using, for instance, the FE shape functions Nn. e at each graded finite element e can be expressed as: Thus, the pseudo-density ȡTOM e ȡTOM x, y
nd
¦ȡ
n TOM i
N i x, y
(7)
i 1
n
where ȡTOMi and Ni are respectively the nodal design variable and shape function for node i (i = 1,…,nd), and nd is the number of nodes at each GFE. This formulation allows a continuous distribution of material along the design domain instead of the traditional piecewise material distribution applied by previous formulation of topology optimization [10]. Finally, it is implemented a new scheme to achieve an explicit gradient control by introducing a layer of nodal variables on top of the existing nodal variables [11]. The variables in the new layer are used as design variables, which are updated by the iterative optimization process. The projection technique employs a function to relate the n
p
nodal design variable ȡTOMi to the nodal material pseudo-density ȡTOMi . That projection function is defined as:
¦ ȡ W r ¦ W r jSi
P TOMi
ȡ
n TOM j
jSi
ij
with: rij
x j xi
(8)
ij
n
p where ȡTOMi is the nodal design variable assumed to be located at node i; ȡTOM j is the material pseudo-density
located at node j; and Si is the set of nodes j in the subdomain under influence of node i. In other words, the subdomain Si corresponds to a circle with its center located at the node i and user-defined radius equal to rmin. The
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W.M. Rubio et al. / Physics Procedia 3 (2010) 891–896 Rubio et al./ Physics Procedia 00 (2010) 000–000
terms xi and xj represent the Cartesian coordinate of nodes i and j, respectively. Last, W represents a weight function [11].
5. Results To illustrate the proposed method, a uni-modal and a multi-modal FGPUT are designed considering plane strain assumption. The design domain used is shown in Fig.1. The design domain is specified as a 20 mm by 5 mm rectangle with two fixed supports at the end of the left and right-hand sides. The idea is to simultaneously distribute two types of materials into the design domain. The material type 1 is represented by a PZT-5A piezoelectric ceramic and the material type 2 is a PZT-5H. Initially, the design domain contains only PZT-5A material and a material gradation along thickness direction is assumed. A mesh of 50 x 30 finite elements is adopted. Fig.2 and Fig.3 show the results when a uni-modal and a multi-modal FGPUTs are designed by using TOM. For the uni-modal transducer, the electromechanical coupling (measured by the PMC [8]) value of the piston-like mode is increased by 59% while the PMC values of adjacent modes are decreased approximately by 75%. For the multimodal transducer, the PMC value of the piston-like mode is increased by 15%, while the PMC values of the left and right adjacent modes are increased by 15% and 181%, respectively. In both designs, the piston-like mode is kept, producing the highest mean axial displacements on the top and bottom transducer surfaces. Thus, the TOM has shown to be a general and powerful approach for designing FGPUTs; specifically, to design uni-modal and multimodal transducers.
5 mm
20 mm
V
Design Domain
Fig.1 Design domain for FGPUT design.
0.2
0.3
0.4
PZT-5H
0.5
0.6
0.7
0.8
0.9
PZT-5A
Design variable
35
Layers
30 25 20 15
Piezoelectric Modal Constant
4
x 10
3.5
4
“Movement” Piston-like mode
Initial values Final values
3 2.5 2
“Movement” right mode
1.5 1
“Movement” left mode
0.5
10
0 3.4
5 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3.5
3.6
3.7
3.8
Frequency (Hz)
3.9
4
4.1 5 x 10
Design variable PZT-5H
PZT-5A
(a)
(b)
Fig.2 Design of a uni-modal FGPUT. (a) Final piston-like mode (dashed and solid lines respectively depict non-deformed and deformed structures) and final material distribution; (b) initial and final frequency response.
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W.M. Rubio et al. / Physics Procedia 3 (2010) 891–896 Rubio et al./ Physics Procedia 00 (2010) 000–000
0.05
0.1 0.15
0.2
0.25
0.3 0.35
0.4 0.45
Design variable
PZT-5H
0.5 0.55
Close to PZT-5A
35 30
Layers
25 20 15
Piezoelectric Modal Constant
3
x 10
2.5
4
Initial values Final values
“Movement” Right mode
“Movement” piston-like mode
2 1.5 1 “Movement” left mode
0.5
10
0 3.4
5 0 0
0.1
PZT-5H
0.2
0.3
0.4
0.5
0.6
0.7
3.5
3.6
3.7
3.8
3.9
4
Frequency (Hz)
4.1 5 x 10
Design variable (a)
Close to PZT-5A
(b)
Fig.3 Design of a multi-modal FGPUT. (a) Final piston-like mode (dashed and solid lines respectively depict non-deformed and deformed structures) and final material distribution; (b) initial and final frequency response.
6. Conclusion This paper presents a systematic study of the topology optimized design of FGPUTs. The TOM allows finding the optimal gradation properties to enhance FGPUT performance, aiming to design uni-modal and multi-modal transducers, keeping user-defined eigenmode shapes. Hence, the technique here presented can be successfully applied as a systematic tool for designing these transducers.
Acknowledgements The authors thank the FAPESP (grant 05/01762-5), CNPq, and Petrobras/ANP for the financial support.
References [1] Y. Miyamoto, W.A. Kaysser, B.H. Rabin, A. Kawasaki, R.G. Ford: Functionally graded materials: Design, processing and applications, Kluwer, Dordrecht, 1999. [2] R. Samadhiya, A. Mukherjee. “Functionally graded piezoceramic ultrasonic transducers,” Smart Mater. Struct, vol. 15, n. 6, pp. 1627-1631, 2006. [3] K. Yamada, J-I. Sakamura, K. Nakamura. “Broadband ultrasound transducers using effectively graded piezoelectric materials,” IEEE Ultrasonics Symposium, pp. 1085-1089, 1998. [4] H. Guo, J. Cannata, Q. Zhou, K. Shung. “Design and fabrication of broadband graded ultrasonic transducers with rectangular kerfs,” IEEE Transaction on Ultrasonics, Ferroelectrics, and frequency Control, vol. 52, n. 11, pp. 2096-2102, 2005. [5] A. Almajid, M. Taya, S. Hudnut. “Analysis of out-of-plane displacement and stress field in a piezocomposite plate with functionally graded microstructure,” Int. J. Solids Struct., vol. 38, pp. 3377-3391, 2001. [6] T.S. Kim, Y.Y. Kim. “Mac-based mode-tracking in structural topology optimization,” Computers and Structures, vol. 74, pp. 375-383, 2000. [7] K. Matsui, K. Terada. “Continuum approximation of material distribution for topology optimization,” Int. J. Numer. Meth. Eng., vol. 59, pp. 1925-1944, 2004. [8] N. Guo, P. Cawley, D. Hitchings. “The finite element analysis of the vibration characteristics of piezoelectric discs,” Journal of Sound and Vibration, v.159, n.1, pp. 115-138, 1992. [9] J.K. Kim, G.H. Paulino. “Isoparametric graded finite elements for nonhomogeneous isotropic and orthotropic materials,” ASME Journal of Applied Mechanics, vol. 69, pp. 502-514, 2002. [10] M.P. Bendsøe, O. Sigmund: Topology Optimization: theory, Methods and applications, Springer, Berlin, 2003. [11] J.M. Guest, J.H. Prevost, Belytschko. “Achieving Minimum Length Scale in Topology Optimization Using Nodal Design variables and Projection Functions,” Int. J. Numer. Meth. Eng. vol. 61, pp.238-254, 2004.
Physics Procedia 00 (2009) 000–000 Physics Procedia 3 (2010) 891–896 www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
International Congress on Ultrasonics, Universidad de Santiago de Chile, January 2009
Topology Optimized Design of Functionally Graded Piezoelectric Ultrasonic Transducers Wilfredo Montealegre Rubio*, Flávio Buiochi, Julio Cezar Adamowski, Emílio C. N. Silva Department of Mechatronics and Mechanical Systems Engineering, University of São Paulo, Av. Prof. Mello Moraes, 2231 - Cidade Universitária, São Paulo – SP, CEP: 05508-900, Brazil Elsevier use only: Received date here; revised date here; accepted date here
Abstract This work presents a new approach to systematically design piezoelectric ultrasonic transducers based on Topology Optimization Method (TOM) and Functionally Graded Material (FGM) concepts. The main goal is to find the optimal material distribution of Functionally Graded Piezoelectric Ultrasonic Transducers, to achieve the following requirements: (i) the transducer must be designed to have a multi-modal or uni-modal frequency response, which defines the kind of generated acoustic wave, either short pulse or continuous wave, respectively; (ii) the transducer is required to oscillate in a thickness extensional mode or piston-like mode, aiming at acoustic wave generation applications. Two kinds of piezoelectric materials are mixed for producing the FGM transducer. Material type 1 represents a PZT-5A piezoelectric ceramic and material type 2 represents a PZT-5H piezoelectric ceramic. To illustrate the proposed method, two Functionally Graded Piezoelectric Ultrasonic Transducers are designed. The TOM has shown to be a useful tool for designing Functionally Graded Piezoelectric Ultrasonic Transducers with uni-modal or multi-modal dynamic behavior.
PACS: 07.05.Tp; 43.38.FX Keywords: Piezoelectric transducers; topology optimization method; functionally graded materials; piston-like mode
1. Introduction A new concept has been applied to design piezoelectric transducers: Functionally Graded Material (FGM) concept. In this work, these piezoelectric transducers are named Functionally Graded Piezoelectric Transducers (FGPT). In general, FGMs possess continuous graded properties with gradual change in microstructure [1]. The FGM concept can be applied to piezoelectric ultrasonic transducers to reduce reflection wave inside them [2], and to obtain smaller time waveform (large bandwidth) [3, 4], which is desirable in medical imaging and non-destructive testing applications. Although there exist literature concerning FGPT manufacturing, research that focuses on the systematic design and analysis of FGPTs is rather scarce and deals with analytical techniques, which grade a single material property.
* Corresponding author. Tel.: +55-11-92920133; fax: +55-11-30915461. E-mail address: [email protected].
doi:10.1016/j.phpro.2010.01.114
892
W.M. Rubio et al. / Physics Procedia 3 (2010) 891–896 Rubio et al./ Physics Procedia 00 (2010) 000–000
On the contrary, by using finite element approaches, 2D and/or 3D FGPT can be designed, considering one or more gradation properties. Many of the existing numerical approaches are based on multilayer design, where the material properties vary from layer to layer and are uniform at each layer [5]. Additionally, several works have shown that the gradation function of an FGPUT can influence its performance [5], though the best property gradation function for a specific application has not been demonstrated. This research proposes a new, generic and systematic topology optimization formulation to find the optimum material gradation of FGPTs aiming ultrasonic applications. FGPTs are designed by using the Topology Optimization Method (TOM). The TOM is a generic and systematic optimization technique, which combines optimization algorithms with Finite Element Method to maximize a user-defined objective function. The main goal is to find the optimal material distribution of Functionally Graded Piezoelectric Ultrasonic Transducers (FGPUT) to achieve the following requirements: (i) designing uni-modal transducers; (ii) designing multi-modal transducers. The Modal Assurance Criterion [6] is applied to track the desirable piston-like mode. The applied optimization algorithm is based on Sequential Linear Programming and the concept of the Continuum Approximation of Material Distribution is considered for achieving continuous material gradation [7].
2. FGPUT Modeling The constitutive relations for piezoelectric media may be derived in terms of their associated thermodynamic potentials. Assuming the strain tensor S and the electrical field E as independent variables, the constitutive piezoelectric equations are written (with Einstein’s convention) [8]:
Tij
E x, y S kl ekij x, y Ek Cijkl
Di
eikl x, y S kl İikS x, y Ek
for i,j,k,l = 1,2,3
(1)
where T and D are respectively the stress tensor and the electric displacement vector. The elasticity tensor CE (elastic stiffness at constant electric field), the piezoelectric coefficient tensor e, and the dielectric constant tensor HS (dielectric susceptibility at constant strain) are assumed to vary along the Cartesian coordinates x and y (for a twodimensional model). Completing the piezoelectric model, the mechanical and electrical equilibrium equations are applied. The mechanical balance corresponds to the calculation of the forces expressed by the Newton’s equation and the electrical balance is based on the Gauss’s theorem. After a mathematical manipulation, the equilibrium equations for a piezoelectric medium (including equilibrium balance) are written as: - ȡ x, y Ȧ 2 u i
w E x, y S kl ekij x, y E k ; C ijkl wr j
w eikl x, y S kl İ ikS x, y E k wr j
(2)
0
where U is the density of the material, which varies along the Cartesian coordinates x and y; r is a unit vector in the Cartesian coordinate system; u is the displacement vector; and Z is the circular frequency.
3. Optimization Problem The topology optimization problem is formulated to find the optimal gradation law of a FGPUT. The transducer must be designed to have a multi-modal or uni-modal frequency response, which defines the kind of generated acoustic wave, either short pulse or continuous wave, respectively. Furthermore, the transducer is required to oscillate in a thickness mode (piston-like mode), aiming at acoustic wave generation applications. For uni-modal transducers, the electromechanical coupling of a desirable mode k must be maximized, and the electromechanical coupling of the adjacent modes (mode number k + a1 with a1 = 1, 2,., A1, and/or k – a2 with a2 =
893
W.M. Rubio et al. / Physics Procedia 3 (2010) 891–896 Rubio et al./ Physics Procedia 00 (2010) 000–000
1, 2,…, A2) must be minimized. Additionally, the resonance frequencies of the modes k1 = k + a1 with a1 = 1, 2,…, A1 must be maximized, and the resonance frequencies of the modes k2 = k – a2 with a2 = 1, 2,…, A2 must be minimized. For multi-modal transducers, the electromechanical couplings of a mode set must be maximized, and their resonance frequencies must be approximated. For uni-modal (F1) and multi-modal (F2) transducers, the objective functions are given as:
ª 1 § A1 wk Ark « ¨ w1k1 Ark 1 «¬ Į1 ¨© k1 1
¦
F1
ª 1 § A1 « ¨ w3 k Ȝrk «¬ Į3 ¨© k1 1 1 1
¦
·º n3 ¸ » ¸» ¹¼
1
n3
n1
·º ¸» ¸» ¹¼
ª1 « «¬ Į4
1
n1
ª1 « «¬ Į2
§ A2 ¨ w A ¨ k 1 2 k2 rk2 ©2
¦
§ A2 ¨ w Ȝ ¨ k 1 4 k2 rk2 ©2
¦
·º n4 ¸ » ¸» ¹¼
1
n2
·º ¸» ¸» ¹¼
1
n2
... (3)
n4
with: A1
A2
¦
Į1
¦
w1k1 ; Į2
k1 1
F2
ª1 « «¬ Į 1
A1
¦
w2 k2 ; Į3
k2 1
§ ¨ ¨ ©
m
¦ w A k
n1
rk
k 1
A2
w3 k1 ;
¦w
Į4
4 k2
·º ¸» ¸ ¹ »¼
1
n1
ª1 « «¬ Į 2
m
º 1 2 Ȝr Ȝ02k » Ȝ02k k »¼
¦ k 1
; nm
1, 3, 5, 7... ; m = 1, 2, 3, 4
k2 1
k1 1
1
n2
(4)
with: m
Į1
¦
m
wk ;
Į2
k 1
1
¦Ȝ k 1
2 0k
;
Ȝrk
Ȧr2k ;
n1
1, 3, 5, 7... ; n 2
r 2 , r 4 , r 6 , r 8 ...
where, for uni-modal transducers (see (3)), the terms Ark , Ark1 and Ark2 represent the electromechanical coupling (measured by the Piezoelectric Modal Constant – PMC [8]) of the desirable mode, the left and the right adjacent modes, respectively. The terms w k , wik1 (i = 1, 3) and wjk2 (j = 2, 4) are the weight coefficients for each mode considered in the objective function F1. Finally, the terms Ȝrk1 and Ȝrk2 represent the eigenvalues of the left and the right modes with relation to the desirable one (mode number k), and the term n is a given power. For multimodal transducers (see (4)), the constant m is the number of modes considered, the terms Ȝ rk and Ȝ0 k are the current and desirable (or user-defined mode) eigenvalues for mode k (k = 1, 2,…, m), respectively; and Ȧ rk are the resonance frequencies for mode k (k = 1, 2,…, m). For uni-modal and multi-modal transducers, the proposed optimization problem is expressed as: maximize ȡ
TOM
x, y
subjected to :
Fi
i 1 or 2
³ȡ
TOM
x, y dȍ - ȍs d 0
;
(5)
ȍ
0 d ȡTOM x, y d 1 Equilibriu m and Constituti ve Equation
where U720(x, y) is the design variable (pseudo-density) at Cartesian coordinates. The term :s describes the amount of piezoelectric material at two-dimensional domain :. The second requirement (mode shape tracking) is achieved by using the Modal Assurance Criterion (MAC) [6], which is widely used to compare experimental and theoretical modal analyses. The optimization problem is formulated as finding the material gradation of an FGPUT,
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which maximizes the multi-objective function F subjected to a piezoelectric volume constraint. That constraint is implemented to control the amount of piezoelectric material into the design domain, :. Since FGPUT can be constructed by sintering a layer-structured ceramic green body without using adhesive material, the optimization problem is arranged as a layer-like optimization problem; in other words, the design variables are considered equals at each layer line. This makes the FGPUT manufacturing possible. However, the layer-like optimization implies that the finite element mesh must be homogeneous.
4. Numerical Implementation To treat the gradation in FGPUTs, the material property is continuously interpolated inside each finite element based on property values at each finite element node. This approach is named Graded Finite Element (GFE) formulation [9]. In this research, the piezoelectric GFE employs the same shape functions Nn to interpolate the unknown nodal displacements and electrical potentials, spatial coordinates, and the material constants. Thus, the elastic CE, piezoelectric e, and dielectric HS material properties in (2) are respectively given by: E x, y Cijkl
nd
¦ N x, y C
E ijkl n
n
,
n 1
eikl x, y
nd
¦ N x, y e n
ikl n
, and
(6)
n 1
İikS x, y
nd
¦ N x, y İ
S ik n
n
for i, j, k, l 1, 2, 3
n 1
where nd is the number of nodes per finite element. On the other hand, the Continuum Approximation of Material Distribution (CAMD) concept [7] is used to continuously represent the material distribution. The CAMD considers that the design variables inside the finite element are interpolated by using, for instance, the FE shape functions Nn. e at each graded finite element e can be expressed as: Thus, the pseudo-density ȡTOM e ȡTOM x, y
nd
¦ȡ
n TOM i
N i x, y
(7)
i 1
n
where ȡTOMi and Ni are respectively the nodal design variable and shape function for node i (i = 1,…,nd), and nd is the number of nodes at each GFE. This formulation allows a continuous distribution of material along the design domain instead of the traditional piecewise material distribution applied by previous formulation of topology optimization [10]. Finally, it is implemented a new scheme to achieve an explicit gradient control by introducing a layer of nodal variables on top of the existing nodal variables [11]. The variables in the new layer are used as design variables, which are updated by the iterative optimization process. The projection technique employs a function to relate the n
p
nodal design variable ȡTOMi to the nodal material pseudo-density ȡTOMi . That projection function is defined as:
¦ ȡ W r ¦ W r jSi
P TOMi
ȡ
n TOM j
jSi
ij
with: rij
x j xi
(8)
ij
n
p where ȡTOMi is the nodal design variable assumed to be located at node i; ȡTOM j is the material pseudo-density
located at node j; and Si is the set of nodes j in the subdomain under influence of node i. In other words, the subdomain Si corresponds to a circle with its center located at the node i and user-defined radius equal to rmin. The
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terms xi and xj represent the Cartesian coordinate of nodes i and j, respectively. Last, W represents a weight function [11].
5. Results To illustrate the proposed method, a uni-modal and a multi-modal FGPUT are designed considering plane strain assumption. The design domain used is shown in Fig.1. The design domain is specified as a 20 mm by 5 mm rectangle with two fixed supports at the end of the left and right-hand sides. The idea is to simultaneously distribute two types of materials into the design domain. The material type 1 is represented by a PZT-5A piezoelectric ceramic and the material type 2 is a PZT-5H. Initially, the design domain contains only PZT-5A material and a material gradation along thickness direction is assumed. A mesh of 50 x 30 finite elements is adopted. Fig.2 and Fig.3 show the results when a uni-modal and a multi-modal FGPUTs are designed by using TOM. For the uni-modal transducer, the electromechanical coupling (measured by the PMC [8]) value of the piston-like mode is increased by 59% while the PMC values of adjacent modes are decreased approximately by 75%. For the multimodal transducer, the PMC value of the piston-like mode is increased by 15%, while the PMC values of the left and right adjacent modes are increased by 15% and 181%, respectively. In both designs, the piston-like mode is kept, producing the highest mean axial displacements on the top and bottom transducer surfaces. Thus, the TOM has shown to be a general and powerful approach for designing FGPUTs; specifically, to design uni-modal and multimodal transducers.
5 mm
20 mm
V
Design Domain
Fig.1 Design domain for FGPUT design.
0.2
0.3
0.4
PZT-5H
0.5
0.6
0.7
0.8
0.9
PZT-5A
Design variable
35
Layers
30 25 20 15
Piezoelectric Modal Constant
4
x 10
3.5
4
“Movement” Piston-like mode
Initial values Final values
3 2.5 2
“Movement” right mode
1.5 1
“Movement” left mode
0.5
10
0 3.4
5 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3.5
3.6
3.7
3.8
Frequency (Hz)
3.9
4
4.1 5 x 10
Design variable PZT-5H
PZT-5A
(a)
(b)
Fig.2 Design of a uni-modal FGPUT. (a) Final piston-like mode (dashed and solid lines respectively depict non-deformed and deformed structures) and final material distribution; (b) initial and final frequency response.
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W.M. Rubio et al. / Physics Procedia 3 (2010) 891–896 Rubio et al./ Physics Procedia 00 (2010) 000–000
0.05
0.1 0.15
0.2
0.25
0.3 0.35
0.4 0.45
Design variable
PZT-5H
0.5 0.55
Close to PZT-5A
35 30
Layers
25 20 15
Piezoelectric Modal Constant
3
x 10
2.5
4
Initial values Final values
“Movement” Right mode
“Movement” piston-like mode
2 1.5 1 “Movement” left mode
0.5
10
0 3.4
5 0 0
0.1
PZT-5H
0.2
0.3
0.4
0.5
0.6
0.7
3.5
3.6
3.7
3.8
3.9
4
Frequency (Hz)
4.1 5 x 10
Design variable (a)
Close to PZT-5A
(b)
Fig.3 Design of a multi-modal FGPUT. (a) Final piston-like mode (dashed and solid lines respectively depict non-deformed and deformed structures) and final material distribution; (b) initial and final frequency response.
6. Conclusion This paper presents a systematic study of the topology optimized design of FGPUTs. The TOM allows finding the optimal gradation properties to enhance FGPUT performance, aiming to design uni-modal and multi-modal transducers, keeping user-defined eigenmode shapes. Hence, the technique here presented can be successfully applied as a systematic tool for designing these transducers.
Acknowledgements The authors thank the FAPESP (grant 05/01762-5), CNPq, and Petrobras/ANP for the financial support.
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