Transcript
Composite Structures 54 (2001) 161±167
www.elsevier.com/locate/compstruct
Re®ned models for the optimal design of adaptive structures using simulated annealing Victor M. Franco Correia a, Crist ov~ ao M. Mota Soares b,*, Carlos A. Mota Soares b b
a ENIDH, Escola N autica Infante D. Henrique, Av. Eng. Bonneville Franco, 2780-569, Pacßo de Arcos, Portugal IDMEC, Instituto de Engenharia Mec^ anics, Instituto Superior T ecnico, Av. Rovisco Pais, 1049-001, Lisboa, Portugal
Abstract This paper deals with re®ned ®nite element models based on higher-order displacement ®elds applied to the mechanical and electrical behavior of laminated composite plate structures with embedded and/or surface bonded piezoelectric actuators and sensors. Simulated annealing, a stochastic global optimization technique is implemented to ®nd the optimal location of piezoelectric actuators in order to maximize its eciency. The same technique is also used to solve optimization problems of piezolaminated plate structures where the discrete design variables are the ply orientation angles of orthotropic layers. The implemented scheme helps to recover from the premature convergence to a local optimum, without the need of reinitiating the optimal design process, as it is the case of the gradient-based methods with continuous design variables. To show the performance of the proposed optimization methods, two illustrative and simple examples are presented and discussed. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Higher order plate models; Piezoelectric materials; Actuators; Structural optimization; Simulated annealing
1. Introduction Piezoelectric materials are able to produce an electrical response when mechanically stressed (sensors) and inversely high precision motion can be obtained with the application of an electrical ®eld (actuators). Developments in smart composite structures incorporating integrated piezoelectric sensors and actuators oer potential bene®ts in a wide range of engineering applications such as structural health monitoring, vibration and noise suppression, shape control and precision positioning. Accurate and reliable numerical tools able to model the structural behavior of this type of materials is very important. Crawley [1] presented an overview of the technology leading to the development of adaptive structures incorporating actuators and sensors. Many of the developed models for laminated composite plates are based in classical laminate and ®rstorder shear deformation plate theories [2]. It has been demonstrated that these theories can lead to substantial errors in the prediction of stresses of highly anisotropic and/or moderately thick composite plates [3±5]. The same behavior has been observed with embedded or *
Corresponding author. Tel.: +351-21-841-7455; fax: +351-21-8417915. E-mail address:
[email protected] (C.M. Mota Soares).
surface bonded piezoelectric layers [6,7]. A good compromise between accuracy and computational eciency can be achieved by using single layer models based on high-order displacement ®elds [8,9] involving higher order expansions of the displacement ®eld in powers of the thickness coordinate. These models can accurately account for the eects of transverse shear deformation yielding quadratic variation of out-of-plane strains and therefore do not require the use of arti®cial shear correction factors and are suitable for the analysis of highly anisotropic plates ranging from high to low length-tothickness ratios. Several researchers [4,5,10,11] have carried out critical reviews of higher-order models applied to the analysis of laminated composite structures. Recently, Tauchert et al. [12] presented a review of theoretical developments in thermopiezoelasticity with relevance to smart composite beams, plates and shells. A comprehensive survey on the advances and trends on ®nite element formulations and applications to smart structural elements has been recently published by Benjeddou [13], where 113 papers are brie¯y discussed. Optimization and sensitivity analysis techniques have been applied to improve the eciency of piezoelectric actuators and sensors to control the behavior of adaptive structures. Batra and Liang [14] used a threedimensional linear theory of elasticity to ®nd the optimal location of an actuator on a simply supported
0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 0 8 5 - X
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rectangular laminated plate with embedded PZT layers that leads to the maximum out-of-plane displacement for a given distribution of the applied voltage. The optimal design is obtained by ®xing the applied voltage and the size of the actuator and moving it around in order to ®nd the maximum out-of-plane displacement. Liang et al. [15] proposed a model for the optimization of the induced-strain actuator location and con®guration for active vibration control based on the actuator power factor that guarantees the highest energy eciency for single frequency and broad-band applications. Leeks and Weisshaar [16] studied a self-strained actuator mounted on one side of a panel to ®nd the features of the actuator that produces the largest de¯ection of simply supported rectangular panels using a Rayleigh±Ritz model associated to classical plate theory. Seeley and Chattopadhyay [17] presented a multiobjective optimization procedure to address the combined problems of structures/control synthesis and actuator locations for the design of intelligent structures where multiple design objectives such as vibration reduction, dissipated energy, power and a vibration performance index are combined in a multiobjective optimization formulation using the simulated annealing method. Abdullah [18] presented a method for the optimal design of the placement and gains of actuators and sensors at discrete locations in output feedback control systems to reduce structural vibrations. Kang et al. [19] investigated the optimization of the sensor/actuator placement for structural vibration control of laminated composite plates for various ®ber orientations in the plates, where damping and stiness of the adhesive layer and the piezoceramics are taken into account in the ®nite element formulation. Varadarajan et al. [20] studied the shape control of laminated composite plates with integrated piezoelectric actuators using ®nite element models based in the ®rstorder shear deformation theory where the eectiveness of piezoelectric actuators and position sensors were investigated for shape control under the in¯uence of quasi-statically varying unknowns. Padula and Kincaid [21] presented a survey of optimization strategies for sensor and actuator placement in adaptive structures. Han and Lee [22] used genetic algorithms to ®nd the ecient locations of piezoelectric sensors and actuators in composite plates using controllability, observability and spillover prevention criteria. Sadri et al. [23] used the Rayleigh±Ritz method to analyze the behavior of a thin plate excited by patches of piezoelectric actuators and used two controllability criteria for the optimal placement of piezoelectric actuators using genetic algorithms. Recently, Botello et al. [24] studied the performance of genetic algorithms and simulated annealing, applied to the optimization of pin-jointed steel bar structures showing the possibility to join these two optimization schemes into a single parametric family of
algorithms, and that optimal performance could be obtained in a parallel machine by a hybrid scheme. In this paper, the simulated annealing [25±27], a stochastic global optimization method, is used to ®nd the optimal location of piezoelectric actuator patches, in a discrete set of possible locations related to the ®nite element discretization of a given problem, in order to maximize the actuator performance, subject to behavior constraints. Also, the simulated annealing method is used as an alternative to a gradient-based procedure [5,28] where the objective is to maximize the natural frequency of a composite plate structure with integrated piezoelectric layers and the design variables are the orientation angles of the reinforcement ®bers in the composite substrate layers. In this last case the computational eciency and the ability to avoid premature convergence to local optimum of the simulated annealing method is compared to the gradient-based optimization technique. 2. Re®ned ®nite element models The displacement components u, v and w at any point in the laminate space in the x, y and z directions, respectively, are expanded in a Taylor's series powers of the thickness coordinate z. The following higher-order displacement ®eld can be written as [8,9] u
x; y; z; t u0
x; y; t zhx
x; y; t z2 u0
x; y; t z3 ux
x; y; t; v
x; y; z; t v0
x; y; t zhy
x; y; t z2 v0
x; y; t z
3
1
uy
x; y; t;
w
x; y; z; t w0
x; y; t zuz
x; y; t z2 w0
x; y; t; where u0 , v0 , w0 are the displacements of a generic point on the reference surface, hx ,hy are the rotations of normal to the reference surface about the y and x axes, respectively, and uz ; u0 ; v0 ; w0 ; ux ; uy are the higher-order terms in the Taylor's series expansions, de®ned at the reference surface and t is the time. The linear piezoelectric constitutive equations coupling the elastic ®eld and the electric ®eld, can be written as [2,29] r Qe
eE;
2
T
D e e pE;
3 T
where r frxx ryy rzz rxy ryz rxz g is the elastic stress vector T and e fexx eyy ezz cxy cyz cxz g is the elastic strain vector. Q is the elastic constitutive matrix in the laminate
x; y; z coordinate system, e is the piezoelectric stress coecients matrix in the same coordinate system, E the electric ®eld vector, D the electric displacement vector and p the dielectric matrix. The electric ®eld vector E is the negative gradient of the electric potential, i.e.,
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E
r/:
4
The stresses for an arbitrary ply k, written in the laminate
x; y; z coordinate system are evaluated by rk Qk ek
ek Ek
5
or alternatively, rk Qk ek
dk Ek ;
6
where dk is the piezoelectric strain coecient matrix in the
x; y; z coordinate system, for the kth layer, given by ek Qk dk . The terms of constitutive matrix Qk of the kth layer referred to the laminate axes
x; y; z are also obtained from the Qk matrix referred to the material coordinate system (1,2,3) with the usual coordinates transformation given explicitly in Reddy [2]. Within the context of the plate theory, it is assumed that the electric ®eld is applied through the thickness direction and thus the non-zero element of Ek vector for kth layer, in the z direction, is given by Ekz
/k =tk ;
7
where /k is the electric voltage applied across the kth layer and tk is the thickness of the kth layer. The ®nite element model based on the displacement ®eld represented by Eq. (1) is referred to as Q9-HSDT 11P. This model has 11 degrees of freedom at each node for the elastic behavior and one additional electric potential degree of freedom for each piezoelectric layer for the piezoelectric behavior. By deleting the higher-order terms in the w component of the displacement ®eld we can obtain the Q9-HSDT 9P higher-order model with 9 degrees of freedom per node plus the electric potential degrees of freedom. Also the ®rst-order model Q9FDSDT 5P with 5 degrees of freedom per node, plus the electric potential degrees of freedom, can be obtained by deleting all high-order terms and introducing the shear correction factors on the transverse shear terms of constitutive matrix. Details regarding the development of the above ®nite numerical models for the mechanical behavior are fully described in Mota Soares et al. [5]. In the present models, there is one electric potential degree of freedom for each piezoelectric layer to represent the piezoelectric behavior [30] and the vector of electrical degrees of freedom is n oT /e . . . /ej . . . ; j 1; . . . ; NPL;
8 where NPL is the number of piezoelectric layers in a given element. The vector of degrees of freedom for the element, qe is T
9 qe ue1 . . . ue9 /e ; where uei is the mechanical displacements vector for node i. Further details regarding the development of the ®nite element models for the electric behavior can be found in Franco et al. [29].
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3. Equilibrium equations The equilibrium equations of the laminate composite plate with embedded piezoelectric layers can be written as [31] u Kuu Ku/ u Muu 0 K/u K// 0 0 / / Fu
t ;
10 F/
t where Muu is the mass matrix Kuu the elastic stiness matrix, K// is the dielectric `stiness' matrix and Ku/ KT/u are the coupling matrices between elastic mechanical T and electrical eects. fu; /g is the vector of the unknown electromechanical response of the structure, that is, the system generalized displacements and voltages at sensors. fFu
t; F/
tgT is the force vector of the structure including the mechanical loads and the applied electrical charges and t is the time. For static type situations we have Kuu Ku/ Fu u :
11 K/u K// F/ / In practice, voltage may also be speci®ed as input to the actuators. In this case the following equilibrium equation, for static type situations, can be written as Kuu u Fu
Fact ;
12
where Fact Ku/ / is an additional force vector due to the voltage / applied to the actuators. The system matrices Kuu , Ku/ , K// and M// are obtained by assembling the corresponding element matrices Keuu , Keu/ , Ke// and Meuu of the ®nite elements in the domain. For free harmonic vibrations and considering the nth mode, Eq. (10) can be written as Kuu x2n Muu u Ku/ / 0;
13 K/u u K// / 0; where xn is the natural frequency of mode n. By performing the condensation of the electrical potential degrees of freedom we can write
14 K x2n Muu u 0; where K Kuu
Ku/ K//1 K/u :
15
Once the boundary conditions are introduced, Eqs. (11), (12) and (14) can be easily solved.
4. Optimal design using simulated annealing A general structural optimization problem can be stated as
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min
fX
bg
subject to :
bli 6 bi 6 bui ;
i 1; . . . ; ndv;
Wj
q; b 6 0;
j 1; . . . ; m;
16
where X
b is the objective function, b is the vector of design variables bi , Wj
q; b are the m inequality behavioral constraint equations, bli and bui are respectively, the lower and upper limits of the design variables and ndv is the total number of design variables. For very small combinatory problems, complete enumeration of all possible design variable combinations could be an option and ®nding the global optimum can be guaranteed. For larger problems, a variety of methods including simulated annealing are available [21,25±27]. This method cannot guarantee convergence to the global optimum, but its main advantage over gradient-based methods [32] is the ability to overcome the premature convergence towards a local optimum. The main disadvantage is related with the computational eciency, because of the high number of objective function evaluations usually required to reach the optimal solution. This is especially important when the objective function evaluation is computationally expensive. The algorithm was ®rst proposed by Kirkpatrick et al. [25,26]. The implemented simulated annealing procedure employs a random search that generates feasible sets of design variables, accepting not only changes in the design variables that decrease the objective function but also changes that increase it. The latter changes are accepted with a certain probability. The basic functioning of the simulated annealing algorithm can de easily described as follows [27,33,34]: (a) Generate a random perturbation on the design variables and obtain the change in the objective function dX
b. (b) If dX
b 6 0, the new design is better and the new design is accepted. (c) If dX
b > 0, accept the new design with a probability given by p exp
dX=T :
17
In Eq. (17), T is a control parameter, which is called the system temperature, based on the analogy to the physical process of annealing a metal [33,34]. The initial temperature T0 and the rate at which it is lowered, usually referred to as the cooling schedule, have great in¯uence on the performance of the algorithm. This temperature must be set high enough so that, at least initially, all states proposed are accepted. Some authors use the acceptance ratio l of accepted moves to proposed moves, obtaining the initial temperature from the relation dX
b l exp ;
18 T0
where l is set to a chosen value. The temperature decrement can be given by Tk1 a Tk ;
19
where Tk and Tk1 represent the system temperatures at k and k 1 successive iterations and a is the cooling parameter usually taken in the range 0.8±0.95. Note that with decreasing temperatures the probability given by Eq. (17) will decrease, hence as the simulated annealing iterations proceed, it becomes increasingly unlikely that a worse design can be accepted. The search is halted when no improvement in the objective function is found combined with the acceptance ratio falling below a speci®ed value. For the maximization of the objective function, i.e., minf X
bg is used in Eq. (16).
5. Numerical applications 5.1. Optimal location of piezoelectric actuator patches for maximum actuator performance This example illustrates the application of the simulated annealing method to ®nd the optimal location of the piezoelectric actuators patches in order to maximize the actuator performance for maximum plate de¯ection. The 4-ply S-Glass/Epoxy simply supported rectangular plate illustrated in Fig. 1 is considered. The objective is to ®nd the optimal location of 8 pairs of square piezoelectric actuators having the dimensions 20 mm 20 mm with 0.5 mm thickness, bonded to the lower and upper surfaces of the substrate plate, in order to maximize the transversal de¯ection in the midpoint of the plate. The possible locations of piezoelectric actuator patches will be restricted to the discrete positions corresponding to the ®nite element mesh. The size of each element in the mesh has to be equal to the size of the piezoelectric actuator patches. A ®nite element mesh having 4 14 elements is considered. The plate is simply supported along the smaller edges and free along the longer edges. The Homan ®rst ply failure criterion is introduced as a constraint equation [5].
Fig. 1. Optimal location of the piezoelectric actuator pairs in a rectangular simply supported plate.
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The material properties of the S-Glass/Epoxy layers are: E1 55 GPa, E2 E3 16 GPa, G12 G23 G13 7:6 GPa, m12 m23 m13 0:28. The strength properties of the S-Glass/Epoxy for the Homan failure criterion, are: XT 760 MPa, YT ZT 28 MPa, XC 690 MPa, YC ZC 170 MPa, R S T 70 MPa, where XT , YT , ZT are the lamina normal strengths in tension along the 1,2,3 directions, R, S, T are the shear strengths in the 23, 13, and 12 planes, and XC , YC , ZC are the lamina normal strengths in compression along the 1,2,3 directions. The material and electric properties of the PC5H piezoelectric actuators patches are: E1 E2 60:24 GPa, E3 49:02 GPa, G12 G13 G23 23:0 GPa, m12 m13 m23 0:31, d31 d32 306 10 12 m=V, d33 800 10 12 m=V, p33 5:04 10 8 F=m. In the initial design, the 8 pairs of piezoelectric actuators are positioned as shown in Fig. 2(a). For this initial situation the mid-point transversal de¯ection is 0.09511 mm (value obtained with Q9-HSDT 9P model). The electric potentials applied to the 8 pairs of actuators are +200 V to the actuators bonded to the upper surface of the plate and )200 V to the actuators bonded to the lower surface. The optimal location for maximum mid-point transversal displacement is shown in Fig. 2(b). The ®nal value of the mid-point transversal de¯ection is 0.18995 mm. The cooling schedule parameters used in the simulated
Fig. 2. (a) Location of the 8 piezoelectric actuator pairs in the initial design. (b) Location of the piezoelectric actuator pairs in the optimal design, corresponding to the maximum transversal de¯ection in the mid-point of the plate.
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annealing process were for the initial temperature, T0 200 and for the rate of temperature reduction, a 0:95. 5.2. Optimal design of skew plate for maximum natural frequency This example is used to compare a gradient-based optimization scheme and the simulated annealing technique in an unconstrained optimization problem where the design objective is the maximization of the fundamental natural frequency and the design variables are the ®ber angles in each ply. The example consists of a skewed plate clamped in the boundary y 0, as represented in Fig. 3. The plate is made of 4-ply substrate Graphite/Epoxy layers with 1 mm thickness each and two surface bonded piezoelectric PZT 5H layers with 0.5 mm thickness each. A 4 4 ®nite element mesh is used. The optimization problem consists of ®nding the ply angles h2 =h3 =h4 =h5 in the substrate layers that maximize the fundamental natural frequency. The piezoelectric layers are not included as design variables because the piezoceramic PZT 5H is isotropic in the (1,2) plane. In what concerns to the electric boundary conditions, an open circuit condition is assumed, where the electric potentials in the piezoelectric layers remains free. The material properties of the Graphite/Epoxy layers are: E1 145 GPa, E2 E3 10 GPa, G12 G23 G13 4:8 GPa, m12 m23 m13 0:25, q 1580 Kg=m3 . The material and electrical properties of the piezoceramic PZT 5H layers, in open circuit conditions are: E1 E2 69 GPa, E3 106 GPa, G12 G23 G13 26:3 GPa, m12 m23 m13 0:31, q 7700 Kg=m3 , d31 d32 171 10 12 m=V, d33 374 10 12 m=V, p33 1:505 10 8 F=m. The initial and optimal design data obtained with the simulated annealing optimization method are summarized in Table 1. For the simulated annealing optimization problem, the design variables are chosen from a discrete set of possible ply angles de®ned as follows Sdv f0; 10°; 20°; 30°; 40°; 50°; 60°; 70°; 80°; 90°g. For the gradient-based optimization the design variables can vary continuously with any
Fig. 3. Clamped skewed plate with piezoelectric surface bonded layers (dimensions in m).
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Table 1 Optimal ply angles for maximum natural frequency of ®rst vibration mode, obtained by using the simulated annealing method Natural frequencies (rad/s) Q9-HSDT 11P
Q9-HSDT 9P
Q9-FSDT 5P
Initial design
1st mode 2nd mode Angles
206.894 449.867 [0°/0°]s
209.849 450.524 [0°/0°]s
209.880 450.769 [0°/0°]s
Simulated Annealing Optimal design
1st mode 2nd mode Angles
312.960 497.724 [70°/70°/70°/70°]
314.493 498.180 [70°/70°/70°/70°]
314.614 498.621 [70°/70°/70°/70°]
Table 2 Optimal ply angles for maximum natural frequency of ®rst vibration mode, obtained by using a gradient-based optimization method with four trials having dierent initial designs Initial design
Trial
Final design
1st natural frequency (rad/s)
Angles
1st natural frequency (rad/s)
Angles
1
Q9-HSDT 11P Q9-HSDT 9P Q9-FSDT 5P
206.894 209.849 209.880
[0°/0°]s [0°/0°]s [0°/0°]s
303.062 304.652 304.706
[71.7°/10.6°/10.6°/71.7°] [71.6°/10.8°/10.8°/71.6°] [71.0°/10.0°/10°/71.0°]
2
Q9-HSDT 11P Q9-HSDT 9P Q9-FSDT 5P
264.065 266.357 266.446
[45°/45°]s [45°/45°]s [45°/45°]s
310.538 312.079 312.184
[72.1°/48.9°/48.9°/72.1°] [72.0°/48.9°/48.9°/72.0°] [72.0°/48.9°/48.9°/72.0°]
3
Q9-HSDT 11P Q9-HSDT 9P Q9-FSDT 5P
310.554 312.123 312.225
[70°/50°]s [70°/50°]s [70°/50°]s
311.793 313.366 312.451
[74.1°/69.7°/69.7°/74.1°] [74.0°/69.8°/69.8°/74.0°] [71.7°/50.1°/50.1°/71.7°]
4
Q9-HSDT 11P Q9-HSDT 9P Q9-FSDT 5P
312.960 314.493 314.614
[70°/70°]s [70°/70°]s [70°/70°]s
312.962 314.494 314.616
[70.2°/70.0°/70.0°/70.2°] [70.1°/70.0°/70.0°/70.1°] [70.1°/70.5°/70.5°/70.1°]
restrictions from )90° to +90°. With the initial design chosen as [0°/0°]s, the gradient-based optimization converged to a local optimum as shown in Table 2. In order to overcome the problem of the premature convergence to local optimum, four trials with dierent initial designs have been considered and the ®nal designs achieved are presented in Table 2. The fourth trial corresponds to the global optimum and agrees very well with the optimum design obtained with the simulated annealing method. In what concerns to computational eciency, it is important to note that the simulated annealing method took about 20 times the computer time spent in each trial of the gradient-based method. The cooling schedule parameters used in the simulated annealing process were: T0 200, a 0:9. From the results shown, a signi®cant increase in the natural frequencies (about 50% increase in the fundamental mode) is achieved, and a also good agreement between all ®nite element models is found for the lamination sequence in the ®nal design. 6. Conclusions In this work, ®nite element models based on highorder and ®rst-order displacement ®elds, applied to the
analysis of laminated plates with embedded and/or surface bonded piezoelectric laminae have been presented. It has been shown elsewhere [5,6,29] that ®rstorder models can lead to signi®cant errors in the stress prediction of highly anisotropic and/or moderately thick composite laminated plates and also when piezoelectric laminae are involved. The simple illustrative examples presented have shown that the simulated annealing method is a useful tool for discrete variable structural optimization problems and can be used successfully for the optimal location of piezoelectric actuator patches in adaptive structures. For problems of optimal design of composite adaptive structures where the design variables are the ®ber angles in orthotropic layers, the traditional gradientbased optimization methods usually presents the problem of premature convergence to a local optimum. The use of the simulated annealing method in such cases has shown its ability to overcome the premature convergence towards a local optimum. The main disadvantage of this method is related with the computational eciency, because of the high number of objective function evaluations required to reach the optimal solution. This is especially important when the objective function
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evaluation is computationally expensive. Using parallel computing as shown in [27] can attenuate this disadvantage. Acknowledgements ~o para This work was supported by F.C.T. ± Fundacßa a Ci^encia e Tecnologia, Ministerio da Ci^encia e Tecnologia ± Project PRAXIS/P/EME/12028/1998. The ®rst author wishes also to thank Fundacß~ ao Calouste Gulbenkian, for their ®nancial support. References [1] Crawley EF. Intelligent structures for aerospace: A technology overview and assessment. AIAA J 1994;32(8):1689±99. [2] Reddy JN. Mechanics of laminated composite plates ± theory and applications. Boca Raton: CRC Press; 1997. [3] Pagano NJ, Hat®eld SJ. Elastic behavior of multilayered bidirectional composites. AIAA J 1972;10:931±3. [4] Mallikarjuna, Kant T. A critical review and some results of recently developed re®ned theories of ®ber-reinforced laminated composites and sandwiches. Compos Struct 1993;23:293±312. [5] Mota Soares CM, Mota Soares CA, Franco Correia VM. Optimization of multilaminated structures using higher-order deformation models. Comput Meth Appl Mech Eng 1997; 149:133±52. [6] Chattopadhyay A, Seeley CE. A higher order theory for modeling composite laminates with induced strain actuators. Composites Part B 1997;28B:243±52. [7] Robbins DH, Reddy JN. Analysis of piezoelectrically actuated beams using a layer-wise displacement theory. Comput Struct 1991;41(2):265±79. [8] Lo KH, Christensen RM, Wu EM. A high-order theory of plate deformation. Part 1: Homogeneous plates. J Appl Mech 1977; 663±8. [9] Lo KH, Christensen RM, Wu EM. A high-order theory of plate deformation. Part 2: Laminated plates, J Appl Mech 1977;669±76. [10] Reddy JN. An evaluation of equivalent single-layer and layerwise theories of composite laminates. Compos Struct 1993;25:31±5. [11] Reddy JN, Robbins Jr DH. Theories and computational models for composite laminates. Appl Mech Rev 1994;47(6):147±69. [12] Tauchert TR, Ashida F, Noda N, Adali S, Verijenko V. Developments in thermopiezoelasticity with relevance to smart composite structures. Compos Struct 2000;48:31±8. [13] Benjeddou A. Advances in piezoelectric ®nite element modeling of adaptive structural elements: a survey. Comput Struct 2000; 76:347±63. [14] Batra RC, Liang XQ. The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators. Comput Struct 1997;63(2):203±16. [15] Liang C, Sun FP, Rogers CA. Determination of design of optimal actuator location and con®guration based on actuator power factor. J Intell Mater Syst Struct 1995;6:456±64.
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[16] Leeks TJ, Weisshaar TA. Optimizing induced strain actuators for maximum panel de¯ection. In: Proceedings of AIAA/ASME Adaptive Structures Forum, AIAA-94-1774-CP; 1994. p. 378±87. [17] Seely CE, Chattopadhyay A. Development of intelligent structures using multiobjective optimization and simulated annealing. In: Proceedings of AIAA/ASME Adaptive Structures Forum, AIAA-94-1777-CP; 1994, p. 411±21. [18] Abdullah MM. Optimal location and gains of feedback controllers at discrete locations. AIAA J 1998;36(11):2109±16. [19] Kang YK, Park HC, Agrawal B. Optimization of piezoceramic sensor/actuator placement for vibration control of laminated plates. AIAA J 1998;36(9):1763±5. [20] Varadarajan S, Chandrashekhara K, Agarwal S. Adaptive shape control of laminated composite plates using piezoelectric materials. AIAA J 1998;36(9):1694±8. [21] Padula SL, Kincaid RK. Optimization strategies for sensor and actuator placement. NASA/TM-1999-209126, NASA, Langley Research Center, 1999. [22] Han J, Lee I. Optimal placement of piezoelectric sensors and actuators for vibration control of a composite plate using genetic algorithms. Smart Mater Struct 1999;8:257±67. [23] Sadri AM, Wright JR, Wynne RJ. Modelling and optimal placement of piezoelectric actuators in isotropic plates using genetic algorithms. Smart Mater Struct 1999;8:490±8. [24] Botello S, Marroquin JL, O~ nate E, Horebeek JV. Solving structural optimization problems with genetic algorithms and simulated annealing. Int J Numer Meth Eng 1999;45:1069±84. [25] Kirkpatrick S, Gelatt Jr CD, Vecchi MP. Optimization by simulated annealing. Science 1983;220:671±80. [26] Kirkpatrick S. Optimisation by simulated annealing: Quantitative studies. J Stat Phys 1984;34:975±86. [27] Atiqullah MM, Rao SS. Simulated annealing and parallel processing: an implementation for constrained global design optimization. Eng Opt 2000;32:659±85. [28] Mota Soares CM, Mota Soares CA, Franco Correia VM. Optimal design of piezolaminated structures. Compos Struct 1999;47:625± 34. [29] Franco Correia VM, Gomes MA, Suleman A, Mota Soares CM, Mota Soares CA. Modeling and design of adaptive composite structures. Comput Meth Appl Mech Eng 2000;185(2±4):325±46. [30] Suleman A, Venkayya VB. A simple ®nite element formulation for a laminated composite plate with piezoelectric layers. J Intell Mater Syst Struct 1995;6:776±82. [31] Allik H, Hughes TJR. Finite element method for piezoelectric vibration. Int J Numer Meth Eng 1970;2:151±7. [32] Vanderplaats GN. Numerical optimization techniques for engineering design ± with applications. New York: McGraw-Hill; 1994. [33] Press W, Teukolsky S, Vetterling W, Flannery B. Numerical recipes in FORTRAN ± the art of scienti®c computing. 2nd ed. Cambridge: Cambridge University Press; 1992. [34] Hajela P. Stochastic search in discrete structural optimization. Lecture notes on Discrete Structural Optimization, Advanced school coordinated by W. Gutkowski. International Centre for Mechanical Sciences, Udine, Italy; 1996.
