Topology Optimization for Static Shape Control of

Zhan Kang1 State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China e-mail: [email protected]

Xiaoming Wang National Engineering Laboratory for System Integration of High Speed Train, CSR Qingdao Sifang Locomotive & Rolling Stock Co., Ltd., Qingdao 266111, China

Zhen Luo School of Electrical, Mechanical and Mechatronic Systems, Faculty of Engineering and Information Technology, University of Technology, Sydney, NSW 2007, Australia

Topology Optimization for Static Shape Control of Piezoelectric Plates With Penalization on Intermediate Actuation Voltage This paper investigates the simultaneous optimal distribution of structural material and trilevel actuation voltage for static shape control applications. In this optimal design problem, the shape error between the actuated and the desired shapes is chosen as the objective function. The energy and the material volume are taken as constraints in the optimization problem formulation. The discrete-valued optimization problem is relaxed using element-wise continuous design variables representing the relative material density and the actuation voltage level. Artificial interpolation models which relate the mechanical/piezoelectrical properties of the material and the actuation voltage to the design variables are employed. Therein, power-law penalization functions are used to suppress intermediate values of both the material densities and the control voltage. The sensitivity analysis procedure is discussed, and the design variables are optimized by using the method of moving asymptotes (MMA). Finally, numerical examples are presented to demonstrate the applicability and effectiveness of the proposed method. It is shown that the proposed method is able to yield distinct material distribution and to suppress intermediate actuation voltage values as required. [DOI: 10.1115/1.4006527] Keywords: optimal design, topology optimization, piezoelectric, shape control, actuation voltage

1

Introduction

It is desirable in some engineering applications of piezoelectric shape control to simultaneously optimize the piezoelectric material and discrete-level actuation voltage in order to alleviate the complexity of electrical implementation and to achieve the best overall performance. Piezoelectric ceramics have been widely used as actuator elements, since they can convert electric energy into mechanical forces/motion. Smart structures with piezoelectric actuators are capable of changing their geometrical shape or performing vibration control to meet different practical requirements at a high response rate with a high precision. The concept of piezoelectric actuation has been explored in design of structural systems including MEMS, active morphing wings [1–5], and antenna reflectors [6]. For instance, in atomic force microscopy, a piezoelectric bimorph cantilever structure is used to deliver a very small tip deflection so as to obtain a high resolution in the nanometer range. In view of wide engineering applications, static shape control of smart structures incorporating piezoelectric materials as actuators has received much attention. Various design methodologies have also been developed for optimization of piezoelectric devices in shape control applications (see, e.g., Refs. Irschik [7] and Frecker [8]). However, combined optimization of the layout of piezoelectric material and the discrete-valued actuation voltages has been rarely addressed in the literature. With the purpose of improving the shape control performance, many studies have been carried out on design optimization of the applied control voltages or the piezoelectric material in piezoelectric 1 Corresponding author. Contributed by the Design Innovation and Devices of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 1, 2011; final manuscript received March 28, 2012; published online April 24, 2012. Assoc. Editor: Diann Brei.

Journal of Mechanical Design

structures. Liew et al. [9] applied the genetic algorithm (GA) to determine the optimal voltage distribution and displacement control gain values for the shape control of the functionally graded material plates containing piezoelectric patches acting as sensors and actuators. Wang et al. [10] used the GA in the optimal placement of curved piezoelectric beams for plate shape control applications. Tong et al. [11] described an integer programming-based analytical approach for optimizing the actuator locations in the shape control of thin plates. Sun and Tong [12] proposed a design optimization method for piezoelectric actuator patterns in static shape control of laminated plates. In their work, some piezoelectric patches were gradually removed from the structure based on the voltage values obtained from a heuristic optimality criterion. Mukherjee and Joshi [13] suggested a heuristic criterion method based on the residual voltages for the shape design of piezoelectric actuators. In order to achieve the desired shape of the structure, a quadratic function measuring the global displacement error between the desired and the acquired structural configuration is considered as the objective function to be minimized. Luo and Tong [14] used the least square method to determine the optimal electric potentials of piezoelectric stiffeners bonded to a composite plate in order to achieve a desired shape. Moita et al. [15] applied the simulated annealing (SA) algorithm to optimize the positions of piezoelectric patches in the vibration control of composite material structures. As a powerful automated design tool, topology optimization [16–18] has been employed in dealing with a wide range of structural and multidisciplinary design problems, including the shape control design of smart structures. Donoso and Sigmund [19] and Drenckhan et al. [20] studied the optimal layout of bimorph actuators attached to bending beams for reducing their tip deflections. Ko¨gl and Silva [21] investigated topology optimization of piezoelectric plate actuators for generating the displacement at a given output point. An artificial material model with penalization of

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piezoelectric constants and polarization was employed. Wein et al. [22] achieved maximization of the dynamic displacement by optimizing the material layout of piezoelectric layer using Solid Isotropic Material with Penalization (SIMP) method. Wang et al. [23] applied the GA to topology optimization of piezoelectric sensors and actuators for vibration control of composite plates. Recently, Luo et al. [24] presented a multiphase level set method for topology optimization of piezoelectric actuators. A ground structure approach was developed by Bharti and Frecker [25] to optimize the size, location, and number of active elements in a compliant mechanism for achieving maximum output deflection. Nakasone and Silva [26] developed a topology optimization method to determine the piezoelectric material distribution together with the polarization sign for dynamic control of laminated plates. Topology optimization method was also employed in structural design of piezoelectric energy harvesting devices by Zheng et al. [27] and Rupp et al. [28] to achieve the maximum energy converting efficiency. In a recent study by Rupp et al. [29], the optimal spatial pattern of piezoelectric polarization in design of acoustic energy harvesters and actuators is studied by using topology optimization techniques. Although there are many methods available for optimal distribution of piezoelectric materials, only a few studies on simultaneous optimization of material distribution and control voltage have been reported. Gao et al. [30] and Li et al. [31] studied the optimal placement of piezoelectric actuators and their voltage for minimizing the acoustic radiation of vibrating plates. In their study, the positions of the actuators are limited to specified discrete locations. Zhu et al. [32] investigated shape control of plate structures by simultaneous optimal design of structural topology, actuator locations, and control parameters. Quan and Tong [33] presented a multicriteria design method which gradually removes active piezoelectric materials in static shape control of smart plate structures. Both the active piezoelectric actuator configuration and the applied voltage were simultaneously optimized. In the work of Liu et al. [34], both control and geometrical parameters are treated as design variables, and the difference between the actuated and the desired shape of a plate is chosen as the objective function. Kang and Tong investigated integrated layout optimization of piezoelectric material and control intensity [35], as well as distribution optimization of single-channel actuation voltage [36], for static shape control applications by using the topology optimization method. Most of the aforementioned studies allow that the voltage applied to different piezopatches can take continuous values, which may increase complexity of the electrical system implementation in real applications. This paper investigates topology optimization of bimorph-type piezoelectric plates for static shape control applications. For ease of implementation of electric circuits and electrode arrangement, the electrode layers are divided into positive voltage zones and negative voltage zones, and only actuation voltages with a specified absolute value are allowed to be applied to the electrode patches, as schematically shown in Fig. 1. This means that the allowable actuation voltage to be applied on any location must be either 6V or 0 (in void zones). The electrode patches with different sign of electrical potential need to be isolated from each other in practical implementations. All the layers of the laminate plate are assumed to be perfectly bonded together and the stiffness of

Fig. 1 Schematic illustration of a laminated plate with a host layer and two piezoelectric surface layers

051006-2 / Vol. 134, MAY 2012

the adhesive layer as well as the electrode layers is neglected. The polarization of the piezoelectric material is aligned along the layer thickness direction. In the optimization problem formulation, both the actuation voltage and piezoelectric material densities are taken as design variables and optimized simultaneously, so that the shape error between the desired and the actuated shape is minimized. The MMA algorithm is employed to solve the optimization problem on the basis of design sensitivity analysis. For penalizing intermediate design values, power-law relationships are assumed between the design variables and the material mechanical/piezoelectrical properties, as well as the applied control voltage. Numerical examples are given to demonstrate the validity and effectiveness of the proposed method.

2

Finite Element Modeling

2.1 Field Variable Discretization. It is well recognized that use of higher-order elements is more favorable in topology optimization for alleviating numerical instabilities such as the checkerboard patterns and mesh dependency [37]. Hence, in this study, the design domain is discretized with eight-node quadrilateral plate element as shown in Fig. 2. The elemental DOFs ue is defined as ue ¼ fw1 ; hx1 ; hy1 ; w2 ; hx2 ; hy2 ; :::; w8 ; hx8 ; hx8 gT

(1)

where wi is the transverse displacement of the ith elemental node, hxi and hyi are the rotations. The displacement within an element can be expressed in terms of nodal displacements using shape functions as w¼

8 X

Ni wi ;

i¼1

hx ¼

8 X

Ni hxi ;

hy ¼

i¼1

8 X

Ni hyi

(2)

i¼1

where Ni ði ¼ 1; 2; :::; 8Þ are the usual interpolation functions for eight-node elements. Based on Mindlin’s plate assumptions, the displacements u, v, and w at an arbitrary point ðx; y; zÞ are expressed as function of the midplane transverse displacements w and rotations hx and hy as uðx; y; z; tÞ ¼ zhx ðx; y; tÞ vðx; y; z; tÞ ¼ zhy ðx; y; tÞ wðx; y; z; tÞ ¼ wðx; y; tÞ

(3)

Under small deformation assumption, the strain components are given by ex ¼ zhx;x ;

ey ¼ zhy;y ;

czx ¼ w;x þ hx

cxy ¼ zhx;y  zhy;x ;

cyz ¼ w;y  hy ; (4)

Fig. 2 Eight-node laminated plate element for the considered configuration

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Thus, the strain-displacement relations for the eth element can be symbolically expressed by e ¼ Bue

(5)

where B is the strain-displacement matrix. Assuming that the actuation voltage is element-wise constant, and the electrical potential varies linearly along the thickness directions of the piezoelectric layers, the vectors of the electric fields across both piezoelectric layers are ðE1 Þe ¼

  Ve T 0; 0; ; h

ðE2 Þe ¼

  Ve T 0; 0;  h

(6)

where the subscripts 1 and 2 denote the layer number, Ve represents the electric voltage applied to the eth element, and h is the thickness of the piezoelectric layers. 2.2 Constitutive Equations. The linear constitutive equations of piezoelectric material are expressed by the direct and converse piezoelectric equations, respectively. These equations are written in matrix form as [38] (7)

where r and e are the stress tensor and strain tensor, respectively; E and D are the electric field vector and the electric displacement vector, respectively; C, e, and f are the elasticity tensor, the piezoelectric constant tensor, and the dielectric constant tensor, respectively. 2.3 Finite Element Equations. Based on the variational principal and above discretization method, the system equilibrium equations can be obtained as  Kuu u þ Kuu u ¼ F (8) Kuu u  Kuu u ¼ q In the above coupled equations, u is the mechanical displacement vector, u is the electrical potential vector, F is the applied external force vector, and q is the electrical charge vector; Kuu , Kuu ¼ KTuu , and Kuu are the stiffness matrix, the piezoelectric matrix and the dielectric matrix, respectively. In this study, piezoelectric patches are used as voltage-driven actuators and the applied voltage can thus be regarded as input to the piezoelectric actuators. Considering the piezoelectric effect as induced forces, the coupled equations thus reduce to pure mechanical ones. Hence, the global equilibrium equation can now be written in terms of nodal displacements as Kuu u ¼ F þ Fp

(9)

where Fp represents the force contributed by the piezoelectric layers under application of the actuation voltage. The global stiffness matrix Kuu and the piezoelectric force vector Fp can be obtained by assembling elemental contributions. For the eth element, the elemental stiffness matrix Keuu is expressed by ð Xhe

BT Che BdX þ 2

ð Xpe

ðBp ÞT Cpe Bp dX

(10)

and the actuation force produced by both piezoelectric layers is given by ð p Bp ee Ee dX (11) Fe ¼ 2 Xpe

Journal of Mechanical Design

3

Formulation of the Optimization Problem

3.1 Design Variables. In the material distribution conceptbased topology optimization problem, the structural layout is represented by presence/absence of the material in each element. Moreover, in this study, the actuation voltage applied on each element must be chosen from three distinct values fv0 ; v0 ; 0g, where v0 is the prescribed magnitude of the actuation voltage. Therefore, the considered design optimization problem is in nature a discrete-valued one, which is very difficult to be directly solved. A common practice in treating such a problem is to relax it into a continuous-valued optimization problem. To this end, we define element-wise density design variables q ¼ fq1 ; q2 ; :::; qn gT (n is the total number of elements) for material layout of the piezoelectric layers and v ¼ fv1 ; v2 ; :::; vn gT for elemental actuation voltages to be applied. These design variables may take continuous values and their bound limits are qe 2 ½q; 1

r ¼ Ce  eT E D ¼ ee þ fE

Keuu ¼

where ee and Ee are the elemental piezoelectric stress tensor and the elemental electric field vector of the upper piezoelectric layer, respectively. The superscripts h and p denote the quantities related to the host layer and the piezoelectric layers, respectively.

for e 2 f1; 2; :::; ng

ve 2 ½1; 1

for e 2 f1; 2; :::; ng

(12) (13)

where q is a small positive real number. In this study, we set q ¼ 0:001 so as to avoid numerical difficulties associated with zero elemental stiffness matrices. Analogously as in the conventional SIMP approach for the minimum compliance problem, artificial mapping relations are established between these design variables and the mechanical properties of the material, as well as the actuation voltage for each element. In order to suppress the intermediate values of the density variables in the optimal solution, penalizations on both material properties and actuation voltage are employed. The interpolation relation between the mechanical/electrical properties and the elemental density variables is expressed by ðCijkl Þpe ¼ ðqe Þpm ðC0ijkl Þp ðekij Þe ¼

ðqe Þpp e0kij

for e 2 f1; 2; :::; ng

(14)

for e 2 f1; 2; :::; ng

(15)

where ðC0ijkl Þp represents the elastic constants of the piezoelectric material with unit density. The penalty factor pm is used for penalization of elastic constants of the piezoelectric material, and pp is used for penalization of the piezoelectric constants. The mapping relation between the elemental actual applied voltage Ve and the elemental voltage variable ve is assumed to be [36] Ve ðve Þ ¼ ðð1  gÞvpe v þ gve Þv0

e 2 f1; 2; :::; ng

for

(16)

where pv 2 f1; 3; 5; :::g is a penalty factor and g is a small positive number, which is set as g ¼ 0:001 in this study. For the cases of pv ¼ 3; 5; :::, the exponential penalization makes intermediate voltages expensive and thus favors discrete values Ve ¼ v0 ; 0 or v0 for e 2 f1; 2; :::; ng. It has been shown that pv ¼ 3 or pv ¼ 5 are suitable values for penalization of intermediate voltage values [36]. The small linear term in Eq. (16) is included to ensure a positive slope of the function Ve ðve Þ at ve ¼ 0. 3.2 Objective Function. In the considered shape control optimization problem, the design objective is defined as the shape error between the desired and the actuated shape. For a fixed finite element mesh, the shape error is measured by d¼

m X

2

ðwi  wdi Þ

(17)

i¼1

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In Eq. (17), wi is the ith concerned nodal displacement, wdi is the desired value, and m denotes the total number of the concerned displacements. 3.3 Constraints. The piezoelectric material volume constraint is defined as n  X

n X  qe ðV0p Þe  fV ðV0p Þe

e¼1

(18)

Assuming that the external mechanical force is independent of the design variables, we have @F=@qe ¼ 0. The above equation is then rewritten as    p  dg @g @u @F @Kuu  kTg Kuu ¼ þ kTg  u (23) dqe @u @qe @qe @qe After solving the adjoint variable equation

e¼1

Kuu kg ¼

ðV0p Þe

denotes the volume occupied by the piezoelectric where layers of the eth element, fV is the prescribed volume fraction ratio. Control energy function represented by the weighted square of the applied voltage is defined as a constraint condition for restricting the maximum energy required by the piezoelectric actuation [12]. The energy constraint is expressed as n X

ðVe ðve ÞÞ2 Ae  E

where E is the prescribed upper bound limit of the control energy. 3.4 Optimization Problem Formulation. Using the above mentioned continuous design variables, the topology optimization problem of interest is mathematically stated as min f  d ¼ q;v

subject to:

m X i¼1

Kuu u ¼ F þ Fp n  n X X  qe ðV0p Þe  fV ðV0p Þe e¼1

e¼1

(20)

(25)



In Eq. (25), the partial derivatives @Fpe @qe and @Keuu @qe can be easily calculated at the elemental level. For instance, from Eqs. (10) and (14), one obtains the derivative of the elemental stiffness matrix with respect to the density design variable as ð @Keuu ¼ 2pm ðqe Þpm 1 ðBp ÞT Cp0 Bp dX (26) @qe Xpe where Cp0 denotes the elastic matrix of the fully solid piezoelectric material. The sensitivity of the objective function with respect to the control voltage design variables can be evaluated in a similar way as described above, except that the term containing the derivative of the stiffness matrix vanishes.

ðVe ðve ÞÞ2 Ae  E

e¼1

qe 2 ½q; 1 for e 2 f1; 2; :::; ng ve 2 ½1; 1 for e 2 f1; 2; :::; ng Thus, the discrete-valued topology optimization problem is converted into a continuous one, which can be more easily treated with gradient-based mathematical programming methods.

4

 p  dg @F @Kuu ¼ kTg  u dqe @qe @qe   T @Fp @Ke uu e  ue ¼ keg @qe @qe

2

ðwi  wdi Þ

n X

(24)

We have

(19)

e¼1

@g @u

Solution Techniques

4.1 Sensitivity Analysis. The sensitivity analysis of the objective function and the constraints with respect to the design variables is required by gradient-based optimization algorithms. In general, two sensitivity analysis methods are available, namely, the direct variable method and the adjoint variable method. Since the number of design variables is much greater than that of the structural behavior functions in the present optimization problem, the latter method is applied in the sensitivity analysis in view of its computational efficiency. In what follows, the sensitivity evaluation scheme of a general functional gðuðq; vÞÞ with respect to the material density variables will be derived. The general functional gðuðq; vÞÞ can be rewritten by incorporating Eq. (9) as g¼gþ

kTg ðF

p

þ F  Kuu uÞ

(21)

where kg is an arbitrary vector known as the adjoint variable vector. Differentiating gðuðq; vÞÞ with respect to the design variable qe , it yields   dg @g @u @Fp @Kuu @u T @F (22) ¼ þ kg þ  u  Kuu dqe @u @qe @qe @qe @qe @qe 051006-4 / Vol. 134, MAY 2012

4.2 Solution Procedures. The optimization problem (20) can be solved in an iterative manner by using mathematical programming methods. The MMA [39] is employed in this paper, because it is well suited for dealing with large-scale topology optimization problems. The filter technique [40], with a filter radius of 1.5 times the element side length, is employed in each iteration step in order to avoid the checkerboard pattern and to generate mesh-independent optimal topologies. Figure 3 shows the flow chart of the optimization procedure. At the beginning, the material constants, boundary conditions, and loading conditions are specified. Then, the design domain is discretized, and the finite element analysis is performed. Subsequently, the objective function and the constraint functions as well as their sensitivities with respect to the design variables are calculated and fed into the MMA optimizer. The iteration procedure terminates when the prescribed convergence criterion is met. Obviously, the optimization problem (20) is nonconvex and therefore it is not guaranteed that the global optimum can be found. However, numerical experiences confirm that the MMA algorithm behaves well in solving the problem and can usually converge and yield meaningful optimal results within 20–50 iteration steps.

5

Numerical Examples

The design optimization method described above can be applied to shape control of bimorph-type piezoelectric plate structures. In this section, three examples will be given to validate the proposed formulation and numerical techniques. In the first example, the design objective is to nullify the bending deformation of a loaded plate by optimizing the piezoelectric material distribution and the control voltage distribution. In the second example, the design objective is to achieve a desired twisting deformation. Since pseudomodal control has potential importance in vibration Transactions of the ASME

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Fig. 4 Design domain of a clamped square plate

Fig. 3

Flowchart of the optimization procedure

suppression applications, a given mode shape is required to be achieved in the third example. The Young’s moduli, shear moduli, Poisson’s ratio, and piezoelectric constants of the materials used in these examples are given in Table 1. The actuation voltage to be applied to an element is required to be either 1000 V or 1000 V. 5.1 Example 1: Optimization for Minimizing Deflection Under External Load. In this example, we study the optimal layout of the actuation layers in static shape control of a clamped square plate subject to a concentrated force, as shown in Fig. 4. The plate consists of a host layer, which will be kept unchanged during the optimization, and two piezoelectric surface layers. Its side length is 0:40 m. Each layer has a thickness of 5  104 m, and thus the total thickness of the plate is H ¼ 1:5  103 m. The magnitude of the concentrated force is F ¼ 1 N. The plate is meshed into 40  40 eight-node elements. The design objective is to achieve a deformed shape with zero transverse nodal displacements along the three free edges. As described in Eq. (17), the shape error is defined by P d 2 d¼ m i¼1 ðwi  wi Þ , where m ¼ 239 is the total number of finite element nodes at the three free edges. The piezoelectric material volume is restricted to 30% of the original volume occupied by the layers, and the energy constraint is Pn piezoelectric 2 V A  7:84  104 V2 m2 . The penalty factors are set to be e e¼1 e pm ¼ pp ¼ pv ¼ 3. In the initial design, we set the material density variables xe ¼ 0:30 for e 2 f1; 2; :::; ng and the actuation voltage variables ve ¼ 0 for e 2 f1; 2; :::; ng. The deformation of the initial structure is shown in Fig. 5, and the corresponding shape error before application of actuation voltage is 0:1957 m2 . The optimization procedure will be stopped if the change of the objective function value between two adjacent iterations is less than 1  107 m2 . Table 1 Properties E1 ; E2 ðN=m2 ) G12 , G23 , G31 (N=m2 ) 12 , 21 e31 , e32 (C=m2 )

Material properties

Piezoelectric material

Host-layer material

6:1  1010 2:11  1010 0.35 9.3

7:1  1010 2:63  1010 0.35 —

Journal of Mechanical Design

Fig. 5 Deformation before shape control (displacement scale factor: 25)

Fig. 6

Optimal material layout (obtained with pm 5 pp 5 pv 5 3)

By using the proposed method, the optimal solution is obtained after 25 iteration steps. The optimal material densities of the piezoelectric layers and the optimal distribution of control voltage are given in Figs. 6 and 7, respectively. Figure 6 suggests a fairly distinct black-and-white material layout. Moreover, as can be seen from Fig. 7, there exist few areas of intermediate values of actuation voltage in the optimal solution. That is to say, the nonzero voltage zone is dominated by voltage values of 1000 V and 1000 V. This indicates that a distinct trileveled voltage distribution can be obtained by using the present penalization scheme for the voltage design variables. It is also noted that the spatial distribution pattern of nonzero voltage is closely correlated to the resulting topology of the piezoelectric layers. Such a solution basically suggests three bimorph-type actuators be positioned on the plate. The deformation of the optimal structure is shown in Fig. 8, which reveals that the deflection has been significantly reduced. Actually, the shape error for the optimal design is d¼ 4:69  107 m2 . Here, a normalized 1

1

measure for the shape error is also given as d2 =m2 H¼ 0:0295. For illustrating the effects of the penalization on intermediate voltage values, the optimal design with another set of penalty factors (pm ¼ pp ¼ 3 and pv ¼ 1) is also performed. In this case, no penalization is applied on voltage design variables. The obtained optimal layout of the piezoelectric layers and the optimal MAY 2012, Vol. 134 / 051006-5

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Fig. 7 Optimal distribution of actuation voltage (V) (obtained with pm 5 pp 5 pv 5 3): (a) distribution of positive actuation voltage; (b) distribution of negative actuation voltage

Fig. 10 Optimal distribution of actuation voltage (V) (obtained with pm 5 pp 5 3; pv 5 1): (a) distribution of positive actuation voltage; (b) distribution of negative actuation voltage

Fig. 8 Deformation after shape control (displacement scale factor: 25)

Fig. 11

Fig. 9 Optimal material layout (obtained with pm 5 pp 5 3; pv 5 1)

distribution of control voltage are shown in Figs. 9 and 10, respectively. The minimum value of shape error becomes 4:10 107 m2 , slightly lower than that obtained with pm ¼ pp ¼ pv ¼ 3. However, there are much more “gray” areas in the material layout and the voltage distribution. This implies that the proposed penalization on voltage is necessary for achieving a clear trileveled voltage distribution, which facilitates an easier electrical implementation. From the iteration histories plotted in Fig. 11, a basically steady decrease of the shape error can be observed for both sets of the penalty factors. 5.2 Example 2: Optimization for Achieving a Desired Twisting Shape. Consider the optimal design of a clamped laminated plate for minimizing the difference between the actuated 051006-6 / Vol. 134, MAY 2012

Iteration histories for both sets of penalty factors

deformation and the desired twisting shape. The plate is 0:24 m  0:24 m in dimension and consists of a host layer and two piezoelectric surface layers. All the three layers are 5  104 m in thickness, and thus the total thickness of the plate is H ¼ 1:5  103 m. The desired structural shape is described by the function wd ðx; yÞ ¼ x2 y, corresponding to a tip displacement of 6:912  103 m. Here, the plate deflection is still relatively small compared with the side length, which justifies the small deformation assumption. Similarly, as in the first example, the host layer is to be kept unchanged and the material layout of the piezoelectric layers is to be optimized during the optimization. The design domain is discretized with 24  24 finite elements, as shown in Fig. 12. In this example, the shape error is computed with the displacements of the corner nodes of all the eight-node elements, and thus the number of the concerned displacements is m ¼ 600. The volume fractionPratio is set to be 0.4, and the control energy constraint n 2 4 2 2 is given as e¼1 Ve Ae  3:69  10 V m . The shape error before application of actuation voltage reaches 2:22  103 m2 . The optimization process will be terminated when the difference between the objective function values of two adjacent iterations is less than 1  106 m2 . Transactions of the ASME

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Fig. 12 Design domain of a clamped square plate (discretized with 24324 eight-node elements) Fig. 15 Iteration history

Fig. 13 Optimal material layout

Fig. 16 Comparison between the desired shape and the actuated shape: (a) desired shape (displacement scale factor: 10); (b) actuated shape in the optimal design (displacement scale factor: 10)

5.3 Example 3: Optimization for Achieving a Specified Modal Shape. In the last example, the same structure as in the second example is considered. The third vibration mode of the square plate is chosen as the desired shape. The shape error is measured by the function defined as

Fig. 14 Optimal distribution of actuation voltage (V): (a) distribution of positive actuation voltage; (b) distribution of negative actuation voltage

With the penalty factors pm ¼ pp ¼ pv ¼ 5, the optimization process converges after 29 iterations. The optimal material layout and the optimal distribution of actuation voltage are shown in Figs. 13 and 14, respectively. It is found that the shape error decreases steadily during the course of the optimization process (see Fig. 15), and it is finally reduced to d ¼ 8:79  106 m2 , with 1

1

a normalized value d2 =m2 H¼ 0:0807, in the optimal design. Thus, a satisfying shape control accuracy is achieved though the optimization. This can also be observed by comparing the desired shape and the actuated shape depicted in Fig. 16. Journal of Mechanical Design

0 m X B d¼ @w i  i¼1

12 di C    0:0015 mA   maxð dj Þ m

(27)

j¼1

where m ¼ 1776 is the total number of nodes and di is the ith transverse degree of freedom in the third vibration mode. The volume fraction ratio of the piezoelectric material is set to be 0.3. In P addition, the energy constraint ne¼1 Ve2 Ae  1:44  104 V2 m2 is also imposed. The optimization process starts with initial guesses xe ¼ 0:30 for e 2 f1; 2; :::; ng and ve ¼ 0 for e 2 f1; 2; :::; ng. The termination condition of the optimization procedure is the same as in the first example. Using the penalty factors pm ¼ pp ¼ pv ¼ 3, we obtained the optimal material distribution as plotted in Fig. 17 and the optimal actuation voltage distribution as shown in Fig. 18. MAY 2012, Vol. 134 / 051006-7

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Fig. 17 Optimal distribution of piezoelectric material

Fig. 20 Comparison between the desired shape and the actuated shape: (a) third mode shape; (b) actuated shape in the optimal design (displacement scale factor: 25)

6

Fig. 18 Optimal distribution of actuation voltage (V): (a) distribution of positive actuation voltage; (b) distribution of negative actuation voltage

Fig. 19 Iteration history

Figure 19 presents the iteration history of the objective function, from which we can see that the shape error reduces monotonically till the optimization process converges. In the optimal solution, the shape error decreases dramatically from 6:74  104 m2 to 1:65  106 m2 , with a normalized value of 1

1

d2 =m2 H ¼ 0:0203. In the optimal design, the structural deformation achieved under bending forces generated by the actuator elements is almost exactly the required one, as can be seen from Fig. 20. 051006-8 / Vol. 134, MAY 2012

Concluding Remarks

The topology optimization method is applied to static shape control applications for determining the best distribution of piezoelectric material and the trileveled actuation voltage. Element-wise material densities and voltage parameters are taken as design variables. They are to be simultaneously optimized so as to minimize the shape error under a given control energy constraint. Artificial power-law interpolation models, which relate the material mechanical/piezoelectrical properties and the actuation voltages to theses design variables, are employed for suppressing intermediate design variable values. Based on the design sensitivity analysis, the optimization problem is solved with the MMA algorithm. The effectiveness of the proposed method is demonstrated by three numerical examples. In particular, it is shown that nearly distinct black-and-white distributions of the piezoelectric material can be obtained through the optimal design. Moreover, compared with the design with continuous voltage values, the optimal solution obtained with the suggested voltage penalization results in a distinctly trileveled voltage distribution, but without much worsening the objective function value. In this study, discrete-valued voltage distribution optimization is performed on the whole design domain (rather than on a given number of piezopatches). This presents a challenging task for conventional discrete optimization methods, including the GA and the (SA) method due to the large number of design variables, especially when the relative densities of the piezoelectric material are to be simultaneously optimized. Thus, a continuous-valued mathematical programming algorithm in conjunction with intermediate voltage penalization is employed for this purpose. The MMA optimizer is shown to behave well in finding the optimal solutions acceptable from engineering point of view at a reasonable cost. However, if the global optimum is of major concern, some other mathematical programming methods such as the GA and the SA should be resorted.

Acknowledgment The supports of Key Project of Chinese National Programs for Fundamental Research and Development (Grant No. 2010CB832703) and Natural Science Foundation of China (11072047 and 91130025) are gratefully acknowledged.

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