Topology optimization of piezoelectric layers in plates with active vibration control

Abstract

This article investigates topology optimization of the piezoelectric actuator and sensor layers in a plate for achieving the best vibration control performance. Therein, the actuator patches and sensor patches are symmetrically attached to the host layer, and the classical negative velocity feedback control strategy is adopted for reducing the vibration level of the structure. In the optimization model, the dynamic compliance under a specific excitation frequency or the aggregated dynamic compliance in a given frequency range is taken as the objective function. The relative densities of the elements in the actuator layer and the sensor layer are considered as topological design variables. The optimization problem is then formulated by using an artificial material model with penalization for both mechanical and piezoelectric properties. It is pointed out that the global-level damping property, consisting of the structural damping and the active damping effects, is a nonproportional one. For alleviating the computational burden involved in the frequency response analysis, the dynamic equations are solved with the complex mode superposition in the state space after a model reduction transformation. In this context, the sensitivity analysis scheme is also derived. The effectiveness and efficiency of the proposed method are demonstrated by numerical examples.

Keywords

Topology optimization smart structure piezoelectric active control dynamic optimization complex mode superposition

Introduction

Piezoelectric materials are characterized by the capability of delivering a mechanical strain under applied electrical field and vice versa. They have been extensively used in various engineering applications, including static/dynamic shape control, ultrasonic transducers, micro electromechanical systems (MEMS), and energy-harvesting devices (Irschik, 2002; Sodano et al., 2005). In particular, the integration of laminated plates with piezoelectric material layers is generally considered very effective for active vibration control.

To achieve the desired active control performance of piezoelectric structures, a suitable control algorithm must be implemented. The classical constant gain velocity feedback (CGVF) control and its performance have been studied by, for example, Tzou and Tseng (1990), Hwang and Park (1993), and Wang et al. (2001). For improving the overall active control performance, it is highly desirable to optimize the locations and geometries of the piezoelectric actuator/sensor elements. This just raises the topology optimization problem of the piezoelectric actuator/sensor layers in conjunction with a specific control strategy for the design of these smart structures.

Topology optimization is considered as a powerful automated tool for improving the structural performance at the conceptual design stage. Over the last two decades, several popular approaches have been developed, including the homogenization optimization method (Bendsøe and Kikuchi, 1988), the solid isotropic material with penalization (SIMP) approach (Bendsøe, 1989; Zhou and Rozvany, 1991), the level set-based topology optimization methods (Allaire et al., 2004; Wang et al., 2003), and the evolutionary structural optimization (ESO) method (Xie and Steven, 1993). Many studies show that the performance of piezoelectric structures can be effectively improved with these techniques (Buehler et al., 2004; Luo et al., 2009; Nakasone and Silva, 2010; Silva et al., 1997).

Layout design optimization of piezoelectric smart structures has been considered in many studies. Genetic algorithms (Xu et al., 2007), particle swarm optimization algorithms (Zorić et al., 2013), or other heuristic algorithms (Sun and Tong, 2005) have been used to seek the optimal size and location of piezoelectric actuators. Kögl and Silva (2005) introduced a piezoelectric material model with penalization on piezoelectric constant and polarization for topology optimization of piezoelectric actuators. Carbonari et al. (2007) employed the bimaterial SIMP method in the topology optimization of a flexible structure actuated by piezoceramics for generating output displacement and force at a certain specified direction with a multiobjective function. Kang et al. (2011) and Luo et al. (2010) studied the optimal distribution of the compliant supporting structure and the embedded piezoelectric material with a multiphase material model. Kang and Tong (2008b) developed an integrated topology optimization method for simultaneous design of actuator layouts and control voltage in laminated piezoelectric shells for static shape control. Kiyono et al. (2012) demonstrated that the energy conversion rate can be further increased by including polarization parameters and ply angles into the design of piezoelectric shells.

A significant number of studies have also been devoted to dynamic optimization of piezoelectric structures. Lau et al. (2000) and Maddisetty and Frecker (2004) considered the topology optimization for mechanical amplifiers consisting of compliant mechanisms and piezoelectric actuators under different excitation frequencies. Ha and Cho (2006) proposed the design sensitivity analysis and topology optimization of eigenvalue problems for piezoelectric resonators. Trindade (2007) studied the optimal design for minimizing the structural vibration level by embedding piezoelectric actuators and viscoelastic damping treatments into the structure. Wein et al. (2009) presented a topology optimization study of piezoelectric actuator patches for maximizing the resonance response under given harmonic excitations. Donoso and Bellido (2009) considered the topology optimization for piezoelectric modal sensors and actuators by optimizing the electrode surface shape and polarization profile simultaneously. Optimization of piezoelectric structures with passive control was recently studied by Vasques (2012) and Rosi et al. (2012). Some works have also been devoted to topology optimization of piezoelectric energy harvesters for maximizing the energy conversion factor (Zheng et al., 2009), output electrical power (Rupp et al., 2009), or the electromechanical coupling factor (EMCC) (Chen et al., 2010; Noh and Yoon, 2012; Silva and Kikuchi, 1999; Vatanabe et al., 2012).

Some authors also considered topology optimization of piezoelectric structures under active control. Wang et al. (2006) optimized the distribution of piezoelectric actuators/sensors for suppressing the structural vibration with CGVF control by using a genetic algorithm. Drenckhan et al. (2008) considered the topology optimization of the piezoelectric actuator layers attached to a beam under proportional feedback control with a full coverage of sensor layer. Donoso and Sigmund (2009) proposed a parametric optimization model for determining the thickness and width profile of piezoelectric bimorph actuators with active damping for minimizing the tip deflection. Up to now, however, there are few studies on optimizing the topologies of piezoelectric actuators and sensors simultaneously under active control with gradient-based mathematical programming algorithms.

This article presents a topology optimization formulation for topological layout optimization of the piezoelectric actuator layer and sensor layer in a plate under active vibration control. Here, the actuator layer and the sensor layer are attached to the top and bottom surfaces of the host structure (as schematically illustrated in Figure 1) and they have the same layout but opposite poling directions. The structure is subjected to harmonic excitations, and only the steady-state response is of concern in this study. As can be seen in Figure 1, the sensor output is converted into the feedback voltage, which is fed into the actuator layers. The classical CGVF control algorithm is adopted as the active control algorithm. In the proposed optimization model, two types of objective functions are considered, namely, the dynamic compliance at a single excitation frequency and the aggregated dynamic compliance within an excitation frequency range. For a vibrating structural system under the considered control strategy, the active damping matrix cannot be expressed as a combination of the global stiffness matrix and mass matrix. This means the structure has a nonproportional damping property at the global level. In such a circumstance, a suitable strategy for the dynamic response analysis is to employ the complex mode superposition technique in the state space (Igusa et al., 1984). For the purpose of model reduction, the original dynamic equations are first mapped into a reduced-order modal space with the mode superposition method before the complex eigenvalue analysis. In this context, the adjoint variable scheme for the design sensitivity analysis is derived, which facilitates efficient solution of the optimization problem with gradient-based mathematical programming algorithms.

Figure 1.

Schematic illustration of topology optimization for the actuator layer and sensor layer of piezoelectric plates with active control.

Finite element modeling and governing equations

Constitutive equations of laminated plate with piezoelectric layers

The mechanical properties of the host layer, the sensor layer, and the actuator layer are considered linear elastic in this study. Thus, the constitutive equation for the host layer is given as

T = c^{H} S

(1)

while the constitutive equation for the piezoelectric layers is written as (Sodano et al., 2004, 2005)

T = c^{E} S - e^{T} E

(2)

In equations (1) and (2), T and S are the mechanical stress vector and the mechanical strain vector, respectively; $c^{H}$ and $c^{E}$ are the elasticity matrices of host layer and piezoelectric layers, e is the piezoelectricity matrix, and E is the vector of applied electric field.

Based on the Mindlin’s plate theory under the small strain assumptions, the displacements at point $(x, y, z)$ for the time instant t take the form

\begin{matrix} u (x, y, z, t) = u_{0} (x, y, t) - z θ_{x} (x, y, t) \\ v (x, y, z, t) = v_{0} (x, y, t) - z θ_{y} (x, y, t) \\ w (x, y, z, t) = w_{0} (x, y, t) \end{matrix}

(3)

Here, u, v, and w are the displacement components along the x-, y-, and z-axis, respectively; u₀, v₀, and w₀ are the displacement components of the midplane (z = 0); $θ_{x}$ and $θ_{y}$ are the rotations of the transverse normal about the y- and x-axis, respectively.

The electric potential is assumed to vary linearly across the thickness of the piezoelectric actuator and sensor layers. The electric voltage in the actuator layer $Φ$ can thus be expressed as

Φ = {Φ^{1}, Φ^{2}, \dots, Φ^{N_{e}}}^{T}

(4)

where $N_{e}$ is the total number of actuator patches, and $Φ^{i} (i = 1, 2, \dots, N_{e})$ is the electric voltage of the actuator layer in the ith actuator patch.

In practice of active vibration control of a plate with piezoelectric layers, the voltage is applied only in the thickness direction. We assume that the electric potential varies linearly across the actuator layer. Thus, the electric field vector $E^{i} (i = 1, 2, \dots, N_{e})$ for the ith actuator can be expressed by the voltage $Φ$ across the actuator layer as

E^{i} = {0, 0, - \frac{Φ^{i}}{h}}^{T} (i = 1, 2, \dots, N_{e})

(5)

where h is the thickness of the actuator layer.

Governing equations of motion

We consider the optimal placement of a given volume of piezoelectric material in the actuator and sensor layers. The governing equations for the finite element model of a structure with bonded piezoelectric actuator/sensor patches under external harmonic excitations are given as

M \overset{\cdot\cdot}{x} (t) + C \overset{\cdot}{x} (t) + Kx (t) = f (t) + f_{a} (t)

(6)

where $M \in R^{n \times n}$ , $C \in R^{n \times n}$ , and $K \in R^{n \times n}$ (n is the number of degrees of freedom (DOFs)) are the mass matrix, damping matrix, and stiffness matrix of the structural model, respectively; $f (t) = F e^{i θ t}$ is the vector of excitation force with the frequency $θ$ and amplitude $F$ ; $x (t) \in R^{n \times 1}$ , $\overset{\cdot}{x} (t) \in R^{n \times 1}$ , and $\overset{\cdot\cdot}{x} (t) \in R^{n \times 1}$ are the vectors of time-varying displacement, velocity, and acceleration, respectively. The vector of actuation force $f_{a} (t)$ can be further expressed as

f_{a} (t) = K_{u Φ} Φ_{a} (t)

(7)

Here, $Φ_{a} (t)$ is the vector of actuation voltage, which depends on the sensor voltage $Φ_{s} (t)$ as will be described later; $K_{u Φ}$ is the structural electromechanical matrix of the actuator layer and it is expressed as (Kang and Tong, 2008a)

K_{u Φ} = \int_{Ω} B_{u}^{T} e^{T} B_{Φ} d Ω

(8)

where $B_{u}$ and $B_{Φ}$ are, respectively, the strain–displacement matrix and the matrix relating the electrical field with the actuation voltage, and they both depend only on the geometry of the actuator layer.

Active control model and response analysis

Active control model

Sensor voltage

In this study, each sensor element is considered as an individual sensor patch. Thus, the charge output of the ith sensor $Q^{i}$ (under the assumption that $E \equiv 0$ ) can be expressed by the integration of electric displacement $D_{3} = e_{31} S_{1} + e_{32} S_{2} + e_{34} S_{4} + e_{35} S_{5} + e_{36} S_{6}$ in the thickness direction as

Q^{i} (t) = \frac{1}{2} \int_{A_{s}^{i +} + A_{s}^{i -}} D_{3} d A

(9)

where $A_{s}^{i +}$ and $A_{s}^{i -}$ are the upper and bottom surfaces of the ith sensor element, and equation (9) can be rewritten in matrix form as

Q^{i} (t) = K_{Φ u}^{i} x^{i} (t)

(10)

Here, $K_{Φ u}^{i}$ is the ith row of the matrix $K_{Φ u}$ and represents the mechanoelectronic coupling effects of the ith sensor element

K_{Φ u}^{i} = \int_{Ω_{i}} B_{Φ}^{T} e B_{u} d Ω

(11)

Then output charge $Q^{i} (t)$ of the ith sensor can be transformed to the sensor voltage as follows. The current $i_{s}^{i} (t)$ on the surface of the ith piezoelectric sensor is given by

i_{s}^{i} (t) = \frac{d Q^{i} (t)}{d t}

(12)

The current is converted into the open circuit sensor voltage output $Φ_{s}^{i} (t)$ as

Φ_{s}^{i} (t) = G_{s}^{i} i_{s}^{i} (t) = G_{s}^{i} K_{Φ u}^{i} {\overset{\cdot}{x}}^{i} (t)

(13)

where $G_{s}^{i}$ is the gain of the charge amplifier for the ith sensor. The sensor voltage output vector $Φ_{s} (t)$ is then expressed by

Φ_{s} (t) = G_{s} K_{Φ u} \overset{\cdot}{x} (t)

(14)

where $G_{s} = diag (G_{s}^{1}, G_{s}^{2}, \dots, G_{s}^{N_{e}})$ is a diagonal matrix.

Negative feedback control

For the CGVF control strategy, the actuator voltage $Φ_{a} (t)$ is determined by the output sensor voltage $Φ_{s} (t)$ as

Φ_{a} (t) = - G_{a} Φ_{s} (t)

(15)

where $G_{a}$ is a constant gain diagonal matrix. Substituting equation (14) into equation (15) gives

Φ_{a} (t) = - G_{a} G_{s} K_{Φ u} \overset{\cdot}{x} (t)

(16)

Substituting equations (16) and (7) into equation (6) yields the system equations with active damping

M \overset{\cdot\cdot}{x} (t) + (C + C_{A}) \overset{\cdot}{x} (t) + Kx (t) = f (t)

(17)

where $C_{A}$ is the active damping matrix expressed by

C_{A} = K_{u Φ} G_{a} G_{s} K_{Φ u}

(18)

It is noted that $C_{A}$ is not positive definite nor symmetric. However, we assume that the overall damping matrix $C + C_{A}$ (the damping matrix of the host structure also included) is positive definite, which is usually the case in real applications.

Response analysis with complex mode superposition method

In this study, only the steady-state response of the vibrating structure is of interest. The steady-state displacement response $x (t)$ can be expressed as $x (t) = X e^{i θ t}$ with the complex amplitude of vibration $X \in R^{n \times 1}$ , and thus the governing equation (17) is rewritten as (Kang et al., 2012)

SX = F

(19)

where $S = - θ^{2} M + i θ (C + C_{A}) + K$ .

For a structure with a large number of DOFs, it is an expensive task to solve the dynamic equations directly. An effective way to improve the numerical efficiency is to transform the original system equations into reduced-order equations by using certain model reduction techniques (Guyan, 1965). To achieve this, the displacement $x (t)$ is approximated by

x (t) = Ψ u

(20)

where $Ψ = {ψ_{1}, ψ_{2}, \dots, ψ_{n_{b}}}$ denotes the first $n_{b} (n_{b} << n)$ normalized real eigenvectors of the undamped system; $u \in R^{n_{b} \times 1}$ is the vector of generalized displacements. Therein, the matrix $Ψ$ can be obtained by solving the eigenvalue problem of the original finite element model.

Substituting equation (20) into equation (19) and premultiplying $Ψ^{T}$ to both sides of the equation, a reduced-order dynamic equation can be obtained as

M^{*} \overset{\cdot\cdot}{u} + C^{*} \overset{\cdot}{u} + K^{*} u = F^{*} e^{i θ t}

(21)

where the reduced-order system matrices are $M^{*} = {I_{n_{b} \times}}_{n_{b}}$ , $C^{*} = Ψ^{T} (C + C_{A}) Ψ,$ $K^{*} = Ψ^{T} K Ψ$ , and $F^{*} = Ψ^{T} F$ .

With the above transformation, the reduced-order stiffness matrix $K^{*}$ and mass matrix $M^{*}$ both become diagonal matrices. However, as pointed out in the previous section, the active damping matrix $C_{A}$ is asymmetric and cannot be diagonalized with the real eigenvectors. This means the reduced-order equation (20) is not a decoupled one.

For efficient response analysis of the nonproportionally damped system in equation (20), the complex mode superposition method in the state space is employed in this study. To this end, equation (21) is further transformed into the state space as

H \overset{\cdot}{z} (t) + Gz (t) = \hat{F} e^{i θ t}

(22)

with

z = {\begin{matrix} u \\ \overset{\cdot}{u} \end{matrix}}, G = [\begin{matrix} K^{*} & 0 \\ 0 & - M^{*} \end{matrix}], H = [\begin{matrix} C^{*} & M^{*} \\ M^{*} & 0 \end{matrix}], \hat{F} = {\begin{matrix} F^{*} \\ 0 \end{matrix}} .

(23)

Here, the generalized displacement vector $z (t)$ can be obtained by solving the generalized eigenvalue problem of the dynamic system (22). For more details of the complex mode superposition method, the readers are referred to Igusa et al. (1984) and Kang et al. (2012).

Topology optimization problem formulation

Dynamic compliance

Dynamic compliance under a single excitation frequency

We first consider a vibrating structure under a harmonic excitation with specified excitation frequency. The dynamic compliance (defined as the inner product of the external forces and displacement response) is frequently used as an effective measure of vibration level for a given excitation frequency, see Ma et al. (1993). Here, we take the dynamic compliance defined by Ma et al. (1993, 1995) and Yoon (2010) as the first type of objective function in the optimization model

f = c = \sqrt{{(F^{T} X^{R})}^{2} + {(F^{T} X^{I})}^{2}}

(24)

where $X^{R}$ and $X^{I}$ are real and imagination parts of the displacement vector $X$ , respectively.

Maximum dynamic compliance within a frequency range aggregated with Kreisselmeier-Steinhauser (KS) function

The first type of objective function in equation (24) is effective in optimizing the frequency response at a particular excitation frequency. However, in many applications the excitation frequency may vary within an interval range. For an optimized structure under a specific frequency, a small variation in the excitation frequency may drive the dynamic response far away from its optimal values. This thus raises the problem of topology optimization in an excitation frequency range.

Giving $N_{p}$ sampling points $θ_{i} (i = 1, 2, \dots, N_{p})$ within the frequency range of interest $Ω_{f}$ ( $θ_{i}$ is usually uniformly distributed in $Ω_{f}$ ), we now define the objective function as the maximum dynamic compliance for these discrete excitation frequencies

f = max (c_{1}, c_{2}, \dots, c_{N_{p}})

(25)

Such an objective function is not continuous and this poses a major difficulty when solving the optimization problem with a gradient-based algorithm. To circumvent this, we employ the KS function (Kreisselmeier and Steinhauser, 1979) to form an envelope function, which is sufficiently smooth and provides a conservative approximation of the maximum function values (Wrenn, 1989). By this means, the objective function for the maximum dynamic compliance in a frequency range $Ω_{f}$ is rewritten as

f = KS [c_{1}, c_{2}, \dots, c_{N_{p}}] = \frac{1}{η} \ln [\sum_{i = 1}^{N_{p}} e^{η c_{i}}]

(26)

where $η$ is the aggregation parameter of the KS function. The value of $η$ should be appropriately chosen to achieve a reasonable compromise between the computational efficiency and approximation accuracy of the envelop surface. A discussion on this issue can be found in, for example, Poon and Martins (2007).

Optimization problem formulation

We consider the optimal distribution problem of piezoelectric actuator layer and sensor layer for minimizing the structural frequency response with a given amount of piezoelectric material in a prescribed design domain, as schematically illustrated in Figure 1. The topology optimization problem can be formulated as

\begin{matrix} \underset{ρ}{min .} f (ρ) \\ s . t . (- θ^{2} M + i θ (C + C_{A}) + K) X = F \\ \sum_{e = 1}^{N_{e}} ρ_{e} V_{e} - f_{v} \sum_{e = 1}^{N_{e}} V_{e} \leq 0 \\ 0 < \underline{ρ} \leq ρ_{e} \leq 1 (e = 1, 2, \dots, N_{e}) \end{matrix}

(27)

Here, the vector $ρ = {ρ_{1}, ρ_{2}, \dots, ρ_{N_{e}}}^{T}$ collects the element-wise density design variables describing the material layout of the piezoelectric actuator and sensor layers, and $N_{e}$ denotes the total number of finite elements in the design domain. As usual, $ρ_{e} = 0$ and $ρ_{e} = 1$ indicate absence and presence of the piezoelectric material in the eth element, respectively. The symbol $f_{v}$ denotes the volume fraction ratio, and $V_{e}$ is the volume of the eth element. The constraint $\sum_{e = 1}^{N_{e}} ρ_{e} V_{e} - f_{v} \sum_{e = 1}^{N_{e}} V_{e} \leq 0$ imposes a restriction on the volume of the piezoelectric material. The lower bound of the relative density $\underline{ρ}$ is set as $10^{- 6}$ in this study.

The matrices $M$ , $K$ , and $C$ in optimization problem (27) and governing equation (19) are obtained by assembling contributions of the host structure, the sensor layer, and the actuator layer. In the context of the SIMP approach, the matrices $M$ , $K$ , and $C$ are written as

M = \sum_{e = 1}^{N_{e}} M_{h}^{e} + \sum_{e = 1}^{N_{e}} ρ_{e} (M_{a}^{e} + M_{s}^{e})

(28)

K = \sum_{e = 1}^{N_{e}} K_{h}^{e} + \sum_{e = 1}^{N_{e}} {(ρ_{e})}^{p_{1}} (K_{a}^{e} + K_{s}^{e})

(29)

C = α \sum_{e = 1}^{N_{e}} K_{h}^{e} + β \sum_{e = 1}^{N_{e}} M_{h}^{e}

(30)

where $K_{h}^{e}$ and $M_{h}^{e}$ are the element stiffness matrix and the element mass matrix of the host structure, and they remain unchanged during the optimization process; $K_{s}^{e}$ , $K_{a}^{e}$ , $M_{s}^{e}$ , and $M_{a}^{e}$ denote the element stiffness and mass matrices of the actuator/sensor layers with fully solid piezoelectric material; $p_{1} > 1$ is the penalization factor for Young’s modulus of the artificial piezoelectric material model. The symbols $α$ and $β$ denote the structural damping coefficients of the host layer. Due to the fact that the piezoelectric layers are very thin as compared with the host layer, the structural damping effect of the piezoelectric layers is neglected in this study.

By introducing penalization on the piezoelectric property, we express the piezoelectricity matrix as (Kang and Tong, 2008b)

e^{piezo} (ρ_{e}) = ρ_{e}^{p_{2}} e

(31)

where $p_{2} > 1$ is the penalization factor.

Thus, the active damping matrix $C_{A}$ in equation (18) becomes

C_{A} = \sum_{e = 1}^{N_{e}} {(ρ_{e})}^{p_{2}} K_{u Φ}^{e} G_{a}^{e} G_{s}^{e} K_{Φ u}^{e}

(32)

The penalization for the material properties is included for suppressing intermediate density values. It is a usual treatment for topology optimization of piezoelectric structure. The choice of the penalization factors in topology optimization with piezoelectric material has been discussed in the literature (Kim et al., 2010; Kögl and Silva, 2005; Noh and Yoon, 2012). Following the suggestion by Noh and Yoon (2012) and Kögl and Silva (2005), we set $p_{1} = 3$ and $p_{2} = 3$ .

Design sensitivity analysis

The optimization problem (27) is solved by a gradient-based mathematical programming algorithm, which requires sensitivity analysis of the objective function with respect to the design variables. Kang et al. (2012) derived the sensitivity equations with the adjoint variable method (AVM) for frequency response of structures with nonproportional material damping. Here, these sensitivity equations are adapted to the considered system with active control.

Consider the sensitivity analysis for a generalized structural behavior function $f (X (ρ))$ , which explicitly depends on the amplitudes of the steady-state displacement response $X$ . By using the vibration equation (19) and its conjugate equation, the function $f (X)$ can be rewritten as

L = f (X) + μ_{1}^{T} (WX - F) + μ_{2}^{T} (\bar{W} \bar{X} - \bar{F})

(33)

where $\bar{W}$ and $\bar{F}$ denote the conjugates of the dynamic stiffness matrix $W = - θ^{2} M + i θ (C + C_{A}) + K$ and the excitation amplitude vector $F$ , respectively; $μ_{1}$ and $μ_{2}$ are the adjoint vectors.

Differentiating equation (33) with respect to the eth design variable gives

\begin{matrix} \frac{d L}{d ρ_{e}} = μ_{1}^{T} \frac{\partial S}{\partial ρ_{e}} X + μ_{2}^{T} \frac{\partial \bar{S}}{\partial ρ_{e}} \bar{X} + \frac{\partial X^{R}}{\partial ρ_{e}} (\frac{\partial f}{\partial X^{R}} + μ_{1}^{T} W + μ_{2}^{T} \bar{W}) \\ + \frac{\partial X^{I}}{\partial ρ_{e}} (\frac{\partial f}{\partial X^{I}} + i μ_{1}^{T} W - i μ_{2}^{T} \bar{W}) \end{matrix}

(34)

Let the adjoint variables satisfy the following equations

μ_{1}^{T} S = \frac{1}{2} (- \frac{\partial f}{\partial X^{R}} + i \frac{\partial f}{\partial X^{I}}), μ_{2}^{T} \bar{S} = \frac{1}{2} (- \frac{\partial f}{\partial X^{R}} - i \frac{\partial f}{\partial X^{I}})

(35)

By comparing the two equations in equation (35), one sees that $μ_{1} = {\bar{μ}}_{2}$ . With the adjoint vector solutions to equation (35), the derivative of the objective function can be determined by

\frac{d f}{d ρ_{e}} = \frac{d L}{d ρ_{e}} = 2 Re ({μ_{1}}^{T} (- θ^{2} \frac{\partial M}{\partial ρ_{e}} + i θ \frac{\partial (C + C_{A})}{\partial ρ_{e}} + \frac{\partial K}{\partial ρ_{e}}) X)

(36)

in which

\frac{\partial M}{\partial ρ_{e}} = M_{a}^{e} + M_{s}^{e},

\frac{\partial K}{\partial ρ_{e}} = p_{1} {(ρ_{e})}^{(p_{1} - 1)} (K_{a}^{e} + K_{s}^{e}),

(37)

\frac{\partial (C + C_{A})}{\partial ρ_{e}} = \frac{\partial C_{A}}{\partial ρ_{e}} = p_{2} {(ρ_{e})}^{(p_{2} - 1)} K_{u Φ}^{e} G_{a}^{e} G_{s}^{e} K_{Φ u}^{e}

If the objective function f is taken as the dynamic compliance c under a single excitation frequency in equation (24), one can obtain $\partial c / \partial X^{R}$ and $\partial c / \partial X^{I}$ as

\frac{\partial c}{\partial X^{R}} = \frac{\partial \sqrt{{(F^{T} X^{R})}^{2} + {(F^{T} X^{I})}^{2}}}{\partial X^{R}} = \frac{F^{T} X^{R} F^{T}}{c}

(38)

\frac{\partial c}{\partial X^{I}} = \frac{\partial \sqrt{{(F^{T} X^{R})}^{2} + {(F^{T} X^{I})}^{2}}}{\partial X^{I}} = \frac{F^{T} X^{I} F^{T}}{c}

(39)

Substituting equations (38) and (39) into equation (35) leads to

μ_{1}^{T} S = - \frac{1}{2 c} F^{T} \bar{X} F^{T}

(40)

Thus, the derivative of the behavior function c can be obtained from equation (36) with the adjoint vector $μ_{1}$ as

\frac{d c}{d ρ_{e}} = \frac{1}{c} Re {S^{- 1} F^{T} \bar{X} F^{T} [- θ^{2} (M_{a}^{e} + M_{s}^{e}) + i θ p_{2} {(ρ_{e})}^{(p_{2} - 1)} K_{u Φ}^{e} G_{a}^{e} G_{s}^{e} K_{Φ u}^{e} + p_{1} {(ρ_{e})}^{(p_{1} - 1)} (K_{a}^{e} + K_{s}^{e})]}

(41)

For the case of the aggregated function of the dynamic compliance in a frequency range $(f = KS [c_{1}, c_{2}, \dots, c_{N_{p}}])$ , the derivative becomes

\frac{d f}{d ρ_{e}} \frac{= \sum_{i = 1}^{N_{p}} (e^{η c_{i}} \frac{d c_{i}}{d ρ_{e}})}{\sum_{i = 1}^{N_{p}} e^{η c_{i}}}

(42)

Numerical examples

In this section, several numerical examples are presented to illustrate the validity of the optimization formulation and numerical techniques. The proposed method is implemented on the MATLAB platform. The globally convergent method of moving asymptotes (GCMMA) (Svanberg, 2002) is employed as the optimizer in view of its ability to treat highly nonlinear behavior functions in the considered dynamic optimization problem. The optimization process will be terminated when the relative difference of the objective function values between two adjacent iteration steps satisfies the convergence criterion $‖ (f_{new} - f_{old}) / f_{old} ‖ < 10^{- 5}$ .

Topology optimization under a specified excitation frequency

Topology optimization of the actuator/sensor layers attached to a cantilever plate

The first example considers the topology optimization of the actuator and sensor layers attached to a cantilever laminate plate as shown in Figure 2. The host layer of the plate has a geometrical dimension $a = 1.6 m$ , $b = 0.8 m$ , and $t_{h} = 4 \times 10^{- 3} m$ ; the sensor layer and the actuator layer, both with the thickness $t_{s} = t_{a} = 0.5 \times 10^{- 3} m$ , are attached to the bottom surface and the top surface of the host layer, respectively. A time-harmonic external force $f (t) = F e^{i θ t}$ (with $F = 200 N$ , $θ = 2 π f_{p}$ and $f_{p} = 43 Hz$ ) is applied at the midpoint of the free edge. The host layer (aluminum) has Young’s modulus $E^{h} = 6.9 \times 10^{10} N / m^{2}$ , Poisson’s ratio $ν^{h} = 0.3$ , and the mass density $ρ^{h} = 2700 kg / m^{3}$ ; and the properties of the piezoelectric material (PZT) are $E^{piezo} = 7.1 \times 10^{10} N / m^{2}$ , $ν^{piezo} = 0.35$ , and $ρ^{piezo} = 5000 kg / m^{3}$ . The piezoelectric stress coefficients are $e_{31} = e_{32} = - 5.2 C / m^{2}$ , $e_{33} = 15.1 C / m^{2}$ , and $e_{15} = e_{25} = 12.7 C / m^{2}$ . The damping coefficients of the host structure are $α = β = 5 \times 10^{- 4}$ . The gain value of the charge amplifier is $G_{c} = 1 \times 10^{6} V / A$ , and the negative feedback control gain is $G_{a} = 50$ .

Figure 2.

A cantilever plate with active control under a time-harmonic load $f (t) = F e^{i θ t}$ applied at the midpoint of the free edge.

The design domain is discretized by 80 × 40 uniform-sized four-node Mindlin shell elements with the total number of DOFs $n = 16, 605$ . The volume fraction ratio of the piezoelectric material in the design domain is restricted by $f_{v} = 0.5$ and the initial values of the design variables are set to be $ρ_{e} = 0.5 (e = 1, 2, \dots, 3200)$ . The first 40 eigenmodes are involved in the computation (i.e. $n_{b} = 40$ ). The well-known sensitivity filter technique (Sigmund and Torquato, 1997), with a filter radius of $r_{min} = 0.04 m$ , is employed for preventing checkerboard formation and mesh dependency of the optimal solution. The optimization procedure converged after 21 iterations. From the iteration histories shown in Figure 3, a stable decrease of the objective function value can be observed. The optimal design is presented in Figure 4. The vibration amplitude under active control for the initial design and optimal design is shown in Figure 5. The figure shows that the maximum vibration amplitude decreases from 0.0066 m for the initial design to 0.0014 m for the optimal design. The norm (amplitude) of the applied actuator voltage is shown in Figure 6, and it can be observed that the actuator voltage is nearly zero in the regions not covered by the piezoelectric material. In this study, element-wise constant voltage distribution is assumed. In practical implementations, the optimized sensor/actuator layouts can be viewed as a guidance to the conceptual design of positions and shapes of the piezoelectric patches and the electrodes can be designed accordingly with manufacturing and electric circuit considerations.

Figure 3.

Iteration histories of objective function value and volume fraction ratio under excitation frequency $f_{p} = 43 Hz$ with the control gain $G_{a} = 50$ .

Figure 4.

Optimal layout of actuator/sensor layers for the cantilever plate: (a) material density contour and (b) optimal distribution of piezoelectric material (with low-density elements hidden).

Figure 5.

Vibration amplitude for the initial and optimal design: (a) initial design and (b) optimal design (with low-density elements hidden).

Figure 6.

Actuator voltage amplitude for optimal design (the left edge is clamped).

The dynamic compliance for the excitation frequency range 28–60 Hz of the initial design with/without control and that of the optimal design with control (obtained under the 43 Hz excitation) are all shown in Figure 7. It is seen that the dynamic compliance of the optimal design with control is much smaller than that of the initial design (with and without control) around the excitation frequency (40–50 Hz). While for the excitation frequency far away from 43 Hz (e.g. 53–60 Hz), the design obtained under 43 Hz excitation is no longer a good design. This just highlights the importance of topology optimization under a given frequency range rather than a specific frequency, as will be further discussed later.

Figure 7.

Comparison of dynamic compliance under different excitation frequencies for the initial design with/without control and the optimal design with control.

Influence of excitation frequency on optimal solutions

First, we study the excitation frequency dependence of the optimal topology. Here, the same structure and boundary conditions as above are considered but under six different excitation frequencies f_p = 1, 20, 43, 50, 70, and 90 Hz. The obtained optimal solutions are shown in Figure 8. It is obvious that as the excitation frequency f_p increases, the optimal layout of the piezoelectric material becomes spatially more complex and exhibits more isolated areas. This is natural since a higher excitation frequency will excite higher order of eigenmodes, which typically has more localized features.

Figure 8.

Optimization results obtained under different excitation frequencies with control gain $G_{a} = 50$ (the left edge is clamped): (a) f_p = 1 Hz, (b) f_p = 20 Hz, (c) f_p = 43 Hz, (d) f_p = 50 Hz, (e) f_p = 70 Hz, and (f) f_p = 90 Hz.

Influence of control gain on optimal solutions

Now we consider the influence of the feedback control gain under a fixed frequency f_p = 20 Hz. Four different cases of the control gain ( $G_{a} = 100, 200, 500, and 1000$ ) are studied and the corresponding optimal topologies are presented in Figure 9(b) to (e). For comparison, the optimal design for a structure without control ( $G_{a} = 0$ ) is also given in Figure 9(a). It is seen that the optimal design changes gradually as the control gain increases. This also reflects the trend that for relatively large control gain, the differences between the optimal topologies under different excitation frequencies are not so obvious as in the case of small control gain (see Figure 9). The structural stiffness plays a more important role than the active damping in the optimization when the control gain is small, while the active damping effect plays a decisive role for the structure with a relatively large control gain. For illustrating this further, topology optimization for the electrode layer only (the laminated structure with fully covered piezoelectric actuator/sensor layers but designable electrode layout) is performed under four different control gains ( $G_{a} = 100, 200, 500, and 1000$ ) and the optimal solutions are shown in Figure 10. It is clear that the control gains have only slight influence on the overall electrode layout since the active damping effect actually plays the decisive role in this case. It can also be observed that the optimal design in Figure 9(e) has similar tendency as those in Figure 10.

Figure 9.

Optimization results for actuator/sensor layers obtained under 20 Hz excitation with different control gains (the left edge is clamped): (a) $G_{a} = 0$ , (b) $G_{a} = 100$ , (c) $G_{a} = 200$ , (d) $G_{a} = 500$ , and (e) $G_{a} = 1000$ .

Figure 10.

Optimization results for the electrode layout only obtained under 20 Hz excitation with different control gains: (a) $G_{a} = 100$ , (b) $G_{a} = 200$ , (c) $G_{a} = 500$ , and (d) $G_{a} = 1000$ .

Topology optimization in a frequency range

Topology optimization of the actuator/sensor layers in an excitation frequency range

We now consider the topology optimization of the actuator layer and the sensor layer attached to a clamped square plate as shown in Figure 11. The host layer of the plate has a geometrical dimension of $a = 3.0 m$ and $t_{h} = 3 \times 1 0^{- 3} m$ ; the thicknesses of both the sensor layer and the actuator layer are $t_{s} = t_{a} = 0.5 \times 1 0^{- 3} m$ . The material properties are the same as in the previous example. The damping coefficients of the host structure are $α = β = 1 \times 10^{- 4}$ . The negative feedback control gain is taken as $G_{a} = 20$ with the gain value of the charge amplifier $G_{c} = 1 \times 10^{5} V / A$ .

Figure 11.

A four-edge clamped laminated plate with active control under a time-harmonic load $f (t) = F e^{i θ t}$ applied at the center.

A harmonic point load $f (t) = F e^{i θ t}$ is applied at the center of the plate, with the force amplitude $F = 200 N$ and the frequency $f_{p} = θ / 2 π$ falling within a frequency range 29–41 Hz. For computing the KS function in equation (26), 13 specified excitation frequencies (f_p = 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, and 41) are considered and the objective function is taken as the aggregated dynamic compliance at these excitation frequencies with the penalty factor η = 10. The design domain is discretized by 60 × 60 uniform-sized four-node Mindlin shell elements with the total number of DOFs $n = 18, 605$ . The volume fraction ratio of the piezoelectric material is restricted by $f_{v} = 0.5$ and the initial values of the design variables are set to be $ρ_{e} = 0.5 (e = 1, 2, \dots, 3600)$ . The first 40 eigenmodes are involved in the computation (i.e. $n_{b} = 40$ ).

The optimization process converged after 45 iterations, and the iteration histories of the objective function/volume constraint are shown in Figure 12. It can be seen that the value of the objective function has reduced steadily from 1.25 to 0.57 N m. The optimal design of the actuator/sensor layer is illustrated in Figure 13. Figure 14 shows that the structural vibration level has been remarkably reduced in the optimal design as compared with that in the initial design. Actually, the maximum value of the average vibration amplitude ( $A_{average} = \sum_{i = 1}^{n_{fp}} A_{f_{p}^{i}} / n_{fp}$ , $n_{fp} = 13$ is the number of excitation frequencies considered) has been decreased from 0.0066 m in the initial design to 0.0015 m in the optimal design. The norm of the average applied actuator voltage in 29–41 Hz is also shown in Figure 15.

Figure 12.

Iteration histories of objective function value and volume fraction ratio with the control gain $G_{a}$ = 20 for the excitation frequency range 29–41 Hz.

Figure 13.

Optimal layout of actuator/sensor layers for the four-edge clamped plate in the excitation frequency range 29–41 Hz: (a) material density contour and (b) optimal distribution of piezoelectric material (with low-density elements hidden).

Figure 14.

Average vibration amplitude for the initial design and the optimal design in the excitation frequency range 29–41 Hz: (a) initial design and (b) optimal design (with low-density elements hidden).

Figure 15.

Average actuator voltage amplitude for optimal design in the excitation frequency range 29–41 Hz.

The dynamic compliance of the initial design with/without control and the optimal design with control (obtained under the excitation frequency range 29–41 Hz) are shown in Figure 16. It is seen that the dynamic compliance in the optimal design with control is smaller than those of the initial design with/without control for most excitation frequencies except around the trough of the curve for the initial design without control. Furthermore, in the optimal design, the fluctuation of the dynamic compliance within the range 29–41 Hz has also been reduced.

Figure 16.

Comparison of dynamic compliance under different excitation frequencies for the initial design with/without control and the optimal design (obtained under excitation frequency range 29–41 Hz) with control.

Comparison of optimal solutions for a single excitation frequency and in a frequency range

In this example, comparisons are made between the topology optimization results obtained under a single excitation frequency and within a frequency range for the same clamped square plate as in the previous example. First, the optimal topologies obtained under 31 and 38 Hz excitation are given in Figure 17. Obvious differences can be observed from Figures 13(a) and 17(a) and (b). For comparing the dynamic properties of these optimal designs, the frequency response sweeps of dynamic compliance for the optimal solution obtained under 31 Hz, 38 Hz, and frequency range 29–41 Hz are shown in Figure 18. Obviously, although the optimal design obtained under a single excitation frequency has the smallest dynamic compliance around that specific frequency, the optimal design achieved with the aggregated objective function has a smaller peak value of the dynamic compliance in the whole frequency range 29–41 Hz, indicating a remarkable reduction of the overall vibration level within the concerned frequency range.

Figure 17.

Topology optimization results of the actuator/sensor layers for the four-edge clamped plate obtained under specific excitation frequencies: (a) f_p = 31 Hz and (b) f_p = 38 Hz.

Figure 18.

Comparison of dynamic compliance under different excitation frequencies for the optimal designs with active control at a single excitation frequency and in a frequency range.

In addition, the optimal designs in different frequency ranges are also considered and shown in Figure 19. In different frequency ranges, different orders of vibration modes play the decisive roles in the corresponding local frequency range. Thus, it is natural that the optimal designs show obvious differences.

Figure 19.

Optimization results with active control obtained in different excitation frequency ranges: (a) 0.1–2.0 Hz, (b) 4–7 Hz, (c) 15–18 Hz, (d) 21–25 Hz, (e) 29–41 Hz, and (f) 46–50 Hz.

Conclusion

This study focuses on the topology optimization of piezoelectric plates with active control under harmonic excitation. The CGVF control algorithm is adopted for the active vibration control. The topology optimization problem is formulated by employing an artificial material model with penalization for both mechanical and piezoelectric properties. The design objective is to minimize the dynamic compliance under a specific excitation frequency or the maximum dynamic compliance within an excitation frequency range. The method of complex mode superposition method in conjunction with model reduction technique is employed for solving the dynamic systems with nonproportional damping. In this context, the sensitivity analysis is derived by using the AVM. Numerical examples confirm the validity of the proposed method. It is seen that the present approach can significantly reduce the structural dynamic compliance under specified excitation frequency as well as the peak value of dynamic compliance within a concerned frequency range.

In our numerical examples, the induced stress of the PZT elements under applied voltage is well below the admissible stress (usually in the order of magnitude of 100 MPa) of the PZT material. For practical engineering problems, the strength constraint needs to be observed in detailed designs.

Obviously, due to the nonconvex nature of the considered topology optimization problems, global optima cannot be guaranteed by a gradient-based optimization algorithm. However, the proposed approach can be employed to provide useful guidance to the conceptual design of active vibration control structures based on piezoelectric effects.

Footnotes

Acknowledgements

The authors would like to thank Prof. Krister Svanberg for providing the source code of the GCMMA algorithm.

Funding

The authors gratefully acknowledge the support of the Key Project of Chinese National Programs for Fundamental Research and Development (grant 2010CB832703) and Natural Science Foundation of China (grant 11072047, 91130025).

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