Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2010 August 15-18, 2010, Montreal, Quebec, Canada DETC2010-28856 TOPOLOGY OPTIMIZATION OF PIEZOELECTRIC ENERGY HARVESTING DEVICES SUBJECTED TO STOCHASTIC EXCITATION Zheqi Lin * Hae Chang Gea Shutian Liu * Department of Mechanical and Aerospace Engineering Rutgers, The State University of New Jersey Piscataway, NJ 08854 * State Key Laboratory of Structural Analysis of Industrial Equipment Dalian University of Technology, Dalian, 116024, China ABSTRACT Converting ambient vibration energy into electrical energy using piezoelectric energy harvester has attracted much interest in the past decades. In this paper, topology optimization is applied to design the optimal layout of the piezoelectric energy harvesting devices. The objective function is defined as to maximize the energy harvesting performance over a range of ambient vibration frequencies. Pseudo excitation method (PEM) is applied to analyze structural stationary random responses. Sensitivity analysis is derived by the adjoint method. Numerical examples are presented to demonstrate the validity of the proposed approach. Keywords topology optimization; material energy harvesting; 1 piezoelectric INTRODUCTION The search of everlasting energy sources for micro devices has been intensified in the past years due to the massive applications of micro devices in various fields and the short lifespan of the conventional electrochemical power sources. Furthermore, a completely autonomous energy source is particularly advantageous in low power micro-systems with restricted accessibility, such as biomedical implants, remote micro-sensors and wireless devices. Therefore, the ultimate Please send all correspondence to this author: Hae Chang Gea, Email: gea@rci.rutgers.edu renewable energy source for micro devices should equip with an energy-harvesting mechanism that continually to replenish the energy consumed. Although there are many different approaches to converting the ambient energy into usable energy, such as thermoelectrics, photovoltaics and piezoelectrics, the drawbacks of low thermal gradient for thermoelectrics, and small surface area for photovoltaics on a micro device leave the piezoelectric energy-harvesting an attractive option due to their high-energy densities and relatively small form-factors. To this end, converting ambient vibration energy into electrical energy using piezoelectric energy harvester has attracted much interest. Many theoretical and experimental works are available on modeling and applications of piezoelectric energy harvesters [1-4]. An overview of research in this field has been given by Anton and Sodano [5]. Piezoelectric materials can be configured in different ways to improve the efficiency of piezoelectric power harvesting devices. Better geometric designs may lead to a better utilization of the piezoelectric materials. Goldschmidtboeing and Woias [6] studied a family of beam shapes ranging from rectangular beams to triangular beams in terms of efficiency and maximum tolerable excitation amplitude. They showed that triangular-shaped beams are more effective than rectangularshaped ones. A parametric study was performed by Mo et al. [7] to investigate the effects of various configurations on the produced energy of the piezoelectric generator. Lee et al. [8] proposed a stochastic design optimization to determine the 1 Copyright © 2010 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use optimal configuration of the energy harvest in terms of energy efficiency and durability. In contrast to the traditional size and shape design optimization, topology optimization method is a powerful design tool to determine the optimal design layout. There have been a few applications of topology optimization method for the design of piezoelectric devices, such as piezoelectric transducers [9], piezoelectric actuators [10-12] and piezoelectric renators [13]. Most recently, topology optimization method is applied to design piezoelectric energy harvesting devices. Zheng et al. [14] proposed a topological optimum design to maximize energy conversion of the energy harvesting devices. Topology optimization formulations are developed by Nakasone and Silva [15] to maximize energy conversion of the piezoelectric device and control resonance frequencies of the structure. Design of piezoelectric harvesting devices under harmonic excitation are studied by Rupp et al. [16] In most of the work reported, the vibration source is typically represented by a single harmonic signal. Consequently, the energy harvester obtained is tailored to operate only at a single resonant frequency of the driving source. However, the ambient vibration sources where the energy harvester is deployed to are always in a range of frequencies and the performance of the single-frequency energy harvesters will suffer greatly in such an environment. To address this performance issue, a topology optimization based approach is proposed to determine the optimal configurations of the piezoelectric energy harvesting devices which maximizes the energy conversion efficiency from a specific vibration input within a defined range of frequency in this paper. The Pseudo Excitation Method (PEM) developed by Lin [17] is used for structural stationary random response analysis and the adjoint method is used to evaluate the sensitivity analysis. 2 PROBLEM FORMULATION In this section, problem formulation for the design of piezoelectric energy harvesting devices under ambient random vibrations is presented. First, a finite element model with coupled mechanical and electrical fields is introduced. Then, the optimization formulation of maximizing the energy harvesting performance under a prescribed frequency range is presented. 2.1 FINITE ELEMENT MODELING Applications of piezoelectric materials are always involved in both mechanical and electric fields. A coupled mechanical and electrical finite element formulation is used to model piezoelectric energy harvesting devices. A schematic diagram of a piezoelectric energy harvesting device is shown in Figure 1. Figure 1. Schematic diagram of piezoelectric energy harvesters In the finite element formulation, piezoelectric systems under external excitations and charges can be modeled as: ⎡ M uu ⎢ 0 ⎣ 0 ⎤ ⎡ u⎤ ⎡ K uu +⎢ 0 ⎥⎦ ⎢⎣φ⎥⎦ ⎣ K φ u K uφ ⎤ ⎡u ⎤ ⎡ F ⎤ = K φφ ⎥⎦ ⎢⎣φ ⎥⎦ ⎢⎣Q ⎥⎦ (1) where Kφ u = (K uφ )t represents the piezoelectric coupling matrix, K uu and Kφφ denote the structural stiffness and dielectric conductivity matrices, M uu is the structural mass matrix. Since topology optimization method is used in the design of energy harvesting devices, a material model such as the homogenization method or the Solid Isotropic Materials with Penalization (SIMP) model, should be defined first. For simplicity of derivations, the SIMP model used to describe the material properties in the design domain as: c = ρ ( x) p1 c0 , m = ρ ( x)m0 , e = ρ ( x) e0 , ε = ρ ( x) p ε 0 p2 (2) 2 where c , m , e and ε denote the stiffness, mass, piezoelectric and dielectric properties, respectively; ρ ( x) is the volume density of each element and 0 ≤ ρ ( x ) ≤ 1 ; p1 and p2 are penalty coefficients to the material, and they are set as 3 and 1 in the study, respectively. 2.2 ENERGY HARVESTING OPTIMIZATION To measure the performance of an energy harvester under random vibration as shown in Figure 1, we define an average harvested power in the time domain as: ⎡ v 2 (t ) ⎤ E [ P (t ) ] = E ⎢ ⎥ ⎣ R ⎦ (3) where, E [•] is the expectation operator; P (t ) is the harvested power; v(t ) is the voltage; R is resistance. For a prescribed frequency range, f1 ≤ f ≤ f 2 , the average power in the frequency domain can be given as, E [ P(t )] = 2 E ⎡⎣v 2 (t ) ⎤⎦ R = 1 ω2 Svv (ω )d ω R ∫ω1 (4) Copyright © 2010 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use where, ω1 = 2π f1 , ω2 = 2π f 2 , Svv (ω ) represents the Power Spectral Density (PSD) of the output voltage. Thus, we can formulate the topology optimization problem as to maximize the average harvested power within the prescribed frequency range with a given amount of piezoelectric materials as follows: 1 ω2 E [ P(t )] = ∫ Svv (ω )d ω R ω1 Maximize ρ N ∑ ρi vi ≤ V Subject to (5) If the pseudo responses are two arbitrary harmonic response vectors, { y ( t )} and { z ( t )} , it can be similarly verified that the corresponding PSD matrices would be { y}*{ y}T = [ S yy (ω )] (9) { y}*{z}T = [ S yz (ω )] (10) Now, consider a structure subjected to a single random excitation, its equation of motion can be expressed as: i =1 0 ≤ ρi ≤ 1, i = 1,..., N [ M ]{ y} + [C ]{ y} + [ K ]{ y} = { p}x(t ) where vi is the volume of the i th element. V is the upper bound of the piezoelectric materials. PSEUDO EXCITATION METHOD The PEM for stationary single excitation problem can be explained by means of Figure 2. A linear system is subjected to a zero-mean-valued stationary random excitation whose Power Spectral Density (PSD), S xx (ω ) , has been specified. If y = S xx (ω ) H1 (ω )eiωt and z = S xx (ω ) H 2 (ω )eiωt are two arbitrary stationary harmonic responses due to the pseudo harmonic excitation (11) where x ( t ) is a stationary random process and { p} is a given constant vector. Set x(t ) = S xx (ω )eiωt , then Eq. (11) becomes a simple harmonic equation: 3 [ M ]{ y} + [C ]{ y} + [ K ]{ y} = { p} S xx (ω )eiωt (12) The stationary solution can obtained as: { y (t )} = {Y (ω )}eiωt (13) Using the first q modes for mode-superposition, we have, x = S xx (ω )eiωt (6) q {Y (ω )} = ∑ γ j H j {φ j } S xx (ω ) (14) (7) H j = (ω 2j − ω 2 + 2iς j ωω j ) , γ j = {φ j }T { p} (15) (8) where ω j , {φ j } , ς j , γ j are the j th natural angular frequency, mode, damping ratio and mode-participation factor, respectively. The PSD matrix of { y} can be written as: then the following expressions can be readily verified that y * y = S xx (ω ) H1* (ω )eiωt ⋅ S xx (ω ) H1 (ω )eiωt 2 = H1 (ω ) S xx (ω ) = S yy (ω ) y * z = S xx (ω ) H1* (ω )eiωt ⋅ S xx (ω ) H 2 (ω )eiωt = H1* (ω ) S xx (ω ) H 2 (ω ) = S yz (ω ) j =1 −1 where the asterisk * represents complex conjugate. Therefore, the auto- and cross-PSD functions of the two random responses y ( t ) and z ( t ) can be computed using the corresponding pseudo harmonic responses y and z . [ S yy (ω )] = { y}*{ y}T = {Y (ω )}*{Y (ω )}T (16) Substituting Eq. (14) into Eq. (16) and expanding it gives q q [ S yy (ω )] = ∑∑ γ j γ k H *j H k {φ j }{φk }T S xx (ω ) (17) j =1 k =1 Figure 2. Pseudo excitation and responses for an SIMO system This is well-known CQC formula [17-18]. Since Eq. (17) involves the cross-correlation terms between all participant modes, it requires considerable computational efforts. The PEM algorithm based on Eqs. (14) and (16) is used because only one vector multiplication operation is needed to achieve the same precision as Eq. (17). The comparisons of efficiency for PEM and conventional CQC can be found in Lin et al. [19] 3 Copyright © 2010 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use 3.1 PEM FOR COUPLE PIEZOELECTRIC SYSTEMS For a piezoelectric structure subjected to acceleration excitations, x , with a power spectrum density S x (ω ) and the auto-PSD function of voltage Svv (ω ) can be obtained by using the PEM. First, we can define the pseudo acceleration excitation using power spectrum density S x (ω ) as: x = S x (ω )eiωt (18) and the external force has a harmonic form of F = [M uu ]{E} S x (ω )eiωt , where {E} denotes the direction of the acceleration excitation. The structural responses can then be expressed as u = ue eiωt and φ = φe eiωt , where ω is the excitation frequency. Introducing these expressions to Eq. (1) yields the frequency response equation as: ⎡ −ω 2 M uu ⎢ ⎣ 0 0 ⎤ ⎡ue ⎤ ⎡ K uu ⎥⎢ ⎥+⎢ 0 ⎦ ⎣φe ⎦ ⎣ K φ u K uφ ⎤ ⎡ue ⎤ ⎡ F0 ⎤ = K φφ ⎦⎥ ⎢⎣φe ⎥⎦ ⎢⎣ 0 ⎥⎦ (19) ∂Svv ∂φ = 2φ ∂xi ∂xi where φ is the potential response when a pseudo excitation is applied to the structure. To derive the sensitivity of potential response, ∂φ ∂xi , the adjoint method is applied. A new function, φ # , is defined as the following: φ # = ψ Tφ + α T (−ω 2 M uu ue + K uu ue + K uφ φe − M uu E S x (ω )) Svv = φ φ = φv 2 (22) + β T (K φ u ue + K φφ φe ) where ψ is a unit vector, and α , β are two arbitrary adjoint displacement and potential vectors. Taking the first derivative of Eq. (22), the sensitivity of the potential response with respect to design variable xi can be written as: ∂M uu ∂K uu ∂φ # ue + ue = α T (−ω 2 ∂xi ∂xi ∂xi The response of voltage can be solved from the above system by using the model superposition method and the auto-PSD function of voltage can be obtained according to pseudo excitation method as: * v v (21) + ∂K uφ ∂xi +β T ( (20) φe − ∂K φ u ∂xi ∂M uu E S x (ω )) ∂xi ue + ∂K φφ ∂xi φe ) +(−α Tω 2 M uu + α T K uu + β T K uφ ) 4 SENSITIVITY ANALYSIS The sensitivity of Svv respect to design variable can be obtained by taking the first derivative of Eq. (20) with respect to xi as: +(α T K uφ + β T K φφ + ψ ) (23) ∂ue ∂xi ∂φ ∂xi Table 1. Material Properties of PZT C11 =115.65GPa e13=-12.31 C/m2 ε11 =8.93×10-9 F/m C12 =64.89 GPa e23=-12.31 C/m2 ε 22 =8.93×10-9 F/m C13 =62.29 GPa e33= 20.76 C/m2 ε 33 =6.92×10-9 F/m C33 =92.98 GPa e52= 17.04 C/m2 ρ =7640 kg/m3 C44 =17.86 GPa e61= 17.04 C/m2 C55 =17.86 GPa C66 =17.86 GPa 4 Copyright © 2010 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use By defining and solving the adjoint system as the following coupled system for the adjoint displacement and potential vectors, α and β , ⎡ −ω 2 M uu ⎢ ⎣ 0 0 ⎤ ⎡α ⎤ ⎡ K uu ⎥⎢ ⎥+⎢ 0⎦ ⎣ β ⎦ ⎣K φu K uφ ⎤ ⎡α ⎤ ⎡ 0 ⎤ = K φφ ⎦⎥ ⎢⎣ β ⎥⎦ ⎢⎣ −ψ ⎥⎦ (24) a very simple form of the sensitivity of the potential response can be obtained from Eq. (23) and it can be readily solved with the SIMP formulation. Therefore, adding piezoelectric material to the supporting area will increase the natural frequency of the structure and decrease the deformation. Furthermore, since the excitation frequencies are lower, the piezoelectric materials tend to distribute at the free end to decrease the natural frequency. At the same time, the piezoelectric materials placed at the free end also increase the moment induced by the gravity, which help to produce more electric energy. 5 NUMERICAL EXAMPLE Optimal designs of piezoelectric energy harvesters subjected to random acceleration excitations with a spectral density function using the proposed topology optimization method are presented in this section. PZT is employed for the piezoelectric layers considered as design domain and its material properties are given in Table 1. The elastic layer is made of Aluminum with Young’s modulus 71 GPa, Poisson’s ratio 0.33, density 2700 kg/m3. In the finite element model, 8-node brick element is used and equal-potential constraints are applied to the nodes on the electrodes. The configurations of the structures at the top and bottom piezoelectric layers are assumed to be same in the optimization process. The frequency spectrum of the acceleration excitation applied to the structure is constant in the frequency ranges of interest. Objective function Figure 3. Schematic example of a piezoelectric bimorph generator mounted as a cantilever 5.1 EXAMPLE 1: In the first example, a ground acceleration excitation is applied to the piezoelectric bimorph generator, as shown in Figure 3. This model is modified from the piezoelectric bimorph generator by Roundy et al. [20]. The left end of the structure is fixed and the right end is free. A proof mass which has the same weight with the metal layer is mounted at the middle of the free end. A band of excitation frequencies ranged from 10 to 400 Hz which is lower than the first order natural frequency of the structure is applied to the piezoelectric bimorph generator. The external load is set as 10 kΩ . The piezoelectric layers are considered as design domain of 20×10×0.25 mm3. The upper bound of volume constraint is set to be 40 % of the total designable volume. A 40 by 20 finite element model is constructed for each layer to design the piezoelectric energy harvester. The iteration history is showed in Figure 4 and the optimal layout is showed in Figure 5. The piezoelectric material tends to distribute to the supporting area and the free end. Intermediate design variable appear in the supporting area. As shown in Figure 6, piezoelectric materials should be placed at the supporting area to produce more electric energy because the strains on these areas are higher. However, these additional piezoelectric materials also increase the structural stiffness and mass of the structure thereby changing the structural dynamic response. 20.0 19.0 18.0 17.0 16.0 15.0 14.0 0 20 40 60 80 Iteration Figure 4. Iteration history of the optimization design of the piezoelectric generator Figure 5. Optimized piezoelectric bimorph generator under a band of excitation frequencies ranged from 10 to 400 Hz 5 Copyright © 2010 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Objective function 40 30 20 10 0 0 20 40 60 80 100 Iteration Figure 6. Stain distribution of the structure without piezoelectric layer under gravity load Figure 7. Iteration history of the optimization design of the piezoelectric generator 5.2 EXAMPLE 2: In the second example, a band of higher excitation frequencies excitation is applied to the same piezoelectric bimorph generator as case 1. However, the frequencies of external excitations are chosen from 800 to 1000 Hz, which is higher than the first order natural frequency of the structure. The iteration history is showed in Figure 7 and the optimal layout is showed in Figure 8. In this example, a U-shape distribution of piezoelectric materials is generated, which is similar to part of the previous design in the same area. However, no piezoelectric material is placed at the free end. This is because the excitation frequencies are higher in this example and the optimized design should increase its natural frequency by moving piezoelectric materials to the support area instead of the free end. As mentioned in the previous example, the strain is greater at the support area, so the piezoelectric materials distributed here is also helpful to generate more electric energy. Figure 8. Optimized piezoelectric bimorph generator under a band of excitation frequencies ranged from 800 to 1000 Hz 5.3 EXAMPLE 3: An energy generator model modified from Roundy et al. [21] is taken as the third example. The piezoelectric bimorph is simply supported at four corners and the proof mass is mounted at the center of the structure as shown in Figure 9. A ground acceleration excitation is applied to the piezoelectric bimorph generator with a band of excitation frequencies ranged from 10 to 400 Hz. The iteration history of the objective function is shown in Figure 10, and the final configuration for the simply supported piezoelectric generator is showed in Figure 11. Figure 12 plots the strain distribution of the simply supported plate without piezoelectric layer under gravity load. Compare Figure 11 with Figure 12, one can easily found that the piezoelectric materials tend to be placed at the area with higher strain. Objective function Figure 9. Schematic example of a piezoelectric bimorph generator simply supported 2.5 2 1.5 0 10 20 30 40 50 60 Iteration Figure 10. Iteration history of the optimization design of the piezoelectric generator 6 Copyright © 2010 by ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Figure 11. Optimized piezoelectric bimorph generator simply supported Figure 12. Strain distribution of the simply supported plate without piezoelectric layer under gravity load 6 CONCLUSION In this paper, design of piezoelectric energy harvesting devices under stochastic excitation is studied using a topology optimization method. The PEM is used to analyze structural stationary random responses. The energy harvester obtained is designed to maximize energy conversion efficiency within a range of ambient excitations. The adjoint method is derived to evaluate the sensitivity analysis. A few numerical examples are given to demonstrate the effectiveness of the proposed method. REFERENCES [1] S. Roundy, P. K. Wright, and J. Rabaey, "A Study of Low Level Vibrations as a Power Source for Wireless Sensor Nodes," Computer Communications, vol. 26, pp. 11311144, 2003. [2] F. Lu, H. P. Lee, and S. P. 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