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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
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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)
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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
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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
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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
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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
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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.
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8
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