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.

References

References
1.
Koratkar
,
N. A.
, and
Chopra
,
I.
, 2001, “
Wind Tunnel Testing of a Mach-Scaled Rotor Model With Trailing-Edge Flaps
,”
Smart Mater. Struct.
,
10
, pp.
1
14
.
2.
Lockyer
,
A. J.
,
Martin
,
C. A.
,
Lindner
,
D. K.
, and
Walia
,
P. S.
, 2004, “
Power Systems and Requirements for Integration of Smart Structures Into Aircraft
,”
J. Intell. Mater. Syst. Struct.
,
15
, pp.
305
315
.
3.
Kudva
,
J. N.
, 2004, “
Overview of the DARPA Smart Wing Project
,”
J. Intell. Mater. Syst. Struct.
,
15
, pp.
261
268
.
4.
Rabinovitch
,
O.
, and
Vinson
,
J. R.
, 2003, “
On the Design of Piezoelectric Smart Fins for Flight Vehicles
,”
Smart Mater. Struct.
,
12
, pp.
686
695
.
5.
Bae
,
J. S.
,
Kyong
,
N. H.
,
Seigler
,
T. M.
, and
Inman
,
D. J.
, 2005, “
Aeroelastic Considerations on Shape Control of an Adaptive Wing
,”
J. Intell. Mater. Syst. Struct.
,
16
, pp.
1051
1056
.
6.
Tabata
,
M.
, and
Natori
,
M. C.
, 1996, “
Active Shape Control of a Deployable Space Antenna Reflector
,”
J. Intell. Mater. Syst. Struct.
,
7
(
2
), pp.
235
240
.
7.
Irschik
,
H.
, 2002, “
A Review on Static and Dynamic Shape Control of Structures by Piezoelectric Actuation
,”
Eng. Struct.
,
24
, pp.
5
11
.
8.
Frecker
,
M. I.
, 2003, “
Recent Advances in Optimization of Smart Structures and Actuators
,”
J. Intell. Mater. Syst. Struct.
,
14
, pp.
207
216
.
9.
Liew
,
K. M.
,
He
,
X. Q.
, and
Meguid
,
S. A.
, 2004, “
Optimal Shape Control of Functionally Graded Smart Plates Using Genetic Algorithms
,”
Comput. Mech.
,
33
, pp.
245
253
.
10.
Wang
,
J.
,
Zhao
,
G.
, and
Zhang
,
H.
, 2009, “
Optimal Placement of Piezoelectric Curve Beams in Structural Shape Control
,”
Smart Struct. Syst.
,
5
, pp.
241
260
.
11.
Tong
,
D.
,
Williams
,
R. L.
, and
Agrawal
,
S. K.
, 1998, “
Optimal Shape Control of Composite Thin Plates With Piezoelectric Actuators
,”
J. Intell. Mater. Syst. Struct.
,
9
, pp.
458
467
.
12.
Sun
,
D. C.
, and
Tong
,
L. Y.
, 2005, “
Design Optimization of Piezoelectric Actuator Patterns for Static Shape Control of Smart Plates
,”
Smart Mater. Struct.
,
14
, pp.
1353
1362
.
13.
Mukherjee
,
A.
, and
Joshi
,
S.
, 2002, “
Piezoelectric Sensor and Actuator Spatial Design for Shape Control of Piezolaminated Plates
,”
AIAA J.
,
40
, pp.
1204
1210
.
14.
Luo
,
Q. T.
, and
Tong
,
L. Y.
, 2006, “
High Precision Shape Control of Plates Using Orthotropic Piezoelectric Actuators
,”
Finite Elem. Anal. Design
,
42
, pp.
1009
1020
.
15.
Moita
,
J. M. S.
,
Correia
,
V. M. F.
,
Martins
,
P. G.
,
Soares
,
C. M. M.
, and
Soares
,
C. A. M.
, 2006, “
Optimal Design in Vibration Control of Adaptive Structures Using a Simulated Annealing Algorithm
,”
Compos. Struct.
,
75
, pp.
79
87
.
16.
Eschenauer
,
H. A.
, and
Olhoff
,
N.
, 2001, “
Topology Optimization of Continuum Structures: A Review
,”
Appl. Mech. Rev.
54
(
4
), pp.
331
389
.
17.
Rozvany
,
G.
, 2001, “
Aims, Scope, Method, History and Unified Terminology of Computer-Aided Topology Optimization in Structural Mechanics
,”
Struct. Multidiscip. Optim.
,
21
(
2
), pp.
90
108
.
18.
Bendsoe
,
M. P.
, and
Sigmund
,
O.
, 2003,
Topology Optimization: Theory, Methods and Application
,
Springer
,
Berlin
.
19.
Donoso
,
A.
, and
Sigmund
,
O.
, 2009, “
Optimization of Piezoelectric Bimorph Actuators With Active Damping for Static and Dynamic Loads
,”
Struct. Multidiscip. Optim.
,
38
(
2
), pp.
171
183
.
20.
Drenckhan
,
J.
,
Lumsdaine
,
A.
, and
Parsons
,
M.
, 2008, “
Topology Optimization of a Piezoelectric Actuator on an Elastic Beam
,”
J. Intell. Mater. Syst. Struct.
,
19
, pp.
445
455
.
21.
Kögl
,
M.
, and
Silva
,
E. C. N.
, 2005, “
Topology Optimization of Smart Structures: Design of Piezoelectric Plate and Shell Actuators
,”
Smart Mater. Struct.
,
14
, pp.
387
399
.
22.
Wein
,
F.
,
Kaltenbacher
,
M.
,
Bansch
,
E.
,
Leugering
,
G.
, and
Schury
,
F.
, 2009, “
Topology Optimization of a Piezoelectric-Mechanical Actuator With Single-and Multiple-Frequency Excitation
,”
Int. J. Appl. Electromagn. Mech.
,
30
, pp.
201
221
.
23.
Wang
,
S. Y.
,
Tai
,
K.
, and
Quek
,
S. T.
, 2006, “
Topology Optimization of Piezoelectric Sensors/Actuators for Torsional Vibration Control of Composite Plates
,”
Smart Mater. Struct.
,
15
, pp.
253
269
.
24.
Luo
,
Z.
,
Tong
,
L.
,
Luo
,
J.
,
Wei
,
P.
, and
Wang
,
M. Y.
, 2009, “
Design of Piezoelectric Actuators Using a Multiphase Level Set Method of Piecewise Constants
,”
J. Comput. Phys.
,
228
(
7
), pp.
2643
2659
.
25.
Bharti
,
S.
, and
Frecker
,
M.
, 2007, “
Compliant Mechanical Amplifier Design Using Multiple Optimally Placed Actuators
,”
J. Intell. Mater. Syst. Struct.
,
18
(
3
), pp.
209
218
.
26.
Nakasone
,
P. H.
, and
Silva
,
E. C. N.
, 2010, “
Dynamic Design of Piezoelectric Laminated Sensors and Actuators Using Topology Optimization
,”
J. Intell. Mater. Syst. Struct.
,
21
, pp.
1627
1652
.
27.
Zheng
,
B.
,
Chang
,
C.
, and
Gea
,
H.
, 2009, “
Topology Optimization of Energy Harvesting Devices Using Piezoelectric Materials
,”
Struct. Multidiscip. Optim.
,
38
, pp.
17
23
.
28.
Rupp
,
C. J.
,
Evgrafov
,
A.
,
Maute
,
K.
, and
Dunn
,
M. L.
, 2009, “
Design of Piezoelectric Energy Harvesting Systems: A Topology Optimization Approach Based on Multilayer Plates and Shells
,”
J. Intell. Mater. Syst. Struct.
,
20
(
16
), pp.
1923
1939
.
29.
Rupp
,
C. J.
,
Dunn
,
M. L.
, and
Maute
,
K.
, 2010, “
Switchable Phononic Wave Filtering, Guiding, Harvesting, and Actuating in Polarization-Patterned Piezoelectric Solids
,”
Appl. Phys. Lett.
,
96
(
11
), p.
111902
.
30.
Gao
,
F.
,
Shen
,
Y. P.
, and
Li
,
L. X.
, 2000, “
The Optimal Design of Piezoelectric Actuators for Plate Vibroacoustic Control Using Genetic Algorithms With Immune Diversity
,”
Smart Mater. Struct.
,
9
, pp.
485
491
.
31.
Li
,
L. X.
,
Shen
,
Y. P.
, and
Gao
,
F.
, 2001, “
The Optimal Design of Piezoelectric Actuators for Acoustic Control
,”
Smart Mater. Struct.
,
10
, pp.
421
426
.
32.
Zhu
,
Y.
,
Qiu
,
J.
,
Du
,
H.
, and
Tani
,
J.
, 2002, “
Simultaneous Optimal Design of Structural Topology, Actuator Locations and Control Parameters for a Plate Structure
,”
Comput. Mech.
,
29
, pp.
89
97
.
33.
Quan
,
N.
, and
Tong
,
L.
, 2007, “
Voltage and Evolutionary Piezoelectric Actuator Design Optimization for Static Shape Control of Smart Plate Structures
,”
Mater. Des.
,
28
, pp.
387
399
.
34.
Liu
,
S.
,
Tong
,
L.
, and
Lin
,
Z.
, 2008, “
Simultaneous Optimization of Control Parameters and Configurations of PZT Actuators for Morphing Structural Shapes
,”
Finite Elem. Anal. Design
,
44
, pp.
417
424
.
35.
Kang
,
Z.
, and
Tong
,
L.
, 2008, “
Integrated Optimization of Material Layout and Control Voltage for Piezoelectric Laminated Plates
,”
J. Intell. Mater. Syst. Struct.
,
19
(
8
), pp.
889
904
.
36.
Kang
,
Z.
, and
Tong
,
L.
, 2008, “
Topology Optimization-Based Distribution Design of Actuation Voltage in Static Shape Control of Plates
,”
Comput. Struct.
,
86
(
19–20
), pp.
1885
1893
.
37.
Diaz
,
A.
, and
Sigmund
,
O.
, 1995, “
Checkerboard Patterns in Layout Optimization
,”
Struct. Optim.
,
10
, pp.
40
45
.
38.
Kögl
,
M.
, and
Bucalem
,
M. L.
, 2005, “
A Family of Piezoelectric MITC Plate Elements
,”
Comput. Struct.
,
83
, pp.
1277
1297
.
39.
Svanberg
,
K.
, 1987, “
The Method of Moving Asymptotes—A New Method for Structural Optimization
,”
Int. J. Numer. Methods Eng.
,
24
, pp.
359
373
.
40.
Sigmund
,
O.
, 2001, “
A 99 Line Topology Optimization Code Written in MATLAB
,”
Struct. Multidiscip. Optim.
,
21
, pp.
120
127
.
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