research article integrated topology optimization of...
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Research ArticleIntegrated Topology Optimization ofStructureVibration Control for Piezoelectric CylindricalShell Based on the Genetic Algorithm
Jianjun He and Xiangzi Chen
School of Automobile and Mechanical Engineering Changsha University of Science and Technology Changsha 410004 China
Correspondence should be addressed to Jianjun He hezhengde8163com
Received 26 November 2014 Revised 29 March 2015 Accepted 10 April 2015
Academic Editor Quoc Hung Nguyen
Copyright copy 2015 J He and X Chen This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A hybrid optimization strategy for integrated topological optimization design of piezoelectric cylindrical flat shell structureis proposed The method combines the genetic algorithm (GA) and linear-quadratic-regulator (LQR) theory to optimize theperformance of coupling structurecontrol system The GA is used to choose the optimal structure topology and numberand placements of actuators and control parameters meanwhile the LQR is used to design control system to suppressvibration of optimal structure under sinusoidal excitation which is based on the couple-mode space control In addition themathematical morphology operators are used for repairs of disconnected structure topology The results of numerical simulationand computations show that the proposed method is effective and feasible with good performance for the optimal and couplingpiezoelectric cylindrical shell structurecontrol system
1 Introduction
The cylindrical shell structure is widely used as main partsof the large structure design in many fields such as civilengineering and architecture marine and aerospace Forinstance the roofs of many stadiums and solid fuel tank ofrocket are this type of structure due to their excellentmechan-ical properties especially their high specific stiffness andstrength [1] However the vibrations of those structuresunder complex and extreme dynamic loads are inevitableas they are excited by strong wind and wave So it is alsonecessary to design control system to suppress the amplitudeof vibration for those structures in engineering practice [2 3]
The optimization design can effectively improve variousperformances of structure for example it can make theoptimal structure more lightweight and controllable In par-ticular the topology optimization design has huge potentialof saving material in structure design which is verifiedby many applications of this technique in aerospace andautomobile industry For example the application of topologyoptimization in the design of A380 leading edge ribs ledto the forty-five percentage weight reduction compared to
traditional design and so forth At present smart or adaptivestructures are widely used in industrial or engineering fieldsso engineers can monitor and control the normal work ofstructures in real-time However because of the complexityof the adaptive structure the structure and control system iscoupled For instance the vibration controllerrsquos coefficientsare strongly dependent on the topology of structure theplacements and number of the actuatorssensors whichdetermine the effectiveness of structural vibration controlTherefore researching the structure and control of integratedoptimization strategies for smart structures is theoreticallyand practically significant It can simultaneously realize theoptimal design of structure and control system and simplifydesign links of huge and complex structurecontrol inte-grated system So the integrated optimization design greatlyimproves the efficiency of traditional sequential optimizationdesign (namely the designer firstly optimizes the structureparameters and then optimizes the controller parameters)
Presently for the active control of truss or plate andshell structures piezoelectric patches and stacks or piezo-electric ceramic transducer (PZT) is used as one of the mainactuators For the optimization strategy the GA is the main
Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 456147 10 pageshttpdxdoiorg1011552015456147
2 Shock and Vibration
optimization algorithm There have been some works pub-lished on intelligent piezoelectric structure integrated opti-mization For instance Li and Huang [4] combined theGA and the linear quadratic Gaussian (LQG) studied theintegrated optimization of actuator placement and vibrationcontrol for piezoelectric adaptive trusses and verified that theresult of experiment is consistent with numerical results Zhuet al [5] investigated simultaneous optimization with respectto the structural topology actuator locations and controlparameters of an actively controlled plate structure its resultsshowed that it can produce systems with clear structuraltopology and high control performance by the proposedapproach Kang and Tong [6] investigated the integratedoptimization of structural topology and control voltage ofpiezoelectric laminated plates Xu et al [7] discussed someissues associated with integrated optimization of structuraltopology the number and placement of actuators and controlparameters for piezoelectric smart trusses some optimalstrategies based on genetic algorithms were adopted in theirstudy Xu et al [8] studied the integrated optimizationof structural topology and number and positions of theactuators and control parameters for piezoelectric smartplates which is based on the optimal control effect in theindependentmode control and singular value decompositionof the distributed matrix of total performance index for allphysical control forces with the GA as optimization strategyHowever almost all the efforts described in the literaturesinvolve the integrated optimization design of piezoelectrictruss or plate not including shell structure In generalit is more complicated to construct the piezoelectric shellfinite element for mathematical modeling and finite elementanalysis (FEA) for shell structure Shell structure usually hasmore DOF and more complicated structural characteristicandmechanical behavior than truss and plate structure Shellstructure hasmore complicated geometric shape which leadsto its greater difficulties for modeling and analysis If the shellstructure is in vibration its vibration suppression techniqueis usually invalid andmore complicated than that of truss andplate structure Thus the main concern of this paper is toextend the integrated topology optimization design into shellstructure
In the present paper the integrated topology optimizationof structurecontrol for piezoelectric cylindrical flat shell isstudied Firstly the piezoelectric coupling finite element (FE)model was built Then the integrated optimization modelingincluding the design variables the objective function andthe constraint functions was proposed and the LQR wasused for the design of control system in coupling modalspace In addition in order to ensure the realization of inte-grated optimization the GA and mathematical morphologytreatment was applied to solution of the above problems Toverify the validity of the integrated optimization design ofshell structure and promote engineering applications of thisapproach the numerical simulations and computations wereconductedThe results show that the integrated optimizationsof piezoelectric cylindrical shell structure are successful andfeasible
t1
t2
t3
z
Actuator layer
Base layer
Sensor layer
Figure 1 Four-node piezoelectric coupling rectangular flat shell ele-ment
x
y
z
x998400
y998400
z998400
Figure 2 The coordinate system of cylindrical flat shell
2 The Piezoelectric Coupling Dynamic Modelof Cylindrical Flat Shell Structure
In general it is more complicated to construct the piezo-electric shell finite element for mathematical modeling andfinite element analysis (FEA) of shell structure Shell structureusually has more DOF and more complicated structuralcharacteristic and mechanical behavior than truss and platestructure For simplification we utilized the four-node rect-angular flat shell element with 24 displacement degrees offreedom (DOF) and 2 electrical DOF to build the piezoelec-tric coupling dynamic model of cylindrical shell structure [38] The ideal element is shown in Figure 1 The piezoelectricflat shell element is composed of middle isotropic baselayer with upper and lower surfaces of base layer that arecompletely covered by PZT patches where 119905
1and 1199052sim3
are thethickness of base layer and piezoelectric patches respectivelyThe approximate cylindrical flat shell structure is shown inFigure 2
In finite element derivation the node displacement vectorof flat shell structural element is defined as
d1015840119890119894= (1199061015840
119894V1015840119894
1199081015840
1198941205791015840
1199091198941205791015840
1199101198941205791015840
119911119894)T
119894 = 1 2 3 4 (1)
where 119894 is denoted by the four nodes of the flat shell structuralelement
Shock and Vibration 3
Assume that the element stiffness matrix correspondingto the plane stress state is expressed as
k119898119890
=
[[[[[[[
[
k11989811
k11989812
k11989813
k11989814
k11989821
k11989822
k11989823
k11989824
k11989831
k11989832
k11989833
k11989834
k11989841
k11989842
k11989843
k11989844
]]]]]]]
]
node 1
node 2
node 3
node 4
(2)
where superscript ldquo119898rdquo is corresponding to the stiffnessmatrixof plane stress k119898
119894119895is denoted by 2 times 2 submatrix
Similarly assume that the element stiffness matrix corre-sponding to the bending stress state is expressed as
k119887119890=
[[[[[[[[
[
k11988711
k11988712
k11988713
k11988714
k11988721
k11988722
k11988723
k11988724
k11988731
k11988732
k11988733
k11988734
k11988741
k11988742
k11988743
k11988744
]]]]]]]]
]
node 1
node 2
node 3
node 4
(3)
where superscript ldquo119887rdquo is corresponding to the stiffness matrixof bending stress k119887
119894119895is denoted by 3 times 3 submatrix
By combining with (2) and (3) the element stiffnessmatrix of flat shell structural element in local coordinatesystem is obtained as
k1015840119890
=
[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
[
k11989811
0 0 k11989812
0 0 k11989813
0 0 k11989814
0 0
0 k11988711
0 0 k11988712
0 0 k11988713
0 0 k11988714
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989821
0 0 k11989822
0 0 k11989823
0 0 k11989824
0 0
0 k11988721
0 0 k11988722
0 0 k11988723
0 0 k11988724
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989831
0 0 k11989832
0 0 k11989833
0 0 k11989834
0 0
0 k11988731
0 0 k11988732
0 0 k11988733
0 0 k11988734
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989841
0 0 k11989842
0 0 k11989843
0 0 k11989844
0 0
0 k11988741
0 0 k11988742
0 0 k11988743
0 0 k11988744
0
0 0 0 0 0 0 0 0 0 0 0 0
]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
]
(4)
where k1015840119890is derived in local coordinate system which is a 24times
24matrixBy eliminating the singularity of the whole stiffness
matrix of shell structure we should transform all the localelement stiffness matrices into global coordinate system bycoordinate transformation
The transformation matrix is defined as follows
Τ =
[[[[[[[[[[[[[[[[[[[[
[
120593 0 0 0 0 0 0 0
0 120593 0 0 0 0 0 0
0 0 120593 0 0 0 0 0
0 0 0 120593 0 0 0 0
0 0 0 0 120593 0 0 0
0 0 0 0 0 120593 0 0
0 0 0 0 0 0 120593 0
0 0 0 0 0 0 0 120593
]]]]]]]]]]]]]]]]]]]]
]
120593 =
[[[[
[
cos (1199091015840 119909) cos (1199091015840 119910) cos (1199091015840 119911)
cos (1199101015840 119909) cos (1199101015840 119910) cos (1199101015840 119911)
cos (1199111015840 119909) cos (1199111015840 119910) cos (1199111015840 119911)
]]]]
]
(5)
where f is denoted by direction cosine matrix of local coordi-nate system versus global coordinate system
After coordinate transformation by (6) the local elementstiffness matrix is transformed into global element stiffnessmatrix
k119890= Τ
Tk1015840119890ΤT (6)
Finally we can get the whole stiffness matrix in globalcoordinate system through elements assembly process
By using the Hamilton principle we can obtain theformula as follows
120575int
1199052
1199051
(119879119890minus 119880119890+ 119882119890) 119889119905 = 0 (7)
where119879119890119880119890 and119882119890 are the kinetic energy potential energy
and external work of the piezoelectric coupling flat shellelement shown in Figure 1 respectively
Substituting the kinetic energy potential energy andexternal work of the piezoelectric coupling flat shell elementinto (7) and calculating the variation of (7) the piezoelectriccoupling equation is obtained as follows
M119890 q119890 + (Κ119890
119901+ Κ119890
119898) q119890 minus Κ119890
119898119886119881119890
119886119911+ Κ119890
119898119904119881119890
119904119911
= 119865119890
119891
(8)
Κ119890
119886119898q119890 + Κ119890
119886119886119881119890
119886119911= minus∬
119860119890
120590119890
119886119889119909 119889119910 (9)
Κ119890
119904119898q119890 + Κ119890
119904119904119881119890
119904119911= 0 (10)
M119890 = sum
ℎ=119898119904119886
int
119881119890
ℎ
120588ℎ [N]
T[N] 119889119881 (11)
Κ119890
119898= int
119881119890
119898
1199112[B119906]T[D119898] [B119906] 119889119881 (12)
4 Shock and Vibration
Κ119890
119901= int
119881119890
119904+119881119890
119886
1199112[B119906]T[S119864] [B
119906] 119889119881 (13)
Κ119890
119898119886= (Κ119890
119886119898)T= int
119881119890
119886
119911 [B119906]T[e]T
1199052
119889119881 (14)
Κ119890
119898119904= (Κ119890
119904119898)T= int
119881119890
119904
119911 [B119906]T[e]T
1199052
119889119881 (15)
Κ119890
119886119886= Κ119890
119904119904= int
119881119890
119886
120576119864
33
1199052
2
119889119881 (16)
F119890119891 = ∬
119860119890
NTf 119889119909 119889119910 (17)
where q119890 is denoted by node displacement vector of thepiezoelectric coupling flat shell element and the meanings ofthe other symbols are similar to the utilizations of these in theliterature [3 9]
By reducing the 119881119890
119904119911and 119881
119890
119886119911terms in (8)ndash(10) the
piezoelectric coupling finite element equation is obtained asfollows
M119890 q119890 + Κ119890 q119890 = F119890119891 minus F119890control (18)
where
Κ119890= Κ119890
119898+ Κ119890
119901+ (Κ119890
119898119886)T(Κ119890
119886119886)minus1
Κ119886119898
+ (Κ119890
119898119904)T(Κ119890
119904119904)minus1
Κ119904119898
(19)
The equivalent control force of input voltage in actuatorlayer is calculated as
F119890control = Κ119890
119898119886(Κ119890
119886119886)minus1
∬
119860119890
1205890
1199052
119881119890
119886119911119889119909 119889119910 (20)
Similarly we can finally establish the piezoelectric cou-pling dynamic FE equation for the global coupled piezo-electric structurecontrol system through elements stiffnessmatrix assembly process as follows
Mq (119905) + Kq (119905) = Fload (119905) + Fcontrol (119905) (21)
where M and K are the coupled system mass matrix andsystem stiffness matrix respectively q and q are the structurenodal displacement vector and acceleration vector respec-tively Fload(119905) and Fcontrol(119905) are the load vector and controlforce vector respectively Fcontrol(119905) can be expressed asBu(119905)where B and u(119905) are the actuator distributing matrix and thecontrol force vector respectively In (21) we did not considerthe damping term in order to emphasize the vibration sup-pression by active control This section is used for modelingand finite element analysis of piezoelectric coupling flat shellstructure and is followed by the LQR control design in theintegrated optimization procedure
3 The Mathematical Modeling forIntegrated Optimization Designof Piezoelectric Cylindrical Shell Structure
The standard form of the integrated optimization problemfor piezoelectric smart cylindrical flat shell structure can bedescribed as
find 119889119904
119894 119889119888
119896(119894 = 1 2 119873 119896 = 1 2 119872)
min 119878 =
119873
sum
119894=1
4120588119894119886119894119887119894119905119894
st 119892 (119889119904
119894 119889119888
119896)119895le 0 (119895 = 1 2 119901)
(22)
where 119889119904
119894and 119889
119888
119896are the structural design variable and
control design variable respectively to minimize the struc-turecontrol system mass 119878 is the objective function 119873 120588
119894
119886119894 119887119894 and 119905
119894are the number of elements (including ordinary
elements and piezoelectric coupling elements) the densityhalf-length of long side and short side for rectangular flat shellelement and thickness of structure element or piezoelectricelement respectively 119898 119899 are the number of structuraldesign variable and control design variable respectively119892(119889119904
119894 119889119888
119896)119895is coupled constraint functions which means it is
simultaneously in relation to the structural design variableand control design variable
31 Design Variables The topology design variable 119889119888
119894indi-
cates the presence or absence of structure material that is
119889119904
119894isin 0 1 (23)
if 119889119904119894= 1 indicates that the corresponding element exists or
else it is removed in optimization process On the contrary119889119904
119894= 0 indicates that it is a void element without materialThe elements of 119889119888
119896may be the control design parameters
and the number and positions of actuators (PZT patches)
119889119888
119896=
119899119886 119899119886isin [1 119878]
119901119899119886
119901119899119886
isin [1119873]
119877119899119886
119877119899119886
isin [119877119897 119877119906]
(24)
where 119899119886 119901119899119886
and 119877119899119886
are the number of actuators the posi-tions of actuators and weight factor of control weight matrixrespectively 119878 119873 119877119897 and 119877
119906 indicate the allowable maximalnumber of actuators the allowablemaximal element positionof actuators and the lower bound and upper bound of weightfactor respectively
32 Constraint Function Here the constraint functions con-tain the first natural frequency constraint function voltageconstraint function and weight factor constraint function
321 Natural Frequency Constraint Function In engineeringpractice structure usually suffers from dynamic load whichmay cause resonance or coupled effect between the struc-ture and external excitation These cases should be avoided
Shock and Vibration 5
because of their huge destruction to structure In generalwe impose natural frequency constraint to the optimizationmathematical modeling especially for the first natural fre-quency So the natural frequency constraint is expressed as
119891119868
1minus 119891119877
1le 0 (25)
where 119891119868
1and 119891
119877
1are the first anticipant minimal natural
frequency of optimization structure and the first actualnatural frequency of optimal structure respectively
322 Voltage Constraint Function Presently piezoelectricpatches or PZT are used as one of themain actuators in activevibration control of structure However there is a break-over voltage for the piezoelectric patches and the appliedvoltage by the voltage amplifier is limited the active controlsystem cannot provide the unlimited and necessary controlforce beyond the break-over voltage Then the constrainton the active control force can be transformed into thecorresponding voltage constraint that is
119881119886
119896minus 119881119886
119896le 0 (26)
where 119881119886
119896and 119881
119886
119896are the actual applied voltage and its
corresponding allowed upper limit respectively
323Weight Factor Constraint Function In LQR control theselections of control weight matrixes have great impact onthe velocity of response suppression and size of control forceSo we choose weight factor as control design parameter andmake it confined to a specified range that is
119877119897
119896le 119877119896le 119877119906
119896 (27)
where 119877119897
119896and 119877
119906
119896are the corresponding lower bound and
upper bound of control weight factor respectively
4 LQR Control Modeling and Design
In modern control theory LQR is one of the mature andeffective optimal controllerrsquos design methods which is basedon state space expression and is applied in many literaturessuch as in the literature [8] In general the vibration suppres-sion of flat shell structure is more complicated than truss andplate structure For simplifications we still utilize the LQR forcontrol modeling and design here In the following the LQRapproach for design of control system is given out
By considering the first several waiting controlmodes andcombining with the initial conditions aftermodal coordinatetransformation (21) can be rewritten as in the modal space
y119888(119905)
y119903(119905)
+ [
Λ119888
0
0 Λ119903
]
y119888(119905)
y119903(119905)
=
120601T119888
120601T119903
Bu (119905) (28)
120601119888= [1206011 120601
2sdot sdot sdot 120601119899119898]
120593T119894119872120593119895= 120575119894119895
Λ119888= diag (1205962
1 1205962
2 120596
2
119899119898
)
(29)
where y119888(119905) and y
119903(119905) are denoted by the modal displacement
vector for waiting control and the rest modal displacementvector respectively Λ
119888is expressed as the eigenvalues diago-
nal matrix for waiting control modesEquation (28) can be expressed as in state space with
matrix form
Z (119905) = AZ (119905) + B1015840u (119905) (30)
Z (119905) = [
y119888(119905)
y119888(119905)
]
A = [
0119899119898times119899119898
I119899119898times119899119898
minusΛc 0119899119898times119899119898
]
B1015840 = [
0119899119898times119899119886
120593T119888B
]
(31)
To ensure that the waiting control system is stableand controllable the allocation number of actuators 119899
119898in
coupled structurecontrol system should be met with thecondition as
rank [B1015840 AB1015840 A2B1015840 sdot sdot sdot A2119899119898minus1B1015840] = 2119899119898 (32)
where rank[] is denoted as the rank computation of matrixFor obtaining the quadratic form optimal control design
u(119905) is to minimize the objective function 119869
119869 =1
2int
infin
0
(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905
997888rarr min(33)
Q = [
Λ119888
0119899119898times119899119898
0119899119898times119899119898
Ι119899119898times119899119898
]
R =
[[[
[
1199031
0
d
0 119903119899119886
]]]
]
(34)
whereR is expressed as the control design parametersmatrixTo minimize (33) we can obtain u(119905) as
u (119905) = minusRminus1 (B1015840)TPZ (119905) (35)
where P is a 2119899119898
times 2119899119898semidefinite and symmetric matrix
which satisfied the Riccati equation as follows
PA + ATP +Q minus PB1015840Rminus1 (B1015840)TP = 0 (36)
By substituting (35) into (30) we can obtain the stateequation of closed-loop system as
Z (119905) = AZ (119905)
A = A minus B1015840Rminus1 (B1015840)TP
(37)
Thus this section is used for calculating the optimalcontrol force vector u(119905) by (35) and determining the optimalcontrol design parameters for (33)
6 Shock and Vibration
(a) Original topology (b) Final topology
Figure 3 Effect for deletions of some low-stress elements
5 Integrated OptimizationAlgorithm and Strategy
51 Optimization Algorithm Theproblem in (22) has contin-uous and discrete design variables which lead to its solutionbeing more difficult and complex than ordinary optimizationproblem with single typersquos design variable For the sakeof coupling between the structure and control system andcomplexion of constraint functions it is very difficult tofind the optimum solution for combined optimization designproblem with structural topology and control parameterGenerally there exist several equal or not equal suboptimalsolutions For the saving of computational cost and simpli-fying of optimization design we can find the suboptimalsolution instead Because of wide adaptability of the GA tothe different types of optimization problems then it may bean effective method for the solution of the above problem Sothe GA is applied for the integrated optimization design inthis paper The binary is selected for coding of chromosomewith structural design variable and control parameter designvariable [8]Meanwhile the individual fitness forminimizingthe objective function is expressed as
eval (119889119904119894 119889119888
119896) =
1
119878times (1 + penal times
sum119901
119895=1119863119895
119901) (38)
where 119901 is the number of constraints functions and penal isthe penalty factor For natural frequency constraint function1198631
= (119891119877
1minus 119891119868
1)119891119868
1 the other 119863
119895are obtained by similar
method as1198631
52 Deletions of Some Low-Stress Elements There generallyexist massive low-stress or inefficient material elements intopological optimizing structure at each iteration becauseof control requirements realization for integrated design ofcoupled structurecontrol system Those elements make themaximal lightweight potential of structure be not achievedby topology optimization Here we delete a part of low-stress
Figure 4 Morphology bridge operator
elements at each iteration to make the topology structure bemore clear and lightweight The steps of the above strategyare summarized as follows firstly according to results ofvon Mises stress computation we sort the whole elementsof structure in ascending order secondly we delete certainpercentages of low-stress elements The deletion percentagesare usually about 20 For example let us see a comparisonbetween the original topology and final topology treated bydeletions of some low-stress elements which is shown inFigure 3
53 Repair Strategy for Disconnected Topology Based on Mor-phology Operator The literature [10] points out that somedisconnected topologies created by genetic algorithm com-monly appear The disconnected topology configuration isinvalid and uncontrollable for integrated optimization designof structurecontrol system So the repair strategy based onimage mathematical morphology operators is applied to thedisconnected topology in integrated optimization design inthis paper Firstly topology of structure or topology variablematrix is considered as a gray imagewith solid and void areasThenweuse basicmorphology operators ldquobridgerdquo and ldquodilaterdquoto filter the gray image that is obtained from each iterationThe detailed morphology operators and related formulationsare expressed in the literature [11] At first let us simplyunderstand the effects of two morphology operators whichare shown in Figures 4 and 5
Shock and Vibration 7
Figure 5 Morphology dilate operator for a piezoelectric smart plate with four simple supported corners
Begin k = 1
Generate the initial population
Binary decode
The finite element analysis (FEA)
Sort elements for deleting some low-stress elements
Morphology operations by bridge and dilateoperators for disconnected topology
Control design by LQR
Evaluate the fitness of individual
Execute genetic operators selectioncrossover and mutation
k = k + 1
Satisfy the terminationcriterion
Yes End
No
Figure 6 Flowchart for integrated optimization design of flat shell
6 The Integrated OptimizationDesign Steps of StructureControl
Thedetailed steps of integrated topology optimization designfor piezoelectric cylindrical flat shell structurecontrol areshown in Figure 6
The general genetic algorithm LQRandFEAwas utilizedfor integrated topology optimization design for piezoelectriccylindrical flat shell structurecontrol in this paper whichappeared in many literatures [7ndash9] Some steps in flowchartsuch as deletions of some low-stress elements and repair
03
02
01
0
02 0 minus02 0
02
04
06
Figure 7 The original flat shell structure and the appointed node(labeled with red color)
strategy for disconnected topology based on morphologyoperator are described in Sections 52 and 53
7 Numerical Examples
The simulation computations for two piezoelectric cylin-drical flat shell structures are carried out by the proposedmethod Here the base material for two simulated structuresis used by steel and its material properties are as followsYoungrsquos elastic modulus 119864 = 21 times 10
11Nm2 Poissonrsquosratio V = 03 and density 120588 = 7800 kgm3 Meanwhile thematerial properties of PZT patches are as follows Youngrsquoselasticmodulus119864
119901= 63times10
10Nm2 Poissonrsquos ratio V119901= 03
and density 120588119901= 7600 kgm3 Piezoelectric strain constants
11988931
= 11988932
= minus171 times 10minus12 CN and dielectric constant
120576 = 15 nfm
71 Semicylindrical Piezoelectric Shell Structure The semi-cylindrical shell structure is shown in Figure 7 The structureis discretized to 10 times 16 elements with the left and rightedges being fixed and the other edges are free The sizes of
8 Shock and Vibration
03
02
01
0
02 0 minus02 0
02
04
06
Figure 8 The optimal number and positions of PZT patches(labeled with black color)
Table 1The results for integrated optimization design of semicylin-drical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
1900 39208 39208 62244 62244 73553
structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887
= 0004m The thickness of PZT patches is 119905119901
=
0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively
The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)
that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10
minus12 and 10 times 10minus13 respectively The
optimal number and positions of PZT patches are shown inFigure 8
In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1
The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1
The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid
elements and void elements respectivelyThe optimal control
03
02
01
0
020 minus02
0
02
04
06
Figure 9 The optimal structure topology
14
12
10
8
60 20 40 60 80 100 120 140 160 180 200
S(k
g)
Evolutionary iterations
Figure 10 The iteration history of mass index
weight parameters are 11987711988612
= 09789 times 10minus12 and 119877
11988634
=
04155 times 10minus12 respectively The iteration history of mass
index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-
control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11
72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905
119887= 0001m The thickness of
PZT patches is 119905119901
= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively
The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast
119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
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2 Shock and Vibration
optimization algorithm There have been some works pub-lished on intelligent piezoelectric structure integrated opti-mization For instance Li and Huang [4] combined theGA and the linear quadratic Gaussian (LQG) studied theintegrated optimization of actuator placement and vibrationcontrol for piezoelectric adaptive trusses and verified that theresult of experiment is consistent with numerical results Zhuet al [5] investigated simultaneous optimization with respectto the structural topology actuator locations and controlparameters of an actively controlled plate structure its resultsshowed that it can produce systems with clear structuraltopology and high control performance by the proposedapproach Kang and Tong [6] investigated the integratedoptimization of structural topology and control voltage ofpiezoelectric laminated plates Xu et al [7] discussed someissues associated with integrated optimization of structuraltopology the number and placement of actuators and controlparameters for piezoelectric smart trusses some optimalstrategies based on genetic algorithms were adopted in theirstudy Xu et al [8] studied the integrated optimizationof structural topology and number and positions of theactuators and control parameters for piezoelectric smartplates which is based on the optimal control effect in theindependentmode control and singular value decompositionof the distributed matrix of total performance index for allphysical control forces with the GA as optimization strategyHowever almost all the efforts described in the literaturesinvolve the integrated optimization design of piezoelectrictruss or plate not including shell structure In generalit is more complicated to construct the piezoelectric shellfinite element for mathematical modeling and finite elementanalysis (FEA) for shell structure Shell structure usually hasmore DOF and more complicated structural characteristicandmechanical behavior than truss and plate structure Shellstructure hasmore complicated geometric shape which leadsto its greater difficulties for modeling and analysis If the shellstructure is in vibration its vibration suppression techniqueis usually invalid andmore complicated than that of truss andplate structure Thus the main concern of this paper is toextend the integrated topology optimization design into shellstructure
In the present paper the integrated topology optimizationof structurecontrol for piezoelectric cylindrical flat shell isstudied Firstly the piezoelectric coupling finite element (FE)model was built Then the integrated optimization modelingincluding the design variables the objective function andthe constraint functions was proposed and the LQR wasused for the design of control system in coupling modalspace In addition in order to ensure the realization of inte-grated optimization the GA and mathematical morphologytreatment was applied to solution of the above problems Toverify the validity of the integrated optimization design ofshell structure and promote engineering applications of thisapproach the numerical simulations and computations wereconductedThe results show that the integrated optimizationsof piezoelectric cylindrical shell structure are successful andfeasible
t1
t2
t3
z
Actuator layer
Base layer
Sensor layer
Figure 1 Four-node piezoelectric coupling rectangular flat shell ele-ment
x
y
z
x998400
y998400
z998400
Figure 2 The coordinate system of cylindrical flat shell
2 The Piezoelectric Coupling Dynamic Modelof Cylindrical Flat Shell Structure
In general it is more complicated to construct the piezo-electric shell finite element for mathematical modeling andfinite element analysis (FEA) of shell structure Shell structureusually has more DOF and more complicated structuralcharacteristic and mechanical behavior than truss and platestructure For simplification we utilized the four-node rect-angular flat shell element with 24 displacement degrees offreedom (DOF) and 2 electrical DOF to build the piezoelec-tric coupling dynamic model of cylindrical shell structure [38] The ideal element is shown in Figure 1 The piezoelectricflat shell element is composed of middle isotropic baselayer with upper and lower surfaces of base layer that arecompletely covered by PZT patches where 119905
1and 1199052sim3
are thethickness of base layer and piezoelectric patches respectivelyThe approximate cylindrical flat shell structure is shown inFigure 2
In finite element derivation the node displacement vectorof flat shell structural element is defined as
d1015840119890119894= (1199061015840
119894V1015840119894
1199081015840
1198941205791015840
1199091198941205791015840
1199101198941205791015840
119911119894)T
119894 = 1 2 3 4 (1)
where 119894 is denoted by the four nodes of the flat shell structuralelement
Shock and Vibration 3
Assume that the element stiffness matrix correspondingto the plane stress state is expressed as
k119898119890
=
[[[[[[[
[
k11989811
k11989812
k11989813
k11989814
k11989821
k11989822
k11989823
k11989824
k11989831
k11989832
k11989833
k11989834
k11989841
k11989842
k11989843
k11989844
]]]]]]]
]
node 1
node 2
node 3
node 4
(2)
where superscript ldquo119898rdquo is corresponding to the stiffnessmatrixof plane stress k119898
119894119895is denoted by 2 times 2 submatrix
Similarly assume that the element stiffness matrix corre-sponding to the bending stress state is expressed as
k119887119890=
[[[[[[[[
[
k11988711
k11988712
k11988713
k11988714
k11988721
k11988722
k11988723
k11988724
k11988731
k11988732
k11988733
k11988734
k11988741
k11988742
k11988743
k11988744
]]]]]]]]
]
node 1
node 2
node 3
node 4
(3)
where superscript ldquo119887rdquo is corresponding to the stiffness matrixof bending stress k119887
119894119895is denoted by 3 times 3 submatrix
By combining with (2) and (3) the element stiffnessmatrix of flat shell structural element in local coordinatesystem is obtained as
k1015840119890
=
[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
[
k11989811
0 0 k11989812
0 0 k11989813
0 0 k11989814
0 0
0 k11988711
0 0 k11988712
0 0 k11988713
0 0 k11988714
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989821
0 0 k11989822
0 0 k11989823
0 0 k11989824
0 0
0 k11988721
0 0 k11988722
0 0 k11988723
0 0 k11988724
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989831
0 0 k11989832
0 0 k11989833
0 0 k11989834
0 0
0 k11988731
0 0 k11988732
0 0 k11988733
0 0 k11988734
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989841
0 0 k11989842
0 0 k11989843
0 0 k11989844
0 0
0 k11988741
0 0 k11988742
0 0 k11988743
0 0 k11988744
0
0 0 0 0 0 0 0 0 0 0 0 0
]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
]
(4)
where k1015840119890is derived in local coordinate system which is a 24times
24matrixBy eliminating the singularity of the whole stiffness
matrix of shell structure we should transform all the localelement stiffness matrices into global coordinate system bycoordinate transformation
The transformation matrix is defined as follows
Τ =
[[[[[[[[[[[[[[[[[[[[
[
120593 0 0 0 0 0 0 0
0 120593 0 0 0 0 0 0
0 0 120593 0 0 0 0 0
0 0 0 120593 0 0 0 0
0 0 0 0 120593 0 0 0
0 0 0 0 0 120593 0 0
0 0 0 0 0 0 120593 0
0 0 0 0 0 0 0 120593
]]]]]]]]]]]]]]]]]]]]
]
120593 =
[[[[
[
cos (1199091015840 119909) cos (1199091015840 119910) cos (1199091015840 119911)
cos (1199101015840 119909) cos (1199101015840 119910) cos (1199101015840 119911)
cos (1199111015840 119909) cos (1199111015840 119910) cos (1199111015840 119911)
]]]]
]
(5)
where f is denoted by direction cosine matrix of local coordi-nate system versus global coordinate system
After coordinate transformation by (6) the local elementstiffness matrix is transformed into global element stiffnessmatrix
k119890= Τ
Tk1015840119890ΤT (6)
Finally we can get the whole stiffness matrix in globalcoordinate system through elements assembly process
By using the Hamilton principle we can obtain theformula as follows
120575int
1199052
1199051
(119879119890minus 119880119890+ 119882119890) 119889119905 = 0 (7)
where119879119890119880119890 and119882119890 are the kinetic energy potential energy
and external work of the piezoelectric coupling flat shellelement shown in Figure 1 respectively
Substituting the kinetic energy potential energy andexternal work of the piezoelectric coupling flat shell elementinto (7) and calculating the variation of (7) the piezoelectriccoupling equation is obtained as follows
M119890 q119890 + (Κ119890
119901+ Κ119890
119898) q119890 minus Κ119890
119898119886119881119890
119886119911+ Κ119890
119898119904119881119890
119904119911
= 119865119890
119891
(8)
Κ119890
119886119898q119890 + Κ119890
119886119886119881119890
119886119911= minus∬
119860119890
120590119890
119886119889119909 119889119910 (9)
Κ119890
119904119898q119890 + Κ119890
119904119904119881119890
119904119911= 0 (10)
M119890 = sum
ℎ=119898119904119886
int
119881119890
ℎ
120588ℎ [N]
T[N] 119889119881 (11)
Κ119890
119898= int
119881119890
119898
1199112[B119906]T[D119898] [B119906] 119889119881 (12)
4 Shock and Vibration
Κ119890
119901= int
119881119890
119904+119881119890
119886
1199112[B119906]T[S119864] [B
119906] 119889119881 (13)
Κ119890
119898119886= (Κ119890
119886119898)T= int
119881119890
119886
119911 [B119906]T[e]T
1199052
119889119881 (14)
Κ119890
119898119904= (Κ119890
119904119898)T= int
119881119890
119904
119911 [B119906]T[e]T
1199052
119889119881 (15)
Κ119890
119886119886= Κ119890
119904119904= int
119881119890
119886
120576119864
33
1199052
2
119889119881 (16)
F119890119891 = ∬
119860119890
NTf 119889119909 119889119910 (17)
where q119890 is denoted by node displacement vector of thepiezoelectric coupling flat shell element and the meanings ofthe other symbols are similar to the utilizations of these in theliterature [3 9]
By reducing the 119881119890
119904119911and 119881
119890
119886119911terms in (8)ndash(10) the
piezoelectric coupling finite element equation is obtained asfollows
M119890 q119890 + Κ119890 q119890 = F119890119891 minus F119890control (18)
where
Κ119890= Κ119890
119898+ Κ119890
119901+ (Κ119890
119898119886)T(Κ119890
119886119886)minus1
Κ119886119898
+ (Κ119890
119898119904)T(Κ119890
119904119904)minus1
Κ119904119898
(19)
The equivalent control force of input voltage in actuatorlayer is calculated as
F119890control = Κ119890
119898119886(Κ119890
119886119886)minus1
∬
119860119890
1205890
1199052
119881119890
119886119911119889119909 119889119910 (20)
Similarly we can finally establish the piezoelectric cou-pling dynamic FE equation for the global coupled piezo-electric structurecontrol system through elements stiffnessmatrix assembly process as follows
Mq (119905) + Kq (119905) = Fload (119905) + Fcontrol (119905) (21)
where M and K are the coupled system mass matrix andsystem stiffness matrix respectively q and q are the structurenodal displacement vector and acceleration vector respec-tively Fload(119905) and Fcontrol(119905) are the load vector and controlforce vector respectively Fcontrol(119905) can be expressed asBu(119905)where B and u(119905) are the actuator distributing matrix and thecontrol force vector respectively In (21) we did not considerthe damping term in order to emphasize the vibration sup-pression by active control This section is used for modelingand finite element analysis of piezoelectric coupling flat shellstructure and is followed by the LQR control design in theintegrated optimization procedure
3 The Mathematical Modeling forIntegrated Optimization Designof Piezoelectric Cylindrical Shell Structure
The standard form of the integrated optimization problemfor piezoelectric smart cylindrical flat shell structure can bedescribed as
find 119889119904
119894 119889119888
119896(119894 = 1 2 119873 119896 = 1 2 119872)
min 119878 =
119873
sum
119894=1
4120588119894119886119894119887119894119905119894
st 119892 (119889119904
119894 119889119888
119896)119895le 0 (119895 = 1 2 119901)
(22)
where 119889119904
119894and 119889
119888
119896are the structural design variable and
control design variable respectively to minimize the struc-turecontrol system mass 119878 is the objective function 119873 120588
119894
119886119894 119887119894 and 119905
119894are the number of elements (including ordinary
elements and piezoelectric coupling elements) the densityhalf-length of long side and short side for rectangular flat shellelement and thickness of structure element or piezoelectricelement respectively 119898 119899 are the number of structuraldesign variable and control design variable respectively119892(119889119904
119894 119889119888
119896)119895is coupled constraint functions which means it is
simultaneously in relation to the structural design variableand control design variable
31 Design Variables The topology design variable 119889119888
119894indi-
cates the presence or absence of structure material that is
119889119904
119894isin 0 1 (23)
if 119889119904119894= 1 indicates that the corresponding element exists or
else it is removed in optimization process On the contrary119889119904
119894= 0 indicates that it is a void element without materialThe elements of 119889119888
119896may be the control design parameters
and the number and positions of actuators (PZT patches)
119889119888
119896=
119899119886 119899119886isin [1 119878]
119901119899119886
119901119899119886
isin [1119873]
119877119899119886
119877119899119886
isin [119877119897 119877119906]
(24)
where 119899119886 119901119899119886
and 119877119899119886
are the number of actuators the posi-tions of actuators and weight factor of control weight matrixrespectively 119878 119873 119877119897 and 119877
119906 indicate the allowable maximalnumber of actuators the allowablemaximal element positionof actuators and the lower bound and upper bound of weightfactor respectively
32 Constraint Function Here the constraint functions con-tain the first natural frequency constraint function voltageconstraint function and weight factor constraint function
321 Natural Frequency Constraint Function In engineeringpractice structure usually suffers from dynamic load whichmay cause resonance or coupled effect between the struc-ture and external excitation These cases should be avoided
Shock and Vibration 5
because of their huge destruction to structure In generalwe impose natural frequency constraint to the optimizationmathematical modeling especially for the first natural fre-quency So the natural frequency constraint is expressed as
119891119868
1minus 119891119877
1le 0 (25)
where 119891119868
1and 119891
119877
1are the first anticipant minimal natural
frequency of optimization structure and the first actualnatural frequency of optimal structure respectively
322 Voltage Constraint Function Presently piezoelectricpatches or PZT are used as one of themain actuators in activevibration control of structure However there is a break-over voltage for the piezoelectric patches and the appliedvoltage by the voltage amplifier is limited the active controlsystem cannot provide the unlimited and necessary controlforce beyond the break-over voltage Then the constrainton the active control force can be transformed into thecorresponding voltage constraint that is
119881119886
119896minus 119881119886
119896le 0 (26)
where 119881119886
119896and 119881
119886
119896are the actual applied voltage and its
corresponding allowed upper limit respectively
323Weight Factor Constraint Function In LQR control theselections of control weight matrixes have great impact onthe velocity of response suppression and size of control forceSo we choose weight factor as control design parameter andmake it confined to a specified range that is
119877119897
119896le 119877119896le 119877119906
119896 (27)
where 119877119897
119896and 119877
119906
119896are the corresponding lower bound and
upper bound of control weight factor respectively
4 LQR Control Modeling and Design
In modern control theory LQR is one of the mature andeffective optimal controllerrsquos design methods which is basedon state space expression and is applied in many literaturessuch as in the literature [8] In general the vibration suppres-sion of flat shell structure is more complicated than truss andplate structure For simplifications we still utilize the LQR forcontrol modeling and design here In the following the LQRapproach for design of control system is given out
By considering the first several waiting controlmodes andcombining with the initial conditions aftermodal coordinatetransformation (21) can be rewritten as in the modal space
y119888(119905)
y119903(119905)
+ [
Λ119888
0
0 Λ119903
]
y119888(119905)
y119903(119905)
=
120601T119888
120601T119903
Bu (119905) (28)
120601119888= [1206011 120601
2sdot sdot sdot 120601119899119898]
120593T119894119872120593119895= 120575119894119895
Λ119888= diag (1205962
1 1205962
2 120596
2
119899119898
)
(29)
where y119888(119905) and y
119903(119905) are denoted by the modal displacement
vector for waiting control and the rest modal displacementvector respectively Λ
119888is expressed as the eigenvalues diago-
nal matrix for waiting control modesEquation (28) can be expressed as in state space with
matrix form
Z (119905) = AZ (119905) + B1015840u (119905) (30)
Z (119905) = [
y119888(119905)
y119888(119905)
]
A = [
0119899119898times119899119898
I119899119898times119899119898
minusΛc 0119899119898times119899119898
]
B1015840 = [
0119899119898times119899119886
120593T119888B
]
(31)
To ensure that the waiting control system is stableand controllable the allocation number of actuators 119899
119898in
coupled structurecontrol system should be met with thecondition as
rank [B1015840 AB1015840 A2B1015840 sdot sdot sdot A2119899119898minus1B1015840] = 2119899119898 (32)
where rank[] is denoted as the rank computation of matrixFor obtaining the quadratic form optimal control design
u(119905) is to minimize the objective function 119869
119869 =1
2int
infin
0
(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905
997888rarr min(33)
Q = [
Λ119888
0119899119898times119899119898
0119899119898times119899119898
Ι119899119898times119899119898
]
R =
[[[
[
1199031
0
d
0 119903119899119886
]]]
]
(34)
whereR is expressed as the control design parametersmatrixTo minimize (33) we can obtain u(119905) as
u (119905) = minusRminus1 (B1015840)TPZ (119905) (35)
where P is a 2119899119898
times 2119899119898semidefinite and symmetric matrix
which satisfied the Riccati equation as follows
PA + ATP +Q minus PB1015840Rminus1 (B1015840)TP = 0 (36)
By substituting (35) into (30) we can obtain the stateequation of closed-loop system as
Z (119905) = AZ (119905)
A = A minus B1015840Rminus1 (B1015840)TP
(37)
Thus this section is used for calculating the optimalcontrol force vector u(119905) by (35) and determining the optimalcontrol design parameters for (33)
6 Shock and Vibration
(a) Original topology (b) Final topology
Figure 3 Effect for deletions of some low-stress elements
5 Integrated OptimizationAlgorithm and Strategy
51 Optimization Algorithm Theproblem in (22) has contin-uous and discrete design variables which lead to its solutionbeing more difficult and complex than ordinary optimizationproblem with single typersquos design variable For the sakeof coupling between the structure and control system andcomplexion of constraint functions it is very difficult tofind the optimum solution for combined optimization designproblem with structural topology and control parameterGenerally there exist several equal or not equal suboptimalsolutions For the saving of computational cost and simpli-fying of optimization design we can find the suboptimalsolution instead Because of wide adaptability of the GA tothe different types of optimization problems then it may bean effective method for the solution of the above problem Sothe GA is applied for the integrated optimization design inthis paper The binary is selected for coding of chromosomewith structural design variable and control parameter designvariable [8]Meanwhile the individual fitness forminimizingthe objective function is expressed as
eval (119889119904119894 119889119888
119896) =
1
119878times (1 + penal times
sum119901
119895=1119863119895
119901) (38)
where 119901 is the number of constraints functions and penal isthe penalty factor For natural frequency constraint function1198631
= (119891119877
1minus 119891119868
1)119891119868
1 the other 119863
119895are obtained by similar
method as1198631
52 Deletions of Some Low-Stress Elements There generallyexist massive low-stress or inefficient material elements intopological optimizing structure at each iteration becauseof control requirements realization for integrated design ofcoupled structurecontrol system Those elements make themaximal lightweight potential of structure be not achievedby topology optimization Here we delete a part of low-stress
Figure 4 Morphology bridge operator
elements at each iteration to make the topology structure bemore clear and lightweight The steps of the above strategyare summarized as follows firstly according to results ofvon Mises stress computation we sort the whole elementsof structure in ascending order secondly we delete certainpercentages of low-stress elements The deletion percentagesare usually about 20 For example let us see a comparisonbetween the original topology and final topology treated bydeletions of some low-stress elements which is shown inFigure 3
53 Repair Strategy for Disconnected Topology Based on Mor-phology Operator The literature [10] points out that somedisconnected topologies created by genetic algorithm com-monly appear The disconnected topology configuration isinvalid and uncontrollable for integrated optimization designof structurecontrol system So the repair strategy based onimage mathematical morphology operators is applied to thedisconnected topology in integrated optimization design inthis paper Firstly topology of structure or topology variablematrix is considered as a gray imagewith solid and void areasThenweuse basicmorphology operators ldquobridgerdquo and ldquodilaterdquoto filter the gray image that is obtained from each iterationThe detailed morphology operators and related formulationsare expressed in the literature [11] At first let us simplyunderstand the effects of two morphology operators whichare shown in Figures 4 and 5
Shock and Vibration 7
Figure 5 Morphology dilate operator for a piezoelectric smart plate with four simple supported corners
Begin k = 1
Generate the initial population
Binary decode
The finite element analysis (FEA)
Sort elements for deleting some low-stress elements
Morphology operations by bridge and dilateoperators for disconnected topology
Control design by LQR
Evaluate the fitness of individual
Execute genetic operators selectioncrossover and mutation
k = k + 1
Satisfy the terminationcriterion
Yes End
No
Figure 6 Flowchart for integrated optimization design of flat shell
6 The Integrated OptimizationDesign Steps of StructureControl
Thedetailed steps of integrated topology optimization designfor piezoelectric cylindrical flat shell structurecontrol areshown in Figure 6
The general genetic algorithm LQRandFEAwas utilizedfor integrated topology optimization design for piezoelectriccylindrical flat shell structurecontrol in this paper whichappeared in many literatures [7ndash9] Some steps in flowchartsuch as deletions of some low-stress elements and repair
03
02
01
0
02 0 minus02 0
02
04
06
Figure 7 The original flat shell structure and the appointed node(labeled with red color)
strategy for disconnected topology based on morphologyoperator are described in Sections 52 and 53
7 Numerical Examples
The simulation computations for two piezoelectric cylin-drical flat shell structures are carried out by the proposedmethod Here the base material for two simulated structuresis used by steel and its material properties are as followsYoungrsquos elastic modulus 119864 = 21 times 10
11Nm2 Poissonrsquosratio V = 03 and density 120588 = 7800 kgm3 Meanwhile thematerial properties of PZT patches are as follows Youngrsquoselasticmodulus119864
119901= 63times10
10Nm2 Poissonrsquos ratio V119901= 03
and density 120588119901= 7600 kgm3 Piezoelectric strain constants
11988931
= 11988932
= minus171 times 10minus12 CN and dielectric constant
120576 = 15 nfm
71 Semicylindrical Piezoelectric Shell Structure The semi-cylindrical shell structure is shown in Figure 7 The structureis discretized to 10 times 16 elements with the left and rightedges being fixed and the other edges are free The sizes of
8 Shock and Vibration
03
02
01
0
02 0 minus02 0
02
04
06
Figure 8 The optimal number and positions of PZT patches(labeled with black color)
Table 1The results for integrated optimization design of semicylin-drical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
1900 39208 39208 62244 62244 73553
structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887
= 0004m The thickness of PZT patches is 119905119901
=
0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively
The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)
that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10
minus12 and 10 times 10minus13 respectively The
optimal number and positions of PZT patches are shown inFigure 8
In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1
The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1
The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid
elements and void elements respectivelyThe optimal control
03
02
01
0
020 minus02
0
02
04
06
Figure 9 The optimal structure topology
14
12
10
8
60 20 40 60 80 100 120 140 160 180 200
S(k
g)
Evolutionary iterations
Figure 10 The iteration history of mass index
weight parameters are 11987711988612
= 09789 times 10minus12 and 119877
11988634
=
04155 times 10minus12 respectively The iteration history of mass
index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-
control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11
72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905
119887= 0001m The thickness of
PZT patches is 119905119901
= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively
The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast
119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
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International Journal of
Shock and Vibration 3
Assume that the element stiffness matrix correspondingto the plane stress state is expressed as
k119898119890
=
[[[[[[[
[
k11989811
k11989812
k11989813
k11989814
k11989821
k11989822
k11989823
k11989824
k11989831
k11989832
k11989833
k11989834
k11989841
k11989842
k11989843
k11989844
]]]]]]]
]
node 1
node 2
node 3
node 4
(2)
where superscript ldquo119898rdquo is corresponding to the stiffnessmatrixof plane stress k119898
119894119895is denoted by 2 times 2 submatrix
Similarly assume that the element stiffness matrix corre-sponding to the bending stress state is expressed as
k119887119890=
[[[[[[[[
[
k11988711
k11988712
k11988713
k11988714
k11988721
k11988722
k11988723
k11988724
k11988731
k11988732
k11988733
k11988734
k11988741
k11988742
k11988743
k11988744
]]]]]]]]
]
node 1
node 2
node 3
node 4
(3)
where superscript ldquo119887rdquo is corresponding to the stiffness matrixof bending stress k119887
119894119895is denoted by 3 times 3 submatrix
By combining with (2) and (3) the element stiffnessmatrix of flat shell structural element in local coordinatesystem is obtained as
k1015840119890
=
[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
[
k11989811
0 0 k11989812
0 0 k11989813
0 0 k11989814
0 0
0 k11988711
0 0 k11988712
0 0 k11988713
0 0 k11988714
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989821
0 0 k11989822
0 0 k11989823
0 0 k11989824
0 0
0 k11988721
0 0 k11988722
0 0 k11988723
0 0 k11988724
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989831
0 0 k11989832
0 0 k11989833
0 0 k11989834
0 0
0 k11988731
0 0 k11988732
0 0 k11988733
0 0 k11988734
0
0 0 0 0 0 0 0 0 0 0 0 0
k11989841
0 0 k11989842
0 0 k11989843
0 0 k11989844
0 0
0 k11988741
0 0 k11988742
0 0 k11988743
0 0 k11988744
0
0 0 0 0 0 0 0 0 0 0 0 0
]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
]
(4)
where k1015840119890is derived in local coordinate system which is a 24times
24matrixBy eliminating the singularity of the whole stiffness
matrix of shell structure we should transform all the localelement stiffness matrices into global coordinate system bycoordinate transformation
The transformation matrix is defined as follows
Τ =
[[[[[[[[[[[[[[[[[[[[
[
120593 0 0 0 0 0 0 0
0 120593 0 0 0 0 0 0
0 0 120593 0 0 0 0 0
0 0 0 120593 0 0 0 0
0 0 0 0 120593 0 0 0
0 0 0 0 0 120593 0 0
0 0 0 0 0 0 120593 0
0 0 0 0 0 0 0 120593
]]]]]]]]]]]]]]]]]]]]
]
120593 =
[[[[
[
cos (1199091015840 119909) cos (1199091015840 119910) cos (1199091015840 119911)
cos (1199101015840 119909) cos (1199101015840 119910) cos (1199101015840 119911)
cos (1199111015840 119909) cos (1199111015840 119910) cos (1199111015840 119911)
]]]]
]
(5)
where f is denoted by direction cosine matrix of local coordi-nate system versus global coordinate system
After coordinate transformation by (6) the local elementstiffness matrix is transformed into global element stiffnessmatrix
k119890= Τ
Tk1015840119890ΤT (6)
Finally we can get the whole stiffness matrix in globalcoordinate system through elements assembly process
By using the Hamilton principle we can obtain theformula as follows
120575int
1199052
1199051
(119879119890minus 119880119890+ 119882119890) 119889119905 = 0 (7)
where119879119890119880119890 and119882119890 are the kinetic energy potential energy
and external work of the piezoelectric coupling flat shellelement shown in Figure 1 respectively
Substituting the kinetic energy potential energy andexternal work of the piezoelectric coupling flat shell elementinto (7) and calculating the variation of (7) the piezoelectriccoupling equation is obtained as follows
M119890 q119890 + (Κ119890
119901+ Κ119890
119898) q119890 minus Κ119890
119898119886119881119890
119886119911+ Κ119890
119898119904119881119890
119904119911
= 119865119890
119891
(8)
Κ119890
119886119898q119890 + Κ119890
119886119886119881119890
119886119911= minus∬
119860119890
120590119890
119886119889119909 119889119910 (9)
Κ119890
119904119898q119890 + Κ119890
119904119904119881119890
119904119911= 0 (10)
M119890 = sum
ℎ=119898119904119886
int
119881119890
ℎ
120588ℎ [N]
T[N] 119889119881 (11)
Κ119890
119898= int
119881119890
119898
1199112[B119906]T[D119898] [B119906] 119889119881 (12)
4 Shock and Vibration
Κ119890
119901= int
119881119890
119904+119881119890
119886
1199112[B119906]T[S119864] [B
119906] 119889119881 (13)
Κ119890
119898119886= (Κ119890
119886119898)T= int
119881119890
119886
119911 [B119906]T[e]T
1199052
119889119881 (14)
Κ119890
119898119904= (Κ119890
119904119898)T= int
119881119890
119904
119911 [B119906]T[e]T
1199052
119889119881 (15)
Κ119890
119886119886= Κ119890
119904119904= int
119881119890
119886
120576119864
33
1199052
2
119889119881 (16)
F119890119891 = ∬
119860119890
NTf 119889119909 119889119910 (17)
where q119890 is denoted by node displacement vector of thepiezoelectric coupling flat shell element and the meanings ofthe other symbols are similar to the utilizations of these in theliterature [3 9]
By reducing the 119881119890
119904119911and 119881
119890
119886119911terms in (8)ndash(10) the
piezoelectric coupling finite element equation is obtained asfollows
M119890 q119890 + Κ119890 q119890 = F119890119891 minus F119890control (18)
where
Κ119890= Κ119890
119898+ Κ119890
119901+ (Κ119890
119898119886)T(Κ119890
119886119886)minus1
Κ119886119898
+ (Κ119890
119898119904)T(Κ119890
119904119904)minus1
Κ119904119898
(19)
The equivalent control force of input voltage in actuatorlayer is calculated as
F119890control = Κ119890
119898119886(Κ119890
119886119886)minus1
∬
119860119890
1205890
1199052
119881119890
119886119911119889119909 119889119910 (20)
Similarly we can finally establish the piezoelectric cou-pling dynamic FE equation for the global coupled piezo-electric structurecontrol system through elements stiffnessmatrix assembly process as follows
Mq (119905) + Kq (119905) = Fload (119905) + Fcontrol (119905) (21)
where M and K are the coupled system mass matrix andsystem stiffness matrix respectively q and q are the structurenodal displacement vector and acceleration vector respec-tively Fload(119905) and Fcontrol(119905) are the load vector and controlforce vector respectively Fcontrol(119905) can be expressed asBu(119905)where B and u(119905) are the actuator distributing matrix and thecontrol force vector respectively In (21) we did not considerthe damping term in order to emphasize the vibration sup-pression by active control This section is used for modelingand finite element analysis of piezoelectric coupling flat shellstructure and is followed by the LQR control design in theintegrated optimization procedure
3 The Mathematical Modeling forIntegrated Optimization Designof Piezoelectric Cylindrical Shell Structure
The standard form of the integrated optimization problemfor piezoelectric smart cylindrical flat shell structure can bedescribed as
find 119889119904
119894 119889119888
119896(119894 = 1 2 119873 119896 = 1 2 119872)
min 119878 =
119873
sum
119894=1
4120588119894119886119894119887119894119905119894
st 119892 (119889119904
119894 119889119888
119896)119895le 0 (119895 = 1 2 119901)
(22)
where 119889119904
119894and 119889
119888
119896are the structural design variable and
control design variable respectively to minimize the struc-turecontrol system mass 119878 is the objective function 119873 120588
119894
119886119894 119887119894 and 119905
119894are the number of elements (including ordinary
elements and piezoelectric coupling elements) the densityhalf-length of long side and short side for rectangular flat shellelement and thickness of structure element or piezoelectricelement respectively 119898 119899 are the number of structuraldesign variable and control design variable respectively119892(119889119904
119894 119889119888
119896)119895is coupled constraint functions which means it is
simultaneously in relation to the structural design variableand control design variable
31 Design Variables The topology design variable 119889119888
119894indi-
cates the presence or absence of structure material that is
119889119904
119894isin 0 1 (23)
if 119889119904119894= 1 indicates that the corresponding element exists or
else it is removed in optimization process On the contrary119889119904
119894= 0 indicates that it is a void element without materialThe elements of 119889119888
119896may be the control design parameters
and the number and positions of actuators (PZT patches)
119889119888
119896=
119899119886 119899119886isin [1 119878]
119901119899119886
119901119899119886
isin [1119873]
119877119899119886
119877119899119886
isin [119877119897 119877119906]
(24)
where 119899119886 119901119899119886
and 119877119899119886
are the number of actuators the posi-tions of actuators and weight factor of control weight matrixrespectively 119878 119873 119877119897 and 119877
119906 indicate the allowable maximalnumber of actuators the allowablemaximal element positionof actuators and the lower bound and upper bound of weightfactor respectively
32 Constraint Function Here the constraint functions con-tain the first natural frequency constraint function voltageconstraint function and weight factor constraint function
321 Natural Frequency Constraint Function In engineeringpractice structure usually suffers from dynamic load whichmay cause resonance or coupled effect between the struc-ture and external excitation These cases should be avoided
Shock and Vibration 5
because of their huge destruction to structure In generalwe impose natural frequency constraint to the optimizationmathematical modeling especially for the first natural fre-quency So the natural frequency constraint is expressed as
119891119868
1minus 119891119877
1le 0 (25)
where 119891119868
1and 119891
119877
1are the first anticipant minimal natural
frequency of optimization structure and the first actualnatural frequency of optimal structure respectively
322 Voltage Constraint Function Presently piezoelectricpatches or PZT are used as one of themain actuators in activevibration control of structure However there is a break-over voltage for the piezoelectric patches and the appliedvoltage by the voltage amplifier is limited the active controlsystem cannot provide the unlimited and necessary controlforce beyond the break-over voltage Then the constrainton the active control force can be transformed into thecorresponding voltage constraint that is
119881119886
119896minus 119881119886
119896le 0 (26)
where 119881119886
119896and 119881
119886
119896are the actual applied voltage and its
corresponding allowed upper limit respectively
323Weight Factor Constraint Function In LQR control theselections of control weight matrixes have great impact onthe velocity of response suppression and size of control forceSo we choose weight factor as control design parameter andmake it confined to a specified range that is
119877119897
119896le 119877119896le 119877119906
119896 (27)
where 119877119897
119896and 119877
119906
119896are the corresponding lower bound and
upper bound of control weight factor respectively
4 LQR Control Modeling and Design
In modern control theory LQR is one of the mature andeffective optimal controllerrsquos design methods which is basedon state space expression and is applied in many literaturessuch as in the literature [8] In general the vibration suppres-sion of flat shell structure is more complicated than truss andplate structure For simplifications we still utilize the LQR forcontrol modeling and design here In the following the LQRapproach for design of control system is given out
By considering the first several waiting controlmodes andcombining with the initial conditions aftermodal coordinatetransformation (21) can be rewritten as in the modal space
y119888(119905)
y119903(119905)
+ [
Λ119888
0
0 Λ119903
]
y119888(119905)
y119903(119905)
=
120601T119888
120601T119903
Bu (119905) (28)
120601119888= [1206011 120601
2sdot sdot sdot 120601119899119898]
120593T119894119872120593119895= 120575119894119895
Λ119888= diag (1205962
1 1205962
2 120596
2
119899119898
)
(29)
where y119888(119905) and y
119903(119905) are denoted by the modal displacement
vector for waiting control and the rest modal displacementvector respectively Λ
119888is expressed as the eigenvalues diago-
nal matrix for waiting control modesEquation (28) can be expressed as in state space with
matrix form
Z (119905) = AZ (119905) + B1015840u (119905) (30)
Z (119905) = [
y119888(119905)
y119888(119905)
]
A = [
0119899119898times119899119898
I119899119898times119899119898
minusΛc 0119899119898times119899119898
]
B1015840 = [
0119899119898times119899119886
120593T119888B
]
(31)
To ensure that the waiting control system is stableand controllable the allocation number of actuators 119899
119898in
coupled structurecontrol system should be met with thecondition as
rank [B1015840 AB1015840 A2B1015840 sdot sdot sdot A2119899119898minus1B1015840] = 2119899119898 (32)
where rank[] is denoted as the rank computation of matrixFor obtaining the quadratic form optimal control design
u(119905) is to minimize the objective function 119869
119869 =1
2int
infin
0
(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905
997888rarr min(33)
Q = [
Λ119888
0119899119898times119899119898
0119899119898times119899119898
Ι119899119898times119899119898
]
R =
[[[
[
1199031
0
d
0 119903119899119886
]]]
]
(34)
whereR is expressed as the control design parametersmatrixTo minimize (33) we can obtain u(119905) as
u (119905) = minusRminus1 (B1015840)TPZ (119905) (35)
where P is a 2119899119898
times 2119899119898semidefinite and symmetric matrix
which satisfied the Riccati equation as follows
PA + ATP +Q minus PB1015840Rminus1 (B1015840)TP = 0 (36)
By substituting (35) into (30) we can obtain the stateequation of closed-loop system as
Z (119905) = AZ (119905)
A = A minus B1015840Rminus1 (B1015840)TP
(37)
Thus this section is used for calculating the optimalcontrol force vector u(119905) by (35) and determining the optimalcontrol design parameters for (33)
6 Shock and Vibration
(a) Original topology (b) Final topology
Figure 3 Effect for deletions of some low-stress elements
5 Integrated OptimizationAlgorithm and Strategy
51 Optimization Algorithm Theproblem in (22) has contin-uous and discrete design variables which lead to its solutionbeing more difficult and complex than ordinary optimizationproblem with single typersquos design variable For the sakeof coupling between the structure and control system andcomplexion of constraint functions it is very difficult tofind the optimum solution for combined optimization designproblem with structural topology and control parameterGenerally there exist several equal or not equal suboptimalsolutions For the saving of computational cost and simpli-fying of optimization design we can find the suboptimalsolution instead Because of wide adaptability of the GA tothe different types of optimization problems then it may bean effective method for the solution of the above problem Sothe GA is applied for the integrated optimization design inthis paper The binary is selected for coding of chromosomewith structural design variable and control parameter designvariable [8]Meanwhile the individual fitness forminimizingthe objective function is expressed as
eval (119889119904119894 119889119888
119896) =
1
119878times (1 + penal times
sum119901
119895=1119863119895
119901) (38)
where 119901 is the number of constraints functions and penal isthe penalty factor For natural frequency constraint function1198631
= (119891119877
1minus 119891119868
1)119891119868
1 the other 119863
119895are obtained by similar
method as1198631
52 Deletions of Some Low-Stress Elements There generallyexist massive low-stress or inefficient material elements intopological optimizing structure at each iteration becauseof control requirements realization for integrated design ofcoupled structurecontrol system Those elements make themaximal lightweight potential of structure be not achievedby topology optimization Here we delete a part of low-stress
Figure 4 Morphology bridge operator
elements at each iteration to make the topology structure bemore clear and lightweight The steps of the above strategyare summarized as follows firstly according to results ofvon Mises stress computation we sort the whole elementsof structure in ascending order secondly we delete certainpercentages of low-stress elements The deletion percentagesare usually about 20 For example let us see a comparisonbetween the original topology and final topology treated bydeletions of some low-stress elements which is shown inFigure 3
53 Repair Strategy for Disconnected Topology Based on Mor-phology Operator The literature [10] points out that somedisconnected topologies created by genetic algorithm com-monly appear The disconnected topology configuration isinvalid and uncontrollable for integrated optimization designof structurecontrol system So the repair strategy based onimage mathematical morphology operators is applied to thedisconnected topology in integrated optimization design inthis paper Firstly topology of structure or topology variablematrix is considered as a gray imagewith solid and void areasThenweuse basicmorphology operators ldquobridgerdquo and ldquodilaterdquoto filter the gray image that is obtained from each iterationThe detailed morphology operators and related formulationsare expressed in the literature [11] At first let us simplyunderstand the effects of two morphology operators whichare shown in Figures 4 and 5
Shock and Vibration 7
Figure 5 Morphology dilate operator for a piezoelectric smart plate with four simple supported corners
Begin k = 1
Generate the initial population
Binary decode
The finite element analysis (FEA)
Sort elements for deleting some low-stress elements
Morphology operations by bridge and dilateoperators for disconnected topology
Control design by LQR
Evaluate the fitness of individual
Execute genetic operators selectioncrossover and mutation
k = k + 1
Satisfy the terminationcriterion
Yes End
No
Figure 6 Flowchart for integrated optimization design of flat shell
6 The Integrated OptimizationDesign Steps of StructureControl
Thedetailed steps of integrated topology optimization designfor piezoelectric cylindrical flat shell structurecontrol areshown in Figure 6
The general genetic algorithm LQRandFEAwas utilizedfor integrated topology optimization design for piezoelectriccylindrical flat shell structurecontrol in this paper whichappeared in many literatures [7ndash9] Some steps in flowchartsuch as deletions of some low-stress elements and repair
03
02
01
0
02 0 minus02 0
02
04
06
Figure 7 The original flat shell structure and the appointed node(labeled with red color)
strategy for disconnected topology based on morphologyoperator are described in Sections 52 and 53
7 Numerical Examples
The simulation computations for two piezoelectric cylin-drical flat shell structures are carried out by the proposedmethod Here the base material for two simulated structuresis used by steel and its material properties are as followsYoungrsquos elastic modulus 119864 = 21 times 10
11Nm2 Poissonrsquosratio V = 03 and density 120588 = 7800 kgm3 Meanwhile thematerial properties of PZT patches are as follows Youngrsquoselasticmodulus119864
119901= 63times10
10Nm2 Poissonrsquos ratio V119901= 03
and density 120588119901= 7600 kgm3 Piezoelectric strain constants
11988931
= 11988932
= minus171 times 10minus12 CN and dielectric constant
120576 = 15 nfm
71 Semicylindrical Piezoelectric Shell Structure The semi-cylindrical shell structure is shown in Figure 7 The structureis discretized to 10 times 16 elements with the left and rightedges being fixed and the other edges are free The sizes of
8 Shock and Vibration
03
02
01
0
02 0 minus02 0
02
04
06
Figure 8 The optimal number and positions of PZT patches(labeled with black color)
Table 1The results for integrated optimization design of semicylin-drical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
1900 39208 39208 62244 62244 73553
structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887
= 0004m The thickness of PZT patches is 119905119901
=
0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively
The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)
that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10
minus12 and 10 times 10minus13 respectively The
optimal number and positions of PZT patches are shown inFigure 8
In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1
The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1
The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid
elements and void elements respectivelyThe optimal control
03
02
01
0
020 minus02
0
02
04
06
Figure 9 The optimal structure topology
14
12
10
8
60 20 40 60 80 100 120 140 160 180 200
S(k
g)
Evolutionary iterations
Figure 10 The iteration history of mass index
weight parameters are 11987711988612
= 09789 times 10minus12 and 119877
11988634
=
04155 times 10minus12 respectively The iteration history of mass
index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-
control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11
72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905
119887= 0001m The thickness of
PZT patches is 119905119901
= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively
The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast
119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
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4 Shock and Vibration
Κ119890
119901= int
119881119890
119904+119881119890
119886
1199112[B119906]T[S119864] [B
119906] 119889119881 (13)
Κ119890
119898119886= (Κ119890
119886119898)T= int
119881119890
119886
119911 [B119906]T[e]T
1199052
119889119881 (14)
Κ119890
119898119904= (Κ119890
119904119898)T= int
119881119890
119904
119911 [B119906]T[e]T
1199052
119889119881 (15)
Κ119890
119886119886= Κ119890
119904119904= int
119881119890
119886
120576119864
33
1199052
2
119889119881 (16)
F119890119891 = ∬
119860119890
NTf 119889119909 119889119910 (17)
where q119890 is denoted by node displacement vector of thepiezoelectric coupling flat shell element and the meanings ofthe other symbols are similar to the utilizations of these in theliterature [3 9]
By reducing the 119881119890
119904119911and 119881
119890
119886119911terms in (8)ndash(10) the
piezoelectric coupling finite element equation is obtained asfollows
M119890 q119890 + Κ119890 q119890 = F119890119891 minus F119890control (18)
where
Κ119890= Κ119890
119898+ Κ119890
119901+ (Κ119890
119898119886)T(Κ119890
119886119886)minus1
Κ119886119898
+ (Κ119890
119898119904)T(Κ119890
119904119904)minus1
Κ119904119898
(19)
The equivalent control force of input voltage in actuatorlayer is calculated as
F119890control = Κ119890
119898119886(Κ119890
119886119886)minus1
∬
119860119890
1205890
1199052
119881119890
119886119911119889119909 119889119910 (20)
Similarly we can finally establish the piezoelectric cou-pling dynamic FE equation for the global coupled piezo-electric structurecontrol system through elements stiffnessmatrix assembly process as follows
Mq (119905) + Kq (119905) = Fload (119905) + Fcontrol (119905) (21)
where M and K are the coupled system mass matrix andsystem stiffness matrix respectively q and q are the structurenodal displacement vector and acceleration vector respec-tively Fload(119905) and Fcontrol(119905) are the load vector and controlforce vector respectively Fcontrol(119905) can be expressed asBu(119905)where B and u(119905) are the actuator distributing matrix and thecontrol force vector respectively In (21) we did not considerthe damping term in order to emphasize the vibration sup-pression by active control This section is used for modelingand finite element analysis of piezoelectric coupling flat shellstructure and is followed by the LQR control design in theintegrated optimization procedure
3 The Mathematical Modeling forIntegrated Optimization Designof Piezoelectric Cylindrical Shell Structure
The standard form of the integrated optimization problemfor piezoelectric smart cylindrical flat shell structure can bedescribed as
find 119889119904
119894 119889119888
119896(119894 = 1 2 119873 119896 = 1 2 119872)
min 119878 =
119873
sum
119894=1
4120588119894119886119894119887119894119905119894
st 119892 (119889119904
119894 119889119888
119896)119895le 0 (119895 = 1 2 119901)
(22)
where 119889119904
119894and 119889
119888
119896are the structural design variable and
control design variable respectively to minimize the struc-turecontrol system mass 119878 is the objective function 119873 120588
119894
119886119894 119887119894 and 119905
119894are the number of elements (including ordinary
elements and piezoelectric coupling elements) the densityhalf-length of long side and short side for rectangular flat shellelement and thickness of structure element or piezoelectricelement respectively 119898 119899 are the number of structuraldesign variable and control design variable respectively119892(119889119904
119894 119889119888
119896)119895is coupled constraint functions which means it is
simultaneously in relation to the structural design variableand control design variable
31 Design Variables The topology design variable 119889119888
119894indi-
cates the presence or absence of structure material that is
119889119904
119894isin 0 1 (23)
if 119889119904119894= 1 indicates that the corresponding element exists or
else it is removed in optimization process On the contrary119889119904
119894= 0 indicates that it is a void element without materialThe elements of 119889119888
119896may be the control design parameters
and the number and positions of actuators (PZT patches)
119889119888
119896=
119899119886 119899119886isin [1 119878]
119901119899119886
119901119899119886
isin [1119873]
119877119899119886
119877119899119886
isin [119877119897 119877119906]
(24)
where 119899119886 119901119899119886
and 119877119899119886
are the number of actuators the posi-tions of actuators and weight factor of control weight matrixrespectively 119878 119873 119877119897 and 119877
119906 indicate the allowable maximalnumber of actuators the allowablemaximal element positionof actuators and the lower bound and upper bound of weightfactor respectively
32 Constraint Function Here the constraint functions con-tain the first natural frequency constraint function voltageconstraint function and weight factor constraint function
321 Natural Frequency Constraint Function In engineeringpractice structure usually suffers from dynamic load whichmay cause resonance or coupled effect between the struc-ture and external excitation These cases should be avoided
Shock and Vibration 5
because of their huge destruction to structure In generalwe impose natural frequency constraint to the optimizationmathematical modeling especially for the first natural fre-quency So the natural frequency constraint is expressed as
119891119868
1minus 119891119877
1le 0 (25)
where 119891119868
1and 119891
119877
1are the first anticipant minimal natural
frequency of optimization structure and the first actualnatural frequency of optimal structure respectively
322 Voltage Constraint Function Presently piezoelectricpatches or PZT are used as one of themain actuators in activevibration control of structure However there is a break-over voltage for the piezoelectric patches and the appliedvoltage by the voltage amplifier is limited the active controlsystem cannot provide the unlimited and necessary controlforce beyond the break-over voltage Then the constrainton the active control force can be transformed into thecorresponding voltage constraint that is
119881119886
119896minus 119881119886
119896le 0 (26)
where 119881119886
119896and 119881
119886
119896are the actual applied voltage and its
corresponding allowed upper limit respectively
323Weight Factor Constraint Function In LQR control theselections of control weight matrixes have great impact onthe velocity of response suppression and size of control forceSo we choose weight factor as control design parameter andmake it confined to a specified range that is
119877119897
119896le 119877119896le 119877119906
119896 (27)
where 119877119897
119896and 119877
119906
119896are the corresponding lower bound and
upper bound of control weight factor respectively
4 LQR Control Modeling and Design
In modern control theory LQR is one of the mature andeffective optimal controllerrsquos design methods which is basedon state space expression and is applied in many literaturessuch as in the literature [8] In general the vibration suppres-sion of flat shell structure is more complicated than truss andplate structure For simplifications we still utilize the LQR forcontrol modeling and design here In the following the LQRapproach for design of control system is given out
By considering the first several waiting controlmodes andcombining with the initial conditions aftermodal coordinatetransformation (21) can be rewritten as in the modal space
y119888(119905)
y119903(119905)
+ [
Λ119888
0
0 Λ119903
]
y119888(119905)
y119903(119905)
=
120601T119888
120601T119903
Bu (119905) (28)
120601119888= [1206011 120601
2sdot sdot sdot 120601119899119898]
120593T119894119872120593119895= 120575119894119895
Λ119888= diag (1205962
1 1205962
2 120596
2
119899119898
)
(29)
where y119888(119905) and y
119903(119905) are denoted by the modal displacement
vector for waiting control and the rest modal displacementvector respectively Λ
119888is expressed as the eigenvalues diago-
nal matrix for waiting control modesEquation (28) can be expressed as in state space with
matrix form
Z (119905) = AZ (119905) + B1015840u (119905) (30)
Z (119905) = [
y119888(119905)
y119888(119905)
]
A = [
0119899119898times119899119898
I119899119898times119899119898
minusΛc 0119899119898times119899119898
]
B1015840 = [
0119899119898times119899119886
120593T119888B
]
(31)
To ensure that the waiting control system is stableand controllable the allocation number of actuators 119899
119898in
coupled structurecontrol system should be met with thecondition as
rank [B1015840 AB1015840 A2B1015840 sdot sdot sdot A2119899119898minus1B1015840] = 2119899119898 (32)
where rank[] is denoted as the rank computation of matrixFor obtaining the quadratic form optimal control design
u(119905) is to minimize the objective function 119869
119869 =1
2int
infin
0
(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905
997888rarr min(33)
Q = [
Λ119888
0119899119898times119899119898
0119899119898times119899119898
Ι119899119898times119899119898
]
R =
[[[
[
1199031
0
d
0 119903119899119886
]]]
]
(34)
whereR is expressed as the control design parametersmatrixTo minimize (33) we can obtain u(119905) as
u (119905) = minusRminus1 (B1015840)TPZ (119905) (35)
where P is a 2119899119898
times 2119899119898semidefinite and symmetric matrix
which satisfied the Riccati equation as follows
PA + ATP +Q minus PB1015840Rminus1 (B1015840)TP = 0 (36)
By substituting (35) into (30) we can obtain the stateequation of closed-loop system as
Z (119905) = AZ (119905)
A = A minus B1015840Rminus1 (B1015840)TP
(37)
Thus this section is used for calculating the optimalcontrol force vector u(119905) by (35) and determining the optimalcontrol design parameters for (33)
6 Shock and Vibration
(a) Original topology (b) Final topology
Figure 3 Effect for deletions of some low-stress elements
5 Integrated OptimizationAlgorithm and Strategy
51 Optimization Algorithm Theproblem in (22) has contin-uous and discrete design variables which lead to its solutionbeing more difficult and complex than ordinary optimizationproblem with single typersquos design variable For the sakeof coupling between the structure and control system andcomplexion of constraint functions it is very difficult tofind the optimum solution for combined optimization designproblem with structural topology and control parameterGenerally there exist several equal or not equal suboptimalsolutions For the saving of computational cost and simpli-fying of optimization design we can find the suboptimalsolution instead Because of wide adaptability of the GA tothe different types of optimization problems then it may bean effective method for the solution of the above problem Sothe GA is applied for the integrated optimization design inthis paper The binary is selected for coding of chromosomewith structural design variable and control parameter designvariable [8]Meanwhile the individual fitness forminimizingthe objective function is expressed as
eval (119889119904119894 119889119888
119896) =
1
119878times (1 + penal times
sum119901
119895=1119863119895
119901) (38)
where 119901 is the number of constraints functions and penal isthe penalty factor For natural frequency constraint function1198631
= (119891119877
1minus 119891119868
1)119891119868
1 the other 119863
119895are obtained by similar
method as1198631
52 Deletions of Some Low-Stress Elements There generallyexist massive low-stress or inefficient material elements intopological optimizing structure at each iteration becauseof control requirements realization for integrated design ofcoupled structurecontrol system Those elements make themaximal lightweight potential of structure be not achievedby topology optimization Here we delete a part of low-stress
Figure 4 Morphology bridge operator
elements at each iteration to make the topology structure bemore clear and lightweight The steps of the above strategyare summarized as follows firstly according to results ofvon Mises stress computation we sort the whole elementsof structure in ascending order secondly we delete certainpercentages of low-stress elements The deletion percentagesare usually about 20 For example let us see a comparisonbetween the original topology and final topology treated bydeletions of some low-stress elements which is shown inFigure 3
53 Repair Strategy for Disconnected Topology Based on Mor-phology Operator The literature [10] points out that somedisconnected topologies created by genetic algorithm com-monly appear The disconnected topology configuration isinvalid and uncontrollable for integrated optimization designof structurecontrol system So the repair strategy based onimage mathematical morphology operators is applied to thedisconnected topology in integrated optimization design inthis paper Firstly topology of structure or topology variablematrix is considered as a gray imagewith solid and void areasThenweuse basicmorphology operators ldquobridgerdquo and ldquodilaterdquoto filter the gray image that is obtained from each iterationThe detailed morphology operators and related formulationsare expressed in the literature [11] At first let us simplyunderstand the effects of two morphology operators whichare shown in Figures 4 and 5
Shock and Vibration 7
Figure 5 Morphology dilate operator for a piezoelectric smart plate with four simple supported corners
Begin k = 1
Generate the initial population
Binary decode
The finite element analysis (FEA)
Sort elements for deleting some low-stress elements
Morphology operations by bridge and dilateoperators for disconnected topology
Control design by LQR
Evaluate the fitness of individual
Execute genetic operators selectioncrossover and mutation
k = k + 1
Satisfy the terminationcriterion
Yes End
No
Figure 6 Flowchart for integrated optimization design of flat shell
6 The Integrated OptimizationDesign Steps of StructureControl
Thedetailed steps of integrated topology optimization designfor piezoelectric cylindrical flat shell structurecontrol areshown in Figure 6
The general genetic algorithm LQRandFEAwas utilizedfor integrated topology optimization design for piezoelectriccylindrical flat shell structurecontrol in this paper whichappeared in many literatures [7ndash9] Some steps in flowchartsuch as deletions of some low-stress elements and repair
03
02
01
0
02 0 minus02 0
02
04
06
Figure 7 The original flat shell structure and the appointed node(labeled with red color)
strategy for disconnected topology based on morphologyoperator are described in Sections 52 and 53
7 Numerical Examples
The simulation computations for two piezoelectric cylin-drical flat shell structures are carried out by the proposedmethod Here the base material for two simulated structuresis used by steel and its material properties are as followsYoungrsquos elastic modulus 119864 = 21 times 10
11Nm2 Poissonrsquosratio V = 03 and density 120588 = 7800 kgm3 Meanwhile thematerial properties of PZT patches are as follows Youngrsquoselasticmodulus119864
119901= 63times10
10Nm2 Poissonrsquos ratio V119901= 03
and density 120588119901= 7600 kgm3 Piezoelectric strain constants
11988931
= 11988932
= minus171 times 10minus12 CN and dielectric constant
120576 = 15 nfm
71 Semicylindrical Piezoelectric Shell Structure The semi-cylindrical shell structure is shown in Figure 7 The structureis discretized to 10 times 16 elements with the left and rightedges being fixed and the other edges are free The sizes of
8 Shock and Vibration
03
02
01
0
02 0 minus02 0
02
04
06
Figure 8 The optimal number and positions of PZT patches(labeled with black color)
Table 1The results for integrated optimization design of semicylin-drical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
1900 39208 39208 62244 62244 73553
structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887
= 0004m The thickness of PZT patches is 119905119901
=
0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively
The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)
that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10
minus12 and 10 times 10minus13 respectively The
optimal number and positions of PZT patches are shown inFigure 8
In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1
The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1
The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid
elements and void elements respectivelyThe optimal control
03
02
01
0
020 minus02
0
02
04
06
Figure 9 The optimal structure topology
14
12
10
8
60 20 40 60 80 100 120 140 160 180 200
S(k
g)
Evolutionary iterations
Figure 10 The iteration history of mass index
weight parameters are 11987711988612
= 09789 times 10minus12 and 119877
11988634
=
04155 times 10minus12 respectively The iteration history of mass
index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-
control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11
72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905
119887= 0001m The thickness of
PZT patches is 119905119901
= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively
The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast
119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
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Shock and Vibration 5
because of their huge destruction to structure In generalwe impose natural frequency constraint to the optimizationmathematical modeling especially for the first natural fre-quency So the natural frequency constraint is expressed as
119891119868
1minus 119891119877
1le 0 (25)
where 119891119868
1and 119891
119877
1are the first anticipant minimal natural
frequency of optimization structure and the first actualnatural frequency of optimal structure respectively
322 Voltage Constraint Function Presently piezoelectricpatches or PZT are used as one of themain actuators in activevibration control of structure However there is a break-over voltage for the piezoelectric patches and the appliedvoltage by the voltage amplifier is limited the active controlsystem cannot provide the unlimited and necessary controlforce beyond the break-over voltage Then the constrainton the active control force can be transformed into thecorresponding voltage constraint that is
119881119886
119896minus 119881119886
119896le 0 (26)
where 119881119886
119896and 119881
119886
119896are the actual applied voltage and its
corresponding allowed upper limit respectively
323Weight Factor Constraint Function In LQR control theselections of control weight matrixes have great impact onthe velocity of response suppression and size of control forceSo we choose weight factor as control design parameter andmake it confined to a specified range that is
119877119897
119896le 119877119896le 119877119906
119896 (27)
where 119877119897
119896and 119877
119906
119896are the corresponding lower bound and
upper bound of control weight factor respectively
4 LQR Control Modeling and Design
In modern control theory LQR is one of the mature andeffective optimal controllerrsquos design methods which is basedon state space expression and is applied in many literaturessuch as in the literature [8] In general the vibration suppres-sion of flat shell structure is more complicated than truss andplate structure For simplifications we still utilize the LQR forcontrol modeling and design here In the following the LQRapproach for design of control system is given out
By considering the first several waiting controlmodes andcombining with the initial conditions aftermodal coordinatetransformation (21) can be rewritten as in the modal space
y119888(119905)
y119903(119905)
+ [
Λ119888
0
0 Λ119903
]
y119888(119905)
y119903(119905)
=
120601T119888
120601T119903
Bu (119905) (28)
120601119888= [1206011 120601
2sdot sdot sdot 120601119899119898]
120593T119894119872120593119895= 120575119894119895
Λ119888= diag (1205962
1 1205962
2 120596
2
119899119898
)
(29)
where y119888(119905) and y
119903(119905) are denoted by the modal displacement
vector for waiting control and the rest modal displacementvector respectively Λ
119888is expressed as the eigenvalues diago-
nal matrix for waiting control modesEquation (28) can be expressed as in state space with
matrix form
Z (119905) = AZ (119905) + B1015840u (119905) (30)
Z (119905) = [
y119888(119905)
y119888(119905)
]
A = [
0119899119898times119899119898
I119899119898times119899119898
minusΛc 0119899119898times119899119898
]
B1015840 = [
0119899119898times119899119886
120593T119888B
]
(31)
To ensure that the waiting control system is stableand controllable the allocation number of actuators 119899
119898in
coupled structurecontrol system should be met with thecondition as
rank [B1015840 AB1015840 A2B1015840 sdot sdot sdot A2119899119898minus1B1015840] = 2119899119898 (32)
where rank[] is denoted as the rank computation of matrixFor obtaining the quadratic form optimal control design
u(119905) is to minimize the objective function 119869
119869 =1
2int
infin
0
(ZT(119905)QZ (119905) + uT (119905)Ru (119905)) 119889119905
997888rarr min(33)
Q = [
Λ119888
0119899119898times119899119898
0119899119898times119899119898
Ι119899119898times119899119898
]
R =
[[[
[
1199031
0
d
0 119903119899119886
]]]
]
(34)
whereR is expressed as the control design parametersmatrixTo minimize (33) we can obtain u(119905) as
u (119905) = minusRminus1 (B1015840)TPZ (119905) (35)
where P is a 2119899119898
times 2119899119898semidefinite and symmetric matrix
which satisfied the Riccati equation as follows
PA + ATP +Q minus PB1015840Rminus1 (B1015840)TP = 0 (36)
By substituting (35) into (30) we can obtain the stateequation of closed-loop system as
Z (119905) = AZ (119905)
A = A minus B1015840Rminus1 (B1015840)TP
(37)
Thus this section is used for calculating the optimalcontrol force vector u(119905) by (35) and determining the optimalcontrol design parameters for (33)
6 Shock and Vibration
(a) Original topology (b) Final topology
Figure 3 Effect for deletions of some low-stress elements
5 Integrated OptimizationAlgorithm and Strategy
51 Optimization Algorithm Theproblem in (22) has contin-uous and discrete design variables which lead to its solutionbeing more difficult and complex than ordinary optimizationproblem with single typersquos design variable For the sakeof coupling between the structure and control system andcomplexion of constraint functions it is very difficult tofind the optimum solution for combined optimization designproblem with structural topology and control parameterGenerally there exist several equal or not equal suboptimalsolutions For the saving of computational cost and simpli-fying of optimization design we can find the suboptimalsolution instead Because of wide adaptability of the GA tothe different types of optimization problems then it may bean effective method for the solution of the above problem Sothe GA is applied for the integrated optimization design inthis paper The binary is selected for coding of chromosomewith structural design variable and control parameter designvariable [8]Meanwhile the individual fitness forminimizingthe objective function is expressed as
eval (119889119904119894 119889119888
119896) =
1
119878times (1 + penal times
sum119901
119895=1119863119895
119901) (38)
where 119901 is the number of constraints functions and penal isthe penalty factor For natural frequency constraint function1198631
= (119891119877
1minus 119891119868
1)119891119868
1 the other 119863
119895are obtained by similar
method as1198631
52 Deletions of Some Low-Stress Elements There generallyexist massive low-stress or inefficient material elements intopological optimizing structure at each iteration becauseof control requirements realization for integrated design ofcoupled structurecontrol system Those elements make themaximal lightweight potential of structure be not achievedby topology optimization Here we delete a part of low-stress
Figure 4 Morphology bridge operator
elements at each iteration to make the topology structure bemore clear and lightweight The steps of the above strategyare summarized as follows firstly according to results ofvon Mises stress computation we sort the whole elementsof structure in ascending order secondly we delete certainpercentages of low-stress elements The deletion percentagesare usually about 20 For example let us see a comparisonbetween the original topology and final topology treated bydeletions of some low-stress elements which is shown inFigure 3
53 Repair Strategy for Disconnected Topology Based on Mor-phology Operator The literature [10] points out that somedisconnected topologies created by genetic algorithm com-monly appear The disconnected topology configuration isinvalid and uncontrollable for integrated optimization designof structurecontrol system So the repair strategy based onimage mathematical morphology operators is applied to thedisconnected topology in integrated optimization design inthis paper Firstly topology of structure or topology variablematrix is considered as a gray imagewith solid and void areasThenweuse basicmorphology operators ldquobridgerdquo and ldquodilaterdquoto filter the gray image that is obtained from each iterationThe detailed morphology operators and related formulationsare expressed in the literature [11] At first let us simplyunderstand the effects of two morphology operators whichare shown in Figures 4 and 5
Shock and Vibration 7
Figure 5 Morphology dilate operator for a piezoelectric smart plate with four simple supported corners
Begin k = 1
Generate the initial population
Binary decode
The finite element analysis (FEA)
Sort elements for deleting some low-stress elements
Morphology operations by bridge and dilateoperators for disconnected topology
Control design by LQR
Evaluate the fitness of individual
Execute genetic operators selectioncrossover and mutation
k = k + 1
Satisfy the terminationcriterion
Yes End
No
Figure 6 Flowchart for integrated optimization design of flat shell
6 The Integrated OptimizationDesign Steps of StructureControl
Thedetailed steps of integrated topology optimization designfor piezoelectric cylindrical flat shell structurecontrol areshown in Figure 6
The general genetic algorithm LQRandFEAwas utilizedfor integrated topology optimization design for piezoelectriccylindrical flat shell structurecontrol in this paper whichappeared in many literatures [7ndash9] Some steps in flowchartsuch as deletions of some low-stress elements and repair
03
02
01
0
02 0 minus02 0
02
04
06
Figure 7 The original flat shell structure and the appointed node(labeled with red color)
strategy for disconnected topology based on morphologyoperator are described in Sections 52 and 53
7 Numerical Examples
The simulation computations for two piezoelectric cylin-drical flat shell structures are carried out by the proposedmethod Here the base material for two simulated structuresis used by steel and its material properties are as followsYoungrsquos elastic modulus 119864 = 21 times 10
11Nm2 Poissonrsquosratio V = 03 and density 120588 = 7800 kgm3 Meanwhile thematerial properties of PZT patches are as follows Youngrsquoselasticmodulus119864
119901= 63times10
10Nm2 Poissonrsquos ratio V119901= 03
and density 120588119901= 7600 kgm3 Piezoelectric strain constants
11988931
= 11988932
= minus171 times 10minus12 CN and dielectric constant
120576 = 15 nfm
71 Semicylindrical Piezoelectric Shell Structure The semi-cylindrical shell structure is shown in Figure 7 The structureis discretized to 10 times 16 elements with the left and rightedges being fixed and the other edges are free The sizes of
8 Shock and Vibration
03
02
01
0
02 0 minus02 0
02
04
06
Figure 8 The optimal number and positions of PZT patches(labeled with black color)
Table 1The results for integrated optimization design of semicylin-drical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
1900 39208 39208 62244 62244 73553
structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887
= 0004m The thickness of PZT patches is 119905119901
=
0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively
The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)
that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10
minus12 and 10 times 10minus13 respectively The
optimal number and positions of PZT patches are shown inFigure 8
In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1
The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1
The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid
elements and void elements respectivelyThe optimal control
03
02
01
0
020 minus02
0
02
04
06
Figure 9 The optimal structure topology
14
12
10
8
60 20 40 60 80 100 120 140 160 180 200
S(k
g)
Evolutionary iterations
Figure 10 The iteration history of mass index
weight parameters are 11987711988612
= 09789 times 10minus12 and 119877
11988634
=
04155 times 10minus12 respectively The iteration history of mass
index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-
control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11
72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905
119887= 0001m The thickness of
PZT patches is 119905119901
= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively
The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast
119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Shock and Vibration
(a) Original topology (b) Final topology
Figure 3 Effect for deletions of some low-stress elements
5 Integrated OptimizationAlgorithm and Strategy
51 Optimization Algorithm Theproblem in (22) has contin-uous and discrete design variables which lead to its solutionbeing more difficult and complex than ordinary optimizationproblem with single typersquos design variable For the sakeof coupling between the structure and control system andcomplexion of constraint functions it is very difficult tofind the optimum solution for combined optimization designproblem with structural topology and control parameterGenerally there exist several equal or not equal suboptimalsolutions For the saving of computational cost and simpli-fying of optimization design we can find the suboptimalsolution instead Because of wide adaptability of the GA tothe different types of optimization problems then it may bean effective method for the solution of the above problem Sothe GA is applied for the integrated optimization design inthis paper The binary is selected for coding of chromosomewith structural design variable and control parameter designvariable [8]Meanwhile the individual fitness forminimizingthe objective function is expressed as
eval (119889119904119894 119889119888
119896) =
1
119878times (1 + penal times
sum119901
119895=1119863119895
119901) (38)
where 119901 is the number of constraints functions and penal isthe penalty factor For natural frequency constraint function1198631
= (119891119877
1minus 119891119868
1)119891119868
1 the other 119863
119895are obtained by similar
method as1198631
52 Deletions of Some Low-Stress Elements There generallyexist massive low-stress or inefficient material elements intopological optimizing structure at each iteration becauseof control requirements realization for integrated design ofcoupled structurecontrol system Those elements make themaximal lightweight potential of structure be not achievedby topology optimization Here we delete a part of low-stress
Figure 4 Morphology bridge operator
elements at each iteration to make the topology structure bemore clear and lightweight The steps of the above strategyare summarized as follows firstly according to results ofvon Mises stress computation we sort the whole elementsof structure in ascending order secondly we delete certainpercentages of low-stress elements The deletion percentagesare usually about 20 For example let us see a comparisonbetween the original topology and final topology treated bydeletions of some low-stress elements which is shown inFigure 3
53 Repair Strategy for Disconnected Topology Based on Mor-phology Operator The literature [10] points out that somedisconnected topologies created by genetic algorithm com-monly appear The disconnected topology configuration isinvalid and uncontrollable for integrated optimization designof structurecontrol system So the repair strategy based onimage mathematical morphology operators is applied to thedisconnected topology in integrated optimization design inthis paper Firstly topology of structure or topology variablematrix is considered as a gray imagewith solid and void areasThenweuse basicmorphology operators ldquobridgerdquo and ldquodilaterdquoto filter the gray image that is obtained from each iterationThe detailed morphology operators and related formulationsare expressed in the literature [11] At first let us simplyunderstand the effects of two morphology operators whichare shown in Figures 4 and 5
Shock and Vibration 7
Figure 5 Morphology dilate operator for a piezoelectric smart plate with four simple supported corners
Begin k = 1
Generate the initial population
Binary decode
The finite element analysis (FEA)
Sort elements for deleting some low-stress elements
Morphology operations by bridge and dilateoperators for disconnected topology
Control design by LQR
Evaluate the fitness of individual
Execute genetic operators selectioncrossover and mutation
k = k + 1
Satisfy the terminationcriterion
Yes End
No
Figure 6 Flowchart for integrated optimization design of flat shell
6 The Integrated OptimizationDesign Steps of StructureControl
Thedetailed steps of integrated topology optimization designfor piezoelectric cylindrical flat shell structurecontrol areshown in Figure 6
The general genetic algorithm LQRandFEAwas utilizedfor integrated topology optimization design for piezoelectriccylindrical flat shell structurecontrol in this paper whichappeared in many literatures [7ndash9] Some steps in flowchartsuch as deletions of some low-stress elements and repair
03
02
01
0
02 0 minus02 0
02
04
06
Figure 7 The original flat shell structure and the appointed node(labeled with red color)
strategy for disconnected topology based on morphologyoperator are described in Sections 52 and 53
7 Numerical Examples
The simulation computations for two piezoelectric cylin-drical flat shell structures are carried out by the proposedmethod Here the base material for two simulated structuresis used by steel and its material properties are as followsYoungrsquos elastic modulus 119864 = 21 times 10
11Nm2 Poissonrsquosratio V = 03 and density 120588 = 7800 kgm3 Meanwhile thematerial properties of PZT patches are as follows Youngrsquoselasticmodulus119864
119901= 63times10
10Nm2 Poissonrsquos ratio V119901= 03
and density 120588119901= 7600 kgm3 Piezoelectric strain constants
11988931
= 11988932
= minus171 times 10minus12 CN and dielectric constant
120576 = 15 nfm
71 Semicylindrical Piezoelectric Shell Structure The semi-cylindrical shell structure is shown in Figure 7 The structureis discretized to 10 times 16 elements with the left and rightedges being fixed and the other edges are free The sizes of
8 Shock and Vibration
03
02
01
0
02 0 minus02 0
02
04
06
Figure 8 The optimal number and positions of PZT patches(labeled with black color)
Table 1The results for integrated optimization design of semicylin-drical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
1900 39208 39208 62244 62244 73553
structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887
= 0004m The thickness of PZT patches is 119905119901
=
0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively
The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)
that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10
minus12 and 10 times 10minus13 respectively The
optimal number and positions of PZT patches are shown inFigure 8
In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1
The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1
The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid
elements and void elements respectivelyThe optimal control
03
02
01
0
020 minus02
0
02
04
06
Figure 9 The optimal structure topology
14
12
10
8
60 20 40 60 80 100 120 140 160 180 200
S(k
g)
Evolutionary iterations
Figure 10 The iteration history of mass index
weight parameters are 11987711988612
= 09789 times 10minus12 and 119877
11988634
=
04155 times 10minus12 respectively The iteration history of mass
index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-
control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11
72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905
119887= 0001m The thickness of
PZT patches is 119905119901
= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively
The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast
119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 7
Figure 5 Morphology dilate operator for a piezoelectric smart plate with four simple supported corners
Begin k = 1
Generate the initial population
Binary decode
The finite element analysis (FEA)
Sort elements for deleting some low-stress elements
Morphology operations by bridge and dilateoperators for disconnected topology
Control design by LQR
Evaluate the fitness of individual
Execute genetic operators selectioncrossover and mutation
k = k + 1
Satisfy the terminationcriterion
Yes End
No
Figure 6 Flowchart for integrated optimization design of flat shell
6 The Integrated OptimizationDesign Steps of StructureControl
Thedetailed steps of integrated topology optimization designfor piezoelectric cylindrical flat shell structurecontrol areshown in Figure 6
The general genetic algorithm LQRandFEAwas utilizedfor integrated topology optimization design for piezoelectriccylindrical flat shell structurecontrol in this paper whichappeared in many literatures [7ndash9] Some steps in flowchartsuch as deletions of some low-stress elements and repair
03
02
01
0
02 0 minus02 0
02
04
06
Figure 7 The original flat shell structure and the appointed node(labeled with red color)
strategy for disconnected topology based on morphologyoperator are described in Sections 52 and 53
7 Numerical Examples
The simulation computations for two piezoelectric cylin-drical flat shell structures are carried out by the proposedmethod Here the base material for two simulated structuresis used by steel and its material properties are as followsYoungrsquos elastic modulus 119864 = 21 times 10
11Nm2 Poissonrsquosratio V = 03 and density 120588 = 7800 kgm3 Meanwhile thematerial properties of PZT patches are as follows Youngrsquoselasticmodulus119864
119901= 63times10
10Nm2 Poissonrsquos ratio V119901= 03
and density 120588119901= 7600 kgm3 Piezoelectric strain constants
11988931
= 11988932
= minus171 times 10minus12 CN and dielectric constant
120576 = 15 nfm
71 Semicylindrical Piezoelectric Shell Structure The semi-cylindrical shell structure is shown in Figure 7 The structureis discretized to 10 times 16 elements with the left and rightedges being fixed and the other edges are free The sizes of
8 Shock and Vibration
03
02
01
0
02 0 minus02 0
02
04
06
Figure 8 The optimal number and positions of PZT patches(labeled with black color)
Table 1The results for integrated optimization design of semicylin-drical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
1900 39208 39208 62244 62244 73553
structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887
= 0004m The thickness of PZT patches is 119905119901
=
0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively
The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)
that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10
minus12 and 10 times 10minus13 respectively The
optimal number and positions of PZT patches are shown inFigure 8
In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1
The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1
The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid
elements and void elements respectivelyThe optimal control
03
02
01
0
020 minus02
0
02
04
06
Figure 9 The optimal structure topology
14
12
10
8
60 20 40 60 80 100 120 140 160 180 200
S(k
g)
Evolutionary iterations
Figure 10 The iteration history of mass index
weight parameters are 11987711988612
= 09789 times 10minus12 and 119877
11988634
=
04155 times 10minus12 respectively The iteration history of mass
index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-
control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11
72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905
119887= 0001m The thickness of
PZT patches is 119905119901
= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively
The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast
119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Shock and Vibration
03
02
01
0
02 0 minus02 0
02
04
06
Figure 8 The optimal number and positions of PZT patches(labeled with black color)
Table 1The results for integrated optimization design of semicylin-drical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
1900 39208 39208 62244 62244 73553
structure are as follows the length along generatrix directionis 119871 = 06m the ring radius is 119877 = 03m and the thicknessis 119905119887
= 0004m The thickness of PZT patches is 119905119901
=
0001m The static load condition is that the load of 1000Nis imposed in the negative 119911-direction of the appointed nodewhich is shown in Figure 7The genetic algorithmparametersare as follows population size the crossover rate and themutation rate are 40 08 and 002 respectivelyThe allowablemaximal iterations and the penalty factor are 200 and 1000respectively
The vibration of the cylindrical shell structure is causedby the sinusoidal excitation 119865 = 1000 lowast sin(2 lowast 119901119894 lowast 15 lowast 119905)
that is imposed in the negative 119911-direction of the same nodeas the static load conditionThe amplitude of applied voltagesis not more than 100V the allowable maximal number ofpiezoelectric elements or actuators is not more than 6 Wehope that we can use the several PZT actuators that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 15Hz The upper and lower bounds for control weightparameters are 10 times 10
minus12 and 10 times 10minus13 respectively The
optimal number and positions of PZT patches are shown inFigure 8
In Figure 8 the black blocks are piezoelectric couplingelements as expressed in Figure 1
The results of integrated topology optimization design forsemicylindrical shell structurecontrol are shown in Table 1
The optimal structure topology is shown in Figure 9In Figure 9 the black and gray block areas are the solid
elements and void elements respectivelyThe optimal control
03
02
01
0
020 minus02
0
02
04
06
Figure 9 The optimal structure topology
14
12
10
8
60 20 40 60 80 100 120 140 160 180 200
S(k
g)
Evolutionary iterations
Figure 10 The iteration history of mass index
weight parameters are 11987711988612
= 09789 times 10minus12 and 119877
11988634
=
04155 times 10minus12 respectively The iteration history of mass
index for the coupled structurecontrol is shown in Figure 10The vibration responses comparisons between under un-
control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 11
72 Cantilever Piezoelectric Cylindrical Shell Structure Thecantilever cylindrical shell structure is shown in Figure 12The structure is discretized to 10 times 32 elements with thefront ring edges being fixed and the other edges are freeThesizes of structure are as follows the length along generatrixdirection is 119871 = 08m the ring radius is 119877 = 04mand the base thickness is 119905
119887= 0001m The thickness of
PZT patches is 119905119901
= 00015m The static load conditionis that the load of 10000N is imposed in the negative 119911-direction of the specified node which is shown in Figure 11The genetic algorithm parameters are as follows populationsize the crossover rate and the mutation rate are 40 08 and002 respectively The allowable maximal iterations and thepenalty factor are 100 and 10000 respectively
The vibration of the cantilever cylindrical shell structureis caused by the sinusoidal excitation 119865 = 10000 lowast sin(2 lowast
119901119894 lowast 175 lowast 119905) that is imposed in the negative 119911-direction ofthe same node as the static load condition The amplitude
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 9
15
1
05
0
minus05
minus1
minus150 005 01 015 02502 03 035 04 045 05
t (s)
The t
ime r
espo
nse (
m)
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
times10minus4
Figure 11 The time response of vibration structure at the excitationnode along negative 119911-direction
0
0
02
04
06
08
Figure 12 The original flat shell structure and the specified node(labeled with red color)
of applied voltages is not more than 100V the allowablemaximal number of PZT patches is not more than 6 Wehope that we can use the several PZT patches that are notmore than 6 to control the six lower-order natural modes ofthe optimization structure The frequency constraint is thatthe first natural frequency of the optimal structure is not lessthan 2Hz The upper and lower bounds for control weightparameters are 10 times 10
minus8 and 10 times 10minus9 respectively The
optimal number and positions of PZT patches are shown inFigure 13The four piezoelectric flat shell elements are 55 6696 and 105 respectively
The results of integrated optimization design for can-tilever cylindrical shell structurecontrol are shown inTable 2
The optimal structure topology is shown in Figure 14
0
02
04
06
08
Figure 13 The optimal number and positions of PZT patches
0
02
04
06
08
Figure 14 The optimal structure topology
Table 2The results for integrated optimization design of cantilevercylindrical shell
1198911Hz 119881
1V 119881
2V 119881
3V 119881
4V 119878 (Kg)
2886 8842 8842 24806 24806 78205
The optimal control weight parameters are 11987711988612
=
09561times10minus8 and 119877
11988634
= 07574times10minus8 respectivelyThe iter-
ation history of mass index for the coupling structurecontrolis shown in Figure 15
The vibration responses comparisons between under un-control condition and under integrated optimization controlcondition for this shell structure is shown in Figure 16
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 Shock and VibrationS
(kg)
9
85
8
750 20 40 60 80 100
Evolutionary iterations
Figure 15 The iteration history of mass index
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus250 01 02 03 04 05
t (s)
The t
ime r
espo
nse (
m)
times10minus7
Unc
ontro
lIn
tegr
ated
optim
izat
ion
cont
rol
Figure 16 The time response comparisons at the excitation nodealong negative 119911-direction
8 Conclusions
In order to enlarge the scope of application of the integratedoptimization design of structurecontrol in engineering andpractice the integrated topology optimization design ofpiezoelectric cylindrical flat shell structurecontrol is studiedin this paper Firstly the piezoelectric flat shell elementis used for dynamic modeling of piezoelectric cylindricalshell structure Secondly based on the LQR control incoupling modal space the integrated topology optimizationdesign modeling with the GA is presented Meanwhile thestrategies with the deletions of some low-stress elements andmorphology operators are used for realization of integratedoptimization design and smooth structure topology Finallynumerical simulation computations are carried out for twocylindrical flat shell structures
The results of numerical examples show that the proposedmethod can produce clear structure topology and highcontrol performance In two examples the number and thepositions of PZT patches are reasonable and the amplitude ofapplied voltages is small which are valuable to the realizationand efficiency of integrated topology optimization design ofpiezoelectric cylindrical flat shell structure in engineeringpractice
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the support byNatural Science Foundation of China under Grant 51305048and Scientific Research Fund of Hunan Provincial EducationDepartment under Grant 11C0045
References
[1] P Vidal L Gallimard and O Polit ldquoShell finite element basedon the Proper Generalized Decomposition for the modeling ofcylindrical composite structuresrdquo Computers amp Structures vol132 pp 1ndash11 2014
[2] Y Zhang H Niu S Xie and X Zhang ldquoNumerical and exper-imental investigation of active vibration control in a cylindricalshell partially covered by a laminated PVDF actuatorrdquo SmartMaterials and Structures vol 17 no 3 Article ID 035024 2008
[3] X T Cao L Shi X S Zhang and G Jiang ldquoActive control ofacoustic radiation from laminated cylindrical shells integratedwith a piezoelectric layerrdquo Smart Materials and Structures vol22 no 6 Article ID 065003 pp 3ndash18 2013
[4] W P Li and H Huang ldquoIntegrated optimization of actuatorplacement and vibration control for piezoelectric adaptivetrussesrdquo Journal of Sound andVibration vol 332 no 1 pp 17ndash322013
[5] Y Zhu J Qiu H Du and J Tani ldquoSimultaneous optimaldesign of structural topology actuator locations and controlparameters for a plate structurerdquoComputationalMechanics vol29 no 2 pp 89ndash97 2002
[6] Z Kang and L Tong ldquoIntegrated optimization of materiallayout and control voltage for piezoelectric laminated platesrdquoJournal of Intelligent Material Systems and Structures vol 19 no8 pp 889ndash904 2008
[7] B Xu J S Jiang and J P Ou ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart trussesusing genetic algorithmrdquo Journal of Sound and Vibration vol307 no 3ndash5 pp 393ndash427 2007
[8] B Xu J P Ou and J S Jiang ldquoIntegrated optimization ofstructural topology and control for piezoelectric smart platebased on genetic algorithmrdquo Finite Elements in Analysis andDesign vol 64 pp 1ndash12 2013
[9] W-S Hwang and H C Park ldquoFinite element modeling ofpiezoelectric sensors and actuatorsrdquo AIAA journal vol 31 no5 pp 930ndash937 1993
[10] S Y Wang K Tai and S T Quek ldquoTopology optimization ofpiezoelectric sensorsactuators for torsional vibration control ofcomposite platesrdquo Smart Materials and Structures vol 15 no 2pp 253ndash269 2006
[11] O Sigmund ldquoMorphology-based black and white filters fortopology optimizationrdquo Structural and Multidisciplinary Opti-mization vol 33 no 4-5 pp 401ndash424 2007
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of