Research ArticleOn the Topology Optimization of Elastic SupportingStructures under Thermomechanical Loads

Jie Hou Ji-Hong Zhu and Qing Li

Engineering Simulation and Aerospace Computing (ESAC) The Key Laboratory of Contemporary Design andIntegrated Manufacturing Technology Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Ji-Hong Zhu jhzhunwpueducn

Received 21 November 2015 Revised 11 May 2016 Accepted 2 June 2016

Academic Editor Haibo Wang

Copyright copy 2016 Jie Hou et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is to present a thermomechanical topology optimization formulation By designing structures that support specificnondesignable domain optimization is to suppress the stress level in the nondesignable domain and maintain global stiffnesssimultaneously A global stress measure based on 119901-norm function is then utilized to reduce the number of stress constraintsin topology optimization Sensitivity analysis employs adjoint method to derive the global stress measure with respect to thetopological pseudodensity variables Some particular behaviors in thermomechanical topology optimization of elastic supportssuch as the influence of different thermomechanical loads and the existence of intermediate material are also analyzed numericallyFinally examples of elastic supports on a cantilever beam and a nozzle flap under different thermomechanical loads are tested withreasonable optimized design obtained

1 Introduction

Optimization design of thermoelastic structures is an impor-tant problem in aeronautics and aerospace products such asturbine engine components and thermal protection systems(TPS) As one of themost challenging topics in topology opti-mization many difficulties are to be settled before effectiveengineering application

One the one hand literatures dealing with structuralconfigurations design for different heat conduction purposeshave been present using topology optimization where thetotal potential energy was usually minimized as the designobjective For example Li et al [1] optimized the heatconduction path using the typical evolutionary structuraltopology optimization The ineffective materials were grad-ually removed from the design domain iteratively The finaldesigns turned out to be some tree-like structures for volume-to-point heart conduction Later density based topologyoptimization methods were used to obtain similar optimizeddesigns [2 3] Gao et al [4] recognized the heat sourcesas design-dependent thermal loads some more effective

and clear optimized designs were finally obtained Later thedesign-dependent effect of heat conduction and convectionin homogenization based topology optimization was furtherinvestigated [5] Recently Dirker and Meyer [6] extendedthe heat conduction topology optimization to the internalcooling system design

One the other hand topology optimization of coupledthermoelastic structures was generally considered as meancompliance minimization problems with different solutionsin earlier works (eg [7ndash13]) Gao and Zhang [14] thenextended such formulation to thermoelastic structures ofmultiphase materials The concept of thermal stress coef-ficient (TSC) defined by Youngrsquos modulus and thermalexpansion coefficients was proposed to characterize thedependence of thermal stress and thermal expansion coef-ficient upon element pseudodensities Deaton and Grandhi[15] investigated the compliance design of the supportingstructures in thermoelastic environment Nonintuitivenessin design space and the design dependency that occurswith thermal loading were presented and different topologyoptimization formulations were used to solve the problem

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2016 Article ID 7372603 12 pageshttpdxdoiorg10115520167372603

2 International Journal of Aerospace Engineering

A recent literature survey by Deaton and Grandhi [16] hassummarized the advances and applications of thermoelastictopology optimization

Actually eligible design of thermoelastic structuresshould meet both requirements of structural stiffness andstress to prevent structural failure As a result recent effortshave been continuously devoted to highlighting thermoe-lastic problems with the consideration of compliance strainenergy and maximum stress P Pedersen and N L Ped-ersen [17 18] proposed strength optimization minimizingthe maximum Von-Mises stress for thermoelastic structureswhere thermal stress was generated by thermal expansion ofclamped structures Zhang et al [19] presented a topologyoptimization formulation of elastic supports for thermoelas-tic structures and investigated the strength design by directlyminimizing the maximum Von-Mises stress in the nondes-ignable subregion They further analyzed the differences ofelastic strain energy and mean compliance of thermoelasticstructures Recently Deaton and Grandhi [20] presented atopology optimization method with combined mechanicaland thermal loads A relaxation technique and a modified119901-norm function were utilized to remove the singularityphenomenon and aggregate the large number of stressconstraints

In this paper we continue to study the thermoelasticstructures topology optimization suppressing stress leveland maintaining global stiffness simultaneously As shownin Figure 1 a typical thermoelastic continuum structureundergoes coupled thermal andmechanical loads SubregionΩ is nondesignable andΩ

119904is the elastic supporting structure

assigned as the design domain Large stress level appearsin the nondesignable domain due to the coupled effect ofthermal expansion and mechanical loads Design objectivehere is to find proper structural configuration of the elasticsupports that suppress the stress in the supported nondes-ignable domain and simultaneously maintain global stiffnessevaluated by structural compliance As the stress constraintsare only applied on the nondesignable domain the singularityproblem in stress-based topology optimization is actually notinvolved here

To have an in-depth understanding of the thermoelasticproblems with elastic supports we perform a detailed anal-ysis of a three-bar truss analytical model to present someillustrative phenomena during optimizationThebehaviors ofstructural compliance and stress are discussed respectivelyNumerical examples of elastic supports on a 2D cantileverbeam and 3D nozzle flap are finally presented to verify thevalidity of the presented formulation

2 Three-Bar Truss Analytical Model

The investigation on a truss-frame structure can reveal theunderlying schemeof continuum topology optimization (eg[21 22]) In this section the analytical solutions of a three-bartruss system are derived under different thermomechanicalloads with regard to the stress constraints The structuralcompliance of thermoelastic problems is evaluated withfurther details discussed here

Elastic support

Nondesignabledomain

ΔT

ΔT Ωs

Ω

Fm

Figure 1 Topology optimization of thermoelastic structure withelastic support

1

2

3

4

Elasticsupport

Nondesignable domain

ΔT

E(2)

E(1)

x2

E(3)fmy

fmx

x3x1

x

y 60∘ 60∘

Figure 2 Three-bar truss system with elastic support

As sketched in Figure 2 the truss system is comprisedof three equal-length bars that is 119864

(1) 119864(2) and 119864

(3) The

barsrsquo cross-sectional areas are assigned as 1199091 1199092 and 119909

3

respectively One end of those bars is fixed and the other endsare hinged together to node 4 The mechanical loads appliedon node 4 can be decomposed into two components 119891

119898119909

and 119891119898119910 Meanwhile all three bars undergo a temperature

increase Δ119879 and the corresponding thermal expansion willcause thermal stresses 119864

(1)and 119864

(2)constitute the elastic

support and the cross-sectional areas that is 1199091and 119909

2

which are assigned as the design variables 119864(3)

is chosen asthe nondesignable domain which means 119909

3is fixed In this

example the continuous design variables are used to evaluatethe behaviors of the structural response

According to the thermoelastic theory the thermal loadin each bar can be calculated as

119865119894th = 119864120572Δ119879119909119894 (1)

where 119864 is Youngrsquos modulus 120572 is the thermal expansioncoefficientΔ119879 is the temperature increase and119909

119894is the cross-

sectional area of the corresponding bar

International Journal of Aerospace Engineering 3

The strain and stress of the bar 119864(119894)can be expressed as

120576119894=119880119894119909

119871

120590119894= 119864120576119894119898= 119864 (120576

119894minus 120576119894th) = 119864(

119880119894119909

119871minus 120572Δ119879)

(2)

where 119880119894119909is the displacement with respect to the elemental

119909-axis and 120576119894119898

and 120576119894th are corresponding mechanical and

thermal strainThe elastic strain energy 120601 is defined as the potential

mechanical energy in the elastic body which is written as

120601 =1

2int (120576119898)119879D120576119898119889119881

=1

2int (120576 minus 120576th)

119879D (120576 minus 120576th) 119889119881

=1

2int 120576119879D120576 119889119881 minus int 120576119879D120576th119889119881 +

1

2int 120576119879

thD120576th119889119881

(3)

where D is the elastic matrix and 120576 is the total strain vectorconsisting of mechanical and thermal items 120576

119898and 120576th

Then we consider the mean compliance of structure

119862 =1

2F119879U = 1

2F119879Kminus1F (4)

where F and U are nodal load and displacement vectors K isglobal stiffness matrix

According to the existing works the strain energy mini-mization is more beneficial for the stress reduction while themean compliance reflects the structural overall stiffness [19]The purpose of this study is to find the optimized configura-tion of elastic supports preventing large deformations result-ing from thermal and mechanical load Naturally the meancompliance minimization is selected as the design objectivethroughout this paper Besides two design constraints that isvolume constraint and stress constraint are involved in thisstudy The formulation of the optimization can be expressedas

Find 1199091 1199092

Min 119862

st (1199091+ 1199092) 119871 le 119881

100381610038161003816100381612059031003816100381610038161003816 le |120590|

(5)

To demonstrate the relation between the design variables andthe global compliance parameters listed in Table 1 are used

At the first place the mechanical load is fixed with 119891119898119909=

119891119898119910

= 7000N Structural compliance with different tem-perature increases is plotted by sets of contours in Figure 3Thick solid line and dashed line denote stress and volumeconstraints respectively Along with the lower and upperlimits of the design variables these constraints define thefeasible regions of optimizationwhich is indicated as the darkarea in the figure The black spot denotes the optimal point

Table 1 Constant parameters list

Youngrsquos modulus 119864 (Pa) 2 times 1011

Thermal expansion coefficient 120572 (∘Cminus1) 1 times 10minus5

Length of bar 119871 (m) 1

Cross-sectional area 1199093(m2) 1 times 10

minus4

Upper limit of stress in nondesignable domain |120590| (MPa) 80

Upper limit of design domainrsquos volume 119881 (m3) 25times10minus4

Lower limit of design variables 119909 (m2) 5 times 10minus5

Upper limit of design variables 119909 (m2) 2 times 10minus4

where the compliance reaches a minimum Those symbolsigns are applied to Figures 3 and 4

As illustrated in Figure 3 when the structure undergoesthermomechanical loads a small temperature increase willlead to nonmonotone compliance with respect to the designvariables Consequently the volume constraint may not beactive as shown in Figures 3(b) to 3(d) In this case usingmore material could weaken the structural mechanical per-formance due to the thermomechanical loading condition

In another case fixing Δ119879 = 20∘C the mechanical loadsare proportionally raisedThe contour lines of compliance areshown in Figure 4 At the beginning the stress is primarilycaused by thermal load Very compliant structures are usedto offset thermal stress Typically when both mechanical andthermal loads are small as in Figure 4(a) the optimal pointlies at the lower left corner of the feasible region At this pointthe stress constraint is inactive Asmechanical loads increasemore materials are required to strengthen the structure Thevolume constraint is finally active in Figure 4 We can foreseethat as themechanical loads increase there will be no feasibledesign for this problem

Notably there exist some critical optimal points forthermomechanical loads for example in Figures 3(a) and4(d) The volume constraint is exactly active there A slightincrease of temperature or decrease of mechanical loads willlead to inactive volume constraint These phenomena alsohappen in topology optimization of continuum structureswith thermomechanical loads where inactive volume con-straints and elements with intermediate densities are alwaysfound (see [20])

3 Formulation of ThermomechanicalTopology Optimization

Based on the analyses in the previous section we propose touse the topology optimization formulation as

Find X = (1199091 1199092 119909

119899)

Min 119862 =1

2F119879U = 1

2(Fth + F119898)

119879U

st 119881 (X) =119899

sum

119894=1

119909119894V119894le 119881

max119895=12119898

(120590VM119895) le 120590

4 International Journal of Aerospace Engineering

138

141

144

141

144

144

147

147

147

150

150

150

153

153

156

159

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 16∘C 119891119898119909= 119891119898119910= 7000N

160158

158

156

156

154

154

154

154152

152

152

150150

150148

148 146144

146

156

156

158

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 18∘C 119891119898119909= 119891119898119910= 7000N

153 153156 156

159 159

159

162162

162

165

165

165

168

168

168 171

171

171

174

174177

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7000N

248240

240

232

232

224

224

224

216

216

216

208

208

208

200

200

200192

192

192

184

184176

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 25∘C 119891119898119909= 119891119898119910= 7000N

Figure 3 Optimizations under different temperature increases

0 lt 120575 le 119909119894le 1

119894 = 1 2 119899

(6)

where 119899 and119898 are the numbers of elements in elastic supportand nondesignable domain respectively X is the vector ofdesign variables that is the pseudodensities which varybetween 0 and 1 to describe the material distribution over thedesign domain 119862 is the global compliance of the structureFth and F

119898are the thermal and mechanical load vectors

composing the nodal load vector F 120590VM119895

is the Von-Misesstress of the 119895th element in the nondesignable domain 119881(X)is the volume of the design domain and V

119894is the volume of the

119894th element 120575 is a small constant set as 0001 in this paper toavoid singularity of stiffness matrix

31 Design-Dependent Thermal Load Unlike the staticmechanical load thermal load is a typical design-dependentload depending upon the material layout over the design

International Journal of Aerospace Engineering 5

115

120

120

120

125

125

125

130

130

130

135

135

135

140

140

140145

145

145

150

150 155

160

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 5500N

136

140140

140 144

144

144

148

148

148152

152

152156

156

156 160

160

160164

164

168

172

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 6500N

180

180

180

180

177

177

177

174

174174

171 171168

183

183

186

186

189

192

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7500N

195

195

192

192

192192

192

195

195

198

189189

189

186

186

186183183

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7800N

Figure 4 Optimizations under different mechanical loads

domain Thus the matching relation between the thermalload and stiffness should be carefully handled

According to the thermoelastic theory the thermal loadover the 119894th element is expressed as

F119894th = int

Ω119894

B119879119894D119894120576119894th119889Ω (7)

where B119894and D

119894are the element strain-displacement matrix

and elasticity matrix respectively 120576119894th is the thermal strain of

the 119894th element D119894is evaluated in terms of Youngrsquos modulus

119864119894 Here a polynomial interpolation is used to link the

pseudodensity variables to element elastic modulus [23]

119864119894= ((1 minus 120573) 119909

119901

119894+ 120573119909119894) 1198641198940 (8)

where 119901 = 3 and 120573 = 116 This polynomial model workswell in topology optimization with design-dependent loadssuch as inertial loads dynamic loads and thermal loads Itmaintains a positive gradient when the pseudodensity is zero

Suppose the thermal expansion coefficient is tempe-rature-independent and only the steady-state temperature

6 International Journal of Aerospace Engineering

field is taken into account The thermal strain vector can bewritten as follows

120576119894th = 120572119894Δ119879119894120572

119879

(9)

Here 120572119894is the thermal expansion coefficient Δ119879

119894denotes the

temperature increase120572 is a constant vector for the calculationof strain vector which is [1 1 1 0 0 0] in this paper [17]

Referring to the conception of thermal stress coefficient(TSC) [14] the element stiffness and the thermal stress loadshould be penalized independently in terms of element pseu-dodensity A linear interpolation to the thermal expansioncoefficient reads

120572119894= 1199091198941205721198940 (10)

where 1205721198940

is the original thermal expansion coefficientConsequently the elemental thermal expansion coefficientnow varies with the pseudodensities in coordination withYoungrsquos modulus

With the substitution of (8)ndash(10) into (7) the thermal loadcan be expressed as

F119894th = 120574119894 (119909119894) F1198940th

F1198940th = int

Ω119894

B119879119894D1198940120572 119889Ω

120574119894= 119864119894120572119894

(11)

where F1198940th is the thermal load when the element is solid with

unit Youngrsquos modulus unit thermal expansion coefficientand unit temperature increase which is always constant in theoptimization process 120574

119894denotes the thermal stress coefficient

(TSC) which can be treated as an inherent material propertyWith the introduction of TSC the thermal load can beexplicitly expressed as the function of design variables

32 The Global Stress Measure Difficulties of min-maxproblems are generally involved in stress-based topologyoptimization Naturally stress is evaluated and constrained ineach element In this way the overall stress level is restrictedbelow the prescribed limit But the scale of the optimizationproblem and the computational cost increase dramaticallyTo remedy the drawback aggregationmethods combining alllocal stresses into one or several Kreisselmeier-Steinhauser(KS) or 119901-norm global functions [24 25] are generally usedA tradeoff between the global and local stress approachesis that global stress measure reduces the computational costwhile it introduces high nonlinearity in global measurefunctions and loses the control over local stresses Differentaggregation strategies have been brought in For examplethe active set approach [26] aggregates the active constraintsinto one set which highly reduces the number of stressconstraints However the changes of the active set can leadto poor convergence Le et al [27] introduced an adaptivelyupdated 119901-norm aggregation method which can preciselyapproximate the maximum local stress A clustered method[28] was proposed as a compromise between global andlocal methods The constraints were aggregated into several

clustersThismethod was proven to be efficient and robust bynumerical examples but the results strongly depended uponthe number of clusters

Concerning the optimization of elastic support structurein this paper the thermal stress constraints are imposedonly in the nondesignable domain Compared with thetotal elements number the number of evaluated stresses iscomparatively small Without loss of generality the standard119901-norm measure is utilized which reads

120590119875119873= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875

(12)

where 120590119875119873

is the 119901-norm of the elemental Von-Mises stressand 119875 is the aggregation parameter Apparently the 119901-normdegenerates to the summation of the stress components when119875 is assigned as 1 Moreover it can be proven mathematicallythat

lim119901rarrinfin

(

119898

sum

119895=1

(120590VM119895)119875

)

1119875

= max119895=12119898

120590VM119895 (13)

Obviously with a sufficiently large value for 119875 the globalmeasure can exactly match the maximum stress Howeverthe large value may lead to oscillation and ill-posed problems[29] Consequently to balance the effect of constraintsrsquoquality and convergence efficiency the value of 119875 should beproperly chosen which is assigned as 15 in this paper

33 Adjoint Sensitivity Analysis Firstly the sensitivity of theglobal compliance with the design-dependent thermal load isderived and written as

120597119862

120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot120597U120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

= U119879 120597Fth120597119909119894

minus1

2U119879 120597K120597119909119894

U

(14)

Sensitivities of the thermal load with respect to the designvariables can be obtained by differentiating its definitionformulated in (11) The derivative of the stiffness matrix canbe easily evaluated according to the interpolation functionsin (8)

Sensitivity analysis is then carried out to evaluate theglobal stress measure in the nondesignable domain with

International Journal of Aerospace Engineering 7

respect to the pseudodensity variables which can be writtenas120597120590119875119873

120597119909119894

= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875minus1

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

(15)

Here the Von-Mises equivalent stress is defined as

120590VM119895radic120590119879119895V120590119895

= radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus120576119895th)

(16)

where 120590119895is the element stress vector U

119895and 120576

119895th are theelement nodal displacement vector and thermal strain vectorThe matrix V is a constant matrix that is

V2D =[[[[

[

1 minus1

20

minus1

21 0

0 0 3

]]]]

]

V3D =

[[[[[[[[[[

[

(

2 minus1 minus1

minus1 2 minus1

minus1 minus1 2

) 03times3

03times3

(

6

6

6

)

]]]]]]]]]]

]

(17)

As only the Von-Mises stresses in the nondesignable domainare concerned we have

120597120590VM119895

120597119909119894

=(B119895U119895minus 120576119895th)119879

D119879119895VD119895

radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus 120576119895th)

B119895

120597U119895

120597119909119894

=1

120590VM119895

sdot 120590119879

119895VD119895B119895

120597U119895

120597119909119894

(18)

where 120597D119895120597119909119894= 120597120576119895th120597119909119894 = 0 In this case the singularity

phenomenon in stress-based topology optimization is actu-ally not involved in this optimization problem

We can also defineU119895= A119895U

120597U119895

120597119909119894

= A119895

120597U120597119909119894

(19)

Table 2 Material properties used in the optimization

Properties Elasticsupport

Nondesignabledomain

Elastic modulus(MPa) 71000 3500

Coefficient of thermalexpansion (10minus6∘C) 23 7

Poissonrsquos ratio 033 04

The matrix A transforms the global nodal displacementvector to an element one

Based on the differentiation of finite element equilibriumequation we have the derivative of the nodal displacementvector

120597U120597119909119894

= Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U) (20)

The substitution of the above equation into the derivative of119901-norm aggregation function produces

120597120590119875119873

120597119909119894

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

(21)

Suppose

Q119879 = (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)Kminus1 (22)

is adjoint displacement the derivative of the119901-norm functioncan be obtained by an additional finite element calculation

4 Numerical Study

In this section a 2D cantilever beam and a 3D nozzle flap of aturbine engine are tested to study the topology optimizationof elastic supports under thermomechanical loads

41 2D Cantilever Beam As shown in Figure 5 a 2D can-tilever beam consists of a nondesignable top surface andan elastic support assigned as design domain The materialproperties are listed in Table 2 Since the thermal expansioncoefficient of elastic support is higher than that of thenondesignable domain significant thermal stresseswill occuras a result of temperature increase

In accordance with Section 3 the global compliance ofthe whole structure is minimized as the design objectiveThe volume fraction is constrained to 50 and the upperbound of the stress constraints in the nondesignable domainis 200MPa

The optimized results with various load combinations arelisted and compared in Figure 6 The result in Figure 6(a)is set as a benchmark with clear configuration Firstly themechanical load is fixed at 20Nmm in Figures 6(a)ndash6(d)

8 International Journal of Aerospace Engineering

q

Elastic support

Nondesignable domain

ΔT

50mm 5mm

20mm

100mm

Figure 5 Definition of the 2D cantilever beam

(a) Δ119879 = 50∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum Stress 154MPa

(b) Δ119879 = 200∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum stress 1664MPa

(c) Δ119879 = 500∘C 119902 = 20Nmm Optimized volumefraction 417 Maximum stress 200MPa

(d) Δ119879 = 1000∘C 119902 = 20Nmm Optimized volumefraction 358 Maximum stress 200MPa

(e) Δ119879 = 500∘C 119902 = 40Nmm Optimized volumefraction 50 Maximum stress 200MPa

(f) Δ119879 = 500∘C 119902 = 200Nmm Optimized volumefraction 50 Maximum stress 8124MPa

Figure 6 Optimized results of minimizing compliance under different loads

As the temperature increases the volume constraint startsto be inactive More elements with intermediate densitiesarise This is due to the nature of thermal stress constraintwhere the elements with compliant materials can offsetthe thermal stress better Later considering the changes inFigures 6(c) 6(e) and 6(f) clear structure patterns gradu-ally appear when larger mechanical loads are applied Thusthe elastic support is able to undergo the pressure with goodstiffness However when the thermomechanical loads are toohigh the optimization finds no feasible solution where thestress constraints are violated as shown in Figure 6(f)

The optimized results have shown that the relative mag-nitude of the thermal and mechanical load greatly influencesthe optimized results which have good consistency with theanalytical solution of the three-bar truss model in Section 3Moreover the existence of a large amount of intermediatedensity material is reasonable when the thermal load isdominant over the mechanical load Eliminating the greyelements directly or using some numerical schemes mayimprove the global stiffness but will unfortunately lead tohigher stress level

42 3D Nozzle Flap As shown in Figure 7 a nozzle flapof a turbine engine is composed of titanium stiffeners and

Table 3 Material properties used in the optimization

Material properties Titanium CeramicElastic modulus(MPa) 11000 3500

Coefficient of thermalexpansion (10minus6∘C) 10 7

Poissonrsquos ratio 033 04

a ceramic plate Material properties are listed in Table 3Figure 7 also illustrates the thermoelastic loads applied onthe model including a uniform pressure of 1MPa and globaltemperature increase of 500∘C Significant thermal stressesare generated in the plate due to the different thermalexpansion coefficient In practical design the stiffeners areassigned as the elastic support design domain and the plateis nondesignable The design objective is to minimize theglobal compliance with a 25MPa stress constraint on thenondesignable plate and a 30material volume constraint onthe design domain

The optimized design of elastic support as shown inFigure 8 is obtained by topology optimization using theproposed formulation The optimized design is presented by

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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International Journal of

2 International Journal of Aerospace Engineering

A recent literature survey by Deaton and Grandhi [16] hassummarized the advances and applications of thermoelastictopology optimization

Actually eligible design of thermoelastic structuresshould meet both requirements of structural stiffness andstress to prevent structural failure As a result recent effortshave been continuously devoted to highlighting thermoe-lastic problems with the consideration of compliance strainenergy and maximum stress P Pedersen and N L Ped-ersen [17 18] proposed strength optimization minimizingthe maximum Von-Mises stress for thermoelastic structureswhere thermal stress was generated by thermal expansion ofclamped structures Zhang et al [19] presented a topologyoptimization formulation of elastic supports for thermoelas-tic structures and investigated the strength design by directlyminimizing the maximum Von-Mises stress in the nondes-ignable subregion They further analyzed the differences ofelastic strain energy and mean compliance of thermoelasticstructures Recently Deaton and Grandhi [20] presented atopology optimization method with combined mechanicaland thermal loads A relaxation technique and a modified119901-norm function were utilized to remove the singularityphenomenon and aggregate the large number of stressconstraints

In this paper we continue to study the thermoelasticstructures topology optimization suppressing stress leveland maintaining global stiffness simultaneously As shownin Figure 1 a typical thermoelastic continuum structureundergoes coupled thermal andmechanical loads SubregionΩ is nondesignable andΩ

119904is the elastic supporting structure

assigned as the design domain Large stress level appearsin the nondesignable domain due to the coupled effect ofthermal expansion and mechanical loads Design objectivehere is to find proper structural configuration of the elasticsupports that suppress the stress in the supported nondes-ignable domain and simultaneously maintain global stiffnessevaluated by structural compliance As the stress constraintsare only applied on the nondesignable domain the singularityproblem in stress-based topology optimization is actually notinvolved here

To have an in-depth understanding of the thermoelasticproblems with elastic supports we perform a detailed anal-ysis of a three-bar truss analytical model to present someillustrative phenomena during optimizationThebehaviors ofstructural compliance and stress are discussed respectivelyNumerical examples of elastic supports on a 2D cantileverbeam and 3D nozzle flap are finally presented to verify thevalidity of the presented formulation

2 Three-Bar Truss Analytical Model

The investigation on a truss-frame structure can reveal theunderlying schemeof continuum topology optimization (eg[21 22]) In this section the analytical solutions of a three-bartruss system are derived under different thermomechanicalloads with regard to the stress constraints The structuralcompliance of thermoelastic problems is evaluated withfurther details discussed here

Elastic support

Nondesignabledomain

ΔT

ΔT Ωs

Ω

Fm

Figure 1 Topology optimization of thermoelastic structure withelastic support

1

2

3

4

Elasticsupport

Nondesignable domain

ΔT

E(2)

E(1)

x2

E(3)fmy

fmx

x3x1

x

y 60∘ 60∘

Figure 2 Three-bar truss system with elastic support

As sketched in Figure 2 the truss system is comprisedof three equal-length bars that is 119864

(1) 119864(2) and 119864

(3) The

barsrsquo cross-sectional areas are assigned as 1199091 1199092 and 119909

3

respectively One end of those bars is fixed and the other endsare hinged together to node 4 The mechanical loads appliedon node 4 can be decomposed into two components 119891

119898119909

and 119891119898119910 Meanwhile all three bars undergo a temperature

increase Δ119879 and the corresponding thermal expansion willcause thermal stresses 119864

(1)and 119864

(2)constitute the elastic

support and the cross-sectional areas that is 1199091and 119909

2

which are assigned as the design variables 119864(3)

is chosen asthe nondesignable domain which means 119909

3is fixed In this

example the continuous design variables are used to evaluatethe behaviors of the structural response

According to the thermoelastic theory the thermal loadin each bar can be calculated as

119865119894th = 119864120572Δ119879119909119894 (1)

where 119864 is Youngrsquos modulus 120572 is the thermal expansioncoefficientΔ119879 is the temperature increase and119909

119894is the cross-

sectional area of the corresponding bar

International Journal of Aerospace Engineering 3

The strain and stress of the bar 119864(119894)can be expressed as

120576119894=119880119894119909

119871

120590119894= 119864120576119894119898= 119864 (120576

119894minus 120576119894th) = 119864(

119880119894119909

119871minus 120572Δ119879)

(2)

where 119880119894119909is the displacement with respect to the elemental

119909-axis and 120576119894119898

and 120576119894th are corresponding mechanical and

thermal strainThe elastic strain energy 120601 is defined as the potential

mechanical energy in the elastic body which is written as

120601 =1

2int (120576119898)119879D120576119898119889119881

=1

2int (120576 minus 120576th)

119879D (120576 minus 120576th) 119889119881

=1

2int 120576119879D120576 119889119881 minus int 120576119879D120576th119889119881 +

1

2int 120576119879

thD120576th119889119881

(3)

where D is the elastic matrix and 120576 is the total strain vectorconsisting of mechanical and thermal items 120576

119898and 120576th

Then we consider the mean compliance of structure

119862 =1

2F119879U = 1

2F119879Kminus1F (4)

where F and U are nodal load and displacement vectors K isglobal stiffness matrix

According to the existing works the strain energy mini-mization is more beneficial for the stress reduction while themean compliance reflects the structural overall stiffness [19]The purpose of this study is to find the optimized configura-tion of elastic supports preventing large deformations result-ing from thermal and mechanical load Naturally the meancompliance minimization is selected as the design objectivethroughout this paper Besides two design constraints that isvolume constraint and stress constraint are involved in thisstudy The formulation of the optimization can be expressedas

Find 1199091 1199092

Min 119862

st (1199091+ 1199092) 119871 le 119881

100381610038161003816100381612059031003816100381610038161003816 le |120590|

(5)

To demonstrate the relation between the design variables andthe global compliance parameters listed in Table 1 are used

At the first place the mechanical load is fixed with 119891119898119909=

119891119898119910

= 7000N Structural compliance with different tem-perature increases is plotted by sets of contours in Figure 3Thick solid line and dashed line denote stress and volumeconstraints respectively Along with the lower and upperlimits of the design variables these constraints define thefeasible regions of optimizationwhich is indicated as the darkarea in the figure The black spot denotes the optimal point

Table 1 Constant parameters list

Youngrsquos modulus 119864 (Pa) 2 times 1011

Thermal expansion coefficient 120572 (∘Cminus1) 1 times 10minus5

Length of bar 119871 (m) 1

Cross-sectional area 1199093(m2) 1 times 10

minus4

Upper limit of stress in nondesignable domain |120590| (MPa) 80

Upper limit of design domainrsquos volume 119881 (m3) 25times10minus4

Lower limit of design variables 119909 (m2) 5 times 10minus5

Upper limit of design variables 119909 (m2) 2 times 10minus4

where the compliance reaches a minimum Those symbolsigns are applied to Figures 3 and 4

As illustrated in Figure 3 when the structure undergoesthermomechanical loads a small temperature increase willlead to nonmonotone compliance with respect to the designvariables Consequently the volume constraint may not beactive as shown in Figures 3(b) to 3(d) In this case usingmore material could weaken the structural mechanical per-formance due to the thermomechanical loading condition

In another case fixing Δ119879 = 20∘C the mechanical loadsare proportionally raisedThe contour lines of compliance areshown in Figure 4 At the beginning the stress is primarilycaused by thermal load Very compliant structures are usedto offset thermal stress Typically when both mechanical andthermal loads are small as in Figure 4(a) the optimal pointlies at the lower left corner of the feasible region At this pointthe stress constraint is inactive Asmechanical loads increasemore materials are required to strengthen the structure Thevolume constraint is finally active in Figure 4 We can foreseethat as themechanical loads increase there will be no feasibledesign for this problem

Notably there exist some critical optimal points forthermomechanical loads for example in Figures 3(a) and4(d) The volume constraint is exactly active there A slightincrease of temperature or decrease of mechanical loads willlead to inactive volume constraint These phenomena alsohappen in topology optimization of continuum structureswith thermomechanical loads where inactive volume con-straints and elements with intermediate densities are alwaysfound (see [20])

3 Formulation of ThermomechanicalTopology Optimization

Based on the analyses in the previous section we propose touse the topology optimization formulation as

Find X = (1199091 1199092 119909

119899)

Min 119862 =1

2F119879U = 1

2(Fth + F119898)

119879U

st 119881 (X) =119899

sum

119894=1

119909119894V119894le 119881

max119895=12119898

(120590VM119895) le 120590

4 International Journal of Aerospace Engineering

138

141

144

141

144

144

147

147

147

150

150

150

153

153

156

159

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 16∘C 119891119898119909= 119891119898119910= 7000N

160158

158

156

156

154

154

154

154152

152

152

150150

150148

148 146144

146

156

156

158

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 18∘C 119891119898119909= 119891119898119910= 7000N

153 153156 156

159 159

159

162162

162

165

165

165

168

168

168 171

171

171

174

174177

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7000N

248240

240

232

232

224

224

224

216

216

216

208

208

208

200

200

200192

192

192

184

184176

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 25∘C 119891119898119909= 119891119898119910= 7000N

Figure 3 Optimizations under different temperature increases

0 lt 120575 le 119909119894le 1

119894 = 1 2 119899

(6)

where 119899 and119898 are the numbers of elements in elastic supportand nondesignable domain respectively X is the vector ofdesign variables that is the pseudodensities which varybetween 0 and 1 to describe the material distribution over thedesign domain 119862 is the global compliance of the structureFth and F

119898are the thermal and mechanical load vectors

composing the nodal load vector F 120590VM119895

is the Von-Misesstress of the 119895th element in the nondesignable domain 119881(X)is the volume of the design domain and V

119894is the volume of the

119894th element 120575 is a small constant set as 0001 in this paper toavoid singularity of stiffness matrix

31 Design-Dependent Thermal Load Unlike the staticmechanical load thermal load is a typical design-dependentload depending upon the material layout over the design

International Journal of Aerospace Engineering 5

115

120

120

120

125

125

125

130

130

130

135

135

135

140

140

140145

145

145

150

150 155

160

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 5500N

136

140140

140 144

144

144

148

148

148152

152

152156

156

156 160

160

160164

164

168

172

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 6500N

180

180

180

180

177

177

177

174

174174

171 171168

183

183

186

186

189

192

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7500N

195

195

192

192

192192

192

195

195

198

189189

189

186

186

186183183

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7800N

Figure 4 Optimizations under different mechanical loads

domain Thus the matching relation between the thermalload and stiffness should be carefully handled

According to the thermoelastic theory the thermal loadover the 119894th element is expressed as

F119894th = int

Ω119894

B119879119894D119894120576119894th119889Ω (7)

where B119894and D

119894are the element strain-displacement matrix

and elasticity matrix respectively 120576119894th is the thermal strain of

the 119894th element D119894is evaluated in terms of Youngrsquos modulus

119864119894 Here a polynomial interpolation is used to link the

pseudodensity variables to element elastic modulus [23]

119864119894= ((1 minus 120573) 119909

119901

119894+ 120573119909119894) 1198641198940 (8)

where 119901 = 3 and 120573 = 116 This polynomial model workswell in topology optimization with design-dependent loadssuch as inertial loads dynamic loads and thermal loads Itmaintains a positive gradient when the pseudodensity is zero

Suppose the thermal expansion coefficient is tempe-rature-independent and only the steady-state temperature

6 International Journal of Aerospace Engineering

field is taken into account The thermal strain vector can bewritten as follows

120576119894th = 120572119894Δ119879119894120572

119879

(9)

Here 120572119894is the thermal expansion coefficient Δ119879

119894denotes the

temperature increase120572 is a constant vector for the calculationof strain vector which is [1 1 1 0 0 0] in this paper [17]

Referring to the conception of thermal stress coefficient(TSC) [14] the element stiffness and the thermal stress loadshould be penalized independently in terms of element pseu-dodensity A linear interpolation to the thermal expansioncoefficient reads

120572119894= 1199091198941205721198940 (10)

where 1205721198940

is the original thermal expansion coefficientConsequently the elemental thermal expansion coefficientnow varies with the pseudodensities in coordination withYoungrsquos modulus

With the substitution of (8)ndash(10) into (7) the thermal loadcan be expressed as

F119894th = 120574119894 (119909119894) F1198940th

F1198940th = int

Ω119894

B119879119894D1198940120572 119889Ω

120574119894= 119864119894120572119894

(11)

where F1198940th is the thermal load when the element is solid with

unit Youngrsquos modulus unit thermal expansion coefficientand unit temperature increase which is always constant in theoptimization process 120574

119894denotes the thermal stress coefficient

(TSC) which can be treated as an inherent material propertyWith the introduction of TSC the thermal load can beexplicitly expressed as the function of design variables

32 The Global Stress Measure Difficulties of min-maxproblems are generally involved in stress-based topologyoptimization Naturally stress is evaluated and constrained ineach element In this way the overall stress level is restrictedbelow the prescribed limit But the scale of the optimizationproblem and the computational cost increase dramaticallyTo remedy the drawback aggregationmethods combining alllocal stresses into one or several Kreisselmeier-Steinhauser(KS) or 119901-norm global functions [24 25] are generally usedA tradeoff between the global and local stress approachesis that global stress measure reduces the computational costwhile it introduces high nonlinearity in global measurefunctions and loses the control over local stresses Differentaggregation strategies have been brought in For examplethe active set approach [26] aggregates the active constraintsinto one set which highly reduces the number of stressconstraints However the changes of the active set can leadto poor convergence Le et al [27] introduced an adaptivelyupdated 119901-norm aggregation method which can preciselyapproximate the maximum local stress A clustered method[28] was proposed as a compromise between global andlocal methods The constraints were aggregated into several

clustersThismethod was proven to be efficient and robust bynumerical examples but the results strongly depended uponthe number of clusters

Concerning the optimization of elastic support structurein this paper the thermal stress constraints are imposedonly in the nondesignable domain Compared with thetotal elements number the number of evaluated stresses iscomparatively small Without loss of generality the standard119901-norm measure is utilized which reads

120590119875119873= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875

(12)

where 120590119875119873

is the 119901-norm of the elemental Von-Mises stressand 119875 is the aggregation parameter Apparently the 119901-normdegenerates to the summation of the stress components when119875 is assigned as 1 Moreover it can be proven mathematicallythat

lim119901rarrinfin

(

119898

sum

119895=1

(120590VM119895)119875

)

1119875

= max119895=12119898

120590VM119895 (13)

Obviously with a sufficiently large value for 119875 the globalmeasure can exactly match the maximum stress Howeverthe large value may lead to oscillation and ill-posed problems[29] Consequently to balance the effect of constraintsrsquoquality and convergence efficiency the value of 119875 should beproperly chosen which is assigned as 15 in this paper

33 Adjoint Sensitivity Analysis Firstly the sensitivity of theglobal compliance with the design-dependent thermal load isderived and written as

120597119862

120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot120597U120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

= U119879 120597Fth120597119909119894

minus1

2U119879 120597K120597119909119894

U

(14)

Sensitivities of the thermal load with respect to the designvariables can be obtained by differentiating its definitionformulated in (11) The derivative of the stiffness matrix canbe easily evaluated according to the interpolation functionsin (8)

Sensitivity analysis is then carried out to evaluate theglobal stress measure in the nondesignable domain with

International Journal of Aerospace Engineering 7

respect to the pseudodensity variables which can be writtenas120597120590119875119873

120597119909119894

= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875minus1

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

(15)

Here the Von-Mises equivalent stress is defined as

120590VM119895radic120590119879119895V120590119895

= radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus120576119895th)

(16)

where 120590119895is the element stress vector U

119895and 120576

119895th are theelement nodal displacement vector and thermal strain vectorThe matrix V is a constant matrix that is

V2D =[[[[

[

1 minus1

20

minus1

21 0

0 0 3

]]]]

]

V3D =

[[[[[[[[[[

[

(

2 minus1 minus1

minus1 2 minus1

minus1 minus1 2

) 03times3

03times3

(

6

6

6

)

]]]]]]]]]]

]

(17)

As only the Von-Mises stresses in the nondesignable domainare concerned we have

120597120590VM119895

120597119909119894

=(B119895U119895minus 120576119895th)119879

D119879119895VD119895

radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus 120576119895th)

B119895

120597U119895

120597119909119894

=1

120590VM119895

sdot 120590119879

119895VD119895B119895

120597U119895

120597119909119894

(18)

where 120597D119895120597119909119894= 120597120576119895th120597119909119894 = 0 In this case the singularity

phenomenon in stress-based topology optimization is actu-ally not involved in this optimization problem

We can also defineU119895= A119895U

120597U119895

120597119909119894

= A119895

120597U120597119909119894

(19)

Table 2 Material properties used in the optimization

Properties Elasticsupport

Nondesignabledomain

Elastic modulus(MPa) 71000 3500

Coefficient of thermalexpansion (10minus6∘C) 23 7

Poissonrsquos ratio 033 04

The matrix A transforms the global nodal displacementvector to an element one

Based on the differentiation of finite element equilibriumequation we have the derivative of the nodal displacementvector

120597U120597119909119894

= Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U) (20)

The substitution of the above equation into the derivative of119901-norm aggregation function produces

120597120590119875119873

120597119909119894

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

(21)

Suppose

Q119879 = (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)Kminus1 (22)

is adjoint displacement the derivative of the119901-norm functioncan be obtained by an additional finite element calculation

4 Numerical Study

In this section a 2D cantilever beam and a 3D nozzle flap of aturbine engine are tested to study the topology optimizationof elastic supports under thermomechanical loads

41 2D Cantilever Beam As shown in Figure 5 a 2D can-tilever beam consists of a nondesignable top surface andan elastic support assigned as design domain The materialproperties are listed in Table 2 Since the thermal expansioncoefficient of elastic support is higher than that of thenondesignable domain significant thermal stresseswill occuras a result of temperature increase

In accordance with Section 3 the global compliance ofthe whole structure is minimized as the design objectiveThe volume fraction is constrained to 50 and the upperbound of the stress constraints in the nondesignable domainis 200MPa

The optimized results with various load combinations arelisted and compared in Figure 6 The result in Figure 6(a)is set as a benchmark with clear configuration Firstly themechanical load is fixed at 20Nmm in Figures 6(a)ndash6(d)

8 International Journal of Aerospace Engineering

q

Elastic support

Nondesignable domain

ΔT

50mm 5mm

20mm

100mm

Figure 5 Definition of the 2D cantilever beam

(a) Δ119879 = 50∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum Stress 154MPa

(b) Δ119879 = 200∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum stress 1664MPa

(c) Δ119879 = 500∘C 119902 = 20Nmm Optimized volumefraction 417 Maximum stress 200MPa

(d) Δ119879 = 1000∘C 119902 = 20Nmm Optimized volumefraction 358 Maximum stress 200MPa

(e) Δ119879 = 500∘C 119902 = 40Nmm Optimized volumefraction 50 Maximum stress 200MPa

(f) Δ119879 = 500∘C 119902 = 200Nmm Optimized volumefraction 50 Maximum stress 8124MPa

Figure 6 Optimized results of minimizing compliance under different loads

As the temperature increases the volume constraint startsto be inactive More elements with intermediate densitiesarise This is due to the nature of thermal stress constraintwhere the elements with compliant materials can offsetthe thermal stress better Later considering the changes inFigures 6(c) 6(e) and 6(f) clear structure patterns gradu-ally appear when larger mechanical loads are applied Thusthe elastic support is able to undergo the pressure with goodstiffness However when the thermomechanical loads are toohigh the optimization finds no feasible solution where thestress constraints are violated as shown in Figure 6(f)

The optimized results have shown that the relative mag-nitude of the thermal and mechanical load greatly influencesthe optimized results which have good consistency with theanalytical solution of the three-bar truss model in Section 3Moreover the existence of a large amount of intermediatedensity material is reasonable when the thermal load isdominant over the mechanical load Eliminating the greyelements directly or using some numerical schemes mayimprove the global stiffness but will unfortunately lead tohigher stress level

42 3D Nozzle Flap As shown in Figure 7 a nozzle flapof a turbine engine is composed of titanium stiffeners and

Table 3 Material properties used in the optimization

Material properties Titanium CeramicElastic modulus(MPa) 11000 3500

Coefficient of thermalexpansion (10minus6∘C) 10 7

Poissonrsquos ratio 033 04

a ceramic plate Material properties are listed in Table 3Figure 7 also illustrates the thermoelastic loads applied onthe model including a uniform pressure of 1MPa and globaltemperature increase of 500∘C Significant thermal stressesare generated in the plate due to the different thermalexpansion coefficient In practical design the stiffeners areassigned as the elastic support design domain and the plateis nondesignable The design objective is to minimize theglobal compliance with a 25MPa stress constraint on thenondesignable plate and a 30material volume constraint onthe design domain

The optimized design of elastic support as shown inFigure 8 is obtained by topology optimization using theproposed formulation The optimized design is presented by

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

International Journal of Aerospace Engineering 3

The strain and stress of the bar 119864(119894)can be expressed as

120576119894=119880119894119909

119871

120590119894= 119864120576119894119898= 119864 (120576

119894minus 120576119894th) = 119864(

119880119894119909

119871minus 120572Δ119879)

(2)

where 119880119894119909is the displacement with respect to the elemental

119909-axis and 120576119894119898

and 120576119894th are corresponding mechanical and

thermal strainThe elastic strain energy 120601 is defined as the potential

mechanical energy in the elastic body which is written as

120601 =1

2int (120576119898)119879D120576119898119889119881

=1

2int (120576 minus 120576th)

119879D (120576 minus 120576th) 119889119881

=1

2int 120576119879D120576 119889119881 minus int 120576119879D120576th119889119881 +

1

2int 120576119879

thD120576th119889119881

(3)

where D is the elastic matrix and 120576 is the total strain vectorconsisting of mechanical and thermal items 120576

119898and 120576th

Then we consider the mean compliance of structure

119862 =1

2F119879U = 1

2F119879Kminus1F (4)

where F and U are nodal load and displacement vectors K isglobal stiffness matrix

According to the existing works the strain energy mini-mization is more beneficial for the stress reduction while themean compliance reflects the structural overall stiffness [19]The purpose of this study is to find the optimized configura-tion of elastic supports preventing large deformations result-ing from thermal and mechanical load Naturally the meancompliance minimization is selected as the design objectivethroughout this paper Besides two design constraints that isvolume constraint and stress constraint are involved in thisstudy The formulation of the optimization can be expressedas

Find 1199091 1199092

Min 119862

st (1199091+ 1199092) 119871 le 119881

100381610038161003816100381612059031003816100381610038161003816 le |120590|

(5)

To demonstrate the relation between the design variables andthe global compliance parameters listed in Table 1 are used

At the first place the mechanical load is fixed with 119891119898119909=

119891119898119910

= 7000N Structural compliance with different tem-perature increases is plotted by sets of contours in Figure 3Thick solid line and dashed line denote stress and volumeconstraints respectively Along with the lower and upperlimits of the design variables these constraints define thefeasible regions of optimizationwhich is indicated as the darkarea in the figure The black spot denotes the optimal point

Table 1 Constant parameters list

Youngrsquos modulus 119864 (Pa) 2 times 1011

Thermal expansion coefficient 120572 (∘Cminus1) 1 times 10minus5

Length of bar 119871 (m) 1

Cross-sectional area 1199093(m2) 1 times 10

minus4

Upper limit of stress in nondesignable domain |120590| (MPa) 80

Upper limit of design domainrsquos volume 119881 (m3) 25times10minus4

Lower limit of design variables 119909 (m2) 5 times 10minus5

Upper limit of design variables 119909 (m2) 2 times 10minus4

where the compliance reaches a minimum Those symbolsigns are applied to Figures 3 and 4

As illustrated in Figure 3 when the structure undergoesthermomechanical loads a small temperature increase willlead to nonmonotone compliance with respect to the designvariables Consequently the volume constraint may not beactive as shown in Figures 3(b) to 3(d) In this case usingmore material could weaken the structural mechanical per-formance due to the thermomechanical loading condition

In another case fixing Δ119879 = 20∘C the mechanical loadsare proportionally raisedThe contour lines of compliance areshown in Figure 4 At the beginning the stress is primarilycaused by thermal load Very compliant structures are usedto offset thermal stress Typically when both mechanical andthermal loads are small as in Figure 4(a) the optimal pointlies at the lower left corner of the feasible region At this pointthe stress constraint is inactive Asmechanical loads increasemore materials are required to strengthen the structure Thevolume constraint is finally active in Figure 4 We can foreseethat as themechanical loads increase there will be no feasibledesign for this problem

Notably there exist some critical optimal points forthermomechanical loads for example in Figures 3(a) and4(d) The volume constraint is exactly active there A slightincrease of temperature or decrease of mechanical loads willlead to inactive volume constraint These phenomena alsohappen in topology optimization of continuum structureswith thermomechanical loads where inactive volume con-straints and elements with intermediate densities are alwaysfound (see [20])

3 Formulation of ThermomechanicalTopology Optimization

Based on the analyses in the previous section we propose touse the topology optimization formulation as

Find X = (1199091 1199092 119909

119899)

Min 119862 =1

2F119879U = 1

2(Fth + F119898)

119879U

st 119881 (X) =119899

sum

119894=1

119909119894V119894le 119881

max119895=12119898

(120590VM119895) le 120590

4 International Journal of Aerospace Engineering

138

141

144

141

144

144

147

147

147

150

150

150

153

153

156

159

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 16∘C 119891119898119909= 119891119898119910= 7000N

160158

158

156

156

154

154

154

154152

152

152

150150

150148

148 146144

146

156

156

158

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 18∘C 119891119898119909= 119891119898119910= 7000N

153 153156 156

159 159

159

162162

162

165

165

165

168

168

168 171

171

171

174

174177

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7000N

248240

240

232

232

224

224

224

216

216

216

208

208

208

200

200

200192

192

192

184

184176

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 25∘C 119891119898119909= 119891119898119910= 7000N

Figure 3 Optimizations under different temperature increases

0 lt 120575 le 119909119894le 1

119894 = 1 2 119899

(6)

where 119899 and119898 are the numbers of elements in elastic supportand nondesignable domain respectively X is the vector ofdesign variables that is the pseudodensities which varybetween 0 and 1 to describe the material distribution over thedesign domain 119862 is the global compliance of the structureFth and F

119898are the thermal and mechanical load vectors

composing the nodal load vector F 120590VM119895

is the Von-Misesstress of the 119895th element in the nondesignable domain 119881(X)is the volume of the design domain and V

119894is the volume of the

119894th element 120575 is a small constant set as 0001 in this paper toavoid singularity of stiffness matrix

31 Design-Dependent Thermal Load Unlike the staticmechanical load thermal load is a typical design-dependentload depending upon the material layout over the design

International Journal of Aerospace Engineering 5

115

120

120

120

125

125

125

130

130

130

135

135

135

140

140

140145

145

145

150

150 155

160

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 5500N

136

140140

140 144

144

144

148

148

148152

152

152156

156

156 160

160

160164

164

168

172

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 6500N

180

180

180

180

177

177

177

174

174174

171 171168

183

183

186

186

189

192

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7500N

195

195

192

192

192192

192

195

195

198

189189

189

186

186

186183183

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7800N

Figure 4 Optimizations under different mechanical loads

domain Thus the matching relation between the thermalload and stiffness should be carefully handled

According to the thermoelastic theory the thermal loadover the 119894th element is expressed as

F119894th = int

Ω119894

B119879119894D119894120576119894th119889Ω (7)

where B119894and D

119894are the element strain-displacement matrix

and elasticity matrix respectively 120576119894th is the thermal strain of

the 119894th element D119894is evaluated in terms of Youngrsquos modulus

119864119894 Here a polynomial interpolation is used to link the

pseudodensity variables to element elastic modulus [23]

119864119894= ((1 minus 120573) 119909

119901

119894+ 120573119909119894) 1198641198940 (8)

where 119901 = 3 and 120573 = 116 This polynomial model workswell in topology optimization with design-dependent loadssuch as inertial loads dynamic loads and thermal loads Itmaintains a positive gradient when the pseudodensity is zero

Suppose the thermal expansion coefficient is tempe-rature-independent and only the steady-state temperature

6 International Journal of Aerospace Engineering

field is taken into account The thermal strain vector can bewritten as follows

120576119894th = 120572119894Δ119879119894120572

119879

(9)

Here 120572119894is the thermal expansion coefficient Δ119879

119894denotes the

temperature increase120572 is a constant vector for the calculationof strain vector which is [1 1 1 0 0 0] in this paper [17]

Referring to the conception of thermal stress coefficient(TSC) [14] the element stiffness and the thermal stress loadshould be penalized independently in terms of element pseu-dodensity A linear interpolation to the thermal expansioncoefficient reads

120572119894= 1199091198941205721198940 (10)

where 1205721198940

is the original thermal expansion coefficientConsequently the elemental thermal expansion coefficientnow varies with the pseudodensities in coordination withYoungrsquos modulus

With the substitution of (8)ndash(10) into (7) the thermal loadcan be expressed as

F119894th = 120574119894 (119909119894) F1198940th

F1198940th = int

Ω119894

B119879119894D1198940120572 119889Ω

120574119894= 119864119894120572119894

(11)

where F1198940th is the thermal load when the element is solid with

unit Youngrsquos modulus unit thermal expansion coefficientand unit temperature increase which is always constant in theoptimization process 120574

119894denotes the thermal stress coefficient

(TSC) which can be treated as an inherent material propertyWith the introduction of TSC the thermal load can beexplicitly expressed as the function of design variables

32 The Global Stress Measure Difficulties of min-maxproblems are generally involved in stress-based topologyoptimization Naturally stress is evaluated and constrained ineach element In this way the overall stress level is restrictedbelow the prescribed limit But the scale of the optimizationproblem and the computational cost increase dramaticallyTo remedy the drawback aggregationmethods combining alllocal stresses into one or several Kreisselmeier-Steinhauser(KS) or 119901-norm global functions [24 25] are generally usedA tradeoff between the global and local stress approachesis that global stress measure reduces the computational costwhile it introduces high nonlinearity in global measurefunctions and loses the control over local stresses Differentaggregation strategies have been brought in For examplethe active set approach [26] aggregates the active constraintsinto one set which highly reduces the number of stressconstraints However the changes of the active set can leadto poor convergence Le et al [27] introduced an adaptivelyupdated 119901-norm aggregation method which can preciselyapproximate the maximum local stress A clustered method[28] was proposed as a compromise between global andlocal methods The constraints were aggregated into several

clustersThismethod was proven to be efficient and robust bynumerical examples but the results strongly depended uponthe number of clusters

Concerning the optimization of elastic support structurein this paper the thermal stress constraints are imposedonly in the nondesignable domain Compared with thetotal elements number the number of evaluated stresses iscomparatively small Without loss of generality the standard119901-norm measure is utilized which reads

120590119875119873= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875

(12)

where 120590119875119873

is the 119901-norm of the elemental Von-Mises stressand 119875 is the aggregation parameter Apparently the 119901-normdegenerates to the summation of the stress components when119875 is assigned as 1 Moreover it can be proven mathematicallythat

lim119901rarrinfin

(

119898

sum

119895=1

(120590VM119895)119875

)

1119875

= max119895=12119898

120590VM119895 (13)

Obviously with a sufficiently large value for 119875 the globalmeasure can exactly match the maximum stress Howeverthe large value may lead to oscillation and ill-posed problems[29] Consequently to balance the effect of constraintsrsquoquality and convergence efficiency the value of 119875 should beproperly chosen which is assigned as 15 in this paper

33 Adjoint Sensitivity Analysis Firstly the sensitivity of theglobal compliance with the design-dependent thermal load isderived and written as

120597119862

120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot120597U120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

= U119879 120597Fth120597119909119894

minus1

2U119879 120597K120597119909119894

U

(14)

Sensitivities of the thermal load with respect to the designvariables can be obtained by differentiating its definitionformulated in (11) The derivative of the stiffness matrix canbe easily evaluated according to the interpolation functionsin (8)

Sensitivity analysis is then carried out to evaluate theglobal stress measure in the nondesignable domain with

International Journal of Aerospace Engineering 7

respect to the pseudodensity variables which can be writtenas120597120590119875119873

120597119909119894

= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875minus1

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

(15)

Here the Von-Mises equivalent stress is defined as

120590VM119895radic120590119879119895V120590119895

= radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus120576119895th)

(16)

where 120590119895is the element stress vector U

119895and 120576

119895th are theelement nodal displacement vector and thermal strain vectorThe matrix V is a constant matrix that is

V2D =[[[[

[

1 minus1

20

minus1

21 0

0 0 3

]]]]

]

V3D =

[[[[[[[[[[

[

(

2 minus1 minus1

minus1 2 minus1

minus1 minus1 2

) 03times3

03times3

(

6

6

6

)

]]]]]]]]]]

]

(17)

As only the Von-Mises stresses in the nondesignable domainare concerned we have

120597120590VM119895

120597119909119894

=(B119895U119895minus 120576119895th)119879

D119879119895VD119895

radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus 120576119895th)

B119895

120597U119895

120597119909119894

=1

120590VM119895

sdot 120590119879

119895VD119895B119895

120597U119895

120597119909119894

(18)

where 120597D119895120597119909119894= 120597120576119895th120597119909119894 = 0 In this case the singularity

phenomenon in stress-based topology optimization is actu-ally not involved in this optimization problem

We can also defineU119895= A119895U

120597U119895

120597119909119894

= A119895

120597U120597119909119894

(19)

Table 2 Material properties used in the optimization

Properties Elasticsupport

Nondesignabledomain

Elastic modulus(MPa) 71000 3500

Coefficient of thermalexpansion (10minus6∘C) 23 7

Poissonrsquos ratio 033 04

The matrix A transforms the global nodal displacementvector to an element one

Based on the differentiation of finite element equilibriumequation we have the derivative of the nodal displacementvector

120597U120597119909119894

= Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U) (20)

The substitution of the above equation into the derivative of119901-norm aggregation function produces

120597120590119875119873

120597119909119894

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

(21)

Suppose

Q119879 = (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)Kminus1 (22)

is adjoint displacement the derivative of the119901-norm functioncan be obtained by an additional finite element calculation

4 Numerical Study

In this section a 2D cantilever beam and a 3D nozzle flap of aturbine engine are tested to study the topology optimizationof elastic supports under thermomechanical loads

41 2D Cantilever Beam As shown in Figure 5 a 2D can-tilever beam consists of a nondesignable top surface andan elastic support assigned as design domain The materialproperties are listed in Table 2 Since the thermal expansioncoefficient of elastic support is higher than that of thenondesignable domain significant thermal stresseswill occuras a result of temperature increase

In accordance with Section 3 the global compliance ofthe whole structure is minimized as the design objectiveThe volume fraction is constrained to 50 and the upperbound of the stress constraints in the nondesignable domainis 200MPa

The optimized results with various load combinations arelisted and compared in Figure 6 The result in Figure 6(a)is set as a benchmark with clear configuration Firstly themechanical load is fixed at 20Nmm in Figures 6(a)ndash6(d)

8 International Journal of Aerospace Engineering

q

Elastic support

Nondesignable domain

ΔT

50mm 5mm

20mm

100mm

Figure 5 Definition of the 2D cantilever beam

(a) Δ119879 = 50∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum Stress 154MPa

(b) Δ119879 = 200∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum stress 1664MPa

(c) Δ119879 = 500∘C 119902 = 20Nmm Optimized volumefraction 417 Maximum stress 200MPa

(d) Δ119879 = 1000∘C 119902 = 20Nmm Optimized volumefraction 358 Maximum stress 200MPa

(e) Δ119879 = 500∘C 119902 = 40Nmm Optimized volumefraction 50 Maximum stress 200MPa

(f) Δ119879 = 500∘C 119902 = 200Nmm Optimized volumefraction 50 Maximum stress 8124MPa

Figure 6 Optimized results of minimizing compliance under different loads

As the temperature increases the volume constraint startsto be inactive More elements with intermediate densitiesarise This is due to the nature of thermal stress constraintwhere the elements with compliant materials can offsetthe thermal stress better Later considering the changes inFigures 6(c) 6(e) and 6(f) clear structure patterns gradu-ally appear when larger mechanical loads are applied Thusthe elastic support is able to undergo the pressure with goodstiffness However when the thermomechanical loads are toohigh the optimization finds no feasible solution where thestress constraints are violated as shown in Figure 6(f)

The optimized results have shown that the relative mag-nitude of the thermal and mechanical load greatly influencesthe optimized results which have good consistency with theanalytical solution of the three-bar truss model in Section 3Moreover the existence of a large amount of intermediatedensity material is reasonable when the thermal load isdominant over the mechanical load Eliminating the greyelements directly or using some numerical schemes mayimprove the global stiffness but will unfortunately lead tohigher stress level

42 3D Nozzle Flap As shown in Figure 7 a nozzle flapof a turbine engine is composed of titanium stiffeners and

Table 3 Material properties used in the optimization

Material properties Titanium CeramicElastic modulus(MPa) 11000 3500

Coefficient of thermalexpansion (10minus6∘C) 10 7

Poissonrsquos ratio 033 04

a ceramic plate Material properties are listed in Table 3Figure 7 also illustrates the thermoelastic loads applied onthe model including a uniform pressure of 1MPa and globaltemperature increase of 500∘C Significant thermal stressesare generated in the plate due to the different thermalexpansion coefficient In practical design the stiffeners areassigned as the elastic support design domain and the plateis nondesignable The design objective is to minimize theglobal compliance with a 25MPa stress constraint on thenondesignable plate and a 30material volume constraint onthe design domain

The optimized design of elastic support as shown inFigure 8 is obtained by topology optimization using theproposed formulation The optimized design is presented by

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

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DistributedSensor Networks

International Journal of

4 International Journal of Aerospace Engineering

138

141

144

141

144

144

147

147

147

150

150

150

153

153

156

159

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 16∘C 119891119898119909= 119891119898119910= 7000N

160158

158

156

156

154

154

154

154152

152

152

150150

150148

148 146144

146

156

156

158

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 18∘C 119891119898119909= 119891119898119910= 7000N

153 153156 156

159 159

159

162162

162

165

165

165

168

168

168 171

171

171

174

174177

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7000N

248240

240

232

232

224

224

224

216

216

216

208

208

208

200

200

200192

192

192

184

184176

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 25∘C 119891119898119909= 119891119898119910= 7000N

Figure 3 Optimizations under different temperature increases

0 lt 120575 le 119909119894le 1

119894 = 1 2 119899

(6)

where 119899 and119898 are the numbers of elements in elastic supportand nondesignable domain respectively X is the vector ofdesign variables that is the pseudodensities which varybetween 0 and 1 to describe the material distribution over thedesign domain 119862 is the global compliance of the structureFth and F

119898are the thermal and mechanical load vectors

composing the nodal load vector F 120590VM119895

is the Von-Misesstress of the 119895th element in the nondesignable domain 119881(X)is the volume of the design domain and V

119894is the volume of the

119894th element 120575 is a small constant set as 0001 in this paper toavoid singularity of stiffness matrix

31 Design-Dependent Thermal Load Unlike the staticmechanical load thermal load is a typical design-dependentload depending upon the material layout over the design

International Journal of Aerospace Engineering 5

115

120

120

120

125

125

125

130

130

130

135

135

135

140

140

140145

145

145

150

150 155

160

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 5500N

136

140140

140 144

144

144

148

148

148152

152

152156

156

156 160

160

160164

164

168

172

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 6500N

180

180

180

180

177

177

177

174

174174

171 171168

183

183

186

186

189

192

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7500N

195

195

192

192

192192

192

195

195

198

189189

189

186

186

186183183

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7800N

Figure 4 Optimizations under different mechanical loads

domain Thus the matching relation between the thermalload and stiffness should be carefully handled

According to the thermoelastic theory the thermal loadover the 119894th element is expressed as

F119894th = int

Ω119894

B119879119894D119894120576119894th119889Ω (7)

where B119894and D

119894are the element strain-displacement matrix

and elasticity matrix respectively 120576119894th is the thermal strain of

the 119894th element D119894is evaluated in terms of Youngrsquos modulus

119864119894 Here a polynomial interpolation is used to link the

pseudodensity variables to element elastic modulus [23]

119864119894= ((1 minus 120573) 119909

119901

119894+ 120573119909119894) 1198641198940 (8)

where 119901 = 3 and 120573 = 116 This polynomial model workswell in topology optimization with design-dependent loadssuch as inertial loads dynamic loads and thermal loads Itmaintains a positive gradient when the pseudodensity is zero

Suppose the thermal expansion coefficient is tempe-rature-independent and only the steady-state temperature

6 International Journal of Aerospace Engineering

field is taken into account The thermal strain vector can bewritten as follows

120576119894th = 120572119894Δ119879119894120572

119879

(9)

Here 120572119894is the thermal expansion coefficient Δ119879

119894denotes the

temperature increase120572 is a constant vector for the calculationof strain vector which is [1 1 1 0 0 0] in this paper [17]

Referring to the conception of thermal stress coefficient(TSC) [14] the element stiffness and the thermal stress loadshould be penalized independently in terms of element pseu-dodensity A linear interpolation to the thermal expansioncoefficient reads

120572119894= 1199091198941205721198940 (10)

where 1205721198940

is the original thermal expansion coefficientConsequently the elemental thermal expansion coefficientnow varies with the pseudodensities in coordination withYoungrsquos modulus

With the substitution of (8)ndash(10) into (7) the thermal loadcan be expressed as

F119894th = 120574119894 (119909119894) F1198940th

F1198940th = int

Ω119894

B119879119894D1198940120572 119889Ω

120574119894= 119864119894120572119894

(11)

where F1198940th is the thermal load when the element is solid with

unit Youngrsquos modulus unit thermal expansion coefficientand unit temperature increase which is always constant in theoptimization process 120574

119894denotes the thermal stress coefficient

(TSC) which can be treated as an inherent material propertyWith the introduction of TSC the thermal load can beexplicitly expressed as the function of design variables

32 The Global Stress Measure Difficulties of min-maxproblems are generally involved in stress-based topologyoptimization Naturally stress is evaluated and constrained ineach element In this way the overall stress level is restrictedbelow the prescribed limit But the scale of the optimizationproblem and the computational cost increase dramaticallyTo remedy the drawback aggregationmethods combining alllocal stresses into one or several Kreisselmeier-Steinhauser(KS) or 119901-norm global functions [24 25] are generally usedA tradeoff between the global and local stress approachesis that global stress measure reduces the computational costwhile it introduces high nonlinearity in global measurefunctions and loses the control over local stresses Differentaggregation strategies have been brought in For examplethe active set approach [26] aggregates the active constraintsinto one set which highly reduces the number of stressconstraints However the changes of the active set can leadto poor convergence Le et al [27] introduced an adaptivelyupdated 119901-norm aggregation method which can preciselyapproximate the maximum local stress A clustered method[28] was proposed as a compromise between global andlocal methods The constraints were aggregated into several

clustersThismethod was proven to be efficient and robust bynumerical examples but the results strongly depended uponthe number of clusters

Concerning the optimization of elastic support structurein this paper the thermal stress constraints are imposedonly in the nondesignable domain Compared with thetotal elements number the number of evaluated stresses iscomparatively small Without loss of generality the standard119901-norm measure is utilized which reads

120590119875119873= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875

(12)

where 120590119875119873

is the 119901-norm of the elemental Von-Mises stressand 119875 is the aggregation parameter Apparently the 119901-normdegenerates to the summation of the stress components when119875 is assigned as 1 Moreover it can be proven mathematicallythat

lim119901rarrinfin

(

119898

sum

119895=1

(120590VM119895)119875

)

1119875

= max119895=12119898

120590VM119895 (13)

Obviously with a sufficiently large value for 119875 the globalmeasure can exactly match the maximum stress Howeverthe large value may lead to oscillation and ill-posed problems[29] Consequently to balance the effect of constraintsrsquoquality and convergence efficiency the value of 119875 should beproperly chosen which is assigned as 15 in this paper

33 Adjoint Sensitivity Analysis Firstly the sensitivity of theglobal compliance with the design-dependent thermal load isderived and written as

120597119862

120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot120597U120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

= U119879 120597Fth120597119909119894

minus1

2U119879 120597K120597119909119894

U

(14)

Sensitivities of the thermal load with respect to the designvariables can be obtained by differentiating its definitionformulated in (11) The derivative of the stiffness matrix canbe easily evaluated according to the interpolation functionsin (8)

Sensitivity analysis is then carried out to evaluate theglobal stress measure in the nondesignable domain with

International Journal of Aerospace Engineering 7

respect to the pseudodensity variables which can be writtenas120597120590119875119873

120597119909119894

= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875minus1

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

(15)

Here the Von-Mises equivalent stress is defined as

120590VM119895radic120590119879119895V120590119895

= radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus120576119895th)

(16)

where 120590119895is the element stress vector U

119895and 120576

119895th are theelement nodal displacement vector and thermal strain vectorThe matrix V is a constant matrix that is

V2D =[[[[

[

1 minus1

20

minus1

21 0

0 0 3

]]]]

]

V3D =

[[[[[[[[[[

[

(

2 minus1 minus1

minus1 2 minus1

minus1 minus1 2

) 03times3

03times3

(

6

6

6

)

]]]]]]]]]]

]

(17)

As only the Von-Mises stresses in the nondesignable domainare concerned we have

120597120590VM119895

120597119909119894

=(B119895U119895minus 120576119895th)119879

D119879119895VD119895

radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus 120576119895th)

B119895

120597U119895

120597119909119894

=1

120590VM119895

sdot 120590119879

119895VD119895B119895

120597U119895

120597119909119894

(18)

where 120597D119895120597119909119894= 120597120576119895th120597119909119894 = 0 In this case the singularity

phenomenon in stress-based topology optimization is actu-ally not involved in this optimization problem

We can also defineU119895= A119895U

120597U119895

120597119909119894

= A119895

120597U120597119909119894

(19)

Table 2 Material properties used in the optimization

Properties Elasticsupport

Nondesignabledomain

Elastic modulus(MPa) 71000 3500

Coefficient of thermalexpansion (10minus6∘C) 23 7

Poissonrsquos ratio 033 04

The matrix A transforms the global nodal displacementvector to an element one

Based on the differentiation of finite element equilibriumequation we have the derivative of the nodal displacementvector

120597U120597119909119894

= Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U) (20)

The substitution of the above equation into the derivative of119901-norm aggregation function produces

120597120590119875119873

120597119909119894

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

(21)

Suppose

Q119879 = (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)Kminus1 (22)

is adjoint displacement the derivative of the119901-norm functioncan be obtained by an additional finite element calculation

4 Numerical Study

In this section a 2D cantilever beam and a 3D nozzle flap of aturbine engine are tested to study the topology optimizationof elastic supports under thermomechanical loads

41 2D Cantilever Beam As shown in Figure 5 a 2D can-tilever beam consists of a nondesignable top surface andan elastic support assigned as design domain The materialproperties are listed in Table 2 Since the thermal expansioncoefficient of elastic support is higher than that of thenondesignable domain significant thermal stresseswill occuras a result of temperature increase

In accordance with Section 3 the global compliance ofthe whole structure is minimized as the design objectiveThe volume fraction is constrained to 50 and the upperbound of the stress constraints in the nondesignable domainis 200MPa

The optimized results with various load combinations arelisted and compared in Figure 6 The result in Figure 6(a)is set as a benchmark with clear configuration Firstly themechanical load is fixed at 20Nmm in Figures 6(a)ndash6(d)

8 International Journal of Aerospace Engineering

q

Elastic support

Nondesignable domain

ΔT

50mm 5mm

20mm

100mm

Figure 5 Definition of the 2D cantilever beam

(a) Δ119879 = 50∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum Stress 154MPa

(b) Δ119879 = 200∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum stress 1664MPa

(c) Δ119879 = 500∘C 119902 = 20Nmm Optimized volumefraction 417 Maximum stress 200MPa

(d) Δ119879 = 1000∘C 119902 = 20Nmm Optimized volumefraction 358 Maximum stress 200MPa

(e) Δ119879 = 500∘C 119902 = 40Nmm Optimized volumefraction 50 Maximum stress 200MPa

(f) Δ119879 = 500∘C 119902 = 200Nmm Optimized volumefraction 50 Maximum stress 8124MPa

Figure 6 Optimized results of minimizing compliance under different loads

As the temperature increases the volume constraint startsto be inactive More elements with intermediate densitiesarise This is due to the nature of thermal stress constraintwhere the elements with compliant materials can offsetthe thermal stress better Later considering the changes inFigures 6(c) 6(e) and 6(f) clear structure patterns gradu-ally appear when larger mechanical loads are applied Thusthe elastic support is able to undergo the pressure with goodstiffness However when the thermomechanical loads are toohigh the optimization finds no feasible solution where thestress constraints are violated as shown in Figure 6(f)

The optimized results have shown that the relative mag-nitude of the thermal and mechanical load greatly influencesthe optimized results which have good consistency with theanalytical solution of the three-bar truss model in Section 3Moreover the existence of a large amount of intermediatedensity material is reasonable when the thermal load isdominant over the mechanical load Eliminating the greyelements directly or using some numerical schemes mayimprove the global stiffness but will unfortunately lead tohigher stress level

42 3D Nozzle Flap As shown in Figure 7 a nozzle flapof a turbine engine is composed of titanium stiffeners and

Table 3 Material properties used in the optimization

Material properties Titanium CeramicElastic modulus(MPa) 11000 3500

Coefficient of thermalexpansion (10minus6∘C) 10 7

Poissonrsquos ratio 033 04

a ceramic plate Material properties are listed in Table 3Figure 7 also illustrates the thermoelastic loads applied onthe model including a uniform pressure of 1MPa and globaltemperature increase of 500∘C Significant thermal stressesare generated in the plate due to the different thermalexpansion coefficient In practical design the stiffeners areassigned as the elastic support design domain and the plateis nondesignable The design objective is to minimize theglobal compliance with a 25MPa stress constraint on thenondesignable plate and a 30material volume constraint onthe design domain

The optimized design of elastic support as shown inFigure 8 is obtained by topology optimization using theproposed formulation The optimized design is presented by

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

International Journal of

International Journal of Aerospace Engineering 5

115

120

120

120

125

125

125

130

130

130

135

135

135

140

140

140145

145

145

150

150 155

160

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(a) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 5500N

136

140140

140 144

144

144

148

148

148152

152

152156

156

156 160

160

160164

164

168

172

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(b) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 6500N

180

180

180

180

177

177

177

174

174174

171 171168

183

183

186

186

189

192

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(c) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7500N

195

195

192

192

192192

192

195

195

198

189189

189

186

186

186183183

180

times10minus4

times10minus4

05 15 2010x1

05

10

15

20

x2

(d) Δ119879 = 20∘C 119891119898119909= 119891119898119910= 7800N

Figure 4 Optimizations under different mechanical loads

domain Thus the matching relation between the thermalload and stiffness should be carefully handled

According to the thermoelastic theory the thermal loadover the 119894th element is expressed as

F119894th = int

Ω119894

B119879119894D119894120576119894th119889Ω (7)

where B119894and D

119894are the element strain-displacement matrix

and elasticity matrix respectively 120576119894th is the thermal strain of

the 119894th element D119894is evaluated in terms of Youngrsquos modulus

119864119894 Here a polynomial interpolation is used to link the

pseudodensity variables to element elastic modulus [23]

119864119894= ((1 minus 120573) 119909

119901

119894+ 120573119909119894) 1198641198940 (8)

where 119901 = 3 and 120573 = 116 This polynomial model workswell in topology optimization with design-dependent loadssuch as inertial loads dynamic loads and thermal loads Itmaintains a positive gradient when the pseudodensity is zero

Suppose the thermal expansion coefficient is tempe-rature-independent and only the steady-state temperature

6 International Journal of Aerospace Engineering

field is taken into account The thermal strain vector can bewritten as follows

120576119894th = 120572119894Δ119879119894120572

119879

(9)

Here 120572119894is the thermal expansion coefficient Δ119879

119894denotes the

temperature increase120572 is a constant vector for the calculationof strain vector which is [1 1 1 0 0 0] in this paper [17]

Referring to the conception of thermal stress coefficient(TSC) [14] the element stiffness and the thermal stress loadshould be penalized independently in terms of element pseu-dodensity A linear interpolation to the thermal expansioncoefficient reads

120572119894= 1199091198941205721198940 (10)

where 1205721198940

is the original thermal expansion coefficientConsequently the elemental thermal expansion coefficientnow varies with the pseudodensities in coordination withYoungrsquos modulus

With the substitution of (8)ndash(10) into (7) the thermal loadcan be expressed as

F119894th = 120574119894 (119909119894) F1198940th

F1198940th = int

Ω119894

B119879119894D1198940120572 119889Ω

120574119894= 119864119894120572119894

(11)

where F1198940th is the thermal load when the element is solid with

unit Youngrsquos modulus unit thermal expansion coefficientand unit temperature increase which is always constant in theoptimization process 120574

119894denotes the thermal stress coefficient

(TSC) which can be treated as an inherent material propertyWith the introduction of TSC the thermal load can beexplicitly expressed as the function of design variables

32 The Global Stress Measure Difficulties of min-maxproblems are generally involved in stress-based topologyoptimization Naturally stress is evaluated and constrained ineach element In this way the overall stress level is restrictedbelow the prescribed limit But the scale of the optimizationproblem and the computational cost increase dramaticallyTo remedy the drawback aggregationmethods combining alllocal stresses into one or several Kreisselmeier-Steinhauser(KS) or 119901-norm global functions [24 25] are generally usedA tradeoff between the global and local stress approachesis that global stress measure reduces the computational costwhile it introduces high nonlinearity in global measurefunctions and loses the control over local stresses Differentaggregation strategies have been brought in For examplethe active set approach [26] aggregates the active constraintsinto one set which highly reduces the number of stressconstraints However the changes of the active set can leadto poor convergence Le et al [27] introduced an adaptivelyupdated 119901-norm aggregation method which can preciselyapproximate the maximum local stress A clustered method[28] was proposed as a compromise between global andlocal methods The constraints were aggregated into several

clustersThismethod was proven to be efficient and robust bynumerical examples but the results strongly depended uponthe number of clusters

Concerning the optimization of elastic support structurein this paper the thermal stress constraints are imposedonly in the nondesignable domain Compared with thetotal elements number the number of evaluated stresses iscomparatively small Without loss of generality the standard119901-norm measure is utilized which reads

120590119875119873= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875

(12)

where 120590119875119873

is the 119901-norm of the elemental Von-Mises stressand 119875 is the aggregation parameter Apparently the 119901-normdegenerates to the summation of the stress components when119875 is assigned as 1 Moreover it can be proven mathematicallythat

lim119901rarrinfin

(

119898

sum

119895=1

(120590VM119895)119875

)

1119875

= max119895=12119898

120590VM119895 (13)

Obviously with a sufficiently large value for 119875 the globalmeasure can exactly match the maximum stress Howeverthe large value may lead to oscillation and ill-posed problems[29] Consequently to balance the effect of constraintsrsquoquality and convergence efficiency the value of 119875 should beproperly chosen which is assigned as 15 in this paper

33 Adjoint Sensitivity Analysis Firstly the sensitivity of theglobal compliance with the design-dependent thermal load isderived and written as

120597119862

120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot120597U120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

= U119879 120597Fth120597119909119894

minus1

2U119879 120597K120597119909119894

U

(14)

Sensitivities of the thermal load with respect to the designvariables can be obtained by differentiating its definitionformulated in (11) The derivative of the stiffness matrix canbe easily evaluated according to the interpolation functionsin (8)

Sensitivity analysis is then carried out to evaluate theglobal stress measure in the nondesignable domain with

International Journal of Aerospace Engineering 7

respect to the pseudodensity variables which can be writtenas120597120590119875119873

120597119909119894

= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875minus1

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

(15)

Here the Von-Mises equivalent stress is defined as

120590VM119895radic120590119879119895V120590119895

= radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus120576119895th)

(16)

where 120590119895is the element stress vector U

119895and 120576

119895th are theelement nodal displacement vector and thermal strain vectorThe matrix V is a constant matrix that is

V2D =[[[[

[

1 minus1

20

minus1

21 0

0 0 3

]]]]

]

V3D =

[[[[[[[[[[

[

(

2 minus1 minus1

minus1 2 minus1

minus1 minus1 2

) 03times3

03times3

(

6

6

6

)

]]]]]]]]]]

]

(17)

As only the Von-Mises stresses in the nondesignable domainare concerned we have

120597120590VM119895

120597119909119894

=(B119895U119895minus 120576119895th)119879

D119879119895VD119895

radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus 120576119895th)

B119895

120597U119895

120597119909119894

=1

120590VM119895

sdot 120590119879

119895VD119895B119895

120597U119895

120597119909119894

(18)

where 120597D119895120597119909119894= 120597120576119895th120597119909119894 = 0 In this case the singularity

phenomenon in stress-based topology optimization is actu-ally not involved in this optimization problem

We can also defineU119895= A119895U

120597U119895

120597119909119894

= A119895

120597U120597119909119894

(19)

Table 2 Material properties used in the optimization

Properties Elasticsupport

Nondesignabledomain

Elastic modulus(MPa) 71000 3500

Coefficient of thermalexpansion (10minus6∘C) 23 7

Poissonrsquos ratio 033 04

The matrix A transforms the global nodal displacementvector to an element one

Based on the differentiation of finite element equilibriumequation we have the derivative of the nodal displacementvector

120597U120597119909119894

= Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U) (20)

The substitution of the above equation into the derivative of119901-norm aggregation function produces

120597120590119875119873

120597119909119894

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

(21)

Suppose

Q119879 = (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)Kminus1 (22)

is adjoint displacement the derivative of the119901-norm functioncan be obtained by an additional finite element calculation

4 Numerical Study

In this section a 2D cantilever beam and a 3D nozzle flap of aturbine engine are tested to study the topology optimizationof elastic supports under thermomechanical loads

41 2D Cantilever Beam As shown in Figure 5 a 2D can-tilever beam consists of a nondesignable top surface andan elastic support assigned as design domain The materialproperties are listed in Table 2 Since the thermal expansioncoefficient of elastic support is higher than that of thenondesignable domain significant thermal stresseswill occuras a result of temperature increase

In accordance with Section 3 the global compliance ofthe whole structure is minimized as the design objectiveThe volume fraction is constrained to 50 and the upperbound of the stress constraints in the nondesignable domainis 200MPa

The optimized results with various load combinations arelisted and compared in Figure 6 The result in Figure 6(a)is set as a benchmark with clear configuration Firstly themechanical load is fixed at 20Nmm in Figures 6(a)ndash6(d)

8 International Journal of Aerospace Engineering

q

Elastic support

Nondesignable domain

ΔT

50mm 5mm

20mm

100mm

Figure 5 Definition of the 2D cantilever beam

(a) Δ119879 = 50∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum Stress 154MPa

(b) Δ119879 = 200∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum stress 1664MPa

(c) Δ119879 = 500∘C 119902 = 20Nmm Optimized volumefraction 417 Maximum stress 200MPa

(d) Δ119879 = 1000∘C 119902 = 20Nmm Optimized volumefraction 358 Maximum stress 200MPa

(e) Δ119879 = 500∘C 119902 = 40Nmm Optimized volumefraction 50 Maximum stress 200MPa

(f) Δ119879 = 500∘C 119902 = 200Nmm Optimized volumefraction 50 Maximum stress 8124MPa

Figure 6 Optimized results of minimizing compliance under different loads

As the temperature increases the volume constraint startsto be inactive More elements with intermediate densitiesarise This is due to the nature of thermal stress constraintwhere the elements with compliant materials can offsetthe thermal stress better Later considering the changes inFigures 6(c) 6(e) and 6(f) clear structure patterns gradu-ally appear when larger mechanical loads are applied Thusthe elastic support is able to undergo the pressure with goodstiffness However when the thermomechanical loads are toohigh the optimization finds no feasible solution where thestress constraints are violated as shown in Figure 6(f)

The optimized results have shown that the relative mag-nitude of the thermal and mechanical load greatly influencesthe optimized results which have good consistency with theanalytical solution of the three-bar truss model in Section 3Moreover the existence of a large amount of intermediatedensity material is reasonable when the thermal load isdominant over the mechanical load Eliminating the greyelements directly or using some numerical schemes mayimprove the global stiffness but will unfortunately lead tohigher stress level

42 3D Nozzle Flap As shown in Figure 7 a nozzle flapof a turbine engine is composed of titanium stiffeners and

Table 3 Material properties used in the optimization

Material properties Titanium CeramicElastic modulus(MPa) 11000 3500

Coefficient of thermalexpansion (10minus6∘C) 10 7

Poissonrsquos ratio 033 04

a ceramic plate Material properties are listed in Table 3Figure 7 also illustrates the thermoelastic loads applied onthe model including a uniform pressure of 1MPa and globaltemperature increase of 500∘C Significant thermal stressesare generated in the plate due to the different thermalexpansion coefficient In practical design the stiffeners areassigned as the elastic support design domain and the plateis nondesignable The design objective is to minimize theglobal compliance with a 25MPa stress constraint on thenondesignable plate and a 30material volume constraint onthe design domain

The optimized design of elastic support as shown inFigure 8 is obtained by topology optimization using theproposed formulation The optimized design is presented by

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 International Journal of Aerospace Engineering

field is taken into account The thermal strain vector can bewritten as follows

120576119894th = 120572119894Δ119879119894120572

119879

(9)

Here 120572119894is the thermal expansion coefficient Δ119879

119894denotes the

temperature increase120572 is a constant vector for the calculationof strain vector which is [1 1 1 0 0 0] in this paper [17]

Referring to the conception of thermal stress coefficient(TSC) [14] the element stiffness and the thermal stress loadshould be penalized independently in terms of element pseu-dodensity A linear interpolation to the thermal expansioncoefficient reads

120572119894= 1199091198941205721198940 (10)

where 1205721198940

is the original thermal expansion coefficientConsequently the elemental thermal expansion coefficientnow varies with the pseudodensities in coordination withYoungrsquos modulus

With the substitution of (8)ndash(10) into (7) the thermal loadcan be expressed as

F119894th = 120574119894 (119909119894) F1198940th

F1198940th = int

Ω119894

B119879119894D1198940120572 119889Ω

120574119894= 119864119894120572119894

(11)

where F1198940th is the thermal load when the element is solid with

unit Youngrsquos modulus unit thermal expansion coefficientand unit temperature increase which is always constant in theoptimization process 120574

119894denotes the thermal stress coefficient

(TSC) which can be treated as an inherent material propertyWith the introduction of TSC the thermal load can beexplicitly expressed as the function of design variables

32 The Global Stress Measure Difficulties of min-maxproblems are generally involved in stress-based topologyoptimization Naturally stress is evaluated and constrained ineach element In this way the overall stress level is restrictedbelow the prescribed limit But the scale of the optimizationproblem and the computational cost increase dramaticallyTo remedy the drawback aggregationmethods combining alllocal stresses into one or several Kreisselmeier-Steinhauser(KS) or 119901-norm global functions [24 25] are generally usedA tradeoff between the global and local stress approachesis that global stress measure reduces the computational costwhile it introduces high nonlinearity in global measurefunctions and loses the control over local stresses Differentaggregation strategies have been brought in For examplethe active set approach [26] aggregates the active constraintsinto one set which highly reduces the number of stressconstraints However the changes of the active set can leadto poor convergence Le et al [27] introduced an adaptivelyupdated 119901-norm aggregation method which can preciselyapproximate the maximum local stress A clustered method[28] was proposed as a compromise between global andlocal methods The constraints were aggregated into several

clustersThismethod was proven to be efficient and robust bynumerical examples but the results strongly depended uponthe number of clusters

Concerning the optimization of elastic support structurein this paper the thermal stress constraints are imposedonly in the nondesignable domain Compared with thetotal elements number the number of evaluated stresses iscomparatively small Without loss of generality the standard119901-norm measure is utilized which reads

120590119875119873= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875

(12)

where 120590119875119873

is the 119901-norm of the elemental Von-Mises stressand 119875 is the aggregation parameter Apparently the 119901-normdegenerates to the summation of the stress components when119875 is assigned as 1 Moreover it can be proven mathematicallythat

lim119901rarrinfin

(

119898

sum

119895=1

(120590VM119895)119875

)

1119875

= max119895=12119898

120590VM119895 (13)

Obviously with a sufficiently large value for 119875 the globalmeasure can exactly match the maximum stress Howeverthe large value may lead to oscillation and ill-posed problems[29] Consequently to balance the effect of constraintsrsquoquality and convergence efficiency the value of 119875 should beproperly chosen which is assigned as 15 in this paper

33 Adjoint Sensitivity Analysis Firstly the sensitivity of theglobal compliance with the design-dependent thermal load isderived and written as

120597119862

120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot120597U120597119909119894

=1

2U119879 120597Fth120597119909119894

+1

2(F119898+ Fth)

119879

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

= U119879 120597Fth120597119909119894

minus1

2U119879 120597K120597119909119894

U

(14)

Sensitivities of the thermal load with respect to the designvariables can be obtained by differentiating its definitionformulated in (11) The derivative of the stiffness matrix canbe easily evaluated according to the interpolation functionsin (8)

Sensitivity analysis is then carried out to evaluate theglobal stress measure in the nondesignable domain with

International Journal of Aerospace Engineering 7

respect to the pseudodensity variables which can be writtenas120597120590119875119873

120597119909119894

= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875minus1

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

(15)

Here the Von-Mises equivalent stress is defined as

120590VM119895radic120590119879119895V120590119895

= radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus120576119895th)

(16)

where 120590119895is the element stress vector U

119895and 120576

119895th are theelement nodal displacement vector and thermal strain vectorThe matrix V is a constant matrix that is

V2D =[[[[

[

1 minus1

20

minus1

21 0

0 0 3

]]]]

]

V3D =

[[[[[[[[[[

[

(

2 minus1 minus1

minus1 2 minus1

minus1 minus1 2

) 03times3

03times3

(

6

6

6

)

]]]]]]]]]]

]

(17)

As only the Von-Mises stresses in the nondesignable domainare concerned we have

120597120590VM119895

120597119909119894

=(B119895U119895minus 120576119895th)119879

D119879119895VD119895

radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus 120576119895th)

B119895

120597U119895

120597119909119894

=1

120590VM119895

sdot 120590119879

119895VD119895B119895

120597U119895

120597119909119894

(18)

where 120597D119895120597119909119894= 120597120576119895th120597119909119894 = 0 In this case the singularity

phenomenon in stress-based topology optimization is actu-ally not involved in this optimization problem

We can also defineU119895= A119895U

120597U119895

120597119909119894

= A119895

120597U120597119909119894

(19)

Table 2 Material properties used in the optimization

Properties Elasticsupport

Nondesignabledomain

Elastic modulus(MPa) 71000 3500

Coefficient of thermalexpansion (10minus6∘C) 23 7

Poissonrsquos ratio 033 04

The matrix A transforms the global nodal displacementvector to an element one

Based on the differentiation of finite element equilibriumequation we have the derivative of the nodal displacementvector

120597U120597119909119894

= Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U) (20)

The substitution of the above equation into the derivative of119901-norm aggregation function produces

120597120590119875119873

120597119909119894

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

(21)

Suppose

Q119879 = (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)Kminus1 (22)

is adjoint displacement the derivative of the119901-norm functioncan be obtained by an additional finite element calculation

4 Numerical Study

In this section a 2D cantilever beam and a 3D nozzle flap of aturbine engine are tested to study the topology optimizationof elastic supports under thermomechanical loads

41 2D Cantilever Beam As shown in Figure 5 a 2D can-tilever beam consists of a nondesignable top surface andan elastic support assigned as design domain The materialproperties are listed in Table 2 Since the thermal expansioncoefficient of elastic support is higher than that of thenondesignable domain significant thermal stresseswill occuras a result of temperature increase

In accordance with Section 3 the global compliance ofthe whole structure is minimized as the design objectiveThe volume fraction is constrained to 50 and the upperbound of the stress constraints in the nondesignable domainis 200MPa

The optimized results with various load combinations arelisted and compared in Figure 6 The result in Figure 6(a)is set as a benchmark with clear configuration Firstly themechanical load is fixed at 20Nmm in Figures 6(a)ndash6(d)

8 International Journal of Aerospace Engineering

q

Elastic support

Nondesignable domain

ΔT

50mm 5mm

20mm

100mm

Figure 5 Definition of the 2D cantilever beam

(a) Δ119879 = 50∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum Stress 154MPa

(b) Δ119879 = 200∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum stress 1664MPa

(c) Δ119879 = 500∘C 119902 = 20Nmm Optimized volumefraction 417 Maximum stress 200MPa

(d) Δ119879 = 1000∘C 119902 = 20Nmm Optimized volumefraction 358 Maximum stress 200MPa

(e) Δ119879 = 500∘C 119902 = 40Nmm Optimized volumefraction 50 Maximum stress 200MPa

(f) Δ119879 = 500∘C 119902 = 200Nmm Optimized volumefraction 50 Maximum stress 8124MPa

Figure 6 Optimized results of minimizing compliance under different loads

As the temperature increases the volume constraint startsto be inactive More elements with intermediate densitiesarise This is due to the nature of thermal stress constraintwhere the elements with compliant materials can offsetthe thermal stress better Later considering the changes inFigures 6(c) 6(e) and 6(f) clear structure patterns gradu-ally appear when larger mechanical loads are applied Thusthe elastic support is able to undergo the pressure with goodstiffness However when the thermomechanical loads are toohigh the optimization finds no feasible solution where thestress constraints are violated as shown in Figure 6(f)

The optimized results have shown that the relative mag-nitude of the thermal and mechanical load greatly influencesthe optimized results which have good consistency with theanalytical solution of the three-bar truss model in Section 3Moreover the existence of a large amount of intermediatedensity material is reasonable when the thermal load isdominant over the mechanical load Eliminating the greyelements directly or using some numerical schemes mayimprove the global stiffness but will unfortunately lead tohigher stress level

42 3D Nozzle Flap As shown in Figure 7 a nozzle flapof a turbine engine is composed of titanium stiffeners and

Table 3 Material properties used in the optimization

Material properties Titanium CeramicElastic modulus(MPa) 11000 3500

Coefficient of thermalexpansion (10minus6∘C) 10 7

Poissonrsquos ratio 033 04

a ceramic plate Material properties are listed in Table 3Figure 7 also illustrates the thermoelastic loads applied onthe model including a uniform pressure of 1MPa and globaltemperature increase of 500∘C Significant thermal stressesare generated in the plate due to the different thermalexpansion coefficient In practical design the stiffeners areassigned as the elastic support design domain and the plateis nondesignable The design objective is to minimize theglobal compliance with a 25MPa stress constraint on thenondesignable plate and a 30material volume constraint onthe design domain

The optimized design of elastic support as shown inFigure 8 is obtained by topology optimization using theproposed formulation The optimized design is presented by

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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

International Journal of Aerospace Engineering 7

respect to the pseudodensity variables which can be writtenas120597120590119875119873

120597119909119894

= (

119898

sum

119895=1

(120590VM119895)119875

)

1119875minus1

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus1

sdot120597120590

VM119895

120597119909119894

)

(15)

Here the Von-Mises equivalent stress is defined as

120590VM119895radic120590119879119895V120590119895

= radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus120576119895th)

(16)

where 120590119895is the element stress vector U

119895and 120576

119895th are theelement nodal displacement vector and thermal strain vectorThe matrix V is a constant matrix that is

V2D =[[[[

[

1 minus1

20

minus1

21 0

0 0 3

]]]]

]

V3D =

[[[[[[[[[[

[

(

2 minus1 minus1

minus1 2 minus1

minus1 minus1 2

) 03times3

03times3

(

6

6

6

)

]]]]]]]]]]

]

(17)

As only the Von-Mises stresses in the nondesignable domainare concerned we have

120597120590VM119895

120597119909119894

=(B119895U119895minus 120576119895th)119879

D119879119895VD119895

radic(B119895U119895minus 120576119895th)119879

D119879119895VD119895(B119895U119895minus 120576119895th)

B119895

120597U119895

120597119909119894

=1

120590VM119895

sdot 120590119879

119895VD119895B119895

120597U119895

120597119909119894

(18)

where 120597D119895120597119909119894= 120597120576119895th120597119909119894 = 0 In this case the singularity

phenomenon in stress-based topology optimization is actu-ally not involved in this optimization problem

We can also defineU119895= A119895U

120597U119895

120597119909119894

= A119895

120597U120597119909119894

(19)

Table 2 Material properties used in the optimization

Properties Elasticsupport

Nondesignabledomain

Elastic modulus(MPa) 71000 3500

Coefficient of thermalexpansion (10minus6∘C) 23 7

Poissonrsquos ratio 033 04

The matrix A transforms the global nodal displacementvector to an element one

Based on the differentiation of finite element equilibriumequation we have the derivative of the nodal displacementvector

120597U120597119909119894

= Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U) (20)

The substitution of the above equation into the derivative of119901-norm aggregation function produces

120597120590119875119873

120597119909119894

= (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)

sdot Kminus1 (120597Fth120597119909119894

minus120597K120597119909119894

U)

(21)

Suppose

Q119879 = (120590119875119873)1minus119875

119898

sum

119895=1

((120590VM119895)119875minus2

120590119879

119895VDB119895A119895)Kminus1 (22)

is adjoint displacement the derivative of the119901-norm functioncan be obtained by an additional finite element calculation

4 Numerical Study

In this section a 2D cantilever beam and a 3D nozzle flap of aturbine engine are tested to study the topology optimizationof elastic supports under thermomechanical loads

41 2D Cantilever Beam As shown in Figure 5 a 2D can-tilever beam consists of a nondesignable top surface andan elastic support assigned as design domain The materialproperties are listed in Table 2 Since the thermal expansioncoefficient of elastic support is higher than that of thenondesignable domain significant thermal stresseswill occuras a result of temperature increase

In accordance with Section 3 the global compliance ofthe whole structure is minimized as the design objectiveThe volume fraction is constrained to 50 and the upperbound of the stress constraints in the nondesignable domainis 200MPa

The optimized results with various load combinations arelisted and compared in Figure 6 The result in Figure 6(a)is set as a benchmark with clear configuration Firstly themechanical load is fixed at 20Nmm in Figures 6(a)ndash6(d)

8 International Journal of Aerospace Engineering

q

Elastic support

Nondesignable domain

ΔT

50mm 5mm

20mm

100mm

Figure 5 Definition of the 2D cantilever beam

(a) Δ119879 = 50∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum Stress 154MPa

(b) Δ119879 = 200∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum stress 1664MPa

(c) Δ119879 = 500∘C 119902 = 20Nmm Optimized volumefraction 417 Maximum stress 200MPa

(d) Δ119879 = 1000∘C 119902 = 20Nmm Optimized volumefraction 358 Maximum stress 200MPa

(e) Δ119879 = 500∘C 119902 = 40Nmm Optimized volumefraction 50 Maximum stress 200MPa

(f) Δ119879 = 500∘C 119902 = 200Nmm Optimized volumefraction 50 Maximum stress 8124MPa

Figure 6 Optimized results of minimizing compliance under different loads

As the temperature increases the volume constraint startsto be inactive More elements with intermediate densitiesarise This is due to the nature of thermal stress constraintwhere the elements with compliant materials can offsetthe thermal stress better Later considering the changes inFigures 6(c) 6(e) and 6(f) clear structure patterns gradu-ally appear when larger mechanical loads are applied Thusthe elastic support is able to undergo the pressure with goodstiffness However when the thermomechanical loads are toohigh the optimization finds no feasible solution where thestress constraints are violated as shown in Figure 6(f)

The optimized results have shown that the relative mag-nitude of the thermal and mechanical load greatly influencesthe optimized results which have good consistency with theanalytical solution of the three-bar truss model in Section 3Moreover the existence of a large amount of intermediatedensity material is reasonable when the thermal load isdominant over the mechanical load Eliminating the greyelements directly or using some numerical schemes mayimprove the global stiffness but will unfortunately lead tohigher stress level

42 3D Nozzle Flap As shown in Figure 7 a nozzle flapof a turbine engine is composed of titanium stiffeners and

Table 3 Material properties used in the optimization

Material properties Titanium CeramicElastic modulus(MPa) 11000 3500

Coefficient of thermalexpansion (10minus6∘C) 10 7

Poissonrsquos ratio 033 04

a ceramic plate Material properties are listed in Table 3Figure 7 also illustrates the thermoelastic loads applied onthe model including a uniform pressure of 1MPa and globaltemperature increase of 500∘C Significant thermal stressesare generated in the plate due to the different thermalexpansion coefficient In practical design the stiffeners areassigned as the elastic support design domain and the plateis nondesignable The design objective is to minimize theglobal compliance with a 25MPa stress constraint on thenondesignable plate and a 30material volume constraint onthe design domain

The optimized design of elastic support as shown inFigure 8 is obtained by topology optimization using theproposed formulation The optimized design is presented by

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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 International Journal of Aerospace Engineering

q

Elastic support

Nondesignable domain

ΔT

50mm 5mm

20mm

100mm

Figure 5 Definition of the 2D cantilever beam

(a) Δ119879 = 50∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum Stress 154MPa

(b) Δ119879 = 200∘C 119902 = 20Nmm Optimized volumefraction 50 Maximum stress 1664MPa

(c) Δ119879 = 500∘C 119902 = 20Nmm Optimized volumefraction 417 Maximum stress 200MPa

(d) Δ119879 = 1000∘C 119902 = 20Nmm Optimized volumefraction 358 Maximum stress 200MPa

(e) Δ119879 = 500∘C 119902 = 40Nmm Optimized volumefraction 50 Maximum stress 200MPa

(f) Δ119879 = 500∘C 119902 = 200Nmm Optimized volumefraction 50 Maximum stress 8124MPa

Figure 6 Optimized results of minimizing compliance under different loads

As the temperature increases the volume constraint startsto be inactive More elements with intermediate densitiesarise This is due to the nature of thermal stress constraintwhere the elements with compliant materials can offsetthe thermal stress better Later considering the changes inFigures 6(c) 6(e) and 6(f) clear structure patterns gradu-ally appear when larger mechanical loads are applied Thusthe elastic support is able to undergo the pressure with goodstiffness However when the thermomechanical loads are toohigh the optimization finds no feasible solution where thestress constraints are violated as shown in Figure 6(f)

The optimized results have shown that the relative mag-nitude of the thermal and mechanical load greatly influencesthe optimized results which have good consistency with theanalytical solution of the three-bar truss model in Section 3Moreover the existence of a large amount of intermediatedensity material is reasonable when the thermal load isdominant over the mechanical load Eliminating the greyelements directly or using some numerical schemes mayimprove the global stiffness but will unfortunately lead tohigher stress level

42 3D Nozzle Flap As shown in Figure 7 a nozzle flapof a turbine engine is composed of titanium stiffeners and

Table 3 Material properties used in the optimization

Material properties Titanium CeramicElastic modulus(MPa) 11000 3500

Coefficient of thermalexpansion (10minus6∘C) 10 7

Poissonrsquos ratio 033 04

a ceramic plate Material properties are listed in Table 3Figure 7 also illustrates the thermoelastic loads applied onthe model including a uniform pressure of 1MPa and globaltemperature increase of 500∘C Significant thermal stressesare generated in the plate due to the different thermalexpansion coefficient In practical design the stiffeners areassigned as the elastic support design domain and the plateis nondesignable The design objective is to minimize theglobal compliance with a 25MPa stress constraint on thenondesignable plate and a 30material volume constraint onthe design domain

The optimized design of elastic support as shown inFigure 8 is obtained by topology optimization using theproposed formulation The optimized design is presented by

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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

International Journal of Aerospace Engineering 9

design domain

Pressure

Nondesignabledomain

Elastic support

ΔT

Figure 7 Nozzle flap model and its design domain loads and boundary conditions

Figure 8 Topological optimized design with a temperature increase of 500∘C

hiding the elements with their pseudodensities under 05to show a clear structural configuration The two strongeststiffeners are composed of solid elements while the detailedstructural branches are using intermediate material withpseudodensities between 05 and 09

To further emphasize the effect of the thermal loads twomore designs are obtained by using different temperatureincreases that is 100∘C and 1000∘C with identical mechani-cal loadsThe optimized designs are shown in Figure 9 Com-pared with the structural topology in Figure 8 the optimizeddesign in Figure 9(a) has shown a much clearer load carryingpath with less intermediate material as the mechanical loadis dominant In Figure 9(b) an extremely high temperatureincrease of 1000∘C is used The optimized design is mostlycomposed of intermediate material as expected No clearstructural configuration is achieved

CAD model as shown in Figure 10(a) is then rebuiltaccording to the optimized design in Figure 8 An existingdesign of the nozzle flap is shown in Figure 10(b) forcomparison To verify the effect of topology optimization thetwo models are analyzed respectively with refined finite ele-ment mesh Twomodels share identical boundary conditions

and thermomechanical loads the stress distribution in thebottom plate is shown in Figure 11 The overall comparisonof the two designs is shown in Table 4

Compared with the existing design the optimized designreduces the maximum stress in the bottom plate significantlyfrom 2847MPa to 25MPa The global compliance decreasesfrom 2719 KJ to 2546 KJ Meanwhile material of 0129 times107mm3 is saved that is 5676 kg lighter than before

5 Conclusion

In this paper topology optimization of elastic supportingstructures under thermomechanical loads is investigated Athree-bar truss model is firstly employed to reveal the par-ticularity of thermoelastic problems that is nonmonotonouscompliance inactive volume constraint with high tempera-ture increase and so forth Similar appearances also havebeen found in the topology optimization of a 2D cantileverbeam structure presented in this paper On account ofstress-based topology optimization with large numbers ofdesign constraints global stress measure approach based on

10 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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 International Journal of Aerospace Engineering

Table 4 Comparison of the optimal design and the existing design

Topological optimized design Rebuilt optimized design Existing designMaximum stress in the bottom plate (MPa) 2500 2500 2847Compliance of the whole structure (KJ) 3022 2546 2719Volume of the elastic support (mm3) 2482 times 107 2078 times 107 2207 times 107

(a) 100∘C (b) 1000∘C

Figure 9 Topological optimized design with different temperature increases

(a) Rebuilt optimized design (b) Existing design

Figure 10 Rebuilt optimized design and existing design to be compared

1252 9169 1709 2500

(a) Optimized design

2722 9673 1907 2847

(b) Existing design

Figure 11 Stress distribution in the bottom plate (MPa)

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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

International Journal of Aerospace Engineering 11

119901-norm function is used to aggregate the stress constraintsinvolved in each iterationMeanwhile formulation of design-dependent thermal load is presented with different inter-polation functions assigned for material elastic modulusand thermal expansion coefficients Sensitivity analysis isthen carried out to evaluate the global stress measure innondesignable domain and the compliance with respect tothe pseudodensity variables In the Numerical Study a 2Dcantilever beam model and a 3D nozzle flap are optimizedComparedwith the existing design the optimized designs notonly use fewer materials but also are both stiffer and betterin reducing the maximum stress in nondesignable domain

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (11432011 51521061) the 111 Project(B07050) and the Fundamental Research Funds for theCentral Universities (3102014JC02020505)

References

[1] Q Li G P Steven O M Querin and Y M Xie ldquoShape andtopology design for heat conduction by evolutionary structuraloptimizationrdquo International Journal of Heat and Mass Transfervol 42 no 17 pp 3361ndash3371 1999

[2] A Gersborg-Hansen M P Bendsoslashe and O Sigmund ldquoTopol-ogy optimization of heat conduction problems using the finitevolumemethodrdquo Structural andMultidisciplinaryOptimizationvol 31 no 4 pp 251ndash259 2006

[3] Y Zhang and S Liu ldquoDesign of conducting paths based ontopology optimizationrdquoHeat and Mass Transfer vol 44 no 10pp 1217ndash1227 2008

[4] T Gao W H Zhang J H Zhu Y J Xu and D H BassirldquoTopology optimization of heat conduction problem involvingdesign-dependent heat load effectrdquo Finite Elements in Analysisand Design vol 44 no 14 pp 805ndash813 2008

[5] A Iga S Nishiwaki K Izui and M Yoshimura ldquoTopol-ogy optimization for thermal conductors considering design-dependent effects including heat conduction and convectionrdquoInternational Journal of Heat and Mass Transfer vol 52 no 11-12 pp 2721ndash2732 2009

[6] J Dirker and J P Meyer ldquoTopology optimization for an internalheat-conduction cooling scheme in a square domain for highheat flux applicationsrdquo Journal of Heat Transfer vol 135 no 11Article ID 111010 2013

[7] H Rodrigues and P Fernandes ldquoA material based model fortopology optimization of thermoelastic structuresrdquo Interna-tional Journal for Numerical Methods in Engineering vol 38 no12 pp 1951ndash1965 1995

[8] D A Tortorelli G Subramani S C Y Lu and R B HaberldquoSensitivity analysis for coupled thermoelastic systemsrdquo Inter-national Journal of Solids and Structures vol 27 no 12 pp 1477ndash1497 1991

[9] Q Li G P Steven and Y M Xie ldquoDisplacement minimizationof thermoelastic structures by evolutionary thickness designrdquo

Computer Methods in Applied Mechanics and Engineering vol179 no 3-4 pp 361ndash378 1999

[10] S Cho and J-Y Choi ldquoEfficient topology optimization ofthermo-elasticity problems using coupled field adjoint sensitiv-ity analysis methodrdquo Finite Elements in Analysis andDesign vol41 no 15 pp 1481ndash1495 2005

[11] Q Xia andM YWang ldquoTopology optimization of thermoelas-tic structures using level setmethodrdquoComputationalMechanicsvol 42 no 6 pp 837ndash857 2008

[12] J Yan G Cheng and L Liu ldquoA uniform optimum materialbased model for concurrent optimization of thermoelasticstructures and materialsrdquo International Journal for SimulationandMultidisciplinaryDesignOptimization vol 2 no 4 pp 259ndash266 2008

[13] S P Sun andWH Zhang ldquoTopology optimal design of thermo-elastic structuresrdquo Chinese Journal of Theoretical and AppliedMechanics vol 41 no 6 pp 878ndash887 2009

[14] T Gao and W Zhang ldquoTopology optimization involvingthermo-elastic stress loadsrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 725ndash738 2010

[15] J D Deaton and R V Grandhi ldquoStiffening of restrainedthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 48 no 4 pp 731ndash745 2013

[16] J D Deaton and R V Grandhi ldquoA survey of structuraland multidisciplinary continuum topology optimization post2000rdquo Structural amp Multidisciplinary Optimization vol 49 no1 pp 1ndash38 2014

[17] P Pedersen and N L Pedersen ldquoStrength optimized designsof thermoelastic structuresrdquo Structural and MultidisciplinaryOptimization vol 42 no 5 pp 681ndash691 2010

[18] P Pedersen and N L Pedersen ldquoInterpolationpenalizationapplied for strength design of 3D thermoelastic structuresrdquoStructural andMultidisciplinary Optimization vol 45 no 6 pp773ndash786 2012

[19] W H Zhang J G Yang Y J Xu and T Gao ldquoTopologyoptimization of thermoelastic structures mean complianceminimization or elastic strain energy minimizationrdquo Structuraland Multidisciplinary Optimization vol 49 no 3 pp 417ndash4292014

[20] J D Deaton and R V Grandhi ldquoStress-based design ofthermal structures via topology optimizationrdquo Structural andMultidisciplinary Optimization vol 53 no 2 pp 253ndash270 2016

[21] G I N Rozvany ldquoExact analytical solutions for some popularbenchmark problems in topology optimizationrdquo StructuralOptimization vol 15 no 1 pp 42ndash48 1998

[22] G I N Rozvany ldquoBasic geometrical properties of exact optimalcomposite platesrdquo Computers amp Structures vol 76 no 1 pp263ndash275 2000

[23] J H Zhu W H Zhang and P Beckers ldquoIntegrated layoutdesign of multi-component systemrdquo International Journal forNumerical Methods in Engineering vol 78 no 6 pp 631ndash6512009

[24] P Duysinx and O Sigmund ldquoNew developments in handlingstress constraints in optimal material distributionrdquo in Pro-ceedings of the 7th AIAAUSAFNASAISSMO Symposium onMultidisciplinary Analysis and Optimization MultidisciplinaryAnalysis Optimization Conferences vol 1 pp 1501ndash1509 1998

[25] R J Yang and C J Chen ldquoStress-based topology optimizationrdquoStructural Optimization vol 12 no 2-3 pp 98ndash105 1996

[26] P Duysinx and M P Bendsoslashe ldquoTopology optimization of con-tinuum structures with local stress constraintsrdquo International

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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

12 International Journal of Aerospace Engineering

Journal for Numerical Methods in Engineering vol 43 no 8 pp1453ndash1478 1998

[27] C Le J Norato T Bruns C Ha and D Tortorelli ldquoStress-based topology optimization for continuardquo Structural andMultidisciplinary Optimization vol 41 no 4 pp 605ndash620 2010

[28] E Holmberg B Torstenfelt and A Klarbring ldquoStress con-strained topology optimizationrdquo Structural and Multidisci-plinary Optimization vol 48 no 1 pp 33ndash47 2013

[29] R T Haftka and Z Gurdal Elements of Structural OptimizationKluwer Academic Dordrecht Netherlands 1992

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

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