Research Article A Structural Optimization Framework for...

Research ArticleA Structural Optimization Framework forMultidisciplinary Design

Mohammad Kurdi

Department of Engineering, The Pennsylvania State University, Altoona, PA 16601, USA

Correspondence should be addressed to Mohammad Kurdi; [email protected]

Received 29 September 2014; Revised 23 December 2014; Accepted 6 January 2015

Academic Editor: Qingsong Xu

Copyright © 2015 Mohammad Kurdi. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This work describes the development of a structural optimization framework adept at accommodating diverse customerrequirements. The purpose is to provide a framework accessible to the optimization research analyst. The framework integratesthe method of moving asymptotes into the finite element analysis program (FEAP) by exploiting the direct interface capabilityin FEAP. Analytic sensitivities are incorporated to provide a robust and efficient optimization search. User macros are developedto interface the design algorithm and analytic sensitivity with the finite element analysis program. To test the optimization tooland sensitivity calculations, three sizing and one topology optimization problems are considered. In addition, flutter analysis of aheated panel is analyzed as an example of coupling to nonstructural discipline. In sizing optimization, the calculated semianalyticsensitivities match analytic and finite difference calculations. Differences between analytic designs and numerical ones are lessthan 2.0% and are attributed to discrete nature of finite elements. In the topology problem, quadratic elements are found robust atresolving checkerboard patterns.

1. Introduction

Increasingly engineering products are required to main-tain high efficiency but yet perform well under conflictingcustomer requirements. Examples of these include thermalloads [1], combination of aeroelastic, thermal, and bucklingloads, and acoustic and impact loads. Structural optimizationstrives to improve performance by interrogatingmultiple dis-ciplines simultaneously. However, there is continuous need toconsider new disciplines within existing optimization tools.

Industry recognized potential for structural optimizationsince the 1980s. Now there aremultiple commercial structuraloptimization tools that address product design. These toolshave emerged primarily from the academic research com-munity and then commercialized to focus on a particularapplication. Early structural optimization implementationsinclude adding sizing and shape optimization to finite ele-ment software such as NASTRAN, ANSYS, and ABAQUS [2,page 243]. Thomas et al. [3] indicates that early version ofAltair OptiStruct is based on application of homogenizationmethod for topology optimization [4]. GENESIS [5] focusedon employing advanced approximation techniques [6, 7] andsensitivity tools to provide an efficient optimization search

[8]. To obtain design sensitivities in ABAQUS, Yi et al. [9]application utilized response surface technique to approxi-mate design derivatives for structural optimization of geardamper. The author utilized python to access the ABAQUSkernel and formulate the noise criterion.

Thomas et al. [3] highlights important requirementsof structural optimization in commercial tools. Of theserequirements, the ease of use of the interface and narrowapplication focus of the tool provide a disincentive to expandthe tools to wider range of applications. For example, untilnow topology optimization is limited to structural problemsand does not consider important physics [10] critical tothe structure. On the other hand, research tools frequentlyutilize in-house codes to model a narrow physical problemand then couple to structural optimization. Extension of themethodology into commercial tools may require significantdevelopment.

It is desired to have a design optimization tool that offersdevelopment environment intermediate between researchand commercial applications.The tool should be amenable tothe use of robust optimizationmethods, analytic sensitivities,advanced approximate methods, and inclusion of all impor-tant physics.The development of analytic sensitivities for new

Hindawi Publishing CorporationJournal of OptimizationVolume 2015, Article ID 345120, 14 pageshttp://dx.doi.org/10.1155/2015/345120

2 Journal of Optimization

criteria is particularly important for effectiveness of gradient-based optimization tools. In addition and as suggested byThomas et al. [3], structural optimization software shouldhave a broad analysis capability by containing different typesof elements and are amenable to integration of new physicsand material types. The finite element analysis program(FEAP) [11], developed by Professor Taylor at UC Berkeley,is a research finite-element code available in FORTRAN. Itoffers wide range of elements and encompasses structuralanalysis and thermal capabilities in linear and nonlinearand static and transient domains. The code is designed toincorporate new elements, physics, materials, meshes, andsolution procedures by development of user macros. Inaddition, the macros lend an interfacing capability to anoptimization program.

Several works exist on the utilization of FEAP in struc-tural optimization considering aeroelastic constraints. Moon[12] conducted an aerostructural optimization ofwings underdivergence constraint. Alonso et al. [13, 14] demonstrated anoptimization framework by coupling high-fidelity CFD codeto FEAP. In particular in [13] finite difference tools are usedto compute the structural sensitivities. In the above studies,a Python-based programming interface is used to wrap thefluid code to structural analysis tool.More recently, Schafer etal. [15] carried out a derivative-free shape optimization studyby coupling a finite volume flow solver (FASTEST) to FEAPusing the coupling interface (MpCCI).

In this work a structural optimization and analyticsensitivity framework is developed. The framework is builtfrom within the FEAP command structure by accessingthe user interface. The advantage of this is direct access tofinite element matrices necessary for formulation of analyticgradients, flexibility in the choice of the numerical optimiza-tion method, and incorporation of new physics. Analyticgradients are derived and provided to the method of movingasymptotes [16] to expedite the optimization search (thismethod is utilized here as an example, and the user canselect other gradient-based optimization techniques depend-ing on the application). To handle competing objectives, aPareto approach [17] multiobjective is utilized to address themultiobjective problem. The Pareto front is computed viathe 𝜖-constraint method. The paper proceeds in Section 2to provide the general form of nonlinear structural systemof equations. In Section 3, the analytic sensitivity equationsare summarized. In Section 4, the implementation of thestructural optimization framework and sensitivity analysisis described. In Section 5, numerical studies are carried outto demonstrate application of the framework to structuralproblems with sizing and density design variables. FinallySection 6 demonstrates incorporation of new physics byanalyzing the buckling and flutter of a heated panel.

2. Structural Analysis

In static nonlinear analysis of structures, the finite elementmethod gives a set of equilibrium equations [2, page 274]:

F (D, 𝑥) = 𝜇P (D, 𝑥) , (1)

where 𝑥 is a design variable, D is a global response defor-mation, P(D, 𝑥) is the external applied load, F(D, 𝑥) is theinternal load in structure due to the deformation, and 𝜇 isa load continuation parameter.

Newtonmethod is used to compute the solution to systemof equations. A Taylor series approximation of the residual is

R (D + 𝛿D, 𝑥) = R (D, 𝑥) + Kt𝛿D ≈ 0, (2)

where Kt is the tangent stiffness matrix due to internal andexternal loads as follows:

𝐾𝑡𝑖𝑗=

𝜕𝑅𝑖(D, 𝑥)𝜕𝐷𝑗

=

𝜕𝐹𝑖

𝜕𝐷𝑗

− 𝜇

𝜕𝑃𝑖

𝜕𝐷𝑗

. (3)

The tangent stiffness matrix is calculated at the element leveland then assembled for the global system. The matrix issymmetric when external loads are independent of deforma-tions. In this case, direct solver is used to solve the system ofequations for 𝛿D.The deformationD is updated with the newsolution 𝛿D until convergence to specified accuracy.

3. Sensitivity Analysis

For performance functions 𝑔(D, 𝑥) with dependence ondeformation, one can write the sensitivity of 𝑔 to designvariable 𝑥:

d𝑔d𝑥

=

𝜕𝑔

𝜕𝑥

+

𝜕𝑔

𝜕DdDd𝑥. (4)

In order to compute the sensitivity using finite differences,one needs to recompute the response D at perturbed valueof design variable 𝑥. In addition to computational cost, thissensitivity is prone to inaccuracies in the finite elementsolution.There are two alternative options for computing thesensitivity semianalytically [2, page 274]. The direct methodcomputes the sensitivity from equilibrium equations (1) at aconverged responseD:

KtdDd𝑥

= −

𝜕R (D, 𝑥)𝜕𝑥

. (5)

Forward finite difference is used to compute sensitivity ofthe residual. Note that this only requires build-up of theresidual at newdesign.The factored tangent stiffnessmatrix isavailable from final iteration in (2). Equation (5) is repeatedlysolved for all design variables. The cost of computationincreases for optimization problems with large number ofdesign variables.The adjointmethod avoids this repetition bypremultiplying (5) with an adjoint vector and adding to (4).The adjoint vector 𝜆 is computed by setting the coefficient ofdD/d𝑥 to zero:

Kt𝑇𝜆 = −

𝜕𝑔

𝜕D, (6)

where 𝑇 refers to transpose of matrix. For structural systemswhere the external load is independent of deformation, thetangent stiffness matrix is symmetric and this operation canbe skipped. Sensitivity of response is then computed as

d𝑔d𝑥

=

𝜕𝑔

𝜕𝑥

+ 𝜆𝑇 𝜕Rd𝑥. (7)

Journal of Optimization 3

Table 1: Description of FEAP input file for structural optimization.

Commands DescriptionMesh definition Define nodes, elements, loading, and boundary conditionsMaterial definition Define distinct material for each design variableProgram command Calculate response, form residual, etc.User commands Call subroutines to compute initial value of objective function and constraintsBegin loopProgram command Calculate response, form residual, etc.User commands Call subroutines to compute objective, constraints, and sensitivities and carry out optimization searchEnd loop

Note that (6) is repeatedly solved for each deformationdependent performance function. In optimization problemswith large number of such functions, the direct method ismore appropriate.

4. FEAP Implementation

The optimization framework is applied in the problemsolution stage of the FEAP input file [11, page 3]. Herea direct interface (referred to as user macro [18]) to theprogram is developed to carry out the optimization search,sensitivity analysis, and definitions of the objective functionand constraints. The macro allows access to the FEAPprogram pointers and defines new user pointers for usein the framework. New user commands are interweavedwith existing solution commands in FEAP to achieve thealgorithm shown in Figure 1, where a user command calls theuser macro; see Table 1.

The user macro for the optimization search integrates theFortran source code of the method of moving asymptotes(MMA) [16]. In addition to its availability, this method is wellsuited for structural optimization.

Due to complexity of data management in the sensitivityanalysis portion of the algorithm, the basic structure of directsensitivity calculation is explained in the following steps.

(i) Calculate response D at a specified design variablex. This is implemented using Newton method. FEAPcommand consists of “TANG,,1” for linear analysis ora loop of same command for nonlinear analysis.

(ii) The user macro saves responseD and design variablevariable vector x and residual R into user designatedpointers.

(iii) Loop on design variable 𝑥𝑖.

(iv) Usermacro assigns saved user pointers ofD and x intoFEAP program pointers.

(v) User macro perturbs a design variable 𝑖, 𝑥𝑖= 𝑥𝑖+Δ𝑥.

(vi) Build the residual for the perturbed design. This isimplemented by the FEAP command “FORM.”

(vii) User macro calculates sensitivity of residual usingforward finite difference. The saved residual in theuser pointer is recalled here.

(viii) User macro calculates the sensitivity of response dueto change in 𝑥

𝑖, (5).

(ix) Go to step (iii) for another design variable.

Initialize MMAparameters

Set material parameters inFEA to design variables

Calculate structural response

Compute initial objec-tive and constraints

Start optimization loop

Calculate structural response

Compute objective and constraints

Sensitivity of objec-tive and constraints

MMA search

Converged

Output

Set materialparametersin FEA to

design variables

Stop

Yes

No

Figure 1: Optimization algorithm.

In case one desires employing the adjoint sensitivity, thenfirst the adjoint vector is calculated for a performance func-tion using (6). For this purpose, a usermacro is constructed tocalculate sensitivity performance to response −(𝜕𝑔/𝜕D). Thissensitivity can be assigned as a fictitious load in the inputfile or in the corresponding pointer of FEAP memory. Thelatter is used here. The adjoint vector is then computed by (1)forming tangent stiffness matrix (“TANG”); (2) constructingresidual (“FORM”); and (3) solving system of equations(“SOLVE”). Now the direct sensitivity algorithm is carriedout but with step (viii) above being replaced with a usermacro that implements (7). Note that a new adjoint vectoris computed for each new design.


Table 2: Length and material of beam studied in uniformly taperedbeams.

Length, m 𝐸, N/m2 ]10 200 × 109 0.3

1 2 3 4 5 1 2 3 4 5

i: design variable number

1500N

10m10m

150N/m

Ai

(a) Tip deflection objective (b) Compliance objectiveand tip load with distributed load

Figure 2: Loading schematic for uniform tapered beam design.

5. Numerical Studies

Three problems are studied regarding the sizing design ofa cantilever beam with different loading conditions. Twoproblems have analytic solutions and serve as validation ofthe optimization framework [19, 20].Then, a topology designproblem is demonstrated.

The sizing optimization problems are arranged withregard to variation of beam cross-section along its axis.First the area is restricted to vary uniformly, maintaining aconstant aspect ratio; second the beam depth is designed,maintaining constant width [2, page 38].

5.1. Uniformly Tapered Beam. An Euler-Bernoulli beam ele-ment is used to represent the deformation. The deformationvector d = {𝑢, 𝑤, 𝜃} consists of 𝑢 axial, 𝑤 lateral, and 𝜃 slopeat each node. The design variables are the cross-sectionalarea of the beam elements. Because separate variables definethe cross-sectional area and moment of inertia in FEAP, themoment of inertia is recalculated after each design update.Table 2 provides properties common to the uniformly taperedbeam.

5.1.1. Tip Deflection Objective and Tip Load. Consider acantilever beam subjected to a static tip load; see Figure 2(a).The objective problem of minimizing the tip deflection andweight is formulated as

min𝐴

𝛿 (𝐴)

𝛿0

, (8a)

subject to

𝑔1(𝐴) =

𝑉 (𝐴)

𝑉0

− 𝑒𝑖≤ 0, 𝑒

𝑖= {0.1, . . . , 1.0} , (8b)

𝑔2(𝐴) =

𝜎 (𝐴)

𝜎0

− 𝑒𝜎≤ 0, (8c)

𝐴𝑙≤ 𝐴 ≤ 𝐴

𝑢, (8d)

where 𝐴 is the cross-section area of each beam element, 𝐴𝑙

and 𝐴𝑢are the lower and upper limits on the area design

variables, 𝛿 is the lateral deflection at the free end of the beam,𝑉 is the volume of beam, 𝜎 is the axial stress, and subscript0 refers to a scaling value for either objective function orconstraints. Scaling value is computed at initial uniformbeam cross-section. 𝑒

𝑖are set of limits imposed on volume

constraint to generate the efficiency curve and 𝑒𝜎is a factor

for setting the maximum stress. Scaling is employed here toachieve an effective optimization search. Due to this scaling,the reported results are generally dimensionless.

Sensitivity of the objective is computed from (4), wherefirst explicit term is zero and

𝜕𝛿

𝜕D= {0, 0, 0, . . . , 0, 1.0, 0}

{1×𝑛}. (9)

Therefore, the objective sensitivity consists only of the deriva-tive of the response at tip in lateral direction. Computedsensitivity is checked by analytic derivation for one or twoelements [2, page 265]. Sensitivity of the constraints iscomputed explicitly.

The beam has an initial uniform cross-sectional area of𝐴𝑖= 0.03m2 and is allowed to change in the interval of

[0.001, 0.05]m2. The stress factor in (8c) is set to 3.0. Tendesign variables are assigned to ten elements in the beam.For 𝑒𝑖= 0.9, the optimization search converges to 0.8𝛿

0

within 8 iterations. Only the volume constraint is foundactive; see Figure 3(a). The efficiency curve of displacementand volume is reported in Figure 3(b). At the right end ofthe curve, the displacement is minimized with the volumeconstrained to be lower than initial one. At left end thedisplacement is minimized with volume constrained to 30%of the initial volume. At same volume as initial design 𝑒

𝑖=

1.0 the displacement is reduced by 19%. Additionally a 20%reduction in volume is achieved for tip displacement equal toinitial design.

5.1.2. Compliance Objective and Distributed Load. The can-tilever beam considered here is subjected to a uniformdistributed load; see Figure 2(b). The objective is to decreasetotal area under deformation curve without increasing thevolume:

min𝐴

𝐽 (𝐴)

𝐽0

, (10a)

subject to

𝑔1(𝐴) =

𝑉 (𝐴)

𝑉0

− 𝑒𝑖≤ 0, 𝑒

𝑖= {0.1, . . . , 1.0} , (10b)

𝑔2(𝐴) =

𝜎 (𝐴)

𝜎0

− 𝑒𝜎≤ 0, (10c)

𝐴𝑙≤ 𝐴 ≤ 𝐴

𝑢. (10d)

The compliance objective 𝐽 is defined as

𝐽 = ∫

𝐿

0

𝑤 (𝑥) d𝑥. (11)


0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

1.2

Iterations

Func

tion

Displacement objectiveVolume constraint

−0.2

(a) Iterations of displacement objective for volume constraint set at 90%of initial volume

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

1

2

3

4

5

6

7

8

Normalized volumeN

orm

aliz

ed d

ispla

cem

ent

Initial uniform beam

(b) Efficiency curve of tip displacement and total volume

Figure 3: Tip displacement objective for beam with tip load.

The integral is evaluated by approximating 𝑤(𝑥) with cubicHermite shape functions and summing over all elements:

𝐽 =

nel∑

𝑖=1

𝐿𝑒(

𝑤(𝑖)

1+ 𝑤(𝑖)

2

2

+ 𝐿𝑒

𝜃(𝑖)

1− 𝜃(𝑖)

2

12

) , (12)

where 𝐿𝑒is the length of each element. The sensitivity of the

objective to the deformation for element 𝑖 is

𝜕𝐽(𝑖)

𝜕d= {0,

𝐿𝑒

2

,

𝐿2

𝑒

12

, 0,

𝐿𝑒

2

, −

𝐿2

𝑒

12

} . (13)

Sensitivity to design variables is evaluated by summing (13)over all elements then applying (4). The efficiency curve iscomputed starting from beam of uniform cross-section areaof 𝐴𝑖= 0.025m2. Lower and upper limits of the area design

variables are 0.001m2 to 0.05m2, respectively. Ten elementsmodel the deformation and one design variable is assignedto each element. Constraint on the volume is varied between𝑒3= 0.3 and 𝑒

10= 1.0 in. increments of 0.1; see Figure 4. For

the same weight, the stiffness of the beam improved by 2.5times (𝐽 = 0.40). This is within 2.5% of the analytic resultreported in [19], where the authors calculated 𝐽 = 135/343.The difference decreases with increase in the number ofdesign variables.

5.2. Depth Tapered Beam. The cantilever beam consideredhas a tip moment, geometry, and dimensions as indicatedin Figure 5(a). Use of U.S. customary units is favored herefor direct comparison to the analytical results. Haug and

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

1

2

3

4

5

Normalized volume

Initial uniform design

Nor

mal

ized

com

plia

nce

Figure 4: Efficiency curve of area under lateral deformation andtotal volume for beam with uniform distributed load.

Kirmser [20] derived analytic thickness design for minimumvolume under stress and tip displacement constraints. A tipdisplacement objective is retained as

minℎ

𝛿 (ℎ) , (14a)


1 2 3 4 5 x

y

z

y

i: design variable number

hi

1 in. (0.025m)10 in. (0.25m)

450 lb·in. (50.85N·m)

(a) Loading and geometry of depth tapered beam

0.1 0.2 0.3 0.4

0 0.49Y

ZX

Displacements (in.)

2 23 31 51 55

573733 53

5935 39 43 47

494541

27

2521 29

3

2 3

7 11

11 12 13 1410

15

15 16 17 18

19

19

13 17

70 1

0 1

63 67

696561

71 75 79

777395

54 6 8 9

(b) Displacement contours of designed beam. The mesh shows rectan-gular elements. A design variable is assigned per two elements

Figure 5: Depth tapered beam example.

0 5 10 15 20

0

Design variable

Sens

itivi

ty

Finite differenceAnalytic

−1

−0.8

−0.6

−0.4

−0.2

(a) 1 element

Finite differenceAnalytic

0 5 10 15 20

0

Design variable

Sens

itivi

ty

−1

−0.8

−0.6

−0.4

−0.2

(b) 2 elements

Figure 6: Comparison between analytic and finite difference sensitivities when number of elements per design variable is increased. 𝑡 = 0.3in and ] = 0.3.

subject to

𝑔1(ℎ) =

𝑉 (ℎ)

𝑉0

− 𝑒 ≤ 0, (14b)

𝑔2(ℎ) = 𝜎 (ℎ) − 𝜎

0≤ 0, (14c)

ℎ𝑙≤ ℎ ≤ ℎ

𝑢. (14d)

ℎ is the plate thickness design variables, ℎ𝑙is the lower limit,

and ℎ𝑢is the upper limit. Plate elements are utilized to model

the deformation [21]. The mesh of the plate is generatedusing block command in FEAP; see Figure 5(b). Each blockcorresponds to a distinct material number. In Figure 5(b), atotal of twenty materials or design variables are generated.Thematerials are then merged together to form the completestructure. Coincident nodes are removed in this process.

Sensitivity of the tip deflection with respect to thicknessdesign variables is reported in Figure 6. The sensitivity isseen to be affected by fixed end boundary condition; seeFigure 6(a) in the plate element. To improve accuracy of thesensitivity, more elements need to be used per design variable[22, 23]. The sensitivity is recomputed with more elementsper design variable; see Figure 6(b). Smoother sensitivity isseen near the fixed end. Alternatively one can eliminate thisby setting ] = 0 to retrieve the beam stiffness [24, page 533].

The analytic minimum beam volume reported in [20]is specified for a maximum tip displacement of 0.5 in.(0.0127m), maximum stress of 30,000 psi (207MPa), andlower and upper limit on the design variables of 0.3 and0.5 in., respectively.Thematerial has𝐸 = 10×106 psi (69GPa)and ] = 0.3. A numerical design is also calculated in [6]with fewer optimization iterations by exploiting improved


Table 3: Final design of cantilever beam with end moment.

Thickness at center of element

Design variable Reference [20]in inches

FEAP/MMA designin inches

1 0.431 0.4222 0.426 0.4193 0.420 0.4144 0.414 0.4085 0.407 0.4016 0.400 0.3957 0.393 0.3888 0.386 0.3809 0.378 0.37310 0.369 0.36411 0.360 0.35512 0.350 0.34513 0.340 0.33514 0.328 0.32315 0.314 0.31016 0.300 0.30017 0.300 0.30018 0.300 0.30019 0.300 0.30020 0.300 0.300Total volume in.3 3.608 3.566Tip deflection in. 0.500 0.495

approximation techniques.Themaximumvolume is set to theminimum analytic one (𝑒 = 0.9453). Comparison of results isprovided in Table 3. Good agreement is found.

5.3. Topology of Cantilever Structure. The topology designof the extensively studied MBB problem is analyzed [25].The beam has a length-to-height ratio of 6 and is simplysupported and loaded at its center. Due to symmetry, half ofthe beam is considered in the optimization; see Figure 7. Theoptimization problem is to minimize the compliance of thestructure at specified limit of material volume:

min𝑥𝑐 (𝑥) , (15a)

subject to

𝑔 (𝑥) =

𝑉 (𝑥)

𝑉0

− 𝑒 ≤ 0. (15b)

The solid isotropic material with penalization (SIMP) isutilized [26] in the topology design. In this approach, thedesign variables 𝑥 correspond to the volume density ofa material element. The volume densities are linked tomechanical properties of interest. The elastic modulus 𝐸 isnormally chosen for compliance objective:

𝐸 (𝑥) = 𝐸min + 𝑥𝑟(𝐸0− 𝐸min) , (16)

where 𝐸0is the elastic modulus of the material and 𝐸min is

minimum stiffness to avoid singularity. Or 𝐸min may be set as

1 2

3 4

7 8

10

12

16

18

20

24

26

28

32

34

36

40

42

44

48

1

2

3

4

5

6

7

8

9

10

11

12Q

Figure 7: Half of MBB beam discretized into 12 square elements.Node 42 restrained in vertical direction. Nodes 1, 3, and 7 restrainedin horizontal direction. Load is applied at node 7.

the modulus of a secondary material. 𝑟 is a penalty parameterselected to yield binary values of the densities (0 or 1). Thecompliance is calculated according to

𝑐 (𝑥) = P𝑇D (𝑥) . (17)

In anticipation of large number of design variables intopology optimization, the adjoint method (7) is utilizedto compute sensitivity of compliance objective. The adjointvector is found from (6) by setting 𝜕𝑔/𝜕D = P. Althoughthe finite difference step used in computing the residualsensitivity must be selected carefully depending on the meshsize and penalty parameter 𝑟, sensitivity of volume constraintis computed explicitly.

The design domain is discretized using two dimensionalsquare blocks. Each block corresponds to a distinct materialor design variable. An example of design domain is con-structed with 12 blocks; see Figure 7. Displacement of thebeam is computed under plane stress assumption. Parametersof the beam considered are 𝐸

0= 1, 𝐸min = 1 × 10

−5, 𝑄 = −1,and ] = 0.3.

The strict application of (16) in conjunction with 4-nodeelements per design variable often results in designs that aremarred by checkerboard patterns. Dıaz and Sigmund [27]showed that designed structures possess a fictitiously higherstiffness when the patterns are present. An example of this isdemonstrated in Figure 8.

Standard methods used to circumvent checkerboards areuse of higher-order elements and filters [28]. Here advantageis made of elements library in FEAP to resolve the numericaldifficulty. 8-node (Q8) and 9-node (Q9) elements are studied,providing extra degrees of freedom per design variable; seeFigure 9.

In contrast to the linear 4-node elements, these elementsprovide quadratic interpolation of the displacement field.The presence of additional degrees of freedom per designvariable allows for better estimate of the displacement fieldand its sensitivity, especially for penalized formulation ofthe SIMP method. To arrive at designs with high efficiency,continuation method is employed on the penalty parameter.Topologies are computed formesh of 26 × 78 design variablesfor Q8 and Q9 elements; see Figures 10 and 11, respectively.

The higher-order elements are found to almost eliminatecheckerboards from the design. A topology with finer designvariable mesh is computed; see Figure 12. The objectivereaches a minimum in around 20 iterations with remain-ing iterations devoted to removing intermediate densities.


10 20 30 40 50 60

5

10

15

20

00.20.40.60.81

(a) 4-node element. 𝑟 = 2

10 20 30 40 50 60

5

10

15

20

00.20.40.60.81

(b) 4-node element. 𝑟 = 3

Figure 8: Designed topology with a volume fraction of 0.5. A mesh of design variables 20 × 60 is utilized. Each design variable correspondsto a 4-node element. 29 iterations of optimization are carried out.

x

y

(a) 8-node element (b) 9-node element

Figure 9: Schematic of high order elements.

The use of sensitivity filter [26] also eliminates the checker-board problem and provides a design independent of refine-ment in the mesh size; see Figure 13. Intermediate densitiesare now transferred to near the boundary of the material andcan pose interpretation difficulties for manufacturing.

6. Buckling and Flutter Analysis ofa Heated Panel

Panel flutter is studied by many researchers in aerospaceapplications. See, for example, Stanford and Beran [29] andreferences therein. A heated panel is studied here that is fixedat both ends and subjected to aerodynamic flow on the uppersurface; see Figure 14.

The upper and lower surfaces are heated from an initiallyunstressed reference temperature. The heating results inthermal stresses that may cause buckling and/or decreaseresistance to lateral aerodynamic load. To analyze the panel,it is necessary to account for the aerodynamic and thermalloads in the structural model. Once these are developed,an eigenvalue analysis of the dynamic system is utilized tocalculate the flutter and buckling metrics.

6.1. Panel Model. Equations of motion are derived based onprinciple of virtual work by balancing external work with

elastic, inertial, and structural dissipation [24, page 375]. Thework balance for an element reduces to

[m] { 𝑑} + [c] { 𝑑} + ([k] + [k𝜎]) {𝑑} = {𝑟

aero} . (18)

Here [m], [c], and [k] are the element mass, damping, andstiffness matrices, [k

𝜎] is the stress stiffness matrix resulting

from thermal stress [24, page 642], and {𝑑} is the nodaldeformation consisting of axial 𝑢, lateral 𝑤, and rotational𝜃 degrees of freedom. {𝑟aero} is the external nodal load dueto aerodynamic flow. In the following, the aerodynamic loadand stress stiffness matrix are derived for an Euler-Bernoullibeam element.

6.2. Aerodynamic Load. The structural panel is exposed toquasi-steady supersonic flow. Piston theory is used to modelthe pressure on the top surface of the panel. The pressure isgiven by Dowell [30]:

𝑝 =

2𝑞𝑎

𝛽

[

𝜕𝑤

𝜕𝑥

+ (

𝑀2

∞− 2

𝑀2

∞− 1

)

1

𝑈∞

𝜕𝑤

𝜕𝑡

] , (19)

where 𝑈∞

is the flow velocity, 𝑀∞

is Mach number, 𝑞𝑎=

𝜌2

∞𝑈∞/2 is the dynamic pressure, 𝜌

∞is the air density, and


10 20 30 40 50 60 70

10

20

00.20.40.60.81

(a) 𝑟 = 2.0. Iterations 60

00.20.40.60.81

10 20 30 40 50 60 70

10

20

(b) 𝑟 = 2.3. Restart. Iterations 81

Figure 10: Volume fraction of 0.5. Mesh 26 × 78. Q8 elements. 𝑐 = 195.8.

10 20 30 40 50 60 70

10

20

00.20.40.60.81

(a) 𝑟 = 2.0. Iterations 80

10 20 30 40 50 60 70

10

20

00.20.40.60.81

(b) 𝑟 = 2.3. Restart. Iterations 80

Figure 11: Volume fraction of 0.5. Mesh 26 × 78. Q9 elements. 𝑐 = 197.2.

0 10 20 30 40

0

0.2

0.4

0.6

0.8

1

1.2

Iterations

Func

tion

Compliance objectiveVolume constraint

−0.2

(a) Design iterations

20 40 60 80

10

20

30

00.20.40.60.81

(b) Topology

Figure 12: Volume fraction of 0.5. Mesh 33 × 99. Q9 elements. 𝑟 = 2.0. Iterations 168. 𝑐 = 194.8.

5 10 15 20 25 30

2468

10

00.20.40.60.81

(a) Mesh 10 × 30

20 40 60 80 100 120

10

20

30

40

00.20.40.60.81

(b) Mesh 40 × 120

Figure 13: Volume fraction of 0.5. Q4 elements. 𝑟 = 2.5.


y

z

l

h

x

U∞

To Ti

Figure 14: Panel geometry showing applied loads.

𝛽 = √(𝑀2

∞− 1).The surface pressure is resolved into a work-

equivalent nodal load via the finite element shape functions𝑁 [24, page 111]:

raero = ∫𝑆

𝑁𝑇Φ (𝜁, 𝑤, 𝜃, ��,

𝜃) d𝑆. (20)

Here 𝑁 are composed of linear shape functions thatdescribe axial displacements and Hermitian shape func-tions that describe lateral deflection 𝑤. The scalar pressureΦ(𝜁, 𝑤, 𝜃, ��,

𝜃) is derived by inserting the approximate lateral

deformation𝑤 into (19). Decomposing the aerodynamic loadinto elastic and damping components,

Φ =

1

𝐿𝑒

2𝑞

𝛽

𝜕 [𝑁𝑤]

𝜕𝜁

{{{

{{{

{

𝑤1

𝜃1

𝑤2

𝜃2

}}}

}}}

}

+

2𝑞

𝛽

(

𝑀2− 2

𝑀2− 1

)

1

𝑈

[𝑁𝑤]

{{{

{{{

{

��1

𝜃1

��2

𝜃2

}}}

}}}

}

.

(21)

The pressure acts normally to panel in opposite direction tothe lateral deformation. For linear analysis, this direction canbe assumed constant:

Φ = −Φ𝑘. (22)

The equivalent elastic nodal load becomes

raero𝑒

= −∫

𝑆

[𝑁𝑤]𝑇

Φ1(𝜁, 𝑤, 𝜃) d𝑆, (23)

and aerodynamic damping load becomes

raero𝑑

= −∫

𝑆

[𝑁𝑤]𝑇

Φ2(𝜁, ��,

𝜃) d𝑆. (24)

The aerodynamic stiffness and damping matrices are foundby taking derivative of aerodynamic forces to deformations[31]:

k𝑎𝑖𝑗= −

𝜕𝑟aero𝑒𝑖

𝜕𝑑𝑗

,

c𝑎𝑖𝑗= −

𝜕𝑟aero𝑑𝑖

𝜕𝑑𝑗

.

(25)

UsingMATLAB symbolic toolbox, the aerodynamicmatricesare computed:

k𝑎= −

2𝑞

𝛽

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 −

1

2

𝐿𝑒

10

0

1

2

−

𝐿𝑒

10

0 −

𝐿𝑒

10

0 0

𝐿𝑒

10

−

𝐿2

𝑒

60

0 0 0 0 0 0

0 −

1

2

−

𝐿𝑒

10

0

1

2

𝐿𝑒

10

0

𝐿𝑒

10

𝐿2

𝑒

60

0 −

𝐿𝑒

10

0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

,

c𝑎= −

2𝑞

𝛽

𝑀2

∞− 2

𝑈∞(𝑀2

∞− 1)

⋅

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0

13𝐿𝑒

35

11𝐿2

𝑒

210

0

9𝐿𝑒

70

−

13𝐿2

𝑒

420

0

11𝐿2

𝑒

210

𝐿3

𝑒

105

0

13𝐿2

𝑒

420

−

𝐿3

𝑒

140

0 0 0 0 0 0

0

9𝐿𝑒

70

13𝐿2

𝑒

420

0

13𝐿𝑒

35

−

11𝐿2

𝑒

210

0 −

13𝐿2

𝑒

420

−

𝐿3

𝑒

140

0 −

11𝐿2

𝑒

210

𝐿3

𝑒

105

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

.

(26)

Here zeros are added to the axial degree of freedom to matchsize of matrices to the element stiffness matrix.

6.3. Thermal Load. The effect of the thermal load is toreduce the overall stiffness of the panel by developing axialcompressive stresses. This has adverse effect on bucklingand flutter constraints. The net compressive stress in panelis computed by accounting for thermal strain panel [24,page 52]. The process of computing the thermal stress issummarized by (1) calculating a consistent nodal thermalload for an element; (2) assembling thermal nodal loads andconventional elastic stiffness matrix; (3) calculating thermalnodal deformations by solving

[K] {𝐷} = {𝑅thermal} , (27)

where K is the global stiffness matrix and 𝑅thermal is thermalload vector; (4) updating stresses in each element withthermal deformations; and (5) finally computing an elementstress stiffness matrix.

The thermal load is computed assuming a prescribedsteady state thermal boundary condition [29]. Then byapplying Fourier law in one dimension, the temperatureprofile in the panel is computed:

𝑇 (𝑧) =

𝑇𝑖+ 𝑇𝑜

2

+

𝑇𝑜− 𝑇𝑖

ℎ

𝑧. (28)

Here 𝑇(𝑧) refers to the increase in temperature from areference temperature with no thermal stress. The thermal


axial force is found by integrating the initial thermal stressover the cross-sectional area [32]:

𝑃𝑇= ∫

𝐴

𝜎0d𝐴 = −𝐸𝛼𝐴

𝑇𝑖+ 𝑇𝑜

2

, (29)

where 𝛼 is the coefficient of thermal expansion, and thethermal bendingmoment is found by integrating the bending

stress over the cross-sectional area [33, page 406]:

𝑀𝑇= ∫

𝐴

𝐸𝛼𝑇 (𝑧) d𝐴 = −𝐸𝛼𝐴𝐼𝑦


ℎ

. (30)

A consistent nodal force andmoment vector are computed byinserting the axial and bending loads into [24, page 90]

{𝑟thermal

} = −∫

𝑆

[𝐵]𝑇{𝜎0} d𝑉, (31)

where [𝐵] is the strain displacement matrix, giving

{𝑟thermal

} = [−𝐸𝐴𝛼(

𝑇𝑖+ 𝑇𝑜

2

) 0 −𝐸𝛼𝐼𝑦(


ℎ

) 𝐸𝐴𝛼(

𝑇𝑖+ 𝑇𝑜

2

) 0 𝐸𝛼𝐼𝑦(


ℎ

)]

𝑇

. (32)

Thermal deformations are computed from (27) and theaxial thermal stress in an element with nodes (𝑖, 𝑗) is thencomputed as

𝜎 = 𝜎0+ 𝐸

𝑢𝑗− 𝑢𝑖

𝐿𝑒

. (33)

A stress stiffness matrix is calculated based on the in-planemembrane stresses [24, page 642]:

k𝜎=

𝜎𝐴

30𝐿𝑒

[

[

[

[

[

[

[

[

0 0 0 0 0 0

0 36 3𝐿𝑒0 −36 3𝐿

𝑒

0 3𝐿𝑒4𝐿2

𝑒0 −3𝐿

𝑒−𝐿2

𝑒

0 0 0 0 0 0

0 −36 −3𝐿𝑒0 36 −3𝐿

𝑒

0 3𝐿𝑒−𝐿2

𝑒0 −3𝐿

𝑒4𝐿2

𝑒

]

]

]

]

]

]

]

]

. (34)

6.4. Equations of Motion. The element equation of motion(18) can be written in terms of aerodynamic damping andstiffness matrices:

[m] { 𝑑} + ([c] + [c𝑎]) {

𝑑}

+ ([k] + [k𝜎] + [k

𝑎]) {𝑑} = {0} .

(35)

Assembling (35) for all elements, the equation of motion forthe panel becomes

[M] {��} + ([C] + [C𝑎]) {��}

+ ([K] + [K𝜎] + [K

𝑎]) {𝐷} = {0} .

(36)

To facilitate stability analysis, (36) is written in first orderform:

[

I 00 M]{

{��}

{��}

} = [

0 I−K𝑡−C𝑡

]{

{𝐷}

{��}} , (37)

whereK𝑡andC

𝑡are the total stiffness and damping matrices,

respectively. One may obtain an eigenvalue problem of (37)by assuming an exponential solution:

𝛽 [A] {V𝑅} = [B] {V𝑅} , (38)

where V𝑅 is the right eigenvector corresponding to theeigenvalue 𝛽. Solution of the eigenvalue problem of thissystem determines its stability characteristics [34, page 248].The eigenvalue solver must be selected to handle nonsym-metry of [B] due to the aerodynamic stiffness matrix. Thenonsymmetric eigenvalue driver DGGEVX.F is utilized here[35].

For the unheated panel and zero flow condition, solutionof (38) for the uniform panel gives the natural modes of thesystem with the first mode given as

𝜔0= 22.373√

𝐷𝑝

𝜌𝑚ℎ𝐿4, (39)

where 𝐷𝑝= 𝐸ℎ

3/12 is the flexural rigidity of the panel.

For the heated panel and zero flow condition, solution of(38) results in the critical buckling temperature 𝑇cr. Thistemperature is nondimensionlized as

𝑅cr =𝛼𝐸𝑇crℎ𝐿

2

𝐷𝑝

. (40)

For a thermal condition of 𝑇𝑖= 𝑇𝑜= 1, the first buckling

temperature yields 𝑅cr = 4.0𝜋2.

6.5. Flutter and Buckling Results. The onset of flutter istypified by emergence of a Hopf bifurcation [36] as a flowspeed parameter is increased. Defining the parameter as in[30]:

𝜆 =

𝜌∞𝑈2

∞𝐿3

𝐷𝑝√𝑀2

∞− 1

. (41)

At the Hopf point, a pair of complex conjugate eigenvalues ofthe systemcross the imaginary axis resulting in a destabilizingpositive damping. To locate the flutter boundary, the dampingof the least stable mode,

𝐺 = max (Re (𝛽)) , (42)

is tracked to identify when 𝐺 = 0.


0 200 400 6000

5

10

𝜆

Imag

(𝛽k/𝜔

0)

(a)

0 200 400 600

0

0.2

−0.2

−0.4

Real

(𝛽k/𝜔

0)

𝜆

(b)

Figure 15: Depiction of system eigenvalues as flow parameter is increased for the unheated panel. (a) Frequencies. (b) Damping.

0 200 400 6000

5

10

𝜆

Imag

(𝛽k/𝜔

0)

(a)

0 200 400 600

0

0.5

1

−0.5

−1

Real

(𝛽k/𝜔

0)

𝜆

(b)

Figure 16: Depiction of buckling and flutter of a heated panel. (a) Frequencies. (b) Damping.

An example of flutter computation is shown in Figure 15.The panel has a mass ratio 𝜇 = 𝜌

∞𝐿/𝜌𝑚ℎ = 0.1. Five elements

are used to analyze the structure and apply the aerodynamicload. No structural damping is considered. For the unheatedpanel and at 𝜆 = 0, the eigenvalues yield the naturalfrequencies of the panel. As the flow parameter 𝜆 is increased,the frequencies of the first and secondmodes are seen to comecloser. Near 𝜆 = 657.14, their frequencies coalesce into onefrequency at which point the damping of first mode becomespositive. This flutter point is slightly different from 𝜆 = 636

reported in [37] as the author discounted the aerodynamicdamping term reducing model to static strip theory [38].

Tracing the eigenvalues of a heated panel at a uniformtemperature of 𝑅/𝜋2 = 6.3 larger than 𝑅cr, one can observethat the panel buckles due to the thermal stress at winds off.This is identified by zero buckling frequency and the positive

damping of first mode; see Figure 16. As the aerodynamicload is increased beyond 𝜆 = 151.10, the panel is blown stableas in [29], until a critical 𝜆 = 186.28 is reached, where thepanel loses stability to flutter.

In a similar manner, the flutter and buckling boundariescan be traced in the domain of (𝜆, 𝑅); see Figure 17. At eachheating load𝑅 the flutter and buckling points are located.Theboundary studied in [29] is calculated here.

The process of identifying the flutter point can be madefaster with use of a root finding technique. The false positionis a root bracketing method that can be used to facilitatelocating of the flutter point by iterating [39, page 129]

𝜆𝑟= 𝜆𝑢−

𝐺 (𝜆𝑢) (𝜆𝑙− 𝜆𝑢)

𝐺 (𝜆𝑙) − 𝐺 (𝜆

𝑢)

, (43)


0 2 4 6 80

200

400

600

Aero

dyna

mic

load

𝜆

Heating load R/𝜋2

𝜆crRcr

Figure 17: Flutter and buckling boundaries.

where 𝜆𝑙and 𝜆

𝑢are the lower and upper values of the flow

parameter, respectively. The method is based on locatingpositive and negative values of a function at opposite sides ofthe root. For zero structural damping, the systems damping iseither positive or zero. Therefore, a small structural dampingmust be introduced for the method to work. In this work,Rayleigh type structural damping is used:

[C] = 𝑎0[M] + 𝑎

1[K] , (44)

where constants 𝑎0and 𝑎

1are evaluated by assuming a

constant damping factor of 0.01% for the first and 4th modes[40, page 455]. Recalculation of the flutter point of theunheated panel gives a nearly identical result as above (0.3%difference).

7. Conclusions

A structural optimization framework is developed that com-bines finite element analysis capability of FEAP with agradient-based optimization search. Analytic sensitivities areincorporated into the framework to provide design gradients.The optimization algorithm is integrated directly with FEAPmemorymanagement system through the use of usermacros,providing efficient execution of the optimization search.

Four test problems are analyzed using the frameworkwithregard to sizing and topology optimization. In area of sizingdesign variables, the problems are found to give solutionsidentical to analytic designs with 1-2% difference due todiscrete nature of finite element. In topology optimization, itis found that 8-node and 9-node square elements are robustin handling checkerboard patterns in the design, with slightadvantage of the latter element. Finally buckling and flutteranalysis of a heated panel is developed to demonstrate capa-bility of introducing new physics into the structural analysis.

There are many advantages to the framework. First theframework provides easy access to all capabilities of FEAP

program, which include thermal and mechanical analysiswith a variety of elements, in addition to capability of interfac-ing with a parallel sparse matrix solver. Second because of thedirect integration, the structural models can be extended tolarge problems. Third the user may expand on existing toolsin FEAP by deriving his own elements and macros to analyzenew physics.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The author would like to thank Professor Robert L. Taylor atUniversity of California, Berkeley, for valuable insights on theuse of FEAP.

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