Materials and Design 23(2002) 657–666

0261-3069/02/$ - see front matter� 2002 Elsevier Science Ltd. All rights reserved.PII: S0261-3069Ž02.00048-1

Bi-objective optimization design of functionally gradient materials

Jinhua Huang *, George M. Fadel , Vincent Y. Blouin , Mica Grujicica, b b b

Department of Aerospace Engineering and Mechanics and Engineering, University of Florida, Gainesville, FL 32611, USAa

Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USAb

Received 5 September 2001; accepted 10 May 2002

Abstract

In this paper, a procedure for bi-objective optimization design of functionally gradient materials(FGM) is presented. Differentmicrostructures formed by two primary materials are evaluated by a micromechanical analysis method. Macroscopically, FGMsare optimally designed by using these microstructures. Instead of using conventional simply assumed power law materialdistribution functions, a generic material distribution function is used. The bi-objective FGM optimization design procedure ishighlighted by a flywheel example. A parametric formulation is used for both the geometric representation and the optimizationprocedure.� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Functionally gradient material(FGM); Micromechanical analysis; Effective material properties; Numerical analysis; Bi-objectiveoptimization; Weighting method; Tchebycheff method

1. Introduction

A functionally gradient material(FGM) is a compos-ite, consisting of two or more phases, which is designedsuch that its composition varies in some spatial direction.This design is intended to take advantage of certaindesirable features of each of the constituent phasesw1,2x.For example, one constituent may be a ceramic whichoffers good high-temperature behavior but is mechani-cally brittle. Another may be a metal that exhibits bettermechanical and heat-transfer properties but cannot with-stand exposure to high temperatures. An FGM couldthus be predominantly ceramic within the hotter regionand metal within the cooler regionw3x.A major problem in the design of an FGM, aside

from that of primary material selection, lies in determin-ing the optimal spatial dependence on the composition.This can be regarded as the composition profile thatbest accomplishes the intended purposes(design objec-tives) of the materials while all constraints are satisfied.Different design objectives and constraints will lead todifferent optimal composition profiles. Another problemlies in computing the effective material properties at

*Corresponding author.E-mail address: david_jinhuahuang@yahoo.com(J. Huang).

different composition points and evaluating the FGMperformance under given working conditions.Few published articles considered multiple objectives

in the design of functionally gradient materials. Forpractical problems, several design objectives may berequired. For example, in a flywheel design, bothstrength and energy storage capability need to be con-sidered. In addition, in conventional FGM design meth-ods, a simple power law function is generally assumedfor possible material distribution as described by Hiranoet al. w4x. However, the optimal material distributionpattern is difficult to predict and may not be representedby the power law formulation.In the calculation of effective material properties,

heuristic rules and micromechanical analysis methodswere widely usedw2,4x and most recently finite elementbased methods were developedw2,5x. Over the lastdecade, a family of multimaterial component(FGM)fabrication techniques such as layered manufacturingand self-propagating high-temperature synthesis havebeen established, which makes it quite practical todesign functionally gradient materials with arbitrarymicrostructures. It is important to have a method thatcan compute the effective material properties of arbitrarymicrostructures. Our objective is to develop a generic

658 J. Huang et al. / Materials and Design 23 (2002) 657–666

bi-objective optimization design procedure for function-ally gradient engineering products with specifiedmicrostructures.In what follows, we first present the bi-objective

optimization design procedure for functionally gradientmaterials. Then, we describe a micromechanical analysismethod for computing effective material properties andaddress the modeling of material spatial one-dimensionaldistribution. Finally, we highlight the bi-objective FGMoptimization design procedure by a flywheel exampleand draw conclusions.

2. Bi-objective FGM optimization design

We consider two primary materialsA and B. Let vA

andv be the volume fractions ofA andB, respectively.B

Both v andv are space dependent and satisfy,A B

v qv s1 (1)A B

Let F andF be the two selected design objectives1 2

dependent on the volume fraction distribution. UsingEq. (1), both F andF can be written as functions of1 2

B’s volume fraction distribution . The general bi-vBobjective FGM optimization design problem can bestated as:

w zx |˜ ˜Ž . Ž .min F v ,F v1 B 2 By ~

˜Ž . Ž .s.t. G v F0 js1,...,l (2)j B

where G ,«,G are l design constraints. The implicit1 l

objective functions and constraints in the above problemare computed by a numerical analysis methodw2x.The solution for problem(2) is a Pareto set and

regardless of the convexity of the problem, any pointon the Pareto set can be found by applying the Tche-bycheff method. If the problem is convex, the weightingmethod may be used.The Tchebycheff method usesL -metrics to probe thep

possible Pareto point in the objective space. It usesdifferent Tchebycheff metrics to find the closest pointsto the ideal(Utopia) point in the objective space. Themetric is the distance between two points in spaceR2

of the two objective functions. TheL -norm of FgR2p

is defined as the length of a vector,

2 1ypw zp≤ ≤ Z Z µ ∂ µ ∂F s F pg 1,2,3... j ` (3)x |i8p y ~is1

The L -metric of two pointsx, ygR is defined as:2p

2 1ypw zp≤ ≤ Z Z µ ∂ µ ∂xyy s xyy pg 1,2,3,... j ` (4)x |i i8p y ~is1

For ps`:

≤ ≤ Z ZF smax F ,µ ∂i` is1,2

≤ ≤ Z Z µ ∂ µ ∂xyy smax xyy pg 1,2,3,... j ` (5)µ ∂i i` is1,2

The ideal criterion vector(Utopia point) F used in*

the Tchebycheff method is computed by:

S W˜Ž .F vi BT T* U X Ž .F smin is1,2 (6)i

T T˜Ž . Ž .V YG v F0 js1,...,lj B

where is the function values obtained by minimizing*Fi

the ith objective function.Given an ideal criterion vectorF and weight*

, the weighted Tchebycheff2S W

T T2U Xl: lgR l G0, l s1i iT TZ 8V Yis1

metric is defined as:l* *≤ ≤ Z ZFyF smax l FyF (7)µ ∂i i i` is1,2

Let . The weighted Tchebycheff meth-l*≤ ≤bs FyF`

od can be written as follows:

min b*w z

x |˜Ž . Ž .s.t. bGl F v yF is1,2i i B iy ~

˜Ž . Ž .G v F0 js1,...,l (8)j B

By varying weight l and l systematically and1 2

solving a series of optimization problems, differentpoints on the Pareto set can be obtained.The weighting method uses weights to combine the

two objective functions in problem(2) to form thefollowing new problem:

˜ ˜Ž . Ž .min w F v qw F v1 1 B 2 2 B

˜Ž . Ž .s.t. G v F0 js1,...,lj B

w qw s11 2

w G0, w G0 (9)1 2

wherew andw are the weighting coefficients for the1 2

corresponding objective functions. Different points onthe Pareto set can be obtained by varyingw andw .1 2

To avoid spurious local optima caused by numericalanalysis accuracy, problem(8) and (9) are both solvedby alternatively using a gradient based method and agenetic algorithmw6,7x. Starting from a feasible point

, b is first minimized by a gadient based method0( )vB

using the software DOTw8x. A 1% random mutation onthe optimal solution from the gradient method is1( )v

B

used to create an initial population for the geneticalgorithm w9x. After b is further reduced at by the2( )v

B

genetic algorithm, the gradient method is used againwith as the new start point. The process is repeated2( )v

B

until both genetic algorithm and gradient based methodfail to give further reduction onb.

3. Computation of effective material properties

The effective material properties are affected by thevariation in both material composition and microstruc-

659J. Huang et al. / Materials and Design 23 (2002) 657–666

Fig. 1. Arbitrary single inclusion composite(a) single arbitrary inclusion unit cell(b) meshed triangular elements for finite element analysis.

Fig. 2. One-dimensional Euclidean space and material distributiondiagram.

ture. Herein, a micromechanical analysis method is usedto calculate the effective material properties of compos-ites with arbitrary microstructures. In this method, theeffective material properties at a specific volume fractionand microstructure are obtained by relating the corre-sponding average physical components within a unitinclusion cell as shown in Fig. 1a. The average physicalcomponents are obtained through a finite element meth-od. Taking the computation of elastic effective materialproperties as an example, a unit cell with one or severalspecified inclusions is first discretized into a number ofsmall elements(Fig. 1b). Each of the discretized smallelements has either matrix or inclusion material.Under certain load and boundary conditions, the stress

s(x) and strain´(x) at an arbitrary pointx within theunit cell are computed through a finite element method.The corresponding average stress, and strain, for¯ ¯s ´the unit cell are the variables of interest for calculatingelastic effective material properties and are defined as:

1¯ Ž .ss s x dv (10)|Z ZV

V

1¯ Ž .´s ´ x dv (11)|Z ZV

V

whereV is the domain of the representative unit cell.The average strain and stress can be related by

e¯ ¯ssC ´ (12)

e¯ ¯´sS s (13)

whereC is the effective 6=6 symmetric stiffness tensore

and S is the effective 6=6 symmetric compliancee

tensor. Thermal effective material properties can becomputed through a similar unit microstructure analysismethod.The method presented here is used in Section 5 for

the design of a functionally graded flywheel made oftwo materials, Sn and aluminum-based alloy2124_T851.

4. Modeling of one-dimensional material distribution

During the optimization process, any arbitrary mate-rial spatial distribution might be generated. ForvBanalysis purpose, a mathematical expression for generic

distribution has to be constructed. This expressionvBmust be as accurate as possible and satisfy continuityrequirements. For one-dimensional problems, severalmethods may be used including linear interpolation, B-spline, and Bezier curvesw2x. Assume the spatialvBdirection is in thex-axis as shown in Fig. 2. For Bezierrepresentation, the design geometry domain CD is dis-cretized inton segments with each node being attacheda control composition (is0,«,n). With ,i C( ) ( )Ž .v x vB C, B

,«, , , the Bezier represen-1 ny1 D( ) ( ) ( )Ž . Ž . Ž .x ,v x ,v x ,v1 B ny1 B D B

tation for is expressed asvB

ny1Si nn nyinŽ . Ž . Ž . Ž .xsx 1yu q x u 1yu q x ql uC i i C 08T is1

U (14)n

i i nyin( )T˜ w xŽ . Ž .v s v u 1yu ug 0,1B B i8V is0

660 J. Huang et al. / Materials and Design 23 (2002) 657–666

Table 1Physical material properties of Sn and 2124_T851w10x

Density Young’s Poisson’s Tensile(=10 kgym )3 3 modulus(GPa) ratio strength(MPa)

Sn 7.29 41.4 0.33 2202124_T851 2.78 73 0.33 485

Fig. 3. A face-centered Snq2124_T851 bimaterial unit microstructure(a) configuration of the unit microstructure and(b) discretization of one-quarter of the unit cell withx- andy-symmetry constraints.

Table 2Effective Poisson’s ratios with respect to different volume fractions of Sn

v _ af Sn 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y bSn 0.33 0.331 0.333 0.335 0.336 0.336 0.335 0.333 0.331 0.330 0.330

v _ is the volume fraction of Sn.af Sn

y is the effective Poisson’s ratio.bSn

where , ,«, , arenq1 con-0 C 1 ny1 n D( ) ( ) ( ) ( ) ( ) ( )v sv v v v svB B B B B B

trol compositions and .n!nŽ .si Ž .i! nyi !

The material distribution control compositions ,C( )vB,«, , in Eq. (14) are selected as the1 ny1 D( ) ( ) ( )v v vB B B

material design variables for problem(2).

5. Application to the design of functionally gradientflywheels

Two primary materials, Tin(Sn) and aluminum-basedalloy 2124_T851 are selected for the FGM flywheeldesign. The former is selected because of its highdensity, which is favored from the standpoint of maxi-mizing the kinetic energy stored by the flywheel. Thelatter material, on the other hand, is selected for its highyield stress, which is desired from the standpoint ofpreventing failure of the flywheel. The physical proper-ties for the two materials are listed in Table 1. The(Snq2124_T851) bimaterial microstructure is assumedto consist of congruent, face-filling squares, each con-taining a smaller concentric square(Fig. 3). The innersquare is assumed to consist of Sn, while the remainedof the larger square contains 2124_T851. To deriveeffective material properties, a uniformly distributed loadis applied to the top and bottom edges of the unit cell.

Due to symmetry, only one-quarter is discretized(Fig.3b). The one-quarter cell hasx- and y-symmetry con-straints. The effective physical properties with respectto the volume fraction of Sn are next obtained using themicromechanical analysis method described in Section3. The effective Poisson’s ratios as a function of thevolume fraction of Sn obtained using this method arelisted in Table 2. Similarly, Fig. 4 shows the values ofthe effective Young’s moduli at different volume frac-tions of Sn.The surface norms(the localz-axis direction in Fig.

3) of all microstructures are supposed to be parallel tothe flywheel axis. It is also assumed that sides of themicrostructure squares are aligned with the local radialdirection. Due to the continuous nature and the axisym-metric shape of the flywheel as shown schematically inFig. 5, the volume fraction, , of Sn is taken to bevf Sny

a function of r, the radial distance from the rotatingaxis. This assumption is consistent with the typicalmicrostructure. Axisymmetric heterogeneous parts canbe produced by rapid tooling processes such as laserengineered net shaping in which each layer of a part isproduced one concentric ring(with constant Sn volumefraction) at a time.Bezier curves are used to represent a flywheel’s

geometry and material variations in the radial direction

661J. Huang et al. / Materials and Design 23 (2002) 657–666

Fig. 4. Effective Young’s moduli in they-axis direction with respect to different volume fractions for the microstructure in Fig. 3.

Fig. 5. The alignment of a typical microstructure in the FGM flywheelto be designed. Fig. 6. Geometry and material configuration of an FGM flywheel.

(Fig. 6). Let (r , h ),«, (r , h ),«, (r , h ) in a plane0 0 i i n n

passing through thez-axis be thenq1 geometry controlpoints and( ), «, ( ), «, ( ) be the0 i n( ) ( ) ( )r ,v r ,v r ,v0 f i f n fSn Sn Sny y y

nq1 material control points. The Bezier representationof the FGM flywheel can then be expressed asw11x:

ny1Si nn nyinŽ . Ž . Ž .rsr 1yu q u 1yu rqr uinner i i outer8

is1Tus2pv

U (15)nw z

i nyinŽ . Ž . Ž .zs u 1yu h w 0Fu,v,wF1x |i i8y ~is0T n

i i nyin ( )˜ Ž . Ž .v s v u 1yuf i fSn Sny y8V is0

wherer and r are the flywheel’s inner and outerinner outer

radii, respectively.

The unknownsh , h , «, h in Eq. (15) are selected0 1 n

as the geometry design variables and the unknowns, , «, in the same equation are selected0 1 n( ) ( ) ( )v v vf f fSn Sn Sny y y

as the material design variables.The volume, mass, and kinetic energy of the FGM

flywheel are all computed by a numerical analysismethodw11x.Shear stresses are neglected, therefore, the radial and

tangential stresses,s and s , respectively, are therr uu

principal stresses and assumed to be uniform across thethickness and circumference. A numerical analysis meth-od is used to computes ands w11x. The Von Misesrr uu

stress is calculated from

1y2S WT T1 2 22U XŽ .s s s ys qs qs (16)w xVM rr uu rr uuT T2V Y

Two criteria for the optimal design of FGM flywheelsare selected. Since the function of flywheels is to storeenergy, maximization of the storage of the kinetic energyis selected as one of the two objectives. In addition,since the flywheel is not allowed to fail by plastic

662 J. Huang et al. / Materials and Design 23 (2002) 657–666

Table 3Design parameters and desired targets(constraints)

n rInner router v hmin hmax Max Ma Max sVM Min Enb

(m) (m) (radsys) (m) (m) (kg) (MPa) (kJ)

20 0.02 0.2 630 0.02 0.1 75 45 50

M is the mass of the flywheel.a

En is the kinetic energy of the flywheel.b

deformation or fracture, minimization of the maximumequivalent stress within the flywheel is chosen as thesecond design objective. The equivalent stress is basedon the corresponding failure criterion. For the complexSnq2124_T851 microstructure, both Sn and 2124_T851are ductile materials, it is reasonable for us to assumethe failure criterion of distortion energy and take theVon Mises stress as the equivalent stress. All flywheelsshould be able to store a certain amount of energy andbecause of strength requirement, the maximum Misesstress should be kept below a certain value. In Table 3,the related design parameters and desired targets arelisted.From Section 2, the bi-objective FGM flywheel

optimization problem can be stated as:

w xŽ . Ž .min yEnergyX ,Max s Xy VM

Ž .s.t. EnergyX G50 kJŽ .Max s X F45 MPay VM

Ž .MassX F75 kg0.02 mFh ,...,h F0.1 m0 20

0 1 20( ) ( ) ( )0Fv ,v ,...,v F1.0 (17)f f fSn Sn Sny y y

where . The mini-0 1 20 T( ) ( ) ( )w xXs v ,v ,...,v ,h ,h ,...,hf f f 0 1 20Sn Sn Sny y y

mum energy and the overall maximum Max_s haveVM

been given in Table 3. The maximum energy and theoverall minimum Max_s are obtained by separatelyVM

solving the following two single objective optimizationproblems:

Ž .max EnergyXŽ .s.t. EnergyX G50 kJ

Ž .Max s X F45 MPay VM

Ž .MassX F75 kg0.02 mFh ,...,h F0.1 m0 20

0 1 20( ) ( ) ( )0Fv ,v ,...,v F1.0 (18)f f fSn Sn Sny y y

Ž .Min Max s Xy VM

Ž .s.t. EnergyX G50 kJŽ .Max s X F45 MPAy VM

Ž .MassX F75 kg0.02 mFh ,...,h F0.1 m0 20

0 1 20( ) ( ) ( )0Fv v ...,v F1.0 (19)f f fSn, Sn, Sny y y

Let S(X)syEnergy(X) and .Ž . Ž .P X sMax s Xy VM

From Table 3, and the solutions of problem(18)and (19), S syEnergy sy248277 J, S smin max max

yEnergy sy50000 J, P sMax_s smin min VM_min

18.7188 MPa, andP sMax_s s45 MPa. Aftermax VM_max

being normalizedw2x, problem(17) becomes:

w zŽ . Ž .S X q248277 P X y18.7188x |min ,y50000q248277 45y18.7188y ~

Ž .s.t. EnergyX G50 kJŽ .Max s X F45 MPay VM

Ž .MassX F75 kg0.02 mFh ,...,h F0.1 m0 20

0 1 20( ) ( ) ( )0Fv ,v ,...,v F1.0 (20)f f fSn Sn Sny y y

Applying the weighting method, problem(20)becomes:

Ž . Ž .S X q248277 P X y18.7188Ž .min w q 1yw1 1198277 26.2812

Ž .s.t. EnergyX G50 kJŽ .Max s X F45 MPay VM

Ž .MassX F75 kg0.02 mFh ,...,h F0.1 m0 20

0 1 20( ) ( ) ( )0Fv ,v ,...,v F1.0f f fSn Sn Sny y y

0Fw F1 (21)1

Applying the weighted Tchebycheff method, problem(20) becomes:

min b

B EŽ . Ž .S X q248277 P X y18.7188C FŽ .bsmax l , 1yl1 1D G198277 26.2812Ž .S X q248277

s.t. bGl1 198277Ž .P X y18.7188

Ž .bG 1yl1 26.2812Ž .EnergyX G50 kJ

Ž .Max s X F45 MPay VM

Ž .MassX F75 kg0.02 mFh ,...,h F0.1 m0 20

0 1 20( ) ( ) ( )0Fv ,v ,...,v F1.0f f fSn Sn Sny y y

0Fl F1 (22)1

By varyingw andl from 0 to 1, a number of points1 1

on the Pareto set are thus obtained and listed in Table 4and 5. In Table 6, the optimal results atw s1.0 orl s1 1

1.0 for the FGM flywheel and two homogeneous fly-wheels (with the same design parameters and desiredtargets as shown in Table 3) are presented. In addition,two alternate FGM flywheels are obtained by addingactive constraints on the mass and the maximum Misesstress. Based on data from Table 4 and Table 5, thePareto set is plotted in Fig. 7. The optimal flywheelprofiles, material distributions, and Mises stress distri-butions corresponding to weightl s0.0, 0.25, 0.5, 0.75,1

and 1.0 with the Tchebycheff method are drawn in Figs.8 and 9, and Fig. 10, respectively. Fig. 11 shows both

663J. Huang et al. / Materials and Design 23 (2002) 657–666

Table 4Optimal results with weighting method

w1 F a1 F b

2 En (kJ) Max_s (MPa)VM

0.0 0.0 0.0 50 18.71880.125 0.0 0.0 50 18.71880.25 0.0 0.0 50 18.71880.375 0.0 0.0 50 18.71880.5 0.4563 0.5527 140.465 33.02580.625 1.0 1.0 248.277 45.00.75 1.0 1.0 248.277 45.00.875 1.0 1.0 248.277 45.01.0 1.0 1.0 248.277 45.0

aŽ .EnergyX y50000

F s1 198277

bŽ .Max_s X y18.7188VM

F s2 26.2812

Table 5Optimal results with Tchebycheff method

l1 F1 F2 En (kJ) Max_s (MPa)VM

0.0 0.0 0.0 50 18.71880.125 0.0726 0.1326 64.394 22.20420.25 0.1824 0.2723 86.161 25.8750.375 0.3210 0.4116 113.646 29.53680.5 0.4563 0.5527 140.465 33.02580.625 0.5927 0.6716 168.418 36.36920.75 0.7381 0.7933 196.341 39.56830.875 0.8715 0.9011 222.796 42.40121.0 1.0 1.0 248.277 45.0

Fig. 7. Energy vs. maximum Von Mises stress Pareto set separately obtained with Tchebycheff and weighting methods.

Table 6Comparison of optimal results(obtained with Tchebycheff method atl s1.0) for FGM and homogeneous flywheels1

Optimum Material Max Energy Max sVM Massresults (kJ) (MPa) (kg)

1 Homogeneous 96.91 45.0F45.0 27.0F75.0(2124_T815)

2 Homogeneous(Sn) 137.03 36.8F45.0 34.4F75.03 FGM 248.28 45.0F45.0 50.6F75.04 FGM 195.30 45.0F45.0 34.4F34.45 FGM 183.07 36.8F36.8 34.4F34.4

the optimal FGM flywheel profile and material distri-bution contours atl s0.5.1

The energy-maximum stress Pareto curve seems con-vex and is very close to a two-linear-segment curve.The higher the optimal energy, the higher the optimalmaximum stress(Fig. 7). The Tchebycheff method

664 J. Huang et al. / Materials and Design 23 (2002) 657–666

Fig. 8. Optimal FGM flywheel profiles with respect to different Tchebycheff weightsl1.

Fig. 9. Optimal FGM flywheel material distributions of Sn with respect to different Tchebycheff weightsl1.

generates a better solution than the weighting method.When the Pareto set is a linear segment, only one Paretopoint can be found through the weighting method.However, with the Tchebycheff method, each point onthe Pareto set can always be found regardless of theshape of the Pareto set.The optimal flywheel geometry and material distri-

bution, and hence Mises stress distribution, and kinetic

energy storage abilities are dependent on the weightl1

or w . In general, the thickness of an optimal FGM1

flywheel is highest near the inner edge and smallestsomewhere between the inner and outer edges, and thecorresponding Sn volume fraction has the highest valuenear the outer edge and the smallest value near the inneredge(Figs. 8 and 9, and Fig. 11). Such optimal thicknessand material distribution patterns allow the flywheel to

665J. Huang et al. / Materials and Design 23 (2002) 657–666

Fig. 10. Optimal FGM flywheel Von Mises stress distributions with respect to different Tchebycheff weightsl1.

Fig. 11. The optimal profile and material distribution contours of anFGM flywheel obtained with Tchebycheff method atl s0.5.1

have the desired high kinetic energy while the stress iskept small.Whenl is close to zero, the bi-objective optimization1

problem becomes a minimization of the maximum Misesstress under a small energy constraint. Usually, thehighest value of Mises stress over the flywheel thicknessoccurs near the inner edge. It is quite straightforward toreduce the maximum Mises stress by increasing thick-ness and putting stiff and light material(2124_T851)near the inner edge, while a little amount of heavymaterial (Sn) near the outer edge makes the smallenergy requirement satisfied. The multipeak Sn distri-bution (Fig. 9) for small l is more effective for1

reducing the stress when the maximum stress movesaway from the inner edge(Fig. 10). Whenl is close1

to one, the bi-objective optimization problem becomes

a maximization of the kinetic energy under a large stressconstraint. Directly putting more heavy material(Sn)near the outer edge leads to the increase of the kineticenergy while the large stress constraint is still satisfied.For l between zero and one, the optimal material and1

thickness distributions are trade-offs between energymaximization and stress minimization.For the optimal homogeneous flywheel with

2124_T851 as the single material, because of the limitof the design space, the kinetic energy is only 55% theenergy of the corresponding optimal FGM flywheel andfor the Sn optimal homogeneous flywheel, the stresslimit is reached and the energy is only 39% of thecorresponding optimal FGM flywheel’s energy(Table6). The advantage of the FGM flywheel, however, iscounterbalanced by a significantly higher mass. Con-straining first the mass and then the maximum Misesstress to the levels of the homogeneous flywheels(optimal results 4 and 5 of Table 6) leads to highperformance flywheels that can store 42% and 34%more energy than the corresponding homogeneous2124_T851 flywheel of the same weight, respectively.

6. Conclusions

The optimal volume fraction design for maximizingboth desired performances combines the numericaloptimizations of geometry and material distributionswith a micromechanical analysis of the microstructure–properties relations. With the generic Bezier represen-

666 J. Huang et al. / Materials and Design 23 (2002) 657–666

tation of volume fraction and thickness distribution, itis quite flexible to adjust product performances accord-ing to requirements. The Tchebycheff method is moreeffective than the weighting method in finding pointson the Pareto set since that set may be non-convex. Themicromechanical analysis approach is easy to implementand can be used to derive effective material propertiesof arbitrary microstructures. We expect the new materialmodeling and design methods presented in this paper tobring a fruitful future for developing advanced smartand functionally gradient materials.

References

w1x Markworth AJ, Ramesh KS, Parks WP. Review—modelingstudies applied to functionally graded materials. J Mater Sci1995;30:2183–2193.

w2x Huang, J., Heterogeneous Component Modeling and OptimalDesign for Manufacturing, Ph.D. Dissertation, Department ofMechanical Engineering, Clemson University, Clemson, SC,2000.

w3x Markworth AJ, Saunders JH. A model of structure optimizationfor a functionally graded material. Mater Lett 1995;22:103–107.

w4x Hirano, T., Teraki, J., Yamada, T., On the Design of Function-ally Gradient Materials, Proceedings of the First InternationalSymposium on FGM, Sendai, 1990, pp. 5–10.

w5x Grujicic M, Cao G, Fadel GM. Effective materials properties:determination and application in mechanical design and optim-ization. J Mater Des Appl 2001;215:225–234.

w6x Huang, J., Venkataraman, S., Rapoff, A.J., Haftka, R.T., Optim-ization Design of Inhomogeneous Isotropic Plates with Holesby Mimicking Bones, Proceedings of 43rd AIAAyASMEyASCEyAHS Structures, Structural Dynamics and MaterialsConference, AIAA, Denver, Colorado, 2002.

w7x Huang, J., Venkataraman, S., Rapoff, A.J., Haftka, R.T., Optim-ization of Axisymmetric Elastic Modulus Distributions Arounda Hole for Increased Strength, submitted to J Struct Optimi-zation for publication.

w8x DOT Users Manual, Vanderplaats, Miura and Associates, Inc.,1993.

w9x http:yylancet.mit.eduygayw10x http:yywww.matweb.comw11x Huang, J., Fadel, G., Heterogeneous flywheel modeling and

optimization, J. Mater. Des., Special Issue for Rapid Prototyp-ing, April 2000, Volume 2(2), pp. 112–125.