Bi-objective optimization design of functionally gradient materials

Materials and Design 23 (2002) 657–666

Bi-objective optimization design of functionally gradient materials Jinhua Huanga,*, George M. Fadelb, Vincent Y. Blouinb, Mica Grujicicb a

Department of Aerospace Engineering and Mechanics and Engineering, University of Florida, Gainesville, FL 32611, USA b Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA 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. Different microstructures formed by two primary materials are evaluated by a micromechanical analysis method. Macroscopically, FGMs are optimally designed by using these microstructures. Instead of using conventional simply assumed power law material distribution functions, a generic material distribution function is used. The bi-objective FGM optimization design procedure is highlighted by a flywheel example. A parametric formulation is used for both the geometric representation and the optimization procedure. 䊚 2002 Elsevier Science Ltd. All rights reserved. Keywords: Functionally gradient material (FGM); Micromechanical analysis; Effective material properties; Numerical analysis; Bi-objective optimization; Weighting method; Tchebycheff method

1. Introduction A functionally gradient material (FGM) is a composite, consisting of two or more phases, which is designed such that its composition varies in some spatial direction. This design is intended to take advantage of certain desirable features of each of the constituent phases w1,2x. For example, one constituent may be a ceramic which offers good high-temperature behavior but is mechanically brittle. Another may be a metal that exhibits better mechanical and heat-transfer properties but cannot withstand exposure to high temperatures. An FGM could thus be predominantly ceramic within the hotter region and metal within the cooler region w3x. A major problem in the design of an FGM, aside from that of primary material selection, lies in determining the optimal spatial dependence on the composition. This can be regarded as the composition profile that best accomplishes the intended purposes (design objectives) of the materials while all constraints are satisfied. Different design objectives and constraints will lead to different optimal composition profiles. Another problem lies in computing the effective material properties at *Corresponding author. E-mail address: [email protected] (J. Huang).

different composition points and evaluating the FGM performance under given working conditions. Few published articles considered multiple objectives in the design of functionally gradient materials. For practical problems, several design objectives may be required. For example, in a flywheel design, both strength and energy storage capability need to be considered. In addition, in conventional FGM design methods, a simple power law function is generally assumed for possible material distribution as described by Hirano et al. w4x. However, the optimal material distribution pattern is difficult to predict and may not be represented by the power law formulation. In the calculation of effective material properties, heuristic rules and micromechanical analysis methods were widely used w2,4x and most recently finite element based methods were developed w2,5x. Over the last decade, a family of multimaterial component (FGM) fabrication techniques such as layered manufacturing and self-propagating high-temperature synthesis have been established, which makes it quite practical to design functionally gradient materials with arbitrary microstructures. It is important to have a method that can compute the effective material properties of arbitrary microstructures. Our objective is to develop a generic

0261-3069/02/$ - see front matter 䊚 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 1 - 3 0 6 9 Ž 0 2 . 0 0 0 4 8 - 1

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bi-objective optimization design procedure for functionally gradient engineering products with specified microstructures. In what follows, we first present the bi-objective optimization design procedure for functionally gradient materials. Then, we describe a micromechanical analysis method for computing effective material properties and address the modeling of material spatial one-dimensional distribution. Finally, we highlight the bi-objective FGM optimization design procedure by a flywheel example and draw conclusions.

The ideal criterion vector (Utopia point) F* used in the Tchebycheff method is computed by: FiŽv˜ B.

S

W

T F*i smin T

T T

U

X

Žis1,2.

(6)

GjŽv˜ B.F0Ž js1,...,l.Y

V

* i

where F is the function values obtained by minimizing the ith objective function. Given an ideal criterion vector F* and weight S T

Z

2

W T

l:UTlgR2 liG0,8lis1XT, the weighted Tchebycheff

2. Bi-objective FGM optimization design

V

Y

is1

metric is defined as: We consider two primary materials A and B. Let vA and vB be the volume fractions of A and B, respectively. Both vA and vB are space dependent and satisfy, (1)

vAqvBs1

Let F1 and F2 be the two selected design objectives dependent on the volume fraction distribution. Using Eq. (1), both F1 and F2 can be written as functions of B’s volume fraction distribution v˜ B. The general biobjective FGM optimization design problem can be stated as: min wyF1Žv˜ B.,F2Žv˜ B.z~ s.t. GjŽv˜ B.F0 Ž js1,...,l. x

|

(2)

where G1,«,Gl are l design constraints. The implicit objective functions and constraints in the above problem are computed by a numerical analysis method w2x. The solution for problem (2) is a Pareto set and regardless of the convexity of the problem, any point on the Pareto set can be found by applying the Tchebycheff method. If the problem is convex, the weighting method may be used. The Tchebycheff method uses Lp-metrics to probe the possible Pareto point in the objective space. It uses different Tchebycheff metrics to find the closest points to the ideal (Utopia) point in the objective space. The metric is the distance between two points in space R 2 of the two objective functions. The Lp-norm of FgR 2 is defined as the length of a vector, w 2

z1yp

y is1

~

≤F≤psx8ZFiZp|

pgµ1,2,3...∂jµ`∂

(3)

The Lp-metric of two points x, ygR 2 is defined as: w 2

z1yp

y is1

~

≤xyy≤psx8ZxiyyiZp|

pgµ1,2,3,...∂jµ`∂

(4)

≤FyF*≤l`smaxµliZFiyF*i Z∂ is1,2

(7)

l

Let bs≤FyF*≤`. The weighted Tchebycheff method can be written as follows: min b s.t. bGliwyFiŽv˜ B.yF*i z~ Žis1,2. GjŽv˜ B.F0 Ž js1,...,l. x

|

(8)

By varying weight l1 and l2 systematically and solving a series of optimization problems, different points on the Pareto set can be obtained. The weighting method uses weights to combine the two objective functions in problem (2) to form the following new problem: min w1F1Žv˜ B.qw2F2Žv˜ B. s.t. GjŽv˜ B.F0 Ž js1,...,l. w1qw2s1 w1G0, w2G0

(9)

where w1 and w2 are the weighting coefficients for the corresponding objective functions. Different points on the Pareto set can be obtained by varying w1 and w2. To avoid spurious local optima caused by numerical analysis accuracy, problem (8) and (9) are both solved by alternatively using a gradient based method and a genetic algorithm w6,7x. Starting from a feasible point v˜ (B0), b is first minimized by a gadient based method using the software DOT w8x. A 1% random mutation on the optimal solution v˜ (B1) from the gradient method is used to create an initial population for the genetic algorithm w9x. After b is further reduced at v˜ (B2) by the genetic algorithm, the gradient method is used again with v˜ (B2) as the new start point. The process is repeated until both genetic algorithm and gradient based method fail to give further reduction on b.

For ps`:

≤F≤`smaxµZFiZ∂, is1,2 ≤xyy≤`smaxµZxiyyiZ∂ is1,2

3. Computation of effective material properties pgµ1,2,3,...∂jµ`∂

(5)

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

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

ture. Herein, a micromechanical analysis method is used to calculate the effective material properties of composites with arbitrary microstructures. In this method, the effective material properties at a specific volume fraction and microstructure are obtained by relating the corresponding average physical components within a unit inclusion cell as shown in Fig. 1a. The average physical components are obtained through a finite element method. Taking the computation of elastic effective material properties as an example, a unit cell with one or several specified inclusions is first discretized into a number of small elements (Fig. 1b). Each of the discretized small elements has either matrix or inclusion material. Under certain load and boundary conditions, the stress s(x) and strain ´(x) at an arbitrary point x within the unit cell are computed through a finite element method. ¯ and strain, ´¯ for The corresponding average stress, s the unit cell are the variables of interest for calculating elastic effective material properties and are defined as: 1 ¯ ss ZVZ

´¯ s

1

| Z|

ZV

sŽx.dv

(10)

V

4. Modeling of one-dimensional material distribution During the optimization process, any arbitrary material spatial distribution v˜ B might be generated. For analysis purpose, a mathematical expression for generic vB distribution has to be constructed. This expression must be as accurate as possible and satisfy continuity requirements. For one-dimensional problems, several methods may be used including linear interpolation, Bspline, and Bezier curves w2x. Assume the v˜ B spatial direction is in the x-axis as shown in Fig. 2. For Bezier representation, the design geometry domain CD is discretized into n segments with each node being attached a control composition v(Bi) (is0,«,n). With ŽxC,vB(C)., Žx1,v(B1). ,«, Žxny1,vB(ny1)., ŽxD,vB(D)., the Bezier representation for v˜ B is expressed as xsxCŽ1yu.nq 8 xiŽin.uiŽ1yu.nyiqŽxCql0.un

T Tv˜ s8v

is1

(14)

U

n

B

V

´Žx.dv

ny1

S

(i) n

i

Ž .u Ž1yu.

B i

nyi

ugw0,1x

is0

(11)

V

where V is the domain of the representative unit cell. The average strain and stress can be related by e ¯ ssC ´¯

(12)

¯ ´¯ sSes

(13)

where C e is the effective 6=6 symmetric stiffness tensor and S e is the effective 6=6 symmetric compliance tensor. Thermal effective material properties can be computed through a similar unit microstructure analysis method. The method presented here is used in Section 5 for the design of a functionally graded flywheel made of two materials, Sn and aluminum-based alloy 2124_T851.

Fig. 2. One-dimensional Euclidean space and material distribution diagram.

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Table 1 Physical material properties of Sn and 2124_T851 w10x

Sn 2124_T851

Density (=103 kgym3)

Young’s modulus (GPa)

Poisson’s ratio

Tensile strength (MPa)

7.29 2.78

41.4 73

0.33 0.33

220 485

Fig. 3. A face-centered Snq2124_T851 bimaterial unit microstructure (a) configuration of the unit microstructure and (b) discretization of onequarter of the unit cell with x- and y-symmetry constraints. Table 2 Effective Poisson’s ratios with respect to different volume fractions of Sn vf_Sna ySnb a b

0.0 0.33

0.1 0.331

0.2 0.333

0.3 0.335

0.4 0.336

0.5 0.336

0.6 0.335

0.7 0.333

0.8 0.331

0.9 0.330

1.0 0.330

vf_Sn is the volume fraction of Sn. ySn is the effective Poisson’s ratio.

where v(B0)sv(BC), v(B1),«, vB(ny1), vB(n)svB(D) are nq1 conn! trol compositions and Žni .s . i!Žnyi.! The material distribution control compositions v(BC), (1) vB ,«, vB(ny1), vB(D) in Eq. (14) are selected as the material design variables for problem (2). 5. Application to the design of functionally gradient flywheels Two primary materials, Tin (Sn) and aluminum-based alloy 2124_T851 are selected for the FGM flywheel design. The former is selected because of its high density, which is favored from the standpoint of maximizing the kinetic energy stored by the flywheel. The latter material, on the other hand, is selected for its high yield stress, which is desired from the standpoint of preventing failure of the flywheel. The physical properties for the two materials are listed in Table 1. The (Snq2124_T851) bimaterial microstructure is assumed to consist of congruent, face-filling squares, each containing a smaller concentric square (Fig. 3). The inner square is assumed to consist of Sn, while the remained of the larger square contains 2124_T851. To derive effective material properties, a uniformly distributed load is 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 has x- and y-symmetry constraints. The effective physical properties with respect to the volume fraction of Sn are next obtained using the micromechanical analysis method described in Section 3. The effective Poisson’s ratios as a function of the volume fraction of Sn obtained using this method are listed in Table 2. Similarly, Fig. 4 shows the values of the effective Young’s moduli at different volume fractions of Sn. The surface norms (the local z-axis direction in Fig. 3) of all microstructures are supposed to be parallel to the flywheel axis. It is also assumed that sides of the microstructure squares are aligned with the local radial direction. Due to the continuous nature and the axisymmetric shape of the flywheel as shown schematically in Fig. 5, the volume fraction, vfySn, of Sn is taken to be a function of r, the radial distance from the rotating axis. This assumption is consistent with the typical microstructure. Axisymmetric heterogeneous parts can be produced by rapid tooling processes such as laser engineered net shaping in which each layer of a part is produced one concentric ring (with constant Sn volume fraction) at a time. Bezier curves are used to represent a flywheel’s geometry and material variations in the radial direction

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

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

(Fig. 6). Let (r0, h0),«, (ri, hi),«, (rn, hn) in a plane passing through the z-axis be the nq1 geometry control points and (r0,v(f0y)Sn), «, (ri,vf(iy) Sn), «, (rn,vf(ny)Sn) be the nq1 material control points. The Bezier representation of the FGM flywheel can then be expressed as w11x: ny1

S

rsrinnerŽ1yu.nq 8 Žin.uiŽ1yu.nyiriqrouterun

T T

is1

us2pv

U w n

zsx8Ž .u Ž1yu. n i

y is0

i

z

(15)

hi|w Ž0Fu,v,wF1.

nyi

~

W1y2 1 T wŽsrrysuu.2qs2rrqs2uuxXT V2 Y S T

n

v˜ fySns8Žni .vf(iy) SnuiŽ1yu.nyi

V

The unknowns h0, h1, «, hn in Eq. (15) are selected as the geometry design variables and the unknowns v(f0y)Sn, vf(1y)Sn, «, vf(ny)Sn in the same equation are selected as the material design variables. The volume, mass, and kinetic energy of the FGM flywheel are all computed by a numerical analysis method w11x. Shear stresses are neglected, therefore, the radial and tangential stresses, srr and suu, respectively, are the principal stresses and assumed to be uniform across the thickness and circumference. A numerical analysis method is used to compute srr and suu w11x. The Von Mises stress is calculated from

is0

where rinner and router are the flywheel’s inner and outer radii, respectively.

Fig. 5. The alignment of a typical microstructure in the FGM flywheel to be designed.

sVMsUT

(16)

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

Fig. 6. Geometry and material configuration of an FGM flywheel.

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Table 3 Design parameters and desired targets (constraints)

18.7188 MPa, and PmaxsMax_sVM_maxs45 MPa. After being normalized w2x, problem (17) becomes:

rInner router v hmin hmax Max Ma Max sVM Min Enb (m) (m) (radsys) (m) (m) (kg) (MPa) (kJ)

n

20 0.02 0.2 a b

630

0.02 0.1

75

45

s.t.

deformation or fracture, minimization of the maximum equivalent stress within the flywheel is chosen as the second design objective. The equivalent stress is based on the corresponding failure criterion. For the complex Snq2124_T851 microstructure, both Sn and 2124_T851 are ductile materials, it is reasonable for us to assume the failure criterion of distortion energy and take the Von Mises stress as the equivalent stress. All flywheels should be able to store a certain amount of energy and because of strength requirement, the maximum Mises stress should be kept below a certain value. In Table 3, the related design parameters and desired targets are listed. From Section 2, the bi-objective FGM flywheel optimization problem can be stated as:

EnergyŽX. EnergyŽX.G50 kJ MaxysVMŽX.F45 MPa MassŽX.F75 kg 0.02 mFh0,...,h20F0.1 m ) 0Fv(f0y)Sn,vf(1y)Sn,...,vf(20 F1.0 ySn

Min MaxysVMŽX. s.t. EnergyŽX.G50 kJ MaxysVMŽX.F45 MPA MassŽX.F75 kg 0.02 mFh0,...,h20F0.1 m ) 0Fv(f0y)Sn,vf(1y)Sn,...,vf(20 F1.0 ySn

PŽX.y18.7188 z 45y18.7188

| ~

EnergyŽX.G50 kJ MaxysVMŽX.F45 MPa MassŽX.F75 kg 0.02 mFh0,...,h20F0.1 m ) 0Fv(f0y)Sn,vf(1y)Sn,...,vf(20 F1.0 ySn

(20)

Applying the weighting method, problem (20) becomes: min w1 s.t.

SŽX.q248277

qŽ1yw1.

PŽX.y18.7188

198277 EnergyŽX.G50 kJ MaxysVMŽX.F45 MPa MassŽX.F75 kg 0.02 mFh0,...,h20F0.1 m ) 0Fv(f0y)Sn,vf(1y)Sn,...,vf(20 F1.0 ySn 0Fw1F1

26.2812

(21)

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

bsmaxCl1 (17)

(18)

(19)

Let S(X)syEnergy(X) and PŽX.sMaxysVMŽX.. From Table 3, and the solutions of problem (18) and (19), SminsyEnergymaxsy248277 J, Smaxs yEnergyminsy50000 J, PminsMax_sVM_mins

SŽX.q248277

198277 SŽX.q248277 D

) where Xswv(f0y)Sn,v(f1y)Sn,...,vf(20 ,h0,h1,...,h20xT. The miniySn mum energy and the overall maximum Max_sVM have been given in Table 3. The maximum energy and the overall minimum Max_sVM are obtained by separately solving the following two single objective optimization problems:

max s.t.

SŽX.q248277

x y50000q248277 , y

50

M is the mass of the flywheel. En is the kinetic energy of the flywheel.

min wyEnergyŽX.,MaxysVMŽX.x s.t. EnergyŽX.G50 kJ MaxysVMŽX.F45 MPa MassŽX.F75 kg 0.02 mFh0,...,h20F0.1 m ) 0Fv(f0y)Sn,vf(1y)Sn,...,vf(20 F1.0 ySn

w

min

s.t.

,Ž1yl1.

PŽX.y18.7188 E 26.2812

F G

bGl1

198277 PŽX.y18.7188 bGŽ1yl1. 26.2812 EnergyŽX.G50 kJ MaxysVMŽX.F45 MPa MassŽX.F75 kg 0.02 mFh0,...,h20F0.1 m ) 0Fv(f0y)Sn,vf(1y)Sn,...,vf(20 F1.0 ySn 0Fl1F1

(22)

By varying w1 and l1 from 0 to 1, a number of points on the Pareto set are thus obtained and listed in Table 4 and 5. In Table 6, the optimal results at w1s1.0 or l1s 1.0 for the FGM flywheel and two homogeneous flywheels (with the same design parameters and desired targets as shown in Table 3) are presented. In addition, two alternate FGM flywheels are obtained by adding active constraints on the mass and the maximum Mises stress. Based on data from Table 4 and Table 5, the Pareto set is plotted in Fig. 7. The optimal flywheel profiles, material distributions, and Mises stress distributions corresponding to weight l1s0.0, 0.25, 0.5, 0.75, and 1.0 with the Tchebycheff method are drawn in Figs. 8 and 9, and Fig. 10, respectively. Fig. 11 shows both

J. Huang et al. / Materials and Design 23 (2002) 657–666 Table 4 Optimal results with weighting method

663

Table 5 Optimal results with Tchebycheff method

w1

F1 a

F2 b

En (kJ)

Max_sVM (MPa)

l1

F1

F2

En (kJ)

Max_sVM (MPa)

0.0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1.0

0.0 0.0 0.0 0.0 0.4563 1.0 1.0 1.0 1.0

0.0 0.0 0.0 0.0 0.5527 1.0 1.0 1.0 1.0

50 50 50 50 140.465 248.277 248.277 248.277 248.277

18.7188 18.7188 18.7188 18.7188 33.0258 45.0 45.0 45.0 45.0

0.0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1.0

0.0 0.0726 0.1824 0.3210 0.4563 0.5927 0.7381 0.8715 1.0

0.0 0.1326 0.2723 0.4116 0.5527 0.6716 0.7933 0.9011 1.0

50 64.394 86.161 113.646 140.465 168.418 196.341 222.796 248.277

18.7188 22.2042 25.875 29.5368 33.0258 36.3692 39.5683 42.4012 45.0

a

b

F1s

F2s

EnergyŽX.y50000 198277

the optimal FGM flywheel profile and material distribution contours at l1s0.5. The energy-maximum stress Pareto curve seems convex and is very close to a two-linear-segment curve. The higher the optimal energy, the higher the optimal maximum stress (Fig. 7). The Tchebycheff method

Max_sVMŽX.y18.7188 26.2812

Fig. 7. Energy vs. maximum Von Mises stress Pareto set separately obtained with Tchebycheff and weighting methods. Table 6 Comparison of optimal results (obtained with Tchebycheff method at l1s1.0) for FGM and homogeneous flywheels Optimum results

Material

Max Energy (kJ)

Max sVM (MPa)

Mass (kg)

1

Homogeneous (2124_T815) Homogeneous (Sn) FGM FGM FGM

96.91

45.0F45.0

27.0F75.0

137.03 248.28 195.30 183.07

36.8F45.0 45.0F45.0 45.0F45.0 36.8F36.8

34.4F75.0 50.6F75.0 34.4F34.4 34.4F34.4

2 3 4 5

664

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

Fig. 8. Optimal FGM flywheel profiles with respect to different Tchebycheff weights l1.

generates a better solution than the weighting method. When the Pareto set is a linear segment, only one Pareto point can be found through the weighting method. However, with the Tchebycheff method, each point on the Pareto set can always be found regardless of the shape of the Pareto set. The optimal flywheel geometry and material distribution, and hence Mises stress distribution, and kinetic

energy storage abilities are dependent on the weight l1 or w1. In general, the thickness of an optimal FGM flywheel is highest near the inner edge and smallest somewhere between the inner and outer edges, and the corresponding Sn volume fraction has the highest value near the outer edge and the smallest value near the inner edge (Figs. 8 and 9, and Fig. 11). Such optimal thickness and material distribution patterns allow the flywheel to

Fig. 9. Optimal FGM flywheel material distributions of Sn with respect to different Tchebycheff weights l1.

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

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Fig. 10. Optimal FGM flywheel Von Mises stress distributions with respect to different Tchebycheff weights l1.

have the desired high kinetic energy while the stress is kept small. When l1 is close to zero, the bi-objective optimization problem becomes a minimization of the maximum Mises stress under a small energy constraint. Usually, the highest value of Mises stress over the flywheel thickness occurs near the inner edge. It is quite straightforward to reduce the maximum Mises stress by increasing thickness and putting stiff and light material (2124_T851) near the inner edge, while a little amount of heavy material (Sn) near the outer edge makes the small energy requirement satisfied. The multipeak Sn distribution (Fig. 9) for small l1 is more effective for reducing the stress when the maximum stress moves away from the inner edge (Fig. 10). When l1 is close to one, the bi-objective optimization problem becomes

a maximization of the kinetic energy under a large stress constraint. Directly putting more heavy material (Sn) near the outer edge leads to the increase of the kinetic energy while the large stress constraint is still satisfied. For l1 between zero and one, the optimal material and thickness distributions are trade-offs between energy maximization and stress minimization. For the optimal homogeneous flywheel with 2124_T851 as the single material, because of the limit of the design space, the kinetic energy is only 55% the energy of the corresponding optimal FGM flywheel and for the Sn optimal homogeneous flywheel, the stress limit is reached and the energy is only 39% of the corresponding optimal FGM flywheel’s energy (Table 6). The advantage of the FGM flywheel, however, is counterbalanced by a significantly higher mass. Constraining first the mass and then the maximum Mises stress to the levels of the homogeneous flywheels (optimal results 4 and 5 of Table 6) leads to high performance flywheels that can store 42% and 34% more energy than the corresponding homogeneous 2124_T851 flywheel of the same weight, respectively. 6. Conclusions

Fig. 11. The optimal profile and material distribution contours of an FGM flywheel obtained with Tchebycheff method at l1s0.5.

The optimal volume fraction design for maximizing both desired performances combines the numerical optimizations of geometry and material distributions with a micromechanical analysis of the microstructure– properties relations. With the generic Bezier represen-

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

tation of volume fraction and thickness distribution, it is quite flexible to adjust product performances according to requirements. The Tchebycheff method is more effective than the weighting method in finding points on the Pareto set since that set may be non-convex. The micromechanical analysis approach is easy to implement and can be used to derive effective material properties of arbitrary microstructures. We expect the new material modeling and design methods presented in this paper to bring a fruitful future for developing advanced smart and functionally gradient materials. References w1x Markworth AJ, Ramesh KS, Parks WP. Review—modeling studies applied to functionally graded materials. J Mater Sci 1995;30:2183 –2193. w2x Huang, J., Heterogeneous Component Modeling and Optimal Design for Manufacturing, Ph.D. Dissertation, Department of Mechanical Engineering, Clemson University, Clemson, SC, 2000.

w3x Markworth AJ, Saunders JH. A model of structure optimization for a functionally graded material. Mater Lett 1995;22:103 – 107. w4x Hirano, T., Teraki, J., Yamada, T., On the Design of Functionally Gradient Materials, Proceedings of the First International Symposium on FGM, Sendai, 1990, pp. 5–10. w5x Grujicic M, Cao G, Fadel GM. Effective materials properties: determination and application in mechanical design and optimization. J Mater Des Appl 2001;215:225 –234. w6x Huang, J., Venkataraman, S., Rapoff, A.J., Haftka, R.T., Optimization Design of Inhomogeneous Isotropic Plates with Holes by Mimicking Bones, Proceedings of 43rd AIAAyASMEy ASCEyAHS Structures, Structural Dynamics and Materials Conference, AIAA, Denver, Colorado, 2002. w7x Huang, J., Venkataraman, S., Rapoff, A.J., Haftka, R.T., Optimization of Axisymmetric Elastic Modulus Distributions Around a Hole for Increased Strength, submitted to J Struct Optimization for publication. w8x DOT Users Manual, Vanderplaats, Miura and Associates, Inc., 1993. w9x http:yylancet.mit.eduygay w10x http:yywww.matweb.com w11x Huang, J., Fadel, G., Heterogeneous flywheel modeling and optimization, J. Mater. Des., Special Issue for Rapid Prototyping, April 2000, Volume 2 (2), pp. 112–125.