Computational Materials Science 69 (2013) 435–442

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Computational Materials Science

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Reliability-based design optimization of volume fraction distribution infunctionally graded composites

0927-0256/$ - see front matter � 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2012.12.003

⇑ Corresponding author.E-mail address: romana79@kmu.ac.kr (Y.J. Noh).

Y.J. Noh a,⇑, Y.J. Kang b, S.J. Youn a, J.R. Cho b,c, O.K. Lim b

a Dept. of Mechanical and Automotive Engineering, Keimyung University, Daegu 704-701, Republic of Koreab School of Mechanical Engineering, Pusan National University, Busan 609-735, Republic of Koreac Research and Development Institute of Midas IT, Gyeonggi 463-400, Republic of Korea

a r t i c l e i n f o

Article history:Received 12 July 2012Received in revised form 9 November 2012Accepted 4 December 2012Available online 17 January 2013

Keywords:Functionally graded compositesGraded layerVolume fractionStress ratioReliability-based design optimization

a b s t r a c t

Functionally graded material (FGM) has a continuous and functional distribution of volume fractions ofconstituent particles, which leads to superior thermo-mechanical performance to classical laminatedcomposite materials. Since the thermo-mechanical characteristics of an FGM depend on the volume frac-tion distribution, it is important to tailor appropriate volume fraction distribution that satisfies thedesired performance requirements under given loading and boundary conditions. Even though numericaloptimization technique may serve as an excellent material tailoring tool, the capacity of current manu-facturing techniques of FGM may not yield the target volume fraction. To deal with uncertainty in themanufacturing process, a reliability-based design optimization (RBDO) for FGM composite is proposed.In RBDO, a finite number of volume fractions of homogenized FGM layers and material properties areconsidered as random variables, with statistical information such as mean, standard deviation, and sta-tistical distributions. Design of experiments and response surface models are used to obtain explicitforms of thermal stresses for RBDO formulation. It is observed through the numerical experiment thatthe RBDO finds the optimized volume fraction distribution with high reliability, such that the graded lay-ers do not fail in the presence of manufacturing uncertainty.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

FGM was firstly proposed as a heat-resistant composite mate-rial in the 1980s to suppress the stress concentration and exfoli-ated phenomena at interfaces between distinct homogeneousmaterial layers, which commonly occur in classical bi-material-type composite materials. Unlike the classical composite materials,FGM has a continuous material property distribution over the en-tire material domain achieved by gradually changing the volumefractions of base constituent materials in the graded layers. Thegraded layers with continuous material characteristics reduce thestress concentration occurring at sharp material interfaces in clas-sical bi-material-type composites. Thanks to the spatial continuityand flexibility in the volume fraction, FGMs are widely used in var-ious mechanical and bioengineering fields, such as electronicmaterials, metal cutting tools, artificial bone, dental implants,and thermally resistant composites [1–3].

However, it is not easy to produce the graded layers with grad-ually varying continuous volume fractions, due to the manufactur-ing cost and time required in real applications, so the graded layersare manufactured by discretizing a continuous volume fraction and

laminating a number of heterogeneous material layers with dis-cretized uniform volume fractions. But, in the design stage, the vol-ume fraction of a graded layer can be optimally tailored in either acontinuous or discrete manner. Various optimization methodshave been proposed for the volume fraction optimization. Ootaoet al. used a neural network method [4] and a genetic algorithm[5] for hollow a FGM cylinder and FGM plate, respectively. Choand Ha [6] used an interior penalty function and golden sectionmethods to minimize the thermal stress concentration that occursin Ni–Al2O3 thermally resistant FGM composite. Cho and Shin [7]carried out volume fraction optimization using a neural networkfor improving the efficiency of the optimization process. Goupeeand Vel [8] presented a two-dimensional simulation and optimiza-tion of material distribution in FGM for a thermo-mechanicalprocess.

The proposed optimization techniques assume that all the de-sign variables are deterministic, because the volume fractions orthe material properties of FGM composites have fixed values. Inthis context, such optimization techniques are classified as a deter-ministic optimization method. However, in real applications, notonly the volume fractions but also the material properties haveprobabilistic characteristics due to uncertainties in the manufac-turing process, such as variation in material properties, and manu-facturing tolerance. The various manufacturing processes include

436 Y.J. Noh et al. / Computational Materials Science 69 (2013) 435–442

powder metallurgy, self-propagating high-temperature synthesis,deposition, and plasma spray [9–12]. These methods are incapableof adequately achieving the target volume fraction because the fineand random particles of the base materials are mixed at the microlevel, making the control of volume fraction distribution difficult.The resulting volume fraction could yield material failure, andaccordingly, the uncertainty in materials and manufacturing pro-cess should be considered in the design optimization process.

In traditional structural analysis, the uncertainty has been dealtwith by introducing a safety factor for ensuring structural safety,but design using safety factors could yield excessively conservativeor unsafe designs [13]. To resolve this problem, reliability-baseddesign optimization was introduced [14]. The reliability-based de-sign optimization is done to find an optimum design that mini-mizes an objective to achieve a target reliability, where thereliability indicates the probability that the current design is safe.Thus, unlike the deterministic optimization or safety-factor-basedoptimization, the reliability-based design optimization can achievecost efficiency and target reliability. In order to extend RBDO toFGM composites, the statistical characteristics of the volume frac-tion such as mean, standard deviation, and distribution typesshould be assumed a priori. Since the material property in eachgraded layer is calculated by volume fractions, the uncertainty ofthe material property is accordingly considered in RBDO.

In this paper, the FGM composite models are explained in Sec-tion 2. The basic concept of reliability-based design optimization isintroduced in Section 3. In Section 4, the RBDO process of volumefraction in FGM is explained. A numerical example will be intro-duced in Section 5 to illustrate how the RBDO is applied to FGMcomposite.

2. Metal–ceramic FGM composite material

2.1. Equivalent material properties of dual-phase FGM composite

Fig. 1 shows an FGM composite composed of two homogenousmaterial layers and a functionally graded material layer, where twobase materials possess thermoelastic performance characteristicsthat conflict with each other. The base materials A and B are onthe bottom and top of the graded layer, respectively, and thegraded layer has a spatially continuous volume fraction from mate-rial A to B by increasing the fraction of the material B. The gradedlayer has mixed volume fractions of A and B, and thus the summa-tion of the volume fraction of A and B should satisfy the physicalrelation of VA + VB = 100%.

The continuous volume fraction in the graded layer is ratherimpractical, because it requires elaborate costly and time-consumingfabrication process. Thus, in real applications, discrete-type FGM

Fig. 1. FGM composite material wit

composites are usually used to shorten the fabrication time and re-duce manufacturing costs, for which the graded layer is discretizedand laminated with a number of heterogeneous layers withdifferent uniform volume fractions, as mentioned above. Forexample, Fig. 2 shows a simply supported metal–ceramic FGMcomposite beam with four graded layers with volume fractionsVB of 20%, 40%, 60%, and 80%. Since the material properties ofgraded layers and their thermoelastic behaviors highly dependon the volume fraction, the volume fraction distribution needs tobe appropriately determined to satisfy the required performanceunder consideration.

The change of volume fraction changes the material propertiesof FGM, so the material properties need to be obtained to calculatethe stress in the multi-layers. The material properties of the gradedlayer are influenced by the size, shape, orientation and dispersionstructure as well as the volume fractions of constituent particlesat the microscopic level [1]. The goal of the current study is to pres-ent the RBDO process for seeking an optimum volume fraction dis-tribution and to compare the optimum results between RBDO andDO. Other homogenization methods, such as self-consistent [15],Mori–Tanaka [16], or asymptotic expansion homogenization meth-ods [17] can be used to calculate the material properties withoutthe loss of generality. The final optimum solution will be affectedby the choice of the homogenization method, because the thermalstress distribution obtained by the thermoelastic analysis dependson the equivalent material properties of FGMs. However, for thegoals of the current study, a homogenization method consideringonly the volume fraction at the macro level, a linear rule of mixture[18], is employed to approximate the response surface model ofthe maximum tensile stress that occurs in FGM. For problemsrequiring more sophisticated calculation of material properties,other homogenization methods can also be used by integratingwith the RBDO process. The equivalent elastic modulus Eh, coeffi-cient of thermal expansion ah, Poison’s ratio mh, and tensilestrength rBt,h of the hth graded layer are calculated using the linearrule of mixture method. Let Vm and Vc be the respective volumefractions of a metal and ceramic, which are calculated as follows:

Eh ¼ VmEm þ VcEc ð1Þ

ah ¼ Vmam þ Vcac ð2Þ

mh ¼ Vmmm þ Vcmc ð3Þ

rBt;h ¼ VmrBt;m þ VcrBt;c ð4Þ

where h = 1 � N, and N is the total number of graded layers.

h continuous volume fractions.

Fig. 2. FGM composite beam composed of discrete homogeneous layers.

Fig. 4. Variation of porosity in achieved gradient [16].

Y.J. Noh et al. / Computational Materials Science 69 (2013) 435–442 437

2.2. Characteristics of metal–ceramic FGM composite

Referring to Fig. 2, let us consider three types of FGM compositebeam models: bi-material composite, FGM composite with onegraded layer, and FGM composite with four graded layers. The basematerials A and B are Ni and Al2O3, and their detailed materialproperties are given in Section 5. The dimensions L and d are 50and 10 mm, respectively. All three FGM beam models are subjectedto the same thermal loading of a temperature difference of 200 Kbetween the top and bottom surfaces. Fig. 3 shows three types ofmetal–ceramic composite beam models and the thickness-wisethermal stress distributions, where the FGM composite with fourgraded layers have two different volume fractions, FGM (4 gradedlayers)_01 and FGM (4 graded layers)_02. The FGM composite with1 layer has a volume fraction of 0.5. The volume fractions ofFGM_01 are 0.1, 0.3, 0.7, and 0.9, and the ones of FGM_02 are0.2, 0.4, 0.6, and 0.8.

When the bi-material composite is used, it produces a signifi-cant stress concentration at the interface of the two base materialsdue to the discrepancy between two heterogeneous material char-acteristics. Even though the mesh size is more refined, the stressconcentration is not reduced. When the FGM composite with onelayer is used, the stress concentration is reduced, but it is still large.When the FGM composite with four graded layers is used, thestress concentration is considerably reduced. However, it is worthnoting that the peak thermal stresses are somewhat different,depending on the volume fraction distribution, even though thesame number of graded layers is used. Thus, one can clearly con-firm the fact that the total number of graded layers and the volumefraction distribution significantly affect the thermal stress concen-tration, and the optimal tailoring of these factors could minimizethe peak thermal stresses.

Fig. 3. Thermal stress distributions: (a) bi-material composite, (b) FGM compos

Even though the optimized volume fractions could be obtainedby numerical optimization technique, it is difficult to manufacturean FGM composite that exactly satisfies such an ideal volume frac-tion. Fig. 4 illustrates a porosity gradient in a sintered ceramic sam-ple produced by pressure filtration [19]. Even though a closelycontrolled manufacturing process is used, the grading achieved inFGM is rather different from the target grading. Thus, the random-ness in the volume fraction of each layer and material propertiesneeds to be considered at the design stage. This can be achievedby RBDO, in which the uncertainty in the manufacturing processis reflected by introducing the probability of failure for constraints.

ite with one graded layer, and (c) FGM composite with four graded layers.

Fig. 5. Comparison of deterministic optimization and reliability-based designoptimization.

438 Y.J. Noh et al. / Computational Materials Science 69 (2013) 435–442

3. Reliability-based design optimization

The reliability-based design optimization is done to find a de-sign that is not only economical, but also reliable in the presenceof uncertainty. The objective is to minimize the total cost withoutthe probability of failure for a system exceeding a specified targetprobability, as shown in Eqs. (5)–(7). The cost function can be ex-pressed as a vector of design variables d, which are the mean val-ues of the vector of random variables X. The random variable canbe the design variable or random parameter whose mean valuesare not changed during the RBDO process. For example, in struc-tural problems, geometrical variables such as thickness or lengthcan be design variables, but material properties such as Young’smodulus are random parameters. The function Gi(X) indicates alimit state function, and a system no longer fulfills the ith designcriteria if Gi(X) > 0. The probability that the random variables arewithin a safe region should be less than the specified target prob-ability of failure PTar

Fi.

min : costðdÞ ð5Þ

s:t: PðGiðXÞ > 0Þ 6 PTarFi; i ¼ 1; . . . ;NC ð6Þ

d ¼ lðXÞ; dL6 d 6 dU

; d 2 RNDV and X 2 RNRV ð7Þ

where NC, NDV, and NRV are the number of probabilistic constraints,number of design variables, and number of random variables,respectively.

The probability of failure for each constraint function is esti-mated by a multi-dimensional integral as [17]:

PF � PðGðXÞ > 0Þ ¼Z

GðXÞP0fxðXÞdx ð8Þ

where x is the realization of the random vector X, and fx(X) is thejoint probability density function (PDF) of X. Since it is very difficultto compute the multi-dimensional integral analytically, approxima-tion methods such as the first-order reliability method (FORM) [20–22] or the second-order reliability method (SORM) are used [23,24].The FORM and SORM estimate the probability of failure by approx-imating the constraint function G(X) by first and second-order Tay-lor series expansions, respectively. The FORM mostly providesadequate accuracy, and is much easier to use than the SORM, andhence it has been commonly used in RBDO.

To calculate the probability of failure of the constraint functionusing FORM and SORM, it is necessary to find a contact point on thelimit state function which has the shortest distance between thelimit state function and the origin in the independent Gaussianspace (U-space). The limit sate function is transformed fromX-space to U-space using the Rosenblatt transformation [25],defined as g(u) � G(x(u)) = G(x), and the contact point is referredto as the most probable point (MPP), denoted as u⁄. Using a perfor-mance measure approach (PMA+) [26], the MPP u⁄ can be obtainedby solving the following optimization problem:

max : giðuÞ ð9Þ

s:t: kuk ¼ btið10Þ

where btiis the target reliability index such that PTar

Fi¼ Uð�bti

Þ usingthe FORM. The ith constraint in Eq. (6) can be rewritten as:

P½GiðXÞ > 0� � PTarFi6 0) Giðx�Þ 6 0 ð11Þ

where Gi(x⁄) is the ith constraint function evaluated at the mostprobable point (MPP), x⁄, in X-space. If the constraint function atthe MPP is less than or equal to zero, then the ith constraint inEq. (6) is satisfied for the given target reliability. Thus, Eqs. (5)–(7) can be rewritten using PMA+ as:

min : costðdÞ ð12Þ

s:t: Giðx�Þ 6 0; i ¼ 1; . . . ;NC ð13Þ

dL6 d 6 dU

;d 2 RNDV and X 2 RNRV ð14Þ

The RBDO requires more computational time comparing withthe deterministic optimization due to the reliability analysis. Thus,the PMA + uses a deterministic optimum design as an initial designto save the computational time because the RBDO optimum is usu-ally close to the deterministic optimum.

Fig. 5 shows a graphical comparison between deterministicoptimization and reliability-based design optimization. If X1 andX2 are random variables, the deterministic optimum point onlyhas 50% or even less of reliability ð1� PTar

F Þ%, because the obtainedoptimum might be within the failure region (graded area) due tothe variability of two random variables. On the other hand, theRBDO optimum point is away from the constraints and has a de-sired reliability, e.g. 95% reliability, regardless of the variability oftwo random variables. Thus, RBDO provides a more reliable designthan DO.

4. RBDO of Metal–ceramic heat-resisting FGM composite

4.1. RBDO process

In the RBDO process for FGM composites, the objective functionand constraint function first need to be defined. Since the stressesand stress ratios as objective and constraint functions cannot beexpressed explicitly, their approximate models are obtained usingdesign of experiment through the finite element analysis of FGMstructures. For the finite element analysis, the material propertiesfor graded layers are calculated using a homogenization technique,such as the linear mixture rule as shown in Fig. 6. The accuracy ofthe approximate models for the objective and constraint functionshighly depends on the experimental points, so it is important to se-lect appropriate experimental points that can well approximatereal models. Increasing the number of experimental points can im-prove the accuracy of the approximate model, but it does not al-ways guarantee the accuracy of the model, and increases thecomputational time for the design of experiment. In this paper,interior central composite design (ICCD), which is known as an effi-cient and accurate DOE method, is used to select experimentalpoints and determine their number. Once the experimental pointsare selected, the approximate model can be obtained using

Fig. 6. Flowchart of the volume fraction optimization using RBDO.

Y.J. Noh et al. / Computational Materials Science 69 (2013) 435–442 439

regression or interpolation models. In this paper, a response sur-face model with third-order polynomials is used.

After the response surface models for objective and constraintfunctions are obtained, statistical information such as mean, stan-dard deviation, and distribution types for random variables andtarget reliability for each constraint needs to be defined. Usingthe statistical information of random variables, the RBDO findsan optimum design that satisfies the probabilistic constraints. Toverify whether the RBDO optimum design achieves the target reli-ability, the Monte Carlo Simulation is carried out at the obtainedRBDO design points.

4.2. Deterministic optimization and reliability-based designoptimization

In the deterministic optimization for FGM problems, the volumefractions are determined to minimize the maximum thermal stressthat occurs due to the thermal expansion such that yielding can beprevented in the FGM composite. Thus, the volume fractions ofgraded layers are the design variables, the maximum thermalstress is the objective function, and the constraint functions arethe stress ratios of graded layers, which are the ratios of theX-directional stresses to the tensile strength. Since the thermalstresses in other directions are ignorable in the next numericalexample explained Section 5, they are not considered.

Find d ¼ fVig; i ¼ 1; . . . ;Ng ð15Þ

min: f ðdÞ ¼ jrx-axialmax ðdÞj ð16Þ

s:t: GjðdÞ ¼ r�VjðdÞ � 1 6 0; j ¼ m;1; . . . ;Ng ; c ð17Þ

dLi 6 di 6 dU

i ; i ¼ 1; . . . ;Ng ð18Þ

where V1–V4 are the volume fractions of the ceramic, Ng is the totalnumber of graded layers, and dL

j 6 dj 6 dUj are the lower and upper

bounds of the volume fraction. The stress ratio is defined as:

r�v j¼

�rt ¼ rxxrBt;j

;rxx P 0

�rc ¼ rxxrBc;j

;rxx < 0

(; j ¼ m;1; . . . ;Ng ; c ð19Þ

where rBt,j is the tensile strength of the jth graded layer; rBc,j is thecompressive strength of the jth graded layer; rxx is the X-directionaltensile stress; and �rt and �rc are the ratios of the X-directional ten-sile or compressive stress to the tensile or compressive strength,respectively. If the stress ratios are larger than 1, it means thatthe FGM composite fails. The tensile and compressive strength ofa graded layer are calculated using the linear rule of mixture meth-od, Eq. (4). In this paper, a sequential quadratic programming meth-od, which is suitable for nonlinear problems, is used as adeterministic optimization method [27].

Similarly, the RBDO formulation is done to find the volume frac-tion of each graded layer to minimize the maximum thermal stress,but the probability that the stress ratio is larger than 1.0 should beless than the target probability of failure in all layers.

Find d ¼ fVig; i ¼ 1; . . . ;Ng ð20Þ

min : f ðdÞ ¼ jrx�axialmax ðdÞj ð21Þ

s:t: PðGjðXðdÞÞ > 0Þ 6 PTarFi; j ¼ m;1; . . . ;Ng ; c ð22Þ

dLi 6 di 6 dU

i ; i ¼ 1; . . . ;Ng ð23Þ

For problems with many design variables such as 2-D smoothvolume fraction gradings, it is difficult to find the optimum volumefraction values satisfying given constraints in a vast design spacefor all design variables. However, in real applications, it is rare tofind the FGM composite problems with continuously variablegraded layers due to technical limits and manufacturing costs, sothe current RBDO method can be used in many FGM compositeproblems. For future research, a new RBDO method will be studiedto be applied to various FGM composite problems.

5. Numerical example

A numerical experiment is carried out in this section to explorehow the RBDO yields reliable design for a metal–ceramic heat-resistant FGM composite. Section 5.1 explains the definition ofthe model problem, and Section 5.2 explains the design of experi-ments and approximate models required for the volume fractionoptimization by RBDO. The optimum results of DO and RBDO arecompared in Section 5.3.

5.1. Definition of problem

In this paper, a simply supported metal–ceramic FGM compos-ite beam shown in Fig. 7 is considered for the numerical experi-ment. The bottom and top layers are Ni and Al2O3, respectively,and the material properties for which are shown in Table 1 [6].The length and thickness of each graded layer are 50 and 8 mm,respectively, and the thickness of both the Ni and Al2O3 is 1 mm.The values V1, V2, V3 and V4 indicate the uniform volume fractionsof ceramic in each graded layer. Only the right half of the beam istaken for the numerical simulation according to the problem sym-metry, and the symmetric boundary condition is specified to theleft end, while the right end is simply supported. The thermal load-ing is applied such that the temperature varies linearly from 300 Kat the bottom to 1300 K at the top. The FGM composite domain isdiscretized with a total of 6000 8-node quadrilateral elements, andthe thermoelastic analysis is carried out using a commercial FEMcode, Midas NFX [28].

Fig. 7. Simply supported FGM composite beam subject to thermal loading.

Table 1Thermo-mechanical material properties of Ni and Al2O3.

Items Ni Al2O3

Young’s modulus, E (GPa) 199.5 393Coefficient of thermal expansion, a (�10�6/K) 15.4 7.4Poisson’s ratio, m 0.3 0.25Tensile strength, rBt (MPa) 317 300

440 Y.J. Noh et al. / Computational Materials Science 69 (2013) 435–442

5.2. Design of experiment and response surface models

To generate approximate models for the objective and con-straint functions, the lower and upper bounds of the volume frac-tions in each layer are given in Table 2. With 75 sampling pointsobtained from the design of experiment, the stress ratios are calcu-lated from the stress of the X-direction rxx at x = 5 mm along theY-direction, and rx�axial

max is obtained from the maximum value ofrxx. As is addressed in the numerical experiment of our previouspaper [6], stress components other than the bending stress rxx

are negligible, except for in the region in the vicinity of the rightend, where the edge effect occurs with relatively small stress val-ues. For this reason, the stress distribution of rxx in the thicknessdirection at x = 5 mm is considered. The objective and constraintfunction values at each sampling point are shown in Table 3, where

Table 2Range of design variables.

Design variables Ranges

V1 0.1–0.3V2 0.3–0.5V3 0.5–0.7V4 0.7–0.9

Table 3Thermoelastic results obtained by finite element analysis according to DOE.

Case Design variables Results

V1 V2 V3 V4 r�m

1 0.1375 0.3875 0.6375 0.7875 0.39462 0.1375 0.3875 0.6375 0.8625 0.36113 0.1375 0.3875 0.7125 0.7875 0.463773 0.2000 0.4000 0.6000 0.7000 0.568174 0.2000 0.4000 0.6000 0.9000 0.480675 0.2000 0.4000 0.6000 0.8000 0.5257

r�Vi, r�m and r�c are the stress ratios in the ith graded layer, the metal

layer, and the ceramic layer, respectively. Using the design variableand function values, response surface models with third-orderpolynomials can be obtained as shown in Eqs. (24)–(30). Theexperimental design and response surface models are carried outusing a commercial tool for Process Integration, Automation andOptimization (PIDO), PIAnO [29].

r�m ¼ �1:31881þ 3:05106d1 þ 2:58989d2 þ 1:50208d3

þ 0:214545d4 � 0:603048d21 � 1:45011d2

2

� 1:00462d23 � 0:451386d2

4 þ 2:82056d31 þ 1:86439d3

2

þ 0:731612d33 þ 0:108758d3

4 � 1:09815d1d2

� 0:406924d1d3 � 0:0882187d1d4 � 0:311893d2d3

� 0:19672d2d4 � 0:0710538d3d4 ð24Þ

r�V1¼ �18:3228þ 9:77482d1 þ 27:1205d2 þ 31:6413d3

þ 28:2677d4 � 52:4722d21 � 68:2489d2

2 � 52:7785d23

� 33:8619d24 þ 98:8046d3

1 þ 54:6755d32 þ 27:9583d3

3

þ 13:233d34 � 4:8404d1d2 � 4:01354d1d3

þ 0:68732d1d4 þ 5:59683d2d3 � 0:0239242d2d4

þ 0:974334d3d4 ð25Þ

r�V2¼ �83:7747þ 13:9217d1 þ 106:658d2 þ 63:1457d3

þ 211:513d4 � 62:8199d21 � 222:395d2

2 � 96:4937d23

� 261:78d24 þ 120:018d3

1 þ 221:008d32 þ 54:3814d3

3

þ 110:106d34 � 32:6707d1d2 þ 9:79338d1d3

þ 4:86073d1d4 � 27:5352d2d3 � 18:4658d2d4

þ 4:17587d3d4 ð26Þ

r�V1r�V2

r�V3r�V4

r�c rx-axialmax

0.7380 0.4660 0.7624 0.4903 0.8161 244.84400.7564 0.5585 0.6105 0.9695 0.4206 293.10800.8253 0.5838 1.3101 0.5606 0.6032 399.42000.4862 0.3369 0.5697 0.7161 1.4135 424.04400.5386 0.5843 0.6173 1.2721 0.3689 383.79100.5115 0.4566 0.3800 0.6204 0.9080 272.4140

Y.J. Noh et al. / Computational Materials Science 69 (2013) 435–442 441

r�V3¼ �34:6291� 0:786958d1 þ 14:901d2 � 156:724d3

� 5:9347d4 þ 21:7318d21 � 11:5472d2

2 þ 270:514d23

þ 17:7701d24 � 21:6132d3

1 þ 19:4294d32 � 115:928d3

3

þ 0:133163d34 þ 1:89199d1d2 � 12:9415d1d3

þ 1:98677d1d4 � 23:2493d2d3 � 2:20161d2d4

� 36:9515d3d4 ð27Þ

r�V4¼ 113:275þ 5:42688d1 � 47:8163d2 � 52:8649d3

� 351:014d4 � 7:64572d21 þ 132:122d2

2 þ 117:88d23

þ 430:293d24 þ 11:9796d3

1 � 109:573d32 � 60:5421d3

3

� 160:428d34 þ 1:74405d1d2 � 2:56398d1d3

� 3:85066d1d4 þ 6:07341d2d3 � 11:9286d2d4

� 32:2736d3d4 ð28Þ

r�c ¼ �22:83þ 8:4216d1 � 13:3784d2 � 26:7937d3

þ 133:52d4 � 16:0508d21 þ 28:4377d2

2 þ 32:4613d23

� 180:47d24 þ 26:6383d3

1 � 24:3404d32 � 18:0228d3

3

þ 76:0271d34 � 0:5657d1d2 � 2:66527d1d3

� 3:08544d1d4 þ 1:09703d2d3 þ 1:66669d2d4

þ 6:11871d3d4 ð29Þ

rx�axialmax ¼ �24163:5� 1974:53d1 þ 16031:2d2 þ 37191:4d3

þ 69192:8d4 þ 6294:12d21 � 36505:8d2

2

� 66244:2d23 � 97235:61d2

4 � 4013:67d31

þ 32818:8d32 þ 40949:4d3

3 þ 46096:9d34

� 1588:36d1d2 � 1374:79d1d3 � 3023:89d1d4

� 1379:75d2d3 � 2301:86d2d4 � 2216:4d3d4 ð30Þ

To verify the accuracy of the response surface model, an ad-justed coefficient of determination in Eq. (31) is used [30]. The clo-ser the adjusted coefficient of determination is to 1, the closer theresponse surface is to the real model. As shown in Table 4, becauseall adjusted coefficients of determination are over 0.9, the obtainedresponse surface well describes the real behavior of this FGMexample.

R2adj ¼ 1� ð1� R2Þ n� 1

n� p� 1ð31Þ

in which n is the degrees of freedom, p is the total number of regres-sors, and R2 is the coefficient of determination, which is defined as:

R2 ¼ 1� SSerr

SStot¼ 1�

Piðyi � fiÞ2Piðyi � �yÞ2

ð32Þ

Table 4Adjusted coefficients of determination.

Items R2adj

r�m 0.9944r�V1

0.9773r�V2

0.9384r�V3

0.9600r�V4

0.9429r�c 0.9885rx�axial

max0.9218

Here, SSerr is the sum of squares of residuals; SStot is the totalsum of squares; fi is the associated function value; �y is the meanof the observed data yi. A narrow-range width of volume fractionsin each layer, 0.2, is used to save computational time of carryingout the design of experiments as shown in Table 2. The volumefractions of the ceramic are expected to linearly increase alongY-direction to endure the high temperature on the top of the beam,so the used design bounds for the volume fractions are enough tobuild the DOE-created response surface. For a wide range ofvolume fractions, 0 to 1, many designs of experiments would benecessary. Accordingly, an efficient method of design of experi-ments combined with an accurate response surface model will befurther studied for more general FGM problems.

5.3. Comparison of optimum results between DO and RBDO

For optimization of the FGM example, the initial values of vol-ume fractions are 0.2, 0.4, 0.6, and 0.8 from the bottom to thetop layer. The lower and upper bounds are 0.0 and 1.0 (Table 5).For RBDO, statistical information on random variables is necessary.The statistical information needs to be obtained from experimentaldata of random variables. For example, in Fig. 4, if the porosity isconsidered as a random variable, the porosity gradients need tobe obtained by testing enough number of sintered ceramic sam-ples. However, the experimental data do not exist in the practicalapplication of FGM. Thus, it is assumed that the distribution typesof the volume fractions are normal distributions, and the coeffi-cients of variations, which are the ratios of the standard deviationto mean values, are 5%. The target probability of failure for eachconstraint is 2.275% (target reliability = 97.725%). To estimate thereliabilities at the initial design, DO design, and RBDO design,100,000 sampling points are used for Monte Carlo simulation.

Table 5 shows the DO and RBDO results of the FGM example.Since the volume fraction values of the ceramic are expected to lin-early increase along the Y-direction due to the thermal load, theinitial values are selected as 0.2, 0.4, 0.6, and 0.8. Comparing withthe maximum thermal stress at the initial design, the maximumthermal stress at the DO optimum design is decreased from252.60 to 213.05 MPa. When the RBDO is used, the maximum ther-mal stress is increased slightly from 213.05 to 224.12 MPa. As ex-pected, the optimum volume fractions gradually increase along theY-direction, and they have a larger portion of the ceramic than theinitial ones such that the beam endures the high temperature onthe top. As explained in Section 3, since the RBDO optimum designyields a more reliable design, it leads to somewhat of an increase inthe objective function with high reliability in the RBDO optimumdesign. The increased maximum thermal stress in RBDO might leadto concern about the yielding of the FGM composite, but the tensileor compressive strengths of graded layers are also increased for thevolume fraction distribution obtained from RBDO. Thus, the in-creased maximum thermal stress does not lead to the yielding ofthe FGM composite.

Table 6 shows the reliabilities estimated at the initial design,DO optimum design, and RBDO optimum design. The initial design

Table 5Comparison of optimum results between DO and RBDO.

Items Initial Optimum values by DO Optimum by RBDO

Design variablesV1 0.2 0.208 0.223V2 0.4 0.451 0.482V3 0.6 0.643 0.658V4 0.8 0.834 0.830

Objective function (MPa)jrx-axial

max j 252.60 213.05 224.12

Table 6Reliability of DO and RBDO using Monte Carlo simulation.

Layer Reliability (%)

Initial DO RBDO

Metal 100 100 99.99With V1 100 100 99.97With V2 99.98 99.88 97.20With V3 99.51 98.02 95.77With V4 92.66 88.86 95.55Ceramic 66.99 96.00 97.35

442 Y.J. Noh et al. / Computational Materials Science 69 (2013) 435–442

has violated probabilistic constraints, with reliability less than thetarget, 97.725%, for the fourth layer and the ceramic layer, and thedeterministic optimum design also has one violated probabilisticconstraint for the fourth layer. All probabilistic constraints of theRBDO optimum design are safe. The third and fourth layer seemto have slightly violated constraints, but the RBDO optimum de-signs are still acceptable because a small percentage of reliabilityerror can be occurred due to FORM approximation of constraintfunctions and Monte Carlo simulation. In addition, comparing withthe reliability in the initial design, e.g. 66.9899% in the 4th layer,the reliability at the RBDO design is substantially improved. Thus,the RBDO optimum provides the most reliable volume fraction dis-tribution of FGM composite.

6. Conclusion

FGM composites are widely used, because they reduce stressconcentration, which often occurs at the interface of classical bi-material composites. However, the strength of stress concentrationdepends on the volume fraction in each graded layer, so the opti-mized volume fraction needs to minimize the stress concentration.Even though the optimized volume fractions are numerically ob-tained, the manufacturing techniques of FGM could not achievethe target volume fraction distribution due to uncertainty in themanufacturing process. To deal with this uncertainty, RBDO is car-ried out in this paper using statistical information of the volumefraction distributions. It has been observed from the numericalexample that the volume distribution obtained by RBDO has targetreliability such that the FGM composite does not fail. This researchis the first attempt to apply RBDO for the volume fraction optimi-zation of FGM composite structure considering the uncertainty inthe manufacturing process. However, since the number of gradedlayers and thickness of each layer are thought to affect the maxi-mum thermal stress in FGM composite, the consideration of suchfactors in RBDO deserves future study.

Acknowledgments

This research was supported by the Basic Science Research Pro-gram through the National Research Foundation of Korea (NRF),funded by the Ministry of Education, Science and Technology(Grant No. 2010-0024148). The financial support for this workthrough World Class 300 from the Ministry of Knowledge Economyof Korea is also acknowledged.

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