Random Field Characterization Considering Statistical Dependence for Probability Analysis and Design Zhimin Xi Department of Mechanical Engineering, University of Maryland, College Park, MD 20742 e-mail: [email protected]

Byeng D. Youn1 Assistant Professor School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea e-mail: [email protected]

Chao Hu Department of Mechanical Engineering, University of Maryland, College Park, MD 20742 e-mail: [email protected]

The proper orthogonal decomposition method has been employed to extract the important field signatures of random field observed in an engineering product or process. Our preliminary study found that the coefficients of the signatures are statistically uncorrelated but may be dependent. To this point, the statistical dependence of the coefficients has been ignored in the random field characterization for probability analysis and design. This paper thus proposes an effective random field characterization method that can account for the statistical dependence among the coefficients for probability analysis and design. The proposed approach has two technical contributions. The first contribution is the development of a natural approximation scheme of random field while preserving prescribed approximation accuracy. The coefficients of the signatures can be modeled as random field variables, and their statistical properties are identified using the chi-square goodness-of-fit test. Then, as the paper’s second technical contribution, the Rosenblatt transformation is employed to transform the statistically dependent random field variables into statistically independent random field variables. The number of the transformation sequences exponentially increases as the number of random field variables becomes large. It was found that improper selection of a transformation sequence among many may introduce high nonlinearity into system responses, which may result in inaccuracy in probability analysis and design. Hence, this paper proposes a novel procedure of determining an optimal sequence of the Rosenblatt transformation that introduces the least degree of nonlinearity into the system response. The proposed random field characterization can be integrated with any advanced probability analysis method, such as the eigenvector dimension reduction method or polynomial chaos expansion method. Three structural examples, including a microelectromechanical system bistable mechanism, are used to demonstrate the effectiveness of the proposed approach. The results show that the statistical dependence in the random field characterization cannot be neglected during probability analysis and design. Moreover, it is shown that the proposed random field approach is very accurate and efficient. 关DOI: 10.1115/1.4002293兴 Keywords: random field, proper orthogonal decomposition, probability analysis and design, eigenvector dimension reduction

1

Introduction

Manufacturing variability 共geometries and material properties兲 over samples and the stochastic nature of loads have been modeled using spatially independent random variables 关1–14兴. Although this literature has provided a good foundation for integrating probability analysis into engineering system design, these works lack practical consideration of spatial variability over samples 共or the random field兲. In many engineering applications, the manufacturing and load variability is a function of spatial variables 共x, y, and z兲 and the temporal variable 共t兲. The random field is thus used to reflect spatial and temporal variability. For instance, the thickness of a metal sheet has variation over space and samples 共or sampled time兲. This notion of the random field can also be observed in material properties 共e.g., an elastic modulus兲 and loading conditions. In the 1990s, the random field had already become popular in applications of civil engineering 关15–20兴. It had also been con1 Corresponding author. Contributed by the Design for Manufacturing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 13, 2009; final manuscript received July 29, 2010; published online October 4, 2010. Assoc. Editor: Michael Kokkolaras.

Journal of Mechanical Design

sidered in many other disciplines, including fluid dynamics 关21兴, wind pressure fields 关22兴, coherent structures 关23兴, and pattern recognition 关24兴. Numerous techniques have been developed to represent a discrete or continuous random field. Methods include the midpoint method 关25兴, spatial averaging method 关26兴, shape function method 关27兴, and proper orthogonal decomposition 共POD兲 method 关28兴. However, little effort has been made to consider the random field in probability analysis and design 关29–32兴. This is because spatial variability has been conceived to negligibly affect system responses. However, our preliminary study showed that spatial variability may influence variability in system responses significantly, especially in geometry-sensitive failures 共e.g., buckling, fillet failures兲 and small-scale applications in which tolerance control is more challenging. This research was initially inspired by a random field paper 关32兴 that originally applied the idea of the POD to engineering design problems. In this paper, the POD method was employed to extract the important signatures of the random field observed in an engineering product or process. Our preliminary study found that the parametric representation of the random field in Missoum’s work was not directly related to the available random field data and that the coefficients of the signatures were statistically uncorrelated but dependent on most cases. This paper thus proposes an effec-

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Other than the POD method 关28兴, the midpoint method 关25兴, the spatial averaging method 关26兴, and the shape function method 关27兴 can be used to model a random field. The first two methods can represent a random field in a discrete manner using a random vector. However, they require a large number of random field variables to describe the field with a given accuracy level 关39兴. The shape function method can express a random field using a continuous function. But the accuracy of the method depends on the selection of shape functions 关39兴. The POD method can describe a random field in both continuous and discrete ways. Furthermore, the POD method is much more efficient than the other three methods at a given accuracy level 关20兴. This paper thus uses the POD method for modeling the random field. 2.1 Review of the POD Method. The purpose of the POD method is to find the most significant field signature ␾共x兲 of an ensemble of the random field variation ␯共x , t兲 over the entire sampled time. This turns out to be an optimization problem that maximizes the following objective function 共f兲:

Fig. 1 Random field over samples

tive random field characterization method that can account for the statistical dependence among the coefficients for probability analysis and design. The proposed approach has two technical contributions. The first contribution is the development of a natural approximation scheme of random field while preserving prescribed approximation accuracy. The coefficients of the signatures can be modeled as random field variables, and their statistical properties are identified using the chi-square goodness-of-fit test. As the second technical contribution, the Rosenblatt transformation is employed to transform the statistically dependent random field variables into statistically independent random field variables. The number of the transformation sequences exponentially increases as the number of random field variables becomes large. It was found that improper selection of a transformation sequence among many may introduce high nonlinearity into system responses, which may result in inaccuracy in probability analysis and design. Hence, this paper proposes a novel procedure to determine an optimal sequence of the Rosenblatt transformation that introduces the least degree of nonlinearity into the system response. The proposed random field characterization can be integrated with any advanced probability analysis method, such as the dimension reduction 共DR兲 method 关33–35兴, eigenvector dimension reduction 共EDR兲 method 关36兴, or polynomial chaos expansion 共PCE兲 method 关37,38兴. Three structural examples, including a microelectromechanical system 共MEMS兲 bistable mechanism, are used to demonstrate the effectiveness of the proposed approach. Compared with the Monte Carlo simulation 共MCS兲, the proposed random field approach appears to be very accurate and efficient. Moreover, the results show that the statistical dependence in the random field characterization cannot be neglected during probability analysis and design. Section 2 reviews the POD method, and Sec. 3 presents the procedure of random field modeling and probability analysis. Three examples are used to demonstrate the effectiveness of the proposed approach in Sec. 4.

f = 共␾共x兲 · ␯⬁兲2

maximize

f = 共␾共x兲 · ␯⬁兲2 =

=

=

冕 再冕 冕 再冕 ⍀

⍀

⍀

⍀

冕

␾共x兲␯⬁共x兲dx

⍀

冎

冎

␪共x,tk兲 = ␮共x兲 + ␯共x,tk兲

共1兲

As shown in Figs. 1共a兲 and 1共b兲, geometric quantities 共e.g., thickness兲 can be measured at numerous measurement points over many samples. Examples can be found in various engineering fields: chassis and frames in automotive and aerospace, channel diameters in microfluidics, and beam heights in the MEMS bistable mechanism. Graphically, two sampled snapshots are shown in Fig. 1共c兲 at time t1 and t2, respectively. 101008-2 / Vol. 132, OCTOBER 2010

共3兲

where K共x , x⬘兲 = ␯⬁共x兲␯⬁共x⬘兲 and x⬘ is the dummy variable. Define a positive-definite integral operator as L=

冕

K共x,x⬘兲共 · 兲dx

共4兲

⍀

Then the objective function can be simplified as f = 共␾共x兲 · ␯⬁兲2 =

冕

共L␾共x兲兲共␾共x兲兲dx⬘ = 共L␾共x兲 · ␾共x兲兲

共5兲

⍀

To maximize the objective function, L␾共x兲 should have the same direction as the signature ␾共x兲. Thus it is obvious to observe that L␾共x兲 = ␭␾共x兲

共6兲

From Eq. 共6兲, ␾共x兲 is the eigenfunction of the operator L and ␭ is the corresponding eigenvalue. Thus, the field variation ␯共x , t兲 can be decomposed as

␾i共x兲

兺 ␣ 共t兲 储␾ 共x兲储 i

共7兲

i

i=1

A random field ␪共x , t兲 can be decomposed into mean ␮共x兲 and variation parts ␯共x , t兲. At a specific sample time tk, the random field of a sampled snapshot can be observed as

␾共x⬘兲␯⬁共x⬘兲dx⬘

⍀

K共x,x⬘兲␾共x兲dx ␾共x⬘兲dx⬘

␯共x,t兲 =

Review of the POD Method

冕

␯⬁共x兲␯⬁共x⬘兲␾共x兲dx ␾共x⬘兲dx⬘

⬁

2

共2兲

where ␯⬁ stands for the ensemble of the field variation ␯共x , t兲 and the operator • means an inner product. By definition of the inner product, the objective function can be further expressed as

where ␣i共t兲 is the coefficient of the ith eigenfunction. Its value can be obtained by projecting the field variation ␯共x , t兲 onto the corresponding unit eigenfunction as

␣i共t兲 = ␯共x,t兲 ·

␾i共x兲 储␾i共x兲储

共8兲

Therefore, the random field can be decomposed into Eq. 共9兲 using the POD approach: ⬁

␪共x,t兲 = ␮共x兲 +

␾i共x兲

兺 ␣ 共t兲 储␾ 共x兲储 i

i=1

共9兲

i

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2.2 Discrete Representation of a Random Field [32]. In engineering applications, it is more practical to represent a random field in a discrete manner than in a continuous way because a finite amount of field data is given at discrete field locations. As shown in Fig. 1, each snapshot is assumed to have a finite amount of measurement points 共n兲, and the physical quantities at the measurement points have variability over a finite amount of sampled snapshots 共m兲. The data sets characterizing the random field could be relevant to geometries, material properties, and/or loads. Thus an m ⫻ n matrix 共⌰兲 representing the random field can be constructed as

冤

␪11 ␪12 ␪21 ␪22 ⌰= ] ] ␪m1 ␪m2

¯ ␪1n ¯ ␪2n


]

¯ ␪mn

冥

where ␪il indicates the measured data at the jth measurement point of the lth sampled snapshot. Such a representation works for multidimensional problems. Regardless of the dimension of the random field, the scanned multidimensional data are listed in a onedimensional array from ␪l1 to ␪ln for the lth sampled snapshot. The mean of the random field is estimated as

␮ = 关¯␪•1,¯␪•2, . . . ,¯␪•n兴 where ¯␪•j stands for the average of the jth measured data over the samples. Hence the variation part is expressed as

␯=

冤

␪11 − ¯␪•1 ␪12 − ¯␪•2 ¯ ␪1n − ¯␪•n ␪21 − ¯␪•1 ␪22 − ¯␪•2 ¯ ␪2n − ¯␪•n ]
] ] ␪m1 − ¯␪•1 ␪m2 − ¯␪•2 ¯ ␪mn − ¯␪•n

冥

The signature ␾ can then be obtained by solving an eigenproblem as C␾ = ␭␾

共10兲

where ␭ is the eigenvalue of the covariance matrix C 共m ⫻ m兲 that is defined as C = ␯␯T

3

共11兲

Random Field Characterization

This section presents two technical contributions. In Sec. 3.1, the first contribution is the development of a natural approximation scheme of random field characterization. Section 3.2 presents the second contribution, which is a probability analysis method with statistically dependent random field variables. 3.1

Random Field Approximation Scheme

3.1.1 Important Signatures for Representing a Random Field. Theoretically, an infinite number of signatures are required to represent the random field exactly, as shown in Eq. 共9兲. Practically, only a few important signatures may be vital to approximate the random field accurately. Hence, instead of using all signatures, a small number of important signatures 共r兲 are selected to approximate the random field, as shown in Eq. 共12兲: r

␪共x,t兲 ⬇ ˜␪共x,t兲 = ␮共x兲 +

␾i共x兲

兺 ␣ 共t兲 储␾ 共x兲储 i

i=1

共12兲

i

where ˜␪共x , t兲 is an approximate random field with r number of important signatures. The importance of the signature is indicated by the magnitude of the eigenvalue, as shown in Eq. 共10兲. The larger eigenvalue indicates the greater importance of the corresponding signature in Journal of Mechanical Design

Fig. 2 Flowchart for determining the number of the important signatures

approximating the random field. Therefore, the eigenvalue can be ranked based on the magnitude of the normalized eigenvalue 共␳i兲, defined as

␳i =

␭i , ␭1

0 ⱕ ␳i ⱕ 1

for

i = 1, . . . ,m

共13兲

where ␭1 is the largest eigenvalue and m is the total number of eigenvalues. The number 共r兲 based on the magnitude of the normalized eigenvalue could be determined subjectively. Therefore, a posteriori normalized error ␧ is defined to adaptively determine the minimal number of the most important signatures, which preserves a prescribed accuracy in approximating the random field. The normalized error is defined as m

1 ␧= mn

n

兺 兺 兩␪

ij

− ˜␪ij兩

i=1 j=1

␮max − ␮min

共14兲

where m is the number of the sampled snapshots, n is the number of the measurement points at each snapshot, ˜␪ij is the approximate random field data at the jth measurement point of the ith sampled snapshot with k共ⱕr兲 number of important signatures, ␪ij is the actual random field data, and ␮max and ␮min are the maximum and minimum values of the mean of the random field defined in Sec. 2, respectively. The normalized error indicates an average error between the actual and approximate random fields at all measurement points. A flowchart for adaptively selecting the number of the most important signatures is shown in Fig. 2. Once the random field data sets 共␪m⫻n兲 are obtained, the total m number of the signatures can be ranked based on Eq. 共13兲. The approximate random field is gradually refined by adding one more signature to each iteration until the normalized error in Eq. 共14兲 is smaller than a threshold error value ␧c, which is generally set to 0.1%. The threshold value must be small enough to ensure high accuracy of the random field modeling while using the minimal number of the important signatures. Otherwise, the statistical uncertainty in the random field modeling may be comparable to the physical uncertainty or may even dominate in probability analysis. 3.1.2 Modeling Random Field Variables. The random field variables will be used to characterize a random field observed in an engineering product or process. In Eq. 共12兲, ␣i共t兲 represents a coefficient data set of the ith signature obtained from all sampled snapshots 共t = 1 , . . . , m兲. We define Vi as a random field variable OCTOBER 2010, Vol. 132 / 101008-3

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that statistically models the coefficient data set of the ith signature. By replacing ␣i共t兲 with the random field variable 共Vi兲, Eq. 共12兲 can be rewritten as r

˜␪共x,t兲 = ␮共x兲 +

␾i共x兲

兺 V 储␾ 共x兲储 i

共15兲

i

i=1

The formulation of the random field variable 共Vi兲 is different from the previous study 关32兴, in which a weight function was multiplied by a user-selected coefficient, say, ␣i共1兲, for the parametric representation of a random field. The weight function was used to modify the contribution of each random field signature. However, this parametric representation may fail to represent the actual random field, since it is not directly related to the available random field data. In this paper, the random field variable 共Vi兲 contains the variability over the sampled snapshots that are obtained during the sampled time 共t兲. Once the statistical properties of Vi are characterized, the original random field can be approximated by Eq. 共15兲. Therefore, the random system response in the presence of the random field can be effectively analyzed using any probability analysis method. Accuracy in modeling the random field variable Vi depends on the number of sampled snapshots. This study employs a large amount of sampled snapshots and, thereafter, considers aleatory uncertainty2 only. For epistemic uncertainty3 with the lack of sampled snapshots, Bayesian statistics 关40兴 can be integrated to the proposed framework, which is beyond the scope of this paper. This study uses a large amount of input random data for the construction of aleatory uncertainty. The statistical properties of the random field variable Vi can be characterized with the following three steps. Step 1: Obtain optimum distribution parameters for candidate distributions using the maximum likelihood method, which can be formulated as m

maximize

L共Vi兩␦兲 =

兺 log

10关f共vil兩␦兲兴

l=1

where ␦ is the unknown distribution parameter vector, vil is a realization of Vi from the lth snapshot, L共 • 兲 is the likelihood function, m is the number of snapshots, and f is the probability density function 共PDF兲 of Vi for the given ␦. Step 2: Perform quantitative hypothesis tests for the candidate distribution types with the optimum distribution parameters obtained in step 1. Among the chi-square goodness-of-fit test, the Kolmogorov–Smirnov 共KS兲 test, and the Anderson–Darling 共AD兲 test, the chi-square goodness-of-fit test was selected in this study due to its good performance for both continuous and discrete distributions given a large amount of data. Step 3: Select the distribution type with the maximum p-value as the optimal distribution type for Vi. 3.1.3 Statistical Properties of Random Field Variables. When multiple random field variables are needed to accurately approximate the random field, statistical correlation and statistical dependence of the random field variables become one of the greatest concerns in probability analysis. Using Eqs. 共8兲, 共11兲, and 共15兲, the inner product of any two random field variables can be expressed as V iV j =

␭i␾i共x兲 · ␾ j共x兲 储␾i共x兲储储␾ j共x兲储

共16兲

Since the two signatures 共␾i共x兲 and ␾ j共x兲兲 are orthogonal, E共ViV j兲 becomes zero. Furthermore, the expected value 共or mean兲 of every 2 Aleatory uncertainty is defined as objective and irreducible uncertainty with sufficient information on the random variable. 3 Epistemic uncertainty can be classified as subjective and reducible uncertainty due to the lack of knowledge on the random variable.

101008-4 / Vol. 132, OCTOBER 2010

random field variable is zero because the mean of the variation in Eq. 共8兲 is zero. Thus, Vi and V j must be statistically uncorrelated; that is, E关共Vi − ␮i兲共V j − ␮ j兲兴 = E共ViV j兲 − E共Vi兲E共V j兲 = 0 However, they may not be statistically independent because of f ViV j共vi , v j兲 ⫽ f Vi共vi兲f V j共v j兲. If the random field variables are statistically independent, they are statistically uncorrelated. But the converse is not true. A complicated random field tends to require a large number of random field variables. Such a problem poses a great challenge in handling statistical dependence of the random field variables since little effort has been devoted to handling probability analysis for system responses with statistically dependent random variables. 3.2 Probability Analysis Method With Statistically Dependent Random Field Variables. To handle the statistical dependence of the random field variables for probability analysis of system responses, the statistically dependent random field variables need to be transformed into statistically independent random field variables. Thus, any advanced probability analysis method can be integrated with the proposed random field approach for probability analysis and design. In Sec. 3.2.1, the Rosenblatt transformation is employed to transform the statistically dependent random field variables into statistically independent random field variables. The number of the transformation sequences exponentially increases as the number of random field variables becomes large. It was found that improper selection of a transformation sequence among many may introduce high nonlinearity into system responses, which may result in inaccuracy in probability analysis and design. Section 3.2.2 thus proposes a novel procedure for determining an optimal sequence of the Rosenblatt transformation that introduces the least degree of nonlinearity to the system response. In Sec. 3.2.3, two numerical examples are employed to demonstrate the effectiveness of the proposed approach in handling the statistically dependent random variables for probability analysis. 3.2.1 Incorporation of the Rosenblatt Transformation. In many advanced probability analysis methods, only a few simulations or function evaluations at a set of samples of the input random variables are required for probability analysis if the input random variables are statistically independent. For example, the EDR method demands either 2N + 1 or 4N + 1 samples for probability analysis, where N is the number of the input random variables. In the PCE method, the evaluation of the PCE coefficients requires the response values at the predefined Gaussian quadrature points 关41兴, the collocation points specified by the Smolyak algorithm 关42兴, or the univariate and bivariate sample points 关38兴. Hence, the probability analysis of the system response can be carried out using one of the advanced probability analysis methods if the system response can be evaluated at the required samples in the transformed standard normal space 共or U-space兲. In this section, we attempt to determine the samples in the statistically dependent random space 共or V-space兲 for probability analysis. The samples in the V-space can be obtained through the inverse Rosenblatt transformation from those in the U-space. The overall procedure is detailed as follows. 共j兲 Step 1: Obtain the required sample points 共u1共j兲 , . . . , uN 兲 for j = 1 , . . . , M in the U-space for a given probability analysis method, where M is the total number of the sample points. Step 2: Transform the sample points from the U-space to the V-space using the inverse Rosenblatt transformation as v1共 j兲 = FV−11 关⌽共u1共 j兲兲兴 v2共 j兲 = FV−1兩V 关⌽共u2共 j兲兲兴 2

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vN共 j兲 = FV−1 兩V N

1,V2,. . .,VN−1

关⌽共uN共 j兲兲兴

共17兲

共j兲

共j兲

where 共v1 , . . . , vN 兲 denotes the jth transformed sample point in the V-space, and F−1共 • 兲 is the inverse joint CDF of the random variable in the V-space. It is noted that the choice of a transformation sequence significantly affects the nonlinearity of the system response and will be discussed in Sec. 3.2.2. Step 3: Obtain the system response values at the transformed 共j兲 sample points: Y共v1共j兲 , . . . , vN 兲, for j = 1 , . . . , M and perform probability analysis. To further elaborate step 2 with a two-dimensional problem, let 共u1 , u2兲 = 共−3 , 0兲 be one of the required samples in the U-space. An empirical cumulative density function 共CDF兲 of v1 can be obtained using the data set for v1. The first component value c1 共=v1兲 of the sample in the V-space can be set to F−1 V1 共⌽共u1 = −3兲兲. Given the identified first component value v1 = c1, an empirical conditional CDF of v2 can be constructed using the statistically dependent data set. The second component value c2 共=v2兲 of the sample in the V-space can then be set to FV−1兩V 共⌽共u2 = 0兲兲. This 2 1 process can be continued to determine other samples in the V-space using the available statistically dependent data. 3.2.2 Determination of an Optimal Transformation Sequence. The number of the transformation sequences exponentially increases as the number of random field variables becomes large. It was found that improper selection of a transformation sequence among many may introduce high nonlinearity into system responses, which may result in inaccuracy in probability analysis and design. Hence, it is critical to determine an optimal sequence of the Rosenblatt transformation that introduces the least degree of nonlinearity into the system response. A linear response function is employed to study the nonlinearity introduced by different transformation sequences, as shown in Eq. 共18兲: N

YT =

兺v

共18兲

i

i=1

where u = Tk共v兲

共19兲

where Tk共 • 兲 indicates the Rosenblatt transformation with the kth transformation sequence, Y T,k stands for the nonlinear response obtained through the kth transformation, and k = 1 , . . . , N!. It is apparent that the best sequence must have the least degree of nonlinearity in the system response. Hence, we need to quantify the nonlinearity of Y T,k for all possible sequences. The degree of nonlinearity in Eq. 共19兲 introduced by a particular transformation sequence can be obtained by measuring the degree of deviation from a linear response Y L = ⌺iui. The degree of deviation of Y T,k共 • 兲 from the jth linear response 共Yˆ j兲 through the kth transformation sequence can be defined as Q

S j,k =

兺 关Y

T,k共0,

. . . ,0,u j,l,0, . . . ,0兲 − Yˆ j,l兴2

rithm can be employed to effectively determine the best sequence with the minimum total degree of deviation. An optimization problem can be formulated as N

minimize

兺S

j,k

j=1

S.T.

k 苸 兵1, ¯ ,N!其

3.2.3 Examples. One numerical example is employed here to demonstrate the effectiveness of the proposed approach in handling the statistically dependent random variables for probability analysis. One of the probability analysis methods, the EDR method, is used for probability analysis. 3.2.3.1 Two-dimensional statistical dependence. A mathematical example with non-normally distributed, statistically dependent random variables demonstrates the procedure for determining the best transformation sequence. The system response is expressed as Y=1−

where N is the total number of random variables, and vi is the ith random variable. The linear response function becomes nonlinear after the Rosenblatt transformation and is expressed as Y T,k = f共u1,u2, . . . ,uN兲

Fig. 3 Statistical dependence of v1 and v2

共v1 + v2 − 5兲2 共v1 − v2 − 12兲2 − 30 120

共21兲

where v1 and v2 are the statistically dependent random variables with sufficient data 共say, 1000 sampled data兲, as shown in Fig. 3. To observe the sequence effect, the total degrees of deviation were calculated as 0.0279 and 0.1680 for two sequences 关v1 , v2兴 and 关v2 , v1兴, respectively, where 关v1 , v2兴 means the transformation of v1 and v2 in order. Figures 4 and 5 show the response nonlinearity after two different transformation sequences. The figures confirm that the second sequence 关v2 , v1兴 produced much higher nonlinearity than the first. This study suggests using the first sequence 关v1 , v2兴 for probability analysis. Using the Rosenblatt transformation with the sequence 关v1 , v2兴, the EDR method with 4N + 1 sampling scheme was used for probability analysis of the system response subject to the non-normally distributed, statistically dependent random variables. Figure 6 shows the nine EDR samples mapped in the V-space and the predicted PDF of the response. It was observed in Table 1 that the

where Yˆ j,l = u j,l

l=1

共20兲 where Q is the number of discrete data points along the jth random variable u j. Repeating this for N random variables, the total degree of deviation can be calculated as ⌺ jS j,k for the kth transformation sequence. The sequence with the minimum total degree of deviation will be defined as the optimal sequence of the Rosenblatt transformation. For a small number of random variables 共say, N ⬍ 10兲, the best sequence can be determined by finding the minimum total degree of deviation in all possible sequences. For a large number of random variables 共say, N ⬎ 10兲, a genetic algoJournal of Mechanical Design

Fig. 4 Nonlinearity of YT,k with the transformation sequence †v1 , v2‡

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Fig. 8 Simulation model of a cantilever beam with the random field Fig. 5 Nonlinearity of YT,k with the transformation sequence †v2 , v1‡

EDR method predicted the first four moments very accurately. The predicted PDF using the EDR method agrees well with the normalized histogram using MCS. It was also found in Fig. 7 that the EDR method using an inappropriate transformation sequence 关v2 , v1兴 yielded a relatively large prediction error in probability analysis.

4

Examples

4.1 Beam Example. A cantilever beam is one of the most commonly used structures in engineering applications and has spatial variability to some degree. This variability may influence variability in beam responses. For this investigation, the top and bottom surfaces of the beam were modeled to have a symmetric random field about the midsurface. A mathematical expression of the random field in the top surface was formulated as h共x兲 = 0.1 sin共K␲x/2兲

共22兲

where K ⬃ normal共2 , 0.022兲 and 0 ⱕ x ⱕ 10 关mm兴. The beam height was 2 mm at x = 0. 1000 sampled snapshots were artificially created by generating 1000 random K values from the prescribed normal distribution. 100 measurement points were evenly distributed along the length of the beam. The cantilever beam was fixed at the right end and a concentrate force 共100 N兲 was applied at the

left tip of the beam, as shown in Fig. 8. The maximum beam deflection was considered as the system response. The beam deflection was calculated through a finite element 共FE兲 analysis using OPTISTRUCT in HyperMesh. different FE models were created for different snapshots using HYPERMORPH in HyperMesh. Specifically, a FE basis model was built first based on the mean of the random field. HYPERMORPH was then used to define the perturbation vectors of the measurement points 共or element nodes兲 based on the signatures of the random field. The signature coefficients were defined next as the perturbation coefficients of the perturbation vectors in HYPERMORPH. Then, different FE models were constructed by providing the corresponding set of coefficients. The thickness of the shell elements was set to 1 mm. Both probability analyses of the system response using the random field approach 共RFA兲 and random parameter approach 共RPA兲 were carried out for comparison purposes. For the RPA, the average height of the 100 measurement points obtained in one sampled snapshot was treated as the uniform height over the entire beam length. 4.1.1 Step 1: Determination of the Important Signatures. First, an m ⫻ n matrix 共⌰兲 representing the random field was created to obtain the field signatures. Using the posteriori normalized error in Eq. 共14兲, the two most important signatures were selected to approximate the random field 共see Fig. 9兲. The normalized error of the approximate random field is less than 0.1% with these two signatures. Thus, the random field can be approximated as 2

˜h共x兲 = ␮共x兲 +

i

i=1

Fig. 6 EDR results with the transformation sequence †v1 , v2‡ Table 1 Statistical moments of Y using the EDR method and MCS

EDR MCS

Mean

Std

Skewness

Kurtosis

Fun. eval.

⫺1.0427 ⫺1.0406

0.2183 0.2164

⫺0.0562 ⫺0.0334

2.3132 2.3707

9 1000

Fig. 7 EDR results with the transformation sequence †v2 , v1‡

101008-6 / Vol. 132, OCTOBER 2010

␾i共x兲

兺 V 储␾ 共x兲储

共23兲

i

Figure 10 shows the first random field realization in the region 8 ⱕ x ⱕ 10 mm, which confirms the accuracy of approximate random fields with the two most important signatures. The figure contains one true and two approximate realizations of the random field. The approximate realizations were built using one and two of the most important signatures. The use of the most important signature produced a normalized error of 0.44% in approximating the random realization. The inclusion of the second most important signature decreased the error to 0.03%. Two random field variables 共V1 and V2兲 were thus used to describe the random field.

Fig. 9 The first two normalized signatures

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Fig. 10 Comparison of the exact and approximate random fields „first random field realization…

4.1.2 Step 2: Modeling Random Field Variables and Statistical Dependence. 100 random samples of two random field variables 共V1 and V2兲 were obtained from 100 sampled snapshots. The maximum likelihood estimation 共MLE兲 and chi-square goodnessof-fit tests were used to find the distributions and statistical parameters of two random field variables. They were modeled as V1 ⬃ normal 共0,0.11922兲

and

V2 ⬃ beta 共0,0.018,

− 0.156,0.013兲 As explained in Sec. 3.1.3, no statistical correlation exists between V1 and V2. However, their statistical dependence was clearly observed by plotting 1000 samples of V1 and V2, as shown in Fig. 11. In this special case, V2 is a function of V1. For a given V1 value, the corresponding V2 value was obtained using the moving least squares method. Therefore, Eq. 共23兲 can be reformulated to resolve the difficulty of the statistical dependence. ˜h共x兲 = ␮共x兲 + V ␾ 共x兲 + f共V 兲␾ 共x兲 1 1 1 2

共24兲

4.1.3 Step 3: Probability Analysis Considering a Random Field. Probability analyses using MCS with 1000 samples were conducted using two different approaches: RFA and RPA. The maximum beam deflection was considered as a system response, and the histograms from two different approaches are shown in Fig. 12. The two approaches produced substantially different histograms because the beam with a uniform height in RPA is stiffer than the beam with a varying height in RFA. Because the RPA greatly underestimates the displacement by ignoring the spatial variation, it is very important to consider the RFA for probability analysis and design. The EDR method with 2N + 1 samples 共three analyses兲 was employed for RFA, and the maximum beam deflection was statistically quantified in terms of the four statistical moments. Table 2 shows that the proposed RFA accurately estimates the four statistical moments compared with MCS. The 100共1

Fig. 12 Histograms of the maximum beam deflection using RFA and RPA

− 2␣兲% confidence intervals 关d1 , d2兴 of the four statistical moments can be found by solving the equation of F共m 兩 m = d1兲 = 1 ⬘ − ␣ and F共m⬘ 兩 m = d2兲 = ␣, where m denotes the true statistical moment, m⬘ is a consistent estimator of m, and F共m⬘ 兩 m兲 is the CDF of the estimator m⬘ 关43兴. It is confirmed in Fig. 12 that the proposed RFA can accurately approximate the PDF of the maximum beam deflection. 4.2 MEMS Bistable Mechanism. A MEMS device was used for the second example because spatial variability may significantly influence variability in MEMS device responses. A bistable mechanism is able to remain in stable equilibrium in two distinct positions. MEMS bistable mechanisms are useful as microvalves 关44兴, microrelays 关45兴, and fiber optical switches 关46兴. At the microscale, a monolithic bistable mechanism is necessary to avoid friction, backlash, and wear at joints. One feasible monolithic MEMS bistable mechanism 关47,48兴 was recently developed by rigidly coupling two curved beams together at their midpoints, as shown in Fig. 13. Figure 14 shows the relationship between a typical force and displacement curve for such a bistable mechanism when the force is applied downward at the center of the upper beam. There are three equilibriums during this process. S1 and S3 are the stable equilibriums and S2 is the unstable one. If the force is released before passing the unstable equilibrium S2, the structure returns to the stable equilibrium S1. Otherwise, it moves to the second stable equilibrium S3. Three system responses, the maximum force, minimum force, and distance from the state S1 to S2, are normally important for different applications. The two curved beams were designed to have uniformly distributed thickness, and the top surface of the beam can be modeled as

Table 2 Statistical moments of Y using the proposed RFA and MCS

MCS RFA

Fig. 11 Statistical properties of V1 and V2

Journal of Mechanical Design

Mean

Std

Skewness

Kurtosis

Fun. eval.

⫺0.2749 ⫺0.2749

0.0016 0.0016

⫺2.4728 ⫺2.3244

11.8629 11.3535

1000 3

Fig. 13 Bistable mechanism

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Fig. 16 The first two normalized signatures

Fig. 14 Force displacement curve

冋 冉 冊册

2␲x h w共x兲 = 1 − cos 2 l

共25兲

where h / 2 is the apex of the curved beam, and l is the length. The bottom surface of the beam is described as w共x兲 − t, where t is the thickness of the beam. In the application of such a MEMS bistable mechanism, the thickness commonly lies in the range of a few micrometers, so it is extremely difficult to fabricate a uniformly thick beam. Random field may significantly affect the reliability of the MEMS device, since the device responses are considerably affected by the spatial variability of the thickness. A mathematical expression for the random field in the top surface was formulated as w⬘共x兲 =

冋

冉 冊册

h k 2␲ x k1 − cos 2 l

共26兲

where k1 ⬃ normal共1 , 0.012兲 and k2 ⬃ normal共2 , 0.012兲. Figure 15 displays the top 共w+兲, bottom 共w−兲, and a realization of the randomly field for the top surface 共w⬘兲. This example employs a moderate degree of random field, compared with the former example 共Sec. 4.1兲. Two beams in the bistable mechanism were assumed to share the same random field. The bistable mechanism was modeled with beam 23 elements using 1388 nodes and 4148 DOFs, and the connection between two beams was modeled using a rigid element. ANSYS 10.0 was employed for the FE analysis. To achieve the force-displacement system response, nonlinear FE analyses were performed. Each simulation took about 40 s. The FE model information follows: the length l was 3 mm, the thickness t was 6 ␮m, the apex value was 60 ␮m, the beam depth was 490 ␮m, Young’s modulus was 169 Pa, and the gap between two beams was 90 ␮m. 1000 sampled snapshots were used for characterizing the random field, and each snapshot had 100 measurement points evenly distributed along the length of the beam. 4.2.1 Step 1: Determination of the Important Signatures. An m ⫻ n matrix 共⌰兲 representing the random field was created to

Fig. 15 Creation of the random field for one beam

101008-8 / Vol. 132, OCTOBER 2010

obtain the important field signatures. Using the posteriori normalized error in Eq. 共14兲, the two most important signatures were selected to approximate the random field 共see Fig. 16兲. The criterion for the normalized field characterization error was set to 0.1%. The random field in the top surface of the beam can be approximated as 2

˜ ⬘共x兲 = ␮共x兲 + w

兺 V ␾ 共x兲 i

i

共27兲

i=1

Figure 17 shows the first random field realization in the region 1000ⱕ x ⱕ 2000 ␮m, which confirms the accuracy of approximate random fields with the two most important signatures. The figure contains one true and two approximate realizations of the random field. The approximate realizations were built using one and two of the most important signatures. The use of the most important signature led to a normalized error of 0.29% in approximating the random realization, whereas the inclusion of the second most important signature decreased the error to 0.01%. Two random field variables 共V1 and V2兲 can thus be used to describe the random field. 4.2.2 Step 2: Modeling Random Field Variables and Statistical Dependence. 1000 random values for two random field variables 共V1 and V2兲 can be generated from 1000 sampled snapshots. The MLE and chi-square goodness-of-fit tests were used to find the distributions and statistical parameters of two random field variables. Two random field variables were modeled as V1 ⬃ normal 共0,4.002兲

and

V2 ⬃ normal 共0,2.562兲

Unlike the first example, V1 and V2 are statistically independent, as shown in Fig. 18. 4.2.3 Step 3: Probability Analysis Considering a Random Field. Probability analyses using MCS with 1000 samples were conducted using two different approaches: RFA and RPA. The maximum and minimum forces, and unstable equilibrium distance

Fig. 17 Comparison of exact and approximate random fields „first random field realization…

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in RFA. Thus, ignorance of the spatial variability in RPA may mislead decision-makers on the system responses. The EDR method with 2N + 1 samples 共five analyses兲 was employed to predict the random behavior of the system responses for RFA. Table 3 shows that the proposed RFA accurately estimates the four statistical moments and approximates the PDFs of the system responses, as shown in Fig. 20. 4.3 Beam Example With Statistical Dependence. This example employed the same cantilever beam used in Sec. 4.1 with different spatial variabilities. The top and bottom surfaces of the beam were modeled to have a symmetric random field about the midsurface. A mathematical expression for the random field in the top surface was formulated as 5

Fig. 18 Statistical properties of V1 and V2

Y = 0.1

兺 i=1

were considered as system responses, and the histograms from two different approaches are shown in Fig. 19. Even if this example engages the smaller degree of random field variability, both approaches produced substantially different histograms in three system responses. It was observed that RFA produced relatively narrower distributions for both maximum and minimum forces than RPA, whereas RFA yielded wider unstable equilibrium distance than RPA. This is mainly because the uniformly thick beam in RPA is generally stiffer than the beam with a varying thickness

冋冉 冊 冉 sin

Ki␲共L − x兲 K i␲ x + sin i i

冊册

共28兲

where Ki ⬃ normal共2 , 0.022兲, L 共=10 mm兲 is the length, the beam height is 2 mm at x = 0, and a concentrate force 共100 N兲 is applied at the left tip of the beam. The system response is the maximum beam deflection. 1000 snapshots of the random field can be constructed by generating 1000 random values of Ki. Figure 21 shows the first, 501st and 1000th random field snapshots. The eight most important signatures are required to attain the prescribed accuracy in approximating the random field. Figure 22 displays three approximate random fields 共with one,

Fig. 19 Comparison of RFA and RPA Table 3 Statistical moments of Y using the proposed RFA and MCS Response

Method

Mean

Std

Skewness

Kurtosis

Fun. eval.

Maximum force

MCS RFA

4.0986 4.1253

0.6923 0.6561

0.2367 0.3181

3.2445 3.1545

1000 5

Minimum force

MCS RFA

⫺1.5221 ⫺1.5617

0.3389 0.3004

0.7759 0.5695

5.6786 5.5742

1000 5

Unstable equilibrium distance

MCS RFA

90.6885 91.3631

1.6106 1.7723

0.7327 0.5221

5.1094 4.1532

1000 5

Journal of Mechanical Design

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Fig. 20 Comparison of the proposed RFA and MCS

four, and eight signatures兲 for the 501st and 1000th random field snapshots. It is apparent that the use of the eight most important signatures represents the true random field very accurately. Statistical dependences were observed for the eight random field variables. Among all statistical dependences, three statistical dependences between V1, V4, and V8 are shown in Fig. 23. The optimal

sequence was obtained as 关v1 , v2 , v3 , v4 , v7 , v8 , v6 , v5兴, which presents the minimum total degree of deviation 共=0.0458兲 using the genetic algorithm provided in the MATLAB software. Then, the EDR method with a bivariate decomposition 关33–35兴 was used to predict the statistical properties of the maximum beam deflection for RFA. MCS with 1000 samples was executed for a benchmarking solution. Table 4 shows that the proposed RFA accurately assesses the four statistical moments compared with MCS. Figure 24 compares the PDFs from RFA using the EDR method and MCS. It was found that the PDF of the maximum beam deflection produced by RPA was significantly different from that produced by RFA. Figure 25 shows the reliability errors using RFA with and without considering the statistical dependences between the eight random field variables and underscores the importance of considering statistical dependence in probability analysis. The reliabilities were computed at a set of system response target values.

5

Conclusion

Until now, spatial variability 共or random field兲 has been generally overlooked in most engineering probability analysis and design. This could be due in part to a lack of an effective approach Fig. 21 Three random field snapshots

Fig. 22 Approximation of the random field with different numbers of signatures

101008-10 / Vol. 132, OCTOBER 2010

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Fig. 23 Statistical dependence of random field variables

for random field characterization in probability analysis and design, a misconception of the minor influence of the random field on the system response, or both. Hence, the RPA has been popular in engineering probability analysis and design by simply modeling manufacturing variability without its spatial randomness. To address this issue, this paper proposed an effective random field characterization method that can account for the statistical dependence among the coefficients for probability analysis and design. The proposed approach has two technical contributions. The first contribution is the development of a natural approximation scheme of random field while preserving prescribed approximation accuracy. The coefficients of the signatures can be modeled as random field variables, and their statistical properties are identified using the chi-square goodness-of-fit test. As the second technical contribution, the Rosenblatt transformation is employed Table 4 Statistical moments of Y using the proposed RFA and MCS

MCS RFA

Mean

Std

Skewness

Kurtosis

Fun. eval.

0.3935 0.3935

0.0002 0.0002

0.1242 0.1132

3.2262 3.3143

1000 56

to transform the statistically dependent random field variables into statistically independent random field variables. The number of the transformation sequences exponentially increases as the number of random field variables becomes large. It was found that improper selection of a transformation sequence among many may introduce high nonlinearity into system responses, which may result in inaccuracy in probability analysis and design. Hence, this paper proposes a novel procedure for determining an optimal sequence of the Rosenblatt transformation that introduces the least degree of nonlinearity into the system response. The proposed random field characterization can be integrated with any advanced probability analysis method, such as the EDR method or PCE method. Three structural examples, including a MEMS bistable mechanism, were used to demonstrate the effectiveness of the proposed approach. The results show that the statistical dependence in random field characterization cannot be neglected during probability analysis and design. Moreover, it was shown that the proposed random field approach is very accurate and efficient. It was assumed in this study that the proposed method requires a sufficiently large number of the snapshots. The proposed idea is not applicable to problems with insufficient sampled snapshots. To overcome this limitation, our future work intends to incorporate Bayesian statistics with the proposed idea.

Acknowledgment This research was partially supported by the U.S. Army STAS Contract No. TCN-07215.

Nomenclature ␪共x , t兲 ˜␪共x , t兲 ⌰ ␮共x兲 ␮ Fig. 24 Comparison of RFA and RPA

␯共x , t兲 ␯

␾共x兲 ␾ ␭ ␳i ␣i共t兲 C m n Fig. 25 Reliability error by ignoring statistical dependence of random field variables

Journal of Mechanical Design

␮max

⫽ continuous random field ⫽ approximate random field ⫽ discrete random field ⫽ mean of the continuous random field 共=mean function of the random field兲 ⫽ mean of the discrete random field 共=mean vector of the random field兲 ⫽ variance of the continuous random field 共=variance function of the random field兲 ⫽ variance of the discrete random field 共=variance vector of the random field兲 ⫽ signature of the continuous random field 共=signature function of the random field兲 ⫽ signature of the discrete random field 共=signature vector of the random field兲 ⫽ eigenvalue ⫽ the ith normalized eigenvalue ⫽ coefficient of the ith corresponding signature ⫽ covariance matrix ⫽ number of the sampled snapshots ⫽ number of the measurement points at each snapshot ⫽ maximum value of the mean sampled random field OCTOBER 2010, Vol. 132 / 101008-11

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␮min ⫽ minimum value of the mean sampled random field ␧ ⫽ normalized error ␧c ⫽ threshold error value Vi ⫽ the ith random field variable f Vi共vi兲 ⫽ probability density function of the ith random field variable E ⫽ expectation operator

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