[American Institute of Aeronautics and Astronautics 48th AIAA/ASME/ASCE/AHS/ASC Structures,...

Efficient Moment and Probability Distribution

Estimation Using the Point Estimate Method for High-

dimensional Engineering Problems

Liping Wang∗

GE Global Research Center, K1-2A62B, One Research Circle, NY12309

Don Beeson† and Gene Wiggs

‡

GE Aviation, 1 Neumann Way, Cincinnati, OH45215

The point estimate method (PEM) is an alternative to Monte Carlo Simulation

(MCS) and First Order Second Moments (FOSM) for evaluating the moments and

probability distribution of the system or component performance. Although PEM is

a powerful and simple method, it is often limited by the need to make 2n or even 3

n

evaluations when there are n random variables, which is unaffordable for many

engineering applications. In the previous paper by the authors [15], a variable-point

PEM method was proposed to improve the efficiency and accuracy of the existing

approaches. However, when it applied to the problems with large number of design

variables (number of design variables > 10), the method still requires hundreds or

thousands of simulation runs. This paper further improves the efficiency of the

variable-point PEM based upon two fundamental concepts: 1) The Pareto principle;

and 2) The Central Limit Theorem of Statistics, i.e., under common engineering

conditions, a linear combination of random variables can be approximated to first

order by a normal distribution. The efficiency and accuracy of the proposed method

are validated with three benchmark problems

I. Introduction

Performance function probability distributions and their associated first moments are important

statistical unknowns in engineering probabilistic design activities for robustness and reliability. Methods of

predicting this quantities using computer simulations enable engineers to more accurately model the

physical world and make design changes which will produce engineering systems, components or

processes that are less sensitive to manufacturing and environmental variation while at the same time

achieving requirements for performance, reduced costs and other six sigma goals.

Commonly used methods for moment calculations are: Taylor’s series expansion, transfer function or

meta-model construction, Monte Carlo Simulation (MCS), and First-Order Second-Moment (FOSM).

FOSM and Taylor’s series expansion both require accurate gradient calculations that impose excessive

restrictions on the performance functions (smoothness and existence and continuity of the first or first few

derivatives). On the other hand, MCS and traditional transfer functions generated from DOEs (Design Of

Experiments) do not require gradients calculations, but may also have difficulty because the computational

cost for them may be too expensive if high accuracy is required or if the number of input variables is large

(greater than 10). In the recent years, applying MCS to advanced meta-models generated from DACE

(Design Analysis and Computer Experiments) [17] becomes a popular approach. It is good alternative for

∗ Mechanical Engineer, GE Global Research Center, K1-2A62B, One Research Circle, NY12309. AIAA Member

† Program Manager, GE Aviation, 1 Neumann Way, Cincinnati, OH45215.

‡ Principal Engineer, GE Aviation, 1 Neumann Way, Cincinnati, OH45215

48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br> 15th23 - 26 April 2007, Honolulu, Hawaii

AIAA 2007-1874

Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

2

fast and accurate probabilistic analysis since it can accurately model complex engineering problems with

highly non-linear, non-monotonic and high dimensional design space.

Commonly used methods for probability distribution analysis to generate the Cumulative Distribution

Function (CDF) and Probability Density Function (PDF) are: FOSM, MCS, and MCS with transfer

function or meta-models. Similar to moment estimation, MCS usually requires a minimum of thousands of

runs for CDF and PDF generation. FOSM using AMV (Advanced Mean Value method) can produce CDF

or PDF results with 2-4 orders of magnitude fewer runs than MCS, and the FOSM method usually works

well with linear or almost linear problems. However, FOSM may fail to provide accurate results for any

problem with a noisy performance functions or multiple MPPs (Most Probable failure Points). In addition,

the method becomes less efficient when applied to the problems with multiple performance functions.

The Point Estimate Method (PEM), originally proposed by Rosenblueth [1], is a simple but powerful

technique for evaluating the moments and probability distribution information of performance functions. It

has been widely used in many geo-technical and civil engineering analyses [2-9]. Unlike most FOSM

techniques [10-13], PEM does not require the gradients of the performance functions with respect to the

input variables. Also, it does not use any search techniques. The distribution of performance function is

typically described in terms of a 4-parameter Beta or Lambda distribution. The non-calculus nature of the

algorithm makes it extremely robust to typical engineering simulations that may be noisy, discontinuous or

non-monotonic. Also, several performance functions can be dealt with simultaneously without significant

additional computational effort.

Many PEM methods described in the articles [1-9] used two points per variable. This requires 2n

(n is

the number of input variables) function runs. N. C. Lind [8] developed a face center point method that

requires 4n3

runs and a two-point approximation method that requires 4n2 runs. Furthermore, Milton Harr

and H. P. Hong attempt to reduce the number of runs to 2n [9]. Lind’s, Harr’s and Hong’s methods all

involve replacing the points at the 2n corners of the hypercube with 2n points at or near the intersections of

the circumscribing hypersurface with its principal axes. However, these approaches must move the

evaluation points farther away from the means as n increases since the radius of a hypersphere

circumscribing a unit hypercube of n dimensions is n ; for a bounded variable, the points may easily fall

outside the domain of definition of the variable. Seo and Kwak [14] developed more accurate method for

moment calculations using two points per variable, which requires 3n function runs. This approach becomes

unaffordable if the number of input variables is large (greater than 10).

In the previous paper by the authors [15], a variable-point PEM method proposed two unique and novel

improvements : 1) efficient and accurate moments estimations were obtained by applying a different

number of PEM points for each input variable and 2) a new robust procedure was described for finding the

four parameters of the Beta and Lambda distributions. This approach has been tested and applied to many

engineering applications. It has been demonstrated that the method is able to improve both the efficiency

and accuracy of the 3-point PEM approach given in Ref. [14]. However, when it applied to the problems

with large number of design variables (number of design variables > 100), the method still requires

hundreds of simulation runs which could become very time-consuming and impractical if the engineering

problems require hours or days for a single simulation run.

The objective of this paper is to further improve the efficiency of the 3-point PEM for solving

probabilistic problems with very high number of design variables (100X+). The new method is based upon

two fundamental concepts: 1) The Pareto principle and 2) The Central Limit Theorem of Statistics that

says, under common engineering conditions, a linear combination of random variables can be approximated

to first order by a normal distribution.

3

II. Overview of PEM Techniques

A. 2-point Rosenblueth and 3-point Seo-Kwak PEM Approaches

Many PEM methods, such as Rosenblueth and Seo-Kwak, used the same number of points for each input

variable. For example, the Rosenblueth method used 2 discrete points to represent a continuous distribution

for each variable. By solving the system equations which match the area and first three moments of each

variable,

(1)

the x- locations and probabilities at the two discrete points are computed as

12

2

1

2

2

2

1

1

)2

(1

1

215.0

,)2

(12

,)2

(12

pp

p

x

x

x

x

x

xx

x

x

xx

x

−=

+

−=

+−+=

+++=

υ

υ

συυ

µ

συυ

µ

(2)

where xµ , xσ and xυ are the mean, standard deviation and skewness of the variable x, respectively.

The Seo-Kwak method used three discrete points to represent a continuous distribution for each

variable. By solving the system equations which match the area and first four moments of each variable,

(3)

the x-locations and probabilities at the three points are calculated as

x

x

x

pxpx

pxpx

pxpx

pp

υ

σ

µ

=+

=+

=+

=+

1

3

21

3

1

1

2

21

2

1

1211

21 1

x

x

x

x

pxpxpx

pxpxpx

pxpxpx

pxpxpx

ppp

γ

υ

σ

µ

=++

=++

=++

=++

=++

3

4

31

4

21

4

1

3

3

31

3

21

3

1

3

2

31

2

21

2

1

331211

321 1

4

−++

−−+

=

2

2

2

2

1

3422

3422

xx

x

x

x

x

x

xx

x

x

x

x

x

x

x

υγσ

υσ

µ

µ

υγσ

υσ

µ

(4)

−−

−−−

−

−−

−−

−+−

=

))(34(2

3434

1

))(34(2

3434

22

22

2

2

22

22

3

2

1

xxxx

xxxxx

xx

xx

xxxx

xxxxx

p

p

p

υγυγ

υγυυγ

υγ

υγ

υγυγ

υγυυγ

where xγ is the kurtosis of the variable x. The detailed information on Seo-Kwak approach can be

found in Ref. [14].

B. Variable-point PEM

In order to reduce the number of runs, while at the same time achieving better accuracy, the variable-

point PEM approach [15] applied a different numbers of points for each input variable. The percent

contribution of each variable x to the total variation and the non-linearity of y with respect to x were taken

into account to determine how many points to use for each variable. In general, more points, for example, 3

or 4 points, are used if the variables cause high non-linearity and make significant contributions to y.

Otherwise, fewer points (1 or 2 points) are used if the variables are linear or close to linear and have low

contributions to y. Recently the minimum number of points has been set to 2 to obtain more accurate

standard deviations and higher moments. The percent contribution of the ith

variable to the total variation,

Pcti , is computed as

22

2

2

2

1

1

2

)(...)()(

)(

n

n

i

ii

x

y

x

y

x

y

x

y

Pct

σσσ

σ

∂

∂++

∂

∂+

∂

∂

∂

∂

= (5)

A Two-Point Adaptive Nonlinear Approximation (TANA) [16] is used to determine the non-linearity

index ir . At the pre-selected two points, 1X and 2X , the approximation is given as

∑∑=

−

=

−+−∂

∂+=

n

i

r

i

r

i

r

i

r

i

i

n

in

i

iiii xxxxr

x

x

XgXgXg

1

2

1,1,

1

1,

1 1

1

2 )(2

1)(

)()()(~ ε (6)

The non-linearity index ir is solved by matching the function value and gradients at 2X , that is,

5

nigg

rxxxxX

x

x

xX

i

r

i

r

i

r

i

i

ri

i

i

i

iii ,...,2,1,)()(

)()(

1

2,1,2,

11

1,

2,1 =−+= −− ε∂

∂

∂

∂ (7a)

∑∑==

−

−+−+=n

i

ri

i

ri

i

ri

i

ri

i

n

ii

ri

i

i xxxxx

xX

XX r

g

gg1

2

1,2,1,2,1

1

1,1

12)(

2

1)(

)(

)()( ε∂

∂

(7b)

(See Ref. [16] for more details on TANA)

The number of points for each variable is determined based on the non-linearity and percentage

contribution to Ys of each variable . If the number of the points is one, the point is located at the mean and

probability is 1.0, that is,

0.1== px xµ (8)

where xµ is the mean of variable x. If the number of the points is two, the Rosenblueth equation (2) is

used for the x-location and probability calculations. If the number of the points is three, the Seo-Kwak

equations (4) are applied to find the locations and probabilities of the three points. If the number of points

equals to four, the 4-point locations and probabilities are derived from the area and the first 7 moments

given below, that is,

74

7

43

7

31

7

21

7

1

64

6

43

6

31

6

21

6

1

54

5

43

5

31

5

21

5

1

44

4

43

4

31

4

21

4

1

34

3

43

3

31

3

21

3

1

24

2

43

2

31

2

21

2

1

144331211

4321 1

mpzpzpzpz

mpzpzpzpz

mpzpzpzpz

mpzpzpzpz

mpzpzpzpz

mpzpzpzpz

mpzpzpzpz

pppp

=+++

=+++

=+++

=+++

=+++

=+++

=+++

=+++

(9)

where 43,21 ,, zzzz and 43,21 ,, pppp are the point locations in the standardized space and

corresponding probabilities. 71 ,...,mm are the standardized moments

By taking into account of the fact that all the probabilities )4,3,2,1( =ipi are linear factors in the

above system of equations (9), the probabilities can be simply expressed as shown below

MZZZP TT 1)( −= (10)

where

6

=

4

3

2

1

p

p

p

p

P

=

=

7

6

5

4

3

2

1

7

4

7

3

7

2

7

1

6

4

6

3

6

2

6

1

5

4

5

3

5

2

5

1

4

4

4

3

4

2

4

1

3

4

3

3

3

2

3

1

2

4

2

3

2

2

2

1

4321

11111

m

m

m

m

m

m

m

M

zzzz

zzzz

zzzz

zzzz

zzzz

zzzz

zzzz

Z (11)

The errors for the area and moments can then be calculated as

PZMzzzzE ⋅−=),,( 4,321 (12)

Solving for the 4-point locations and probabilities from Eq. (9) can be expressed as the following

optimization problem,

4321

4,3214,321

..

),,(),,(

zzzztS

zzzzEzzzzEMinT

<<< (13)

The optimum search for the above problem is much more robust and accurate since the problem has

only 4 variables instead of 8 parameters. This can be applied for solving 4-point locations of any

distributions.

For the distributions like normal, lognormal, uniform and exponential, more robust and simpler

methods can be used to find point locations and probabilities. Instead of performing optimization searches

for point locations every time PEM is run, a pre-calculated table that stores the point locations for

standardized distributions can be generated. In order to obtain the pre-calculated table, the transformation

functions of the distributions must be used to convert the first 7 moments to standardized moments, and the

four point locations in the standardized space Z are computed. Based on ,,, 221 ZZZ and ,4Z the 4- point

locations in the original space X can be easily computed from the transformation functions. For example,

for a normal distribution, the transformation function is given as

x

xxxz

σ

µ−=)( (14)

The standardized 7 moments are given as

7

=

0

15

0

3

0

1

0

1

7

6

5

4

3

2

1

1

M

M

M

M

M

M

M

(15)

The four point locations in the standardized space Z and probabilities are computed as

=

99212.33441397

30950.74196315

48530.74196411-

17192.33441421-

4

3

2

1

Z

Z

Z

Z

(16a)

=

62270.04587588

4210.45412444

22670.45412384

32550.04587585

4

3

2

1

P

P

P

P

(16b)

For skewed distributions like weibull and extreme value, a one dimensional table of the point locations and

probabilities as a function of the shape parameter β can be generated by solving the optimization problem

given by Eq. (13).

The performance function values at all the combinations of the point locations for all the variables are

computed by running the simulation code. The total number of runs is

nNNNNofRun ...21= (17)

where iN is the number of points for the th

i variable. The first four moments of the performance

function Y, which is non-linear in general, are then calculated from the following equations,

2

2

4

1

11

4

2

3

2

3

1

11

3

2

1

11

2

11

1

/)),...,((...

/)),...,((...

)),...,((...

),...,(...

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

MMxxyppM

MMxxyppM

MxxyppM

xxyppM

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

i

N

i

ii

N

i

i

i

N

i

ii

N

i

i

i

N

i

ii

N

i

i

i

N

i

ii

N

i

i

−=

−=

−=

=

∑∑

∑∑

∑∑

∑∑

==

==

==

==

(18)

where ,,, 321 MMM and 4M are the mean, variance, skewness and kurtosis of the performance function.

8

SEO-KWAK’s 3-point PEM used the Pearson system to solve the above system equations to find the four

parameters of the Beta distributions. The variable-point PEM found the analytical solution of the four

parameters of Beta distribution and performed optimization search to find the four parameters of Lambda

distribution. The flow-chart of the variable-point PEM approach is shown in Figure 1. (See more details on

the variable-point PEM in [15])

Analytical equations and numerical methods

to ensure robustnessand accuracy

3. Compute four moments of

responses Ys using

,

a b

4. Finding 4 parametersof Beta or Lambda distribution based on

four moments of Ys

. λ1 λ2

Key Steps

1. Determine how many pointsto use for each x

1) Percent contribution of x to the total variation y

2) Non-linearity of y with respect to x

xp1_1

x = location

p = probability

.

(P1) (g1k) + (P2) (Z2K) + (P3) (g3K) + (P4) (g4K) = MK

Mean (k=1), Std dev (K=2), Skewness (K=3), Kurtosis (k=4)

2. Run simulation code to calculate function values g and

corresponding probabilities

gi is Y value at (x1_i,x2)

pi is product of p1_i and p2

Engineering

Simulation

X1_1, x1_2X1_3, x1_4

X2

g1, p1g2, p2g3, p3g4, p4

For example, x1 uses 4 points andx2 uses 1 point

xp2

x = location

p = probability

.

x1

x2

p1_2 p1_3 p1_4

Beta or Lambda

distribution

Figure 1. Flow-Chart of the Variable-point PEM

III. Proposed PEM Approach

Although the variable-point PEM approach is able to improve the efficiency and accuracy of the

existing 3-point PEM approach given in Ref. [14]. However, when it applied to the problems with large

number of design variables (number of design variables > 10), the method still requires hundreds of or

thousands of simulation runs that could become very time-consuming and impractical if the engineering

problems require hours or days for a single simulation run. The objective of this paper is to further

improve the efficiency of the modified PEM for solving probabilistic problems with high number of design

variables ( > 10 or 100X+). The new method is based upon two fundamental concepts: 1) The Pareto

principle and 2) The Central Limit Theorem of Statistics that says, under common engineering conditions, a

linear combination of random variables can be approximated to first order by a normal distribution.

The detailed steps of the proposed approach are:

9

1. Perform a sensitivity analysis centered on the mean values of all the X factors.

2. Based on the Pareto Principle concept, it can be expected that approximately 80% of the

variance will be caused by 20% of the design factors. Therefore, 20% of the design factors

will likely determine the overall shape of the probability distribution of the output variable

3. However, the variation effects of the remaining 80% minor design factors cannot be ignored.

These effects can be approximated by running the simulation at the nominal values of the

minor design factors and rolling the total variation contribution up into a single equivalent

normal distribution that has a mean of zero and a standard deviation that is determined from

the sensitivity analysis..

4. Finally, a reduced PEM analysis is run based upon the significant design factors (20%) with a

single extra normal distribution added on (representing the other 80%).

IV. Numerical Examples

The efficiency and accuracy of the proposed PEM approach are demonstrated with the following three

benchmark problems. The results are compared with Monte Carlo, variable-point and 3-point PEM

approaches.

Example 1

The performance function is

7577.8),( 2

2

121 −+= xxxxg (19)

where 1x and 2x are the random variables with normal distributions (mean=10, standard deviation =

3). The sensitivity analysis information for the output variable g with respect to the two variables (see

Table 1) shows that 99.75% of the variation is contributed by 1x . The remaining variable 2x contributes

only 0.25% to the total variation of the output variable. The significant variation effects of the minor

factor is converted into a normal distribution that has a mean of zero but a non-zero standard deviation of

0.04994 (square root of 0.002494). The modified PEM method can now be run with design factor 1x that

has its original mean and standard deviation, and one extra variable x with a normal distribution N(0,

0.04994) which is added on. That is, the output random, g( 1x , 2x ), which is function of the 1major input

random variable at the mean of 2x , ie., 2xµ , is modified by the x normal random variable as shown by

the equation

xxgxg x += ),()(ˆ211 µ

Table 2 shows that the four moments obtained from the new PEM approach are almost exactly matched

the Monte Carlo results. Figure 2 shows the CDF comparisons from the Monte Carlo (pink), the new PEM

(blue), variable-point PEM (green) and 3-point PEM (black). The new PEM and variable-point PEM

methods match the Monte Carlo’s (pink) exactly, however, 3-point Seo-Kwak PEM is off from the correct

solution at the tail. Since this example only has two design variables, the numbers of function evaluations

for both variable-point and new PEM methods are the same and actually higher than 3-point PEM due to

extra runs required by sensitivity analysis.

10

Table1. Sensitivity Information of Example 1

Table2. Four Moments Comparison of Example 1

Figure 2. CDF Comparison of Example 1

Effect of Factor Variation on Response Variation

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Variable Var Contrib Std Dev %Contribution

X1 3600.0360 60.0003 99.7506%

X2 9.0000 3.0000 0.2494%

-------------

Total: 100%

Method 3-point Seo-

Kwak PEM

Variable-point

PEM

New PEM Monte

Carlo

Mean 110.2423 110.2423 110.2423 110.1703

Std Dev 61.40847 61.40846 61.33516 61.6242

Skewness 0.845778 0.864667 0.86777 0.86804

Kurtosis 3.2433 3.99533 4.00012 4.01879

No. of Runs 9 13 13 100,000

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 100.00 200.00 300.00 400.00 500.00 600.00

g

Pro

bab

ilit

y

New PEM Variable-point PEM Monte Carlo 3-point PEM

11

Example 2- 10 Variables I-Beam Problem

In the example of an I-Beam CDF/PDF analysis shown in Figure 3, 10 input variables are taken into

account (see Figure 3 for detailed inputs, in fact only 8 variables are active since only 1 output Y is

considered), and the performance function is computed as

SY −= maxσ

where S is the strength, which is a random variable, and maxσ is the maximum stress, which is given as

IL

daLap

⋅⋅

⋅−⋅⋅=

2

)(maxσ

where

12

)2)(( 33

fwff tdtbdbI

−−−=

Figure 3. 10-Variable I-Beam

The sensitivity analysis information for the output variable Y with respect to the ten variables (see

Table 1) shows that 51.59% of the variation is contributed by tf, 26.11% is contributed by L and 7.85% is

contributed by a. The remaining variables d, tw, bf, E, rho, P and S contribute only 14.455% to the total

variation of the output variable Y. The significant variation effects of the minor factors is rolled into a

normal distribution that has a mean of zero but a non-zero standard deviation of 0.380197 (square root of

0.144549797). The modified PEM method can now be run with design factors tf, L and a that have their

original means and standard deviations, and one extra variable x with a normal distribution N(0, 0.380197)

which is added on. That is, the output random, Y, which is function of the 3 major input random variables

at the means of d, tw, bf, E, rho, P and S is modified by the x normal random variable as shown by the

equation

aP

L

tw

bf

d

tf

E, rho

Variable Type Mean Std

P Normal 6070 200

L Normal 120 6

a Normal 72 6

S Normal 170E3 4760

E Normal 30E6 3E6

rho Normal 0.28 0.028

d Normal 2.3 1/24

bf Normal 2.3 1/24

tw Normal 0.16 1/24

tf Normal 0.26 1/24

aP

L

tw

bf

d

tf

E, rho

Variable Type Mean Std

P Normal 6070 200

L Normal 120 6

a Normal 72 6

S Normal 170E3 4760

E Normal 30E6 3E6

rho Normal 0.28 0.028

d Normal 2.3 1/24

bf Normal 2.3 1/24

tw Normal 0.16 1/24

tf Normal 0.26 1/24

12

xSY +−= maxˆ σ

Table 4 shows the four moments obtained from the new PEM approach are almost exactly matched the

Monte Carlo results. The new PEM takes 33 runs, however, the variable-point PEM takes 384 runs and 3-

point PEM will need 59049 runs. In [15], the number of runs required by the variable-point PEM was 213

if the minimum number of points for the variables with fewer contributions to the outputs was set to 1. In

order to obtain more accurate standard deviations and higher order moments of the outputs, the minimum

number of points used in the variable-point PEM is set to 2 in this paper. Figure 4 shows the CDF

comparisons from the Monte Carlo (pink), the new PEM (blue) and variable-point PEM (green). The new

PEM and variable-point PEM methods match the Monte Carlo’s (pink) exactly, however the new PEM

only takes 33 runs which reduces the number of runs significantly.

Table3. Sensitivity Information of I-Beam

Table4. Four Moments of I-Beam

Effect of Factor Variation on Response Variation

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Variable Var Contrib Std Dev %Contribution

d 13259100 3641.304 2.66%

tw 4340278 2083.333 0.87%

bf 6617570 2572.464 1.33%

tf 256821200 16025.64 51.59%

E 0 0 0.00%

rho 0 0 0.00%

P 25082440 5008.237 5.04%

L 129960000 11400 26.11%

a 39062500 6250 7.85%

S 22657600 4760 4.55%

-------------

Total: 100%

Method 3-point Seo-

Kwak PEM

Variable-point

PEM

New PEM Monte

Carlo

Mean - -17966.22 -17966.22 -17967.12

Std Dev - 22110.19 22110.97 22111.07

Skewness - 0.55 0.55 0.55

Kurtosis - 3.15 3.15 3.14

No. of Runs 59049 384 33 100000

13

Figure 2. CDF Comparison of I-Beam

Example 3- 18-Variable Torsion Vibration Problem

The new PEM approach is also demonstrated with a conceptual 18-variables torsion vibration problem

shown in Figure 5. The mean values of the eighteen design variables are given in Table 5. The three output

variables include the weight, low and high natural frequencies (see Table 6).

Figure 5. 18-Variables Torsion Vibration Problem

1 2 3

1

2

1 2 3

1

2

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-80000.

00

-60000.

00

-40000.

00

-20000.

00

0.00 20000.

00

40000.

00

60000.

00

80000.

00

New PEM Variable-point PEM Monte Carlo

14

Table5. 18 Design Variables

Table 6. Three Output Variables

... weight density (lb/cubic inch)λ3 0.10=

... modulus or rigidity (lb/sq inch)G3

3.9 106

⋅=

... length (inches)L3

8=

... diameter (inches)d3

2.25=Shaft 3

(Aluminum)



6.2 106

⋅=


12=


1.825=Shaft 2

(Titanium)



11.7 106

⋅=


10=


2=Shaft 1

(Steel)

Design Parameters (18 total)

Disk 1

(Steel)D

112= ...diameter (inches)

t1

3= ...thickness (inches)

ρ 1 0.28= ... weight density (lb/cubic inch)

Disk2

(Aluminum)D


t2





3.9 106

⋅=


8=


2.25=Shaft 3

(Aluminum)



6.2 106

⋅=


12=


1.825=Shaft 2

(Titanium)



11.7 106

⋅=


10=


2=Shaft 1

(Steel)

Design Parameters (18 total)

Disk 1

(Steel)D


t1



Disk2

(Aluminum)D


t2



Kiπ Gi⋅ di⋅

32 Li⋅= Mj

ρ j

gπ⋅ tj⋅

Dj

4⋅=

bK1 K2+

J1

K2 K3+

J2+

−=

cK1 K2⋅ K2 K3⋅+ K3 K1⋅+

J1 J2⋅=

22

1

23

1

3

22

4

2

22

4

1

)2

()2

(

2

2

j

j

j

ji

i

i

i

a

acbb

a

acbb

Dt

dLY

Y

Y

πρπλ

π

π

∑∑==

−+−

−−−

+=

=

=

where

a = 1

Jj1

2Mj⋅

Dj

2

2

⋅=

- Low natural frequency

- High natural frequency

- Weight

(i=1,2,3)

(j=1,2)

15

Table 7. Sensitivity Information Of Low Frequency w.r.t 18 Design Factors

The sensitivity analysis information for the low frequency with respect to the eighteen design variables

(see Table 7) shows that 77.8% of the variation is contributed by D1, D2, T2 and R2. The remaining 14

variables contribute only 22.2% to the total variation of the low frequency. The significant variation

effects of the 14 minor factors are rolled up into a single normal distribution that has a mean of zero but a

non-zero standard deviation. Working through the sensitivity table numbers indicates that a standard

deviation of 6.921373 should be used for that equivalent normal distribution variable. The modified PEM

method can now be run with only the 4 design factors (D1, D2, T2, R2) that have their original means and

standard deviations, and one extra variable x with a normal distribution N(0, 6.921) which is added on.

That is, the output random, y1, which is function of the 4 major input random variables, is modified by the

x normal random variable as shown by the equation

xyy += 11ˆ

The CDF results generated from the new PEM approach (Blue) is compared with the one calculated

from Monte Carlo with 100000 simulations (Red).

EFFECT OF FACTOR VARIATION ON RESPONSE VARIATION

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Factor Partial Deriv St Dev Var Contrib Pct of Total

-------- ------------- ------------ ------------ ------------

DS1 1.205000e+01 2.000000e-01 5.808100e+00 2.6956

L1 -2.530000e+00 1.000000e+00 6.400900e+00 2.9707

G1 2.059829e-06 1.170000e+06 5.808100e+00 2.6956

Lmd1 -1.442288e-11 2.800000e-02 1.630873e-25 0.0000

DS2 1.350000e+00 2.000000e-01 7.290000e-02 0.0338

L2 -2.166667e-01 1.200000e+00 6.760000e-02 0.0314

G2 3.870968e-07 6.200000e+05 5.760000e-02 0.0267

Lmd2 -3.707967e-12 1.600000e-02 3.519749e-27 0.0000

DS3 1.177778e+01 2.250000e-01 7.022500e+00 3.2592

L3 -3.225000e+00 8.000000e-01 6.656400e+00 3.0893

G3 6.794872e-06 3.900000e+05 7.022500e+00 3.2592

Lmd3 -1.791619e-11 1.000000e-02 3.209900e-26 0.0000

D1 -6.000000e+00 1.200000e+00 5.184000e+01 24.0596

T1 -7.066667e+00 3.000000e-01 4.494400e+00 2.0859

R1 -7.571429e+01 2.800000e-02 4.494400e+00 2.0859

D2 -7.725806e+00 1.240000e+00 9.177640e+01 42.5946

T2 -8.650000e+00 4.000000e-01 1.197160e+01 5.5562

R2 -3.460000e+02 1.000000e-02 1.197160e+01 5.5562

------------ -------

Total Var = 2.154650e+02 100.0000

Std Dev = 1.467873e+01

16

Figure 6. CDF Curves from the MCS and New PEM Methods

Table 8 shows the number of simulation runs required by MCS, 3-level PEM, variable-level PEM and

the new PEM approaches. It demonstrates the new PEM significantly reduces the number of simulation

runs for the 18-variable problems. The 78 runs of the new method includes 19 runs for sensitivity analysis

and 59 runs for performing PEM with 4 major factors with 1 extra normal distribution added on.

Table 8. Number of Simulations Required by the

MCS, 3-level PEM, Variable-level PEM and new PEM Approaches

V. Summary

The new PEM provides nearly identical results as Monte Carlo simulation with a large number of runs,

while the number of runs is significantly fewer than the number of Monte Carlo runs, as well as the

previous variable-point PEM. Benchmark examples demonstrate that the new PEM significantly improves

the efficiency of the existing 3-point and variable-point PEM approaches with high accuracy of the four

moments and CDF/PDF results for highly non-linear problems.

In the future work, non-normal distributions of input variables will be taken into account. The

transformation from non-normal to normal distributions need to be considered.

Y1 (Low Freq)

Probability

Method MCS3-Level PEM

(Gen 1)

Variable-Level PEM

(Gen 2)

New PEM

(Gen 3)

No. of Runs 100,000 387,420,489 393,216 78

17

VI. References

1. Emilio Rosenblueth, “Point Estimates for Probability Moments”, Applied Mathematics Modeling,

Vol. 5, October 1981, pp. 329-335, 1981.

2. Miltone Harr, “Reliability Based Design in Civil Engineering”, McGraw-Hill Book Company, New

York, 1987.

3. John T. Christian and Gregory B. Baecher, “The Point-Estimate Method With Large Numbers of

Variables”, International Journal for Numerical and Analytical Methods in Geo-mechanics, 26,

pp1515-1529, 2002.

4. Chengqing Wu, Hong Hao and Yingxin Zhou, “Distinctive and Fuzzy Failure Probability Analysis of

An Anisotropic Rock Mass to Explosion Load”, International Journal for Numerical Methods In

Engineering, 56, pp767-786, 2003.

5. Geethanjali Panchalingam, “Modeling of Many Correlated and Skewed Random Variables”, Appl.

Math. Modeling, Vol. 18, Nov. pp. 635-640, 1994.

6. K.S.Li, “Point Estimate Method for Calculating Statistical Moments, Journal of Engineering

Mechanics, Vol. 118, No. 7, July 1992.

7. Nagaraman Sivakugan and Ali Al-Harthy, “ Probabilistic Solutions to Geotechnical Problems”,

Probabilistic Mechanics & Structural Reliability: Proceeding of the Seventh Specialty Conference,

Worcester Polytechnic Institute, Worcester, Massachusetts, Aug. 7-9, 1996.

8. N. C. Lind, “Modeling of Uncertainty In Discrete Dynamical Systems”, Appl. Math. Modeling, Vol. 7,

June, 1983.

9. H. P. Hong, “An Efficient Point Estimate Method for Probabilistic Analysis”, Reliability Engineering

and System Safety, 1998, 59, pp. 261-267.

10. H. O. Madsen, S. Krenk, N. C. Lind, “Methods of Structural Safety”, Prentice-Hall International

Series in Civil Engineering and Engineering Mechanics, 1986, pp. 44-101.

11. R. E. Melchers , “Structural Reliability Analysis and Prediction”, Ellis Horwood Limited Publishers,

Halsted Press, a Division of John Wiley & Sons, 1987, pp. 104-141.

12. Y. T. Wu and O. H. Burnside , “Efficient Probabilistic Structural Analysis Using An Advanced Mean

Value Method”, Proceeding of 5th

ASCE speciality Conference – Probabilistic Methods in Civil

Engineering, ASCE, New York, pp. 492-495.

13. Liping Wang and Ramana Grandhi, “Efficient Safety Index Calculation for Structural Reliability

Analysis”, Journal of Computers and Structures, Vol. 52, Nov.1, 1994, pp. 103-111.

14. Hyun Seok Seo and Byung Man Kwak, “Efficient Statistical Tolerance Analysis for General

Distributions Using Three-Point Information”, Int. J Prod. Res. 2002, Vol. 40, No. 4, pp931-944.

15. Liping Wang, Don Beeson and Gene Wiggs, “Efficient and Accurate Point Estimate Method for

Moments and Probability Distribution Estimation”, 10th AIAA/ISSMO Symposium on

Multidisciplinary Analysis and Optimization”, August 30 – Sept 1, 2004, Albany, New York.

18

16. Liping Wang, and Ramana Grandhi, “Improved Two-point Function Approximation for Design

Optimization”, AIAA Journal, Vol. 33, No. 9, 1995, pp. 1720-1727

17. Liping Wang, Don Beeson, Gene Wiggs, and Mahidhar Rayasam, “A Comparison Of Meta-modeling

Methods Using Practical Industry Requirements”, AIAA 2006-1811, 47th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 1 - 4 May

2006, Newport, Rhode Island.

18. Liping Wang, Don Beeson and Gene Wiggs , “A Robust and Efficient Probabilistic Approach for

Challenging Industrial Applications with High-dimensional and Non-monotonic Design Spaces”,

AIAA 2006-7014, 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 6 - 8

September 2006, Portsmouth, Virginia

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