A CORRECTED MVFO APPROXIMATION OF THE EXPECTED VALUE
IN STRUCTURAL ANALYSIS WITH PARAMETER UNCERTAINTY∗
Hemanthkumar Brindavan† and Dr. Madara Ogot‡
Dept. of Mechanical and Aerospace Engg.Rutgers, The State University of New Jersey
Piscataway, NJ 08854
ABSTRACT
The rapid developments in the computing indus-try have made it possible to account for uncertaintyin structural analysis through faster and cheapercomputational simulations, with a reduction in phys-ical prototyping. Traditionally, incorporation ofuncertainty in structural analysis models has beenachieved mainly through either Monte Carlo simu-lations (MCS) or Mean Value (MV) approximations.The former, while relatively accurate, can be com-putationally prohibitive. MV approximations on theother hand, trade accuracy for higher computationalspeed. The proposed Corrected Mean Value (CMV)method exploits the correlation between the param-eter coefficients of variation and the fractional errorincurred using the MV approach to determine ap-propriate correction factors. Multiplication of theMV approximation by the correction term signifi-cantly reduces the error, without sacrificing compu-tational speed. The approach therefore presents amore accurate approximation of the expected valueof a function. In this paper correction factors forseveral functions are derived and discussed. Sev-eral structural analysis examples are presented todemonstrate the efficacy of the approach.
INTRODUCTION
Accounting for parameter uncertainty in struc-tural analysis and design has been the focus of re-
†Graduate Student‡Associate Professor, Author of Correspondence
search for the past four decades.1−4 Most of thesemethods can be grouped into Monte Carlo-based andMean Value-based approximations. Monte Carlosimulation based methods provide high accuracy andgreat flexibility in the types of distributions they canhandle, but at great computational expense. De-pending on the non-linearity of the problem thou-sands or tens of thousands of iterations need to beperformed to evaluate the expected value of eachresponse. Astill et al.5 incorporated the MonteCarlo simulation into a finite element method. Themethod has also been incorporated into a commer-cial FEM package DAKOTA.6 Liang et al.7 used theMonte Carlo method to calculate the kinematic errorin spatial linkages with parameter uncertainty.
Methods based on Taylor series expansions aboutthe mean, collectively referred to as Mean Valueapproximations, have emerged as an alternative toMonte Carlo simulations. They provide an approx-imation of the first two moments (mean value andvariance) of the system response functions. MeanValue First Order (MVFO) approximations use afirst order Taylor series expansion to estimate themean value and the variance. The truncation of theTaylor series facilitates the simple evaluation of themean value and variance response functions at rela-tively low computational expense. Researchers whohave used this approach include.8−11
For highly non-linear response functions, however,the MVFO methods have been shown not to be suf-ficiently accurate.12−15 In response, research effortshave focussed on improving the MVFO approxima-tion, for example Wu et al.13 with additional termsor working with mean value second order (MVSO)approximations. The latter provide higher accuracy,
1American Institute of Aeronautics and Astronautics
43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con22-25 April 2002, Denver, Colorado
but often difficulty in evaluating the second-orderderivatives. The Corrected Mean Value (CMV) ap-proximation, presented here, seeks to improve theaccuracy of the MVFO approximation of the re-sponse function expected value while retaining itssimplicity and computational speed.
CORRECTED MEAN VALUE
Let the mean value fractional error, e, be definedas
e =µ− µmv
µ= 1− µmv
µ(1)
where µ and µmv are the exact and the MVFO ap-proximation of the expected value, respectively. Re-arranging terms, Equation 1 becomes
µ = µmv
(1
1− e
)= µmvC (2)
where C is defined as the correction factor and is afunction of the mean value fractional error, e, i.e.,
C =1
1− e(3)
The Corrected Mean Value (CMV) is therefore de-fined as
µcmv = µmvC (4)
The CMV approach was first proposed by the au-thors in.16 In that paper, polynomial expressions de-rived from performing regressions on data collectedfrom thousands of numerical experiments were usedto model the mean value fractional errors. The frac-tional errors for several common mathematical op-erations whose expected values cannot be evaluatedfrom standard statistical identities - e.g., 1/x and xn
- were numerically determined and plotted againstthe corresponding input parameter coefficients ofvariation (where COVx = σx/µx). The aim was totest the hypothesis that:
“ For these mathematical operation thereexists a direct correlation between thestochastic parameter’s coefficient of varia-tion and the mean value fractional errors.”
A correlation would mean that if the COVx ofany parameter was known the corresponding meanvalue fractional error, and therefore the appropri-ate correction factor, C, for a particular mathemat-ical operation could be calculated. The COV is a
non-dimensional measure of the degree of parameteruncertainty. Through several examples the authorswere able to show that indeed a correlation existed,and that the CMV approximations significantly re-duced the error in estimating the mean as comparedto the traditional MVFO approximation.
The present study develops the analytical founda-tion and further generalizes the concepts presentedin.16
ANALYTICAL FORMULATION
In this section the correction factors for the xn
and 1/x mathematical operations will be derived.Presently, a uniform distribution is assumed for allparameters with uncertainty. Future work will in-vestigate other probability distributions and mathe-matical operations.
The goal is to be able to define the correction fac-tors as a function of COVx.
xn, for n > 0
Consider the function
y = xn (5)
where the expected value and variance of x, (µx, vx)are known. Let the polynomial correction factor,C
(n)x , be defined as
C(n)x =
1
1− e(n)x
(6)
where the polynomial mean value fractional error isdefined as (from Equation 1)
e(n)x = 1− µmv y
µy(7)
The superscript refers to the mathematical opera-tion and the subscript to the stochastic variable be-ing operated on.
Assuming the uncertainty in x is uniformly dis-tributed, the expected value of y can be calculatedby direct integration from
E(y) = µy =(
1b− a
) ∫ b
a
xndx. (8)
where a and b are the lower and upper bounds of x,respectively. After integration, Equation 8 becomes
2American Institute of Aeronautics and Astronautics
µy =b(n+1) − a(n+1)
(b− a) (n + 1)(9)
The MVFO approximation of the expected valueof y is
E(y) ' µmv y =(
a + b
2
)n
(10)
Substitution of Equations 9 and 10 into Equation 7gives
enx = 1−
(a + b
2
)n ((b− a) (n + 1)b(n+1) − a(n+1)
)(11)
As x follows a uniform distribution the expectedvalue, µx, and standard deviation, σx, can be definedas 17
µx =a + b
2(12)
σx =b− a√
12(13)
Substituting Equations 12 and 13 into the coefficientof variation definition, COVx = µx/σx,
COVx =2(b− a)
(a + b)√
12(14)
Rearranging terms
b = aKx (15)
where
Kx =2/√
12 + COVx
2/√
12− COVx
(16)
Substituting Equations 15 into Equation 11 yields
e(n)x = 1− (n + 1) (Kx − 1) (Kx + 1)n(
K(n+1)x − 1
)2n
(17)
Therefore, by knowing the COVx, the correspondingpolynomial mean value fractional error (n > 0) canbe calculated. A plot of the variation of en
x withCOVx and the polynomial order, n, is presented inFigure 1. The correction factor for the polynomialMV approximation of xn, Cn
x , is
C(n)x =
1
1− e(n)x
(18)
Figure 1: Representative illustration of the varia-tion of the polynomial mean value approximationfractional error with change in the COV and thepolynomial order, n.
where e(n)x is defined in Equation 17.
A couple of observations can be made. For poly-nomial functions (n > 0), the MVFO approximationis known to underestimate the E(xn). As a resultand with reference to Equation 7, the polynomialmean value fractional error has an upper bound of1. Further, close examination of Equation 16 showsthat Kx and therefore, e
(n)x becomes undefined (ap-
proaches infinity) as COVx approaches 2/√
12 or0.578. This directly supports the results from thenumerical experiments in 16 where the one-to-onecorrelation between the COV and the polynomialmean value fractional error begun to break down atCOVx > 0.5. This would suggest that the poly-nomial correction factor for uniform distributions islimited to parameters with COVx < 0.578. In en-gineering practice, however, COV values typicallyrange from 0.05-0.15. Larger values suggest a lackof understanding about the system being designedor analyzed.17
Reciprocal, 1/x
Consider the following function:
p =1x
(19)
where x maintains the same definition and charac-teristics as the previous section. The expected valueof p, E(p) is therefore
3American Institute of Aeronautics and Astronautics
E(p) = µp =(
1b− a
) ∫ b
a
1x
dx (20)
After integrating,
µp =ln(b)− ln(a)
b− a(21)
The MV approximation of the expected value is
E(p) ' µmv p =1µx
(22)
or from Equation 12
µmv p =2
a + b(23)
The reciprocal mean value fractional error, e(−1)x , is
e(−1)x = 1− µmv p
µp(24)
Substitution of Equations 21 and 23 into Equa-tion 24 yields
e(−1)x = 1− 2(b− a)
(b + a)(ln(b)− ln(a))(25)
Substituting Equation 15 into Equation 25 and re-arranging terms gives
e(−1)x = 1− 2(Kx − 1)
(Kx + 1)(ln(Kx))(26)
the reciprocal mean value fractional error. The cor-rection factor for the reciprocal MV approximation,C
(−1)x , is therefore
C(−1)x =
1
1− e(−1)x
(27)
As the C(−1)x is a function of Kx, the same bounds
on COVx as previously discussed applies.
xm, for 0 > m
Consider the function
q = xm m < 0 (28)
this is equivalent to
q =1xn
n > 0, n = −m (29)
if let y = xn then the problem becomes findingE(xn) and E(1/y) both of which had been previ-ously addressed. From Equations 4, 18 and 27, thecorrected mean value approximation of the expectedvalue of q, E(q) is
E(q) ' µcmv q =1
yC(n)x
C(−1)y (30)
In Equation 30, the only known statistical param-eters are µx and σx hence C
(n)x can be evaluated. To
calculate C(−1)y , however, µy and σy are need. µy can
be evaluated as previously described. To calculateσy consider the statistical identity for the calculationof variance,
vx = E(x2
)− (E(x))2 (31)
therefore
vy = vxn = E(x2n
)− (E(xn))2 (32)
Employing the polynomial correction factor (Equa-tion 18), Equation 32 becomes
vcmv y ' (µx)2nC(2n)
x −(µn
xC(n)x
)2
(33)
One can now obtain a corrected mean value approx-imation of σy, and therefore evaluate C
(−1)y .
The need to calculate the expected value and vari-ance for intermediate terms, such as y, make it im-possible to use direct integration to calculate the cor-rection factors as the bounds on y are unknown. Itthus becomes crucial to define the correction fac-tors in terms of parameters such as COV that canbe readily calculated. Further, for problems withintermediate terms - most problems encountered instructural analysis - there is a slight reduction in theaccuracy of the calculated correction factors. This isbecause the fractional error expressions assume thestochastic parameters are characterized by a uniformdistribution. Intermediate terms such as y, however,may not strictly follow a uniform distribution.
The efficacy of the present approach is demon-strated via a couple of examples presented below.
EXAMPLES
Cantilever Beam
Consider a cantilever beam subjected to a load P atthe free end (see Figure 2). The beam parameters -
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Figure 2: Cantilever beam example
Table 1: Mean values for parameters used in can-tilever beam example. All parameters are assumedto vary ±15% about the mean, corresponding to aCOV = 0.087
P (KN) L (m) w (m) h (m) E (GPa)10 10 0.2 0.4 200
length (L), height (h), width (w) and Young’s Mod-ulus (E) - are assumed to have uncertainty charac-terized by a uniform distribution. Their values aregiven in Table 1. The problem is to calculate themean deflection of the beam along its length.
A corrected mean value approximation of themean response involves three basic steps:Step I: Develop deterministic analysis equa-tions The deterministic beam deflection equation is
ν(x) =P
6EI
(2L3 − 3L2x + x3
)(34)
where x is distance along the length of the beammeasured from the free end and I is the area momentof inertia of the beam. Realizing that I = wh3/12,substituting into Equation 34 yields
ν(x) =2P
Ewh3
(2L3 − 3L2x + x3
)(35)
Step II: Convert deterministic equations toMVFO approximation expressions Employingthe traditional MVFO method the expected valuefor the deflection, µν(x), can be approximated from
µν(x) ' µmv ν(x) =2µP
(2µ3
L − 3µ2Lx + x3
)µEµwµh3
(36)
Step III: Correct the MVFO approximationfor all relevant mathematical operations oninitial and intermediate stochastic parame-ters Let Equation 36 be written as
µmv ν(x) =N
D(37)
whereN = 2µP
(2µ3
L − 3µ2Lx + x3
)(38)
D = µEµwµh3 (39)
Then the CMV approximation of the expected valueof the beam deflection is
µν(x) ' µcmv ν(x) = µcmv N
(1
µcmv D
)C
(−1)D
(40)where
µcmv N = 2µp
(2µ3
LC(3)L − 3µ2
LC(2)L x + x3
)(41)
µcmv D = µEµwµ3hC
(3)h (42)
To evaluate C(−1)D , however, σD is required. The
variance of D can be approximated by
vcmv D = E(D2
)− E(D)2 (43)
or
vcmv D = µ2EC2
Eµ2wC2
wµ6hC6
h − (µcmv D)2 (44)
Equations 42 and 44 can therefore be used to solvefor C
(−1)D .
The problem above was solved using the MonteCarlo simulation (represents the correct answer), thetraditional MVFO and the CMVFO methods. Theresults for the mean displacement, µν(x), generatedby all three methods, as a function of location on thebeam, x, are displayed in 3. Figure 4 displays thepercent error between the MVFO and the CMVFOmethods with the Monte Carlo simulation results.The maximum error obtained from the MVFO is7.9% at x = L. The CMVFO method on the otherhand has a maximum error of 1.2% at the same lo-cation, representing a 85% reduction in error.
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Figure 3: Comparison of mean beam deflection, µν
calculated by the Monte Carlo, Mean Value FirstOrder and Corrected Mean Value First Order Meth-ods. x is the position along the beam as measuredfrom the free end.
Figure 4: Comparison of the percent error betweenthe MVFO and the CMVFO methods with theMonte Carlo simulation results. x is the positionalong the beam as measured from the free end.
Figure 5: Circular cross-section with two circularholes
Area Moment of Inertia Example
Consider the cross-sectional area illustrated in Fig-ure 5. The outer radius, R, has a mean value, µR,of 250mm. The inner circles both have the same ra-dius, r, with a mean value, µr, of 90mm. Both pa-rameters are assumed to vary uniformly ±15% abouttheir mean values, corresponding to a COV = 0.087.The objective is to find the area moment of inertiaabout the x-axis, Ix. As in the previous example,the problem will be solved in three steps.
Step I: Develop deterministic analysis equa-tion
The deterministic area moment of inertia equationfor the shaded area is
Ixx = Iout xx − 2Iin xx (45)
where Iout xx and Iin xx are the area moment of in-ertia for the outer and the inner circles about thex-axis, respectively. Now,
Iout xx =πR4
4(46)
Iin xx =πr4
4+
πr2R2
4(47)
Substituting Equations 46 and 47 into Equation 45yields
Ixx =πR4
4− π
2(r4 + r2R2
)(48)
6American Institute of Aeronautics and Astronautics
Table 2: Comparison of Area Moment InertiaMCS MVFO CMV
µIxx(mm4 x109) 2.31 2.17 2.292
Percent Error - 6.1% 0.78%
Step II: Convert deterministic equations toMVFO approximation expressions Employingthe traditional MVFO the expected value for thearea moment of inertia can be approximated as
µIxx ' µmv Ixx =πµ4
R
4− π
2(µ4
r + µ2rµ
2R
)(49)
Step III : Correct the MVFO approximationfor all relevant mathematical operations oninitial and intermediate stochastic parame-ters Adding correction factors to the appropriateterms in the MVFO equation given in Equation 49,the corrected mean value approximation becomes
Icmv xx =πµ4
RC(4)R
4− π
2
(µ4
rC(4)r + µ2
rC(2)r µ2
RC(2)R
)(50)
The area moment of inertia was then calculatedusing the Monte Carlo simulation, MVFO and CMVmethods. The results are presented in Table 2. Fromthe table one can see that the CMV has reduces theerror from the MVFO approximation by 87% with anegligible increase in computational expense.
CONCLUDING REMARKS
Directly accounting for uncertainty in analysismodels has recently begun to receive more attentionfrom researchers. There is always a constant needto strike a balance between accuracy of the resultsand the computational cost. The corrected meanvalue method presented in this paper aims to pro-vide increased accuracy over the mean value first or-der approximations for highly non-linear problems,yet maintain the latter’s low computational require-ments. The results presented here represent the be-ginning of research in this direction. Future workwill consider other mathematical functions and dis-tributions.
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