Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 325025 13 pageshttpdxdoiorg1011552013325025

Research ArticleGeneralized Neumann Expansion and Its Application inStochastic Finite Element Methods

Xiangyu Wang1 Song Cen123 and Chenfeng Li4

1 Department of Engineering Mechanics School of Aerospace Tsinghua University Beijing 100084 China2High Performance Computing Center School of Aerospace Tsinghua University Beijing 100084 China3 Key Laboratory of Applied Mechanics School of Aerospace Tsinghua University Beijing 100084 China4College of Engineering Swansea University Singleton Park Swansea SA2 8PP UK

Correspondence should be addressed to Song Cen censongtsinghuaeducn

Received 18 July 2013 Accepted 13 August 2013

Academic Editor Zhiqiang Hu

Copyright copy 2013 Xiangyu Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An acceleration technique termed generalized Neumann expansion (GNE) is presented for evaluating the responses of uncertainsystems The GNE method which solves stochastic linear algebraic equations arising in stochastic finite element analysis is easyto implement and is of high efficiency The convergence condition of the new method is studied and a rigorous error estimator isproposed to evaluate the upper bound of the relative error of a given GNE solution It is found that the third-order GNE solutionis sufficient to achieve a good accuracy even when the variation of the source stochastic field is relatively high The relationshipbetween the GNEmethod the perturbationmethod and the standard Neumann expansionmethod is also discussed Based on thelinks between these three methods quantitative error estimations for the perturbationmethod and the standard Neumannmethodare obtained for the first time in the probability context

1 Introduction

Over the past few decades the stochastic finite elementmethod (SFEM) has attracted increasing attention from bothacademia and industries The objective of SFEM is to analyzeuncertainty propagation in practical engineering systems thatare affected by various random factors for example hetero-geneous materials and seismic loading The current researchof SFEM has two main focuses probabilistic modeling ofrandom media and efficient numerical solution of stochasticequations Properties of heterogeneous materials (eg rocksand composites) vary randomly through the medium andare generally modeled with stochastic fields Various proba-bilistic methods [1ndash7] have been developed to quantitativelydescribe the random material properties which includethe spectral representation method [1 2] the Karhunen-Loeve expansion method [3 4] and the Fourier-Karhunen-Loeve expansion method [5 6] to name a few In thesemethods the random material properties are represented bya linear expansion such that they can be directly insertedinto the governing equations of the physical system which

facilitates further numerical analysisThe resulting equationsare stochastic equations whose solution is the second mainfocus of SFEM research and also the focus of this work

For static and steady-state problems the resulting SFEMequations have the following form [8ndash13]

(A0+ A11205721+ A21205722+ sdot sdot sdot + A

119898120572119898) u (120596) = f (120596) (1)

where A1 A

119898are 119899 times 119899 symmetric and deterministic

matrices u(120596) the unknown stochastic nodal vector and f(120596)the nodal force vector If the material properties are modeledas Gaussian fields then 120572

1 1205722 120572

119898are independent Gaus-

sian random variablesThe uncertainties from the right-handside of (1) are relatively easy to handle because they are notdirectly coupled with the random matrix from the left-handside

A number of methods have been developed for thesolution of (1) including Monte Carlo methods the per-turbation method the Neumann expansion method andvarious projection methods The Monte Carlo methods [14ndash19] first generate a number of samples of the randommaterialproperties then solve each sample with the deterministic

2 Mathematical Problems in Engineering

FEM solver and finally evaluate the statistics from thesedeterministic solutions The Monte Carlo methods are sim-ple flexible friendly to parallel computing and not restrictedby the number of random variables [20] The computationalcost of Monte Carlo methods is generally high especiallyfor large-scale problems although there exist a number ofefficient sampling techniques such as important sampling[14 19] subset simulation [15 17] and line sampling [18 19]Owning to its generosity and accuracy when sufficient sam-pling can be ensured the Monte Carlo method is often takenas the reference method to verify other SFEM techniquesThe perturbation method [11 21ndash25] expresses the stiffnessmatrix the load vector and the unknown nodal displacementvector in terms of Taylor expansion with respect to theinput randomvariables and the Taylor expansion coefficientscorresponding to the unknown nodal displacement are thensolved by equating terms of the same order from both sidesof (1) Its accuracy decreases when the random fluctuationincreases and there has not been a quantitative conclusionfor the relationship between the solution error and the levelof perturbation making it difficult to confidently deploy theperturbation method in practical problems The Neumannexpansion method [9 26 27] is an iterative process for thesolution of linear algebraic equations Because the spectralradius of the iterative matrix must be smaller than oneit is often considered not suitable for problems with largerandom variations However the fluctuation range allowedby the Neumann expansion method is generally higher thanthe perturbation methods because it is easier to adopthigher order expansions in the former one It should benoted that the stationary iterative procedure employed inNeumann expansion is not as efficient as the precondi-tioned conjugate gradient procedure employed in MonteCarlo methods Therefore the computational efficiency ofthe Neumann expansion method is inferior to direct MonteCarlo simulations In the projection methods [3 4 12 28ndash35] the unknown nodal displacements are projected ontoa set of orthogonal polynomials or other orthogonal basesafter which the Galerkin approach is employed to solvethe projection coefficients However the projection methodstypically result in an equation system much larger than thedeterministic counterpart and thus these methods are oftenlimited to cases with a small set of base functions and afew independent random variables In order to handle prob-lems with more random variables the stochastic collocationmethods [36ndash42] have been developed in which a reducedrandom basis is used in the projection operation and theprojection coefficients are determined by approximating theprecomputedMonte Carlo solutions However the stochasticcollocationmethods are believed to be less rigorous and theiraccuracy is poorer than the full projectionmethods using theGalerkin approach Other variants of the projection proce-dures can be found in [13 43 44] The fatal drawback of theprojection types of solvers is their low efficiency [45] becausethe size of the base function set grows exponentially with thenumber of randomvariables and the order of the polynomialsused Moreover the accuracy of the projection methods iscriticized by [46 47] For linear stochastic problems a jointdiagonalization solution strategy was proposed in [8ndash10] in

which the stochastic solution is obtained in a closed form byjointly diagonalizing the matrix family in (1) This method isnot sensitive to the number of randomvariables in the systemwhile there has not been a mathematically rigorous proof forits accuracy

We propose a generalized Neumann expansion methodfor the solution of (1) The new GNE method is many timesmore efficient than the standard Neumann method which iscompromised by increased memory demand In the contextof probability a mathematically rigorous error estimation isestablished for the new GNE scheme It is also proved thatfor (1) the GNE scheme the standard Neumann expansionmethod and the perturbation method are equivalent Hencethe stochastic error estimation obtained here also shed alight on the standard Neumannmethod and the perturbationmethod which has remained as an outstanding challenge inthe literatureThe rest of the paper is organized as followsTheGNE scheme and its error estimation are derived in Section2 after which it is validated and examined in Section 3 via acomprehensive numerical test Concluding remarks are givenin Section 4

2 Generalized Neumann Expansion Method

Consider a linear elastic statics system with Youngrsquos modulusmodeled as a random field By using the Karhunen-Loeveexpansion [3 4] or the Fourier-Karhunen-Loeve expansion[5 6] random Youngrsquos modulus can be expressed as

119864 (x 120596) = 1198640(x) + sum

119894

radic120582119894120572119894(120596) 120595119894(x) (2)

where 119864(x 120596) denotes random Youngrsquos modulus at point x1198640(x) is the expectation of 119864(x 120596) 120572

119894(120596) are uncorrelated

random variables and 120582119894and 120595

119894(x) are deterministic eigen

values and eigen functions corresponding to the covariancefunction of the source field 119864(x 120596) If 119864(x 120596) is modeled as aGaussian field 120572

119894(120596) become independent standard Gaussian

random variables For the sake of simplicity we drop therandom symbol120596 and for the rest of the paper use120572

119894to denote

the random variables in (2)Following the standard finite element (FE) discretization

procedure a stochastic system of linear algebraic equationscan be built from (2)

(A0+sum119894

120572119894A119894) u = f (3)

In the above equation A0is the conventional FE stiffness

matrix which is symmetric positive definite and sparsewhile A

119894are symmetric and sparse matrices The determin-

istic matrices A119894share the same entry pattern as the stiffness

matrix A0 but they are usually not positive definite

21 RelatedMethods TheMonteCarlomethods first generatea set of samples of120572

119894in (3) and each sample of120572

119894corresponds

to a deterministic equation which can be solved by thestandard FEMmethod

Mathematical Problems in Engineering 3

In the perturbation procedure the unknown randomvector u is expanded as [21 25 26]

u = u0+sum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895+ sdot sdot sdot (4)

Substituting (4) into (3) yields

(A0+sum119894

120572119894A119894)(u

0+sum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895+ sdot sdot sdot ) = f

(5)

from which the components u0 u119894 u119894119895of u are solved as

follows

u0= Aminus10f

u119894= minusAminus10A119894u0

u119894119895= Aminus10A119894Aminus10A119895u0

(6)

It should be noted that in the stochastic context there has notbeen a rigorous convergence criterion for the perturbationscheme described above

In the Neumann expansion method the solution of (3) iswritten as

u = (A0+ ΔA)minus1f = (I + B)minus1Aminus1

0f (7)

where ΔA = sum119894120572119894A119894 B = Aminus1

0ΔA and I is the identity matrix

According to Neumann expansion

(I + B)minus1 = I minus B + B2 minus B3 + sdot sdot sdot (8)

solution (7) can be simplified as

u = u0minus Bu0+ B2u

0minus B3u

0+ sdot sdot sdot (9)

where u0= Aminus10f The spectral radius of matrix B must be

smaller than 1 to ensure the convergence of (9)

22 The Generalized Neumann Expansion Method TheNeu-mann expansion (8) can be generalized to the inverse ofmultiple matrices

(I +sum119894

120572119894B119894)

minus1

= I minussum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894B119895minus sdot sdot sdot (10)

The generalized Neumann expansion (GNE) in (10) can beproved as follows

(I +sum119894

120572119894B119894)(I +sum

119894

120572119894B119894)

minus1

= I minussum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894B119895minus sdot sdot sdot

+ 1205721B1minussum119894

1205721120572119894B1B119894+ sdot sdot sdot

+ 1205722B2minussum119894

1205722120572119894B2B119894+ sdot sdot sdot

+ 120572119899B119899minussum119894

120572119899120572119894B119899B119894+ sdot sdot sdot

= I

(11)

where 119899 is the total number of B119894

The Neumann expansion (8) and the generalized Neu-mann expansion (10) are equivalent in mathematics and onlydiffer in organizing thematrix operationsThis difference canlead to a significant acceleration in computing the solution of(3) Specifically let B

119894= Aminus10A119894 and applying the GNE in (10)

to (3) gives

u = (A0+sum119894

120572119894A119894)

minus1

f = (I +sum119894

120572119894Aminus10A119894)

minus1

Aminus10f

= u0minussum119894

120572119894Aminus10A119894u0+sum119894

sum119895

120572119895120572119894Aminus10A119895Aminus10A119894u0minus sdot sdot sdot

= u0minussum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895minus sdot sdot sdot

(12)

in which

u119894= Aminus10A119894u0

u119894119895= Aminus10A119894Aminus10A119895u0

(13)

Equation (12) is the GNE solution

23 Convergence Analysis and Error Estimation The conver-gence condition of the GNE (10) is essentially the same as theNeumann expansion (8) that is 120588(sum120572

119894B119894) lt 1 where 120588(sdot)

denotes the spectral radius of a matrix Hence for the GNEsolution in (12) the convergence condition is

120588 (sum120572119894Αminus1

0A119894) lt 1 (14)

It should be noted that

120588 (sum120572119894Αminus1

0A119894) = 120588 (sum120572

119894Cminus1A119894Cminus1) = 10038171003817100381710038171003817sum120572

119894Cminus1A119894Cminus110038171003817100381710038171003817119898

2

(15)

4 Mathematical Problems in Engineering

where sdot 1198982

denotes the spectral norm of a matrix and thesymmetric and positive definite matrix C is defined as

A0= C2 (16)

As a type ofmatrix norm the spectral norm has the followingproperties

P +Q1198982

le P1198982

+ Q1198982

PQ1198982

le P1198982Q119898

2

ℎP1198982

= ℎP1198982

(17)

where P and Q are general matrices and ℎ is a positiveconstant Hence10038171003817100381710038171003817sum120572119894Cminus1A119894Cminus110038171003817100381710038171003817119898

2

le100381710038171003817100381710038171205721Cminus1A1Cminus110038171003817100381710038171003817119898

2

+100381710038171003817100381710038171205722Cminus1A2Cminus110038171003817100381710038171003817119898

2

+ sdot sdot sdot10038171003817100381710038171003817120572119899Cminus1A119899Cminus110038171003817100381710038171003817119898

2

120588 (sum120572119894Cminus1A119894Cminus1)

le 120588 (1205721Cminus1A1Cminus1) + 120588 (120572

2Cminus1A2Cminus1)

+ sdot sdot sdot 120588 (120572119899Cminus1A119899Cminus1)

120588 (sum120572119894Αminus1

0A119894)

le 120588 (1205721Αminus1

0A1) + 120588 (120572

2Αminus1

0A2) + sdot sdot sdot 120588 (120572

119899Αminus1

0A119899)

le 119899max (120572119894)max (120588 (Αminus1

0A119894))

(18)

Based on the above relation a sufficient but not necessarycondition for convergence is

119899max (120572119894)max (120588 (Αminus1

0A119894)) lt 1 (19)

The maximum spectral radius 120588max = max(120588(Αminus10A119894)) can

be obtained from A0and A

119894 Equation (19) can be used

to verify the convergence of GNE in advance When 120572119894

are independent standard Gaussian random variables theirabsolute values are smaller than 3 with the probability of997 In this case (19) can be further simplified

119899max (120572119894)max (120588 (Αminus1

0A119894)) asymp 3119899120588max lt 1

120588max lt1

3119899

(20)

Considering the 119896th-order GNE (10) or Neumann expansion(8) their error can be estimated as follows

10038171003817100381710038171003817(I + B)minus1 minus (I minus B + B2 + sdot sdot sdot + (minus1)119896B119896)10038171003817100381710038171003817119898

2

=10038171003817100381710038171003817B119896+1(I + B)minus110038171003817100381710038171003817119898

2

le10038171003817100381710038171003817B119896+110038171003817100381710038171003817119898

2

10038171003817100381710038171003817(I + B)minus110038171003817100381710038171003817119898

2

=120590119896+1119861

1 minus 120590119861

(21)

where120590119861is the spectral normofB which is assumed to satisfy

120590119861lt 1 (22)

The exact solution of (3) is

u = (I +sum119894

120572119894Aminus10A119894)

minus1

Aminus10f = (I + B)minus1Aminus1

0f

Cu = (I +sum119894

120572119894Cminus1A119894Cminus1)minus1

Cminus1f

= (I +D)minus1Cminus1f

(23)

where B = sum119894120572119894Aminus10A119894 D = sum

119894120572119894Cminus1A119894Cminus1 The 119896th-order

GNE or Neumann expansion solution is given by

u = (I minus B + B2 + sdot sdot sdot + (minus1)119896B119896)Aminus10f

Cu = (I minusD +D2 + sdot sdot sdot + (minus1)119896D119896)Cminus1f (24)

The norm of the solution error is defined asCu minus Cu2

=10038171003817100381710038171003817[(I +D)

minus1 minus (I minusD +D2 + sdot sdot sdot + (minus1)119896D119896)]Cminus1f100381710038171003817100381710038172

=10038171003817100381710038171003817D119896+1(I +D)minus1Cminus1f100381710038171003817100381710038172 =

10038171003817100381710038171003817D119896+1Cu100381710038171003817100381710038172

le10038171003817100381710038171003817D119896+110038171003817100381710038171003817119898

2

Cu2 = 120590119896+1

119863Cu2 = 120588

119896+1

119861Cu2

(25)

where 120588119861is the spectral radius of B

The relative error of the solution is defined as

RE =radic(u minus u)119879A

0(u minus u)

radicu119879A0u

=radic(u minus u)119879C2 (u minus u)

radicu119879C2u=Cu minus Cu2Cu2

le 120588119896+1119861

(26)

Comparing (21) and (26) it can be seen that the accuracy forthe solution vector u is even better than the accuracy for thematrix inverse (I + B)minus1

24 Solutions forMore General Stochastic Algebraic EquationsFor the solution of (3) the GNE solution in (12) sharesthe same form as the perturbation solution in (5) Hencefor solving (3) the GNE method the perturbation methodand the standard Neumann expansion method are identicalThe error estimation obtained in (26) also holds for theperturbation approach

However for the following more general stochastic alge-braic equations [26]

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot ) u

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(27)

Mathematical Problems in Engineering 5

the three methods are not equivalent Specifically ap-plying the perturbation method to the above equationyields

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot )

times (u0+sum119894

120573119894u119894+sum119894

sum119895

120573119894120573119895u119894119895+ sdot sdot sdot )

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(28)

Equation (28) can be reorganized as

A0u0

+sum119894

120572119894A119894u0

+sum119894

sum119895

120572119894120572119895A119894119895u0

sdot sdot sdot

+sum119894

A0120573119894u119894

+sum119894

sum119895

120572119894A119894120573119895u119895

+sum119894

sum119895

sum119896

120572119894120572119895A119894119895120573119896u119896

sdot sdot sdot

+sum119894

sum119895

A0120573119894120573119895u119894119895+sum119894

sum119895

sum119896

120572119894A119894120573119895120573119896u119895119896+sum119894

sum119895

sum119896

sum119897

120572119894120572119895A119894119895120573119896120573119897u119896119897sdot sdot sdot

sdot sdot sdot sdot sdot sdot sdot sdot sdot d

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(29)

If 120572119894= 120573119894= 120574119894 then the perturbation components u

0 u119894 u119894119895

u11989411198942sdotsdotsdot119894119899

of u can be solved as

A0u0= f0

A0u119894+ A119894u0= f119894

A0u119894119895+ A119894u119895+ A119894119895u0= f119894119895

A0u11989411198942sdotsdotsdot119894119899

+ A1198941

u11989421198943sdotsdotsdot119894119899

+ A11989411198942

u11989431198944sdotsdotsdot119894119899

+ A11989411198942sdotsdotsdot119894119899

u0= f11989411198942sdotsdotsdot119894119899

(30)

Solving (27) with the Neumann expansion gives

Aminus1 = (A0+ ΔA)minus1

= (I + B)minus1Aminus10= (I minus B + B2 minus B3 + sdot sdot sdot )Aminus1

0

(31)

where B = Aminus10ΔA ΔA = sum

119894120572119894A119894+ sum119894sum119895120572119894120572119895A119894119895+ sdot sdot sdot Then

u = (I minus B + B2 minus B3 + sdot sdot sdot )Aminus10(f0+ Δf) (32)

in which

Δf = sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot (33)

Because higher order entries are included in B the termBAminus10(f0+ Δf) (the first-order term of the Neumann solution

in (32)) does not correspond to the first-order entries of uTherefore theNeumann solution in (32) and the perturbationsolution in (30) are different for the general stochasticequation (27)

Solving (27) with the GNE approach gives

u = (A0+sum119894119894

1205721198941

A1198941

+sum1198942

sum1198941

1205721198941

1205721198942

A11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

A11989411198942sdotsdotsdot119894119899

)

minus1

f

= (I +sum119894119894

1205721198941

B1198941

+sum1198942

sum1198941

1205721198941

1205721198942

B11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

B11989411198942sdotsdotsdot119894119899

)

minus1

Aminus10f

= (I +sum119894

120573119894C119894)

minus1

Aminus10f

= (I minussum119894

120573119894C119894+sum119894

sum119895

120573119894120573119895C119894C119895

minussum119894

sum119895

sum119896

120573119894120573119895120573119896C119894C119895C119896+ sdot sdot sdot )Aminus1

0f

(34)

where f = f0+ sum119894120574119894f119894+ sum119894sum119895120574119894120574119895f119894119895+ sdot sdot sdot B

119894= Aminus10A119894

B119894119895= Aminus10A119894119895 and other B terms can be similarly formed

Matrices C119894are determined by realignment of B

119894 B119894119895

and random variables 120573119894are formed by rearrangement of

1205721198941

12057211989411198942

12057211989411198942sdotsdotsdot119894119899

In summary when solving (27) the GNEmethod and the

Neumannmethod are identical while they are not equivalentto the perturbation procedure However if the higher orderterms in the GNE or Neumann expansion solutions areomitted they degenerate into the perturbation solution

6 Mathematical Problems in Engineering

x

y

p

Figure 1 Cross section of the random material sample

Two simplified examples are given here to demonstrate thedifference between the GNE and the perturbation method

241 Example I Only first-order entries are included in bothsides of (27) such that the equation becomes

(A0+sum119894

120572119894A119894) u = f

0+sum119894

120572119894f119894 (35)

The first-order perturbation solution is

u0= Aminus10f0

u119894= Aminus10f119894minus Aminus10A119894u0

u = u0+sum119894

120572119894u119894

= Aminus10f0+sum119894

120572119894Aminus10f119894minussum119894

120572119894Aminus10A119894u0

(36)

and the first-order GNE solution is

u = (Aminus10minussum119894

120572119894Aminus10A119894Aminus10)(f0+sum119894

120572119894f119894)

= Aminus10f0minussum119894

120572119894Aminus10A119894u0+sum119894

120572119894Aminus10f119894

minussum119894

sum119895

120572119894120572119895Aminus10A119894f119895

(37)

Comparing (36) and (37) it can be seen that theGNE solutioncontains an extra second-order term minussum

119894sum119895120572119894120572119895Aminus10A119894f119895

242 Example II Up to second-order terms are included inthe left-hand side of (27) while only the constant term isincluded in the right-hand side In this case the equationbecomes

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895) u = f

0 (38)

Denote u0= Aminus10f0 B119894= Aminus10A119894 and B

119894119895= Aminus10A119894119895 then the

first-order perturbation solution can be expressed as

u0= Aminus10f0

u119894= minussum119894

Aminus10A119894u0= minussum119894

B119894u0

u = u0+sum119894

120572119894u119894= u0minussum119894

120572119894B119894u0

(39)

while the first-order GNE solution is

u = (I +sum119894

120572119894Aminus10A119894+sum119894

sum119895

120572119894120572119895Aminus10A119894119895)

minus1

u0

= (I +sum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894119895)

minus1

u0

= (I minussum119894

120572119894B119894minussum119894

sum119895

120572119894120572119895B119894119895) u0

(40)

Comparing (39) and (40) the second-order termsminussum119894sum119895120572119894120572119895Aminus10A119894119895u0in the GNE solution do not appear in

the perturbation solution (39)

3 Numerical Examples

31 Example Description As shown in Figure 1 a plain strainproblem with an orthohexagonal cross section is consideredhere The length of each edge of the hexagon is 04m ThePoisson ratio of thematerial is set as 02 andYoungrsquosmodulusis assumed to be a Gaussian field with amean value of 30GPaand a covariance function defined as

119877 (1205911 1205912) = 1198881119890minus120591121198882minus1205912

21198882 (41)

in which 1205911= 1199091minus 1199092and 1205912= 1199101minus 1199102are the coordinate

differences and 1198881and 1198882are adjustable constants As shown in

Figure 2 the cross section is discretized with triangular linearelements where each side is divided into 8 equal parts Thereare in total 384 elements and 217 nodes

Mathematical Problems in Engineering 7

Figure 2 Finite element mesh

The relative error is defined as

RE =1003817100381710038171003817CuGNE minus CuMC

100381710038171003817100381721003817100381710038171003817CuMC

10038171003817100381710038172times 100 (42)

inwhichuMC is the nodal displacement vector solvedwith theMonte Carlo method and uGNE is the GNE solution Unlessstated explicitly the numerical testing is performed for 4340samples generated from the Gaussian field defined in (41)

The convergence condition of the GNE method has beenrigorously proved The main purpose of this example isto examine the error distribution and its relationship withvarious randomparameters forwhich a number of numericaltests are performed in the following subsections

32 Error Estimation The third-order GNE solution isadoptedThe constants 119888

1and 1198882in (41) are varied to examine

the error estimator in (26) The constant 1198881is the variance of

the Gaussian field and the constant 1198882controls the shape of

the covariance function The effective correlation length 119871effof the covariance function 119877(120591) is defined by

119877 (119871eff)

119877 (0)= 001 (43)

For any two points in the medium if the distance betweenthem is larger than 119871eff their material properties are onlyweakly correlated The effective correlation length of theGaussian field defined in (41) is

119877 (120591)

119877 (0)= 119890minus120591

21198882 = 001 997904rArr 119871eff = 2146radic1198882 (44)

In the first set of tests the constant 1198881is varied to change the

variance of the Gaussian field The parameter settings are

1198881= 001 119888

2= 32119864 minus 2 119871eff = 038

1198881= 004 119888

2= 32119864 minus 2 119871eff = 038

(45)

In the second set of tests the constant 1198882is varied to change

the shape of the covariance function The parameter settingsare

1198881= 001 119888

2= 08119864 minus 2 119871eff = 019

1198881= 001 119888

2= 24119864 minus 2 119871eff = 033

1198881= 001 119888

2= 40119864 minus 2 119871eff = 043

(46)

321 The Influence of 1198881 Corresponding to 119888

1= 001 and

1198881= 004 the standard deviation 120590 of the stochastic field

is 01 and 02 respectively A total number of 4340 samplesare generated for each case For each realization both theMonte Carlo solution and the third-order GNE solution arecomputed The relationship between the spectral radius andthe relative error is drawn in Figures 3 and 4 in which thered crosses show the actual relative errors corresponding toeach sample and the blue line shows the theoretical errorestimation It can been seen that for both cases the theoreticalerror estimator in (26) holds accurately and its effectivenessis not affected by the change of the variance controlled by 119888

1

322 The Influence of 1198882 The constant 119888

2is adjusted to take

three different values 08 times 10minus2 24 times 10minus2 and 40 times 10minus2while the constant 119888

1remains unchanged Following (41)

the effective correlation length grows with the increase of1198882 Again 4340 samples are generated for each stochastic

field and for each realization both the Monte Carlo solutionand the third-order GNE solution are computed Figure 5plots the relationship between spectral radius and the relativeerror which verifies that for all cases the theoretical errorestimation in (26) holds The changing of 119888

2 which changes

the variation frequency of the stochastic field does not affectthe error estimation

As indicated in (44) the effective correlation length 119871effis directly associated with the parameter 119888

2 To investigate

the influence of 119871eff on the relative error the empiricalcumulative distribution functions (ECDF) of relative errorsare plotted in Figure 6 for different 119871eff cases Comparing thethree ECDF curves at different effective correlation lengthsit is found that with the decrease of the effective correlationlength the accuracy of the GNE solution improves

33 The Relative Error for Other Types of Stochastic FieldsIn the last two subsections the error estimation analysis hastaken the Gaussian assumption that is all random variables120572119894in (3) are assumed to be independent Gaussian random

variables In this subsection we investigate the relative errorof the GNE method for other types of stochastic fieldsSpecifically the random variables 120572

119894in (3) are assumed to

have the two-point distribution the binomial distributionand the 120573 distribution respectively For the case of two-pointdistribution values minus03119864

0 031198640 are assigned with equal

probability that is 05 For the case of binomial distributionten times Bernoulli trials with the probability 05 are adopted

8 Mathematical Problems in Engineering

minus14 minus12 minus10 minus8 minus6 minus4minus14

minus12

minus10

minus8

minus6

minus4

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

Solution with c1 = 001

log (RE) = 4 log (120588B)

4 log (120588B)

Figure 3 The relation between the relative error and the spectral radius

minus11 minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 0minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Solution with c1 = 004

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 4 The relation between the relative error and the spectral radius

and the integer values between 0 and 10 are linearly projectedonto [minus03119864

0 031198640] For the case of 120573 distribution the

probability density function is defined as

119891 (119909 | 119886 119887) =1

119861 (119886 119887)119909119886minus1(1 minus 119909)119887minus1119868

(01)(119909) (47)

where 119861(119886 119887) is a 120573 function and the filter function 119868(01)(119909)

is defined as

119868(01) (119909) =

1 119909 isin (0 1)

0 119909 notin (0 1) (48)

Three kinds of 120573 distributions are considered with 119886 = 119887 =075 119886 = 119887 = 1 and 119886 = 119887 = 4 respectivelyThe interval (0 1)is linearly mapped to [minus03119864

0 031198640]

Corresponding to these five distributions the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7 Comparing the ECDF curves at differenteffective correlation lengths the same trend is observed asthe Gaussian case That is the relative error decreases withthe drop of effective correlation length

34 Efficiency Analysis Using the same computing platformthe Monte Carlo method the Neumann expansion method

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

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Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

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ISRN Geometry

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

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Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

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Journal ofApplied Mathematics

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Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

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Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

2 Mathematical Problems in Engineering

FEM solver and finally evaluate the statistics from thesedeterministic solutions The Monte Carlo methods are sim-ple flexible friendly to parallel computing and not restrictedby the number of random variables [20] The computationalcost of Monte Carlo methods is generally high especiallyfor large-scale problems although there exist a number ofefficient sampling techniques such as important sampling[14 19] subset simulation [15 17] and line sampling [18 19]Owning to its generosity and accuracy when sufficient sam-pling can be ensured the Monte Carlo method is often takenas the reference method to verify other SFEM techniquesThe perturbation method [11 21ndash25] expresses the stiffnessmatrix the load vector and the unknown nodal displacementvector in terms of Taylor expansion with respect to theinput randomvariables and the Taylor expansion coefficientscorresponding to the unknown nodal displacement are thensolved by equating terms of the same order from both sidesof (1) Its accuracy decreases when the random fluctuationincreases and there has not been a quantitative conclusionfor the relationship between the solution error and the levelof perturbation making it difficult to confidently deploy theperturbation method in practical problems The Neumannexpansion method [9 26 27] is an iterative process for thesolution of linear algebraic equations Because the spectralradius of the iterative matrix must be smaller than oneit is often considered not suitable for problems with largerandom variations However the fluctuation range allowedby the Neumann expansion method is generally higher thanthe perturbation methods because it is easier to adopthigher order expansions in the former one It should benoted that the stationary iterative procedure employed inNeumann expansion is not as efficient as the precondi-tioned conjugate gradient procedure employed in MonteCarlo methods Therefore the computational efficiency ofthe Neumann expansion method is inferior to direct MonteCarlo simulations In the projection methods [3 4 12 28ndash35] the unknown nodal displacements are projected ontoa set of orthogonal polynomials or other orthogonal basesafter which the Galerkin approach is employed to solvethe projection coefficients However the projection methodstypically result in an equation system much larger than thedeterministic counterpart and thus these methods are oftenlimited to cases with a small set of base functions and afew independent random variables In order to handle prob-lems with more random variables the stochastic collocationmethods [36ndash42] have been developed in which a reducedrandom basis is used in the projection operation and theprojection coefficients are determined by approximating theprecomputedMonte Carlo solutions However the stochasticcollocationmethods are believed to be less rigorous and theiraccuracy is poorer than the full projectionmethods using theGalerkin approach Other variants of the projection proce-dures can be found in [13 43 44] The fatal drawback of theprojection types of solvers is their low efficiency [45] becausethe size of the base function set grows exponentially with thenumber of randomvariables and the order of the polynomialsused Moreover the accuracy of the projection methods iscriticized by [46 47] For linear stochastic problems a jointdiagonalization solution strategy was proposed in [8ndash10] in

which the stochastic solution is obtained in a closed form byjointly diagonalizing the matrix family in (1) This method isnot sensitive to the number of randomvariables in the systemwhile there has not been a mathematically rigorous proof forits accuracy

We propose a generalized Neumann expansion methodfor the solution of (1) The new GNE method is many timesmore efficient than the standard Neumann method which iscompromised by increased memory demand In the contextof probability a mathematically rigorous error estimation isestablished for the new GNE scheme It is also proved thatfor (1) the GNE scheme the standard Neumann expansionmethod and the perturbation method are equivalent Hencethe stochastic error estimation obtained here also shed alight on the standard Neumannmethod and the perturbationmethod which has remained as an outstanding challenge inthe literatureThe rest of the paper is organized as followsTheGNE scheme and its error estimation are derived in Section2 after which it is validated and examined in Section 3 via acomprehensive numerical test Concluding remarks are givenin Section 4

2 Generalized Neumann Expansion Method

Consider a linear elastic statics system with Youngrsquos modulusmodeled as a random field By using the Karhunen-Loeveexpansion [3 4] or the Fourier-Karhunen-Loeve expansion[5 6] random Youngrsquos modulus can be expressed as

119864 (x 120596) = 1198640(x) + sum

119894

radic120582119894120572119894(120596) 120595119894(x) (2)

where 119864(x 120596) denotes random Youngrsquos modulus at point x1198640(x) is the expectation of 119864(x 120596) 120572

119894(120596) are uncorrelated

random variables and 120582119894and 120595

119894(x) are deterministic eigen

values and eigen functions corresponding to the covariancefunction of the source field 119864(x 120596) If 119864(x 120596) is modeled as aGaussian field 120572

119894(120596) become independent standard Gaussian

random variables For the sake of simplicity we drop therandom symbol120596 and for the rest of the paper use120572

119894to denote

the random variables in (2)Following the standard finite element (FE) discretization

procedure a stochastic system of linear algebraic equationscan be built from (2)

(A0+sum119894

120572119894A119894) u = f (3)

In the above equation A0is the conventional FE stiffness

matrix which is symmetric positive definite and sparsewhile A

119894are symmetric and sparse matrices The determin-

istic matrices A119894share the same entry pattern as the stiffness

matrix A0 but they are usually not positive definite

21 RelatedMethods TheMonteCarlomethods first generatea set of samples of120572

119894in (3) and each sample of120572

119894corresponds

to a deterministic equation which can be solved by thestandard FEMmethod

Mathematical Problems in Engineering 3

In the perturbation procedure the unknown randomvector u is expanded as [21 25 26]

u = u0+sum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895+ sdot sdot sdot (4)

Substituting (4) into (3) yields

(A0+sum119894

120572119894A119894)(u

0+sum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895+ sdot sdot sdot ) = f

(5)

from which the components u0 u119894 u119894119895of u are solved as

follows

u0= Aminus10f

u119894= minusAminus10A119894u0

u119894119895= Aminus10A119894Aminus10A119895u0

(6)

It should be noted that in the stochastic context there has notbeen a rigorous convergence criterion for the perturbationscheme described above

In the Neumann expansion method the solution of (3) iswritten as

u = (A0+ ΔA)minus1f = (I + B)minus1Aminus1

0f (7)

where ΔA = sum119894120572119894A119894 B = Aminus1

0ΔA and I is the identity matrix

According to Neumann expansion

(I + B)minus1 = I minus B + B2 minus B3 + sdot sdot sdot (8)

solution (7) can be simplified as

u = u0minus Bu0+ B2u

0minus B3u

0+ sdot sdot sdot (9)

where u0= Aminus10f The spectral radius of matrix B must be

smaller than 1 to ensure the convergence of (9)

22 The Generalized Neumann Expansion Method TheNeu-mann expansion (8) can be generalized to the inverse ofmultiple matrices

(I +sum119894

120572119894B119894)

minus1

= I minussum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894B119895minus sdot sdot sdot (10)

The generalized Neumann expansion (GNE) in (10) can beproved as follows

(I +sum119894

120572119894B119894)(I +sum

119894

120572119894B119894)

minus1

= I minussum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894B119895minus sdot sdot sdot

+ 1205721B1minussum119894

1205721120572119894B1B119894+ sdot sdot sdot

+ 1205722B2minussum119894

1205722120572119894B2B119894+ sdot sdot sdot

+ 120572119899B119899minussum119894

120572119899120572119894B119899B119894+ sdot sdot sdot

= I

(11)

where 119899 is the total number of B119894

The Neumann expansion (8) and the generalized Neu-mann expansion (10) are equivalent in mathematics and onlydiffer in organizing thematrix operationsThis difference canlead to a significant acceleration in computing the solution of(3) Specifically let B

119894= Aminus10A119894 and applying the GNE in (10)

to (3) gives

u = (A0+sum119894

120572119894A119894)

minus1

f = (I +sum119894

120572119894Aminus10A119894)

minus1

Aminus10f

= u0minussum119894

120572119894Aminus10A119894u0+sum119894

sum119895

120572119895120572119894Aminus10A119895Aminus10A119894u0minus sdot sdot sdot

= u0minussum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895minus sdot sdot sdot

(12)

in which

u119894= Aminus10A119894u0

u119894119895= Aminus10A119894Aminus10A119895u0

(13)

Equation (12) is the GNE solution

23 Convergence Analysis and Error Estimation The conver-gence condition of the GNE (10) is essentially the same as theNeumann expansion (8) that is 120588(sum120572

119894B119894) lt 1 where 120588(sdot)

denotes the spectral radius of a matrix Hence for the GNEsolution in (12) the convergence condition is

120588 (sum120572119894Αminus1

0A119894) lt 1 (14)

It should be noted that

120588 (sum120572119894Αminus1

0A119894) = 120588 (sum120572

119894Cminus1A119894Cminus1) = 10038171003817100381710038171003817sum120572

119894Cminus1A119894Cminus110038171003817100381710038171003817119898

2

(15)

4 Mathematical Problems in Engineering

where sdot 1198982

denotes the spectral norm of a matrix and thesymmetric and positive definite matrix C is defined as

A0= C2 (16)

As a type ofmatrix norm the spectral norm has the followingproperties

P +Q1198982

le P1198982

+ Q1198982

PQ1198982

le P1198982Q119898

2

ℎP1198982

= ℎP1198982

(17)

where P and Q are general matrices and ℎ is a positiveconstant Hence10038171003817100381710038171003817sum120572119894Cminus1A119894Cminus110038171003817100381710038171003817119898

2

le100381710038171003817100381710038171205721Cminus1A1Cminus110038171003817100381710038171003817119898

2

+100381710038171003817100381710038171205722Cminus1A2Cminus110038171003817100381710038171003817119898

2

+ sdot sdot sdot10038171003817100381710038171003817120572119899Cminus1A119899Cminus110038171003817100381710038171003817119898

2

120588 (sum120572119894Cminus1A119894Cminus1)

le 120588 (1205721Cminus1A1Cminus1) + 120588 (120572

2Cminus1A2Cminus1)

+ sdot sdot sdot 120588 (120572119899Cminus1A119899Cminus1)

120588 (sum120572119894Αminus1

0A119894)

le 120588 (1205721Αminus1

0A1) + 120588 (120572

2Αminus1

0A2) + sdot sdot sdot 120588 (120572

119899Αminus1

0A119899)

le 119899max (120572119894)max (120588 (Αminus1

0A119894))

(18)

Based on the above relation a sufficient but not necessarycondition for convergence is

119899max (120572119894)max (120588 (Αminus1

0A119894)) lt 1 (19)

The maximum spectral radius 120588max = max(120588(Αminus10A119894)) can

be obtained from A0and A

119894 Equation (19) can be used

to verify the convergence of GNE in advance When 120572119894

are independent standard Gaussian random variables theirabsolute values are smaller than 3 with the probability of997 In this case (19) can be further simplified

119899max (120572119894)max (120588 (Αminus1

0A119894)) asymp 3119899120588max lt 1

120588max lt1

3119899

(20)

Considering the 119896th-order GNE (10) or Neumann expansion(8) their error can be estimated as follows

10038171003817100381710038171003817(I + B)minus1 minus (I minus B + B2 + sdot sdot sdot + (minus1)119896B119896)10038171003817100381710038171003817119898

2

=10038171003817100381710038171003817B119896+1(I + B)minus110038171003817100381710038171003817119898

2

le10038171003817100381710038171003817B119896+110038171003817100381710038171003817119898

2

10038171003817100381710038171003817(I + B)minus110038171003817100381710038171003817119898

2

=120590119896+1119861

1 minus 120590119861

(21)

where120590119861is the spectral normofB which is assumed to satisfy

120590119861lt 1 (22)

The exact solution of (3) is

u = (I +sum119894

120572119894Aminus10A119894)

minus1

Aminus10f = (I + B)minus1Aminus1

0f

Cu = (I +sum119894

120572119894Cminus1A119894Cminus1)minus1

Cminus1f

= (I +D)minus1Cminus1f

(23)

where B = sum119894120572119894Aminus10A119894 D = sum

119894120572119894Cminus1A119894Cminus1 The 119896th-order

GNE or Neumann expansion solution is given by

u = (I minus B + B2 + sdot sdot sdot + (minus1)119896B119896)Aminus10f

Cu = (I minusD +D2 + sdot sdot sdot + (minus1)119896D119896)Cminus1f (24)

The norm of the solution error is defined asCu minus Cu2

=10038171003817100381710038171003817[(I +D)

minus1 minus (I minusD +D2 + sdot sdot sdot + (minus1)119896D119896)]Cminus1f100381710038171003817100381710038172

=10038171003817100381710038171003817D119896+1(I +D)minus1Cminus1f100381710038171003817100381710038172 =

10038171003817100381710038171003817D119896+1Cu100381710038171003817100381710038172

le10038171003817100381710038171003817D119896+110038171003817100381710038171003817119898

2

Cu2 = 120590119896+1

119863Cu2 = 120588

119896+1

119861Cu2

(25)

where 120588119861is the spectral radius of B

The relative error of the solution is defined as

RE =radic(u minus u)119879A

0(u minus u)

radicu119879A0u

=radic(u minus u)119879C2 (u minus u)

radicu119879C2u=Cu minus Cu2Cu2

le 120588119896+1119861

(26)

Comparing (21) and (26) it can be seen that the accuracy forthe solution vector u is even better than the accuracy for thematrix inverse (I + B)minus1

24 Solutions forMore General Stochastic Algebraic EquationsFor the solution of (3) the GNE solution in (12) sharesthe same form as the perturbation solution in (5) Hencefor solving (3) the GNE method the perturbation methodand the standard Neumann expansion method are identicalThe error estimation obtained in (26) also holds for theperturbation approach

However for the following more general stochastic alge-braic equations [26]

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot ) u

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(27)

Mathematical Problems in Engineering 5

the three methods are not equivalent Specifically ap-plying the perturbation method to the above equationyields

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot )

times (u0+sum119894

120573119894u119894+sum119894

sum119895

120573119894120573119895u119894119895+ sdot sdot sdot )

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(28)

Equation (28) can be reorganized as

A0u0

+sum119894

120572119894A119894u0

+sum119894

sum119895

120572119894120572119895A119894119895u0

sdot sdot sdot

+sum119894

A0120573119894u119894

+sum119894

sum119895

120572119894A119894120573119895u119895

+sum119894

sum119895

sum119896

120572119894120572119895A119894119895120573119896u119896

sdot sdot sdot

+sum119894

sum119895

A0120573119894120573119895u119894119895+sum119894

sum119895

sum119896

120572119894A119894120573119895120573119896u119895119896+sum119894

sum119895

sum119896

sum119897

120572119894120572119895A119894119895120573119896120573119897u119896119897sdot sdot sdot

sdot sdot sdot sdot sdot sdot sdot sdot sdot d

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(29)

If 120572119894= 120573119894= 120574119894 then the perturbation components u

0 u119894 u119894119895

u11989411198942sdotsdotsdot119894119899

of u can be solved as

A0u0= f0

A0u119894+ A119894u0= f119894

A0u119894119895+ A119894u119895+ A119894119895u0= f119894119895

A0u11989411198942sdotsdotsdot119894119899

+ A1198941

u11989421198943sdotsdotsdot119894119899

+ A11989411198942

u11989431198944sdotsdotsdot119894119899

+ A11989411198942sdotsdotsdot119894119899

u0= f11989411198942sdotsdotsdot119894119899

(30)

Solving (27) with the Neumann expansion gives

Aminus1 = (A0+ ΔA)minus1

= (I + B)minus1Aminus10= (I minus B + B2 minus B3 + sdot sdot sdot )Aminus1

0

(31)

where B = Aminus10ΔA ΔA = sum

119894120572119894A119894+ sum119894sum119895120572119894120572119895A119894119895+ sdot sdot sdot Then

u = (I minus B + B2 minus B3 + sdot sdot sdot )Aminus10(f0+ Δf) (32)

in which

Δf = sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot (33)

Because higher order entries are included in B the termBAminus10(f0+ Δf) (the first-order term of the Neumann solution

in (32)) does not correspond to the first-order entries of uTherefore theNeumann solution in (32) and the perturbationsolution in (30) are different for the general stochasticequation (27)

Solving (27) with the GNE approach gives

u = (A0+sum119894119894

1205721198941

A1198941

+sum1198942

sum1198941

1205721198941

1205721198942

A11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

A11989411198942sdotsdotsdot119894119899

)

minus1

f

= (I +sum119894119894

1205721198941

B1198941

+sum1198942

sum1198941

1205721198941

1205721198942

B11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

B11989411198942sdotsdotsdot119894119899

)

minus1

Aminus10f

= (I +sum119894

120573119894C119894)

minus1

Aminus10f

= (I minussum119894

120573119894C119894+sum119894

sum119895

120573119894120573119895C119894C119895

minussum119894

sum119895

sum119896

120573119894120573119895120573119896C119894C119895C119896+ sdot sdot sdot )Aminus1

0f

(34)

where f = f0+ sum119894120574119894f119894+ sum119894sum119895120574119894120574119895f119894119895+ sdot sdot sdot B

119894= Aminus10A119894

B119894119895= Aminus10A119894119895 and other B terms can be similarly formed

Matrices C119894are determined by realignment of B

119894 B119894119895

and random variables 120573119894are formed by rearrangement of

1205721198941

12057211989411198942

12057211989411198942sdotsdotsdot119894119899

In summary when solving (27) the GNEmethod and the

Neumannmethod are identical while they are not equivalentto the perturbation procedure However if the higher orderterms in the GNE or Neumann expansion solutions areomitted they degenerate into the perturbation solution

6 Mathematical Problems in Engineering

x

y

p

Figure 1 Cross section of the random material sample

Two simplified examples are given here to demonstrate thedifference between the GNE and the perturbation method

241 Example I Only first-order entries are included in bothsides of (27) such that the equation becomes

(A0+sum119894

120572119894A119894) u = f

0+sum119894

120572119894f119894 (35)

The first-order perturbation solution is

u0= Aminus10f0

u119894= Aminus10f119894minus Aminus10A119894u0

u = u0+sum119894

120572119894u119894

= Aminus10f0+sum119894

120572119894Aminus10f119894minussum119894

120572119894Aminus10A119894u0

(36)

and the first-order GNE solution is

u = (Aminus10minussum119894

120572119894Aminus10A119894Aminus10)(f0+sum119894

120572119894f119894)

= Aminus10f0minussum119894

120572119894Aminus10A119894u0+sum119894

120572119894Aminus10f119894

minussum119894

sum119895

120572119894120572119895Aminus10A119894f119895

(37)

Comparing (36) and (37) it can be seen that theGNE solutioncontains an extra second-order term minussum

119894sum119895120572119894120572119895Aminus10A119894f119895

242 Example II Up to second-order terms are included inthe left-hand side of (27) while only the constant term isincluded in the right-hand side In this case the equationbecomes

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895) u = f

0 (38)

Denote u0= Aminus10f0 B119894= Aminus10A119894 and B

119894119895= Aminus10A119894119895 then the

first-order perturbation solution can be expressed as

u0= Aminus10f0

u119894= minussum119894

Aminus10A119894u0= minussum119894

B119894u0

u = u0+sum119894

120572119894u119894= u0minussum119894

120572119894B119894u0

(39)

while the first-order GNE solution is

u = (I +sum119894

120572119894Aminus10A119894+sum119894

sum119895

120572119894120572119895Aminus10A119894119895)

minus1

u0

= (I +sum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894119895)

minus1

u0

= (I minussum119894

120572119894B119894minussum119894

sum119895

120572119894120572119895B119894119895) u0

(40)

Comparing (39) and (40) the second-order termsminussum119894sum119895120572119894120572119895Aminus10A119894119895u0in the GNE solution do not appear in

the perturbation solution (39)

3 Numerical Examples

31 Example Description As shown in Figure 1 a plain strainproblem with an orthohexagonal cross section is consideredhere The length of each edge of the hexagon is 04m ThePoisson ratio of thematerial is set as 02 andYoungrsquosmodulusis assumed to be a Gaussian field with amean value of 30GPaand a covariance function defined as

119877 (1205911 1205912) = 1198881119890minus120591121198882minus1205912

21198882 (41)

in which 1205911= 1199091minus 1199092and 1205912= 1199101minus 1199102are the coordinate

differences and 1198881and 1198882are adjustable constants As shown in

Figure 2 the cross section is discretized with triangular linearelements where each side is divided into 8 equal parts Thereare in total 384 elements and 217 nodes

Mathematical Problems in Engineering 7

Figure 2 Finite element mesh

The relative error is defined as

RE =1003817100381710038171003817CuGNE minus CuMC

100381710038171003817100381721003817100381710038171003817CuMC

10038171003817100381710038172times 100 (42)

inwhichuMC is the nodal displacement vector solvedwith theMonte Carlo method and uGNE is the GNE solution Unlessstated explicitly the numerical testing is performed for 4340samples generated from the Gaussian field defined in (41)

The convergence condition of the GNE method has beenrigorously proved The main purpose of this example isto examine the error distribution and its relationship withvarious randomparameters forwhich a number of numericaltests are performed in the following subsections

32 Error Estimation The third-order GNE solution isadoptedThe constants 119888

1and 1198882in (41) are varied to examine

the error estimator in (26) The constant 1198881is the variance of

the Gaussian field and the constant 1198882controls the shape of

the covariance function The effective correlation length 119871effof the covariance function 119877(120591) is defined by

119877 (119871eff)

119877 (0)= 001 (43)

For any two points in the medium if the distance betweenthem is larger than 119871eff their material properties are onlyweakly correlated The effective correlation length of theGaussian field defined in (41) is

119877 (120591)

119877 (0)= 119890minus120591

21198882 = 001 997904rArr 119871eff = 2146radic1198882 (44)

In the first set of tests the constant 1198881is varied to change the

variance of the Gaussian field The parameter settings are

1198881= 001 119888

2= 32119864 minus 2 119871eff = 038

1198881= 004 119888

2= 32119864 minus 2 119871eff = 038

(45)

In the second set of tests the constant 1198882is varied to change

the shape of the covariance function The parameter settingsare

1198881= 001 119888

2= 08119864 minus 2 119871eff = 019

1198881= 001 119888

2= 24119864 minus 2 119871eff = 033

1198881= 001 119888

2= 40119864 minus 2 119871eff = 043

(46)

321 The Influence of 1198881 Corresponding to 119888

1= 001 and

1198881= 004 the standard deviation 120590 of the stochastic field

is 01 and 02 respectively A total number of 4340 samplesare generated for each case For each realization both theMonte Carlo solution and the third-order GNE solution arecomputed The relationship between the spectral radius andthe relative error is drawn in Figures 3 and 4 in which thered crosses show the actual relative errors corresponding toeach sample and the blue line shows the theoretical errorestimation It can been seen that for both cases the theoreticalerror estimator in (26) holds accurately and its effectivenessis not affected by the change of the variance controlled by 119888

1

322 The Influence of 1198882 The constant 119888

2is adjusted to take

three different values 08 times 10minus2 24 times 10minus2 and 40 times 10minus2while the constant 119888

1remains unchanged Following (41)

the effective correlation length grows with the increase of1198882 Again 4340 samples are generated for each stochastic

field and for each realization both the Monte Carlo solutionand the third-order GNE solution are computed Figure 5plots the relationship between spectral radius and the relativeerror which verifies that for all cases the theoretical errorestimation in (26) holds The changing of 119888

2 which changes

the variation frequency of the stochastic field does not affectthe error estimation

As indicated in (44) the effective correlation length 119871effis directly associated with the parameter 119888

2 To investigate

the influence of 119871eff on the relative error the empiricalcumulative distribution functions (ECDF) of relative errorsare plotted in Figure 6 for different 119871eff cases Comparing thethree ECDF curves at different effective correlation lengthsit is found that with the decrease of the effective correlationlength the accuracy of the GNE solution improves

33 The Relative Error for Other Types of Stochastic FieldsIn the last two subsections the error estimation analysis hastaken the Gaussian assumption that is all random variables120572119894in (3) are assumed to be independent Gaussian random

variables In this subsection we investigate the relative errorof the GNE method for other types of stochastic fieldsSpecifically the random variables 120572

119894in (3) are assumed to

have the two-point distribution the binomial distributionand the 120573 distribution respectively For the case of two-pointdistribution values minus03119864

0 031198640 are assigned with equal

probability that is 05 For the case of binomial distributionten times Bernoulli trials with the probability 05 are adopted

8 Mathematical Problems in Engineering

minus14 minus12 minus10 minus8 minus6 minus4minus14

minus12

minus10

minus8

minus6

minus4

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

Solution with c1 = 001

log (RE) = 4 log (120588B)

4 log (120588B)

Figure 3 The relation between the relative error and the spectral radius

minus11 minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 0minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Solution with c1 = 004

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 4 The relation between the relative error and the spectral radius

and the integer values between 0 and 10 are linearly projectedonto [minus03119864

0 031198640] For the case of 120573 distribution the

probability density function is defined as

119891 (119909 | 119886 119887) =1

119861 (119886 119887)119909119886minus1(1 minus 119909)119887minus1119868

(01)(119909) (47)

where 119861(119886 119887) is a 120573 function and the filter function 119868(01)(119909)

is defined as

119868(01) (119909) =

1 119909 isin (0 1)

0 119909 notin (0 1) (48)

Three kinds of 120573 distributions are considered with 119886 = 119887 =075 119886 = 119887 = 1 and 119886 = 119887 = 4 respectivelyThe interval (0 1)is linearly mapped to [minus03119864

0 031198640]

Corresponding to these five distributions the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7 Comparing the ECDF curves at differenteffective correlation lengths the same trend is observed asthe Gaussian case That is the relative error decreases withthe drop of effective correlation length

34 Efficiency Analysis Using the same computing platformthe Monte Carlo method the Neumann expansion method

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

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Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

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ISRN Geometry

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Discrete Dynamics in Nature and Society

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

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ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

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Journal ofApplied Mathematics

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Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

Mathematical Problems in Engineering 3

In the perturbation procedure the unknown randomvector u is expanded as [21 25 26]

u = u0+sum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895+ sdot sdot sdot (4)

Substituting (4) into (3) yields

(A0+sum119894

120572119894A119894)(u

0+sum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895+ sdot sdot sdot ) = f

(5)

from which the components u0 u119894 u119894119895of u are solved as

follows

u0= Aminus10f

u119894= minusAminus10A119894u0

u119894119895= Aminus10A119894Aminus10A119895u0

(6)

It should be noted that in the stochastic context there has notbeen a rigorous convergence criterion for the perturbationscheme described above

In the Neumann expansion method the solution of (3) iswritten as

u = (A0+ ΔA)minus1f = (I + B)minus1Aminus1

0f (7)

where ΔA = sum119894120572119894A119894 B = Aminus1

0ΔA and I is the identity matrix

According to Neumann expansion

(I + B)minus1 = I minus B + B2 minus B3 + sdot sdot sdot (8)

solution (7) can be simplified as

u = u0minus Bu0+ B2u

0minus B3u

0+ sdot sdot sdot (9)

where u0= Aminus10f The spectral radius of matrix B must be

smaller than 1 to ensure the convergence of (9)

22 The Generalized Neumann Expansion Method TheNeu-mann expansion (8) can be generalized to the inverse ofmultiple matrices

(I +sum119894

120572119894B119894)

minus1

= I minussum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894B119895minus sdot sdot sdot (10)

The generalized Neumann expansion (GNE) in (10) can beproved as follows

(I +sum119894

120572119894B119894)(I +sum

119894

120572119894B119894)

minus1

= I minussum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894B119895minus sdot sdot sdot

+ 1205721B1minussum119894

1205721120572119894B1B119894+ sdot sdot sdot

+ 1205722B2minussum119894

1205722120572119894B2B119894+ sdot sdot sdot

+ 120572119899B119899minussum119894

120572119899120572119894B119899B119894+ sdot sdot sdot

= I

(11)

where 119899 is the total number of B119894

The Neumann expansion (8) and the generalized Neu-mann expansion (10) are equivalent in mathematics and onlydiffer in organizing thematrix operationsThis difference canlead to a significant acceleration in computing the solution of(3) Specifically let B

119894= Aminus10A119894 and applying the GNE in (10)

to (3) gives

u = (A0+sum119894

120572119894A119894)

minus1

f = (I +sum119894

120572119894Aminus10A119894)

minus1

Aminus10f

= u0minussum119894

120572119894Aminus10A119894u0+sum119894

sum119895

120572119895120572119894Aminus10A119895Aminus10A119894u0minus sdot sdot sdot

= u0minussum119894

120572119894u119894+sum119894

sum119895

120572119894120572119895u119894119895minus sdot sdot sdot

(12)

in which

u119894= Aminus10A119894u0

u119894119895= Aminus10A119894Aminus10A119895u0

(13)

Equation (12) is the GNE solution

23 Convergence Analysis and Error Estimation The conver-gence condition of the GNE (10) is essentially the same as theNeumann expansion (8) that is 120588(sum120572

119894B119894) lt 1 where 120588(sdot)

denotes the spectral radius of a matrix Hence for the GNEsolution in (12) the convergence condition is

120588 (sum120572119894Αminus1

0A119894) lt 1 (14)

It should be noted that

120588 (sum120572119894Αminus1

0A119894) = 120588 (sum120572

119894Cminus1A119894Cminus1) = 10038171003817100381710038171003817sum120572

119894Cminus1A119894Cminus110038171003817100381710038171003817119898

2

(15)

4 Mathematical Problems in Engineering

where sdot 1198982

denotes the spectral norm of a matrix and thesymmetric and positive definite matrix C is defined as

A0= C2 (16)

As a type ofmatrix norm the spectral norm has the followingproperties

P +Q1198982

le P1198982

+ Q1198982

PQ1198982

le P1198982Q119898

2

ℎP1198982

= ℎP1198982

(17)

where P and Q are general matrices and ℎ is a positiveconstant Hence10038171003817100381710038171003817sum120572119894Cminus1A119894Cminus110038171003817100381710038171003817119898

2

le100381710038171003817100381710038171205721Cminus1A1Cminus110038171003817100381710038171003817119898

2

+100381710038171003817100381710038171205722Cminus1A2Cminus110038171003817100381710038171003817119898

2

+ sdot sdot sdot10038171003817100381710038171003817120572119899Cminus1A119899Cminus110038171003817100381710038171003817119898

2

120588 (sum120572119894Cminus1A119894Cminus1)

le 120588 (1205721Cminus1A1Cminus1) + 120588 (120572

2Cminus1A2Cminus1)

+ sdot sdot sdot 120588 (120572119899Cminus1A119899Cminus1)

120588 (sum120572119894Αminus1

0A119894)

le 120588 (1205721Αminus1

0A1) + 120588 (120572

2Αminus1

0A2) + sdot sdot sdot 120588 (120572

119899Αminus1

0A119899)

le 119899max (120572119894)max (120588 (Αminus1

0A119894))

(18)

Based on the above relation a sufficient but not necessarycondition for convergence is

119899max (120572119894)max (120588 (Αminus1

0A119894)) lt 1 (19)

The maximum spectral radius 120588max = max(120588(Αminus10A119894)) can

be obtained from A0and A

119894 Equation (19) can be used

to verify the convergence of GNE in advance When 120572119894

are independent standard Gaussian random variables theirabsolute values are smaller than 3 with the probability of997 In this case (19) can be further simplified

119899max (120572119894)max (120588 (Αminus1

0A119894)) asymp 3119899120588max lt 1

120588max lt1

3119899

(20)

Considering the 119896th-order GNE (10) or Neumann expansion(8) their error can be estimated as follows

10038171003817100381710038171003817(I + B)minus1 minus (I minus B + B2 + sdot sdot sdot + (minus1)119896B119896)10038171003817100381710038171003817119898

2

=10038171003817100381710038171003817B119896+1(I + B)minus110038171003817100381710038171003817119898

2

le10038171003817100381710038171003817B119896+110038171003817100381710038171003817119898

2

10038171003817100381710038171003817(I + B)minus110038171003817100381710038171003817119898

2

=120590119896+1119861

1 minus 120590119861

(21)

where120590119861is the spectral normofB which is assumed to satisfy

120590119861lt 1 (22)

The exact solution of (3) is

u = (I +sum119894

120572119894Aminus10A119894)

minus1

Aminus10f = (I + B)minus1Aminus1

0f

Cu = (I +sum119894

120572119894Cminus1A119894Cminus1)minus1

Cminus1f

= (I +D)minus1Cminus1f

(23)

where B = sum119894120572119894Aminus10A119894 D = sum

119894120572119894Cminus1A119894Cminus1 The 119896th-order

GNE or Neumann expansion solution is given by

u = (I minus B + B2 + sdot sdot sdot + (minus1)119896B119896)Aminus10f

Cu = (I minusD +D2 + sdot sdot sdot + (minus1)119896D119896)Cminus1f (24)

The norm of the solution error is defined asCu minus Cu2

=10038171003817100381710038171003817[(I +D)

minus1 minus (I minusD +D2 + sdot sdot sdot + (minus1)119896D119896)]Cminus1f100381710038171003817100381710038172

=10038171003817100381710038171003817D119896+1(I +D)minus1Cminus1f100381710038171003817100381710038172 =

10038171003817100381710038171003817D119896+1Cu100381710038171003817100381710038172

le10038171003817100381710038171003817D119896+110038171003817100381710038171003817119898

2

Cu2 = 120590119896+1

119863Cu2 = 120588

119896+1

119861Cu2

(25)

where 120588119861is the spectral radius of B

The relative error of the solution is defined as

RE =radic(u minus u)119879A

0(u minus u)

radicu119879A0u

=radic(u minus u)119879C2 (u minus u)

radicu119879C2u=Cu minus Cu2Cu2

le 120588119896+1119861

(26)

Comparing (21) and (26) it can be seen that the accuracy forthe solution vector u is even better than the accuracy for thematrix inverse (I + B)minus1

24 Solutions forMore General Stochastic Algebraic EquationsFor the solution of (3) the GNE solution in (12) sharesthe same form as the perturbation solution in (5) Hencefor solving (3) the GNE method the perturbation methodand the standard Neumann expansion method are identicalThe error estimation obtained in (26) also holds for theperturbation approach

However for the following more general stochastic alge-braic equations [26]

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot ) u

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(27)

Mathematical Problems in Engineering 5

the three methods are not equivalent Specifically ap-plying the perturbation method to the above equationyields

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot )

times (u0+sum119894

120573119894u119894+sum119894

sum119895

120573119894120573119895u119894119895+ sdot sdot sdot )

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(28)

Equation (28) can be reorganized as

A0u0

+sum119894

120572119894A119894u0

+sum119894

sum119895

120572119894120572119895A119894119895u0

sdot sdot sdot

+sum119894

A0120573119894u119894

+sum119894

sum119895

120572119894A119894120573119895u119895

+sum119894

sum119895

sum119896

120572119894120572119895A119894119895120573119896u119896

sdot sdot sdot

+sum119894

sum119895

A0120573119894120573119895u119894119895+sum119894

sum119895

sum119896

120572119894A119894120573119895120573119896u119895119896+sum119894

sum119895

sum119896

sum119897

120572119894120572119895A119894119895120573119896120573119897u119896119897sdot sdot sdot

sdot sdot sdot sdot sdot sdot sdot sdot sdot d

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(29)

If 120572119894= 120573119894= 120574119894 then the perturbation components u

0 u119894 u119894119895

u11989411198942sdotsdotsdot119894119899

of u can be solved as

A0u0= f0

A0u119894+ A119894u0= f119894

A0u119894119895+ A119894u119895+ A119894119895u0= f119894119895

A0u11989411198942sdotsdotsdot119894119899

+ A1198941

u11989421198943sdotsdotsdot119894119899

+ A11989411198942

u11989431198944sdotsdotsdot119894119899

+ A11989411198942sdotsdotsdot119894119899

u0= f11989411198942sdotsdotsdot119894119899

(30)

Solving (27) with the Neumann expansion gives

Aminus1 = (A0+ ΔA)minus1

= (I + B)minus1Aminus10= (I minus B + B2 minus B3 + sdot sdot sdot )Aminus1

0

(31)

where B = Aminus10ΔA ΔA = sum

119894120572119894A119894+ sum119894sum119895120572119894120572119895A119894119895+ sdot sdot sdot Then

u = (I minus B + B2 minus B3 + sdot sdot sdot )Aminus10(f0+ Δf) (32)

in which

Δf = sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot (33)

Because higher order entries are included in B the termBAminus10(f0+ Δf) (the first-order term of the Neumann solution

in (32)) does not correspond to the first-order entries of uTherefore theNeumann solution in (32) and the perturbationsolution in (30) are different for the general stochasticequation (27)

Solving (27) with the GNE approach gives

u = (A0+sum119894119894

1205721198941

A1198941

+sum1198942

sum1198941

1205721198941

1205721198942

A11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

A11989411198942sdotsdotsdot119894119899

)

minus1

f

= (I +sum119894119894

1205721198941

B1198941

+sum1198942

sum1198941

1205721198941

1205721198942

B11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

B11989411198942sdotsdotsdot119894119899

)

minus1

Aminus10f

= (I +sum119894

120573119894C119894)

minus1

Aminus10f

= (I minussum119894

120573119894C119894+sum119894

sum119895

120573119894120573119895C119894C119895

minussum119894

sum119895

sum119896

120573119894120573119895120573119896C119894C119895C119896+ sdot sdot sdot )Aminus1

0f

(34)

where f = f0+ sum119894120574119894f119894+ sum119894sum119895120574119894120574119895f119894119895+ sdot sdot sdot B

119894= Aminus10A119894

B119894119895= Aminus10A119894119895 and other B terms can be similarly formed

Matrices C119894are determined by realignment of B

119894 B119894119895

and random variables 120573119894are formed by rearrangement of

1205721198941

12057211989411198942

12057211989411198942sdotsdotsdot119894119899

In summary when solving (27) the GNEmethod and the

Neumannmethod are identical while they are not equivalentto the perturbation procedure However if the higher orderterms in the GNE or Neumann expansion solutions areomitted they degenerate into the perturbation solution

6 Mathematical Problems in Engineering

x

y

p

Figure 1 Cross section of the random material sample

Two simplified examples are given here to demonstrate thedifference between the GNE and the perturbation method

241 Example I Only first-order entries are included in bothsides of (27) such that the equation becomes

(A0+sum119894

120572119894A119894) u = f

0+sum119894

120572119894f119894 (35)

The first-order perturbation solution is

u0= Aminus10f0

u119894= Aminus10f119894minus Aminus10A119894u0

u = u0+sum119894

120572119894u119894

= Aminus10f0+sum119894

120572119894Aminus10f119894minussum119894

120572119894Aminus10A119894u0

(36)

and the first-order GNE solution is

u = (Aminus10minussum119894

120572119894Aminus10A119894Aminus10)(f0+sum119894

120572119894f119894)

= Aminus10f0minussum119894

120572119894Aminus10A119894u0+sum119894

120572119894Aminus10f119894

minussum119894

sum119895

120572119894120572119895Aminus10A119894f119895

(37)

Comparing (36) and (37) it can be seen that theGNE solutioncontains an extra second-order term minussum

119894sum119895120572119894120572119895Aminus10A119894f119895

242 Example II Up to second-order terms are included inthe left-hand side of (27) while only the constant term isincluded in the right-hand side In this case the equationbecomes

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895) u = f

0 (38)

Denote u0= Aminus10f0 B119894= Aminus10A119894 and B

119894119895= Aminus10A119894119895 then the

first-order perturbation solution can be expressed as

u0= Aminus10f0

u119894= minussum119894

Aminus10A119894u0= minussum119894

B119894u0

u = u0+sum119894

120572119894u119894= u0minussum119894

120572119894B119894u0

(39)

while the first-order GNE solution is

u = (I +sum119894

120572119894Aminus10A119894+sum119894

sum119895

120572119894120572119895Aminus10A119894119895)

minus1

u0

= (I +sum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894119895)

minus1

u0

= (I minussum119894

120572119894B119894minussum119894

sum119895

120572119894120572119895B119894119895) u0

(40)

Comparing (39) and (40) the second-order termsminussum119894sum119895120572119894120572119895Aminus10A119894119895u0in the GNE solution do not appear in

the perturbation solution (39)

3 Numerical Examples

31 Example Description As shown in Figure 1 a plain strainproblem with an orthohexagonal cross section is consideredhere The length of each edge of the hexagon is 04m ThePoisson ratio of thematerial is set as 02 andYoungrsquosmodulusis assumed to be a Gaussian field with amean value of 30GPaand a covariance function defined as

119877 (1205911 1205912) = 1198881119890minus120591121198882minus1205912

21198882 (41)

in which 1205911= 1199091minus 1199092and 1205912= 1199101minus 1199102are the coordinate

differences and 1198881and 1198882are adjustable constants As shown in

Figure 2 the cross section is discretized with triangular linearelements where each side is divided into 8 equal parts Thereare in total 384 elements and 217 nodes

Mathematical Problems in Engineering 7

Figure 2 Finite element mesh

The relative error is defined as

RE =1003817100381710038171003817CuGNE minus CuMC

100381710038171003817100381721003817100381710038171003817CuMC

10038171003817100381710038172times 100 (42)

inwhichuMC is the nodal displacement vector solvedwith theMonte Carlo method and uGNE is the GNE solution Unlessstated explicitly the numerical testing is performed for 4340samples generated from the Gaussian field defined in (41)

The convergence condition of the GNE method has beenrigorously proved The main purpose of this example isto examine the error distribution and its relationship withvarious randomparameters forwhich a number of numericaltests are performed in the following subsections

32 Error Estimation The third-order GNE solution isadoptedThe constants 119888

1and 1198882in (41) are varied to examine

the error estimator in (26) The constant 1198881is the variance of

the Gaussian field and the constant 1198882controls the shape of

the covariance function The effective correlation length 119871effof the covariance function 119877(120591) is defined by

119877 (119871eff)

119877 (0)= 001 (43)

For any two points in the medium if the distance betweenthem is larger than 119871eff their material properties are onlyweakly correlated The effective correlation length of theGaussian field defined in (41) is

119877 (120591)

119877 (0)= 119890minus120591

21198882 = 001 997904rArr 119871eff = 2146radic1198882 (44)

In the first set of tests the constant 1198881is varied to change the

variance of the Gaussian field The parameter settings are

1198881= 001 119888

2= 32119864 minus 2 119871eff = 038

1198881= 004 119888

2= 32119864 minus 2 119871eff = 038

(45)

In the second set of tests the constant 1198882is varied to change

the shape of the covariance function The parameter settingsare

1198881= 001 119888

2= 08119864 minus 2 119871eff = 019

1198881= 001 119888

2= 24119864 minus 2 119871eff = 033

1198881= 001 119888

2= 40119864 minus 2 119871eff = 043

(46)

321 The Influence of 1198881 Corresponding to 119888

1= 001 and

1198881= 004 the standard deviation 120590 of the stochastic field

is 01 and 02 respectively A total number of 4340 samplesare generated for each case For each realization both theMonte Carlo solution and the third-order GNE solution arecomputed The relationship between the spectral radius andthe relative error is drawn in Figures 3 and 4 in which thered crosses show the actual relative errors corresponding toeach sample and the blue line shows the theoretical errorestimation It can been seen that for both cases the theoreticalerror estimator in (26) holds accurately and its effectivenessis not affected by the change of the variance controlled by 119888

1

322 The Influence of 1198882 The constant 119888

2is adjusted to take

three different values 08 times 10minus2 24 times 10minus2 and 40 times 10minus2while the constant 119888

1remains unchanged Following (41)

the effective correlation length grows with the increase of1198882 Again 4340 samples are generated for each stochastic

field and for each realization both the Monte Carlo solutionand the third-order GNE solution are computed Figure 5plots the relationship between spectral radius and the relativeerror which verifies that for all cases the theoretical errorestimation in (26) holds The changing of 119888

2 which changes

the variation frequency of the stochastic field does not affectthe error estimation

As indicated in (44) the effective correlation length 119871effis directly associated with the parameter 119888

2 To investigate

the influence of 119871eff on the relative error the empiricalcumulative distribution functions (ECDF) of relative errorsare plotted in Figure 6 for different 119871eff cases Comparing thethree ECDF curves at different effective correlation lengthsit is found that with the decrease of the effective correlationlength the accuracy of the GNE solution improves

33 The Relative Error for Other Types of Stochastic FieldsIn the last two subsections the error estimation analysis hastaken the Gaussian assumption that is all random variables120572119894in (3) are assumed to be independent Gaussian random

variables In this subsection we investigate the relative errorof the GNE method for other types of stochastic fieldsSpecifically the random variables 120572

119894in (3) are assumed to

have the two-point distribution the binomial distributionand the 120573 distribution respectively For the case of two-pointdistribution values minus03119864

0 031198640 are assigned with equal

probability that is 05 For the case of binomial distributionten times Bernoulli trials with the probability 05 are adopted

8 Mathematical Problems in Engineering

minus14 minus12 minus10 minus8 minus6 minus4minus14

minus12

minus10

minus8

minus6

minus4

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

Solution with c1 = 001

log (RE) = 4 log (120588B)

4 log (120588B)

Figure 3 The relation between the relative error and the spectral radius

minus11 minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 0minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Solution with c1 = 004

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 4 The relation between the relative error and the spectral radius

and the integer values between 0 and 10 are linearly projectedonto [minus03119864

0 031198640] For the case of 120573 distribution the

probability density function is defined as

119891 (119909 | 119886 119887) =1

119861 (119886 119887)119909119886minus1(1 minus 119909)119887minus1119868

(01)(119909) (47)

where 119861(119886 119887) is a 120573 function and the filter function 119868(01)(119909)

is defined as

119868(01) (119909) =

1 119909 isin (0 1)

0 119909 notin (0 1) (48)

Three kinds of 120573 distributions are considered with 119886 = 119887 =075 119886 = 119887 = 1 and 119886 = 119887 = 4 respectivelyThe interval (0 1)is linearly mapped to [minus03119864

0 031198640]

Corresponding to these five distributions the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7 Comparing the ECDF curves at differenteffective correlation lengths the same trend is observed asthe Gaussian case That is the relative error decreases withthe drop of effective correlation length

34 Efficiency Analysis Using the same computing platformthe Monte Carlo method the Neumann expansion method

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

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Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

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Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

4 Mathematical Problems in Engineering

where sdot 1198982

denotes the spectral norm of a matrix and thesymmetric and positive definite matrix C is defined as

A0= C2 (16)

As a type ofmatrix norm the spectral norm has the followingproperties

P +Q1198982

le P1198982

+ Q1198982

PQ1198982

le P1198982Q119898

2

ℎP1198982

= ℎP1198982

(17)

where P and Q are general matrices and ℎ is a positiveconstant Hence10038171003817100381710038171003817sum120572119894Cminus1A119894Cminus110038171003817100381710038171003817119898

2

le100381710038171003817100381710038171205721Cminus1A1Cminus110038171003817100381710038171003817119898

2

+100381710038171003817100381710038171205722Cminus1A2Cminus110038171003817100381710038171003817119898

2

+ sdot sdot sdot10038171003817100381710038171003817120572119899Cminus1A119899Cminus110038171003817100381710038171003817119898

2

120588 (sum120572119894Cminus1A119894Cminus1)

le 120588 (1205721Cminus1A1Cminus1) + 120588 (120572

2Cminus1A2Cminus1)

+ sdot sdot sdot 120588 (120572119899Cminus1A119899Cminus1)

120588 (sum120572119894Αminus1

0A119894)

le 120588 (1205721Αminus1

0A1) + 120588 (120572

2Αminus1

0A2) + sdot sdot sdot 120588 (120572

119899Αminus1

0A119899)

le 119899max (120572119894)max (120588 (Αminus1

0A119894))

(18)

Based on the above relation a sufficient but not necessarycondition for convergence is

119899max (120572119894)max (120588 (Αminus1

0A119894)) lt 1 (19)

The maximum spectral radius 120588max = max(120588(Αminus10A119894)) can

be obtained from A0and A

119894 Equation (19) can be used

to verify the convergence of GNE in advance When 120572119894

are independent standard Gaussian random variables theirabsolute values are smaller than 3 with the probability of997 In this case (19) can be further simplified

119899max (120572119894)max (120588 (Αminus1

0A119894)) asymp 3119899120588max lt 1

120588max lt1

3119899

(20)

Considering the 119896th-order GNE (10) or Neumann expansion(8) their error can be estimated as follows

10038171003817100381710038171003817(I + B)minus1 minus (I minus B + B2 + sdot sdot sdot + (minus1)119896B119896)10038171003817100381710038171003817119898

2

=10038171003817100381710038171003817B119896+1(I + B)minus110038171003817100381710038171003817119898

2

le10038171003817100381710038171003817B119896+110038171003817100381710038171003817119898

2

10038171003817100381710038171003817(I + B)minus110038171003817100381710038171003817119898

2

=120590119896+1119861

1 minus 120590119861

(21)

where120590119861is the spectral normofB which is assumed to satisfy

120590119861lt 1 (22)

The exact solution of (3) is

u = (I +sum119894

120572119894Aminus10A119894)

minus1

Aminus10f = (I + B)minus1Aminus1

0f

Cu = (I +sum119894

120572119894Cminus1A119894Cminus1)minus1

Cminus1f

= (I +D)minus1Cminus1f

(23)

where B = sum119894120572119894Aminus10A119894 D = sum

119894120572119894Cminus1A119894Cminus1 The 119896th-order

GNE or Neumann expansion solution is given by

u = (I minus B + B2 + sdot sdot sdot + (minus1)119896B119896)Aminus10f

Cu = (I minusD +D2 + sdot sdot sdot + (minus1)119896D119896)Cminus1f (24)

The norm of the solution error is defined asCu minus Cu2

=10038171003817100381710038171003817[(I +D)

minus1 minus (I minusD +D2 + sdot sdot sdot + (minus1)119896D119896)]Cminus1f100381710038171003817100381710038172

=10038171003817100381710038171003817D119896+1(I +D)minus1Cminus1f100381710038171003817100381710038172 =

10038171003817100381710038171003817D119896+1Cu100381710038171003817100381710038172

le10038171003817100381710038171003817D119896+110038171003817100381710038171003817119898

2

Cu2 = 120590119896+1

119863Cu2 = 120588

119896+1

119861Cu2

(25)

where 120588119861is the spectral radius of B

The relative error of the solution is defined as

RE =radic(u minus u)119879A

0(u minus u)

radicu119879A0u

=radic(u minus u)119879C2 (u minus u)

radicu119879C2u=Cu minus Cu2Cu2

le 120588119896+1119861

(26)

Comparing (21) and (26) it can be seen that the accuracy forthe solution vector u is even better than the accuracy for thematrix inverse (I + B)minus1

24 Solutions forMore General Stochastic Algebraic EquationsFor the solution of (3) the GNE solution in (12) sharesthe same form as the perturbation solution in (5) Hencefor solving (3) the GNE method the perturbation methodand the standard Neumann expansion method are identicalThe error estimation obtained in (26) also holds for theperturbation approach

However for the following more general stochastic alge-braic equations [26]

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot ) u

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(27)

Mathematical Problems in Engineering 5

the three methods are not equivalent Specifically ap-plying the perturbation method to the above equationyields

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot )

times (u0+sum119894

120573119894u119894+sum119894

sum119895

120573119894120573119895u119894119895+ sdot sdot sdot )

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(28)

Equation (28) can be reorganized as

A0u0

+sum119894

120572119894A119894u0

+sum119894

sum119895

120572119894120572119895A119894119895u0

sdot sdot sdot

+sum119894

A0120573119894u119894

+sum119894

sum119895

120572119894A119894120573119895u119895

+sum119894

sum119895

sum119896

120572119894120572119895A119894119895120573119896u119896

sdot sdot sdot

+sum119894

sum119895

A0120573119894120573119895u119894119895+sum119894

sum119895

sum119896

120572119894A119894120573119895120573119896u119895119896+sum119894

sum119895

sum119896

sum119897

120572119894120572119895A119894119895120573119896120573119897u119896119897sdot sdot sdot

sdot sdot sdot sdot sdot sdot sdot sdot sdot d

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(29)

If 120572119894= 120573119894= 120574119894 then the perturbation components u

0 u119894 u119894119895

u11989411198942sdotsdotsdot119894119899

of u can be solved as

A0u0= f0

A0u119894+ A119894u0= f119894

A0u119894119895+ A119894u119895+ A119894119895u0= f119894119895

A0u11989411198942sdotsdotsdot119894119899

+ A1198941

u11989421198943sdotsdotsdot119894119899

+ A11989411198942

u11989431198944sdotsdotsdot119894119899

+ A11989411198942sdotsdotsdot119894119899

u0= f11989411198942sdotsdotsdot119894119899

(30)

Solving (27) with the Neumann expansion gives

Aminus1 = (A0+ ΔA)minus1

= (I + B)minus1Aminus10= (I minus B + B2 minus B3 + sdot sdot sdot )Aminus1

0

(31)

where B = Aminus10ΔA ΔA = sum

119894120572119894A119894+ sum119894sum119895120572119894120572119895A119894119895+ sdot sdot sdot Then

u = (I minus B + B2 minus B3 + sdot sdot sdot )Aminus10(f0+ Δf) (32)

in which

Δf = sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot (33)

Because higher order entries are included in B the termBAminus10(f0+ Δf) (the first-order term of the Neumann solution

in (32)) does not correspond to the first-order entries of uTherefore theNeumann solution in (32) and the perturbationsolution in (30) are different for the general stochasticequation (27)

Solving (27) with the GNE approach gives

u = (A0+sum119894119894

1205721198941

A1198941

+sum1198942

sum1198941

1205721198941

1205721198942

A11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

A11989411198942sdotsdotsdot119894119899

)

minus1

f

= (I +sum119894119894

1205721198941

B1198941

+sum1198942

sum1198941

1205721198941

1205721198942

B11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

B11989411198942sdotsdotsdot119894119899

)

minus1

Aminus10f

= (I +sum119894

120573119894C119894)

minus1

Aminus10f

= (I minussum119894

120573119894C119894+sum119894

sum119895

120573119894120573119895C119894C119895

minussum119894

sum119895

sum119896

120573119894120573119895120573119896C119894C119895C119896+ sdot sdot sdot )Aminus1

0f

(34)

where f = f0+ sum119894120574119894f119894+ sum119894sum119895120574119894120574119895f119894119895+ sdot sdot sdot B

119894= Aminus10A119894

B119894119895= Aminus10A119894119895 and other B terms can be similarly formed

Matrices C119894are determined by realignment of B

119894 B119894119895

and random variables 120573119894are formed by rearrangement of

1205721198941

12057211989411198942

12057211989411198942sdotsdotsdot119894119899

In summary when solving (27) the GNEmethod and the

Neumannmethod are identical while they are not equivalentto the perturbation procedure However if the higher orderterms in the GNE or Neumann expansion solutions areomitted they degenerate into the perturbation solution

6 Mathematical Problems in Engineering

x

y

p

Figure 1 Cross section of the random material sample

Two simplified examples are given here to demonstrate thedifference between the GNE and the perturbation method

241 Example I Only first-order entries are included in bothsides of (27) such that the equation becomes

(A0+sum119894

120572119894A119894) u = f

0+sum119894

120572119894f119894 (35)

The first-order perturbation solution is

u0= Aminus10f0

u119894= Aminus10f119894minus Aminus10A119894u0

u = u0+sum119894

120572119894u119894

= Aminus10f0+sum119894

120572119894Aminus10f119894minussum119894

120572119894Aminus10A119894u0

(36)

and the first-order GNE solution is

u = (Aminus10minussum119894

120572119894Aminus10A119894Aminus10)(f0+sum119894

120572119894f119894)

= Aminus10f0minussum119894

120572119894Aminus10A119894u0+sum119894

120572119894Aminus10f119894

minussum119894

sum119895

120572119894120572119895Aminus10A119894f119895

(37)

Comparing (36) and (37) it can be seen that theGNE solutioncontains an extra second-order term minussum

119894sum119895120572119894120572119895Aminus10A119894f119895

242 Example II Up to second-order terms are included inthe left-hand side of (27) while only the constant term isincluded in the right-hand side In this case the equationbecomes

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895) u = f

0 (38)

Denote u0= Aminus10f0 B119894= Aminus10A119894 and B

119894119895= Aminus10A119894119895 then the

first-order perturbation solution can be expressed as

u0= Aminus10f0

u119894= minussum119894

Aminus10A119894u0= minussum119894

B119894u0

u = u0+sum119894

120572119894u119894= u0minussum119894

120572119894B119894u0

(39)

while the first-order GNE solution is

u = (I +sum119894

120572119894Aminus10A119894+sum119894

sum119895

120572119894120572119895Aminus10A119894119895)

minus1

u0

= (I +sum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894119895)

minus1

u0

= (I minussum119894

120572119894B119894minussum119894

sum119895

120572119894120572119895B119894119895) u0

(40)

Comparing (39) and (40) the second-order termsminussum119894sum119895120572119894120572119895Aminus10A119894119895u0in the GNE solution do not appear in

the perturbation solution (39)

3 Numerical Examples

31 Example Description As shown in Figure 1 a plain strainproblem with an orthohexagonal cross section is consideredhere The length of each edge of the hexagon is 04m ThePoisson ratio of thematerial is set as 02 andYoungrsquosmodulusis assumed to be a Gaussian field with amean value of 30GPaand a covariance function defined as

119877 (1205911 1205912) = 1198881119890minus120591121198882minus1205912

21198882 (41)

in which 1205911= 1199091minus 1199092and 1205912= 1199101minus 1199102are the coordinate

differences and 1198881and 1198882are adjustable constants As shown in

Figure 2 the cross section is discretized with triangular linearelements where each side is divided into 8 equal parts Thereare in total 384 elements and 217 nodes

Mathematical Problems in Engineering 7

Figure 2 Finite element mesh

The relative error is defined as

RE =1003817100381710038171003817CuGNE minus CuMC

100381710038171003817100381721003817100381710038171003817CuMC

10038171003817100381710038172times 100 (42)

inwhichuMC is the nodal displacement vector solvedwith theMonte Carlo method and uGNE is the GNE solution Unlessstated explicitly the numerical testing is performed for 4340samples generated from the Gaussian field defined in (41)

The convergence condition of the GNE method has beenrigorously proved The main purpose of this example isto examine the error distribution and its relationship withvarious randomparameters forwhich a number of numericaltests are performed in the following subsections

32 Error Estimation The third-order GNE solution isadoptedThe constants 119888

1and 1198882in (41) are varied to examine

the error estimator in (26) The constant 1198881is the variance of

the Gaussian field and the constant 1198882controls the shape of

the covariance function The effective correlation length 119871effof the covariance function 119877(120591) is defined by

119877 (119871eff)

119877 (0)= 001 (43)

For any two points in the medium if the distance betweenthem is larger than 119871eff their material properties are onlyweakly correlated The effective correlation length of theGaussian field defined in (41) is

119877 (120591)

119877 (0)= 119890minus120591

21198882 = 001 997904rArr 119871eff = 2146radic1198882 (44)

In the first set of tests the constant 1198881is varied to change the

variance of the Gaussian field The parameter settings are

1198881= 001 119888

2= 32119864 minus 2 119871eff = 038

1198881= 004 119888

2= 32119864 minus 2 119871eff = 038

(45)

In the second set of tests the constant 1198882is varied to change

the shape of the covariance function The parameter settingsare

1198881= 001 119888

2= 08119864 minus 2 119871eff = 019

1198881= 001 119888

2= 24119864 minus 2 119871eff = 033

1198881= 001 119888

2= 40119864 minus 2 119871eff = 043

(46)

321 The Influence of 1198881 Corresponding to 119888

1= 001 and

1198881= 004 the standard deviation 120590 of the stochastic field

is 01 and 02 respectively A total number of 4340 samplesare generated for each case For each realization both theMonte Carlo solution and the third-order GNE solution arecomputed The relationship between the spectral radius andthe relative error is drawn in Figures 3 and 4 in which thered crosses show the actual relative errors corresponding toeach sample and the blue line shows the theoretical errorestimation It can been seen that for both cases the theoreticalerror estimator in (26) holds accurately and its effectivenessis not affected by the change of the variance controlled by 119888

1

322 The Influence of 1198882 The constant 119888

2is adjusted to take

three different values 08 times 10minus2 24 times 10minus2 and 40 times 10minus2while the constant 119888

1remains unchanged Following (41)

the effective correlation length grows with the increase of1198882 Again 4340 samples are generated for each stochastic

field and for each realization both the Monte Carlo solutionand the third-order GNE solution are computed Figure 5plots the relationship between spectral radius and the relativeerror which verifies that for all cases the theoretical errorestimation in (26) holds The changing of 119888

2 which changes

the variation frequency of the stochastic field does not affectthe error estimation

As indicated in (44) the effective correlation length 119871effis directly associated with the parameter 119888

2 To investigate

the influence of 119871eff on the relative error the empiricalcumulative distribution functions (ECDF) of relative errorsare plotted in Figure 6 for different 119871eff cases Comparing thethree ECDF curves at different effective correlation lengthsit is found that with the decrease of the effective correlationlength the accuracy of the GNE solution improves

33 The Relative Error for Other Types of Stochastic FieldsIn the last two subsections the error estimation analysis hastaken the Gaussian assumption that is all random variables120572119894in (3) are assumed to be independent Gaussian random

variables In this subsection we investigate the relative errorof the GNE method for other types of stochastic fieldsSpecifically the random variables 120572

119894in (3) are assumed to

have the two-point distribution the binomial distributionand the 120573 distribution respectively For the case of two-pointdistribution values minus03119864

0 031198640 are assigned with equal

probability that is 05 For the case of binomial distributionten times Bernoulli trials with the probability 05 are adopted

8 Mathematical Problems in Engineering

minus14 minus12 minus10 minus8 minus6 minus4minus14

minus12

minus10

minus8

minus6

minus4

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

Solution with c1 = 001

log (RE) = 4 log (120588B)

4 log (120588B)

Figure 3 The relation between the relative error and the spectral radius

minus11 minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 0minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Solution with c1 = 004

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 4 The relation between the relative error and the spectral radius

and the integer values between 0 and 10 are linearly projectedonto [minus03119864

0 031198640] For the case of 120573 distribution the

probability density function is defined as

119891 (119909 | 119886 119887) =1

119861 (119886 119887)119909119886minus1(1 minus 119909)119887minus1119868

(01)(119909) (47)

where 119861(119886 119887) is a 120573 function and the filter function 119868(01)(119909)

is defined as

119868(01) (119909) =

1 119909 isin (0 1)

0 119909 notin (0 1) (48)

Three kinds of 120573 distributions are considered with 119886 = 119887 =075 119886 = 119887 = 1 and 119886 = 119887 = 4 respectivelyThe interval (0 1)is linearly mapped to [minus03119864

0 031198640]

Corresponding to these five distributions the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7 Comparing the ECDF curves at differenteffective correlation lengths the same trend is observed asthe Gaussian case That is the relative error decreases withthe drop of effective correlation length

34 Efficiency Analysis Using the same computing platformthe Monte Carlo method the Neumann expansion method

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

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International Journal of

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ProbabilityandStatistics

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Stochastic AnalysisInternational Journal of

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ISRN Discrete Mathematics

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Differential EquationsInternational Journal of

Volume 2013

Mathematical Problems in Engineering 5

the three methods are not equivalent Specifically ap-plying the perturbation method to the above equationyields

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895+ sdot sdot sdot )

times (u0+sum119894

120573119894u119894+sum119894

sum119895

120573119894120573119895u119894119895+ sdot sdot sdot )

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(28)

Equation (28) can be reorganized as

A0u0

+sum119894

120572119894A119894u0

+sum119894

sum119895

120572119894120572119895A119894119895u0

sdot sdot sdot

+sum119894

A0120573119894u119894

+sum119894

sum119895

120572119894A119894120573119895u119895

+sum119894

sum119895

sum119896

120572119894120572119895A119894119895120573119896u119896

sdot sdot sdot

+sum119894

sum119895

A0120573119894120573119895u119894119895+sum119894

sum119895

sum119896

120572119894A119894120573119895120573119896u119895119896+sum119894

sum119895

sum119896

sum119897

120572119894120572119895A119894119895120573119896120573119897u119896119897sdot sdot sdot

sdot sdot sdot sdot sdot sdot sdot sdot sdot d

= f0+sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot

(29)

If 120572119894= 120573119894= 120574119894 then the perturbation components u

0 u119894 u119894119895

u11989411198942sdotsdotsdot119894119899

of u can be solved as

A0u0= f0

A0u119894+ A119894u0= f119894

A0u119894119895+ A119894u119895+ A119894119895u0= f119894119895

A0u11989411198942sdotsdotsdot119894119899

+ A1198941

u11989421198943sdotsdotsdot119894119899

+ A11989411198942

u11989431198944sdotsdotsdot119894119899

+ A11989411198942sdotsdotsdot119894119899

u0= f11989411198942sdotsdotsdot119894119899

(30)

Solving (27) with the Neumann expansion gives

Aminus1 = (A0+ ΔA)minus1

= (I + B)minus1Aminus10= (I minus B + B2 minus B3 + sdot sdot sdot )Aminus1

0

(31)

where B = Aminus10ΔA ΔA = sum

119894120572119894A119894+ sum119894sum119895120572119894120572119895A119894119895+ sdot sdot sdot Then

u = (I minus B + B2 minus B3 + sdot sdot sdot )Aminus10(f0+ Δf) (32)

in which

Δf = sum119894

120574119894f119894+sum119894

sum119895

120574119894120574119895f119894119895+ sdot sdot sdot (33)

Because higher order entries are included in B the termBAminus10(f0+ Δf) (the first-order term of the Neumann solution

in (32)) does not correspond to the first-order entries of uTherefore theNeumann solution in (32) and the perturbationsolution in (30) are different for the general stochasticequation (27)

Solving (27) with the GNE approach gives

u = (A0+sum119894119894

1205721198941

A1198941

+sum1198942

sum1198941

1205721198941

1205721198942

A11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

A11989411198942sdotsdotsdot119894119899

)

minus1

f

= (I +sum119894119894

1205721198941

B1198941

+sum1198942

sum1198941

1205721198941

1205721198942

B11989411198942

+ sdot sdot sdot

+sum119894119899

sdot sdot sdotsum1198942

sum1198941

12057211989411198942sdotsdotsdot119894119899

B11989411198942sdotsdotsdot119894119899

)

minus1

Aminus10f

= (I +sum119894

120573119894C119894)

minus1

Aminus10f

= (I minussum119894

120573119894C119894+sum119894

sum119895

120573119894120573119895C119894C119895

minussum119894

sum119895

sum119896

120573119894120573119895120573119896C119894C119895C119896+ sdot sdot sdot )Aminus1

0f

(34)

where f = f0+ sum119894120574119894f119894+ sum119894sum119895120574119894120574119895f119894119895+ sdot sdot sdot B

119894= Aminus10A119894

B119894119895= Aminus10A119894119895 and other B terms can be similarly formed

Matrices C119894are determined by realignment of B

119894 B119894119895

and random variables 120573119894are formed by rearrangement of

1205721198941

12057211989411198942

12057211989411198942sdotsdotsdot119894119899

In summary when solving (27) the GNEmethod and the

Neumannmethod are identical while they are not equivalentto the perturbation procedure However if the higher orderterms in the GNE or Neumann expansion solutions areomitted they degenerate into the perturbation solution

6 Mathematical Problems in Engineering

x

y

p

Figure 1 Cross section of the random material sample

Two simplified examples are given here to demonstrate thedifference between the GNE and the perturbation method

241 Example I Only first-order entries are included in bothsides of (27) such that the equation becomes

(A0+sum119894

120572119894A119894) u = f

0+sum119894

120572119894f119894 (35)

The first-order perturbation solution is

u0= Aminus10f0

u119894= Aminus10f119894minus Aminus10A119894u0

u = u0+sum119894

120572119894u119894

= Aminus10f0+sum119894

120572119894Aminus10f119894minussum119894

120572119894Aminus10A119894u0

(36)

and the first-order GNE solution is

u = (Aminus10minussum119894

120572119894Aminus10A119894Aminus10)(f0+sum119894

120572119894f119894)

= Aminus10f0minussum119894

120572119894Aminus10A119894u0+sum119894

120572119894Aminus10f119894

minussum119894

sum119895

120572119894120572119895Aminus10A119894f119895

(37)

Comparing (36) and (37) it can be seen that theGNE solutioncontains an extra second-order term minussum

119894sum119895120572119894120572119895Aminus10A119894f119895

242 Example II Up to second-order terms are included inthe left-hand side of (27) while only the constant term isincluded in the right-hand side In this case the equationbecomes

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895) u = f

0 (38)

Denote u0= Aminus10f0 B119894= Aminus10A119894 and B

119894119895= Aminus10A119894119895 then the

first-order perturbation solution can be expressed as

u0= Aminus10f0

u119894= minussum119894

Aminus10A119894u0= minussum119894

B119894u0

u = u0+sum119894

120572119894u119894= u0minussum119894

120572119894B119894u0

(39)

while the first-order GNE solution is

u = (I +sum119894

120572119894Aminus10A119894+sum119894

sum119895

120572119894120572119895Aminus10A119894119895)

minus1

u0

= (I +sum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894119895)

minus1

u0

= (I minussum119894

120572119894B119894minussum119894

sum119895

120572119894120572119895B119894119895) u0

(40)

Comparing (39) and (40) the second-order termsminussum119894sum119895120572119894120572119895Aminus10A119894119895u0in the GNE solution do not appear in

the perturbation solution (39)

3 Numerical Examples

31 Example Description As shown in Figure 1 a plain strainproblem with an orthohexagonal cross section is consideredhere The length of each edge of the hexagon is 04m ThePoisson ratio of thematerial is set as 02 andYoungrsquosmodulusis assumed to be a Gaussian field with amean value of 30GPaand a covariance function defined as

119877 (1205911 1205912) = 1198881119890minus120591121198882minus1205912

21198882 (41)

in which 1205911= 1199091minus 1199092and 1205912= 1199101minus 1199102are the coordinate

differences and 1198881and 1198882are adjustable constants As shown in

Figure 2 the cross section is discretized with triangular linearelements where each side is divided into 8 equal parts Thereare in total 384 elements and 217 nodes

Mathematical Problems in Engineering 7

Figure 2 Finite element mesh

The relative error is defined as

RE =1003817100381710038171003817CuGNE minus CuMC

100381710038171003817100381721003817100381710038171003817CuMC

10038171003817100381710038172times 100 (42)

inwhichuMC is the nodal displacement vector solvedwith theMonte Carlo method and uGNE is the GNE solution Unlessstated explicitly the numerical testing is performed for 4340samples generated from the Gaussian field defined in (41)

The convergence condition of the GNE method has beenrigorously proved The main purpose of this example isto examine the error distribution and its relationship withvarious randomparameters forwhich a number of numericaltests are performed in the following subsections

32 Error Estimation The third-order GNE solution isadoptedThe constants 119888

1and 1198882in (41) are varied to examine

the error estimator in (26) The constant 1198881is the variance of

the Gaussian field and the constant 1198882controls the shape of

the covariance function The effective correlation length 119871effof the covariance function 119877(120591) is defined by

119877 (119871eff)

119877 (0)= 001 (43)

For any two points in the medium if the distance betweenthem is larger than 119871eff their material properties are onlyweakly correlated The effective correlation length of theGaussian field defined in (41) is

119877 (120591)

119877 (0)= 119890minus120591

21198882 = 001 997904rArr 119871eff = 2146radic1198882 (44)

In the first set of tests the constant 1198881is varied to change the

variance of the Gaussian field The parameter settings are

1198881= 001 119888

2= 32119864 minus 2 119871eff = 038

1198881= 004 119888

2= 32119864 minus 2 119871eff = 038

(45)

In the second set of tests the constant 1198882is varied to change

the shape of the covariance function The parameter settingsare

1198881= 001 119888

2= 08119864 minus 2 119871eff = 019

1198881= 001 119888

2= 24119864 minus 2 119871eff = 033

1198881= 001 119888

2= 40119864 minus 2 119871eff = 043

(46)

321 The Influence of 1198881 Corresponding to 119888

1= 001 and

1198881= 004 the standard deviation 120590 of the stochastic field

is 01 and 02 respectively A total number of 4340 samplesare generated for each case For each realization both theMonte Carlo solution and the third-order GNE solution arecomputed The relationship between the spectral radius andthe relative error is drawn in Figures 3 and 4 in which thered crosses show the actual relative errors corresponding toeach sample and the blue line shows the theoretical errorestimation It can been seen that for both cases the theoreticalerror estimator in (26) holds accurately and its effectivenessis not affected by the change of the variance controlled by 119888

1

322 The Influence of 1198882 The constant 119888

2is adjusted to take

three different values 08 times 10minus2 24 times 10minus2 and 40 times 10minus2while the constant 119888

1remains unchanged Following (41)

the effective correlation length grows with the increase of1198882 Again 4340 samples are generated for each stochastic

field and for each realization both the Monte Carlo solutionand the third-order GNE solution are computed Figure 5plots the relationship between spectral radius and the relativeerror which verifies that for all cases the theoretical errorestimation in (26) holds The changing of 119888

2 which changes

the variation frequency of the stochastic field does not affectthe error estimation

As indicated in (44) the effective correlation length 119871effis directly associated with the parameter 119888

2 To investigate

the influence of 119871eff on the relative error the empiricalcumulative distribution functions (ECDF) of relative errorsare plotted in Figure 6 for different 119871eff cases Comparing thethree ECDF curves at different effective correlation lengthsit is found that with the decrease of the effective correlationlength the accuracy of the GNE solution improves

33 The Relative Error for Other Types of Stochastic FieldsIn the last two subsections the error estimation analysis hastaken the Gaussian assumption that is all random variables120572119894in (3) are assumed to be independent Gaussian random

variables In this subsection we investigate the relative errorof the GNE method for other types of stochastic fieldsSpecifically the random variables 120572

119894in (3) are assumed to

have the two-point distribution the binomial distributionand the 120573 distribution respectively For the case of two-pointdistribution values minus03119864

0 031198640 are assigned with equal

probability that is 05 For the case of binomial distributionten times Bernoulli trials with the probability 05 are adopted

8 Mathematical Problems in Engineering

minus14 minus12 minus10 minus8 minus6 minus4minus14

minus12

minus10

minus8

minus6

minus4

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

Solution with c1 = 001

log (RE) = 4 log (120588B)

4 log (120588B)

Figure 3 The relation between the relative error and the spectral radius

minus11 minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 0minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Solution with c1 = 004

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 4 The relation between the relative error and the spectral radius

and the integer values between 0 and 10 are linearly projectedonto [minus03119864

0 031198640] For the case of 120573 distribution the

probability density function is defined as

119891 (119909 | 119886 119887) =1

119861 (119886 119887)119909119886minus1(1 minus 119909)119887minus1119868

(01)(119909) (47)

where 119861(119886 119887) is a 120573 function and the filter function 119868(01)(119909)

is defined as

119868(01) (119909) =

1 119909 isin (0 1)

0 119909 notin (0 1) (48)

Three kinds of 120573 distributions are considered with 119886 = 119887 =075 119886 = 119887 = 1 and 119886 = 119887 = 4 respectivelyThe interval (0 1)is linearly mapped to [minus03119864

0 031198640]

Corresponding to these five distributions the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7 Comparing the ECDF curves at differenteffective correlation lengths the same trend is observed asthe Gaussian case That is the relative error decreases withthe drop of effective correlation length

34 Efficiency Analysis Using the same computing platformthe Monte Carlo method the Neumann expansion method

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

6 Mathematical Problems in Engineering

x

y

p

Figure 1 Cross section of the random material sample

Two simplified examples are given here to demonstrate thedifference between the GNE and the perturbation method

241 Example I Only first-order entries are included in bothsides of (27) such that the equation becomes

(A0+sum119894

120572119894A119894) u = f

0+sum119894

120572119894f119894 (35)

The first-order perturbation solution is

u0= Aminus10f0

u119894= Aminus10f119894minus Aminus10A119894u0

u = u0+sum119894

120572119894u119894

= Aminus10f0+sum119894

120572119894Aminus10f119894minussum119894

120572119894Aminus10A119894u0

(36)

and the first-order GNE solution is

u = (Aminus10minussum119894

120572119894Aminus10A119894Aminus10)(f0+sum119894

120572119894f119894)

= Aminus10f0minussum119894

120572119894Aminus10A119894u0+sum119894

120572119894Aminus10f119894

minussum119894

sum119895

120572119894120572119895Aminus10A119894f119895

(37)

Comparing (36) and (37) it can be seen that theGNE solutioncontains an extra second-order term minussum

119894sum119895120572119894120572119895Aminus10A119894f119895

242 Example II Up to second-order terms are included inthe left-hand side of (27) while only the constant term isincluded in the right-hand side In this case the equationbecomes

(A0+sum119894

120572119894A119894+sum119894

sum119895

120572119894120572119895A119894119895) u = f

0 (38)

Denote u0= Aminus10f0 B119894= Aminus10A119894 and B

119894119895= Aminus10A119894119895 then the

first-order perturbation solution can be expressed as

u0= Aminus10f0

u119894= minussum119894

Aminus10A119894u0= minussum119894

B119894u0

u = u0+sum119894

120572119894u119894= u0minussum119894

120572119894B119894u0

(39)

while the first-order GNE solution is

u = (I +sum119894

120572119894Aminus10A119894+sum119894

sum119895

120572119894120572119895Aminus10A119894119895)

minus1

u0

= (I +sum119894

120572119894B119894+sum119894

sum119895

120572119894120572119895B119894119895)

minus1

u0

= (I minussum119894

120572119894B119894minussum119894

sum119895

120572119894120572119895B119894119895) u0

(40)

Comparing (39) and (40) the second-order termsminussum119894sum119895120572119894120572119895Aminus10A119894119895u0in the GNE solution do not appear in

the perturbation solution (39)

3 Numerical Examples

31 Example Description As shown in Figure 1 a plain strainproblem with an orthohexagonal cross section is consideredhere The length of each edge of the hexagon is 04m ThePoisson ratio of thematerial is set as 02 andYoungrsquosmodulusis assumed to be a Gaussian field with amean value of 30GPaand a covariance function defined as

119877 (1205911 1205912) = 1198881119890minus120591121198882minus1205912

21198882 (41)

in which 1205911= 1199091minus 1199092and 1205912= 1199101minus 1199102are the coordinate

differences and 1198881and 1198882are adjustable constants As shown in

Figure 2 the cross section is discretized with triangular linearelements where each side is divided into 8 equal parts Thereare in total 384 elements and 217 nodes

Mathematical Problems in Engineering 7

Figure 2 Finite element mesh

The relative error is defined as

RE =1003817100381710038171003817CuGNE minus CuMC

100381710038171003817100381721003817100381710038171003817CuMC

10038171003817100381710038172times 100 (42)

inwhichuMC is the nodal displacement vector solvedwith theMonte Carlo method and uGNE is the GNE solution Unlessstated explicitly the numerical testing is performed for 4340samples generated from the Gaussian field defined in (41)

The convergence condition of the GNE method has beenrigorously proved The main purpose of this example isto examine the error distribution and its relationship withvarious randomparameters forwhich a number of numericaltests are performed in the following subsections

32 Error Estimation The third-order GNE solution isadoptedThe constants 119888

1and 1198882in (41) are varied to examine

the error estimator in (26) The constant 1198881is the variance of

the Gaussian field and the constant 1198882controls the shape of

the covariance function The effective correlation length 119871effof the covariance function 119877(120591) is defined by

119877 (119871eff)

119877 (0)= 001 (43)

For any two points in the medium if the distance betweenthem is larger than 119871eff their material properties are onlyweakly correlated The effective correlation length of theGaussian field defined in (41) is

119877 (120591)

119877 (0)= 119890minus120591

21198882 = 001 997904rArr 119871eff = 2146radic1198882 (44)

In the first set of tests the constant 1198881is varied to change the

variance of the Gaussian field The parameter settings are

1198881= 001 119888

2= 32119864 minus 2 119871eff = 038

1198881= 004 119888

2= 32119864 minus 2 119871eff = 038

(45)

In the second set of tests the constant 1198882is varied to change

the shape of the covariance function The parameter settingsare

1198881= 001 119888

2= 08119864 minus 2 119871eff = 019

1198881= 001 119888

2= 24119864 minus 2 119871eff = 033

1198881= 001 119888

2= 40119864 minus 2 119871eff = 043

(46)

321 The Influence of 1198881 Corresponding to 119888

1= 001 and

1198881= 004 the standard deviation 120590 of the stochastic field

is 01 and 02 respectively A total number of 4340 samplesare generated for each case For each realization both theMonte Carlo solution and the third-order GNE solution arecomputed The relationship between the spectral radius andthe relative error is drawn in Figures 3 and 4 in which thered crosses show the actual relative errors corresponding toeach sample and the blue line shows the theoretical errorestimation It can been seen that for both cases the theoreticalerror estimator in (26) holds accurately and its effectivenessis not affected by the change of the variance controlled by 119888

1

322 The Influence of 1198882 The constant 119888

2is adjusted to take

three different values 08 times 10minus2 24 times 10minus2 and 40 times 10minus2while the constant 119888

1remains unchanged Following (41)

the effective correlation length grows with the increase of1198882 Again 4340 samples are generated for each stochastic

field and for each realization both the Monte Carlo solutionand the third-order GNE solution are computed Figure 5plots the relationship between spectral radius and the relativeerror which verifies that for all cases the theoretical errorestimation in (26) holds The changing of 119888

2 which changes

the variation frequency of the stochastic field does not affectthe error estimation

As indicated in (44) the effective correlation length 119871effis directly associated with the parameter 119888

2 To investigate

the influence of 119871eff on the relative error the empiricalcumulative distribution functions (ECDF) of relative errorsare plotted in Figure 6 for different 119871eff cases Comparing thethree ECDF curves at different effective correlation lengthsit is found that with the decrease of the effective correlationlength the accuracy of the GNE solution improves

33 The Relative Error for Other Types of Stochastic FieldsIn the last two subsections the error estimation analysis hastaken the Gaussian assumption that is all random variables120572119894in (3) are assumed to be independent Gaussian random

variables In this subsection we investigate the relative errorof the GNE method for other types of stochastic fieldsSpecifically the random variables 120572

119894in (3) are assumed to

have the two-point distribution the binomial distributionand the 120573 distribution respectively For the case of two-pointdistribution values minus03119864

0 031198640 are assigned with equal

probability that is 05 For the case of binomial distributionten times Bernoulli trials with the probability 05 are adopted

8 Mathematical Problems in Engineering

minus14 minus12 minus10 minus8 minus6 minus4minus14

minus12

minus10

minus8

minus6

minus4

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

Solution with c1 = 001

log (RE) = 4 log (120588B)

4 log (120588B)

Figure 3 The relation between the relative error and the spectral radius

minus11 minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 0minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Solution with c1 = 004

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 4 The relation between the relative error and the spectral radius

and the integer values between 0 and 10 are linearly projectedonto [minus03119864

0 031198640] For the case of 120573 distribution the

probability density function is defined as

119891 (119909 | 119886 119887) =1

119861 (119886 119887)119909119886minus1(1 minus 119909)119887minus1119868

(01)(119909) (47)

where 119861(119886 119887) is a 120573 function and the filter function 119868(01)(119909)

is defined as

119868(01) (119909) =

1 119909 isin (0 1)

0 119909 notin (0 1) (48)

Three kinds of 120573 distributions are considered with 119886 = 119887 =075 119886 = 119887 = 1 and 119886 = 119887 = 4 respectivelyThe interval (0 1)is linearly mapped to [minus03119864

0 031198640]

Corresponding to these five distributions the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7 Comparing the ECDF curves at differenteffective correlation lengths the same trend is observed asthe Gaussian case That is the relative error decreases withthe drop of effective correlation length

34 Efficiency Analysis Using the same computing platformthe Monte Carlo method the Neumann expansion method

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

Mathematical Problems in Engineering 7

Figure 2 Finite element mesh

The relative error is defined as

RE =1003817100381710038171003817CuGNE minus CuMC

100381710038171003817100381721003817100381710038171003817CuMC

10038171003817100381710038172times 100 (42)

inwhichuMC is the nodal displacement vector solvedwith theMonte Carlo method and uGNE is the GNE solution Unlessstated explicitly the numerical testing is performed for 4340samples generated from the Gaussian field defined in (41)

The convergence condition of the GNE method has beenrigorously proved The main purpose of this example isto examine the error distribution and its relationship withvarious randomparameters forwhich a number of numericaltests are performed in the following subsections

32 Error Estimation The third-order GNE solution isadoptedThe constants 119888

1and 1198882in (41) are varied to examine

the error estimator in (26) The constant 1198881is the variance of

the Gaussian field and the constant 1198882controls the shape of

the covariance function The effective correlation length 119871effof the covariance function 119877(120591) is defined by

119877 (119871eff)

119877 (0)= 001 (43)

For any two points in the medium if the distance betweenthem is larger than 119871eff their material properties are onlyweakly correlated The effective correlation length of theGaussian field defined in (41) is

119877 (120591)

119877 (0)= 119890minus120591

21198882 = 001 997904rArr 119871eff = 2146radic1198882 (44)

In the first set of tests the constant 1198881is varied to change the

variance of the Gaussian field The parameter settings are

1198881= 001 119888

2= 32119864 minus 2 119871eff = 038

1198881= 004 119888

2= 32119864 minus 2 119871eff = 038

(45)

In the second set of tests the constant 1198882is varied to change

the shape of the covariance function The parameter settingsare

1198881= 001 119888

2= 08119864 minus 2 119871eff = 019

1198881= 001 119888

2= 24119864 minus 2 119871eff = 033

1198881= 001 119888

2= 40119864 minus 2 119871eff = 043

(46)

321 The Influence of 1198881 Corresponding to 119888

1= 001 and

1198881= 004 the standard deviation 120590 of the stochastic field

is 01 and 02 respectively A total number of 4340 samplesare generated for each case For each realization both theMonte Carlo solution and the third-order GNE solution arecomputed The relationship between the spectral radius andthe relative error is drawn in Figures 3 and 4 in which thered crosses show the actual relative errors corresponding toeach sample and the blue line shows the theoretical errorestimation It can been seen that for both cases the theoreticalerror estimator in (26) holds accurately and its effectivenessis not affected by the change of the variance controlled by 119888

1

322 The Influence of 1198882 The constant 119888

2is adjusted to take

three different values 08 times 10minus2 24 times 10minus2 and 40 times 10minus2while the constant 119888

1remains unchanged Following (41)

the effective correlation length grows with the increase of1198882 Again 4340 samples are generated for each stochastic

field and for each realization both the Monte Carlo solutionand the third-order GNE solution are computed Figure 5plots the relationship between spectral radius and the relativeerror which verifies that for all cases the theoretical errorestimation in (26) holds The changing of 119888

2 which changes

the variation frequency of the stochastic field does not affectthe error estimation

As indicated in (44) the effective correlation length 119871effis directly associated with the parameter 119888

2 To investigate

the influence of 119871eff on the relative error the empiricalcumulative distribution functions (ECDF) of relative errorsare plotted in Figure 6 for different 119871eff cases Comparing thethree ECDF curves at different effective correlation lengthsit is found that with the decrease of the effective correlationlength the accuracy of the GNE solution improves

33 The Relative Error for Other Types of Stochastic FieldsIn the last two subsections the error estimation analysis hastaken the Gaussian assumption that is all random variables120572119894in (3) are assumed to be independent Gaussian random

variables In this subsection we investigate the relative errorof the GNE method for other types of stochastic fieldsSpecifically the random variables 120572

119894in (3) are assumed to

have the two-point distribution the binomial distributionand the 120573 distribution respectively For the case of two-pointdistribution values minus03119864

0 031198640 are assigned with equal

probability that is 05 For the case of binomial distributionten times Bernoulli trials with the probability 05 are adopted

8 Mathematical Problems in Engineering

minus14 minus12 minus10 minus8 minus6 minus4minus14

minus12

minus10

minus8

minus6

minus4

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

Solution with c1 = 001

log (RE) = 4 log (120588B)

4 log (120588B)

Figure 3 The relation between the relative error and the spectral radius

minus11 minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 0minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Solution with c1 = 004

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 4 The relation between the relative error and the spectral radius

and the integer values between 0 and 10 are linearly projectedonto [minus03119864

0 031198640] For the case of 120573 distribution the

probability density function is defined as

119891 (119909 | 119886 119887) =1

119861 (119886 119887)119909119886minus1(1 minus 119909)119887minus1119868

(01)(119909) (47)

where 119861(119886 119887) is a 120573 function and the filter function 119868(01)(119909)

is defined as

119868(01) (119909) =

1 119909 isin (0 1)

0 119909 notin (0 1) (48)

Three kinds of 120573 distributions are considered with 119886 = 119887 =075 119886 = 119887 = 1 and 119886 = 119887 = 4 respectivelyThe interval (0 1)is linearly mapped to [minus03119864

0 031198640]

Corresponding to these five distributions the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7 Comparing the ECDF curves at differenteffective correlation lengths the same trend is observed asthe Gaussian case That is the relative error decreases withthe drop of effective correlation length

34 Efficiency Analysis Using the same computing platformthe Monte Carlo method the Neumann expansion method

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

8 Mathematical Problems in Engineering

minus14 minus12 minus10 minus8 minus6 minus4minus14

minus12

minus10

minus8

minus6

minus4

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

Solution with c1 = 001

log (RE) = 4 log (120588B)

4 log (120588B)

Figure 3 The relation between the relative error and the spectral radius

minus11 minus10 minus9 minus8 minus7 minus6 minus5 minus4 minus3 minus2 minus1 0minus11

minus10

minus9

minus8

minus7

minus6

minus5

minus4

minus3

minus2

minus1

0

Solution with c1 = 004

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 4 The relation between the relative error and the spectral radius

and the integer values between 0 and 10 are linearly projectedonto [minus03119864

0 031198640] For the case of 120573 distribution the

probability density function is defined as

119891 (119909 | 119886 119887) =1

119861 (119886 119887)119909119886minus1(1 minus 119909)119887minus1119868

(01)(119909) (47)

where 119861(119886 119887) is a 120573 function and the filter function 119868(01)(119909)

is defined as

119868(01) (119909) =

1 119909 isin (0 1)

0 119909 notin (0 1) (48)

Three kinds of 120573 distributions are considered with 119886 = 119887 =075 119886 = 119887 = 1 and 119886 = 119887 = 4 respectivelyThe interval (0 1)is linearly mapped to [minus03119864

0 031198640]

Corresponding to these five distributions the empiricalcumulative distribution functions of relative errors are plot-ted in Figure 7 Comparing the ECDF curves at differenteffective correlation lengths the same trend is observed asthe Gaussian case That is the relative error decreases withthe drop of effective correlation length

34 Efficiency Analysis Using the same computing platformthe Monte Carlo method the Neumann expansion method

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

Mathematical Problems in Engineering 9

minus14 minus12 minus10 minus8 minus6 minus4

minus14

minus12

minus10

minus8

minus6

minus4

Solution with Leff = 019

Solution with Leff = 033

Solution with Leff = 043

log (RE) = 4 log (120588B)

Spectral radius sim

Rela

tive e

rror

simlo

g (RE

)

4 log (120588B)

Figure 5 The relation between the relative error and the spectral radius

0 1 2 3 4 5 6 7 8 90

08

06

04

02

1

Relative error

ECD

F

Leff = 019

Leff = 033

Leff = 043

times10minus3

Figure 6 ECDF of the relative errors

and the GNE method are employed to solve the sameset of solutions (corresponding to 4340 samples) and thecomputation time is shown in Table 1The number of randomvariables in the stochastic equation system differs dependingon the effective correlation length A fixed mesh is adoptedin all of the computation such that the dimension of a singleequation system is 434 times 434

If the mesh density is doubled then the dimensionof each matrix is 1634 times 1634 The number of randomvariables remains the same in each case The correspondingcomputation time is shown in Table 2

The two data tables indicate that the computationalefficiency of the GNEmethod (up to the third order) is muchhigher than Neumann expansion and Monte Carlo proce-dures However the memory requirement for higher orderGNE schemes increases rapidly causing the computationalefficiency to decrease

4 Conclusions

Anacceleration approach ofNeumann expansion is proposedfor solving stochastic linear equations The new method

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

10 Mathematical Problems in Engineering

0 05 1 15 2 25Relative error

ECD

F

0

08

06

04

02

1

times10minus5

(a) Two-point distribution

0 05 1 15 2 25 3 35 4 45 5Relative error

ECD

F

0

08

06

04

02

1

times10minus7

(b) Binomial distribution

0 05 1 15 2 25 3 35 4

ECD

F

Relative error times10minus6

0

08

06

04

02

1

(c) 120573 distribution with 119886 = 119887 = 075

0 05 1 15 2 25 3 35

ECD

F

Relative error

0

08

06

04

02

1

times10minus6

(d) 120573 distribution with 119886 = 119887 = 1

0 1 2 3 4 5 6Relative error

ECD

F

0

08

06

04

02

1

times10minus7

Leff = 019

Leff = 033

Leff = 043

(e) 120573 distribution with 119886 = 119887 = 4

Figure 7 ECDF of the relative error

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

Mathematical Problems in Engineering 11

Table 1 Computation time for 4340 samples (unit s)

Number of random variables 10 11 16 21 39

Monte Carlo 4078 4326 5402 6555 10713

GNE

First order 044 044 048 053 069Second order 079 083 105 128 221Third order 130 143 229 358 1333

Neumann

First order 2674 2896 4043 5223 9390Second order 2863 3088 4228 5370 9522Third order 3028 3257 4405 5549 9708

Table 2 Computation time for 16340 samples (unit s)

Number of random variables 11 16 21

Monte Carlo 656001 787131 911839

GNEM

First order 1331 1485 1639Second order 1606 1923 2250Third order 2455 4186 7097

Neumann

First order 332530 485150 601791Second order 363938 501079 631489Third order 388143 547371 660992

termed generalized Neumann expansion generalizes Neu-mann expansion from one matrix to multiple matricesThe GNE method is equivalent to Neumann expansion forgeneral stochastic equations and for stochastic linear equa-tions both methods are also equivalent to the perturbationmethod According to this equivalence relation a rigorouserror estimation is obtained which is applicable to all thethree methods Numerical tests show that the third-orderGNE expansion can provide good accuracy even for relativelylarge random fluctuations

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The authors would like to acknowledge the financial sup-ports of the National Natural Science Foundation of China(11272181) the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080)and the National Basic Research Program of China (Projectno 2010CB832701) The authors would also like to thankthe support from British Council and Chinese Scholarship

Council through the support of an award for Sino-UKHigherEducation Research Partnership for PhD Studies

References

[1] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 p 191 1991

[2] M Shinozuka and G Deodatis ldquoSimulation of multi-dimen-sional Gaussian stochastic fields by spectral representationrdquoApplied Mechanics Reviews vol 49 no 1 pp 29ndash53 1996

[3] R G Ghanem and P D Spanos ldquoSpectral techniques for stoch-astic finite elementsrdquo Archives of Computational Methods inEngineering vol 4 no 1 pp 63ndash100 1997

[4] R G Ghanem and P D Spanos Stochastic Finite Elements Aspectral Approach Courier Dover Publications 2003

[5] C F Li Y T Feng D R J Owen and I M Davies ldquoFourierrepresentation of random media fields in stochastic finiteelement modellingrdquo Engineering Computations vol 23 no 7pp 794ndash817 2006

[6] C F Li Y T Feng D R J Owen D F Li and I M Davis ldquoAFourier-Karhunen-Loeve discretization scheme for stationaryrandommaterial properties in SFEMrdquo International Journal forNumericalMethods in Engineering vol 73 no 13 pp 1942ndash19652008

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

12 Mathematical Problems in Engineering

[7] J W Feng C F Li S Cen and D R J Owen ldquoStatistical recon-struction of two-phase randommediardquoComputersamp Structures2013

[8] C F Li Y T Feng and D R J Owen ldquoExplicit solution to thestochastic system of linear algebraic equations (120572

11198601+ 12057221198602+

sdot sdot sdot+120572119898119860119898)119909 = 119887rdquoComputerMethods in AppliedMechanics and

Engineering vol 195 no 44ndash47 pp 6560ndash6576 2006[9] C F Li S Adhikari S Cen Y T Feng and D R J Owen

ldquoA joint diagonalisation approach for linear stochastic systemsrdquoComputers and Structures vol 88 no 19-20 pp 1137ndash1148 2010

[10] F Wang C Li J Feng S Cen and D R J Owen ldquoA noveljoint diagonalization approach for linear stochastic systems andreliability analysisrdquo Engineering Computations vol 29 no 2 pp221ndash244 2012

[11] W K Liu T Belytschko and A Mani ldquoRandom field finiteelementsrdquo International Journal for Numerical Methods in Engi-neering vol 23 no 10 pp 1831ndash1845 1986

[12] R G Ghanem and R M Kruger ldquoNumerical solution of spec-tral stochastic finite element systemsrdquo Computer Methods inApplied Mechanics and Engineering vol 129 no 3 pp 289ndash3031996

[13] M K Deb I M Babuska and J T Oden ldquoSolution of stoch-astic partial differential equations using Galerkin finite elementtechniquesrdquo Computer Methods in Applied Mechanics and Engi-neering vol 190 no 48 pp 6359ndash6372 2001

[14] G Schueller ldquoDevelopments in stochastic structural mechan-icsrdquoArchive of AppliedMechanics vol 75 no 10ndash12 pp 755ndash7732006

[15] S-K Au and J L Beck ldquoEstimation of small failure probabilitiesin high dimensions by subset simulationrdquo Probabilistic Engi-neering Mechanics vol 16 no 4 pp 263ndash277 2001

[16] M Shinozuka ldquoStructural response variabilityrdquo Journal of Engi-neering Mechanics vol 113 no 6 pp 825ndash842 1987

[17] S K Au and J L Beck ldquoSubset simulation and its application toseismic risk based on dynamic analysisrdquo Journal of EngineeringMechanics vol 129 no 8 pp 901ndash917 2003

[18] P S Koutsourelakis H J Pradlwarter and G I SchuellerldquoReliability of structures in high dimensionsmdashpart I algorithmsand applicationsrdquo Probabilistic Engineering Mechanics vol 19no 4 pp 409ndash417 2004

[19] G I Schueller H J Pradlwarter and P S Koutsourelakis ldquoAcritical appraisal of reliability estimation procedures for highdimensionsrdquo Probabilistic Engineering Mechanics vol 19 no 4pp 463ndash474 2004

[20] J E Gentle Elements of Computational Statistics Springer NewYork NY USA 2002

[21] O Cavdar A Bayraktar A Cavdar and S Adanur ldquoPerturba-tion based stochastic finite element analysis of the structuralsystems with composite sections under earthquake forcesrdquo Steeland Composite Structures vol 8 no 2 pp 129ndash144 2008

[22] W K Liu T Belytschko and A Mani ldquoProbabilistic finite ele-ments for nonlinear structural dynamicsrdquo Computer Methodsin Applied Mechanics and Engineering vol 56 no 1 pp 61ndash811986

[23] E Vanmarcke and M Grigoriu ldquoStochastic finite elementanalysis of simple beamsrdquo Journal of EngineeringMechanics vol109 no 5 pp 1203ndash1214 1983

[24] C Papadimitriou L S Katafygiotis and J L Beck ldquoApprox-imate analysis of response variability of uncertain linear sys-temsrdquoProbabilistic EngineeringMechanics vol 10 no 4 pp 251ndash264 1995

[25] H G Matthies C E Brenner C G Bucher and C G SoaresldquoUncertainties in probabilistic numerical analysis of structuresand solidsmdashstochastic finite elementsrdquo Structural Safety vol 19no 3 pp 283ndash336 1997

[26] F Yamazaki M Shinozuka and G Dasgupta ldquoNeumannexpansion for stochastic finite-element analysisrdquo Journal ofEngineering Mechanics vol 114 no 8 pp 1335ndash1354 1988

[27] M Shinozuka and G Deodatis ldquoResponse variability of stoch-astic finite-element systemsrdquo Journal of Engineering Mechanicsvol 114 no 3 pp 499ndash519 1988

[28] R G Ghanem and P D Spanos ldquoSpectral stochastic finite-ele-ment formulation for reliability analysisrdquo Journal of EngineeringMechanics vol 117 no 10 pp 2351ndash2372 1991

[29] M F Pellissetti and R G Ghanem ldquoIterative solution of systemsof linear equations arising in the context of stochastic finiteelementsrdquo Advances in Engineering Software vol 31 no 8 pp607ndash616 2000

[30] D B ChungMAGutierrez L L Graham-Brady and F-J Lin-gen ldquoEfficient numerical strategies for spectral stochastic finiteelement modelsrdquo International Journal for Numerical Meth-ods in Engineering vol 64 no 10 pp 1334ndash1349 2005

[31] A Keese and H G Matthies ldquoHierarchical parallelisation forthe solution of stochastic finite element equationsrdquo Computersand Structures vol 83 no 14 pp 1033ndash1047 2005

[32] A Nouy ldquoA generalized spectral decomposition technique tosolve a class of linear stochastic partial differential equationsrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 45ndash48 pp 4521ndash4537 2007

[33] A Nouy ldquoGeneralized spectral decompositionmethod for solv-ing stochastic finite element equations invariant subspace pro-blem and dedicated algorithmsrdquo Computer Methods in AppliedMechanics and Engineering vol 197 no 51-52 pp 4718ndash47362008

[34] C E Powell and H C Elman ldquoBlock-diagonal preconditioningfor spectral stochastic finite-element systemsrdquo IMA Journal ofNumerical Analysis vol 29 no 2 pp 350ndash375 2009

[35] D Ghosh P Avery and C Farhat ldquoAn FETI-preconditionedconjugate gradient method for large-scale stochastic finite ele-ment problemsrdquo International Journal for Numerical Methods inEngineering vol 80 no 6-7 pp 914ndash931 2009

[36] B Ganapathysubramanian and N Zabaras ldquoSparse grid col-location schemes for stochastic natural convection problemsrdquoJournal of Computational Physics vol 225 no 1 pp 652ndash6852007

[37] N Zabaras and B Ganapathysubramanian ldquoA scalable frame-work for the solution of stochastic inverse problems using asparse grid collocation approachrdquo Journal of ComputationalPhysics vol 227 no 9 pp 4697ndash4735 2008

[38] B Ganapathysubramanian and N Zabaras ldquoA seamless ap-proach towards stochastic modeling sparse grid collocationand data driven input modelsrdquo Finite Elements in Analysis andDesign vol 44 no 5 pp 298ndash320 2008

[39] B Ganapathysubramanian andN Zabaras ldquoModeling diffusionin random heterogeneous media data-driven models stochas-tic collocation and the variational multiscale methodrdquo Journalof Computational Physics vol 226 no 1 pp 326ndash353 2007

[40] M DWebsterM A Tatang andG JMcRaeApplication of theProbabilistic Collocation Method for an Uncertainty Analysis ofa Simple Ocean Model MIT Joint Program on the Science andPolicy of Global Change 1996

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

Mathematical Problems in Engineering 13

[41] I Babuska F Nobile and R Tempone ldquoA stochastic collocationmethod for elliptic partial differential equations with randominput datardquo SIAM Journal on Numerical Analysis vol 45 no 3pp 1005ndash1034 2007

[42] S Huang S Mahadevan and R Rebba ldquoCollocation-basedstochastic finite element analysis for random field problemsrdquoProbabilistic Engineering Mechanics vol 22 no 2 pp 194ndash2052007

[43] I Babuska R Temponet and G E Zouraris ldquoGalerkin finiteelement approximations of stochastic elliptic partial differentialequationsrdquo SIAM Journal on Numerical Analysis vol 42 no 2pp 800ndash825 2004

[44] O P Le Maitre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[45] B J Debusschere H N Najm P P Pebayt O M Knio R GGhanem and O P Le Maıtre ldquoNumerical challenges in the useof polynomial chaos representations for stochastic processesrdquoSIAM Journal on Scientific Computing vol 26 no 2 pp 698ndash719 2005

[46] T Crestaux O Le Maıtre and J-M Martinez ldquoPolynomialchaos expansion for sensitivity analysisrdquo Reliability Engineeringand System Safety vol 94 no 7 pp 1161ndash1172 2009

[47] R V Field Jr and M Grigoriu ldquoOn the accuracy of the polyno-mial chaos approximationrdquoProbabilistic EngineeringMechanicsvol 19 no 1 pp 65ndash80 2004

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Geometry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Advances in

Mathematical Physics

ISRN Algebra

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Mathematical Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Discrete Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2013