Application of the stochastic finite element method forGaussian and non-Gaussian systems

M. Schevenels, G. Lombaert, G. DegrandeK.U.Leuven, Department of Civil Engineering,Kasteelpark Arenberg 40, B-3001, Leuven, Belgiume-mail: mattias.schevenels@bwk.kuleuven.ac.be

AbstractThe stochastic finite element method (Ghanem and Spanos) is illustrated for the solution of mechanicalsystems with Gaussian characteristics. It is shown numerically and analytically that the variance of theresponse of a system with a Gaussian stiffness is infinite. Since this is not physically sound, a randomstiffness should not be assumed to be Gaussian. Therefore, an adaptation of the stochastic finite elementmethod as to calculate the response of non-Gaussian systemsis presented. This adaptation is based on thegeneration of a non-Gaussian process by means of a Hermite polynomial expansion (Sakamoto and Ghanem,Puig et al.) and the projection of the obtained realizationson the Karhunen-Loeve modes of the non-Gaussianprocess (Poirion and Soize).

1 Introduction

This paper illustrates the stochastic finite element method(SFEM) as developed by Ghanem and Spanos[1] for the solution of mechanical problems with spatially varying random characteristics. The probabilitydensity function (PDF) and the lowest order statistical moments of the displacement are calculated for abeam with a spatially varying random bending stiffness under a deterministic static load.

The SFEM is based on the Karhunen-Loeve (KL) decomposition of the random system characteristics. TheKL decomposition of a random process can be regarded as the continuous counterpart of the decorrelationof a set of random variables [10]. It allows to approximate a random process by a linear combination oforthonormal deterministic functions (KL modes) with uncorrelated random coefficients. If the marginalPDF of the process is Gaussian then the uncorrelated KL coefficients are Gaussian as well and as a resultthey are independent. The KL decomposition is discussed in section 2.

A similar decomposition is performed at the level of the finite element equations, which leads to the SFEMequations: a linear combination of deterministic equations with random (KL) coefficients. This decomposi-tion is presented in section 3.

Two techniques are applied to solve the SFEM equations for a system with a Gaussian stiffness. The firsttechnique is based on a Monte Carlo simulation of the independent Gaussian KL coefficients. The secondtechnique is based on the projection of the response on the polynomial chaos, this is a set of Hermite poly-nomials of the KL coefficients. A Galerkin approach is followed to calculate the projections of the responseon these polynomials. The solution of the SFEM equations forsystems with Gaussian properties is treatedin section 4.

The examples in section 4 reveal the divergence of the variance of the response of a system with a Gaussianstiffness. In section 5, it is shown analytically that the variance of the response of such a system is infinite.Since this is not physically sound, a random stiffness should not be assumed to be Gaussian.

Section 6 treats the solution of the SFEM equations for non-Gaussian systems by means of a Monte Carlosimulation of the KL coefficients. In this case, it is impossible to generate these coefficients directly due to

3299

their mutual dependence and their unknown joint PDF. Therefore, realizations of the non-Gaussian processare generated as transformations of an underlying Gaussianprocess [6, 7]. The projection of these real-izations on the KL modes of the non-Gaussian process leads torealizations of the mutually dependent KLcoefficients [4].

2 Discretization of a random process

Consider a mechanical problem where one of the system characteristics is modelled as a scalar randomprocessS(x, θ) : D × Ω → R. This process is defined on the probability space(Ω,Σ, P ) and the index setD ⊂ R

d. The latter coincides with thed-dimensional physical domain of the problem. The processS(x, θ)is characterized by its marginal PDFpS(s) : R → R

+ and its covariance functionCS(x1,x2) : D×D → R.

In order to assemble the SFEM equations for this problem, therandom processS(x, θ) has to be expressedas a deterministic function of a small number of random variables. This discretization is achieved by meansof the Karhunen-Loeve (KL) decomposition [1], presented inthe following section.

2.1 Karhunen-Loeve decomposition

The non-zero mean random processS(x, θ) is decomposed as follows:

S(x, θ) = mS(x) + Y (x, θ) (1)

wheremS(x) = ES(x, θ) is the mean value of the random processS(x, θ) andY (x, θ) is a zero meanrandom process. Both the correlation functionRY (x1,x2) and the covariance functionCY (x1,x2) of thezero mean random processY (x, θ) are equal to the covariance functionCS(x1,x2) of the non-zero meanrandom processS(x, θ). All three are denoted byCS(x1,x2) in the following.

Let Θ be the Hilbert space of random variablesZ(θ) : Ω → R defined on the probability space(Ω,Σ, P ),with the inner product〈Z1(θ), Z2(θ)〉Θ = E Z1(θ)Z2(θ). Let ξj(θ)j be a Hilbert basis ofΘ. The KLdecomposition of the zero mean random processY (x, θ) consists of the projection of the process on theHilbert basisξj(θ)j . This leads to the following expansion:

Y (x, θ) =∞

∑

j=1

cj(x)ξj(θ) (2)

The covariance functionCS(x1,x2) of the zero mean random processY (x, θ) is equal to:

CS(x1,x2) = E Y (x1, θ)Y (x2, θ) =

∞∑

j=1

∞∑

k=1

cj(x1)ck(x2)E ξj(θ)ξk(θ) =

∞∑

j=1

cj(x1)cj(x2) (3)

where the orthonormality of the Hilbert basis vectorsξj(θ)j is taken into account. The covariance functionCS(x1,x2) has the following spectral decomposition:

CS(x1,x2) =∞∑

j=1

λjfj(x1)fj(x2) (4)

Herein,fj(x)j andλjj are the normalized eigenfunctions and the eigenvalues of the covariance functionCS(x1,x2). The eigenfunctions are orthonormal and the eigenvalues are positive sinceCS(x1,x2) is a realsymmetric function. They are obtained as the solution of theeigenvalue problem found by the projection ofequation (4) onfk(x1):

∫

D

CS(x1,x2)fk(x1) dx1 = λkfk(x2) (5)

3300 PROCEEDINGS OFISMA2004

An expression for the functioncj(x) in equation (2) is obtained by the elimination ofCS(x1,x2) fromequations (3) and (4):

cj(x) =√

λjfj(x) (6)

Introduction of equation (6) in equation (2) gives:

Y (x, θ) =∞∑

j=1

√

λjfj(x)ξj(θ) (7)

or equivalently:

S(x, θ) = mS(x) +

∞∑

j=1

√

λjfj(x)ξj(θ) (8)

Equation (7) shows the KL decomposition of the zero mean random processY (x, θ), i.e. the decompositionin terms of a set of normalized uncorrelated random variables ξj(θ)j .

The discretization of the random processS(x, θ) is accomplished by a truncation of the infinite series inequation (8) after the terms corresponding to the highestM eigenvaluesλjj :

S(x, θ) ≈ mS(x) +

M∑

j=1

√

λjfj(x)ξj(θ) (9)

M is called the order of the KL decomposition. As the terms in the decomposition are not correlated (thevariablesξj(θ)j are orthonormal random variables), the KL decomposition isthe most efficient decom-position of a random process: it minimizes the truncation error for a given number of terms. Ghanem andSpanos [1] present a proof of this error minimizing propertyof the KL decomposition.

An expression for the KL coefficientξk(θ) is obtained by the projection of equation (7) onfk(x):

ξk(θ) =1√λk

∫

D

Y (x, θ)fk(x) dx (10)

If the processS(x, θ) has a Gaussian marginal PDF, thenY (x, θ) reduces to a zero mean Gaussian variablefor a fixed positionx and the integral in equation (10) can be interpreted as an infinite series of zero meanGaussian variables. As a result, the integral itself is a zero mean Gaussian variable, and so isξk(θ). Thusthe KL coefficients are uncorrelated standard Gaussian variables and therefore independent. By virtue ofthis independence, realizations of the KL coefficients are easily generated within the frame of a Monte Carlosimulation (MCS). Realizations of the random processS(x, θ) are then obtained according to equation (9).

2.2 Numerical implementation of the Karhunen-Loeve decomp osition

This section covers the solution of the eigenvalue problem (5) by means of a Galerkin procedure presentedby Ghanem and Spanos [1]. The eigenfunctionfk(x) is approximated as:

fk(x) ≈J

∑

j=1

fjkNj(x) (11)

whereNj(x) : D → Rj is a set of shape functions. Introduction of equation (11) inequation (5) and achange of the order of the integration and the summation leads to:

J∑

j=1

fjk

∫

D

CS(x1,x2)Nj(x1) dx1 − λkNj(x2)

≈ 0 (12)

VARIABILITY 3301

There is no exact equality of the LHS and the RHS in equation (12) due to the truncation of the infiniteseries in equation (11). Following a Galerkin approach, thetruncation error in equation (12) is required to beorthogonal to the space spanned by the shape functionsNi(x)i. After changing the order of the integrationand the summation, this leads to:

J∑

j=1

fjk

∫

D

∫

D

CS(x1,x2)Nj(x1)Ni(x2) dx1 dx2 − λk

∫

D

Nj(x2)Ni(x2) dx2

= 0 (13)

Denoting

Cij =

∫

D

∫

D

CS(x1,x2)Nj(x1)Ni(x2) dx1 dx2 Nij =

∫

D

Nj(x2)Ni(x2) dx2 Λlk = δlkλk (14)

equation (13) reduces to:J

∑

j=1

Cijfjk =

J∑

j=1

J∑

l=1

NijfjlΛlk (15)

or in matrix notation:Cf = NfΛ (16)

Equation (16) represents aJ-dimensional generalized algebraic eigenvalue problem that leads to the eigen-valuesΛ and the eigenvectorsf . The eigenfunctionsfk(x)k of the covariance functionCS(x1,x2) areobtained by substitution off in equation (11).

Example Consider a beam with a lengthl = 4m and a random bending stiffnessS(x, θ) = E(x, θ)I(x, θ)whereE(x, θ) denotes the Young’s modulus andI(x, θ) the moment of inertia.S(x, θ) is a one-dimensionalstationary random process with a mean valuemS = 6 × 106 Nm2 and a standard deviationσS = 0.3mS .The covariance function of the random process is assumed to be exponential:

CS(x1, x2) = σ2S exp

(

−∆x

lc

)

(17)

where∆x = x2 − x1. The correlation lengthlc reflects the smoothness of a realization of the process: if∆x is small with respect tolc, then the probability is small thatS(x2, θ) differs substantially fromS(x1, θ).In the present example, the correlation lengthlc = 2m. This leads to the covariance functionCS(x1, x2)shown in figure 1.

Figure 1: Bending stiffness covarianceCS(x1, x2).

3302 PROCEEDINGS OFISMA2004

The random processS(x, θ) is discretized by means of the KL decomposition of the zero mean randomprocessY (x, θ) = S(x, θ)−mS. As to demonstrate the influence of the orderM of the KL decomposition,the discretization is performed withM = 4 andM = 10. The KL decomposition is performed using aset of element based piecewise third order polynomial shapefunctionsNj(x)j . For the caseM = 4, amesh consisting of 10 elements is used, resulting in a set ofJ = 22 shape functions. For the caseM = 10,a mesh consisting of 25 elements is used, resulting in a set ofJ = 52 shape functions. The matricesC andN are computed according to equation (14). The solution of thegeneralized algebraic eigenvalueproblem (16) and the backsubstitution in equation (11) leadto the eigenvaluesλkk and the eigenfunctionsfk(x)k. In the following, only the highestM eigenvalues and the corresponding eigenfunctions are takeninto account. Figure 2 shows these eigenfunctions for both casesM = 4 andM = 10. The restriction

0 1 2 3 4−1

−0.5

0

0.5

1

Distance [m]

KL−

mod

es [

− ]

a.M = 4.

0 1 2 3 4−1

−0.5

0

0.5

1

Distance [m]

KL−

mod

es [

− ]

b. M = 10.

Figure 2: Eigenfunctionsfk(x)k corresponding to the highestM eigenvaluesλkk with (a)M = 4 and(b)M = 10.

to the highestM eigenvalues and eigenfunctions implies that the truncatedspectral representation of thecovariance function is taken into account instead of the exact covariance function. Figure 3 illustrates how

a.M = 4. b. M = 10.

Figure 3:M -th order KL approximation of the covarianceCS(x1, x2) of the random processS(x, θ) with(a)M = 4 and (b)M = 10.

this (non-stationary) KL approximation converges to the (stationary) exact covariance function as the orderM of the KL decomposition increases.

In the following, the processS(x, θ) is assumed to be Gaussian. The highestM eigenvalues and eigenfunc-tions of the covariance function are used to perform a Monte Carlo simulation of the random processS(x, θ).First,Nmcs independent sets ofM independent standard Gaussian variablesξj(θi)j are generated. In the

VARIABILITY 3303

0 1 2 3 40

2

4

6

8

10

12x 10

6

Distance [m]

Ben

ding

stif

fnes

s [N

m2 ]

a.M = 4.

0 1 2 3 40

2

4

6

8

10

12x 10

6

Distance [m]

Ben

ding

stif

fnes

s [N

m2 ]

b. M = 10.

Figure 4:M -th order realizations of the Gaussian processS(x, θ) with (a)M = 4 and (b)M = 10.

present case,Nmcs = 20000. Next, each of these setsξj(θi)j is introduced in equation (9), leading toNmcs realizations of the random processS(x, θ). Figure 4 shows 10 of these realizations. Statistics allowto verify the accuracy of the set of realizations as a representation of the random process: forM = 4, themean value of all of the realizations at all positions is equal to mmcs = 5.99 × 106 Nm2, while the standarddeviation equalsσmcs = 0.28mmcs. ForM = 10, the mean value of all of the realizations at all positions isequal tommcs = 6.00 × 106 Nm2, while the standard deviation equalsσmcs = 0.29mmcs. As the numberNmcs of realizations increases, their covariance converges to theM -th order KL approximation.

3 SFEM system equations

This section covers the assembly of the SFEM equations for a system with random characteristics, based onthe KL decomposition of these characteristics [1].

Consider a stochastic static finite element problem with deterministic mechanical boundary conditions andzero displacements as kinematic boundary conditions. The equilibrium equations for the system are writtenas follows:

K(θ)U(θ) = F (18)

where the stochastic stiffness matrixK(θ) is linearly dependent on a random processS(x, θ) : D×Ω → R.This relation is expressed using the following operator:

K(θ) = K (S(x, θ)) (19)

The random processS(x, θ) is discretized according to equation (9). The stiffness matrix is decomposedaccordingly:

K(θ) = K (S(x, θ)) ≈ K0 +M∑

j=1

Kjξj(θ) (20)

K0 is the stiffness matrix of the mean system. The deterministic matricesKj are given by the operator:

Kj =

K (mS(x)) j = 0

K(√

λjfj(x))

j = 1, . . . ,M(21)

The SFEM equations of the static problem become:

K0 +M∑

j=1

Kjξj(θ)

U(θ) = F (22)

3304 PROCEEDINGS OFISMA2004

4 Gaussian SFEM

This section demonstrates the solution of the SFEM equations for mechanical systems with random charac-teristics modelled as a Gaussian process. The assumption ofa Gaussian PDF for the system characteristicsis frequently encountered in literature [1, 2, 11, 12]. Two techniques are presented for the solution of theSFEM equations. The first technique is based on a Monte Carlo simulation of the independent Gaussian KLcoefficients. The second technique is based on the projection of the response on the polynomial chaos.

4.1 Monte Carlo simulation

Consider the SFEM problem defined by equations (18-22). The KL coefficientsξj(θ)j in equation (22) areindependent standard Gaussian variables sinceS(x, θ) is a Gaussian process. The solution of this equationis obtained as follows by means of a Monte Carlo simulation:

1. assembly of the system matricesKj according to equation (21),

2. generation of the setsξ1(θi), . . . , ξM (θi) with i = 1, . . . , nMCS of independent standard Gaussianvariables,

3. assembly of the stiffness matrixK(θi) according to equation (20) for every realizationi and solutionof the deterministic system of equationsK(θi)U(θi) = F,

4. estimation of the PDF of the response based on the statistics ofU(θi)i.

Example The Monte Carlo simulation is performed for the calculationof the PDF of the displacement ofthe tip of a clamped beam with a random bending stiffness. Thebeam has a lengthl = 4m. The bendingstiffnessS(x, θ) has a Gaussian marginal PDF and an exponential covariance function, in accordance withthe bending stiffness discussed in section 2.1. The beam tipis subjected to a determistic vertical point loadf = 50kN as shown in figure 5.

x

y

f = 50kN

Figure 5: Clamped beam subjected to a point load.

The SFEM equations are assembled taking into account the first M = 4 KL modes of the bending stiffness.A 10 element mesh is used both for the calculation of the KL modes and for the assembly of the systemequations which are solved by means of a Monte Carlo simulation with nMCS = 20000 realizations. Foreach realizationi, the vertical displacementUn(θi) of the beam tip is calculated. Statistics are used toestimate the mean valuemUn and the standard deviationσUn of the displacementUn(θ): this leads to themean valuemUn = −0.19m and the standard deviationσUn = 0.75m. Note that the mean displacementmUn of the random system is larger than the displacement of the mean systemumn = Pl3/3mS = −0.18m.The histogram of the displacementsUn(θi)i leads to the estimation of the PDFpUn(un) shown in figure6. While the Gaussian marginal PDF of the bending stiffness is symmetric with respect to the mean value,the PDF of the response is asymmetric. This is due to the non-linearity of the transformation applied to

VARIABILITY 3305

−1 −0.5 0 0.5 10

2

4

6

8

10

Displacement [m]P

roba

bilit

y de

nsity

[1/m

]

Figure 6: Estimation of the PDFpUn(un) of the vertical displacement of the beam tip.

the bending stiffness as to obtain the response. Furthermore, as the support of the Gaussian marginal PDFextends over the entire set of real numbersR, the direction of the responseUn(θ) does not correspond to thedirection of the forcef almost surely. As a result, the PDFpUn(un) is not exactly equal to zero forun > 0,even though this is not visible in figure 6.

4.2 Projection on the polynomial chaos

Consider the SFEM problem defined by equations (18-22). The stochastic responseU(ξ1(θ), . . . , ξM (θ)) isprojected on the polynomial chaos of orderP :

U(θ) ≈Q

∑

q=1

Aqψq(ξ1(θ), . . . , ξM (θ)) (23)

Herein,Aq is a deterministic vector of the same length as the response vectorU(θ).

The polynomial chaos is a Hilbert basis of the spaceΘ of random variables, consisting of theM -dimensionalHermite polynomialsψq(ξ1(θ), . . . , ξM (θ)) in terms of the KL coefficientsξj(θ)j . The polynomial chaosof orderP is the subset of this basis containing the polynomials up to orderP . The restriction to this subset inequation (23) results in aP -th order polynomial approximation of the response surfaceU(ξ1(θ), . . . , ξM (θ)).TheM -dimensional Hermite polynomials are defined as follows:

ψq(z) =M∏

k=1

hηqk(zk) (24)

Herein, theQ ×M -dimensional matrixη collects all multi-indicesηq ∈ NM satisfying

∑Mk=1 ηqk ≤ P .

hn(z) is then-th order normalized one-dimensional Hermite polynomial,defined as:

hn(z) =1√n!Hn(z) (25)

where the polynomialHn(z) follows from the recurrence relation:

H0(z) = 1 H1(z) = z Hn+1(z) = zHn(z) − nHn−1(z) (26)

TheM -dimensional Hermite polynomialsψq(z)q are orthonormal with respect to the Gaussian probabilitymeasure, so ifz is aM -dimensional standard Gaussian variable then:

E ψq(z)ψr(z) =

∫

RM

ψq(z)ψr(z)1

√

(2π)nexp(−‖z‖

2

2) dz = δqr (27)

3306 PROCEEDINGS OFISMA2004

The introduction of equation (23) in equation (22) leads to:

Q∑

q=1

K0 +

M∑

j=1

Kjξj(θ)

Aqψq ≈ F (28)

The arguments of the Hermite polynomialψq(ξ1(θ), . . . , ξM (θ)) have been omitted as to improve readability.There is no exact equality of the LHS and the RHS of equation (28) due to the restriction of the Hilbert basisto the polynomial chaos of orderP . According to the Galerkin finite element method, orthogonality of thetruncation error in equation (28) and the Hermite polynomials is enforced. This leads to:

Q∑

q=1

M∑

j=0

Kjcjqr

Aq = Fδ0r , r = 1, . . . , Q (29)

wherecjqr is defined as:

cjqr =

E ψqψr j = 0

E ξjψqψr j = 1, . . . ,M(30)

From the recurrence relation (26) and the orthonormality property (27), the following expression forcjqr isderived:

cjqr =

δqr j = 0(√ηrj δηqj+1,ηrj

+√ηqj δηrj+1,ηqj

)

M∏

k=1k 6=j

δηqkηrkj 6= 0 (31)

The set of equations (29) can be written in matrix notation:

KpcApc = Fpc (32)

with:

Kpc =

M∑

j=0

Kjcj11 · · ·M∑

j=0

Kjcj1Q

.... ..

M∑

j=0

KjcjQ1

M∑

j=0

KjcjQQ

Apc =

A1

...AQ

Fpc =

F

0

...0

(33)

The solution of the system of equations (32) leads to the vectorsAqq. A polynomial approximation of theresponse surfaceU(ξ1(θ), . . . , ξM (θ)) is obtained by substitution of these vectorsAqq in equation (23).The mean valuemU of the response is obtained as:

mU =

Q∑

q=1

AqE ψq =

Q∑

q=1

Aqδ1q = A1 (34)

where we assume without loss of generality thatηq = 0 for q = 1. The correlation matrixRU of theresponse is obtained as:

RU =

Q∑

q=1

Q∑

r=1

AqATr E ψqψr =

Q∑

q=1

Q∑

r=1

AqATr δqr =

Q∑

q=1

AqATq

VARIABILITY 3307

Example The SFEM problem described in section 4.1 is solved by a projection of the response on thepolynomial chaos. A number ofM = 4 KL modes is taken into account and the order of the polynomialchaos is chosen equal toP = 4. In this case, the polynomial chaosψq(ξ1(θ), . . . , ξ4(θ))q consists ofQ = 70 four-dimensional Hermite polynomials.

First, the KL decomposition of the bending stiffnessS(x, θ) leads to the matricesKjj according to equa-tion (21). Next,4 × 70 × 70 values ofcjqr are calculated according to equation (31) and used to assembleKpc according to equation (33). The solution of the system of equations (32) gives the vectorsAqq. Fi-nally, equations (34) and (35) are used to calculate the meanvaluemUn and the standard deviationσUn ofthe displacementUn(θ) of the beam tip: this leads to a mean valuemUn = −0.20m and a standard deviationσUn = 0.08m.

Note that this estimation of the standard deviationσUn differs substantially from the standard deviationσUn = 0.75m obtained with the Monte Carlo simulation in section 4.1. As demonstrated in the followingsection 5, this discrepancy is due to the assumption that themarginal PDF of the bending stiffnessS(x, θ) isGaussian.

5 Response of a Gaussian system

In this section, the implications of the assumption of a Gaussian stiffness on the stochastic properties of theresponse of a mechanical system are discussed.

Consider a clamped beam similar as in section 4.1. Let the correlation lengthlc of the bending stiffnessS(x, θ) be infinite so that the Gaussian processS(x, θ) reduces to a Gaussian variableS(θ). The beam issubjected to a deterministic forcef as in figure 5. As to assess the displacementUn(θ) of the beam tip,the beam can be considered as a single degree of freedom (SDOF) system with a Gaussian spring stiffnessK(θ) = 3S(θ)/l3. The second order moment of the responseUn(θ) is equal to:

E

[Un(θ)]2

=

∞∫

−∞

(

f

k

)2

pK(k) dk (35)

wherepK(k) is the Gaussian PDF of the spring stiffness. Since the integrand

(

f

k

)2

pK(k) is non-negative

in the entire integration domain:

E

[Un(θ)]2

≥ǫ

∫

−ǫ

(

f

k

)2

pK(k) dk ∀ ǫ > 0 (36)

The Gaussian PDFpK(k) is such that:

∃b > 0, ǫ > 0 : pK(k) ≥ b ∀ k ∈ [−ǫ, ǫ] (37)

Hence:

E

[Un(θ)]2

≥ǫ

∫

−ǫ

(

f

k

)2

b dk = limα→0−

α∫

−ǫ

(

f

k

)2

b dk + limα→0+

ǫ∫

α

(

f

k

)2

b dk = ∞ (38)

Equation (38) illustrates that the second order moment of the displacement of a system with a Gaussianstiffness due to a deterministic force is infinite, or equivalently, that the response is not a second orderrandom variable [8]. In fact, this is not only true for the Gaussian PDF but for any PDF satisfying equation(37). A stochastic mechanical system leading to a displacement with an infinite second order moment is not

3308 PROCEEDINGS OFISMA2004

physically sound, since it implies that the expected value of the deformation energy is undefined. Therefore,it is impossible that the PDF of the stiffness of a mechanicalsystem satisfies equation (37). Consequently, amechanical system with a Gaussian stiffness does not exist.

This corollary has important implications regarding the application of the SFEM to Gaussian systems. It ne-cessitates the re-interpretation of the results of the example with the clamped beam with a Gaussian bendingstiffness elaborated in sections 4.1 and 4.2. First, the estimation of the standard deviationσUn of the responseUn(θ) can never be accurate since the exact value of the standard deviation is infinite. Indeed, performingdifferent Monte Carlo simulations never reveals convergence of the standard deviationσUn , whatever thenumber of realizationsnMCS is. Second, the central limit theorem cannot be used to evaluate the reliabilityof the estimation of the mean responsemUn as obtained by means of a Monte Carlo simulation, as this the-orem is only valid for second order random variables [8, 9]. As a result, the Monte Carlo simulation can notbe used for a reliable prediction of the mean value of the response of a mechanical system with a Gaussianstiffness.

6 Non-Gaussian SFEM

This section demonstrates a solution technique for the SFEMequations for mechanical systems with non-Gaussian random characteristics. The technique is based ona Monte Carlo simulation of the mutuallydependent KL coefficents.

6.1 Monte Carlo simulation

Consider the SFEM problem defined by equations (18-22), where the random processS(x, θ) has a non-Gaussian marginal PDF. The Monte Carlo simulation as described in section 4.1 is applicable for thisproblem, except for step 2. The KL coefficientsξj(θ)j of the discretized non-Gaussian processS(x, θ)are mutually dependent and have an unknown joint PDF. Hence,it is impossible to generate realizations ofthese coefficients directly [4]. This problem is circumvented by using another method to generate realizationsof the random process. Next, these realizations are projected on the KL modes according to equation (10), sodelivering realizations of the KL coefficients [4]. Given the realizations of the random process, the projectionon the KL modes might seem redundant since it is possible to assemble the FEM equations directly foreach realization. However, the direct assembly necessitates the projection of each realization on all shapefunctions used for the FEM discretization, while the KL based assembly as performed in step 3 in section4.1 only requires the projection on the firstM KL modes.

Poirion and Soize [3, 4] generate realizations of a filtered Poisson process as to simulate a non-Gaussianprocess with given covariance function and lowest order statistical moments. However, this method does notallow to achieve a set of realizations with a proposed marginal PDF. As a result, this method is not fit forthe simulation of an almost surely strictly positive process or a process with a marginal PDF that satisfiesequation (37).

In the present paper, an alternative simulation technique is applied. First, the non-Gaussian processS(x, θ)with marginal PDFpSx

(s) and covariance functionCS(x1,x2) is decomposed in terms of the mean processmS(x) and the zero mean random non-Gaussian processY (x, θ) according to equation (1). Next, the non-Gaussian processY (x, θ) is expressed as a memoryless transformation of an underlying standard GaussianprocessZ(x, θ). The covariance functionCZ(x1,x2) of the underlying Gaussian process is chosen so thatthe transformation leads to a non-Gaussian process with theproposed covariance functionCS(x1,x2). Re-alizations of the Gaussian process are generated by means ofits KL decomposition. The application of thetransformation leads to realizations of the non-Gaussian process [5, 6, 7].

The underlying Gaussian processZ(x, θ) is transformed to the non-Gaussian processY (x, θ) by means of

VARIABILITY 3309

the cumulative density functions (CDF) of both processes:

Y (x, θ) = F−1Yx

(FZ(Z(x, θ))) (39)

whereFYx(y) is the marginal CDF of the non-Gaussian process at positionx andFZ(z) is the standard

Gaussian CDF. An approximation of the transformation (39) is obtained in terms of the one-dimensionalpolynomial chaos of orderP :

Y (x, θ) ≈P

∑

n=0

vn(x)hn(Z(x, θ)) (40)

wherehn(z) denotes the normalized one-dimensional Hermite polynomial of ordern as defined by equation(25). By virtue of the orthonormality of the Hermite polynomials hn(z)n with respect to the Gaussianprobability measure, the coefficientsvn(x)n are obtained as:

vn(x) = E Y (x, θ)hn(Z(x, θ)) =

∞∫

−∞

F−1Yx

(FZ(z))hn(z)pZ(z)dz (41)

wherepZ(z) denotes the standard Gaussian PDF. IfY (x, θ) is a stationary process, then equation (41)reduces to:

vn =

∞∫

−∞

F−1Y (FZ(z))hn(z)pZ(z)dz (42)

Equating the covariance of the LHS and the RHS of equation (40) leads to:

CS(x1,x2) ≈P

∑

n=0

vn(x1)vn(x2) [CZ(x1,x2)]n (43)

If Y (x, θ) is a stationary process, then equation (43) reduces to:

CS(∆x) ≈P

∑

n=0

v2n [CZ(∆x)]n (44)

If this polynomial equation can be solved for every distance∆x and if this leads to a non-negative definitefunctionCZ(∆x), then the covariance function of the underlying Gaussian process is found. Otherwise,a method developed by Puig et al. [6] yielding an approximation of the solution can be applied. Once thecovariance functionCZ(∆x) is obtained, theMZ -th order KL decomposition of the underlying Gaussianprocess is performed. Realizations of this process are thengenerated as in section 2.1. These are transformedinto realizations of the non-Gaussian processY (x, θ) according to equation (39).

The non-Gaussian realizations are projected on the KL modesobtained from theMY -th order KL decompo-sition of the non-Gaussian processY (x, θ). This leads to the realizations of the non-Gaussian KL coefficientsrequired to solve the SFEM equations by means of a Monte Carlosimulation as described in section 4.1.

Example The SFEM is applied to calculate the displacement of a clamped beam with a stationary non-Gaussian bending stiffness. The beam has a lengthl = 4m. The bending stiffness has a Gamma marginalPDF. The parameters of the Gamma PDF areα = 11.111 andθ = 5.4 × 105, so that the mean value of thebending stiffness ismS = 6× 106 Nm2 and the standard deviation is equal toσS = 0.3mS . The covariancefunction of the random process is assumed to be exponential according to equation (17) with a correlationlengthlc = 2m. The beam tip is subjected to a determistic vertical point loadf = 50kN as shown in figure5.

The random process is decomposed into the deterministic mean processmS and the zero mean randomprocessY (x, θ) according to equation (1). This processY (x, θ) is expressed as a transformation of a standard

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−5 0 5−1

−0.5

0

0.5

1

1.5x 10

7

Underlying Gaussian Z(x,θ) [ − ]

Non

−G

auss

ian

Y(x

,θ)

[Nm

2 ]

ExactQ=1Q=2Q=4

Figure 7: Transformation from the underlyingGaussian processZ(x, θ) to the non-Gaussian pro-cessY (x, θ): exact transformation andP -th orderpolynomial chaos approximations.

−1 −0.5 0 0.5 1 1.5

x 107

0

0.5

1

1.5

2

2.5x 10

−7

Non−Gaussian Y(x,θ) [Nm2]

Pro

babi

lity

dens

ity [1

/Nm

2 ] ExactQ=1Q=2Q=4

Figure 8: Marginal PDF for the exact non-GaussianprocessY (x, θ) and for theP -th order polynomialchaos approximations of the process.

Gaussian processZ(x, θ) by means of equation (39). The projection of this transformation on the polynomialchaos of orderP provides an approximation ofY (x, θ) as a function ofZ(x, θ) according to equation(40). The coefficientsvn in equation (40) are obtained according to equation (42). Figure 7 shows theexact transformation and theP -th order polynomial chaos approximation forP = 1, 2 and 4. The exacttransformation and the 4th order approximation are indistinguishable. Figure 8 shows the marginal PDFpY (y) of the zero mean non-Gaussian process as obtained by a transformation of the underlying Gaussianprocess. The 1st order approximation of the transformationleads to a Gaussian process while the 4th orderapproximation leads to a process with a marginal PDF indistinguishable from the target PDF. The 2nd orderapproximation leads to a process with an acceptable marginal PDF and is selected for further calculation.

0 1 2 3 40

0.5

1

1.5

Distance [m]

Cov

aria

nce

[ − ]

CS/σ

S2

CZ

Figure 9: Normalized covarianceCS(∆x)/σ2S of the non-Gaussian processY (x, θ) and covarianceCZ(∆x)

of the underlying Gaussian processZ(x, θ).

For every distance∆x, the value of the covariance functionCZ(∆x) of the underlying Gaussian processZ(x, θ) is obtained as a solution of equation (44). Figure 9 shows thecovarianceCZ(∆x) as comparedto the normalized covarianceCS(∆x)/σ2

S of the non-Gaussian processY (x, θ). In the present case, bothcurves are nearly indistinguishable.

Starting from the covariance functionCZ(∆x), the KL decomposition of orderMZ = 4 is performed asto generatenMCS = 20000 realizations of the underlying Gaussian processZ(x, θ). These realizations aretransformed into realizations of the non-Gaussian processand projected on theMY = 4 first KL modes ofthe non-Gaussian process, so delivering realizations of the non-Gaussian KL coefficientsξj(θij . The KL

VARIABILITY 3311

decomposition of both processes is performed as in section 2.1, using a mesh consisting of 10 elements.

The SFEM equations are solved by means of a Monte Carlo simulation: for each realizationi, the verticaldisplacementUn(θi) of the beam tip is calculated. The estimated mean value of this displacement is equalto mUn = −0.19m and the estimated standard deviation isσUn = 0.05m. According to the central limittheorem, the probability of an absolute difference larger than1mm between the actual and the estimatedmean value of the displacement is equal to 0.0065. The convergence of the estimation of the standarddeviationσUn is verified by performing an increasing number of Monte Carlosimulations. Figure 10 showsan estimation of the PDFpUn(un) of the displacementUn(θ). Since the support of the Gamma marginal

−1 −0.5 0 0.5 10

2

4

6

8

10

Displacement [m]

Pro

babi

lity

dens

ity [1

/m]

Figure 10: Estimation of the PDFpUn(un) of the vertical displacement of the beam tip.

PDF of the bending stiffness is restricted to the set of positive real numbersR+, the direction of the responseUn(θ) corresponds to the direction of the forcef almost surely. Consequently, the PDFpUn(un) is exactlyequal to zero forun > 0, as opposed to the PDF of the displacement of a beam with a Gaussian bendingstiffness.

7 Conclusion

This paper illustrates the application of the SFEM for the solution of mechanical problems with spatiallyvarying random characteristics modelled as random processes. Both Gaussian and non-Gaussian processesare covered.

First, the Gaussian case is considered. Two techniques are applied to solve the SFEM equations for systemswith Gaussian characteristics. The first technique is a Monte Carlo simulation of the random coefficients oc-curring in the KL decomposition of the Gaussian processes. The second technique is based on the projectionof the response on the polynomial chaos in terms of the randomKL coefficients. It is shown both numericallyand analytically that the variance of the response of a system with a Gaussian stiffness is infinite. This is notphysically sound as it implies that the expected value of thedeformation energy of the system is undefined.

Next, the non-Gaussian case is treated. An adaptation of theMonte Carlo solution of the SFEM equations ispresented as to enable the calculation of the response of non-Gaussian systems. As opposed to the Gaussiancase, the direct generation of the KL coefficients is impossible in this case since they are mutually dependentand their joint probability distribution is unknown. Hencethey are obtained by the projection of realizationsof the non-Gaussian process, generated by means of Hermite polynomial expansion.

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References

[1] R. Ghanem and P.D. Spanos.Stochastic finite elements: a spectral approach. Springer-Verlag NewYork Inc., New York, 1991.

[2] C.Z. Karakostas and G.D. Manolis. Dynamic response of tunnels in stochastic soils by the boundaryelement method.Engineering Analysis with Boundary Elements, 26:667–680, 2002.

[3] F. Poirion. Numerical simulation of homogeneous non-Gaussian random vector fields.Journal ofSound and Vibration, 160(1):25–42, 1993.

[4] F. Poirion and C. Soize. Monte Carlo construction of Karhunen Loeve expansion for non-Gaussianrandom fields. In N. Jones and R. Ghanem, editors,13th ASCE Engineering Mechanics DivisionConference, Baltimore, USA, June 13-16 1999.

[5] B. Puig and J.-L. Akian. Non-Gaussian simulation using Hermite polynomial expansion and maximumentropy principle.Probabilistic Engineering Mechanics, 2003. Article in press.

[6] B. Puig, F. Poirion, and C. Soize. Non-Gaussian simulation using Hermite polynomial expansion:convergences and algorithms.Probabilistic Engineering Mechanics, 17:253–264, 2002.

[7] S. Sakamoto and R. Ghanem. Polynomial chaos decomposition for the simulation of non-Gaussiannonstationary stochastic processes.Journal of Engineering Mechanics, Proceedings of the ASCE,128(2):190–201, 2002.

[8] C. Soize. Probabilites et modelisation des incertitudes: elements de base et concepts fondamentaux.Handed out at the seminaire de formation de l’ecole doctorale MODES, Paris, May 2003.

[9] A. Spanos.Probability theory and statistical inference. Cambridge University Press, Cambridge, U.K.,1999.

[10] B. Van den Nieuwenhof.Stochastic finite elements for elastodynamics: random fieldand shape un-certainty modelling using direct and modal perturbation-based approaches. PhD thesis, UniversiteCatholique de Louvain, 2003.

[11] C.H. Yeh and M.S. Rahman. A stochastic finite element analysis for the seismic response variabilitysoil sites. In H.J. Siriwardane and M.M. Zaman, editors,Computer Methods and Advances in Geome-chanics: Proceedings of the Eighth International Conference, pages 1017–1022, Morgantown, USA,May 1994. Balkema Publishers.

[12] A. Zerva and T. Harada. Effect of surface layer stochasticity on seismic ground motion coherence andstrain estimates.Soil Dynamics and Earthquake Engineering, 16:445–457, 1997.

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