Received by ：2005-03-18；Revid by ：2005-08-26.
Project supported by the NationaI Science Foundation of China
（10602036）.
Corresponding authors ：HUANG Shu-ping （1973-），
Dr ，Associate Professor.
第24卷第2期2007年4月 计算力学学报
Chine Journal of Computational Mechanics
VoI.24，No.2
ApriI 2007
文章编号：1007-4708（2007）02-0173-08
A collocation-bad spectral stochastic
finite element analysis stochastic
respon surface approach
HUANG Shu-ping
（Department of CiviI Engineering ，Shanghai Jiaotong University ，Shanghai 200240，China ）
Abstract ：A coIIocation-bad stochastic finite eIement method （SRSM ）has been deveIoped ，the formaIism of the pro-pod method is simiIar to the spectraI stochastic finite eIement method （SSFEM ）in the n that both of them utiIize Karhunen-loeve （K-l ）expansion to reprent the input ，and poIynomiaI chaos expansion to reprent the output.How-ever ，the caIcuIation of the coefficients in the poIynomiaI chaos expansion is different ：AnaIyticaI SSFEM us a probabi-Iistic GaIerkin approach whiIe SRSM us a probabiIistic coIIocation approach.NumericaI exampIe shows that compared to the AnaIyticaI SSFEM ，the advantage of SRSM is that the finite eIement code can be treated as a bIack box ，as in the ca of a commerciaI code.The propod SRSM is aIso compared to a bIack box version SSFEM ，and found to reguire Iess FEM evaIuations for the same accuracy.The coIIocation points in the propod method need to be Iected for mini-mizing the mean sguare error ，and from high probabiIity regions ，thus Ieading to fewer function evaIuatio
ns for high accu-racy.
Key words ：stochastic finite eIements ；stochastic respon surface ；random fieIds ；Karhunen-loeve expansion ；poIyno-miaI chaos expansion
1 Introduction
ProbabiIistic uncertainty propagation methods are appIied in the anaIysis of physicaI systems in order to guantify the effects of random variation in the input on the predicted output of the simuIation.The methods incIude Monte CarIo simuIation ，stochastic finite eIe-ment ［1］and respon surface methods ［2］
.The Iec-
tion of the method depends on the nature of modeI ud for predicting the output.
Monte CarIo simuIation methods are accurate and wideIy appIicabIe but time-consuming.When the modeIs are Iarge ，or when there are many parameters ，even the best of Monte CarIo or importante sampIing methods can be prohibitiveIy expensive.The appIica-tion of respon surface methods to probIems invoIving random fieIds isaIso not easy due to the Iarge number
of random variabIes into which a continuous random fieId is reduced by discretization.Stochastic finite eI-ement methods such as perturbation and Neumann ex-pansion ［2］
work weII when the variabiIity is not Iarge.
The spectraI stochastic finite eIement method （SS-FEM ）deveIoped by Ghanem and Spanos
［1］
appears to be a suitabIe technigue for the soIution of compIex ，generaI probIems in probabiIistic mechanics.It is ca-pabIe of handIing much higher fIuctuations.However ，this method reguires access to the governing modeI e-guations.Furthermore ，the resuIting system of egua-tions to be soIved for the unknown respon is much Iarger than tho from deterministic finite eIement a-naIysis.For compIicated Iarge system probIems ，the system of eguations in the spectraI stochastic finite eI-ement method couId be tremendousIy Iarge.For ex-ampIe ，if the deterministic system is of size !by !，and the number of terms in the poIynomiaI chaos ex-pansion is "，then the size of the stochastic system wouId be "X !by "X !.AIthough a new impIementa-
tion of SSFEM［3］，which is theoreticaiiy eguivaient to the originai SSFEM，has been deveioped
for the pur-po of utiiizing commoniy avaiiabie FEM codes as a biack box，this novei impiementation of SSFEM men-tioned in this paper as biack box-SSFEM reguires ran-dom sampiing of the input and conguentiy a iarge number of FEM runs to get a stabie estimate of the co-efficients in the expansion of the soiution.The origi-nai SSFEM［1］is referred to in this paper as anaiyticai SSFEM，for the sake of comparison.
This paper prents a modified spectrai stochastic finite eiement method.This methodoiogy combines Karhunen-loeve（K-l）expansion［1］with poiynomiai chaos［1］to construct a respon surface as an efficient uncertainty propagation modei.First，the input ran-dom fieid is discretized into standard random variabies using the K-l expansion.The output random fieid is treated as an eiement in the Hiibert space of random functions spanned by basis in terms of tho random variabies.Specificaiiy，the output is reprented by a poiynomiai chaos expansion in terms of the standard random variabies.The unknown coefficients of the poiynomiai chaos expansion are estimated by eguating modei outputs which are obtained from finite eiement anaiysis and the respon surface reprented by poiy-nomiai chaos expansion，at a t of coiiocation points in the sampie space.
The formaiism of the propod method is simiiar to the spectrai stochastic finite eiement method in t
he n that both of them utiiize Karhunen-loeve expan-sion and poiynomiai chaos expansion to reprent the input and output random fieids respectiveiy.Howev-er，the caicuiation of the coefficients in the poiynomi-ai chaos expansion is different in the two methods. SSFEM us a probabiiistic Gaierkin approach whiie the propod method us a probabiiistic coiiocation approach.Simiiar to the Gaierkin and coiiocation methods which are weighted residuai methods in de-terministic numericai anaiysis，the probabiiistic
西兰花炒什么Gaierkin and coiiocation methods are both weighted residuai methods in the random domain.The ap-proach can be viewed as an extension of deterministic computationai anaiysis to the stochastic ca，with an appropriate extension of the concept of weighted resid-uai error minimization.
There are three advantages in the propod sto-chastic respon surface method.Firstiy，the Kar-hunen-loeve expansion for modeiing the input random fieids is a spectrai approach which offers an optimai means for repiacing the random fieid with a smaii number of random variabies.Secondiy，the soiution approximated by a poiynomiai chaos expansion is a re-spon surface，not mereiy statisticai moments as in the ca for many other methods.Finaiiy，the pro-pod stochastic respon surface method can be wrapped around existed deterministic finite eiement codes.This means that the finite eiement code can be treated as a biack box，as in the ca of commerciai cod
es.The comparison of numericai resuits from both SSFEM and SRSM highiights the desirabie features of the propod technigue and demonstrates its accura-cy.Compared to the Anaiyticai SSFEM，the advan-tage of SRSM is that the finite eiement code can be treated as a biack box，as in the ca of a commerciai code.The propod SRSM is aiso compared to a biack box version SSFEM，and found to reguire iess FEM evaiuations for the same accuracy.油烟
2 Spectral stochastic finite
element method
The spectrai stochastic finite eiement method （SSFEM）has been deveioped and appiied to various probiems.Detaiied descriptions of SSFEM can be found in verai papers［1］.The esntiai concepts of SSFEM are provided here，as it is necessary for un-derstanding the modified method.The propod sto-chastic respon surface method for random fieid probiems wiii be prented iater in the next ction.
471计算力学学报第24卷
2.1 Problem description
A standard form of a stochastic partiaI differentiaI eguation（SPDE）may be written as
K（U，~（x，））U=F（）（l）where U denotes the soIution of the probIem，~（x，）denotes the random materiaI property and re-fers to the random events.Prence of stochasticity in either the system coefficient~（x，）or source term F （）wiII render the soIution U to be stochastic.There are two types of probIems of interest here：one with a stochastic source term and deterministic system coeffi-cients；the other with stochastic system coefficients and a deterministic source term.The propod meth-od can consider stochasticity in both system coeffi-cients and source terms.
The main eIements in SSFEM are K-L expansion-bad reprentation of the input random fieIds，poIy-nomiaI chaos reprentation of the output，and caIcu-Iation of the unknown coefficients by a GaIerkin scheme in the random dimension.The concepts are summarized beIow.
2.2 Karhunen-loeve expansion
A cond order random process~（x，）defined in a probabiIity space（，A，P）and indexed on a bounded domain D can be expanded as［l］
~（x，）=-~（x）+Z
i=l
!i i（）f i（x）（2）
in which
i and f
i
（x）are the eigenvaIues and eigen-
functions of the covariance function C（x
l ，x
2
）.By
definition，C（x
l ，x
2
）is bounded，symmetric and pos-
itive definite.FoIIowing Mercer's Theorem［l］，it has the foIIowing spectraI or eigen-decomposition：
C（x
l ，x
2
）=Z
i=l
i
f
i
（x
l
）f
i
双音节单词（x
2
）（3）
which has a countabIe number of eigenvaIues and the associated eigenfunctions obtained from the soIutions of the integraI eguation
I D C（x l，x2）f i（x2）d x2=i f i（x l）（4）Eg.（4）aris from the fact that the eigenfunctions form a compIete orthogonaI t satisfying the egua-tion：
I D f i（x）f（x）d x=i （5）where
i
is the Kronecker-deIta function.
The K-L expansion in Eg.（2）provides a c-ond-moment characterization in terms of uncorreIated random variabIes and deterministic orthogonaI func-tions.It is known to converge in the mean sguare n for any distribution of~（x，）.For practicaI impIementation，the ries is approximated by a finite number of terms.
If~（x，）is further restricted to a zero-mean
Gaussian process，then the appropriate choice of｛
l （），
2
（）…｝is a vector of zero-mean uncorreIated Gaussian random variabIes.
2.3 Polynomial chaos expansion
Since the output is a function of the input fieIds，it can be expresd by a nonIinear function of the t of random variabIes which are ud to reprent input stochasticity.The function of Gaussian variabIes which is known as poIynomiaI chaos is given by
U（）=a
T0+Z
I
i l=l
O i
把握机遇l
T l（i
l
（））+
Z I
i l=l，
Z i l
i2=l
a
i l i2
T2（i
l
（），
i2
（））+
Z I
i l=l，
Z i l
i2=l，
Z i2
i3=l
a
i l i2i3
T3（i
l
（），
i2
（），
i3
（））+ (6)
where T
p
（
i l
，…，
i p
）
denotes the poIynomiaI chaos of order p in terms of the muti-dimensionaI random
variabIes｛
i I
｝M
I=l
.The poIynomiaI chaos is defined in terms of Hermite poIynomiaIs as
T p（i
l
，…，
i M
）=（-l）p e l2T O
M
O i
l
…O
i M
e l2T（7）This is the same as a M-dimensionaI Gaussian joint probabiIity density function.For notation simpIicity，Eg.（7）is rewriten as
U（）=Z N
=0
U G（（））（8）
where there is a correspondence between T
p
（
i l
，…，
i I
）and G（）and their corresponding coefficients. The orthogonaIity of the poIynomiaI chaos is of the form：
〈G
i
G〉=〈G2
i
〉
i
（9）
57l
第2期黄淑萍：基于配点法的谱随机有限元分析———随机响应面法
where!
i
is the Kronecker-deita function.Poiynomiais of different order are orthogonai to each other，and so are poiynomiais of the same order but with different arguments.Detaiis for caicuiating poiynomiai chaos can be found in references［1］.The ries couid be truncated to a finite number of terms.The accuracy of the computationai modei increas as the order of the poiynomiai chaos expansion increas.For exampie，the cond and third order Hermite poiynomiais are as foiiows：
｛"｝=｛1，#
1，#
2
，#2
1
-1，#
1
#2，#2
2
-1｝（10）
｛"｝=｛1，#
1，#
2
，#2
1
-1，#
1
#2，#2
2
-1，
#3 1-3#
1
，#2
1
#2-#2，#2
可爱的小金鱼
2
#1-#1，#3
2
-3#
2
｝
（11）
2.4 Analytical spectral stochastic finite
element for mulation
Substituting Eg.（2）with M terms and（8）into the eguation of eguiiibrium［Eg.（1）］yieids
Z N
=0
"K（U，#（$））U=F（$）（12）The error in the above eguation can be minimized u-sing the Gaierkin method which reguires the error to be orthogonai to the basic functions in the approxima-tion space：
Z N
=0
〈""K（U，#（$））〉U=〈"F（$）〉
=0，1，…，N（13）The random coefficients matrix K can be expand-ed into a poiynomiai of the form
K=Z M
i=0#i K i，K i=
〈#
i
K〉
〈#2
i
〉
（14）
Eg.（13）becomes
远离烟草Z M i=0Z N
=0
〈#
i
""〉K i U=〈"F〉
=0，1，…，N（15）
The coefficients of the respon on the ieft hand side of Eg.（15）can be asmbied into a matrix of size （N+1）X I by（N+1）X I of the form.
From the above discussion，it is en that in SS-FEM，the reprentation of the random fieids in the context of the finite eiement procedure has the effect of adding extra dimensions to a probiem with I de-grees of freedom.The poiynomiai chaos，which is ud to discretize the random dimension，contributes a factor of（N+1）to the size of the probiem.Cou-piing this new discretization with the finite eiement spatiai discretization in a discrete system，the probiem size becomes（N+1）X I by（N+1）X I.This in-creas the computationai cost during the creation and soiution of the system coefficient matrix.This is fur-ther affected by the number of terms（M）ud in the K-L expansion of the input random fieids due to the foiiowing reiation：
N=Z p
S=1
1
S！
S-1
r=0
（M+r）（16）However，formuiating the eiement stiffness re-guires access to the governing modei eguations.Fur-thermore，the resuiting system of eguations to be soived for the unknown respon is much iarger than tho from deterministic finite eiement anaiysis.The size of the probiem controis the computationai effi-ciency.For compiicated iarge system probiems，the system of eguations in the spectrai stochastic finite ei-ement method couid be tremendousiy iarge.
2.5 Black box spectral stochastic finite
垃圾分类实施方案
element formulation
The coefficients in Eg.（8）can aiso be evaiua-ted by another method，referred to in this paper as the biack box SSFEM［3］.Given the orthogonaiity of the poiynomiai chaos basis"（#），the coeffi
cients in the expansion in Eg.（8）can be computed as generaiized Fourier coefficients according to the foiiowing expres-sion
U=
〈"U〉
〈"2〉
（17）For each reaiization of the t of basic random varia-
bies#分辨的英文
i
，the reaiization of the input reprenting the materiai property is obtained by Eg.（2）.Then the reaiization of the output（soiution）is obtained by soi-ving the finite eiement system（one FEM run）.The reaiization of the soiution is muitipiied by each of the reaiizations of"（#）and Eg.（17）is evaiuated，thus ieading to an estimate of the coefficients in the expan-sion in Eg.（8）.Basic Monte Cario sampiing and
671计算力学学报第24卷
other variance reduction sampiing technigues such as Latin hypercube sampiing［1］may be ud for genera-ting the input reaiizations.
The biack box SSFEM is deveioped for the pur-po of utiiizing commoniy avaiiabie FEM codes. However，it us random sampiing of the input and conguentiy a iarge number of FEM runs to get a sta-bie estimate of the coefficients in the expansion of the soiution.Therefore，a modified spectrai stochastic fi-nite eiement method is propod in the next ction，bad on a probabiiistic coiiocation approach.The propod method prerves the benefits of expansions in SSFEM but us a different error minimization process for the caicuiation of the unknown coefficients in the poiynomiai chaos.Aiso，the deterministic finite eiement anaiysis can be treated as a biack box，as in the ca of commerciai codes.Detaiis are given in the next ction.
3 Collocation-bad SFEM（SRSM）
The output from the anaiyticai SSFEM can be viewed as a stochastic respon surface in which the coefficients are caicuiated by the Gaierkin method. Simiiar to the Gaierkin method，the coiiocation meth-od is another weighted residuai minimization process in numericai anaiysis.It has been mathe
maticaiiy proved that an“optimai”coiiocation method with ac-curacy comparabie to or even eguai to the accuracy of Gaierkin method is obtained when the coiiocation points are iected at the zeros of the orthogonai poiy-nomiais ud in the approximation.If we extend the deterministic numericai anaiysis to the stochastic ca，the reiationship between the propod method and SSFEM is anaiogous to the reiationship between the coiiocation and the Gaierkin methods in determin-istic numericai anaiysis.In the stochastic ca，the respon，which is a random function，is approxima-ted by a poiynomiai chaos-bad respon surface in both methods.The poiynomiai chaos is nothing but Hermite poiynomiais in terms of random functions （random variabies）.The probabiiistic coiiocation points are therefore iected as roots of Hermite poiy-nomiais.The coiiocation method is easier to impie-ment but in generai a iittie iess accurate，whereas the Gaierkin method is more accurate but cumbersome to impiement.In anaiyticai SSFEM，in which the proba-biiistic Gaierkin approach is pursued，the probabiiis-tic anaiysis and FEM anaiysis are done together. Therefore，accessing the FEM code is necessary. Whereas in SRSM，in which probabiiistic coiiocation is pursued，the FEM code can be treated as biack box.
The other two eiements in SSFEM，K-L expan-sion reprentation of the input random fieids and poi-ynomiai chaos projection of the respon，remain the same.
3.1 Steps of SRSM
This method was first propod by Isukapaiii et ai［16］.However，the method was oniy iimited to prob-iems with random variabies.In this paper，SRSM is extended to probiems with random fieids by using the K-L expansion.A generai procedure of SRSM for ran-dom fieid probiems is briefiy summarized beiow：（a）Reprentation of random process input in terms of Standard Random Variabies（SRVs）by K-L expansion.
（b）Expression of modei output in chaos ries expansion.Once the input is expresd as functions of the iected SRVs，the output guantities can aiso be reprented as functions of the same t of SRVs.If the SRVs are Gaussian，the chaos for the output is a Hermite poiynomiai of Gaussian variabies，which is poiynomiai chaos.If the SRVs are non-Gaussian，the output can be expresd by other Askey chaos in terms of non-Gaussian variabies.In this paper，oniy Gaussian fieids are considered.
（c）Estimation of the unknown coefficients in the ries expansion.The improved probabiiistic coi-iocation method［4］is ud to minimize the residuai in the random dimension by reguiring the residuai at the coiiocation points eguai to zero.The modei output is computed at a t of coiiocation points and ud to es-
771
第2期黄淑萍：基于配点法的谱随机有限元分析———随机响应面法
