Adaptive Random FieldMesh Reﬁnements in Stochastic...

Copyright c© 2007 Tech Science Press CMES, vol.19, no.1, pp.23-54, 2007

Adaptive Random Field Mesh Refinements in Stochastic Finite ElementReliability Analysis of Structures

M. Manjuprasad1 and C. S. Manohar2

Abstract: A technique for adaptive randomfield refinement for stochastic finite element re-liability analysis of structures is presented in thispaper. Refinement indicator based on global im-portance measures are proposed and used for car-rying out adaptive random field mesh refinements.Reliability index based error indicator is proposedand used for assessing the percentage error in theestimation of notional failure probability. Adap-tive mesh refinement is carried out using hierar-chical graded mesh obtained through bisection ofelements. Spatially varying stochastic system pa-rameters (such as Young’s modulus and mass den-sity) and load parameters are modeled in generalas non-Gaussian random fields with prescribedmarginal distributions and covariance functions inconjunction with Nataf’s models. Expansion opti-mum linear estimation method is used for randomfield discretisation. A framework is developedfor spatial discretisation of random fields for sys-tem/load parameters considering Gaussian/non-Gaussian nature of random fields, multidimen-sional random fields, and multiple-random fields.Structural reliability analysis is carried out us-ing first order reliability method with a few re-fined features, such as, treatment of multiple de-sign points and/or multiple regions of compara-ble importance. The gradients of the performancefunction are computed using direct differentiationmethod. Problems of multiple performance func-tions, either in series, or in parallel, are handledusing the method based on product of conditionalmarginals. The efficacy of the proposed adaptivetechnique is illustrated by carrying out numericalstudies on a set of examples covering linear static,

1 Scientist, Structural Engineering Research Centre, Tara-mani, Chennai, India

2 Professor, Department of Civil Engineering, Indian Insti-tute of Science, Bangalore, India

free vibration and forced vibration problems.

1 Introduction

One of the major developments in the recent yearshas been the extension of deterministic finite el-ement methods to cover problems involving un-certainties in structural properties and in specify-ing loads which has resulted in the developmentof stochastic finite element methods (SFEMs).These developments include treatment of spatiallyvarying structural properties that are modeled asrandom fields, loads that are modeled as space-time random processes, structural behavior cover-ing static and dynamic regimes, linear and nonlin-ear structural mechanical behavior, and character-ization of response variability and determinationof structural reliability measures. Various mathe-matical tools, including, discretisation of randomfields, random eigenvalue analyses, solution ofstochastic boundary value problems, inversion ofrandom matrix and differential operators, compu-tation of reliability measures via nonlinear opti-mization tools, Monte Carlo simulation methodsincluding variance reduction strategies, and appli-cations of response surface modeling have beendeveloped (for example see Ghanem and Spanos,1991; Kleiber and Hien, 1992; Schueller, 1997;Manohar and Ibrahim, 1999; Haldar and Mahade-van, 2000; Jefferson Stroud, Krishnamurthy andSteven Smith, 2002; Manohar and Gupta, 2005).

The finite element method, even when applied todeterministic problems, is already approximate innature, and, when the inputs to the finite elementmodels are additionally treated as being prob-abilistic in nature, the character of approxima-tions involved gets further compounded by inac-curacies realized in treatment of probability mea-sure and uncertainty propagation through the sys-

24 Copyright c© 2007 Tech Science Press CMES, vol.19, no.1, pp.23-54, 2007

tem mechanics. Consequently, questions on er-rors involved in response predictions, vis-à-vis thechoices made in handling of uncertainties, lead tochallenging research problems. In a problem in-volving reliability analysis of structures analyzedusing SFEM, there could be several sources of nu-merical errors. These errors can be grouped asfollows:

Group I:

• errors of discretisation of displacementfields,

• errors of time stepping in direct integration,

• errors in computation of stress fields,

• errors in determination of natural frequen-cies and mode shapes, and

• errors due to truncation of modal expansionof response.

Group II:

• errors associated with discretisation of ran-dom fields,

• truncation errors in series representationsused for random field discretisation,

• treatment of non-Gaussian nature of randomfields via transformation to normal space,

• errors associated with characterization of re-liability measures via linearization and sub-sequent solution of constrained optimizationproblems,

• errors associated with computation of gradi-ents needed in calculation of reliability in-dices,

• errors associated with treatment of multipledesign points and multiple regions of compa-rable importance in reliability calculations,

• errors associated with treatment of multiplefailure modes,

• treatment of time-variant reliability usingasymptotic theories of extremes of randomprocesses, and

• treatment of sampling fluctuations if MonteCarlo simulation methods are employed inreliability assessment.

In any given problem, all the above sources of er-rors are present simultaneously and interact in acomplicated way contributing to the overall errorin reliability estimates. A unified treatment of allthese errors, with a view to select parameters ofnumerical analysis so as to control the errors in re-liability analysis, leads to complicated questions.In the present study, we restrict our aim to addressissues related to impact of decisions made on dis-cretisation of random fields on assessment of reli-ability of structure on hand.

In the context of SFEM based reliability analy-sis of dynamic structures with spatially distributedrandom properties, there could be three differentmeshes that need to be formulated; these are: (a)mesh for discretizing the displacement fields, (b)mesh for discretizing the random fields, and (c)time steps for integrating the equilibrium equa-tions. The accuracy of random response charac-terization depends upon the choices made in for-mulating these meshes. In an adaptive process,attempts are made to estimate the errors with-out excessively increasing the numerical effortand to leave the decision of suitably refining themesh/time steps more or less to a computer pro-gram. Considerable research has been carried outon adaptive mesh/time step refinements, particu-larly in the context of deterministic finite elements(for example see Zienkiewicz and Taylor, 2000;Ainsworth and Oden, 2000; Stein, Ruter and Ohn-imus, 2004; Jin Ma, Hongbing Lu, and Ranga Ko-manduri, 2006; Wacher; and Givoli, 2006). How-ever, literature available on adaptive methods forstochastic finite element analysis is limited. Liu,Belytschko and Mani (1986) have studied the er-ror induced by discretising a random field anddemonstrated that the number of random variablesrequired to represent a random field is controlledby the correlation length of the random field. Liand Der Kiureghian (1993) have carried out stud-ies on the error in different random field discreti-sation methods as a function of element size forselected correlation functions. Liu and Liu (1993)studied the influence of random field mesh on the

Adaptive Random Field Mesh Refinements 25

performance of stochastic finite element reliabil-ity analysis in static problems. They presentedrules for the selection and refinement of randomfield meshes. The error was computed using reli-ability index as the error norm and the refinementof random field mesh was carried using the gra-dients of Limit-state function as error indicators.Deb, Babuska and Oden (2001) have developeda framework for the construction of Galerkin ap-proximations of elliptic boundary-value problemswith stochastic input data. A theory of posteri-ori error estimation and corresponding adaptiveapproaches based on practical experience can beutilized. Pellissetti (2003) has discussed issues re-lated to adaptive data refinement in the spectralstochastic finite element method.

The present paper addresses some of the questionsrelated to the issue of random field mesh refine-ments. Specifically, we consider linear systemshaving non-Gaussian spatially distributed prop-erties subject to static or dynamic loads. Theproblem of refinement of the mesh used to dis-cretize the random fields, in order to achieve anacceptable estimate of the reliability of the sys-tem, is considered. The study proposes a generalframework to achieve this objective. The reliabil-ity analysis procedure used is based on conceptsfrom first order reliability methods involving sev-eral improvements to handle possible existence ofmultiple design points or multiple regions of com-parable importance and subsequent use of meth-ods of system reliability analysis, and, to take intoaccount random dynamic loads. The refinementprocedure itself is based on a global importancemeasure (GIM) assigned primarily to the basicrandom variables in standard normal space. Thesemeasures are subsequently assigned to the orig-inal non-Gaussian random variables using a se-quence of transformations. Adaptive mesh refine-ment is carried out using hierarchical graded meshobtained through bisection of elements. The studyaims at refining the reliability index/failure proba-bility estimates obtained through adaptive refine-ment of finite element mesh and random fieldmesh (RF mesh). Illustrative examples involvingstatic/dynamic problems of structures with one-dimensional Euler beam elements and plane stress

elements are presented.

It should be noted that exact solutions to problemsof reliability analysis of structure modeled us-ing SFEM are seldom available. In the proposedmethod for adaptive refinement of random fieldmesh, the first step consists of solving the prob-lem with a coarse uniform mesh configured basedon user’s judgment. The proposed GIM is derivedbased on this coarse mesh and these measures in-dicate spatial regions where the RF meshes needto be refined. Reliability analysis with the refinedmesh is subsequently performed, which, in turn,leads to the improved plots of GIM. This processis continued till a satisfactory convergence is ob-tained on the reliability measure being estimated.In the present study, to demonstrate the feasibilityof the proposed mesh refinement procedures, wefirst solve the reliability analysis problem usinga fairly fine displacement field mesh (DF mesh)and RF mesh configurations. The result from thisanalysis is treated as a benchmark against whichthe merit of the mesh configuration at a givenstage of refinement is evaluated. It is to be noted,however, that the application of refinement proce-dure developed herein, does not depend upon theavailability of a benchmark solution. It is also im-portant to note that the refinement strategy here isgoal oriented. That is, specifically, we focus onone or more specified performance functions withrespect to which the structural reliability is anal-ysed. The performance functions are typically interms of response at a set of specified points. Inthe present paper, we focus on performance func-tions involving displacements at specified loca-tions or natural frequencies of specified modes ofthe structure.

2 Problem Formulation

Consider the problem of reliability analysis of astructure in which uncertainties enter through oneor more of the following routes:

1. spatially varying structural properties (suchas, material properties, strength characteris-tics, joint characteristics or boundary condi-tions on an edge), which are modeled as a setof non-Gaussian random fields; these are de-


noted collectively by the vector random fieldwww(xxx) with xxx denoting the vector of spatial co-ordinates;

2. loads that act on the structure that need tobe modeled as random processes evolving isspace and/or time; these are denoted collec-tively by fff (xxx, t) with xxx denoting the vector ofspatial co-ordinates and t denoting the time;and,

3. those properties of structure or the appliedloads for which a random variable modelwould suffice; these are denoted collectivelyby the vector of mutually dependent non-Gaussian random variables Θ .

This list could also include uncertainties in mod-eling; this, however, has not been included in thepresent study. It is assumed that the random fieldswww(xxx) and fff (xxx, t) would be discretised suitablyleading to a vector of random variables Ξ , which,in general, would be non-Gaussain and mutuallydependent in nature. Furthermore, the randomvariable vectors Θ and Ξ are collectively denotedby the vector Ψ = [Θ Ξ ]t . We define a set of per-formance functions gi (Ψ) (i = 1,2. . .,n) such thatthe regions in the Ψ -space for which gi (Ψ) < 0and gi (Ψ) > 0, respectively, denote the unsafeand safe regions with the surface gi (Ψ) = 0 rep-resenting the limit surface with respect to the ith

performance function. We denote by Pfi the esti-mate of the probability of failure with respect toith performance function and Pf as the estimate ofthe probability of failure from structural systemreliability analysis. The problem on hand consistsof selecting and modifying the details of randomfield discretisation mesh so that the accuracy ofestimation of Pf is improved.

3 Overview of the Proposed Adaptive Solu-tion Strategy

In this section we present the broad features ofthe proposed adaptive solution strategy. The de-tailed description of the various steps involved isdiscussed in subsequent sections. In the presentstudy, an adaptive technique is proposed for car-rying out stochastic finite element reliability anal-

ysis of structures. It aims at refining the re-liability index/failure probability estimates ob-tained through adaptive refinement of randomfield mesh. The following are the broad featuresof strategy adopted to tackle the problem stated inthe previous section.

A. Random field discretisation: We model one ormore of the structural properties as a vec-tor of mutually dependent non-Gaussian ran-dom fields. These fields are taken to bedifferentiable in the mean square sense andare assumed to be spatially homogeneous andisotropic. We follow the expansion optimumlinear estimation (EOLE) method of discreti-sation combined with Nataf’s transformationtechnique to represent uncertainties throughan equivalent vector of standard normal ran-dom variables.

B. Discretisation of load fields: We handle thespatially distributed surface load as a Gaus-sian random field. This field is taken to bedifferentiable in the mean square sense andis assumed to be spatially homogeneous andisotropic. We again follow the expansion opti-mum linear estimation (EOLE) method of dis-cretisation to represent uncertainties throughan equivalent vector of standard normal ran-dom variables. We model the time varyingstochastic excitations due to nodal loads aszero-mean stationary Gaussian random pro-cess with a specified power spectral density(PSD) function.

C. Mesh refinement and error indicator: Wehave proposed the use of GIM with appro-priate transformations as mesh refinementindicators and this is one of the originalideas developed for carrying out adaptiverandom field mesh refinements proposed inthis paper. A refinement norm based on anotional reliability index is also proposed andused as an error indicator for estimating thepercentage error in the estimation of Pf .

D. Mesh refinement strategy: We focus attentionmainly on placement of nodes for randomfield meshes. The method developed does not


impose any restriction on relative sizes of dif-ferent RF meshes as well as the DF mesh. Allthese meshes could be refined independent ofeach other. The method has provisions for as-sembly of various structural matrices takinginto account differences in the associated RFmeshes. The mesh refinement itself is carriedon by using the method of successive bisec-tion. The study also examines the possibil-ity of derefinement of meshes, wherein, wereduce the number of elements in spatial re-gions considered less important by the com-puted GIMs. The question of using GIM alsoto refine DF mesh and time steps of direct inte-gration problem are also addressed albeit verybriefly.

E. Reliability estimation: Structural reliabilityanalysis is carried out using FORM with thefollowing improvements: (a) treatment ofmultiple design points or multiple regions ofcomparable importance in reliability calcula-tions, (b) system reliability problems involv-ing multiple performance functions, and (c)dynamic problems involving random excita-tions. The performance function is computedusing finite element method and the gradi-ents needed in reliability calculations are eval-uated numerically using direct differentiationmethod.

4 Random Field Discretisation

In this study, we consider the random fields thatmodel the structural properties as a vector of non-Gaussian mutually dependent random fields. Wealso take that these random fields are only par-tially specified with knowledge on their propertieslimited to the first order marginal probability den-sity function (pdf), and the set of auto-covarianceand cross-covariance functions. We discretisethese random fields using the EOLE method (Liand Der Kiureghian, 1993) in conjunction withthe Nataf’s transformation technique (Der Ki-ureghian and Liu, 1986), which leads to the def-inition of a vector of mutually dependent non-Gaussian random variables.

In order to clarify the steps involved in discreti-

sation of random field, and its relation to dis-cretisation of displacement field we consider aone-dimensional problem involving a displace-ment field u(x) and a non-Gaussian random fieldw(x) with 0 < x < L. Let Fw [w;x] and ρww (x,x′)denote, respectively, the first order pdf and auto-correlation function associated with w(x). Letthe displacement field be discretised into N∗ el-ements of length Δ ′x and (N∗ +1) nodes suchthat N∗Δ ′x = L. Let {x′i}N∗

i=1 denote the centroidalvalue of the ith element. In formulating the struc-tural matrix for the ith element we need the valueof w(x′i).To arrive at the value of w(x′i), we begin byfirst transforming w(x) into an equivalent Gaus-sian random field v(x) using the rule v(x) =Φ−1 [Fw (w;x)] (Der Kiureghian and Liu, 1986).The Gaussian random field v(x) is discretised us-ing another mesh with N nodes and (N −1) el-ements of length Δx such that (N −1)Δx = L.Here, {xi}N

i=1represents the nodal coordinates andv(xi), i = 0,1 . . .,N constitutes a set of correlatedGaussian random variables. We represent v(x) us-ing the EOLE expansion such that for any pointx = xi,

v̂(xi) =

c(xi)+N

∑j=1

b j (xi)

(μ (x j)+

r

∑k=1

Γ ′k

√θkφk j

),

i = 0,1 . . ., N̂ (1a)

where, μ (x) represents the mean of v(x), {θk}rk=1

and {ϕk}rk=1 are the r highest eigenvalues and

eigenvectors associated with auto-covariance ma-trix of v(x) evaluated at x = x1,x2 . . . ,xN ; and,{

Γ ′k

}rk=1 are a set of standard normal random

variables. The functions c(x) and{

b j (x)}N

j=1are the interpolation functions used in discretiz-ing v(x), selected such that the variance of er-

ror given by⟨{v(x)− v̂ (x)}2

⟩is minimized for

every x in (0,L) (Li and Der Kiureghian, 1993).Here, 〈〉 is the expectation operator. When c(x)and

{b j (x)

}Nj=1 are so selected, the optimal error


is obtained at any point x = xi as,

⟨ε2

i

⟩=

⟨{v(xi)

−N

∑j=1

b j (xi)

(μ (x j)+

r

∑k=1

Γ ′k

√θkφk j

)}2⟩,

i = 0,1 . . ., N̂ (2a)

After substituting for c(x) and{

b j (x)}N

j=1andfurther simplifying, Eq. (1a) and Eq. (2a) takethe following forms:

v̂(xi) = μ (xi)+r

∑k=1

Γ ′k√θk

N

∑j=1

φk jS ji,

i = 0,1 . . ., N̂ (1b)

⟨ε2

i

⟩= σ2 (xi)−

r

∑k=1

1θk

(N

∑j=1

φk jS ji

)2

,

i = 0,1 . . ., N̂ (2b)

where, SSSNN̂ is a N × N̂ matrix containing the ex-pectations

⟨(v(x j)−μ (x j)) (v̂(xi)−μ (xi))

⟩( j =

0,1 . . .,N and i = 0,1 . . ., N̂) indicating the covari-ance of v(x j) with v̂(xi).Once the details of discretisation are obtained,one can obtain w(x′i) needed for formulation ofstructural matrices for the ith element by notingthat w(x′i) = F−1

w [Φ (v̂(x′i))] with,

v̂(x′i)= μ

(x′i)+

r

∑k=1

Γ ′k√θk

N

∑j=1

φk jS ji,

i = 0,1 . . .,N∗ (1c)

For the purpose of illustration of some of the fea-tures of the EOLE, we consider a Gaussian ran-dom field with unit mean, unit variance and cor-relation function of the form,

ρ(x,x′)

= exp

(−(x−x′)2

a2

)(3)

where, a is the correlation length factor.

In the following studies we fix N∗ = 32. Theinfluence of varying r on variance of error for

a = 0.25L,0.5L,1.0L,2.0L and N = 9,17,33 isshown in Figs. 1a to 1c. In Fig. 2 we fixa = 0.25L, r=5 to show the variation of the vari-ance of error of discretisation as a function of x/Lfor N = 9,17,33. The influence of varying a onthe variance of error for r=5, N=33 is studied inFig. 3. Fig. 4 shows the plot of correlation func-tion for different values of a. If we define thelength over which correlation drops by a givenfactor (say, 0.5) as the correlation length of therandom field, then it can be observed from Fig.4 that for a = 0.25L,0.5L,1.0L the correlationlengths are respectively 0.22L, 0.42L, and 0.82L.For a = 2L, the correlation length is much greaterthan L. Clearly, shorter the correlation length withrespect to L, greater is the need for accounting forthe random spatial variability in the analysis.

In summary, for the given form of ρ (x,x′) andvalue of a it can be noted that the error of dis-cretisation of the random field depends upon thechoice of N and r. In the present studies we taker=5 and N = 8,16,32, a = 0.25L,0.5L,1.0L and2.0L. For the range values of r and N considered,the standard deviation of the error of discretisa-tion is found to be within 0.04 (See Fig. 2). In theadaptive refinement procedures discussed subse-quently, we hold r = 5 and vary N.

It is important however to note that the error ofdiscretisation that has been discussed in this sec-tion has so far been in the context of random fielddiscretisation alone and these errors are not neces-sarily the ones which we wish to eventually con-trol. Indeed, it is the error in the estimation ofstructural reliability that we wish to eventuallycontrol.

We would also note here that in the contextof problems involving two-dimensional randomfields we assume random fields to be homoge-neous and isotropic with correlation function ofthe form,

ρ(x,x′)

= exp

(− l2

a2

)(4)

where, l represents the distance between twopoints in the random field. Consequently, the dis-cussion presented above in the context of one-dimensional random fields remains broadly valid


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 5 10 15 20 25 30 35

No of terms in EOLE, r

Varia

nce

of E

rror

a=0.25La=0.50La=1.00La=2.00L

Figure 1a: Influence of varying the number of terms of expansion (r) on variance error of discretisationusing EOLE; with correlation length factor, a = 0.25L,0.5L,1.0L,2.0L; number of random field elements,N = 9;x/L = 0.5;N∗ = 32.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 5 10 15 20 25 30 35


Varia

nce

of E

rror

a=0.25La=0.50La=1.00La=2.00L

Figure 1b: Influence of varying the number of terms of expansion (r) on variance error of discretisationusing EOLE; with correlation length factor, a = 0.25L,0.5L,1.0L,2.0L; number of random field elements,N = 17;x/L = 0.5;N∗ = 32.


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 5 10 15 20 25 30 35


Varia

nce

of E

rror

a=0.25La=0.50La=1.00La=2.00L

Figure 1c: Influence of varying the number of terms of expansion (r) on variance error of discretisationusing EOLE; with correlation length factor, a = 0.25L,0.5L,1.0L,2.0L; number of random field elements,N = 33;x/L = 0.5;N∗ = 32.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.2 0.4 0.6 0.8 1 1.2

x/L

Varia

nce

of E

rror

N=9N=17N=33

Figure 2: Variation of the variance of error of discretisation using EOLE as a function of x/L for numberof terms of expansion, r = 5, correlation length factor a = 0.25L and number of random field elementsN = 8,16,33;N∗ = 32.


1E-12

1E-11

1E-10

1E-09

1E-08

1E-07

1E-06

1E-05

0.0001

0.001

0.01

0.1

10 0.2 0.4 0.6 0.8 1 1.2

x/L

Var

ianc

e of

Err

or

a=0.25La=0.50La=1.00La=2.00L

Figure 3: Variation of the variance of error of discretisation using EOLE as a function of x/Lfor numberof terms of expansion, r = 5, number of random field elements N = 33 and correlation length factor a =0.25L,0.5L,1.0L,2.0L;N∗ = 32.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

x/L

Cor

rela

tion

Coe

ffici

ent

a=0.25La=0.50La=1.00La=2.00L

Figure 4: Variation of the correlation function for correlation length factor, a = 0.25L,0.5L,1.0L,2.0L;N∗ =32.


in this case also.

5 Adaptive Mesh Refinement

Our objective is to propose a method to adaptivelyrefine the random field mesh, without excessivelyincreasing the number of random field elements.It is expected that the adaptively refined mesh willresult in improved estimation of reliability of theproblem considered. In an adaptive refinementprocess, the following issues will be of interest:

• identification of a refinement indicator whichdetermines regions where the mesh refine-ments are most effective,

• identification of an error indicator whichserves as a figure of merit for the current so-lution, and

• implementation of effective computationalalgorithms for new improved solutions.

In the following sections we discuss the proposedrefinement indicator, error indicator and compu-tational algorithm for adaptive random field meshrefinement in the context of stochastic finite ele-ment analysis.

5.1 Proposed Refinement Indicator

It is noted from literature that the importancemeasures for ranking the random variables havebeen used successfully in conjunction with sam-pling/simulation methods to reduce the compu-tational effort by eliminating the lowly rankedrandom variables from the analysis (Brenner andBucher, 1995; Gupta and Manohar, 2004). How-ever, we realize that, the importance measures ingeneral, and the global importance measures inparticular, possess potential, which could be uti-lized in the context of stochastic finite elementmethod. With appropriate transformations, theseimportance measures offer an effective means toidentify the spatial regions in the stochastic finiteelement model where refinement of random fieldmesh is required. In other words, the global im-portance measures can be used to serve the pur-pose of indicators for carrying out adaptive re-finement of random field mesh. This is analo-gous to the energy based error indicators used in

conventional deterministic adaptive finite elementmethods. Furthermore, this approach works withmultiple random fields and performance functionswith multiple regions of comparable importance.We have used global importance measures withappropriate transformations as mesh refinementindicators.

In the present work, we assume that the perfor-mance function could consist of multiple regionsof comparable importance. Fig. 5 shows a fewperformance functions where there are multipleregions contributing to the failure probability. Thelimit state surfaces in Fig 5(a) and (b), respec-tively, has two and four points, which have thesame minimal distance from the origin O. Theseare called multiple design points, each of whichmakes equally important contributions to the fail-ure probability. Fig. 5(c) shows a performancefunction, where points a and b are situated atdistances β1 and β2 respectively from the origin,where β2−β1 < ε with ε being a very small num-ber. In this case, even though there exists only asingle design point a, it is obvious that point balsomakes significant contribution to the failure prob-ability. Fig. 5(d) shows two performance func-tions, one that is a perfect circle (full line) and theother an ellipse obtained by a minor modificationto the circle (dotted line). For the performancefunction that is a circle, all points on the limit sur-face lie at the same distance from the origin andhence, the number of design points is infinite innumber. On the other hand, for the ellipse, thereexist two design points, c and d, but infinitelymany points that are not strictly the design points,nevertheless, make significant contributions to thefailure probability.

We use the global importance measure proposedby Gupta and Manohar (2004) for performancefunctions with multiple design points for rankingthe discretised random variables in the order oftheir relative importance, and, also, to investigateif such ranking could be used for adaptive meshrefinement in stochastic finite element reliabilityanalysis. If R distinct points are identified in then-dimensional standard normal space, the globalimportance measure for the ith element in the vec-tor of random variables Ψ in non-Gaussian space


Figure 5: Performance functions in standard normal space with multiple design points (Gupta, 2004)

is computed using the expression,

γi =R

∑l=1

wlλ i2l

/ n

∑i=1

R

∑l=1

wlλ i2l (5)

where, the parameters wl and wl are defined later.The vector of correlated non-Gaussian randomvariables Ψ is related to the vector of normal vari-ables ΓΓΓ , through Nataf’s transformation, given byΓΓΓ = Φ−1 [FΨ (Ψ)]. Here, FΨ and Φ (·) are, re-spectively, the marginal probability distributionfunctions of Ψand ΓΓΓ . The vector of correlatednormal random variables ΓΓΓ are related to the stan-dard normal variables ΓΓΓ ′′′ through the expressionΓΓΓ = LLLΓΓΓ ′′′, where LLL is the lower triangular matrix,such that CCCΓΓΓ = LLLLLLT . Here, CCCΓΓΓ is the correla-tion coefficient matrix of ΓΓΓ expressed in termsof the known correlation coefficients CCCΨΨΨ of Ψ(Der Kiureghian and Liu, 1986). The matrix LLLis determined by Cholesky decomposition of CCCΓΓΓ .Since the proposed reliability analysis algorithm

is carried out in the standard normal space, thedirection cosines for the identified design points(

∂G(ΓΓΓ ′′′)∂ΓΓΓ ′′′

)are obtained in the ΓΓΓ ′′′-space. The cor-

responding direction cosines in the Ψ -space are

expressed as,(

∂g(Ψ)∂Ψ

)= JJJ

(∂G(ΓΓΓ ′′′)

∂ΓΓΓ ′′′

), where, JJJ

is the Jacobian matrix, g(Ψ) is the performancefunction in non-Gaussian space and G

(ΓΓΓ ′′′) is the

performance function in standard normal space.The parameter λ i

l in Eq. (5) is now defined as,

λ il =

R

∑j=1

wl

{(∂g(Ψ)

∂Ψj

)i}2

(6)

Here wl are weighting functions. The detailsof computation of gradients ∂g(Ψ)

∂Ψ of the perfor-mance function are discussed in the work by Man-juprasad (2005). To give a greater weightage tothe points closer to the origin in the ΓΓΓ ′′′-space, the


weights wl are defined in terms of probability con-tent associated with each point in the ΓΓΓ ′′′-space,and are expressed as,

wl = Φ (−βl)/ R

∑j=1

Φ (−β j) (7)

where, βl is the Hasofer-Lind reliability index ofthe lth point in the ΓΓΓ ′′′-space. Thus, points in thefailure domain more likely to fail are given greaterweightage.

While carrying out SFEM based reliability analy-sis, the vector Ψ also contains a subset of discre-tised random field variables Ξ . Hence, the set ofglobal importance measures computed using theprocedure discussed above would also contain asubset of GIM values corresponding to the dis-cretised random field variables. In this study, it isproposed to use these global importance measureswith respect to discretised random field variablesas refinement indicators for carrying out adaptiverefinement of random field element meshes.

5.2 Proposed Error Indicator

In the conventional adaptive mesh refinements itis common to use energy-based norms for estima-tion of errors and to arrive at global error indi-cators. In the present method we propose to usethe reliability index based norm for estimation oferror. We begin by describing the random fieldmesh using a coarse mesh and subsequently re-fine the mesh. Let Pf denote the estimate of fail-ure probability with respect to one or more per-formance functions using a specific mesh config-uration. We treat a uniformly refined ‘fine’ meshas the ‘reference mesh’. Since it is not practi-cally possible to estimate the exact value of fail-ure probability in advance, it is assumed that theestimate obtained using the ‘reference mesh’ asthe best solution. Let P∗

f denote the estimate offailure probability using this reference mesh. Therelative error percentage, ηP, in estimation of fail-ure probability obtained using any specified meshcan therefore be defined as,

ηP (%) =

∣∣∣Pf −P∗f

∣∣∣P∗

f×100 (8)

The probabilities Pf and P∗f could also be ex-

pressed in terms of notional reliability indices

β = −Φ−1 (Pf ) and β ∗ = −Φ−1(

P∗f

)leading to

an alternative definition of relative error index,ηβ , given by

ηβ (%) =|β −β ∗||β ∗| ×100 (9)

Fig. 6 shows the relationship between Pf and β . Itis of interest to note that β is zero when Pf is 0.5, βis negative for Pf > 0.5, and, β is positive for Pf <0.5. The presence of modulus in the denominatorof Eq. 9, thus takes into account the probabilityof β ∗ being negative. Also, given the fact that βvaries monotonically with respect to Pf , it is clearthat either ηP or ηβ could equally well serve therole of error indicator. In the present study wehave preferred to use ηβ for discussion.

6 Mesh Refinement Strategy

As has been already pointed out, there exist sep-arate meshes for displacement field and each ofthe random fields that represent different struc-tural properties. In the present study, the configu-ration of all these meshes could be different fromeach other. Each of the structural property randomfields is represented through its own EOLE. Whilethe displacement field mesh could be coarser orfiner than the random field meshes, in the for-mulation of the structural matrices however, thestructural properties, such as, elastic modulus andmass density, within a displacement field elementis taken to be a constant. These constants them-selves are deduced from the respective EOLE ofthe relevant random fields. In the implementationof mesh refinements, it is assumed that the meshfor displacement field has been arrived at inde-pendently and this mesh itself is not necessarilyopen for refinement. The mesh refinement pro-cess begins with the definition of a coarse meshfor all the random fields of interest. Upon com-pleting the reliability analysis of the structure withthis mesh, indicators of relative importance of dif-ferent random field elements are also obtained.Based on this, one could enhance the number ofrandom field elements in the important zones and


Figure 6: Relationship between probability of failure and reliability index

deplete the number of elements in the least impor-tant regions. Throughout the present study how-ever, we adopt structured meshes that are refinedusing successive bisection as and when needed.

7 Finite Element based Reliability Analysis

Through appropriate transformations, the basicrandom field/process variables obtained throughrandom field discretisation are transformed tospatial element random variables/ temporal ran-dom load time history variables. Finite elementanalysis (static/free vibration/forced vibration) iscarried out using suitable element formulations tocompute the performance function and the gradi-ents of the performance function with respect tothe required random variables. Structural relia-bility analysis is carried out using first-order re-liability method (FORM) with improvements ashas been mentioned in Section 3. In reliabilityanalysis using first-order reliability method, opti-mization algorithms are commonly used to obtainthe design point and the corresponding reliabil-ity index or safety index β . In the present work

the Rackwitz and Fiessler method (Rackwitz andFiessler, 1978) is used to solve the optimizationproblem.

In FORM, an approximation to the probability offailure is obtained by linearising the limit-statesurface at the design point. The existence ofmultiple design points may cause the followingproblems in FORM: first, the optimization algo-rithm may converge to a local design point; inthis case, the FORM solutions will miss the re-gion of dominant contribution to the failure prob-ability integral and, hence, the corresponding ap-proximations will be in gross error; second, evenif the global design point is found, there could besignificant contributions to the failure probabilityintegral from the neighborhoods of local pointsapproximating the limit-state surface only at theglobal design point will not account for these con-tributions. In the present research work, searchfor multiple design points of a reliability prob-lem is carried out using the method of artificialbarriers proposed by Der Kiureghian and Dake-sian (1998). The computation of multinormal in-


tegrals is a necessary step for estimating the prob-ability of failure of structural systems. In thepresent work, an approach using the product ofone-dimensional normal integrals is used is usedto get an approximate value of the multinormalintegral. The method is called product of condi-tional marginals (PCM) method and was proposedby Pandey (1998).

Adaptive stochastic finite element reliability anal-ysis of linear structural problems (static/free vi-bration/forced vibration) is carried out using for-mulations to compute the performance functionand the gradients of the performance functionwith respect to the required element random vari-ables or random load time history variables.

For reliability analysis of structures understochastic static loads the performance functiondefined in terms of displacement at a given nodex̂xx is given by,

g = u∗ −u(x̂xx,Ψ) (10)

Here, u∗ is the permissible displacement and Ψ isthe vector of non-Gaussian random variables ob-tained using EOLE method of discretisation of therelevant random field meshes.

In the context of vibration problems, one of thecommonly used design criteria is related to theavoidance of resonance in periodically driven sys-tems. Thus, if we consider a system driven har-monically, at a frequency ω∗, the designer wouldwish to keep the nearest structural natural fre-quency away from ω∗. Or, alternatively, one maylimit the resonant amplitude within prescribedthresholds. In the former case, one can define theperformance function as the region ω∗ − Δω ≤ωi (Ψ) ≤ ω∗ + Δω , where, ωi (Ψ) is the struc-tural natural frequency centered around ω∗ whichneeds to be avoided. Since the natural frequenciesof randomly parametered systems are themselvesrandom, one can define the failure problem as,

Pf = P[(ω∗

1 −ωi (Ψ)≤ 0)⋂

(ωi (Ψ)−ω∗2 ≤ 0)

](11)

where, ω∗1 = ω∗ −Δω and ω∗

2 = ω∗ + Δω . Thisproblem itself can be perceived as a problem in

parallel system reliability with,

g1 = ω∗1 −ωi (Ψ) (12)

g2 = ωi (Ψ)−ω∗2 (13)

Thus one gets,

PF = P [ωi ∈ (ω∗ −Δω ,ω∗ +Δω)] (14)

= P [ω∗ −Δω ≤ ωi ≤ ω∗ +Δω ] (15)

= P [ω∗1 ≤ ωi ≤ ω∗

2 ] (16)

Here, Δω = 0.5∗ half power bandwidth=ηω∗,ω∗=resonance frequency, η=damping ratio corre-sponding to the mode considered.

PF = P

[2⋂

k=1

gk (Ψ)≤ 0

](17)

Next we consider the reliability analysis of struc-tures in forced vibration under stochastic excita-tions. The performance function here is definedin terms of peak displacement at a given node x̂xx ingiven time duration, which can be expressed as,

g = u∗ − max0≤t≤T

|u(x̂xx, t,Ψ)| (18)

The gradients of the performance function arecomputed using direct differentiation method.Details of formulations developed and used forreliability analysis of static, free vibration andforced vibration problems including gradients ofstructural matrices with respect to parameters ofinterest are given in the work by Manjuprasad(2005).

In reliability analysis using FORM,|βk+1−βk|〈0.001 and

∣∣g(uk+1)−g

(uk)∣∣〈0.001

are used as stopping criteria. In addition, max-imum number of iterations is set equal to 25.In the method of bulges, |βk/β1| 〈1.5 is used asa stopping criterion in addition to a maximumnumber of iterations equal to 5.

8 Summary of the Steps

We summarize here the various steps involved inSFEM based reliability analysis using adaptivelyrefined random field meshes:


1. Input data: This includes the following de-tails:

a. Structural geometry and boundary condi-tions.

b. Probabilistic characteristics of structuralproperties, applied loads, and perfor-mance limits. This description could bein terms of types of first order pdf, itsparameters, and the definition of auto-covariance and cross-covariance func-tions.

c. Number of performance functions (NP)and their details.

d. Details of displacement field discretisa-tion including element types, number ofelements and mesh configuration.

e. Details of the initial coarse mesh for thedifferent random fields.

f. Tolerances on computation of reliabil-ity index using Rackwitz and Fiesslermethod, number of iterations and maxi-mum number of design points (ND), num-ber of terms in of expansion using EOLEmethod.

2. Reliability estimation: A loop over perfor-mance function runs at the outset. Withinthis loop a sub-loop over multiple designpoints is implemented. Within each loop fora design point, reliability index is computedusing FORM in conjunction with Rackwitz-Fiessler algorithm. The required perfor-mance function and its gradients are com-puted using SFEM. For the ith performancefunction we obtain βi j and DPi j, with j =1 . . .k j ≤ ND, where βi j is the jth reliabil-ity index associated with the ith design pointDPi j for the ith performance function. Theith performance function is now linearizedaround each of the design point DPi j, j =1 . . .k j ≤ ND and a set of hyperplanes HPi j ,j = 1 . . .k j ≤ ND are obtained. Subsequent tothe completion of the loop over all the per-formance functions we would be left with acollection of hyperplanes each of which rep-resents a plane that is obtained by linearis-ing one of the performance functions around

one of its design points. We now performa system reliability analysis using the PCMmethod paying due consideration to the con-figuration of the hyperplanes to be in seriesor parallel format. It must be noted that theset of performance functions defined throughthe hyperplanes originating from a given per-formance function are all in series while theperformance functions themselves could bein series or parallel.

3. Computation of GIM: The GIMs as given byEq. 5 are now computed with R=total num-ber of design points (DPi j) evaluated in theprevious step. Similarly, in computing theweights as in Eq. 7, the resulting reliabilityindices (βi j) computed earlier are used. Thisleads to assignment of importance measuresto each of the elements in the non-Gaussianvector Ψ .

4. Mesh refinement: The GIMs calculated inthe previous step point spatial regions inwhich each off the elements of the randomfields www(xxx) and fff (xxx, t) need further refine-ment. Based on this we employ a combi-nation of mesh refinement and derefinementto arrive at the revised definition of randomfield mesh configuration.

5. Stopping criteria: Repeat steps (1) to (4) tillsatisfactory convergence on reliability esti-mate is obtained.

9 Numerical Examples and Discussion

The procedures discussed in the previous sectionsare illustrated with the help of the following threeproblems:

1. Failure due to excessive displacements of astatically loaded one span beam. The elasticmodulus of the beam is modeled as a lognor-mal random field.

2. Failure of a cantilever beam with randomelastic modulus due to occurrence of reso-nance.


3. Failure due to excessive displacements of acantilever beam under random dynamic ex-citation. Here, the elastic modulus and massdensity of the beam are treated as lognormalrandom fields and load is considered as a sta-tionary Gaussian random process.

The numerical examples illustrate the followingrange of issues relevant to the problem:

1. Effect of choice of Nin EOLE method of dis-cretisation.

2. Mesh refinement with respect to alternativeperformance functions as well as multipleperformance functions

3. Mesh refinement issues that arise in staticanalysis, free vibration analysis, and forcedvibration analysis using direct integration.

4. Treatment of more than one random fieldwithin the same structure.

5. Use of one-dimensional beam and two-dimensional plane stress finite elements

6. Treatment of one-dimensional and two-dimensional random fields

Example 1

In this example, we consider the beam structureshown in Fig. 7. The data for this beam are asgiven in the paper by Liu and Liu (1993). Thus,we take the span of the beam as 9.76 m (32 ft) andthe load to be static with uniformly distributed in-tensity of 116.7 N/m (8k/ft). The elastic modulusis assumed to be a homogeneous non-Gaussianrandom field with a lognormal first order pdf andauto-covariance function of the form in Eq. 3.Consequently, the flexural rigidity of the beamalso becomes a random field and we take its meanvalue to be 465706.41 N-m2 (1,125,000 k-ft2) anda coefficient of variation (COV) of 0.20. Thestructure is modeled using a set of Euler-Bernoullibeam elements. The following parametric studieshave been conducted to bring out different aspectsof mesh refinement procedure.

Figure 7: Geometry and loading of fixed endbeam for Example 1

Study 1: Adaptive Refinement of Random FieldMesh

Here, we take a = 0.25L,COV = 0.2 and r = 5.We focus attention on the influence of varyingnumber of elements in the random field mesh.We define the performance function in terms ofmidspan deflection namely g(Ψ) = δ ∗−δmid . Weassume that δ ∗=7.62 mm (0.3 inch). We begin bytaking 32 elements (with total degrees of freedom=96) for the displacement field mesh and considerthe effect of varying the number of elements inthe random field mesh with their lengths beingheld uniform. These results are shown in Table 1ain which the number of elements in the randomfield mesh is varied from 8 to 32 and the notionalreliability index β is computed for all the meshconfigurations. To describe the displacement fieldmesh and random field mesh we characterize eachmesh configuration by a pair of numbers (N1,N2),where, N1=number of elements in the displace-ment field mesh and N2=number of elements inthe random field mesh. For the mesh configura-tion (32, 32) we obtain β =2.0087 and this valueof β is taken to be the reference β ∗ in comput-ing ηβ . From Table 1a it can be seen that ηβ re-duces monotonically as the number of elements inthe random field mesh are increased from 8 to 32.The effect of adaptively refining the random fieldmesh is studied in Tables 1b and 1c.

In obtaining the results in Table 1b we begin byanalyzing the GIM for the uniform (32,8) mesh(Fig. 8). The resulting refined random field meshis shown in Fig. 9. For this configuration we ob-tain a value of β =2.0091 and ηβ =0.0199 (Table1b). A comparison of ηβ for non-uniform (32, 12)mesh (Table 1b, ηβ =0.0199) and uniform (32,12)


Table Table 1a: Uniform refinement of RF mesh

NNN1 NNN2 βββ ηβ (%) = |β−β ∗||β ∗| ×100

32

8 2.0126 0.194212 2.0101 0.069714 2.0096 0.044816 2.0093 0.029918 2.0091 0.019120 2.0089 0.010022 2.0088 0.005032 2.0087(β ∗ ) 0.0000

Table Table 1b: Adaptive refinement of RF mesh using bisection of elements

NNN111 NNN222 βββ ηβ (%) = |β−β ∗||β ∗| ×100

3212 2.0091 0.019914 2.0089 0.001016 2.0089 0.0010

Table Table 1c: Adaptive refinement of RF mesh by relocation of nodes

NNN111 NNN222 βββ ηβ (%) = |β−β ∗||β ∗| ×100

328 2.0138 0.2539

16 2.0091 0.0199

Figure 8: GIM plot for (32, 8) uniform mesh Figure 9: Adaptive 12-element RF mesh for elasticmodulus based on GIM plot for (32, 8) uniform mesh

mesh (Table 1a, ηβ =0.0697) clearly shows a re-duction inηβ by a factor of 3.5 due to adaptive re-finement of the random field mesh for the problemconsidered. Based on the refined (32, 12) meshthe GIM obtained is shown in Fig. 10, which, inturn, leads to the refined (32, 14) mesh shown in

Fig. 11. Further refinements based on the GIMfor the (32, 14) mesh (Fig. 12) leads to the re-fined mesh (32, 16) as shown in Fig. 13. At thisstage we observe that ηβ and β have converged toa value of 0.001 and 2.0089, respectively, and onecould stop further refinement at this stage.


Figure 10: GIM plot for (32, 12) adaptive mesh Figure 11: Adaptive 14-element RF mesh based onGIM plot for (32, 12) adaptive mesh

Figure 12: GIM plot for (32, 14) adaptive mesh Figure 13: Adaptive 16-element RF mesh based onGIM plot for (32, 14) adaptive mesh

In studies reported in Table 1b, the mesh re-finement was carried out by successive bisection,without making efforts to remove nodes that arelocated in regions of low GIM. The question ofrelocating nodes is considered in Table 1c. Here,based on GIM for the uniform (32, 8) mesh (Fig.8), we obtain the random field mesh as shownin Fig. 14a. Here, nodes that lie in regions oflow GIM (50 < x <150 and 250 < x < 350)have been removed and shifted to regions of high

GIM. Figure 14b shows the node locations for auniform mesh for comparison with the adaptivemesh. Similar results beginning with the uniform(32, 16) mesh and its GIM plot (Fig. 15) lead-ing to non-uniform (32, 16) mesh with relocatednodes are shown in Fig 16. A comparison of Ta-bles 1a and 1c shows that by relocating the nodesbased on GIM for the uniform mesh, one getsmixed results. Thus, for (32, 8) mesh the refine-ment leads to a deterioration of ηβ from 0.1942


(a) (b)

Figure 14: (a) Adaptive 8-element RF mesh using relocation of nodes based on GIM plot for (32,8) uniformmesh (b) Uniform 8-element RF mesh

to 0.2539 while for (32, 16) mesh one obtains animprovement from 0.1205 to 0.0199. This dispar-ity could be explained if one considers the vari-ance of error of discretisation with N=8+1=9 andN=16+1=17 random field nodes. This variation isshown in Fig. 17 wherein the variance for the caseN=9 (non-uniform relocated nodes) is seen to besubstantially higher than the result for N=9 withuniform placement of nodes. On the other hand,similar results for N=17 (uniform and relocated)do not show as much difference as is observedfor N=9. This points towards the usefulness ofinformation on variance of error in EOLE indicat-ing if a refinement strategy involving relocation ofnodes could be acceptable or not.

Study 2: Adaptive Refinement of DisplacementField Mesh

The formulation of GIM in this study is mainlymotivated by the need to rank the element randomvariables in terms of their relative importance tothe reliability calculations. Thus, it would appearthat this measure would mainly be applicable tothe refinement of random field mesh. However,the reliability calculations itself would cruciallydepend on the details of displacement field mesh.Thus, question would arise on the possible use-fulness of GIM in relocating the nodes associatedwith the displacement field mesh. Limited studies

have been carried out to explore this issue and theresults are summarized in Tables 2a and 2b.

In this study, again we take a = 0.25L,COV = 0.2and r = 5. We define the performance functionin terms of midspan deflection namely g(Ψ) =δ ∗ − δmid . We assume that δ ∗=7.62 mm (0.3inch). We begin by fixing 32 elements for the ran-dom field mesh and consider the effect of vary-ing the number of elements in the displacementfield mesh to 8, 16 and 32, with their lengths be-ing held uniform. The results of reliability anal-ysis are shown in Table 2a. Now beginning fromGIM plots for (8, 32) and (16, 32) meshes (Figs.18 and 19) we relocate the nodes of the displace-ment field meshes as shown in Figs. 20 and 21respectively. The corresponding results of relia-bility analysis are shown in Table 2b. A compar-ison of Tables 2a and 2b shows that the adaptiverefinement of displacement field mesh based onGIM indeed leads to significant improvement inηβ .

Study 3: Adaptive Refinement with Multiple Ran-dom Field Meshes

In this study we consider the same beam prob-lem, but now carrying a spatially varying randomload. We model the load as a stationary Gaus-sian random field and the elastic modulus of thebeam as a lognormal random field. The load is


Figure 15: GIM plot for (32, 16) uniform mesh Figure 16: Adaptive 16-element RF mesh using relo-cation of nodes based on GIM plot for (32, 16) uni-form mesh

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20 25 30 35

Length of Beam in ft

Var

ianc

e of

Err

or

Case 1Case 2Case 3Case 4

Figure 17: Variation of the variance of error of discretisation using EOLE along the length of the beam,Case (1) uniform 8-element RF mesh, Case (2) adaptive 8-element RF mesh with relocation of nodes, Case(3) uniform 16-element RF mesh, Case (4) adaptive 16-element RF mesh with relocation of nodes; numberof terms of expansion r=5; correlation length factor a = 0.25L; number of finite elements N∗ = 32.

Table Table 2a: Uniform refinement of DF mesh

NNN111 NNN222 DOF βββ ηβ (%) = |β−β ∗||β ∗| ×100

832

24 1.9867 1.095216 48 2.0044 0.214132 96 2.0087(βββ ∗) 0.0000


Table Table 2b: Adaptive refinement of DF mesh by relocation of nodes

NNN111 NNN222 DOF βββ ηβ (%) = |β−β ∗||β ∗| ×100

832

24 2.0238 0.751716 48 2.0095 0.0398

Figure 18: GIM plot for (8, 32) uniform mesh Figure 19: GIM plot for (16, 32) uniform mesh

(a) (b)

Figure 20: (a) Adaptive 8-element DF mesh using relocation of nodes based on GIM plot for (8, 32) uniformmesh (b) Uniform 8-element DF mesh


Figure 21: Adaptive 16-element DF mesh using relocation of nodes based on GIM plot for (16, 32) uniformmesh

modeled as q(x) = q0 (1+ε f (x)) with q0=116.7N/m (8k/ft) and ε=0.2. Here, f (x) is a zero meanstationary Gaussian random field with covariancefunction given by Eq. 3, with a=0.25L. Sinceelastic modulus of the beam is a random field,the flexural rigidity of the beam is a random fieldand we take its mean value to be 465706.41 N-m2

(1,125,000 k-ft2) with a coefficient of variation of0.20 and a = 0.25L. The spatially varying elas-tic modulus E (x) and the distributed load q(x)are discretised using EOLE with r=5. We definethe performance function in terms of midspan de-flection namely g(Ψ) = δ ∗ − δmid . We assumethat δ ∗= 6.35 mm (0.25 inch). The problem onhand consists of arriving at refined meshes forthe random fields E (x) and q(x). We introduce anotation (N1,N2,N3) to characterize a mesh withN1=number elements in the displacement fieldmesh, N2=number of elements in the random fieldmesh for E (x) and N3=number of elements in therandom field mesh for q(x).Table 3 shows results for (32, 8, 8), (32, 14, 12)and (32, 32, 32) uniform meshes. Using the (32,32, 32) uniform mesh, the reference notional reli-ability index β ∗ is determined as 1.3595. The lastrow in Table 3 provides results for (32, 14, 12)mesh in which the random fields E (x) and q(x)have been adaptively refined based on GIM for(32, 8, 8) uniform mesh. The plots of GIM forE (x) and q(x) are shown respectively in Figs. 22

and 23. It is clear that the locations needing re-finement for E (x) and q(x) fields are dissimilar.Figs. 24 and 25 respectively show the location ofnodes for adaptively refined random field meshesfor E (x) and q(x). The benefit of refining the ran-dom field mesh is clearly seen in Table 3 in whichηβ is seen to drop from a value of 2.2803 for a(32,14,12) uniform mesh to a value of 0.4634 forthe (32,14,12) adaptively refined mesh.

Study 4: Adaptive Refinement with IndependentMultiple Performance Functions

Studies 1–5 have been conducted with respectto the beam provided with fixed supports at twoends. In this study we consider the same beamproblem, but now provided with hinged supportsat two ends. We consider a performance functionwith respect to admissible rotations at the left sup-port in addition to the performance criteria withrespect to midspan deflection. It must be notedhowever that here the two performance functionsare considered independent of each other. Theelastic modulus is assumed to be homogeneousnon-Gaussian random field with a lognormal firstorder pdf and auto-covariance function of theform in Eq. 3. In the numerical work, we takeCOV = 0.2, r = 5 and N∗=32. We begin by eval-uating β with uniform (32, 32) mesh for the twoperformance functions namely g1 (Ψ) = δ ∗−δmid

and g2 (Ψ) = θ ∗ − θle f t; see Table 4 for the de-


Table 3: Refinement of multiple RF meshes for elastic modulus and distributed load

NNN111 NNN222 (((EEE))) NNN333 (((qqq))) βββ ηβ (%) = |β−β ∗||β ∗| ×100 Remarks

32 8 8 1.2816 5.7301Uniform RF mesh32 14 12 1.3285 2.2803

32 32 32 1.3595(βββ ∗) 0.032 14 12 1.3532 0.4634 Adaptive RF mesh

Figure 22: GIM plot for elastic modulus with (32, 8)uniform mesh

Figure 23: GIM plot for distributed load with (32, 8)uniform mesh

Figure 24: Adaptive 14-element RF mesh for elasticmodulus based on GIM plot for (32, 8) uniform mesh

Figure 25: Adaptive 12-element RF mesh for dis-tributed load based on GIM plot for (32, 8) uniformmesh


Table 4: RF mesh refinement with multiple independent performance functions in terms of midspan deflec-tion and left support rotation

NNN111 NNN222 βββ ηβ (%) = |β−β ∗||β ∗| ×100 Remarks

PF in terms of midspan deflection, g1 (Ψ) = δ ∗ −δmid

32 08 2.72750 0.00073Uniform RF mesh32 12 2.72749 0.00037

32 32 2.72748 (βββ ∗) 0.0000032 12 2.72748 0.00000 Adaptive RF mesh

PF in terms of support rotation, g2 (Ψ) = θ ∗ −θle f t

32 08 2.87714 0.00070Uniform RF mesh32 12 2.87714 0.00070

32 32 2.87712 (βββ ∗) 0.0000032 12 2.87712 0.00000 Adaptive RF mesh

tails.

We assume that δ ∗=45.72 mm (1.8 inch) andθ ∗=0.015 radian. Next we evaluate the GIM forthe two performance functions starting with (32,8) uniform mesh (see Figs. 26 and 27). Upon suc-cessive bisection of important regions we arrive atthe refined mesh as shown in Figs. 28 and 29.The corresponding estimates of β are providedin Table 4. It is of interest to note that the GIMvariation depends upon the performance functioncriteria used: for criteria on midspan deflection,the midspan region of the beam requires adaptiverefinement whereas, for the performance criteriawith respect to the left end rotation the impor-tant regions are skewed to the left. Furthermore,a comparison of Fig. 26 with Fig. 8 reveals theinfluence of changing boundary conditions fromfixed ends to hinged ends on the process of ran-dom field refinement. Thus it emerges that al-though a beam that is fixed at the two ends anda beam that is hinged at two ends are symmetricwith respect to the midspan, the regions for re-fining the random field mesh for the purpose ofreliability analysis turn out to be different for thetwo structures even when there is no asymmetryeither in loading condition or in the definition ofthe performance function. This observation pointstowards the capacity of the refinement proceduredefined herein to unravel the important features ofdiscretisation that are difficult at the outset to pre-dict.

Study5: Adaptive Refinement with Mutually De-pendent Multiple Performance Functions

In this study, we consider the two performancefunctions g1 (Ψ) = δ ∗ − δmid and g2 (Ψ) = θ ∗ −θle f t to be in series. This would mean that thebeam is considered to have failed if either themidspan deflection exceeds a permissible limitor the rotation at the left end support exceeds apermissible limit. It may be noted that the twoperformance functions have been earlier studiedseparately in Study 4. It is important to notethat the two performance functions here are mutu-ally dependent. As has already been noted, whenconsidered separately the two performance func-tions lead to notably different adaptively refinedmesh configurations for the elastic modulus ran-dom field. Thus it would be of interest to under-stand the nature of adaptive refinement of the ran-dom field mesh when the two performance func-tions are considered as being in series. In this nu-merical illustration, we assume that COV = 0.2,r = 5 and N∗=32. We assume that δ ∗= 45.72 mm(1.8 inch) and θ ∗=0.015 radian. As before, weuse the notation (N1,N2) to denote N1 elements inthe displacement field mesh and N2 elements inthe random field mesh. We define β ∗ with respectto (32,32) uniform mesh. Results for (32, 8), (32,12) and (32, 32) uniform mesh are summarized inTable 5. It was found during the numerical workthat for each of the two performance functionsonly one design point was considered relevant.

Starting with GIM for (32,8) uniform mesh (see


Figure 26: GIM plot for (32, 8) uniform mesh withperformance function in terms of midspan deflection

Figure 27: GIM plot for (32, 8) uniform mesh withperformance function in terms of left-support rotation

Figure 28: Adaptive 12-element RF mesh for elas-tic modulus based on GIM plot for (32, 8) uniformmesh with performance function in terms of midspandeflection

Figure 29: Adaptive 12-element RF mesh for elasticmodulus based on GIM plot for (32, 8) uniform meshwith performance function in terms of left-support ro-tation

Table 5: RF mesh refinement with multiple mutually dependent performance functions in terms of midspandeflection and left support rotation

NNN111 NNN222 βββ ηβ (%) = |β−β ∗||β ∗| ×100 Remarks

32 8 2.58011 0.001163Uniform RF mesh32 12 2.58010 0.000775

32 32 2.58008(βββ∗) 0.00000032 12 2.58008 0.000000 Adaptive RF mesh


Figure 30: GIM plot for (32, 8) uniform mesh withmultiple mutually dependent performance functionsin terms of midspan deflection and left support rota-tion

Figure 31: Adaptive 12-element RF mesh for elasticmodulus based on GIM plot for (32, 8) uniform meshwith multiple mutually dependent performance func-tions in terms of midspan deflection and left supportrotation

Fig. 30) we obtain an adaptively refined (32,12)mesh (see Fig 31). The result of reliability anal-ysis using this mesh is also shown in Table 5. Itmay be observed that the GIM shown in Fig. 30closely resembles the plot when only g1 (x) is con-sidered (see Fig. 26). This observation points to-wards stronger influence of the performance func-tion with respect to translation than with respectto rotation. As might be expected, the estimateon system reliability (β =2.58008) is lesser thanthe two reliability indices when the performanceswith respect to displacement and rotation are con-sidered separately (β =2.72748 and β =2.87712 re-spectively; see Table 4).

Example 2

In this example we study the reliability of a can-tilever beam with performance function defined interms of the fundamental natural frequency of thebeam. The beam structure under study is shownin Fig. 32 and, Nagesh Iyer and Appa Rao (2002)have studied this beam problem earlier in the con-text of adaptive mesh refinements for determinis-tic vibration analysis.

We model the beam structure using 4-noded rect-angular plane stress elements. The elastic modu-

lus of the beam is modeled as a 2D homogeneousisotropic random field with a lognormal first or-der pdf and covariance function as given in Eq.4. The mean value of the elastic modulus is takenas1.0E5 MPa and a coefficient of variation of 0.20is assumed. The Poisson’s ratio, mass density anddamping ratio are taken to be 0.3, 1.0E-6 kg/mm3

and 0.02 respectively. The random field discreti-sation is carried out using 4-noded quadrilateralelements and adapting EOLE method in conjunc-tion with Nataf’s transformation technique. Sincethe elastic properties of the beam are random innature, the natural frequencies of the beam turnout to be random in nature. We examine the prob-ability of occurrence of resonance in the beamstructure when it is subjected to a harmonic ex-citation of frequency f ∗=1760 Hz; see Section 7for the basic motivation for defining this perfor-mance criteria. We define the failure as the eventthat the first natural frequency lies in the interval( f ∗1 , f ∗2 ) with f ∗1 = f ∗ − Δ f and f ∗2 = f ∗ + Δ f .It may be noted that in the absence of random-ness in the elastic modulus, the first natural fre-quency of the beam has a deterministic value of1653 Hz. The probability of failure is now givenby, Pf = P [ f ∗1 ≤ f1 ≤ f ∗2 ]. Thus, the problem can


Figure 32: Geometry of cantilever beam for Example 2

be presumed as a problem in parallel system re-liability involving two elements. The ingredientsfor solving this problem have already been sum-marized in Section 3.

Here, we take a = 0.25L and r = 5. We introducea notation (N∗

1 ,N∗2 ,N1,N2) to denote a mesh con-

figuration with N∗1 ×N∗

2 elements for the displace-ment field mesh and N1 ×N2 for the random fieldmesh. Here, N∗

1 and N1 denote the number of ele-ments along the length direction of the beam andN∗

2 and N2 denote the number of elements alongthe depth direction of the beam. We begin by eval-uating the reference notional reliability index β ∗

by using a (10, 2, 20, 2) uniform mesh. Next,by holding the displacement field mesh at 10×2uniform mesh, we evaluate β for different uniformrandom field meshes starting with 5×2 and goingup to 10×2. These results are shown in Table 6a.The GIM for the (10, 2, 5, 2) mesh is shown inFig. 33. Based on the GIM plot we now succes-sively refine the random field mesh adaptively to6×2 and 7×2 as shown in Figs. 34 and 35 re-spectively. The corresponding results of reliabil-ity analysis are shown in Table 6b.

It was found during the numerical work that foreach of the two performance functions only onedesign point was considered relevant. A compar-ison of Tables 6a and 6b reveals the advantagesof adaptive refinement of random field mesh interms of being able to achieve lesser values ηβwith coarser mesh configuration itself.

Example 3

In this example we consider reliability analy-sis problem involving random dynamic excitationand multiple random field models for spatial vari-

ations in structural properties. The example struc-ture considered is a cantilever beam shown in Fig.36.

The load f (t) due a tip concentrated load is mod-eled as a zero mean stationary Gaussian randomprocess with a PSD given by band limited whitenoise with values of variance=104 kg2, lower cut-off frequency=0 rad/s and upper cut-off frequency= 1000 rad/s. The beam structure is modeledusing Euler-Bernoulli beam elements. Viscousdamping with a proportional damping model isconsidered for the beam. The elastic modulus andmass density of the beam are modeled as mutu-ally dependent homogeneous non-Gaussian ran-dom fields with a lognormal first order pdf andauto-covariance function of the form in Eq. 3.Mean value of elastic modulus is taken as 2.1E5MPa and a coefficient of variation 0.20 is as-sumed. Mean value mass density is taken as 7.8E-6 kg/mm3 and a coefficient of variation 0.20 con-sidered. The Poisson’s ratio is taken as 0.3 anddamping ratio of first and second modes are as-sumed to be 0.02 and 0.01 respectively. Further-more, the applied load is taken to be independentof the random fields for elastic modulus E (x) andmass density ρ (x).

The performance function is defined in terms ofthe maximum value of the tip displacement mea-sured over specified time duration after the beamhas reached a steady state. In implementing thereliability analysis procedure, the load f (t) is re-placed by a Fourier series with random coeffi-cients, with number of terms equal to 10. Theset of basic random variables includes the coef-ficients present in the Fourier expansion for theload in addition to the set of basic random vari-ables emerging during the discretisation of E (x)


Table Table 6a: Uniform RF mesh refinement with performance function in terms of fundamental naturalfrequency

NNN∗1 ×NNN∗

2 NNN111 ×NNN2 βββ ηβ (%) = |β−β ∗||β ∗| ×100

10×2

5×2 2.6941 0.86116×2 2.6812 0.37817×2 2.6783 0.2696

10×2 2.6746 0.131020×2 2.6711(((β ∗) 0.0000

Table Table 6b: Adaptive RF mesh refinement for cantilever beam with performance function in terms offundamental natural frequency

NNN∗1×NNN∗

2 NNN111 ×NNN2 βββ ηβ (%) = |β−β ∗||β ∗| ×100

10×26×2 2.6715 0.01507×2 2.6709 0.0075

Figure 33: GIM plot for (10, 2, 5, 2) uniform mesh

and ρ (x). It is important to note here that, in ad-dition to choosing the details of displacement fieldand random field meshes, one needs to select thetime step of integration as well. We adopt a di-rect integration approach for solving this problemusing Newmark’s method with parameters withδ=0.5, γ =0.25, (corresponding to average accel-eration method).

Here again, we introduce the notation (N1,N2,N3)to denote number of displacement field elementsby N1, number of random field elements in E (x)by N2 and number of random field elements inρ (x) by N3. We define β ∗ using a (10, 10, 10)

uniform mesh with time step Δ t=0.05s. This re-sult along with results for (10, 5, 5), (10, 5, 8) and(10, 10, 10) uniform meshes with a coarser timestep of integration Δ t=0.1s are shown in Table 7a.

By holding Δ t at 0.1s we now adaptively refinethe random field meshes using GIM derived from(10, 5, 5) uniform mesh with Δ t=0.1s. Figs. 37,38 and 39 show the GIM plots for the randomfields representing elastic modulus, mass density,and also the random variables generated from theprocess f (t) by fixing time t. A comparison ofFigs. 37 and 38 show that the GIM values associ-ated with E (x) are negligibly small as compared


Figure 34: Adaptive 6×2 RF mesh for elastic modu-lus based on GIM plot for (10, 2, 5, 2) uniform mesh

Figure 35: Adaptive 7×2 RF mesh for elastic modu-lus based on GIM plot for (10, 2, 6, 2) adaptive mesh

Figure 36: Geometry of cantilever beam with tip point load for Example 3

Table Table 7a: Uniform refinement of RF mesh for mass density of cantilever beam subjected to stationaryrandom excitation due to tip point load

NNN111 Δ t sss NNN222 NNN333 βββ ηβ (%) = |β−β ∗||β ∗| ×100

100.10

5 5 2.0392 1.32115 8 2.0396 1.301710 10 2.0402 1.2727

0.05 10 10 2.0665(((β ∗) 0.0000

Table Table 7b: Adaptive refinement of RF mesh for mass density of cantilever beam subjected to stationaryrandom excitation due to tip point load

NNN111 Δ t sss NNN222 NNN333 βββ ηβ (%) = |β−β ∗||β ∗| ×100

10 0.10 5 8 2.0610 0.2661

with GIM for ρ (x). The spatial variation of GIMalso shows notably different trends but this differ-ence itself is of marginal interest given the verylow values of GIM associated with E (x). Theresults on reliability index obtained using adap-tively refined mesh with Δ t=0.1s and (10, 5, 8)

mesh is shown in Table 7b. It should be notedthat the adaptive refinement here is restricted onlyto the mass density field (see Fig. 40 for detailsof refined mesh for ρ (x)). The mesh for elas-tic modulus field is uniform in nature. Also, itneeds to be emphasized that Δ tis held uniform at


Figure 37: GIM plot for elastic modulus with (10, 5,5) uniform mesh and time step =0.1s

Figure 38: GIM plot for mass density with (10, 5, 5)uniform mesh and time step =0.1s

Figure 39: GIM plot for applied force at differenttime instants with (10, 5, 5) uniform mesh and timestep =0.1s

Figure 40: Adaptive 8-element RF mesh for massdensity based on GIM plot for (10, 5, 5) uniformmesh; time step =0.1s

a coarser value of 0.1s. It is of interest to note that,based on GIM shown in Fig. 39 it is possible, atleast in principle, to identify the regions along thetime axis where the step sizes of integration couldbe adaptively refined. This aspect of refinementhowever has not been investigated further in thisstudy. A comparison of Tables 7a and 7b revealsthat the reliability index obtained using adaptiverandom field mesh refinement assuming a coarserΔ t, matches well with the results on reliability in-dex with a finer time step size (Δ t=0.05s) and a

finer displacement field and random field meshes(10,10,10 uniform mesh). This again points to-wards usefulness of mesh refinement proceduresdiscussed here.

10 Conclusions

The problem of reliability analysis of linearstatic/dynamic systems with spatially varying ran-dom structural properties and loads that couldvary randomly in space and time is considered.


The reliability itself is defined in terms of a setof one or more performance functions. The toolsused for reliability analysis are essentially fash-ioned after first order reliability methods with afew refined features and stochastic finite elementanalysis procedures. An important aspect of theproblem is associated with treatment of structuralproperties as a vector of mutually dependent non-Gaussian random fields. For the purpose of re-liability analysis these random fields need to bediscretised into a set of equivalent random vari-ables. The accuracy of reliability analysis cru-cially depends upon details of discretisation pro-cedures used. The present study chiefly addressesthis issue. Specifically, we have developed a pro-cedure to adaptively refine the random field meshso as to obtain improved estimates of structuralreliability. This adaptive procedure is based uponthe definition of global importance measures. Theprocedures developed can take into account pres-ence of multiple design points and structural reli-ability defined in terms of multiple performancefunctions. In order to evaluate the merit of anadaptively refined random field mesh we have uti-lized the result from a fairly refined mesh as abenchmark. It should be emphasized that such abenchmark is useful only to demonstrate the util-ity of refinement procedures proposed herein andthese procedures themselves do not inherently de-pend upon the availability of such a benchmark.

The numerical examples presented cover bothstatic and dynamic behaviour, one- and two-dimensional random fields and displacement fieldelements, simultaneous presence of more than onerandom field, space-time randomness associatedwith loads and multiple performance functions.These examples have clearly illustrated that it ispossible to obtain better estimates of structural re-liability by using adaptively refined random fieldmeshes than by using uniform meshes with highernumber of nodes. The study also points to-wards the possible usefulness of GIMs consideredherein for the purpose of adaptive displacementfield mesh refinement and adaptive time steppingin integrating the equations of motion. This aspecthowever, has not been explored further in this pa-per.

Acknowledgement: The first author wishes tothank the Director, SERC, Chennai for the kindpermission granted to publish this paper.

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