proppe03_equivalent linearization and mc

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Equivalent linearization and Monte Carlo simulationin stochastic dynamics

C. Proppe*, H.J. Pradlwarter, G.I. Schueller

Institute of Engineering Mechanics, University of Innsbruck, Technikerstr. 13, A-6020 Innsbruck, Austria

Abstract

Equivalent linearization (EQL) and Monte Carlo Simulation (MCS) are the most important techniques in analyzing large nonlinear

structural systems under random excitation. This paper reviews the development and the state-of-the-art of EQL and MCS in stochastic

structural dynamics.q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Complex modal analysis; Controlled Monte Carlo simulation; Importance sampling; Linearization criteria; Local linearization; Neumann

expansion

1. General remarks

An extensive benchmark study [113] has revealed that

most of the computational methods developed in stochastic

structural dynamics are quite limited with respect to thedimension of the problem and consequently cannot be

applied to engineering problems of larger size. It appeared

that only two approaches, namely equivalent linearization

(EQL) and Monte Carlo simulation (MCS), are suited to

solve these problems. Other methods, such as numerical

solutions based on the FokkerPlanck equation (FPE) by

path integration[94,136], cell-mapping[60,61]and finite-

elements[14,128], increase in complexity at least exponen-

tially with the state-space dimensionn. In fact, all numerical

solutions of the FPE available in the literature belong to

low-dimensional problems with n # 4[140].

Both MCS and EQL are conceptional simple and easy toapply. There underlying concepts can be viewed as the poles

between which most of the computational methods devel-

oped so far in stochastic dynamics can be settled: EQL, on

the one hand, uses the simplest global approximation of the

distribution of the response, while MCS is the most local

approach without any a priori assumption. Whereas only

biased estimates for the first two moments of the response

are obtainable by the standard EQL procedure, MCS yields

unbiased estimates for the probability density function of

the nonlinear response. Moreover, the information that EQL

provides is limited to the first two moments of the

distribution of the response, while MCS techniques are

able to give additional information on the tails of the

distribution and subsequently credible reliability infor-

mation on the analyzed dynamical systems.

2. Equivalent linearization

2.1. Introduction

An EQL technique for random external excitation aims

to replace the n-dimensional nonlinear system

dXt fX; tdtdFt; 1

where fX; t is the nonlinear system function and Ft theexcitation process, by a linear system

dXt AXadtdFt; 2such that a certain approximation error is minimized. The

n n-dimensional matrixA and then-dimensional vectora

contain the linearization coefficients. A frequently used

error criterion is the difference

efx; t2Ax 2 a; 3between the system functions which is minimized in mean

square sense. After some mathematical manipulations, this

0266-8920/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 6 - 8 9 2 0 (0 2 )0 0 0 3 7 - 1

Probabilistic Engineering Mechanics 18 (2003) 115

www.elsevier.com/locate/probengmech

* Corresponding author. Tel.: 43-512-507-6843; fax: 43-512-546-865.

E-mail address: [email protected] (C. Proppe).
http://www.elsevier.com/locate/probengmechhttp://www.elsevier.com/locate/probengmech


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leads to the equations

aEfX2AEX 4and

EfiXX2 EfiXEX

EX2 EXX2 EXT

ATi ; i1; ; n

5

for the linearization coefficients, where Aiis the ith row of

matrixA.

EQL has been originated independently by Kazakov,

Booton and Caughey. Kazakov and Booton, who were

concerned with problems of control theory and automatiza-

tion, considered the linearization of nonlinear memoryless

transformations of random variables (RVs). While the work

of Booton seemed to have only a small impact, the

linearization technique of Kazakov was applied and

extended by several authors in the Soviet-Union. Both

authors applied the method also to nonlinear systems under

Gaussian white noise excitation. In this case, an additionaldifficulty occurred: in order to calculate the linearization

coefficients, the probability density function of the nonlinear

system is needed. In general, thisprobability density function

is unknown. Therefore, approximations of the true

linearization coefficients were calculated by replacing the

probability density function of the nonlinear system with the

probability density function of the linearized system. This

has been done under the assumption that the expectations

needed for the calculation of the linearization coefficients

will not change very much when they are evaluated with

respect to the probability density function of the linearized

system. This substitution necessitates successive approxi-mations in order to calculate the approximate linearization

coefficients.

Caughey developed EQL when he tried to generalize the

EQL technique invented by Krylov and Bogoliubov. He

investigated the Van-der-Pol oscillator[25], vibrations of a

nonlinear string [24] and a system with hysteresis [26].

Furthermore, within the development of EQL, it was

important to recognize that for a Gaussian probability

density function, the following relation holds:

EfiXX EXXTE7XfiX: 6Using this formula, the linearization coefficients are obtained

from the expectation of the gradient of the nonlinear

function. This result has been published in Refs. [6,76].

Further simplifications of practical importance are possible

using the linearity property of the expectation operator, e.g.

for chain-like systems[48].

The above outlined procedure may be easily extended to

nonstationary problems, where the linearization coefficients

vary with time. EQL is then carried out in successive discrete

points on the time-axis. The expectations must now be

obtained for a linearsystemwith time-varying coefficients. In

case of Gaussian excitation, the expectations may be

obtained from the solution of differential equations for the

mean value and the covariance matrix[111, p. 113].In the

iterative procedure for the determination of the linearization

coefficients, the solution of Eqs. (4) and (5) alternates then

with an integration of the differential equations for each time

interval [100]. The length of the time intervals can be

controlled by comparing results obtained with different step

sizes.For non-Gaussian excitation, the evaluation of the

expectations may be much more involved. In Refs. [44,

52,66,134], EQL has been applied to systems under

external Poisson white noise excitation. In Refs. [52,134],

the Duffing oscillator is considered, in Ref. [44] a uni-

dimensional system with cubic nonlinearity and in Ref.

[66] a SDOF system with Bouc Wen hysteresis. For

polynomial type nonlinearities, the response moments of

the linearized system have to be computed. This has

been done in Ref. [134] for SDOF systems and in Ref.

[52] for MDOF systems. In case of nonpolynomial type

nonlinearities, the probability density function of the

linear response is approximated by a GramCharlier

series A in Ref. [66]. Looking closer at the results of

these papers, one observes that in most of the cases the

stationary standard deviation of the response differs only

slightly from that obtained by a corresponding Gaussian

excitation. Thus, considering a model with Gaussian

excitation might be an interesting alternative in case that

the random impulses are not relatively rare events.

2.2. Theoretical developments

2.2.1. Linearization criteria

Different linearization techniques may be definedaccording to different definitions of the approximation

error. An approximation error envisages often a specific

application and bears some subjectivity. A large part of

the literature on EQL is devoted to the development and

application of new linearization criteria. However, in

practice only the mean square criterion has found

widespread application.

Kazakov [75] introduced the equality of the first and

second order moments of the linearized and the nonlinear

system function as an error criterion. He also proposed to

take the arithmetic mean of the linearization coefficients

that were calculated according to this error criterion and

the mean square criterion. Criteria based on the potential

and dissipative energy were proposed in Ref. [41].

These criteria are compared for the dimensionless

Duffing oscillator under Gaussian white noise excitation

dX1t X2tdt; 7

dX2t 2cX2tdt2X1t1 eX21t ffiffiffi

2cp

dWt:This equation is linearized by

dX1t X2tdt;dX2t 2cX2tdt2X1t1k ffiffiffi2c

p dWt;

8

C. Proppe et al. / Probabilistic Engineering Mechanics 18 (2003) 1152


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where the linearization coefficient k has the following

form

minimization of difference between system functions:

k1eEX41t=EX21t minimization of difference between potential energies:

k2eEX61t=2EX

41t

equality of mean square system functions: k3effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

EX61t=EX21tq

equality of mean square potential energies: k4eEX41t=2EX21t

Especially, it is seen that k12k4: Evaluating theexpectations with respect to the probability density function

of the linearized system, the stationary standard deviation of

the displacement is obtained in the form sX1 d=e1=4: Theasymptotic behavior for large e of the approximated and

exact stationary standard deviation corresponds, as one can

show thatsX1!

0:8222

=e

1=4

[111]. The asymptotic error issummarized in the last column ofTable 1. In this case, EQL

yields good approximations also for large nonlinearities [34].

It is noted that except for the equality of the mean square

potential energies, the approximations of the standard

deviation are unconservative.

Polidori and Beck [99] chose the linear system whose

stationary probability density function minimizes the FPE

operator of the nonlinear system with respect to the L2 norm

and a weighted L2 norm, respectively. For the Duffing

oscillator and the Duffing oscillator with cubic damping

they improved the results obtained with the mean square

criterion.

In order to calculate power spectral density approxi-mations, Apetaur[24]developed linearization techniques

that are based on statistical characteristics of second order,

especially on the correlation function. Iyengar[67]tried to

approximate the nonlinear system by a linear system of

higher order. The additional linearization coefficients are

determined from time-derivatives of the nonlinearity. This

leads to additional differential equations with parametric

excitation. In this way, Iyengar intended to approximate

higher resonances.

Bernard and Taazount [16] investigate systems, where

the nonlinearity is due to springs with a gap. They show that

the usual linearization technique will yield goodapproximations of the power spectral density, if the gap is

very small or very large compared to the standard deviation

of the system response. If this is not true, then the response

spectrum contains a broad band. In this case Bernard and

Taazount propose to approximate the power spectral density

by the power spectral density of an ARMA process. Then, at

least an approximation of the power spectral density must be

given.

With the aim to apply EQL to reliability calculations,Casciati et al.[23]compare the stationary level crossing rate

at some critical level. They obtained a local linearization

technique that has been successfully applied to the Duffing

oscillator and to hysteretic systems, respectively.

Recently, several authors proposed to calculate the mean

square error with respect to the probability density function

of the linearized system [40,124]. In this case, the derivatives

with respect to the parameters in the Gaussian probability

densityfunction has to be taken into consideration. However,

the results that were obtained with this linearization

technique were worse than those obtained from the usual

mean square criterion.The reasonfor this will be explained in

Section 2.2.2. Unfortunately, by calling the proposed method

as corrected linearization[124]and the usual method as

erroneous [40] the authors gave the impression that the

usual method is false. It is once more mentioned that

the minimization procedure is carried out with respect to the

probability density function of the nonlinear system and that

at a last step, this probability density function is approxi-

mated by the probability density function of the linearized

system. The approximation error is thus not minimized with

respect to the probability density function of the linearized

system. For a further discussion of this topic, the reader is

referred toRef. [36].

2.2.2. Approximation error

Only a few theoretical papers are devoted to the

estimation of the approximation error. Most authors study

specific systems and obtain the approximation error by

comparison with exact solutions, simulation results or with

cumulant or quasi-moment closure methods. For external

Gaussian white noise excitation, comparison with results

from closure techniques is in so far justified as Gaussian

closure and EQL may be viewed as equivalent approaches.

Thus, higher order closure schemes can be considered as

higher order corrections of the results obtained by EQL.

However, for MDOF systems, higher order closure schemes

are very time demanding such that in this case, only

simulation techniques may be suited for error estimation.

The results from comparative studies[111], show that EQL

may give accurate approximations of the standard deviation

of the response even if the nonlinearities are not small. For

MDOF systems the error may be higher [92]. However, in a

recent benchmark study [113], itis concluded thatEQLis

at least for the predictionof the variance sufficiently robust

accurate and also applicable with an acceptably low effort.

In general, the standard deviation is underestimated.

The following remarkable result has been pointed out by

Caughey [27], Crandall [35] and Kozin [81]: if

Table 1

Asymptotic behavior of the standard deviation of displacement for different

linearization criteria

Linearization criterion d % Error

Minimization of difference between system functions 0.7598 7.6

Minimization of difference between potential energies 0.7953 3.3

Equality of mean square system functions 0.7128 13.3

Equality of mean square potential energies 0.9036 9.9

C. Proppe et al. / Probabilistic Engineering Mechanics 18 (2003) 115 3


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the linearization coefficients were obtained from expec-

tations with respect to the probability density function of the

response of the nonlinear system, then the first and second

order moments of the linearized and the nonlinear system

would coincide. The error is thus introduced by the

replacement of the probability density function of

the nonlinear system by the probability density functionof the linearized system.

Several authors tried to improve the probability density

function that is used to obtain the linearization coefficients.

Beaman and Hedrick [13] approximate the probability

density function by a GramCharlier series A of fourth

order. Chang [29] assumes that the probability density

function is a sum of weighted Gaussian probability density

functions. Hurtado and Barbat[62]represent the probability

density function by a Gaussian probability density function

and a Dirac delta function. All these methods are character-

ized by a higher computational effort, especially for MDOF

systems. However, the effort that is neededfor thecalculation

of these approximations may not always be justified by the

accuracy of the results.

Kolovskii [79] estimates the standard deviation of the

process Xt2X0t; where X0t is the state vector of thelinearized system. These estimates contain the standard

deviation of the nonlinearity for which a Lipschitz condition

is assumed to hold.

Skrzypczyk[122]utilizes the theory of integral equations

over locally compact abelian groups in order to estimate a

measure of the difference Xt2X0tby a measure of thedifference between the nonlinear and the linear system

function. In this way he justifies the use of some

linearization techniques.Bernard[15]and Bernard and Wu [17], making use of

the large deviation principle, give also a justification of some

linearization techniques. They consider the norm of the

difference between the probability measures of the non-

linear and the linearized system and estimate it by the

relative entropy, which is then minimized.

2.3. Extension of EQL

2.3.1. EQL for parametric excitation

For parametric excitation by Gaussian white noise,

different linearization techniques may be obtained byconsidering different models for the linearized system. The

following linear models were proposed in the literature:

linearized parametric excitation[18,31]

external excitation instead of parametric excitation [43,

80,123,141]

equivalent linear restoring forces instead of parametric

excitation [129]

In this context, it has been shown that models that retain

the parametric excitation may yield even less accurate first

and second response moments than other models, although

only linear parametric excitation is present in the original

nonlinear problem. Bruckner and Lin[18]pointed out that

the linearized system may not represent the stability

properties of the nonlinear system. Especially for systems

with bifurcation points, equivalent nonlinearization may

lead to a better representation of the stability properties[80].

The linearization procedures itself can be based ondifferent arguments:

linearization of the equations of motion[31]

linearization of the Itoequation[18,43,123]

linearization of the Ito formula[43].

Several proposed linearization techniques have the

property that the truly linearized system and the nonlinear

system have the same first and second order moments, see

Ref. [107]for details.

2.3.2. Application to hysteretic systems

For systems with hysteresis, the nonlinear systemfunctions depend on the history of the system response.

Caughey [26] proposed the application of the averaging

principle [82] in order to apply EQL to systems with

hysteresis. The approach is based on the assumption that the

response has a narrow frequency band. After introduction of

an amplitude and a phase process, the expectations can be

obtained after averaging by integration with respect to the

amplitude distribution. This method has been further

developed and investigated inRefs. [50,78,131]. However,

as has been shown inRefs. [64,109,137], the response may

also contain a broad frequency band. In this case, the

standard deviation is considerably underestimated by EQL.Another approach, due to Wen [137], considers the

introduction of an additional nonlinear differential equation

in order to account for the hysteretic behavior. The system

function is given by

fx;_x c_x akx2 12 akzx; 9where the hysteretic part zxis described by the first orderdifferential equation

_z bl_xllzln21z g_xlzln K_x; 10with constants b, g, K and n. These constants may be

adapted such that a wide variety of hysteresis loops may be

described. The equations of motion together with the

differential equations for the hysteretic part form a nonlinear

system of differential equations, which is then linearized.

Extensions of the so-called Bouc Wen model can be found

in Refs. [10,38,47,98,139]. An overview of the obtained

results is contained in the review article [138]. Despite its

broad acceptance, the BoucWen model is not in agreement

with classical plasticity theory. Extensions of the model to

overcome these contradictions can be found inRef. [20].

In order to treat any kind of nonlinearity, including

hysteretic material laws, a simulation procedure is proposed

in Ref. [100]. The distribution of the state vector of



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the linearized system is simulated and the linearization

coefficients are obtained from a least squares problem. For

Gaussian excitation, the approach requires the generation of

a Gaussian distributed RV only. In case of impulsive

excitation (e.g. by Poisson white noise), the state vector can

be simulated easily and without much numerical effort.

2.3.3. Application to continua

In order to analyze continua, a discretization of the

problem, e.g. by the FE method, is necessary. EQL can be

applied to the differential equations of the continuum or to

the differential equations of the discretized problem.

Iwan and Krousgrill [63] consider the first possibility,

which has the advantage that the material laws and the yield

surface can be specified for the continuum[22, S. 287]. In

Refs. [93,121], continua under nonstationary Gaussian

excitation with visco-elastic properties described by the

BoucWen model are investigated. The authors apply EQL,

discretize the structure and neglect higher modes of thelinearized system. Casciati and Faravelli [21] consider

elasto-plastic material properties with yield surfaces, hard-

ening rules and hysteresis. They also employ a BoucWen

model and linearize the equations prior to discretization.

Iwan and Whirley [65] introduce a decomposition for the

field of linearization parameters for nonlinear continua

under nonstationary random excitation. The minimization

conditions are obtained from a variational problem.

The linearization of the discretized equation of motion

has been proposed for beams[19,118]and plates [1]. The

advantage of this approach is that the discretized system is

built only once, which is extremely useful when coupling

EQL with commercial FE programs.

2.3.4. Local linearization

The need for increasing the accuracy of the results

obtained by EQL led many researchers to extensions of

EQL. For external excitation, one can show that EQL and

Gaussian closure of the moment equations are equivalent

approaches. Most closure schemes can therefore be viewed

as higher order extensions of EQL. These extensions suffer,

however, from the fact that they are difficult to apply and

that the computational efforts increase tremendously for

MDOF systems. One of the reasons for these facts lies in the

global approximation concept applied in EQL and its

extensions.

A local linearization approach has been recently

suggested in Ref. [104], where the nonlinearity is locally

linearized. As a result, the probability density function of

the nonlinear system is very accurately approximated by

suitable Gaussian probability density functions. In this way,

a local linearization approach may be suited, e.g. for

reliability problems, where the tails of the probability

density function have to be approximated accurately. In

contrast to closure schemes, the introduction of Gaussian

probability density functions in the phase space provides

a link between linearization methods and sampling

techniques such as MCS.

2.4. Efficient implementation of EQL for FE models

2.4.1. Local coordinate systems

Dealing with nonlinear structures discretized by FE, thenonlinearities are represented by nonlinear elements.

Linearizing these nonlinear elements employing EQL, the

linearized elements must be determined separately for each

nonlinear element as it is also done in the standard case

where all elements are built independently and then

assembled to the global matrices.

Let ug denote the generalized degrees of freedom

(DOF) of the structure and let eug , ug be the DOF

comprising only components associated with a nonlinear

element. The EQL procedure results then in structural

matrices (stiffness, damping, auxiliary variables) associ-

ated with the DOF of eug: Each linearized element must

be free of stress under rigid body motion, and is

therefore singular. For example, the stiffness matrix of a

linearized spring has a rank of one, although the matrix

is generally of size six by six within a three-dimensional

model. These specific properties of elements suggest that

the linearization should not be performed in the global

coordinate system involving all the DOF of eug but in a

local coordinate system eul involving only the minimal

number of DOF [42]. In this way, the efficiency is

enhanced and ill-conditioned problems are avoided. Also,

the nonlinear restoring force of the elements is defined

best in local minimal coordinates, i.e. for the sake of

clarity and uniqueness. Hence, the restoring force can besafely assumed to be defined in local coordinates efleul:A linear transformation

eul e Teug; 11

with constant transformation matrix eT may be intro-

duced to reduce the linearization to a smaller set of

coordinates. For the statistical linearization the first two

moments of the stochastic response will be needed in

local coordinates. The mean eml and covariance matrixeSl are readily determined from the mean

emg and

covariance matrix eSg of the element response in global

coordinates using the transformations eml

e Temg

andeSl e TeSgeTT: The linearization coefficients are thencomputed in the minimal local coordinate system and

subsequently transformed to the global coordinate system

employing the general transformation rules of element

matrices in FE analysis.

2.4.2. Stochastic response of linearized system

Linearizing a structural nonlinear model in context of

general FE-models, some peculiarities must be con-

sidered by computing the linearized response to stochas-

tic excitation. The stiffness and damping elements

resulting from EQL are in general not symmetric and



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hence ruling out the standard equation solvers for linear

systems (which assume symmetry of the structural

matrices). Modeling hysteretic behavior may require

additional auxiliary variables, where the associated

linearization coefficients are again nonsymmetric. More-

over, modeling the excitation by filtered white noise

requires augmented structural matrices.The stochastic analysis determines the mean and

covariance matrix of the stochastic response. Two basic

approaches are currently available. The first employs

complex modal analysis and the second direct step-by-

step integration schemes.

2.4.2.1. Complex modal analysis. Complex modal analysis,

as originally suggested for MDOF-systems [30], is not

well suited to treat the system of equations resulting

from the linearization of FE-models with a large number

of DOF. Since the matrices have not the standard form,

the highly optimized FE-algorithms for symmetric

classical eigensolutions are not applicable. A state

representation is generally used, involving the displace-

ment, velocities and auxiliary variables, leading to

nonsymmetric matrices with a size . 2m; where m

denotes the number of DOF. The solution of the

nonclassical eigenvalue problem is computationally

quite expensive compared with the classical form of

size m. Moreover, dealing with nonstationary problems,

the nonclassical complex eigenvalue problem has to be

solved numerous times, increasing the computational

burden such, that the analysis of larger FE-systems

becomes intractable. Most of the above mentioned

difficulties can be circumvented by the approach shownin Ref. [115]. It is based on the solution of the classical

eigenvalue system for the symmetric part of the

structural matrices and an extended form of component

mode synthesis [33,34]. However, this approach is

certainly not practical for the solution of nonzero mean

problems, particularly when comparing its efficiency with

the recently developed direct step-by-step integration

schemes.

2.4.2.2. Direct step-by-step integration. Stochastic versions

of special integration schemes have been suggested [91,

97,133]. All these proposed procedures operate explicitly

with the covariance matrix and require special integration

schemes. Since the covariance matrix is fully populated

and of a size . 2m; these approaches cannot be applied

efficiently to FE-systems with a large number m of DOF,

because of storage requirements and number of compu-

tational operations.

A solution technique which avoids the storage and

direct integration of the full covariance matrix has been

developed recently in Refs. [105,106], where it is

suggested that the covariance matrix is represented by

the Karhunen Loeve expansion with a reduced number

of Karhunen Loeve vectors. Continuous white noise is

approximated by discrete noise (shot noise, random

impulse) applied at one instant in each time step. The

KarhunenLoeve vectors can be integrated by any

available and appropriate deterministic step-by-step

integration scheme. These vectors are further updated

in each time-step to take the effect of the random

impulse loading into account. Most important for theefficiency of the suggested approach is the fact that the

update can be carried out exactly in a low-dimensional

reduced subspace. Although this approach requires only

deterministic direct integration schemes [11], these

integrations must be performed NK-times, where NKdenotes the number of Karhunen Loeve vectors. Since

NK lies in the range of 20200, depending on the

problem and the required accuracy, the computations

might be still a tedious task, particularly for very large

FE-systems. For nonlinear systems with large linear

subsystems, modal transformations can be used for a

considerable size reduction of the FE-model. Its appli-

cation in context with nonlinear structural stochastic

analysis is shown in Ref. [106] including the effects of

mode truncation.

2.5. Areas of application

As many comparative studies revealed, EQL yields in

general good approximations for the first and second

order moments of the system response. For most

practical applications, e.g. reliability analyses, the

information provided by EQL is not sufficient. Thus, it

has been often argued that EQL should be employed

during the design stage, in order to obtain a first estimateof the system properties. There seems to be no

systematic studies concerning the applicability of EQL

to design problems. Therefore, possibilities of application

for EQL during the design stage are outlined here,

namely sensitivity analyses (e.g. in order to reduce the

simulation efforts for systems containing random par-

ameters) and the design of controlled structures.

2.5.1. Sensitivity analysis

Suppose that in Eq. (1), the state vector and the system

functions dependent on a design parameter b:

dXt; b fXt; b; bdt FbdWt: 12Taking the first derivative of the state vector with respect to

the design parameter leads to

dZt; b fb

X; bdt fxi

ZidtdF

dbdWt; 13

here Zdx=db: The differential equation (12) acts as afilter for Eq. (13). In order to obtain approximations for

the mean and variance of the design sensitivity with

respect to the parameter b, the application of EQL has

been proposed in Ref. [108]. It is observed that the

original system (12) can be linearized independently of



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Eq. (13). The linearization coefficients for Eq. (12) can

be used for the linearization of Eq. (13).

The linearization of Eq. (13) leads to

dZt; b A1bXt; bdtA2bZt; bdt a3bdt

F

bdW

t;

14

with

A1ijb E

2fi

xjb

2fi

xjxkZk

" #

A2ijb E fi

xj

" #Aij; 15

A3ib E fi

b

E fi

xjZj

" #2A1ijEXj2AijEZj:

The solution scheme consists of two fundamental steps:

1. Solve for the linearization coefficients A, a of system

(12).

2. Minimize A1ijXAijZa3i 2 fi=b2 fi=xkZk inmean square sense and calculate the expectations EZ;EZZT:

A further reduction of the computational effort would be

possible, if the linearized system, together with the system

of differential equations for the derivative of the state vector

with respect to the design variables, would lead to a closed

set of equations for higher order moments. However, this is

in general not the case, even if the nonlinearities in theoriginal system are of polynomial type. Thus, the compu-

tational effort can be reduced from the linearization of a

2n 2n-dimensional system to the linearization of two

n n-dimensional systems.

2.5.2. Stochastic structural control

Control methods for linear systems under external

Gaussian excitation have been investigated by many

researchers. In most of the studies, linear feedback control

and mean square cost functionals were considered, for which

the exact solutions have been found [83]. Control of

nonlinear systems under Gaussian excitation has been

studied by approximate methods, such as stochastic aver-

aging[146], moment closure techniques[28]and EQL[12,

56,143145]. As EQL may be applied to relatively large

systems with geometrical as well as material nonlinearities,

combining EQL and linear optimal control theory opens an

avenue towards the control of relatively large nonlinear

structural systems. The general solution procedure is as

follows:

1. Start with a guess of the linearization coefficients.

2. Solve the Riccati equation for the linear control problem

and obtain the gains.

3. Calculate the expectations for the controlled linear

system.

4. Solve the linear system for the linearization coefficients.

5. Continue with step 2 until convergence.

A decoupling of the problem, i.e. the linearization of the

uncontrolled system followed by the calculation of theoptimal gains, would lead to different results, as the control

terms influence the expectations in the linearization

procedure. Recently, it has been shown [125]by means of

examples that the influence of the linearization criterion on

the optimal value of the cost functional seems to be

negligible.

2.6. Monte Carlo simulation

2.6.1. Introduction

For reasons mentioned in the beginning, the evaluation of

the stochastic response by MCS has particular advantages

over analytical approaches for nonlinear systems with largerstate-space dimension. First publications applying MCS in

stochastic structural mechanics appeared 30 years ago[119,

120]. The method has been since then applied and extended

by numerous authors[51,53,117,127]. In its simplest form,

denoted as direct Monte Carlo simulation (D-MCS), each of

the samples is generated independently according to a given

distribution. D-MCS is very robust and relatively easy to

apply. A sample size offew hundred independent realizations

is generally sufficient to obtain a suitable estimate for the

mean and variance and to provide information on the basic

shape of the distribution within the domain covered with

a probability of 1 2 10=N;whereNdenotes the sample size.However, it is not suited to provide information on the

tails of the distribution as often required for reliability

analysis. The reasons for this are quite obvious: dynamical

systems under service must be reliable, where possible

failures or malfunctions should be the exception to the rule;

i.e. failures should be rare, low probability events. This type

of inefficiency has been recognized since the early beginning

of Monte Carlo sampling at the end of the 1950s and

beginning of the 1960s, when so called variance reduction

methods were developed. In fact, books on Monte Carlo

sampling [45,54,112] are largely devoted to this subject,

addressing variance reduction procedures such as import-

ance sampling (IS), control variates and stratified sampling,

to mention the most common methods only. Other well

established approaches in Monte Carlo sampling, related to

nonlinear dynamical systems, are random walk concepts

used in physics, chemistry, economics and engineering[45].

Today, increasing the computational efficiency is the

main concern in research on MCS techniques. Two different

kinds of improvements are generally employed for an

increase in efficiency. The first kind improves the efficiency

in generating the samples. The otherso called variance

reduction techniquesincrease the efficiency by allowing

a reduced sample size for an acceptable variance of



8/15

the estimator. In the next sections, direct MCS is discussed

first, followed by a survey of methods to increase the

computational efficiency.

2.7. Direct Monte Carlo simulation

In D-MCS, one of the available random numbergenerators [58,59,84,96] is generally used to generate a

sequence of uniformly distributed pseudo-random numbers.

The random number generator must fulfill the statistical

tests of generating independent samples according to the

uniform distribution [46,85]. These pseudo-random num-

bers are then further translated according to the specified

distribution by using the inverse of the cumulative

distribution function or special algorithms like, e.g.

BoxMuller and polar Marsaglia [45,77] for specific

distributions. Since random number generators produce

statistically independent random numbers, correlation

between RVs must be constructed from independent RVs

by suitable transformations or algorithms. The generated

random numbers act as an input for the dynamical analysis,

which represents the computational most involved part.

For a sample sizeNthe variance, a point estimator pffor

the probability of failure pfhas the variance

var{pf}pf12pf=N: 16By accepting a toleranceewith the confidence level 1 2 d

Plpf2pfl , e $ 12 d; 17requires by employing Chebyshevs inequality a sample

size greater or equal to Nc

Ncpf12pf

de2 : 18

Since Chebyshevs inequality is valid for any sample size

and does not imply any restricting assumption, Nccustomarily specifies a larger sample size than necessary.

Assuming N large, a minimal sample size can be derived

from the central limit theorem, specified by the number

Nn

Nnpf12pfF2112 d=2

e

" #2; 19

where F denotes the standard normal cumulative

distribution function. Using Eq. (19) one should keep

in mind that Nn has to be sufficiently large to justify the

application of the central limit theorem.

Among all methods that utilize an N-point estimation in

the n-dimensional space, the Monte Carlo method has an

absolute estimation error that decreases with N21/2,

independently of the dimension n as seen from Eq. (16),

whereas all other approaches have errors which decrease

with N21=n at best[45].

As mentioned before, Monte Carlo sampling becomes

increasingly attractive as the dimension n of the problem

increases for n . 2: Hence MCS is often the only feasible

solution for problems with a larger dimension as one usually

encounters in technical applications. Besides being more

efficient than analytical-based approaches, is has the

advantages that the tools of deterministic analysis can be

fully exploited. Moreover, it is the most general applicable

tool available in stochastic mechanics, i.e. there is almost nostochastic problem where MCS cannot be applied.

Although MCS is more efficient than available alterna-

tives (if any), the required computational costs are often

considerably high. Naturally, the computational efforts

increase with the dimensionn, the required sample size N,

the complexity of the nonlinear structural models, and is

especially high when dynamic problems must be analyzed.

Hence, efficient techniques to compute the dynamical

response of the system for the input data should be employed

in order to reduce the computationally extensive tasks.

Moreover, so called variance reduction procedures[45,114]

can be used to further decrease the computational burden.

2.8. Increasing the efficiency of Monte Carlo simulation

2.8.1. Increasing the efficiency for generating samples

A considerable increase of computational efficiency is

often possible by exploiting the fact that the solutions in

different samples do not differ fundamentally, but just

gradually. In such cases, all (deterministic) solutions are

close to each other. Using a reference solution, the solutions

are in the neighborhood of the reference solution. Therefore,

the reference solution, which has to be determined only

once, is used to gain efficiency in the deterministic

computations. This procedure is described in the followingfor two examples, namely the Neumann series technique in

static stochastic structural analysis and the solution of

random eigenvalue problems in linear stochastic structural

dynamics. The Neumann expansion is a typical example for

such an approach, which has been proposed first in Ref.

[142]for static stochastic structural analysis in context with

stochastic finite elements and MCS. In static structural

analysis, the main computational part must be devoted to the

static solution of

ux Kx21fx; 20

whereu denotes the generalized displacement vector, ftheloading vector and K the stiffness matrix. All quantities

might depend on a set of RVs x. Since the factorization

(equivalent to the inversion) of the stiffness matrix requires

the main part of the solution, the Neumann expansion

represents the stiffness matrix by a reference matrixK0and

a deviator matrix DKxKx K0DKx; 21and factorizes only once the reference matrix K0.

The solution is then obtained by the sum

ux u0 2 u1u2 2 u3 ; 22



9/15

with

u0x K210 fx; 23uix K210 DKxui21x fori . 0: 24The Neumann expansion is equivalent to placing the

deviator matrix on the right hand side

K0ux fx2 DKxux; 25and solving the above relation in an iteration which

converges for sufficiently large s

usx K210 {fx2 DKxus21x}; 26usings $ 0 and defining u21x 0:

It should be noted that the above iteration corresponds to

the well known class of NewtonRaphson procedures[11,

37] using the initial stress method. Newton Raphson

procedures and all its variants are applied in nonlinear

structural analysis since the development of finite element

analysis. The fact, that the Neumann expansion is just aspecial case of the NewtonRaphson schemes, indicates,

that the deterministic algorithm like NewtonRaphson can

be used to increase the computational efficiency of the

deterministic structural analyzes in context with MCS. It

seems that the power of available deterministic procedures

has not yet been fully exploited to treat uncertain/

deterministic nonlinear dynamical systems for determinis-

tic/stochastic loading by MCS.

Large linear FE-models subjected to dynamic loading are

usually analyzed by modal analysis. Within this procedure,

the computation of the modal properties like eigenfrequen-

cies and modes requires most of the computationalresources. In case of an uncertain or stochastic structural

system, it is not efficient to determine the modal properties

independently and anew for each sample. An increase in

efficiency can be obtained by using the fact that all solutions

of the random eigenvalue problem are closely related. This

circumstance has been exploited in Ref. [130], reducing

considerably the number of vector-iterations required to

obtain convergent eigensolutions. Other approaches pro-

posed[116]determine accurately a reference solution and

employ further component mode synthesis for a fast modal

update.

Alternatively, instead of solving the random eigenvalue

problem

Kxcix Mxcixlix; 27for each set of RVs x, a reference solution is determined

only once

K0ciM0cili: 28The effect of the deviator matrices for the stiffness,

damping, and mass, respectively

DKx Kx2K0; 29DCx Cx2C0;

DMx Mx2M0;are then treated on the right hand side of the equation of

motion, analogously to Eq. (26), leading with

ux Czx; 30

whereC comprises the set of eigenvectors, to the followingequation of motion in modal coordinatesz:

zi2jiwi_zi2w2izicTift2cTi DMxCz2c

Ti DCxC_z2cTi DKxCz: 31

The above system can be solved in an iterative manner,

where the modal solutions are slightly coupled by the right

hand side.

2.8.2. Importance sampling

For the case when the failure probability pfis very small,

standard Monte Carlo sampling is no longer efficient. This is

typically true when estimating failure probabilities where

pf, 1023:Suppose, for example, in a technical application,

the acceptable failure probability must be less than 1025.

Requiring a probability of 90% that the absolute sampling

error is less than 0.4 1025 would, then, according to Eq.

(18), require a sample size of at least 6.25 106

independent realizations. Such a huge sample size might,

in case the system is not exceptionally small, exceed

available resources even on powerful present day compu-

ters. Even though it might be possible to generate such large

sample sizes, using supercomputers and employing parallel

processing[68,69], the utilization of standard Monte Carlo

might be a waste of computational resources, since only asmall fraction of realizations of the sample contributes to the

desired statistical information.

Variance reduction procedures exploit additional a priory

information to reduce the necessary sample size N for a

specified accuracy. The widely used IS and controlled MCS

belong to this class of procedures and are discussed in this

and in Section 2.8.3.

IS gains its advantage by altering the density function

that generates the random samples. First publications

applying IS go back to Refs. [7072,74] using the first

computers. IS can be applied when an integral

pfR

nkFxdFx; 32

needs to be estimated by MCS, where Fx denotes thecumulative density function. In reliability problems the

kernelkxcan be interpreted as the indicator function Ixwhich assumes the value Ix 1 if x lies in the failuredomain which is usually expressed by the limit state

function gx , 0;and Ix 0 elsewhere. Introducing theimportance function Rx

Rx pxhx ; 33



10/15

as ratio between the original probability density function

and the IS density function hx; the above integral (32) isequivalent to

pfR

nkHxdHx; 34

whereHx is the IS distribution, andkHx kFxRx: 35IS can be used to reduce the variance of the kernelkxandtherefore the variance of the point estimate pf. Comparing

the variance of the two kernels, the following relation holds

var{kFx}2 var{kHx}

R

nk2Fx12RxdFx; 36

which guarantees a variance reduction if the right hand

side of Eq. (36) is positive. Choosing hx

. p

x

when

k2Fxpxis large and hx , px whenk2Fxpxis smallcontributes to the desired effect of reducing the variance

of the estimator, or equivalently reducing the sample size

N. The above relation also indicates that the IS density

should cover in reliability analysis the failure domain where

kFx Ix 1: Moreover, choosing hx kxpx=pfleads to zero variance thereby minimizing the variance

reduction. Since this optimal sampling density involves the

unknown estimator pf, it is merely of academic interest,

showing the existence of such a distribution.

The application of IS in nonlinear structural dynamics is

a relatively new development. On the contrary to the wide

application of IS in static structural mechanics, whichdeveloped in the last two decades to a well-established

methodology, IS seems presently to be in the early

development state. The obstacles in applying the IS concept

stems from the difficulty to establish a suitable IS density

hx for nonlinear structural problems. There are two mainsources for the problems to establish an IS density. One

difficulty stems from the large number of RVs involved to

represent the stochastic excitation. The other arises from the

nonlinear dynamic behavior which makes it hard to

associate a priori the excitation to the response. In stochastic

dynamics, the excitation is often modeled by white noise.

This implies a virtually infinite set of RVs. Since a very

large number of RVs cannot be handled efficiently using IS,

the stochastic excitation is usually expressed by stochastic

processes involving only a moderate number of RVs, say a

few hundred. Modeling the excitation either by a discrete

random pulse sequence[88]or employing the Karhunen

Loeve representation[49], it is usually possible to represent

the stochastic excitation process by a moderate number

of RVs.

IS is applied in context with reliability considerations

which are in stochastic structural dynamics first excursion

probabilities [110]. For linear systems, a quite efficient IS

density and methodology has been recently suggested in

Ref. [9]for solving the first excursion problem by IS. The IS

density is the union of all elementary failure regions Fikwhich are defined as the region in the RV space and which

cause a barrier crossing at instant tkdue to the ith output.

Using the unique linear relations between input and

response, the design points xpik; i.e. the nearest point to the

origin in standard normal space, can be established in astraightforward manner which define uniquely the elemen-

tary failure regions. Since the elementary failure probabil-

itiesPFikare known exactly, the optimal IS density for thecrossing problem is available. For computing the first

excursion probability, a weighted sum of the elementary

optimal IS densities is used, avoiding the inconvenience of a

multi-modal IS density which deviates considerably from

the optimal one.

IS can be extended to nonlinear systems with consider-

able computational effort to establish an IS density. For

cases where the design points of the elementary failure

events can be determined by nonlinear programming tools,

the above-described procedure can be applied to nonlinear

problems. The investigation in Ref. [39], for example,

utilizes the design pointsxpikof the elementary failure event,

to construct such an IS density. Since the search for the

elementary design point involves a constraint optimization

problem which requires the gradients, methods to determine

these gradients are necessary. Such gradients have been

derived in Ref. [87], applicable for the widely used

Newmark integration scheme. Further work following a

similar approach has been published inRef. [135]where a

hysteretic SDOF oscillator is investigated.

Since the IS density is usually rather difficult to estimate

for nonlinear dynamical systems, subset simulation has beensuggested in Ref. [8]. This work is an extension of an

adaptive importance sampling scheme [7] employing the

Metropolis algorithm, which generates a sequence of

samples that converges towards the optimal IS density.

Subset simulation introduces for a given failure event Fa

decreasing sequence of failure events F1 . F2 . F mF so that Fk >ki1Fi: Then, the final failure probability

pf PFm can be expressed as product of conditionalprobabilitiespf PF1

Qm21i1 PFi1lFi:The advantage of

this subset approach is that the failure probabilities PF1;PFi1lFi can be selected such that MC sampling is stilleffective, say in the order of 0.1.

An IS procedure applicable to Markov processes

described by Itos equation has been suggested in Ref.

[132]. This direction has been pursued further by other

authors[89,90,95]. The reported results show solutions for

restricted academic type problems, but the proposed

methods so far failed to analyze more realistic nonlinear

structural models.

2.8.3. Controlled Monte Carlo simulation

In recent years another concept applicable to dynamical

systemsdenoted as controlled MCShas been developed.

Contrary to IS, the methodology is self-adaptingand does not



11/15

require detailed a priori information. It has no restrictions on

the type of structural model regarding linear, nonlinear

hysteretic and state-space dimension. Regarding efficiency,

IS can theoretically be much more efficient than any other

techniqueincluding controlled MCSif an efficient pro-

cedure is available to compute the IS density. However, one

should note that the efficiency of a procedure depends on howmuch information about a given problemthe method utilizes.

In essence, efficiency is generally gainedat the expense of the

loss of generality. As an example, direct MCS is very general

but not efficient in context with reliability problems. For

dynamic analyses, the IS density has been efficiently

determined for linear systems [9] and weakly nonlinear

systems only. Even for linear systems, the effort required to

determine the IS density is by far greater than the actual IS

simulation. The computational effort to estimate a suitable IS

density increases drastically for nonlinear dynamical sys-

tems. Moreover, for general nonlinear systems, the pro-

cedures established so far might even fail to converge to the

design point of the elementary failure events. If applicable at

all, the total amount of computational resources must be

considered. With respect to this total amount, an IS-

procedure cannot be considered as efficient, if the resources

to establish the IS density are orders of magnitude larger than

the IS-procedure itself. Hence, the following approach

should be seen as alternative, where the procedures

mentioned in Section 2.3.2 are not applicable, or cannot be

applied efficiently.

Controlled MCS methods have been developed for the

class of systems with Markovian properties and is designed

to increase the sampling density in the low probability

domain. This is made possible by modifying the weightsassociated with the realizations. In controlled MCS, the

weights are modified dynamically in an adaptive manner

contrary to IS where the weights (ratio between original and

IS density) are specified a priori.

Controlled MCS increases the sample density in the low

probability domain of the phase space and reduces the

sample density in the high probability domain. The increase

in the sample density is achieved by splitting[57,73,126]of

the sample. The technique of splitting is utilized

extensively in the reliability assessment of queues [5,57,

86]. The sample density is reduced by Russian Roulette[74,

103]or alternatively by a procedure denoted as clumping

[102].

Controlled MCS requires a selection criterion to

distinguish between important regions in the phase space

where the sample density needs to be increased and domains

where the sample density can be decreased. Since critical

dynamic responses are usually those which got an increased

energy input, first approaches employed an energy criterion

[101]increasing samples associated with a relatively high

mechanical energy (potential plus kinetic). This energy

criterion is not applicable for general stochastic dynamical

systems where energy is not defined. Moreover, for strongly

dissipative system, the criterion based on the mechanical

energy did not show a convincing performance. For this

reason, an alternative approach has been developed, for

which such energy criteria are no longer needed [55,103].

The key to this newly developed approach is to distribute

dynamically the sample density (realizations with different

weights or probabilities) approximately uniform within the

state space. By this approach, which does not require anyspecific selection criterion, the sample density is increased

automatically in the tails of the distribution.

In Ref. [55], the tendency to a uniform density of

realizations in the state space is achieved by clumping

subsequently the closest state pair to a single realization

until the number of realizations is reduced to half of the

sample size N. The closest pair is determined dynamically

by evaluating the smallest distance among all remaining

samples (including previously clumped realizations) in a

normalized state space. Subsequently, all realizations are

doubled. The approach suggested inRef. [103]also aims at

a uniform sampling distribution and employs Russian

Roulette and splitting. The importance of a sample is

measured by the distance to several closest neighbors which

allows to establish the selection criteria.

3. Conclusions

Due to its simplicity and computational efficiency, EQL

has become a standard tool of stochastic structural

dynamics, that allows for analyses of MDOF and FE-

systems with non-stationary excitation and hysteretic

material laws, respectively. Due to the introduction of

local coordinates, complex modal analysis, componentmode synthesis and the KarhunenLoeve representation,

EQL is now capable to deal with in principle arbitrary large

FE models.

There are however several points that need further

attention. First of all, despite some theoretical efforts, there

is still no easy to use method available for an estimate of the

approximation error. Next, the areas where EQL can be

applied are still to be fully discovered, e.g. a large part of the

nonlinear models available in commercial FE-codes and

employed by practicing engineers seems to be not suitable

for the computation of the linearization coefficients. The

applicability of EQL to practical problems is limited by the

fact that the method provides only second moment

information. In this regard, an extension of EQL by local

linearization techniques may give access to a wider range of

applications.

As EQL is generally not sufficiently accurate to

determine quantitatively the reliability of a nonlinear

structure under stochastic excitation, MCS procedures are

generally applied to estimate the reliability or its

complement the probability of failure for nonlinear

dynamical systems. Contrary to the static counterpart of

nonlinear systems, procedures to improve the efficiency

of MCS for the dynamic case are still in its early state of



12/15

development. Only recently, some progress has been

achieved in the fields of importance sampling and

controlled MCS, respectively. Except for linear or weakly

nonlinear systems, none of the available approaches is

satisfactory regarding the robustness of the approaches and

the computational efforts required. Hence, considerable

research effort might be necessary to arrive at satisfactoryprocedures to estimate the reliability of nonlinear dyna-

mical systems, especially of larger complex FE systems.

Acknowledgements

This section is sponsored by the IASSAR-CSMSE

Subcommittee 2 on Stochastic Dynamics (Y.K. Wen, chair).

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