Computer methods in spplled

meshrnios and englaeerlng

EISEVIER Comput. Methods Appl. Mech. Engrg. 168 (1999) 273-283

Stochastic analysis of dynamical systems by phase-space-controlled Monte Carlo simulation

N. Hampomchai, H.J. Pradlwarter, G.I. SchuEller* Institute of Engineering Mechanics (IfM), Leopold-Franzens University, Innsbruck, Austria

Received 2 March 1998

Abstract

A simulation technique for the stochastic analysis of dynamical systems subjected to stochastic excitation is presented. The proposed technique is applied in conjunction with the Monte Carlo Simulation (MCS) procedure. The concept is based on the manipulation of the generated samples in phase space by utilizing a geometrical criterion, thus denoted as Phase-Space-Controlled (PSC) simulation technique. This criterion is common to the phase space of genera1 dynamical systems. The results of various examples show that MCS utilizing PSC has significantly higher efficiency than direct MCS, especially in case where the tail of the response distribution is required. In addition, PSC shows the potential of its application for larger dynamical systems. 0 1999 Elsevier Science S.A. All rights reserved.

1. Introduction

According to the recent survey on the solution methods of nonlinear stochastic dynamics (see e.g. [ 1 l]), it has been shown that the available analytical procedures have remarkable restriction with respect to dimensionality and type of nonlinearity. In other words, most of the analytical methods are confined to some particular types of low-dimensional academic-oriented problems. Such methods applicable to higher-dimensional systems as Equivalent Linearization or Perturbation method are not suitable for the purpose of reliability analysis where the accurate distribution on the tails is needed. Due to these disadvantages of the analytical procedures, numerical procedures have been developed in order to solve general nonlinear type dynamical systems (see e.g. [2]). Among numerical procedures, Monte Carlo Simulation (MCS) is regarded as the only available method which can solve the problems of significant complexity in higher dimension [ 111. MCS has distinct advantages over numerical method such as Finite Element from a computational point of view for larger dimensions. In the Finite Element Method the computational efforts grow exponentially with the dimensionality of the problem, whereas those required by MCS are only proportional to the dimensionality and the number of realizations. Using n-point evaluation for the solution of m-dimensional space, MCS has absolute error of estimate in the order of nm1’2 , I.e. independent of the dimension m of the problem, whereas all other numerical procedures have errors in the order of n- ‘lm at best [3]. This implies that MCS has higher computational efficiency than the other numerical procedures when the dimensionality of the problem becomes large. However, MCS has a major drawback in context with its application to reliability analysis. In order to estimate the tail of the distribution- low probability regions-the procedure requires a burdensome high number of samples. For example, the estimation of the probabilities of order 10m6 needs the sample size of order lo7 or even greater. Several simulation techniques have been introduced and applied to circumvent the requirement of such large sample sizes, i.e. estimating the low probability regions with the affordable sample sizes. As discussed by Schu&ller et

* Corresponding author.

004%7825/99/$19.00 0 1999 Elsevier Science S.A. All rights reserved PII: SOO45-7825(98)00145-5

al. 1989, the so-called variance reduction techniques ]9] have been utilized successfully in the problems of time-invariant type. Nevertheless, the direct application of the variance reduction techniques is not suitable to the time-variant problems in case of broad band stochastic loading. Firstly, the number of random variables might become infeasible large. Secondly, the probabilistic description can be non-stationary for the case of non-stationary problems. Therefore, any fixed sampling function, as used in most of variance reduction techniques, may not be suitable to the probability density function (pdf) characterizing these quantities. Due to these drawbacks, there has been the development of simulation techniques for time-variant problems. The developed techniques, denoted as controlled simulation techniques, are utilized in connection with the MCS procedure (see e.g. [6,7]). Utilizing the controlled simulation techniques, the statistical weight of each sample, which represents the associated probability content, will be modified during the course of simulation (see e.g. IShI).

Among the controlled simulation techniques, the so-called Double and Clump (D&C) procedure has been applied successfully with problems of larger dimension [S]. D&C provides a means to increase the number of samples which are expected to contribute to the estimation of low probability regions. In analogy to the importance sampling technique, these contributing samples are considered as important samples. The increase of the important samples, is carried out by the so-called doubling procedure. To keep the sample size constant. from the viewpoint of computational efficiency, the number of less important samples is reduced through the so-called clumping procedure. A system energy criterion c,(t), j = 1, . , N, is established for indicating the importance of each sample X,(r) as

c,(t) = E,(t). P,,,,,(t). ~;,(r)~ j = I, 2, . . . , N (1)

where El(t) and P,,,,, denote the system energy and the input power from the excitation, respectively, and wj(r) is the statistical weight of each particular sample. The parameter 0 < p < 0.5 is utilized for controlling the density of samples in the low probability regions. Although D&C shows its capability in estimating the low probability regions in larger dynamical systems, it still has some drawbacks.

Firstly, the energy criterion is rather heuristic based on mechanical considerations and therefore justified only for mechanical systems. Secondly, the criterion may not work well with highly dissipative dynamical systems. Thirdly, additional heuristic measures must be introduced in order to stabilize the procedure for its application to a large number of consecutive time steps. In this paper, an improved simulation technique which avoids the existing drawbacks in D&C is suggested. The developed technique aims to increase the efficiency of MCS for the reliability analysis of dynamical systems, i.e. to estimate low probability regions using feasible sample sizes. The technique is designed to be applicable for a general class of dynamical systems, i.e. not only mechanical or lightly damped systems. In addition, the proposed technique requires no additional heuristic measure for its robust operation.

2. Phase-space-controlled simulation technique

2. I. General concept

The proposed simulation technique will be utilized in connection with the MCS procedure for the dynamical systems described by

at) = a(X(t), t) + &w), t). J(t) (2)

where X(r) is the state vector, a(X(t), t) is a nonlinear vector function and &X(t), t) is a matrix which might be dependent on the state vector and explicitly on time. g(t) is the vector of independent white noise processes. The designed algorithm is based on the distribution of samples in a phase space. As a fact from the MCS procedure, only a small number of samples exists in the low probability regions where the samples are sparsely distributed. To obtain more information about the low probability regions, the number of samples in these regions needs to be increased. In order to maintain the computational efficiency-as in D&C-a certain number of samples, identical to that of the increased ones, has to be reduced. In other words, the sample size is always constant after utilizing the proposed algorithms. The increase and reduction of samples are carried out by the doubling and

N. Hampomchai et al. I Comput. Methods Appl. Mech. Engrg. 168 (1999) 27%28.3 175

clumping procedures, respectively. For the sake of general applicability of the proposed technique, any criterion restricted to specific systems will not be utilized here. Apart from maintaining the constant sample size, the clumping procedure attempts to induce an approximately uniform distribution of the samples in the phase space, i.e. the uniform probabilistic information. With this uniform distribution, the estimation of the probability in any region, automatically including the low probability regions, is easily obtained. The requirement of the information on the system for the purpose of identifying the low probability regions is thus avoided. As it will be seen later, the algorithms induce the change in the distribution of samples locally. Consequently, the simulation results are very stable even if the proposed simulation technique has to be applied to a larger number of consecutive time steps. In other words, the concept of uniformly distributing the samples is the key for the generally stable procedure. In order to distribute the samples uniformly within the phase space, the samples in the dense regions where the samples are more closely spaced will be clumped more often than those in the sparse regions. A consequence of this clumping strategy is that the samples in the low probability regions will be most likely excluded from the clumping procedure. To reduce more samples in the dense regions, the closest pair of samples will be selected first as a pair to be clumped at every sequence of clumping. Consequently, a geometrical criterion which measures the distance between two samples is now used as the clumping criterion. Using only the geometrical criterion, the proposed simulation technique possesses the generality in the application, i.e. applicable to a wider class of dynamical systems. In the following sections, the computational aspects of the algorithms will be presented.

2.2. Clumping procedure

As discussed in the preceding section, the clumping procedure is utilized in order to reduce a certain number of samples. To have the constant computational storage in the algorithms, clumping will be performed before doubling in order to provide free storage for the increased samples due to doubling. Before performing all procedures, i.e. clumping and doubling respectively, the samples in the original phase space T,(T) have to be transformed into the standardized form Ui(~) using the relation

U;,(d = XJd - &u,(T)

q(7) (31

where ,ui(7) and V,(T) are the mean and standard deviation, respectively, of the ith component of the state vector X,(T). The purpose of such transformation is to avoid any dependency on the units of different state components. Once the standardized samples are obtained, the clumping procedure is employed. Considering the samples as the points in the phase space, the Euclidean norm is selected as the distance measure, i.e.

A,,(T) = d

2 (U,,(r) - U,,bN’ k=l

where A,,(T) is the distance between the samples U,(T) and U,(T), m is the number of components of a sample. By clumping two samples X,(T) and xl\(~), it is meant that these samples are replaced by a single sample X,(T) which is the weighted average of the original samples

w< (7) = W,(T) + w,(4 (5)

where MI,(T) and w,~(T) are the statistical weights of the clumped state vectors X,(T) and X,,(r), respectively. W<(T) is the statistical weight of the resulting state vector.

It should be noted from Eqs. (5) and (6) that the first moment of the sample statistics after clumping is automatically the same as before clumping (Fig. 1). Based on the repeated application of the algorithms as described by Eqs. (5) and (6), the complete clumping procedure is established. In other words, the clumping procedure is a sequence of successive clumping, of which each consists of two operations. Firstly, a pair to be clumped is selected which might also include realizations resulting from previous clumping. Secondly, a resulting sample from clumping is computed according to Eqs. (5) and (6). The sequence of successive

276 N. Hampornchai ef al. f Comput. Methods Appl. Mech. Engrg. 168 (1999) 27.3-28.3

k-th component k-th component

Fig. I. Clumping of two realizations.

clumping is discontinued when the number of samples is reduced to one-half of the original size N, i.e. N/2. According to the clumping strategy, the closest pair of samples will be selected first to be clumped. To obtain such a pair in an efficient manner, the so-called sub-domain division procedure is utilized. Firstly, a domain of whole sample size will be divided into two sub-domains. These sub-domains are again further subdivided, which results in the new smaller sub-domains. The subdivision is recursively performed and therefore carried out in hierarchical manner. The subdivision stops when the sub-domain contains only one single realization. A general iterative scheme, denoted as Successive Bisection (SB) method, is suggested for the division of sub-domains. Computationally, such recursive bisection is represented as binary tree, where the ‘leaves’ represent close pairs of realizations. The division starts from choosing two fixed points in the larger sub-domain. Several pairs of points are randomly selected from the sub-domain. The pair with the largest distance is selected (see Fig. 2). These two points form the center of two smaller sub-domains. The rest of the points will be associated with either of these two center points according to the shorter distance to the center points.

This sub-domain division procedure can be formulated by a binary tree structure which shows a hierarchy of sub-domains (see Fig. 3(a)). A sub-domain in the hierarchy can be classified as a parent, a child, a unitary, or a clumpable sub-domain. This classification is established to make the clumping procedure more clearly understood. A sub-domain from which two smaller sub-domains derive is denoted a parent sub-domain. Each of the two smaller sub-domains is called child sub-domain. A unitary sub-domain has only one sample. The unitary sub-domains are also the lowest level sub-domains in the hierarchy of sub-domains. A clumpable sub-domain is a sub-domain which is built from two unitary sub-domains under the same parent sub-domain. The pairs to be clumped will be selected out of these clumpable sub-domains. Fig. 3(a) shows a typical hierarchy of sub-domains, from the level of all samples. i.e. samples domain, down to the level of single sample, i.e. unitary sub-domain.

Based on the distance definition, the dl, values of the pairs in all clumpable sub-domains are compared and then the pair with the smallest value will be selected. Mathematically, a clumping criterion selecting a pair (i, j) to be clumped can be expressed as

(i, j) = {i, jld, c A,,,, i Zj, k # I E [ 1, N]} (7)

A schematic diagram of the clumping procedure is shown in Fig. 3. The aforementioned binary tree structure comes to play an important role in this procedure. Fig. 3 shows that the information which is required at each

Fig. 2. Selection of two fixed points.

N. Hampomchai et al. I Comput. Methods Appl. Mech. Engrg. I68 (1999) 273-283

&,4

AS.6

A Clumpable Subdomain

- A Pair to Be

q A Unitary Subdomain Clumped

277

Fig. 3. Binary tree structure and successive clumping.

clumping is the distance measure A,,v at each clumpable sub-domain. At the same time, the information which is obtained from a particular clumping is the resulting sample UC(r) and its corresponding statistical weight w,(r). This whole information is stored when the binary tree structure is created. It can be seen that the selection of a pair to be clumped at each clumping can be accessed directly from the binary tree structure. Therefore, the clumping procedure is operated in an efficient manner with the aid of the binary tree structure.

2.3. Doubling procedure

After the clumping procedure is completed, the doubling procedure starts. Let UT(r) be a normalized realization resulting from the clumping procedure. Doubling UT(r) whose statistical weight is wT results in the two copies U 5 (7) and U z(r) with the weights equal to half of the original:

wt1(7) = w,J7) =; W?(T) (8)

Once the statistical weights wi,(r) and wi2(7) are determined from Eq. (8), Ui,(r) and U,*(r) will be calculated

278

empTG @J9G@“:*‘),w~2(7)J

(“,,d7),W,,b(7)) (u:(7), y’(7)) cu:, ( 71, w:,c 7)) h = l,...,n,

Fig. 4. Ih~blmg of a sample.

using a condition which is concerned with the statistics of the samples. It can be shown that clumping makes the statistics of the samples after clumping deviate from that before clumping (see e.g. 181). In this doubling procedure, doubling is designed to restore the first and second moment statistics which is the most crucial statistical information. More specifically, the mean and variance of the samples ZJ, , (7) and U,,(7) from doubling has to be equal to that of all n, samples of U,,,, 17 = 1, , n,, which are clumped to be Cl T(r), (see Fig. 4). Accordingly, the equivalent mean and variance in each component can be expressed explicitly for each kth component of the state vectors as follows:

Fig. 4 illustrates schematically doubling of a sample based upon the statistics constraints of the first two moments. Successive doubling will be performed on all the samples remaining after the clumping procedure.

2.4. Applicntinn of PSC Monte Carlo simulation technique

Since the proposed simulation procedure is based on the manipulation of the samples in the phase space, it is denoted as Phase-Space-Controlled (PSC) simulation technique [4]. Controlled MCS utilizing PSC is illustrated in Fig. 5. At a time step rr at which PSC is applied, the response samples Xj(rL), j = 1, . . . , N need to be obtained first from the MCS integration scheme. The set {X,(7,), j = 1, , N} is then used as input for the PSC procedure from which the output XT(rL) is obtained. Finally, the simulation continues by using the samples X~(T~) as an initial condition for integrating the stochastic differential equation for the following time interval (TV, rk+ ,]. Hence, the integration scheme is completely independent from the PSC procedure. The increase of samples in sparse regions implies that more samples will flow into the lower probability regions. To induce the most likely uniform distribution of simulated samples and to maintain the flow in the lower probability regions during the course of simulation, PSC needs to be applied at consecutive time steps.

I 1 AW7),7)

PSC BM7). 71 /

(X(7), 47))

Fig. 5. Controlled MCS utilizing PSC.

N. Hanzponzchui et ul. I Cornput. Methods Appl. Merh. Engrg. 168 (1999) 27.3-Z?.? 279

3. Numerical examples

In this section, the capability of PSC in increasing the efficiency of MCS will be demonstrated by two numerical examples. Both examples are structural dynamical systems of moderate dimension. To examine the applicability of PSC with respect to the type of dynamical nonlinearity, these examples are chosen to be different from each other. One system is a conservative dynamical system and the other is a dissipative hysteretic system.

.?. 1. A Dufjng type IO-DOA .system

A IO-story shear frame as shown in Fig. 6 is considered [I]. The equations of motion of the relative displacements x are derived as

9 + 2&w& .i + w:,E .(x -I cd)= Fax(t)

in which

E= 2 . . . . . . 0

. . . .

(12)

(13)

and a,(r) is the horizontal ground acceleration and is modelled as a zero-mean unit white noise with auto-spectral density 2nS,, = 1 multiplied by the following envelope function, h(t), which is identified from the North-South component of the 1940 El Centro earthquake records.

h(t) = 2.32[exp(-0.09t) - exp( 1.49t)l (14)

The system parameters are chosen as w. = 101~ s- ’ and & = 0.067. The nonlinear parameter E is chosen to be 8.0 which makes the behaviour of the system highly nonlinear. In this example, the CDF of the normalized relative displacement in the first, fifth and tenth story will be shown from time r = 5 to t = 8, which in fact represents the strong motion period. The normalized displacement U, in the ith floor is calculated from

4 q!,

*

Fig. 6. Ten-story shear frame under ground excitation.

280 N. Hunpornchui et al. I Compur. Methods Appl. Mrch. Engrg. 168 (1999) 273-283

X! - w u, = --g- I (15)

where x, is the displacement in the ith floor, ,LL, and cr, are the mean value and the standard deviation of x,, respectively. The results from MCS with PSC will be compared to those from direct MCS. Three thousand realizations are employed in case of MCS with PSC while 10 000 realizations are used in direct MCS.

As shown in Fig. 7, the results from MCS using PSC are in very good agreement with those from direct MCS. The capability of MCS is increased considerably by PSC because the low probability regions can be estimated by using the sample size of 30 times less than required by direct MCS.

3.2. A 6DOF Izysteretic sysretn

A 6-story shear frame structure under base excitation is considered [lo]. In this example elasto-plastic steel properties are assumed for the material properties. The equation of motions for the floor-displacement x can be obtained as in which

M-i +C.X +R.z=G+a,(t)

where the non-zero coefficients of the structural matrices are defined as follows,

(16)

m,, =m,=81 500kg; c,, = O.l7847m, + O.O039155(KT + K,?. , ) ;

c ,.!+I = c,,,,, = -O.O039155K~+, ; r,, = KY : r ,,,+ I = -KI*,, ; g,, = --*I, (17)

IO-DOF DUFFJNG OSCILLATOR IO-DOF DUFFING OSCILLATOR

B

0.1

0.01

0.001

O.oool

1.305

146 L I

B

1

0.1

0.01

0.001

O.ooOl

k-05

le-04

MCS - psc .._..

- -4 -2

standariized x.5 2 4 -4 -3 -2 -I 0

Standardized xl1 2 3 4

B u

(4 1

0.1

0.01

0.001

O.oool

le-05

k-06

IO-DOF DUFFING OSCILLATOR

-10 -5 0 5 10 Standardized x 10

(4

(b)

Fig. 7. Cumulative distribution function of the standardized relative displacement at different time instances, I = S.O-8.0 s. (a) First floor; (b) fifth floor; (c) tenth floor.

N. Hampomchai et al. I Cornput. Methods Appl. Mech. Engrg. 168 (1999) 273-28-T 2x1

Table 1 System parameters for the 6-DOF hysteretic system

Floor i l-1

Stiffness

KT [lOh N/m]

Mass

‘n, [IO” kg]

Yield hysteretic displacement

L Iml

Plastic hysteretic displacement 7 .‘,” [ml

1 24.92 0.08 1s 0.00644 0.01073 2 25.20 0.08 IS 0.00672 0.01 194 3 21.36 0.08 15 0.00794 0.01222 4 21.52 0.08 I5 0.00958 0.01532 5 17.64 0.08 15 0.01020 0.0 1569 6 17.93 0.0815 0.01255 0.0 1725

specified by the quantities in Table 1. The P - il effect as well as the hysteretic behaviour of the steel columns are taken into account. The stiffness K f, therefore, includes the P - A effect. The constitutive law is given by the following nonlinear differential equation

i,(f) = u,(t)-h(z,, u’,) (18)

h(zf’ ‘,) = t-,. - lzil for I_ / > z,,, and u’;z, > 0 z -z &I ,” \I

(19)

where v, =x, -x,-, and ti, are the relative displacement and the relative velocity of the ith floor, respectively. The system parameters are listed in Table 1. The base excitation u,(f) is modelled as a filtered white noise process defined by

n,(r) = j(f) (20)

The filter equation is given by

where 5, and wK rad/s reflects the soil properties. The quantity w(t) represents Gaussian white noise process with auto-correlation function RW.,,(r) = 6(r) and e(t) is a modulation function given as e(r) = 1.0. [exp(-0.2%) - exp(-OSt)].

Since the accumulated plastic deformation (drift) is a primary source of damage which can lead to the failure of the structural system, the probabilistic information about the plastic deformation will be considered in this example. The modelling of the accumulation of the drift is shown in Fig. 8. It is assumed that the structural elements, e.g. columns fail when the drift exceeds a certain critical level. This critical level is expressed by the ratio x of the plastic displacement d,,,,, to the onset of yielding displacement z,, for the ith floor, i.e.

From the modulation function, the strong motion period takes place from t = 4.0 to t = 8.0 s. Since the damage is a monotonically increasing function of time and x is critical in the first and fifth floors, the CDF of 31 and ‘ys will be examined at t = 8.0 s, i.e. at the end of the strong motion period.

The estimates of the low probabilities regions for y, and y5 are obtained from MCS using PSC, with a sample size of 3000. The results are compared to those obtained from direct MCS, using a sample size of 100 000. One can see that MCS using PSC gives satisfactory results, as shown in Fig. 9. The efficiency of MCS using PSC is much higher than that of direct MCS because the estimates of the low probability regions can be obtained by using 33-time less number of samples, i.e. 3000 instead of 100 000.

et 01. I Cornput. Metho& Appl. Mech. Engrg. 168 (1999) 27.3-283

“I “2

t 4

zPi 2. --------

------ / // ---- d,,i =&+Cq -z,i k I

/ //

Fig. 8. Six-story hysteretic base-excited structure and accumulated plastic drift

1-D HYSTERETIC 6-DOF

0.1

0.01

B 0.001

k-06 3 0 I 2 3 4 5 6

gl

(4

1-D HYSTERETIC 6.DOF

04 Fig. 9. Cumulative distribution function of ‘): (a) y,; (b) yi at t = 8.0 s.

4. Concluding remarks

According to the proposed simulation technique, the following conclusions can be drawn. l A concept of controlled Monte Carlo Simulation techniques for the purpose of reliability analysis of

stochastic dynamical systems is established. The developed simulation technique is a MCS-based method, i.e. utilized in conjunction with the MCS procedure. Similar to D&C, this simulation technique consists of two main procedures, i.e. clumping and doubling. However, a new clumping criterion is proposed and used instead of the energy criterion. The clumping criterion is established from the geometrical aspect of the samples distribution in the phase space. With this new geometrical criterion, the realization of the concept-denoted as the Phase-Space-Controlled simulation technique (PSC)-is applicable to a wider class of dynamical systems, i.e. conservative and non-conservative systems.

l With its simple procedures, PSC can increase the efficiency of MCS in estimating low probability regions for general larger dynamical systems. The numerical examples show the potential of PSC to treat complex and large dynamical systems.

l With the same sample size. the efficiency of MC3 using PSC proves to be much higher than that of direct

N. Harnpotnchai et al. I Comput. Methods Appl. Mech. Engrg. 168 (1999) 27%283 283

MCS. In other words, with the same number of samples, MCS using PSC can access lower probability regions than direct MCS does. As a consequence, the low probability regions can be estimated by using feasible numbers of samples.

l Compared to direct MCS, MCS using PSC consumes much less computational efforts in order to predict the same magnitude of low probability regions.

Acknowledgments

This research is partially supported by the Austrian Exchange Service (OAD) through the North-South Dialog Scholarship Program responsible for the first author and the Austrian Research Council (FWF) under contract P11498-MAT which are gratefully acknowledged by the authors.

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