Solution of the First-Passage Problem by Advanced Monte Carlo Simulation Technique
A selective Monte Carlo simulation procedure
 for the evaluation of the reliability of nonlinear
 structures subjected to dynamic loading is presented.
 The proposed “Russian Roulette and
 Splitting” procedure allows generation of important
 low-probability sa...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2003
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| citation_txt | Solution of the First-Passage Problem by Advanced Monte Carlo
 Simulation Technique / M. Labou // Проблемы прочности. — 2003. — № 6. — С. 67-74. — Бібліогр.: 5 назв. — англ. |
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| description | A selective Monte Carlo simulation procedure
for the evaluation of the reliability of nonlinear
structures subjected to dynamic loading is presented.
The proposed “Russian Roulette and
Splitting” procedure allows generation of important
low-probability samples in nonlinear dynamics.
This procedure is quite efficient in
comparison to the straightforward Monte Carlo
simulation method and allows evaluation of the
nonlinear stochastic response in a very low
probability domain.
Предложен модифицированный численный метод Монте-Карло для оценки надежности
нелинейных конструкций, подвергнутых динамическому нагружению. Использование метода
позволяет обобщить важные низковероятностные выборки для нелинейных динамических
задач. Эффективность данного метода по сравнению с прямом методом Монте-Карло
состоит в возможности определения нелинейного стохастического отклика в области
очень низких вероятностей.
Запропоновано модифікований числовий метод Монте-Карло для оцінки
надійності нелінійних конструкцій під дією динамічного навантаження.
Метод дозволяє узагальнити важливі низькоімовірнісні вибірки для нелінійних
динамічних задач. Ефективність даного методу в порівнянні з прямим
методом Монте-Карло полягає в можливості визначення нелінійного сто-
хастичного відгуку в області дуже низьких імовірностей.
|
| first_indexed | 2025-12-07T16:45:01Z |
| format | Article |
| fulltext |
UDC 539.4
Solution of the First-Passage Problem by Advanced Monte Carlo
Simulation Technique
M. Labou
Institute of Structural Mechanics, Kiev State University of Construction and Architecture,
Kiev, Ukraine
УДК 539.4
Решение задачи “первого вы броса” модифицированным
численным методом Монте-Карло
М. Лабу
Институт строительной механики при Киевском национальном университете
строительства и архитектуры, Киев, Украина
Предложен модифицированный численный метод Монте-Карло для оценки надежности
нелинейных конструкций, подвергнутых динамическому нагружению. Использование метода
позволяет обобщить важные низковероятностные выборки для нелинейных динамических
задач. Эффективность данного метода по сравнению с прямом методом Монте-Карло
состоит в возможности определения нелинейного стохастического отклика в области
очень низких вероятностей.
Клю чевы е слова : динамическое нагружение, нелинейная динамическая
задача, метод Монте-Карло.
In tro d u c tio n . In a recent review o f methods currently available to analyze
multidegree-of-freedom systems under stochastic loading, their merits, limitations
and potential to be used in engineering practice have been discussed [1]. From
this study it becomes clear that only num erical procedures such as Monte Carlo
simulation (MCS) techniques (both in their direct and variance reduction form)
and the Response Surface M ethod (RSM) are capable o f treating nonlinearities
without restriction as well as systems o f higher dimension including reliable
information on the response distribution tails.
As direct MCS procedures are quite lim ited with respect to providing
accurate information on the response distribution tails, the variance reduction
techniques such as importance, directional, adaptive sampling, etc. are used for
static structural problems in order to guide the simulation procedures to the region
o f interest and, hence, to reduce the numerical effort. For dynamic problems, the
selection o f this region is more involved than for static problems, since the
importance measure is generally time variant. Based on a suitable criterion for
indicating which realization will m ost likely lead to failure, the samples in the
region o f interest m ay be split, whereas in the region o f less interest they m ay be
discarded or m ay survive. This procedure has been expanded and applied
successfully to M DOF systems.
© M. LABOU, 2003
ISSN 0556-171X. Проблемы прочности, 2003, № 6 67
M. Labou
1. F irst-P assage D istribu tions. A general performance measure for
structures under stochastic dynamic loading is characterized by first-passage
probabilities, where the distribution o f the exit times T into a state o f failure is
considered. The boundary separating the safe state from the unsafe one m ight be
specified by a so-called limit state function g (x ) = 0. The lifetime o f the structure
is then defined as the first exit time T :
T = in f [t: g ( X ( t )) < 0, t > 0], (1)
and the probability o f the first excursion P f ( t ) within the time interval t is
defined as
Pf ( t ) = P[T < t ], (2)
specifying the evolution o f the failure probability w ith respect to time.
Presently, the available solutions to the above first probability are quite rare.
There is a complete absence o f analytical exact solutions in closed form, and
numerical procedures based on the Fokker-Planck equation are confined to
problems o f low dimension (< 4 in the state space). The only solution technique
at hand seems to be the MCS, which is well suited to simulate nonlinear
stochastic structural responses as required in the reliability assessment o f
structures.
2. M eth o d of A nalysis. For the purpose o f assessing the reliability o f a
structure, the domain o f the response is divided into the safe and unsafe parts,
respectively. The so-called limit state function (LSF) serves as a criterion of
separation. The domain is in general m ultidim ensional containing parameters
such as displacements, velocities, plastic deformations, etc. I f x is defined as the
state vector o f all the response components and g (x ) as the LSF, it is necessary
to determine the probability P ( t ) that the response vector x( t ) exceeds the LSF
at least once within a given time period [0, t], where x(0) lies initially in the safe
domain, i.e., P (0) = 0. In other words, the first-passage probability defines the
evolution o f the failure probability w ith respect to time.
It is well known, that the direct MCS is well suited to obtain, w ith a
relatively small num ber o f realizations, reasonably close estimates for the m ean
vector and the covariance matrix. It is not suited, however, to assess the low
probability domain o f the stochastic response and for reliability analysis where_o _4
failure probabilities P f are o f the order 10 _ 10 . I n order to access such
ranges, it is obvious that a sample size (number o f realizations) o f the order
105 _ 1 0 9 will be required. Such sample sizes are possible for small systems
using a m assively-parallel supercomputer [2]. However, the MCS with a sample
size o f the order > 105 are expensive and certainly not efficient in view o f the
fact that only a small fraction o f the num ber o f realizations falls into the domain
o f interest for the reliability analysis.
This drawback o f the direct MCS has been commonly recognized, and the
so-called variance reduction techniques have been developed and utilized
successfully. All these procedures increase the density o f realizations in the
region o f interest, i.e., in the region that contributes m ost to the failure
68 ISSN 0556-171X. n p o 6n.eubi npounocmu, 2003, N 6
Solution o f the First-Passage Problem
probability. These variance reduction methods are used m ainly for nonlinear static
reliability systems, but can be also used for nonlinear dynamic problems. The
main difficulty with these approaches is the fast growing num erical effort w ith the
num ber o f random variables involved. In case loading is represented by a
stochastic process, it m ight be possible in some cases to represent the stochastic
process by a small num ber o f random variables using the Karhunen Loeve
expansion.
M onte Carlo simulation, however, is applicable to stochastic loading
involving more than ju st a small set o f random variables. In such cases, the
traditional variance reduction procedures are no longer applicable.
Advanced M CS methods have been developed for the class o f systems with
M arkovian properties. Before describing the details and im plementation o f such
procedures, we shall briefly discuss the m ain features o f the advanced MCS.
Consider first the M CS and the resulting approximation for the cumulative
distribution function CDF (x ):
N
CDF (x , t ) = 2 1 [ x n ( t ),x ] w n ( t ), (3)
n=1
where x is the state vector, t is time, x n is the state vector o f the nth
realization, w n ( t ) is the w eight and discrete probability o f the nth realization at
time t, and N is the sample size. The indicator function I [X n ( t ),x ] takes the
value 1 i f the components X n k > x k , k = 1,..., M , for all components M o f the
state vector. Otherwise, the indicator function assumes the value 0. In case the
direct MCS is applied, all weights are constant and assume the value w n ( t ) =
= 1/N . For example, for importance sampling, the original probability density
function f (x ), which reflects the direct MCS, is m odified by using the sampling
distribution h(x ). This can be done without violating the original density function
f (x ) by m odify ing the w eight w n o f all realizations X n ( t ) by w n =
= f (x ) / ( Nh(x )). Since the advanced MCS is designed to assess low probability
_8 _4
ranges o f the order 10 P f < 10 as well, it is obvious that m anipulation of
the weights is indispensable.
A selective M onte Carlo simulation technique, namely, “Russian Roulette &
Splitting” simulation technique (RR&S) has been applied successfully in nuclear
physics [3] to solve neutron transport problems. The basic features o f the
algorithm are described and discussed.
The RR&S procedure involves subdivision o f the safe region by several
borders o f splitting (or sub-barriers). Then the process o f splitting can be
introduced as follows. W hen the response sample function x ( t ) upcrosses the ith
sub-barrier, the state vector is split into mt same vectors, where m is the integer
num ber and i is the num ber o f the sub-barrier. The weight o f each new vector is
equal to w new = w ĉid /m i . Due to the m i stochastically independent sample
functions produced by splitting, one simply increases the chances to upcross the
next (more highly placed) sub-barrier. Thus the splitting technique makes it
possible to increase significantly the num ber o f samples which are capable o f
crossing the limit state function.
ISSN 0556-171X. npoôëeMbi npounocmu, 2003, N2 6 69
M. Labou
The Russian Roulette is a well-known game o f chance where the
‘probability o f death’ is high (i.e., 1/6, originally). This game can be played to
reduce the sample size to a required amount. The “Russian Roulette” technique is
applied in an inverse situation, i.e., when the response sample function
downcrosses the ith sub-barrier. In this case with a probability o f 1 — 1 / , the
simulation process is discarded (killed) and w ith a probability o f 1/mt the
simulation is continued. The weight o f such surviving sample is m ultiplied by mi .
Hence, the main aim o f the “Russian Roulette” technique is to diminish
drastically the num ber o f sample functions which are o f “low interest” in the first
passage problem.
From the algorithm described above it becomes obvious that between the ith
and ( i + 1)th sub-barriers the weights o f the samples are equal to each other and
are expressed as
- 1 n -Wi - - n — , (4)n .m , w
] - 1 ]
where n is the num ber o f initially simulated samples and —] is the num ber of
the splitting on the j th sub-barrier.
It is clear that the weights o f the samples Wb after outcrossing the barrier
will be equal to
k
w b - 1 n (5)
j -1 ]
where k is the total num ber o f sub-barriers.
It is noteworthy that the direct Monte Carlo algorithm is a subset o f the
RR&S technique. In particular, the direct MCS can be obtained from the RR&S
for mt - 1 [4].
3. D e te rm ination of the Effectiveness o f the S im ulation Technique.
According to Rubinstein [5], the effectiveness E o f each simulation technique
can be expressed in the following form:
1
E -
v ar[P ]T
(6)
where var [P ] is the variance o f the estimate and T is the computation time of
the estimate P * .
The Bernoulli scheme o f independent trials in the case o f straightforward
MCS allows one to obtain the following well-known formula for the variance of
the estimate:
* P(1 — P )
var [p ] - \ ' , (7)
where N is the sample size.
70 ISSN 0556-171X. n p o 6n.eubi npounocmu, 2003, N2 6
Solution o f the First-Passage Problem
Unfortunately, due to the correlation between samples in the RR&S
technique, there is a difficulty in obtaining an expression for the estimate variance
similar to (7).
Based on the sample statistics, the variance o f estimate for the RR&S
technique can be calculated by the following formula:
* 1 ^ ( li *
v a r [P ] = —-------t > | — - P
n( n — 1) “ V M
\2
/
(8)
where
* 1
P = - >n M
1=1
(9)
k
M = П mj ,
j =1
n is the sample size, lt is the num ber o f barrier outcrossings in the ith
independent trial, and mj is the number o f the splitting on the j th sub-barrier.
As long as variance (7) has already been minimized for a given sample size
N by the straightforward Monte Carlo simulation, a more effective solution can
be obtained using another simulation scheme (8) with a higher variance o f the
estimate and m uch shorter computation time.
4. N um erical Exam ples. Nonlinear vibrations o f a cylindrical panel under
the action o f a uniformly distributed load are analyzed; the time history is
described by a truncated white noise. It is considered that at the initial instant of
time the cylindrical panel is in a quiescent mode. The action o f the applied load
triggers the excitation o f a nonstationary stochastic process.
The following parameters o f the panel are adopted: the dimensions in plan
a = 0.2 m, b = 0.3 m, the height h = 0.002 m, the modulus o f elasticity
3 3E = 70.63 GPa, the density o f the material p = 2.7-10 kg/m , Poisson’s ratio
v = 1/3, and radius o f curvature R = 0.8363 m.
In the problem under study, the frequency band o f the spectral density o f the
loading covers the first, fifth, seventh, and eleventh symmetrical vibration modes.
The equation o f motion is governed by the following differential equation:
x i + C tx i + K ix i + K f x j x k + K f lx j x kxi = ц i Q i ( t)
( 10)
( i, j , k , l = 1 ,..., 4),
where C j are factors o f the damping matrix, p t are the oscillation modes, Q ( t )
is the Gaussian white noise with intensity I = 1000 m 2/s3, and K ijk and K ijkl
( i, j , k , l = 1 ,..., 4) are, respectively, the coefficients o f matrices o f square and
cubic nonlinearity.
ISSN 0556-171X. Проблеми прочности, 2003, № 6 71
M. Labou
2 2The following input data are considered: K 1 = m 1 = 1.00, K 2 = m 2 = 7.64,
K 3 = m 2 = 11.2, K 4 = m 2 = 22.00, C 1 = 0.0159, C 2 = 0.044, C 3 = 0.0532,
C 4 = 0.0747, / i 1 = - 0.0297, ^ 2 = 0.0095, ^ 3 = 0.00979, ^ 4 = 0.00342.
The flexures in the center o f the cylindrical panel are determined by the
formula
4
x c ( 0 = h 2 x i ( t ) . (11)
Z=1
The first passage probabilities o f the system response x c ( t ) were
determined during time [0, t] for double-sided symmetrical barriers located at the
levels ± 3h , ± 4h, and ± 5h and shown in Fig. 1.
xc (t), mm
-1500
t, s t, s
Fig. 1. Sample white noise function Q (t) - a and the system response variation xc (t) - b.
A t the beginning, the first passage probability for symmetrical barriers of
level ± 3h was determined.
Figure 1a shows one realization o f the stochastic process Q ( t ). The
corresponding variation o f the system response x c( t) is presented in Fig. 1b.
This trajectory reaches the limiting barrier at t = 4 s. In the same figure, the
arrangement o f borders o f splitting and the numbers o f the splitting are shown. In
this case, 6 borders o f splitting were located at ±0.15 cm, ±0.3 cm, and ±0.45 cm;
the num ber o f the splitting at each o f the borders are mi = m2 = m3 = 7; the
sample size n = 1000.
Figure 2 (curve a ) presents in a logarithmic scale the cumulative distribution
function o f the first passage probabilities o f the system response x c ( t ) for
barriers located at the level ± 3h. From this figure one can see that at t < 4 s the
probability o f reaching the barrier sharply decreases. The appropriate events are
rather rare.
We also analyzed the problems on the first passage probability o f the system
response x c ( t ) for barriers located at the levels ± 4h and ± 5h. The appropriate
cumulative distribution functions are shown in Fig. 2 (curves b and c,
respectively).
72 ISSN 0556-171X. npoôëeubi npounocmu, 2003, N2 6
Solution o f the First-Passage Problem
T a b l e 1
The Effectiveness and Calculation Time Ratios - MCS to RR&S
Time (s) err &s /EMCS TMCS TRR &S
2.22 169.00 169.0
2.50 4.20 149.0
3.00 1.10 83.0
4.00 0.40 38.0
5.00 0.17 22.4
t, s t, s
Fig. 2 Fig. 3
Fig. 2. Estimates of the first passage probabilities versus time for the double-sided barriers located
at a = ±3h, b =±h, c = ±5h.
Fig. 3. Estimates of the first-passage probabilities and variances of estimates for barriers located at
± 5 h.
In the second problem (barriers at the level ± 4h) 6 borders o f splitting were
located at distances ±0.2 cm, ±0.4 cm, and ±0.6 cm, the numbers o f the splitting
at each o f the borders are mi = m2 = m3 = 7; the cumulative distribution function
for this problem is shown in Fig. 2 (curve b).
In the third problem (barriers at the level ± 5h) 6 borders o f splitting were
located at distances ±0.25 cm, ±0.5 cm, and ±0.75 cm, the num ber o f the splitting
at each o f the borders are mi = m2 = m3 = 7, the appropriate cumulative
distribution function is shown in Fig. 2 (curve c). The sample size n in the
second and third problem equals to 1 0 0 0 .
*
For the barriers located at ± 5h, the variance o f estimates P was calculated
for different techniques. Figure 3 (curve a) represents cumulative distribution
functions obtained successively by the straightforward MCS (dotted lines), and
the RR&S procedure (solid lines) (see Fig. 2, curve c).
Figure 3 (curve b) represents the variance o f estimates P * , obtained w ith the
help o f formula (8); and curve (c) obtained using formula (7). The ratio o f the
effectiveness Err&s / E MCs calculated for different times t in accordance with
(6 ) is presented in Table 1 together w ith the ratio o f the calculation times
tmcs / trr &s ■
ISSN 0556-171X. Проблемы прочности, 2003, № 6 73
M. Labou
From Table 1 it is seen that the RR&S simulation technique offers a
considerable gain in computation time and a higher effectiveness (in the first 3 s)
as compared to the direct MCS technique.
Conclusion. A n advanced M onte Carlo simulation procedure has been
applied to M DOF systems showing its basic applicability to higher dimensional
problems and the ability to assess the low probability range in terms o f the
first-passage probabilities.
Р е з ю м е
Запропоновано модифікований числовий метод М онте-Карло для оцінки
надійності нелінійних конструкцій під дією динамічного навантаження.
М етод дозволяє узагальнити важливі низькоімовірнісні вибірки для неліній
них динамічних задач. Ефективність даного методу в порівнянні з прямим
методом М онте-Карло полягає в можливості визначення нелінійного сто-
хастичного відгуку в області дуже низьких імовірностей.
1. G. I. Schuller, H. J. Pradlwarter, and M. D. Pandey, “M ethods for reliability
o f n o n lin ear system s u n d er stochastic load ing - a rev iew ,” Proc.
EUROD Y N ’93, Balkema (1993) pp. 751-759.
2. E. A Johnson E. A. and L. A. Bergmann, “M onte Carlo simulation of
dynamic systems o f engineering interest in a m assively parallel computing
environment,” in: Proc. IUTAM Symp. “Advances in Nonlinear Stochastic
Mechanics,” (Trondheim, Norway, July 3 -7 , 1995), Kluwer Academic
Publishers, Dordrecht, the Netherlands (1996), pp. 225-234.
3. J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron
Transport Problems, Addison W esley Publishing Company (1969).
4. P. G. M el’nik-M el’nikov, E. S. Dekhtyaruk, and M. Labou, “Application of
the “Russian Roulette and Splitting” simulation technique for the reliability
assessment o f m echanical system s,” Probl. Prochn., No. 3, 131-135 (1997).
5. R. Y. Rubinstein, Simulation and the Monte Carlo M ethod , W iley, N ew
Y ork (1981).
Received 25. 09. 2003
74 ISSN 0556-171X. Проблеми прочности, 2003, № 6
|
| id | nasplib_isofts_kiev_ua-123456789-47022 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T16:45:01Z |
| publishDate | 2003 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Labou, M. 2013-07-08T16:13:55Z 2013-07-08T16:13:55Z 2003 Solution of the First-Passage Problem by Advanced Monte Carlo
 Simulation Technique / M. Labou // Проблемы прочности. — 2003. — № 6. — С. 67-74. — Бібліогр.: 5 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/47022 539.4 A selective Monte Carlo simulation procedure
 for the evaluation of the reliability of nonlinear
 structures subjected to dynamic loading is presented.
 The proposed “Russian Roulette and
 Splitting” procedure allows generation of important
 low-probability samples in nonlinear dynamics.
 This procedure is quite efficient in
 comparison to the straightforward Monte Carlo
 simulation method and allows evaluation of the
 nonlinear stochastic response in a very low
 probability domain. Предложен модифицированный численный метод Монте-Карло для оценки надежности
 нелинейных конструкций, подвергнутых динамическому нагружению. Использование метода
 позволяет обобщить важные низковероятностные выборки для нелинейных динамических
 задач. Эффективность данного метода по сравнению с прямом методом Монте-Карло
 состоит в возможности определения нелинейного стохастического отклика в области
 очень низких вероятностей. Запропоновано модифікований числовий метод Монте-Карло для оцінки
 надійності нелінійних конструкцій під дією динамічного навантаження.
 Метод дозволяє узагальнити важливі низькоімовірнісні вибірки для нелінійних
 динамічних задач. Ефективність даного методу в порівнянні з прямим
 методом Монте-Карло полягає в можливості визначення нелінійного сто-
 хастичного відгуку в області дуже низьких імовірностей. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Solution of the First-Passage Problem by Advanced Monte Carlo Simulation Technique Решение задачи “первого выброса” модифицированным численным методом Монте-Карло Article published earlier |
| spellingShingle | Solution of the First-Passage Problem by Advanced Monte Carlo Simulation Technique Labou, M. Научно-технический раздел |
| title | Solution of the First-Passage Problem by Advanced Monte Carlo Simulation Technique |
| title_alt | Решение задачи “первого выброса” модифицированным численным методом Монте-Карло |
| title_full | Solution of the First-Passage Problem by Advanced Monte Carlo Simulation Technique |
| title_fullStr | Solution of the First-Passage Problem by Advanced Monte Carlo Simulation Technique |
| title_full_unstemmed | Solution of the First-Passage Problem by Advanced Monte Carlo Simulation Technique |
| title_short | Solution of the First-Passage Problem by Advanced Monte Carlo Simulation Technique |
| title_sort | solution of the first-passage problem by advanced monte carlo simulation technique |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/47022 |
| work_keys_str_mv | AT laboum solutionofthefirstpassageproblembyadvancedmontecarlosimulationtechnique AT laboum rešeniezadačipervogovybrosamodificirovannymčislennymmetodommontekarlo |