simulation of high variable random processes through the spectral- representation-based approach
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Simulation of high variable random processes through the spectral- representation-based approach. Pierfrancesco Cacciola. Senior Lecturer in Civil Engineering ( Structural Design ) - PowerPoint PPT PresentationTRANSCRIPT
Pierfrancesco CacciolaSenior Lecturer in Civil Engineering ( Structural Design )School of Environment and Technology, University of Brighton, Cockcroft building, Lewes Road, BN2 4GJ, Brighton, UK
Simulation of high variable random processes through the spectral- representation-based approach
Outline
Simulation of random processes via the spectral representation method
Enhancing the variability of the spectral representation method
Butterworth filter Numerical results Concluding Remarks
• Consider the zero mean one-dimensional and uni-variate Gaussian non-stationary stochastic process defined as
Simulation of random processes via the spectral representation method
( ) 0,E f t
( )f t
• Evolutionary power spectral density function defined (Priestley, 1965)
( , ) ( ) ( )R t t E f t f t
2
2 1( , ) ( , ) ( ) ( )
t Ti
t
S t A t S E f e dT
( , )t t T T t• Where with is a small interval
• the ensemble average in equation is not commonly used to define the evolutionary spectrum due the difficulties in its numerical evaluation related to the Uncertainty Principle. Therefore indirect representation
Simulation of random processes via the spectral representation method
• As a consequence
1( , ) ( , ) ;
2iS t R t t e d
( , ) ( , ) ( , ) ( ) ;iR t t A t A t S e d
• Stationary case: Wiener-Khintchin relationships
Simulation of random processes via the spectral representation method
( ) ( ) ;iR S e d
1( ) ( ) ;
2iS R e d
• For the stationary case the power spectral density function can be also determined directly from experimental data
Simulation of random processes via the spectral representation method
2/2
/2
1( ) lim ( )
2
Ti
TT
S E f e dT
• Once defined the power spectral density function
either through experimental or physical/theoretical approaches the simulation of the sample of the non-stationary random process through the spectral representation method is performed using the following equation (Shinozuka and Deodatis 1988, Deodatis 1996)
1
( ) 2 2 ( , ) cosN
j j jj
f t S t t
• The simulated process is asymptotically Gaussian as N tend to infinity due to the Central Limit Theorem
• The ensemble averaged mean and correlations tends to the target
BUT
Simulation of random processes via the spectral representation method
• The variability of the energy distribution/Fourier spectra is not controlled
Enhancing the variability of the spectral representation method: Recorded Earthquakes in Messina
2
001
(19 / 04 / 2005)
3.2
6.44 /g
AQ
M
a cm sec
time [sec]0 5 10 15 20
a g [cm
/sec2 ]
-4
0
4
2
263
(10 / 06 / 2006)
3.2
6.74 /g
BL
M
a cm sec
time [sec]
a g [cm
/sec2 ]
0 10 20 30-8
-4
0
4
8
2
370
(27 / 02 / 2007)
4.1
7.59 /g
BK
M
a cm sec
0 10 20 30 40time [sec]
a g [cm
/sec2 ]
-10
010
2
266
(26 /10 / 2006)
5.7
14.05 /g
BL
M
a cm sec
Enhancing the variability of the spectral representation method: Recorded Earthquakes in Messina
5 10 15 20
3 2 1
12
5 10 15 20 25 30 35
2
1
1
2
1 2 3 4 5
3
2
1
1
2
5 10 15 20 1
1
2
2 4 6 8 10 12
4 2
246
5 10 15 20
6 4 2
24
1 2 3 4 5 6
10 5
510
15
2 4 6 8 10
2
1
1
2
10 20 30 40 50
4
2
2
4
(1)
2
( )
[ / ]
f t
cm s
[ ]t s [ ]t s[ ]t s
[ ]t s
[ ]t s
[ ]t s
[ ]t s
[ ]t s
[ ]t s
(1)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
Enhancing the variability of the spectral representation method: Recorded Earthquakes in Messina
50 100 150 200
0 .005
0 .010
0 .015
0 .020
0 .025
50 100 150 200
0 .005
0 .010
0 .015
0 .020
0 .025
50 100 150 200
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0 .0010
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0 .0025
0 .0030
0 .0035
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0 .0010
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0 .015
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50 100 150 200
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0 .0010
0 .0015
0 .0020
0 .0025
50 100 150 200
0 .01
0 .02
0 .03
0 .04
0 .05
0 .06
0 .07
2(1)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
Enhancing the variability of the spectral representation method: Recorded Earthquakes in Messina
50 100 150 200
0 .005
0 .010
0 .015
0 .020
0 .025
50 100 150 200
0 .005
0 .010
0 .015
0 .020
0 .025
50 100 150 200
0 .0005
0 .0010
0 .0015
0 .0020
0 .0025
0 .0030
0 .0035
50 100 150 200
0 .0005
0 .0010
0 .0015
0 .0020
0 .0025
0 .0030
0 .0035
50 100 150 200
0 .005
0 .010
0 .015
0 .020
0 .025
0 .030
50 100 150 200
0 .02
0 .04
0 .06
0 .08
50 100 150 200
0 .02
0 .04
0 .06
0 .08
0 .10
0 .12
50 100 150 200
0 .0005
0 .0010
0 .0015
0 .0020
0 .0025
50 100 150 200
0 .01
0 .02
0 .03
0 .04
0 .05
0 .06
0 .07
2/2
/2
1( ) lim ( )
2
Ti
TT
S E f e dT
50 100 150 200
0 .005
0 .010
0 .015
( )S
1
( ) 2 2 ( , ) cosN
j j jj
f t S t t
2(1)
2 3
( )
[ / ]
F
cm s
• To enhance the variability of the simulated random samples in this paper it is proposed to introduce a random filter acting in series with the expected value of the power spectrum
Enhancing the variability of the spectral representation method
is real positive function that satisfies the following equation
( , , ) ( , ) ( , )S t H S t
is the vector collecting the random parameters of the filter
( , )H
( , , ) ( )d ( , ) ( , ) ( )d ( , )A A
A A
S t p H S t p S t
( )Ap is the joint probability density function of random parameter of the filter
• Embedding the proposed random spectrum in the traditional spectral representation method (SRM) the following simulation formula is derived
Enhancing the variability of the spectral representation method
The samples generated by equation are Gaussian as N tends to infinity due to the Central Limit Theorem and converge to the target mean and correlation function (proved in the paper)
1
( ) 2 2 ( , ) ( , ) cosN
H j j j jj
f t H S t t
can be determined considering the distribution of the energy around the expected vale for each frequency (practically unfeasible).
Alternative strategy is to consider synthetic parameters defining the variability of the energy distribution such as the bandwidths and central frequency.
To this aim the following pass-band Butterworth filter will be adopted
Butterworth filter
( , )H
1 2 2
21
1
1( , , )
2 1
B jH
Butterworth filter
1 2 2
21
1
1( , , )
2 1
B jH
0 20 40 60 80 1000
0.010.020.030.04
0 20 40 60 80 1000
0.02
0.04
0.06
2j 4 8
[ / ]rad s [ / ]rad s
1 10 20 40
( , )H
• The distribution of the filter parameters and can be defined through experimental data measuring the central frequency and bandwidth of the squared Fourier spectrum of the recorded samples.
Butterworth filter
• to illustrative purpose the filter parameters will be assumed statistical independent and uniformly distributed. Therefore,
1 21 21 1 2 2
1 1( ) ( ) ( )A A A
u l u l
p p p
1 21
( ) 2 2 ( ) ( , , ) ( , ) cosN
H B j j j jj
f t C H S t t
1 2
1 2
1
1 2 1 21 1 2 2
1 1( ) ( , , )d d
u u
l l
Bu l u l
C H
Therefore,
with
Numerical results
• Stationary case: Kanai-Tajimi spectrum - simulated samples : a) Traditional SRM; b) proposed EV-SRM
-200
0
200
-200
0
200
0 10 20 30
-200
0
200
-200
0
200
-200
0
200
0 10 20 30
-200
0
200
(1)
2
( )
[ / ]
Hf t
cm s
(2)
2
( )
[ / ]
f t
cm s
(3)
2
( )
[ / ]
Hf t
cm s
[ ]t s
(2)
2
( )
[ / ]
Hf t
cm s
(3)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
[ ]t s
Numerical results
• Stationary case: Kanai-Tajimi spectrum – Fourier transform of the simulated samples :a) Traditional SRM; b) proposed EV-SRM
0100200300400
0
100
200
300
0 20 40 60 80 1000
100200300400
0200400600800
040080012001600
0 20 40 60 80 1000
200
400
600
[ / ]rad s
)a )b
[ / ]rad s
2(2)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
HF
cm s
2(3)
2 3
( )
[ / ]
HF
cm s
2(2)
2 3
( )
[ / ]
HF
cm s
2(3)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
Numerical results
• Stationary case: Kanai-Tajimi spectrum – proof of convergence
[ / ]rad s
0 20 40 60 80 1000
50
100
150
200
250
2 3
( )
[ / ]
S
cm s
Numerical results
• Non-Stationary case: Evolutionary Kanai-Tajimi spectrum - simulated samples : a) Traditional SRM; b) proposed EV-SRM
-200-1000
100200
-200
0
200
0 10 20 30
-200
0
200
-200
0
200
-200
0
200
0 10 20 30
-200
0
200
(1)
2
( )
[ / ]
Hf t
cm s
(2)
2
( )
[ / ]
f t
cm s
(3)
2
( )
[ / ]
Hf t
cm s
[ ]t s
)b)a
(2)
2
( )
[ / ]
Hf t
cm s
(3)
2
( )
[ / ]
f t
cm s
(1)
2
( )
[ / ]
f t
cm s
[ ]t s
Numerical results
• Stationary case: Evolutionary Kanai-Tajimi spectrum – Fourier transform of the simulated samples :a) Traditional SRM; b) proposed EV-SRM
0
40
80
120
0
40
80
0 20 40 60 80 1000
40
80
120
04080120160
0
40
80
0 20 40 60 80 1000
40
80
[ / ]rad s
)a )b
[ / ]rad s
2(2)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
HF
cm s
2(3)
2 3
( )
[ / ]
HF
cm s
2(2)
2 3
( )
[ / ]
HF
cm s
2(3)
2 3
( )
[ / ]
F
cm s
2(1)
2 3
( )
[ / ]
F
cm s
Numerical results
• Stationary case: Evolutionary Kanai-Tajimi spectrum – proof of convergence
0 20 40 60 80 1000
10
20
30
40
50
0 10 20 300
2000
4000
6000
2 3
( )
[ / ]
S
cm s
2 4
( )
[ / ]
t
cm s
[ / ]rad s[ ]t s
0
( ) 2 ( , )N
t S t d
0
1( ) 2 ( , )
ft
f
S S t dtt
Numerical results
• The influence of the enhanced spectrum variability is then investigated through the Monte Carlo study of the distribution of peak values
0 100 200 300 400 5000
100
200
300
0 100 200 300 400 500
120
160
200
240
280
max
2
[ ]
[ / ]
E f
cm s
n
)a )b
n
max
2
[ ]
[ / ]
E f
cm s
• Convergence of the mean value of the peak versus the number of samples n for the a) stationary and b) non-stationary process: traditional SRM (solid line), SRM with enhanced variability (dash-dotted line).
Numerical results
• The influence of the enhanced spectrum variability is then investigated through the Monte Carlo study of the distribution of peak values
• Convergence of the variance of the peak value versus the number of samples n for the a) stationary and b) non-stationary process: traditional SRM (solid line), SRM with enhanced variability (dash-dotted line).
0 100 200 300 400 5000
1000
2000
3000
0 100 200 300 400 500
1000
2000
3000
4000
5000
max
2
2 4[ / ]
f
cm s
n
)b
n
max
2
2 4[ / ]
f
cm s
)a
Numerical results
• The influence of the enhanced spectrum variability is then investigated through the Monte Carlo study of the distribution of peak values
• Comparison between the distribution of peaks for the a) stationary and b) non-stationary process: traditional SRM (solid line), SRM with enhanced variability (dash-dotted line).
100 200 300 400 5000
0.004
0.008
0.012
0.016
0.02
100 200 300 400 5000
0.004
0.008
0.012
0.016
0.02
2max [ / ]f cm s 2
max [ / ]f cm s
)a )bmax max( )Fp f
max max( )Fp f
Numerical results
• The influence of the enhanced spectrum variability is then investigated through the Monte Carlo study of the distribution of peak values
• Comparison between the cumulative distribution of peaks for the a) stationary and b) non-stationary process: traditional SRM (solid line), SRM with enhanced variability (dash-dotted line).
100 200 300 4000
0.2
0.4
0.6
0.8
1
150 200 250 300 350 4000
0.2
0.4
0.6
0.8
1
2max [ / ]f cm s 2
max [ / ]f cm s
)b)amax max( )FF fmax max( )FF f
• A modification to the traditional spectral-representation-method aimed to control the variability of the simulated samples of the random process is proposed.
• The Butterworth pass-band filter with random parameters has been included in the simulation formula to generate samples with different Fourier spectra.
• Remarkably the peak distribution is significantly sensible to the spectrum variability and the latter should be carefully considered when reliability analyses are performed.
• It is also expected in general that the spectrum variability influence whereas non-linear transformation of the power spectrum.are involved.
Concluding remarks