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Probabilistic models for computational stochastic mechanics and applications

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HAL Id: hal-00687868

https://hal-upec-upem.archives-ouvertes.fr/hal-00687868

Submitted on 15 Apr 2012

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de

Probabilistic models for computational stochastic mechanics and applications

Christian Soize

To cite this version:

Christian Soize. Probabilistic models for computational stochastic mechanics and applications. 9th

International Conference on Structural Safety and Reliability ICOSSAR’05, Rome, Italy, 19–23 June

2005, Jun 2005, Rome, Italy. pp.23-42 (Plenary Lecture). �hal-00687868�

(2)

5NIVERSITY OF -ARNELA6ALL|EE 0ARIS &RANCE

+EYWORDS COMPTATIONAL STOCHASTIC MECHANICS MODEL UNCERTAINTIES DATA UNCERTAINTIES DYNAMICAL SYSTEMS

(3)

.OTATION

( (

' '

' ' *

((

$ESIGNED SYSTEM

2EAL SYSTEM

-EAN MODEL AS A PREDICTIVE MODEL %RRORS AND UNCERTAINTIES

%RRORS

5NCERTAINTIES

$ATA UNCERTAINTIES

(4)

-ODEL UNCERTAINTIES

0REDICTABILITY OF THE MEAN MODEL

.ONPARAMETRIC PROBABILISTIC APPROACH OF RANDOM UNCERTAINTIES

#ONCEPT OF THE NONPARAMETRIC PROBABILISTIC AP PROACH TO TAKE INTO ACCOUNT MODEL UNCERTAINTIES

4HE OPERATOR OF THE REAL SYSTEM

4HE MEAN MODEL OF THE OPERATOR

0ARAMETRIC PROBABILISTIC MODEL OF THE OPERATOR

2 2

.ONPARAMETRIC PROBABILISTIC MODEL OF THE OPERA TOR

(5)

%VOLUTION OF THE CONCEPTS SOME HISTORY

-EAN lNITE ELEMENT MODEL OF THE DYNAMICAL SYSTEM

( (

( ) ) ) (

&

( ) ) ) (

&

&& ( (

(

2EDUCED MEAN MODEL

( ) ) ) (

#

( ) ) ) ( &

(6)

$ &

( ) ) ) (

$

$

$ $ $ $ $ $ $

#

$ (

$

( ((

( $ $ ( $ )

3TOCHASTIC RESPONSE OF THE NONLINEAR DYNAMICAL SYSTEM WITH THE NONPARAMETRIC PROBABILISTIC MODEL OF RANDOM UNCERTAINTIES

$ )

( ((

#ONSTRUCTION OF THE PROBABILITY MODEL OF THE RAN DOM MATRICES ( (

((

$

+

+

+

( ( (

$

%NSEMBLE OF RANDOM MATRICES

$ElNITION OF ENSEMBLE

((

)

+

)

+ )

+ )

(7)

$ISPERSION PARAMETER OF A RANDOM MATRIX IN EN

SEMBLE 4

4

)

4

+ 4 +

) 0ROBABILITY DISTRIBUTION OF A RANDOM MATRIX IN

ENSEMBLE

(

# (

(

)

4 (

*

3

#

( -

(

!LGEBRAIC REPRESENTATION OF A RANDOM MATRIX IN ENSEMBLE

4

(

(

+

7 74

7

# 7

#

)

#ONVERGENCE PROPERTY OF A RANDOM MATRIX IN EN SEMBLE WHEN DIMENSION GOES TO INlNITY

( + (

4

/THER ENSEMBLES OF RANDOM MATRICES

THE NORMALIZED POSITIVEDElNITE ENSEMBLE

POSITIVEDElNITE ENSEMBLE

(8)

THE PSEUDOINVERSE ENSEMBLE

#ONSTRUCTION AND CONVERGENCE OF THE STOCHASTIC SOLUTION

3TOCHASTIC SOLUTION AS A SECONDORDER STOCHAS

TIC PROCESS -

(

+ +

+

#ONSTRUCTION OF THE STOCHASTIC SOLUTION

&

#ONVERGENCE ANALYSIS

( (

(

( 5

(

( 5( ) ) ) ( ( 5

$ESIGNED SYSTEM

%

& &

)&

8

O

Z

X Y

H

H

H

2EAL SYSTEM

.

8

O

Z

X Y

฀ ฀

฀ ฀ P

(

(9)

-EAN MODEL

%

%

% )&

. .

*& 6 $*&

/ )

2 ( 2 ( 2 ( 2 ( 2 ( 2 ( 2 ( 2 ( 2 ( 2 ( ) ) ) ( 2

.

%

H O

Y

Z

X

G V X

X

.UMERICAL EXPERIMENT OF THE REAL SYSTEM

*& )

$*&

)() )

( .(

)()() ( ( !

(

2

(

2 ( 2 ( 2 ( 2 ( 2 ( ( ( ( (

%STIMATION OF THE DISPERSION PARAMETERS FOR RAN DOM UNCERTAINTIES MODELING

4 4

4

4 ) 4 ) 4 )

0REDICTION WITH RANDOM UNCERTAINTIES AND EXPER IMENTAL COMPARISONS

(

0 500 1000 1500

−11.5

−11

−10.5

−10

(

%((( %( % )& % )&

)

(10)

0 200 400 600 800 1000

−13

−12

−11

−10

−9

−8

−7

−6

0 200 400 600 800 1000

−13

−12

−11

−10

−9

−8

−7

−6

,ACK OF CAPABILITY OF THE PARAMETRIC PROBABILISTIC APPROACH TO TAKE INTO ACCOUNT MODEL UNCERTAINTIES

6

/

4 4 4

444

6 /

6 / 4 4 4

! !

4

4

! !

444 4 4

6

4

4

4

)

444 ) 4 ) 4 ) 4

)

4 44

4 )

0 200 400 600 800 1000

−13

−12

−11

−10

−9

−8

−7

−6

0 200 400 600 800 1000

−13

−12

−11

−10

−9

−8

−7

−6

#ONCLUSION

(

(

(11)

$ESIGNED PANEL

%

)& % )&

)&

%

)&

6 $*&

" ) " 2 ) ) "

)& $*&

" 2

"

฀ ฀

฀ ฀

%XPERIMENTS

-ANUFACTURED PANELS

%XPERIMENTAL FREQUENCY RESPONSE FUNCTIONS

(

)()(&

)()(&

%XPERIMENTAL MODAL ANALYSIS (

( ) ) ) (

0

. 5 / 5

1 5

, 5 . *

. 5 0

.*3 )

) ) )

0 ( ) ) ) ( / * / 5

0 /

*

/

/ )

-EAN MODEL AND ITS UPDATING

(12)

2EDUCED MEAN MODEL

&2& CALCULATION WITH THE REDUCED MEAN MODEL AND EXPERIMENTAL COMPARISONS

( ( (

500 1000 1500 2000 2500 3000 3500 4000 4500

−1

−0.5 0 0.5 1 1.5 2 2.5 3

&*

%XPERIMENTAL IDENTIlCATION OF THE DISPERSION PA RAMETERS OF THE NONPARAMETRIC MODEL

4 4 4

2 + 4 4 4

4 ) 4 ) 4 )

#ONlDENCE REGION PREDICTION FOR THE &2& AND EXPERIMENTAL COMPARISONS

500 1000 1500 2000 2500 3000 3500 4000 4500

−1

−0.5 0 0.5 1 1.5 2 2.5 3

&*

#ONCLUSIONS

(13)

)NDUSTRIAL APPLICATION ,INEAR DYNAMICS OF A BLADED DISK MISTUNED BY MANUFACTURING UNCERTAINTIES

-EAN MODEL

&

&

(

2EDUCED MEAN MODEL

.ONPARAMETRIC PROBABILISTIC MODELING OF UNCER TAINTIES PROBABILITY DENSITY FUNCTION OF THE RANDOM AMPLIlCATION FACTOR DUE TO THE MISTUNING INDUCED BY MANUFACTURING TOLERANCES

4 )

(14)

5150 520 525 530 535 0.5

1 1.5 2 2.5

#ONCLUSION

0 5 10 15 20

0 0.5 1 1.5 2 2.5

1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

)NDUSTRIAL APPLICATION )DENTIlCATION AND QUAN TIlCATION OF THE DESIGN MARGINS IN NONLINEAR DYNAMICS OF A REACTOR COOLANT SYSTEM

2EAL SYSTEM

(15)

-EAN MODEL

2EDUCED MEAN MODEL

.ONPARAMETRIC PROBABILISTIC MODELING OF UNCER TAINTIES QUANTIlCATION OF THE DESIGN MARGINS OF THE REACTOR COOLANT SYSTEM

4 4 4 )

(16)

4 )

"

5

" *

%

#ONCLUSION

)NDUSTRIAL APPLICATION 2OBUSTNESS OF THE NU MERICAL SIMULATION MODEL OF A SPATIAL STRUCTURE WITH RESPECT TO MODEL AND DATA UNCERTAINTIES IN DYNAMICS

-EAN MODEL

%

(

2EDUCED MEAN MODEL

0ARAMETRIC PROBABILISTIC MODELING OF DATA UN CERTAINTIES

)()

.ONPARAMETRIC PROBABILISTIC MODELING OF MODEL AND DATA UNCERTAINTIES

! !

4 4

!

!

4

(17)

4 ) 4 ) 4 )

X

Y Z

obs

X

Y Z

2OBUSTNESS OF THE NUMERICAL SIMULATION MODEL WITH RESPECT TO MODEL AND DATA UNCERTAINTIES

10 20 30 40 50

−140

−130

−120

−110

−100

−90

−80

−70

−60

−50

30 35 40 45 50

−140

−130

−120

−110

−100

−90

−80

−70

−60

30 35 40 45 50

−140

−130

−120

−110

−100

−90

−80

−70

−60

(18)

)NDUSTRIAL APPLICATION 2OBUSTNESS OF NUMERICAL VIBROACOUSTIC &2& OF CARS WITH RESPECT TO MODEL AND DATA UNCERTAINTIES

-EAN MODEL OF THE VIBROACOUSTIC SYSTEM

(

2EDUCED MEAN MODEL

IN VACCUO

.ONPARAMETRIC PROBABILISTIC MODELING OF MODEL AND DATA UNCERTAINTIES

4 44

4 4 4

4

2OBUSTNESS OF THE NUMERICAL SIMULATION MODEL WITH RESPECT TO MODEL AND DATA UNCERTAINTIES

*

"

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

(19)

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

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!3-% *OURNAL OF 6IBRATION AND !COUSTICS

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