structural reliability analysis using object oriented

STRUCTURAL RELIABILITY ANALYSIS

USING OBJECT ORIENTED ENVIRONMENT

STAND

Division of Numerical Methods of

Reliability and Optimization

http://pmnno.ippt.gov.pl

Institute of Fundamental Technological Research

Polish Academy of Sciences

http://www.ippt.gov.pl

J. Knabel, K. Kolanek, V. Nguyen Hoang, R. Stocki, P. Tauzowski

36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008

Outline

Introduction to structural reliability analysis

Methods of structural reliability analysis

MAISM - Multimodal Adaptive Importance Sampling Method

Reliability Analysis Software STAND

Object oriented library structure of STAND

Response surface-based methods in reliability analysis

Parallel computing, development features, user friendly interface

Reliability analysis examples by STAND



reliable failure-free

Probability of failure

Vector of basic random variables

represents basic uncertain quantities that define the state of the structure,

e.g., loads, material property constants, member sizes.

Reliability ?


x1

x2

0

s

f

g ( x ) = 0

f X ( x ) = const.

Limit state function (LSF)

Safe domain

Failure domain

Limit state surface (LSS)



Methods of structural reliability analysis

Simulation methods Monte Carlo, Adaptive Monte Carlo, Importance Sampling

u 2

G ( u ) = 0

s

f

0 u 1

n ( u,0,I ) = const

x1

x2

0

s

f

g ( x ) = 0

f X ( x ) = const.

u 2

G ( u ) = 0

s

f

0 u 1

n ( u,0,I ) = const

u*

Approximation methods FORM, SORM, Response Surface based

u 2

s

f

u 1

f

1 2

3

R

5

4

G ( u ) = 0

u 2

G ( u ) = 0

s

f

l ( u ) = 0

*

u*

0 u 1

region of mostcontribution toprobability integral

n ( u,0,I ) = const

u 2

Gv(v) = f v ( v ) – v n = 0

s

f

0 u 1

v n v n

v ~

~

v n = sv ( v )

v*

~


MAISM – method of structural reliability analysis

Reliability analysis


It consists of the two major phases:

1. The most probable point (MPP) search using the limit state

function approximation by an adaptive response surface

2. Multimodal adaptive importance sampling

MAISM – method of structural reliability analysis


Description of the algorithm

MPP search

u1

u2

h(u)=0

n ( u,0,I ) = const

0

s – safe domain

f – failure domain

The ”omitted” variables

manifest as a noise in

limit state function computation



MPP search

u1

u2

h(u)=0

0

s

f

-3 3

3

-3



MPP search

u1

u2

h(u)=0

0

s

f

-3 3

3

-3

)()()(~ T

1 ubuauh

WhAWAAb T1T )(ˆ

.0,)2/()(exp1

22exn

j

jijii nuuw

0)(~

1 uh



MPP search

u1

u2

h(u)=0

0

s

f

-3 3

3

-3 0)(~

1 uh

find:

that minimizes:

subject to:

u

uuu T2

0)(~

1 uh



MPP search

u1

u2

h(u)=0

0

s

f

0)(~

uh



MPP search

u1

u2

h(u)=0

0

s

f

0)(~

2 uh



MPP search

u1

u2

h(u)=0

0

s

f

0)(~

2 uh

find:

that minimizes:

subject to:

u

uuu T2

0)(~

2 uh



MPP search

u1

u2

h(u)=0

0

s

f



MPP search

u1

u2

h(u)=0

0

s

f

0)(~

3 uh



MPP search

u1

u2

h(u)=0

0

s

f

find:

that minimizes:

subject to:

u

uuu T2

0)(~

3 uh

0)(~

3 uh



MPP search

u1

u2

h(u)=0

0

s

f

Stop criteria:

)()2

)1

*

min

uh

dd

0)(~

3 uh

d

u*

should account for the space

dimension

mind

should account for the NOISE



Multimodal adaptive importance sampling

u1

u2

h(u)=0

0

s

f

Building the multimodal sampling density

cluster radius

)1(* v̂u

A sample point inside the 1st cluster.Ignored in representative points search

The sample point outside the 1st cluster and closest tothe origin. It is adopted as the 2nd representative point.

)2(v̂

)3(v̂

)4(v̂

contours of the multimodalsampling density

),ˆ,(ˆ)( )(

1

)( IvvvV

j

n

k

j

jws

k

r

r

n

j

njw

1

)(

)()(

),,ˆ(

),,ˆ(ˆ

I0v

I0v




u1

u2

h(u)=0

0

s

f

Sampling till convergence

f

f

PP

P

f ˆ

]ˆVar[ˆ

,)(

),,()(

1ˆ0

10 i

ini

m

i

fs

Im

Pf v

I0vv

V

0

1

2

00 )(

),,()(

)1(

1]ˆ[Var

m

i i

inif

sI

mmP

f v

I0vv

V

2.0ˆfP


STAND Reliability Analysis Software

STochastic ANalysis and DesignSTAND – object oriented library

Failure probability computation by crude Monte Carlo sampling for single and

multiple limit state functions.

Most probable point (MPP) search by various algorithms

HLRF, NLPQL, random search by OLH sampling, adaptive RS based algorithm.

Failure probability assessment using MPP

FORM, SORM, classical importance sampling (IS), multimodal adaptive IS.

Reliability assessment by mean value first order method (MVFO) – suitable only for not too

nonlinear limit state functions

Sampling techniques

random sampling, LH, OLH.

Various probability density functions

uniform, normal, log-normal, Gumbel, Frechet, exponential, Weibull, Rayleight.

Nataf transformation for correlated random variables.

Response surface module.


0

ReliabilityAnalysis

GetProbabilityOfFailureGetReliabilityIndex

RandVariablesVectorLimitStateFunctionProbOfFailureReliabilityIndex

DPSReliabilityAnalysis

DesignPointSearch

MonteCarlo

FORM

SORM

MAISM

DesignPointSearch

DP_HLRF

DP_ARF

DP_MAISM

IS

1MMVFO

Object oriented library structure of STAND

Advantages- natural problem modelling

- better source control

- fast code development

- C++ language

- better error diagnostics

Disadvantages- slower start into object

oriented philosphy

ResponseSurface

ComputeValueWithGradient

PatternCollection

SecondOrderRSFirstOrderRS

KrigingRS


STAND and third party FE code – data flow chart

STAND

FE code (ABAQUS)

OutputInput

Task description

• text file input data

• text file output results

• batch mode processing

• Stochastic model

• Limit State Function definition

• RA method parameters

• FE model description


Structural reliability analysis – applications

• FE model description

ABAQUS



Variable Distribution Mean Std. Dev. Unit Description At design

point (MPP)

H_pos normal 7.321 0.1 cm Horizontal position of

the circle center

7.49734

V_pos normal 7.321 0.1 cm Vertical position of

the circle center

7.44789

R normal 5.0 0.1 cm Circle radius 5.06759

P normal 4.0 0.4 kN/cm2 Load magnitude 4.75991

E normal 21000 2100 kN/cm2 Young modulus 20089.67





STAND


Task description






Output

STAND

FE code (ABAQUS)



121.0

Sg X

S - Huber Mises stress

Task description





STAND


Task description

FE code (ABAQUS)






ABAQUS

Pf = 0.001396

= 2.990STAND


S-rail crashworthiness reliability

Global buckling

Poor energy management

Regular folding

Good energy management



Random variables

Limit state function

- minimal admissible value of absorbed energy

equal to 6000J, which is about 80% of the

energy absorbed by the „nominal beam”

- absorbed energy

Description Distribution Mean value Standard deviation

X1 t1 - thickness of the part 1 lognormal 1.5 [mm] 0.075 [mm]




X5 0 - yield stress normal 180 [MPa] 15 [MPa]

X6 E - Young modulus normal 210000 [MPa] 21000 [MPa]

X7 v0y – y component of the initial velocity normal 0 [m/s] 1.5 [m/s]

X8 v0z – z component of the initial velocity normal 0 [m/s] 1.5 [m/s]

),(1),( min

AXAX

e

eg

),( AXe

mine



MPP search

Stop criteria: 1.0)()242.0,15.0,8)1 *2 uhdnnd

Trust region reduction strategy:1

1

25.0 in

i bb

4.61

4.10

4.184.21

4.04

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

1 2 3 4 5

OUT

IN

IN

ININre

liab

ilit

yin

dex

( ||u

*||

)

iteration

4.61

4.10

4.184.21

4.04

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

1 2 3 4 5

OUT

IN

IN

ININre

liab

ilit

yin

dex

( ||u

*||

)

iteration

4.61

1.62

0.91

0.95

0.32

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1 2 3 4 5

0.42

conv

ergen

cecr

iter

ion

iteration

4.61

1.62

0.91

0.95

0.32

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1 2 3 4 5

0.42

conv

ergen

cecr

iter

ion

iteration

I I 54.04, 2.7 10FORM fP

* { 0.98, 0.29, 1.20, 1.26,

2.19, 0.80,1.92, 1.81}

u

213calls LSF of No.

,0038.0)( *uh




0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250

v

number of generated points

fP̂

3.75E-05

4.00E-05

4.25E-05

4.50E-05

4.75E-05

5.00E-05

5.25E-05

5.50E-05

5.75E-05

6.00E-05

6.25E-05

100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250

number of generated points

fP̂

II 5 II 1 II5.37 10 , 3.87f fP P

No. of LSF calls = 1195



Assessment of the influence of spot weld failures on crashworthiness

reliability

With spot weld failures Without spot weld failures

First part, MPP search

I = FORM 4.04 4.02

PfI 2.7 10-5 2.9 10-5

Second part, multimodal adaptive importance sampling

PfII 5.37 10-5 1.9 10-5

II = - (PfII) 3.87 4.12


Conclussions

• STAND is a simple and effective tool for structural reliability analysis.

• Object oriented structure facilitates code development.

• Efficient interfacing technique between STAND and third party FE codes.

• A wide response surface functionally is a key part of the reliabilityanalysis software dealing with implicit (FE computed) limit state functions.

• The presented multimodal adaptive importance sampling technique proves to be well suited for complex reliability analysis applications. However, its performance depends on many arbitrarily selected parameters. They should be carefully chosen to reduce the computational cost and not to impair the accuracy of the method.

structural reliability analysis using object oriented

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