Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Introduction to structural reliability analysis
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 ?
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
x1
x2
0
s
f
g ( x ) = 0
f X ( x ) = const.
Limit state function (LSF)
Safe domain
Failure domain
Limit state surface (LSS)
Introduction to structural reliability analysis
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
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*
~
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
MAISM – method of structural reliability analysis
Reliability analysis
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
-3 3
3
-3
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
-3 3
3
-3
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
0)(~
uh
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
0)(~
uh
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
0)(~
2 uh
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
0)(~
2 uh
find:
that minimizes:
subject to:
u
uuu T2
0)(~
2 uh
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
0)(~
3 uh
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
MPP search
u1
u2
h(u)=0
0
s
f
find:
that minimizes:
subject to:
u
uuu T2
0)(~
3 uh
0)(~
3 uh
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Description of the algorithm
Multimodal adaptive importance sampling
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
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.
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Structural reliability analysis – applications
• FE model description
ABAQUS
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Structural reliability analysis – applications
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
• Stochastic model
• Limit State Function definition
• RA method parameters
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
STAND
Structural reliability analysis – applications
Task description
• Stochastic model
• Limit State Function definition
• RA method parameters
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Structural reliability analysis – applications
Output
STAND
FE code (ABAQUS)
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Structural reliability analysis – applications
121.0
Sg X
S - Huber Mises stress
Task description
• Stochastic model
• Limit State Function definition
• RA method parameters
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
STAND
Structural reliability analysis – applications
Task description
FE code (ABAQUS)
• Stochastic model
• Limit State Function definition
• RA method parameters
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
Structural reliability analysis – applications
ABAQUS
Pf = 0.001396
= 2.990STAND
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
S-rail crashworthiness reliability
Global buckling
Poor energy management
Regular folding
Good energy management
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
S-rail crashworthiness reliability
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]
X2 t2 - thickness of the part 2 lognormal 1.5 [mm] 0.075 [mm]
X3 t3 - thickness of the part 3 lognormal 1.5 [mm] 0.075 [mm]
X4 t4 - thickness of the part 4 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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
S-rail crashworthiness reliability
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
S-rail crashworthiness reliability
Multimodal adaptive importance sampling
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
S-rail crashworthiness reliability
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
36th Solid Mechanics Conference, Gdańsk, Poland, September 9-12, 2008
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.