Research into stochastic dynamic testing and reliability model updating
C S ManoharDepartment of Civil Engineering
IISc, Bangalore
AcknowledgementCollaborator: V S Sundar, PhD studentFunding: BRNS
Symposium: Greek tradition
…where men gather to drink, eat, contemplate, and ponder over life…
3
Multi-physics problems
Organization
• Overview of treatment of uncertainties in structural engineering
• Testing of highly reliable engineering systems functioning under random dynamic environment
• Existing instrumented structures and model updating
5
Hazards
• Earthquake• Wind• Waves• Vehicles• Blast• Impact• Fire
Undesirable consequence
http://betterplan.squarespace.com/todays-special/2011/5/3/5211-blade-failure-and-wind-project-residents-worries-and-tu.html
http://www.ribapix.comhttp://www.ecy.wa.gov
http://www.thehindu.com/news/national/article2900321.ece
Total no. of slides: 75Extremes
6
Irreducible
Limited knowledge Modeling approximat
Aleatoric
Epistemic
Black swan
ions Reducible
"unknown unHuman errors
Bona fide Mal
kno
a
wns"
fide
•
•
•
•
Uncertainties
http://www.geneticsandsociety.org
Alia: rolling of dice
7
Aleatoric or Epistemic?
http://www.eoearth.org/article/Earthquake
A K Chopra, Dynamics of structures.
1 12 23 34 45 56 6
×
When does the nature roll its dice?And, when it rolls, what happens?
Mavroeidis & Papageorgiu, BSSA, 2003
Near fault
Far field
SiteResponseBased model
8
Peak Ground Accelerationcontours with 10% probability of exceedance in 50 years(Type A sites)
Development of probabilsitic seismic hazard map of India, Technical report of WCE, NDMA, 2010
Long rangeuncertainties
Average shear wave velocity in top 30 m greater than 1.5 km/s Total no. of slides: 75
9
A Kareem (1987)
Short rangeuncertainties
Aleatoric uncertainties in art
11
JCSS (2002)
Description COVYield strength 0.07Ultimate tensile strength
0.04
Young’s modulus 0.03Poisson’s ratio 0.03Ultimate strain 0.06
−−
=
10100160.000145.00075.01
ρ
Steel as a 5-dimensional random variable
Distribution: Multivariate lognormal random variable
12
Load effectsMoments in framesAxial forces in framesShear force in framesMoments in platesForces in platesStresses in 2D solidsStresses in 3D solids
DistributionLNLNLNLNLNNN
Mean1.01.01.01.01.000
COV0.10.050.10.20.10.050.05
JCSS recommendations on model uncertainties
For “more or less standard FE models”
13
Pushover Results by Participants
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 100 200 300 400 500 600 700
Displacement (mm)
Lo
ad (
kN)
NPCIL1NPCIL2IITGIITBIITRTyagarajar1Tyagarajar2BARCSERC1CPRI+NITKSERC2SERC3AERBIITDThapar1Thapar2IITM
P
2P
3P
4P
ROUND ROBIN EXERCISE ON PUSHOVER EXPERIMENTS AND ANALYSIS OF PROTOTYPE STRUCTURE (BARC-CPRI Joint work)
1ST AND 2ND MAY, 2008
Data provided by Dr G R Reddy, BARC
14
D J Ewins, 1982 F Fahy, 1995
Total no. of slides: 75
15
Mail online, 12, Sept 2012
The “unknown unknowns” Relative to the observerNo data for prognosis••
Frameworks for modeling uncertainties
• Probability theory• Interval analysis• Convex sets• Fuzzy set• Hybrid models
16
ChallengeHow to combine these tools with structural analysis methods?
Structuralsystems
Uncertain actions
UncertainSystem parameters
Uncertain outputs
Propagation of uncertainties must be consistent with the laws ofmechanics
18
Intensity Measure (IM)Engineering Demand Parameter (EDP)
Treated as a set of random variables Damage Measure (DM)Decision Variable (DV)
Grouping of basic variables
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
, , , ( , , , )
| , , | , |
| | |
DV DM EDP IM
DV DM EDP
DV DM EDP
p dv dm edp im
p dv dm edp im p dm edp im p edp im p im
p dv dm p dm edp p edp im p im
=
≈
FUNDAMENTAL ASSUMPTION
Performance Based Structural Engineering (PBSE)
19
Stress analysisDynamics Fracture FatigueStabilityCorrosion…
Plate tectonicsState of stressFaultingWave propagationSite amplificationLiquefaction[Instrumental data]
RepairRetrofitInjuriesLoss of lifeDelayed damages
Finite element methodMonte Carlo SimulationsStochastic differential equationsLaboratory and field testingNonlinear behavior
Imperfection sensitivityGeometric complexityControls: passive/active
( ) ( ) ( ) ( ) ( )1 || |Vim dm
DMe
Edp
DPp edp id im
DV dv P dv p dm edpdm m dimdimλ
λ > = − ∫ ∫ ∫Client specification Fragility analysis Response analysisLoss analysis
Hazard
20
Mathematical models
Experimental models
Can we combine them?
What are the issues?
Studies on existing structures
21
1962 1966IS 1893
1970 2002
1967 Koyna1988 Bihar-Nepal1991 Uttarakashi1993 Killari1997 Jabalpur1999 Chamoli2001 Bhuj
What happens to existing structures?
Railway bridgesGauge conversionLocomotion Axle loads
22
Sensors• Strain gauges• LVDT-s• Accelerometers (uni-axial / tri-
axial, translation / rotation)• Tilt• TemperatureLoading• Static / Dynamic• Measured / Unmeasured• Diagnosis and Performance
assessment• NDT & acoustic emission• Cores and samples
http://www.goabest.com/WondersOfGoa/Dudhsagar-Falls-WondersofGoa.asp
Condition monitoring of existing railway Bridges
•Heavier axle loads •Longer trains •Higher speeds
Funding: Indian railways. Collaborators: J M Chandra KishenAnanth Ramaswamy
Total no. of slides: 75
Numerical models
Measurements
Data assimilation•Bayes’ theorem•Markov property
Predictive tools
Physical laws
•Synthetics•Laboratory •Field
Impe
rfec
t
•FEM•Spatio-temporal discretization•Limits on scales
Choice?
You cannot doubt everything and function; you cannot believe everything and survive. (Taleb)
24
( ) ( ) ( ) ( ) ( )0 0 0
1 | | |DV DM EDP
d imDV dv P dv dm p dm edp p edp im dim
dimλ
λ∞ ∞ ∞
> = − ∫ ∫ ∫
PBSE format extension to instrumented structures
( ) ( ) ( ) ( ) ( )
Damage model updating Fragility model updating Reliability model updatingLoss model updating Hazard model updat
|| 1 | , | , | ,DV DM EDP
d im DDV dv D P dv dm D p dm edp D p edp im D
dimλ
λ > = −
( ) ( ) ( )Reliability model upd
0 0
ating
0
System identification
ing
| , | , , |EDP EDP Xp edp im D p edp x im D p x D
di
dx
m∞ ∞ ∞
=
∫ ∫ ∫
∫ ∫ ∫
Line of action: Apply Bayesian tools
The PBSE format
The extension: D=measurement data
2525
Estimation of hidden states
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
, , , ; 0 , 0
, , ; 1,2, ,k k k k k
M U t C U t K U t F U t U t t G t t U U
y t H U t U t t k N
θ θ θ θ θ ξ
θ ε
+ + + = Γ + = + =
Given
( ) ( ) 1:, | ; 1,2,k k kp U t U t y k N = To find
Problem I
[ ]1: 1 2k ky y y y=
STRUCTURAL SYSTEM IDENTIFICATION
System
Input Output
Given GivenTo be determined
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
, , , ; 0 , 0
, , ; 1,2, ,k k k k k
M U t C U t K U t F U t U t t G t t U U
y t H U t U t t k N
θ θ θ θ θ ξ
θ ε
+ + + = Γ + = + =
Given
( )| ; 1,2,kp y t k Nθ = To find
Problem II
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( )
0 0Process equation , ; (0) ; 0
Response Measurement , , ; 1, 2, ,
Input measurement ( ) ; 1, 2,... ; 1, 2, ,k k k k k k k k
r k r k r k
MU t F U t U t p t t U U U U
y t f U t U t q U t U t k N
t p t t r NDOF k N
ς
ν
ρ ε
+ = + = = = + = = + = ≤ =
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )0
, , 0 0, | , ; 1,2, ,
max , , 0 | , ; 1,2, ,
0 | , ; 1,2, ,
p k k
k kt T
m k k
P h U t U t t t T y t t k N
h U t U t t y t t k N
h y t t k N
ρ
ρ
ρ< <
= < ∀ ∈ = = < = = < =
Given
To find
Reliability model updatingProblem III
Performance function could involve statesthat are not measured.
Remark
Two themes
• Random vibration testing with controlled samples
• Updating reliability models of instrumented dynamical systems
V S Sundar and C S Manohar• 2013, Int. Jl. of Non-linear Mechanics, 520, 32-40• 2013, Str. Safety, 40,21-30.• 2013, Structural Control & Health Monitoring, to appear
FE Model
•ParticleFiltering•MCMC
samplingMeasurements
FE modeling &Bayesian tools
Matlab
Commercial codes•Nisa•Ansys
•Laboratory•Field•Synthetics
Batch files in MS DOS
30
( )( ) ( )
1Process equation: , ; 0,1, 2,: , , ; 1,Measurement equat 2ion ,
k k k k k
k k k k k k
x h x w ky f x G x k
θ γθ θ ν
+ = + == + =
Nonlinear dynamic state estimation
( ) ( ) ( )
( ) ( ) ( )( ) ( )
1: 1 1 1 1: 1 1
1: 11:
1: 1
Prediction | | |
| |Updating |
| |
k k k k k k k
k k k kk k
k k k k k
p x y p x x p x y dx
p y x p x yp x y
p y x p x y dx
− − − − −
−
−
=
=
∫
∫
Kalman filter. Exact solution to the state estimation problem.
Use Monte Carlo simula
⇒
⇒
Linear systems with additive Gaussian noises
Nonlinear systems, non - Gaussian noises, multiplicative noises, ...tions
Particle filters.
Time• • •
• • •
Mathematicalmodel
Measurements
Chapman-Kolmogorov equationFPK equation
Markov process
Bayes' theorem
1kt − kt 1kt +
1ky − ky 1ky +
3232
( )
( )
Consider the problem of evaluation of the definite
integral ( ) .
This can be re-written as
1( ) ( ) ( ) ( )
1where ; is now interpreted
as the pdf of
b
a
b b
Xa a
X
I f x dx
I b a f x dx b a f x p x dxb a
p x a x bb a
=
= − = − −
= < < −
∫
∫ ∫
a random variable that is uniformly distributed in to . a b
33
( )
( )
1
Following this, the integral is now interpreted as an expectation
( - )
where the expectation is evaluated with respect to . Furthermore, is now approximated by
1ˆ ( )
where -s are u
X
N
ii
i
I
I b a f X
p x I
I f XN
X=
=
= ∑
( )niformly distributed random numbers
samples from .Xp x
34
100
101
102
103
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
sample size *100
Est
imat
e of
Iestimateexact
12
0
Evaluation of 1 / 3I x dx= =∫
3535
0 50 100 150 200 250 300 350 400 450 5000.2
0.25
0.3
0.35
0.4
0.45
0.5
run
Est
imat
e of
I
estimateexact
500 runs with 500 samples
12
0
Evaluation of 1 / 3I x dx= =∫
3636
0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.370
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normal PDFEmpirical PDF
Estimate of PDF
12
0
Evaluation of
1 / 3I x dx= =∫
( )
( ) ( ) ( )2
2
Remarksˆ lim with probability 1 Law of large numbers,
ˆlim 0, Central limit theorem
ˆ is a consistent estimator with minimum variance
In the evaluation of multi-fold inte
N
N
I I
N I I N
IN
σ
σ
→∞
→∞
• =
• − →
• =
• grals, the sampling variance is independent of dimension of the integral.
3838
( ) ( ) ( )( )
( ) ( )
12
01 1 2 2
2
0 0
2
11
.
Here a valid pdf defined over 0 to 1.
1ˆ where are samples drawn from .
.
NNi
i iii
I x dx
x XI x dx x dxx X
x
XI X xN X
π
ππ π
π
ππ =
=
=
= = =
=
=
∫
∫ ∫
∑
revisitedEvaluation of
39
( )
( )
( )
2
1 2
20
2
21
2
Let 3 ;0 1.
3
1 1ˆ = for any value of and hence for =1.33
3 ;0 1 is the ideal ispdf.Catch: the definition of this ispdf requires the knowledge of being evaluate
Ni
i i
x x x
xI x dxx
XI N NN X
x x x
I
π
π
π=
= < ≤
=
=
= < ≤
∫
∑
d.
40
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
x
Given pdfideal ispdf
41
( )
( ) .3111
10;
1
0
2
1
0
21
0
2
==⇒=⇒=
<<=
∫∫∫
dxxdxxdxx
xxx
ααπ
απ
( ) ( ) ( ) ( )
( )
( ) ( )
1
0 0
*
0
0
, ,
, 1, 2, ,
Let it be required to find P max , 0
max , highest response
q
i i ij jj
i i
F t T
t T
dX t A t t dt t t dB t
X t X i d
P h h t t
h t t
σ=
≤ ≤
≤ ≤
= +
= =
= − ≤
=
∑
The idea of "importance sampling" for dynamical systems
X X
X
X
( )
( ) ( )
( )
1
*
*
011
1
over a time period
permissible value of the response.1ˆ max , 0
ˆ
1ˆVar
ˆHow to reduce Var ?
Increase .
Ni
F t Ti
F F
F FF
F
h
P I h h t tN
P P
P PP
N
P
N
≤ ≤=
=
= − ≤
• =
−• =
∑ X
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
1
1
0 0 0 0
*
0
1
*
Alternative: Girsanov's transformation
, ,
, , 1, 2, ,It can be shown that
max
,
, 0 m
q
i i ij jj
q
ij jj
q
j jj
i i
t T
t t u tdX t A t t dt t t dB t
d t t u t dB t
X t X t
d
i d
I h h t t T h
t
I
σ σ==
=
≤ ≤
= + +
Γ = −Γ
= Γ = Γ =
− ≤ = Γ −
∑ ∑
∑
X
X
XX
( ) ( ) ( )
( )
2
00
*
012 0
ax , 0
1 max , 0
: Radon-Nikodym derivative
t T
Ni
F t Ti
h t t
P T I h h t tN
t
≤ ≤
≤ ≤=
≤ Γ
= Γ − ≤ Γ
Γ
∑
X
X
( )
( )
( ) ( )
2
*
1
Variance of the estimator depends upon sample size and the controls
Idea: Use to control the sampling variance.Importantly, there exists an ideal control.
1 1 , , 2
d
j kj
j
k
j
k
N
u t t
u t
t
t jX
u
ψψσ
=
∂ = − ∂∑
X
( ) ( ) ( ) ( )
*
0
*
1, 2, ,
max , 0
Sampling variance goes to zero.t T
j j
q
T I h h t t
u t u t
ψ≤ ≤
=
= Γ − ≤
⇒ = ⇒
X
( )
( ) ( )
( ) ( ) ( ) ( )
*
1
*
0
*
Idea: Use to control the sampling variance.Importantly, there exists an ideal control.
1 1 , , 1, 2, ,2
max , 0
Sampling variance goes to ze
d
j kjk k
t T
j j
j
u t t t j qX
T I h h t t
u t
u
u
t
t
ψσ
ψ
ψ=
≤ ≤
∂ = − = ∂
= Γ − ≤
⇒ = ⇒
∑
X
X
ro.: The construction of the ideal control requires the
knowledge of the very quantity being sought.: construct suboptimal contr
ols.
,ψCatch
Strategy
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( )
0
Idea: find the controls by solving a deterministic problem
Take , & ,
and consider the deterministic system
0
m mi ij j
t t t t t t
d tt t t
dt
V h t t uτ τ α
= =
= +
=
⇒ = −
Distance minimizing suboptimal controlA
A
A X X X
VV u
V X
σ
σ
σ
( )
( )
1 0
1
, 1, 2, , ; 1, 2, ,
where &
, 1, 2, , ; 1, 2, , impulse response functions
mq
jj
r
lj jl
mij
t dt i r j q
h t i r j q
τ
α
τ
=
=
= =
=
− = = =
∑ ∫
∑
σ
( )( )
( ) ( )
( )
2 2
1 0
2 *
1
2 *
1 0 1 1 0
Define
Find by solving the optimization problem:
minimize subject to the constraint 0
0 and
q mm
jkj k
rm m
i ii
q qm r mm
jk i ij k jk jkj k i j k
u
u t
h V
L u h h t u
L L
β τ
β τ ψ τ
λ ψ τ α
λ
= =
=
= = = = =
= ∆
− =
⇒
= ∆ + − − ∆
∂ ∂
=∂ ∂
∑∑
∑
∑∑ ∑∑∑
( )
( ) ( )
*
1
2
1 0 1 1
0, 1, 2, , ; 0,1, ,
,
1, 2, , ; 0,1, ,
jk
rm
jk i ij ki
jk q m r rm m
jk k i i j ki ijj k i i
j q k mu
h h tu
h t h t
j q k m
α ψ τ
α ψ τ ψ τ
=
= = = =
= = =
⇒
− − =
− − − −
= =
∑
∑∑∑∑
Nonlinear systems• In absence of mathematical model: use impulse response function
• If mathematical model is available, use the model to derive the distance minimizing controls.
• Use alternative variance reduction schemes (currently under development)
( )( ) ( ) ( ) ( )( ) ( )
( ) ( )
( )
1 2
2 3
0
0 1
0 0
2
*
1
2
2.31, 0.7 0.2,0 0
0.06, 4 , 1, 10
P ma
, 0 0
x 0F
t
t
tz z z z F t
e t A e eF t F t e t w t
Az
P h z
z
t
α αηω ω
η
α
βα α ω π β σ
≤ ≤
− −
= = =
+ + + = = −+ =
= − = == ==
= −
≤
0 1 2 3 4 5 6 7 8 9 10-2
0
2
4
6
t s
u(t)
m/s2
0 1 2 3 4 5 6 7 8 9 10-2
0
2
4
t s
u(t)
m/s2
0 1 2 3 4 5 6 7 8 9 10-2
0
2
4
t s
u(t)
m/s2
OptimizationApproximation
OptimizationApproximation
OptimizationApproximation
α = 105
α = 106
α = 104
To implement the Girsanov transformation we need to establishSuboptimal controls
Distance minimizing control in terms of impulse response function of the systemThe Radon-
•
•
Nikodym derivative
Important observation
We do not need a mathematical model for the structureto establish these quantities
We could think of a experimental reliability testing method
( )gx t
( )gy t
5252
Kanai Tajimi & Clough and PenzienPower spectral density function modelsfor free field earthquake ground acceleration
−
Bed rock
Ground
Soil layer
( )bx t
( )u t
Local site condtionsare accounted for( ) : White noisebx t
53
( ) ( )( )
( ) ( )( )
( )
( )( )
( )( ) ( )
4 2 2 2
22 2 2 2 2
4 2 2 22
22 2 2 2 2High pass filter
44 2 2 2
2 22 22 2 2 2 2 2
High pass filter
4
4
4| |
4
4 /
4 1 / 4 /
g g g
g g g
g g gf
g g g
g g g f
g g g f f f
S I
S I H
I
ω η ω ωω
ω ω η ω ω
ω η ω ωω ω
ω ω η ω ω
ω η ω ω ω ω
ω ω η ω ω ω ω ς ω ω
+=
− +
+=
− +
+=
− + − +
Clough and Penzien model
545454
0 5 10 15 20 25 30-1.5
-1
-0.5
0
0.5
1
1.5
time t
acce
lera
tion
555555
( )Strategy: Use a deterministic modulaitng function.
( ) ( )
( ) determigX t e t S t
e t
=
=
How to allow for nonstationary nature of ground accelerations?
Nonstationarity : in amplitude modulation & frequency content.
( ) ( )0
0
nistic envelope function( )=zero mean stationary Gaussian random process
(with PSD given by Kanai-Tajimi or Clough and Penzien models)
( ) exp exp ; 0
( )
S t
e t A t t
e t A
α β α β = − − − > > =
Examples
( ) ( )1 expA t tα+ −
56
0 5 10 15 20 25 30-1.5
-1
-0.5
0
0.5
1
1.5
time t
acce
lera
tion
57
( ) ( )
( )( )( )
21 1 1 1 1 1
2 22 2 2 2 2 2 1 1 1 1 1
2
2
2
1 1
2 1
3 2
4 2
2
2 2
Ground displacementGround velocity
Ground acceleration
Introduce
y y y e t s t
y y y y y
y ty ty t
x yx yx yx y
η ω ω
η ω ω η ω ω
+ + =
+ + = +
=
=
( ) ( )
1 12
2 1 1 1 2
3 32 2
4 1 1 1 2 2 2 4
0 1 0 0 02 0 0 1
0 0 0 1 02 2 0
x xx x
e t s tx xx x
ω ηω
ω η ω ω η ω
− − ⇒ = + − −
58
( )
( )
( ) ( )
( ) ( )
2
2
0 1
for 0 4s4
=1 for 4 24s1=exp 24 for 24 s2
( ) exp exp ; 0
( ) exp
te t t
t
t t
e t a t t
e t A At t
α β α β
α
= < < < <
− − >
= − − − > >
= + −
Examples for envelope function
59
0 5 10 15 20 25 30-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
time s
disp
lace
men
t m
60
0 5 10 15 20 25 30-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
time s
velo
city
m/s
61
0 5 10 15 20 25 30-5
-4
-3
-2
-1
0
1
2
3
4
5
time s
acce
lera
tion
m/s
/s
0 2 4 6 8 10-0.5
0
0.5
1
1.5
2
2.5
3
t s
Bas
e di
spla
cem
ent m
m
Determination of impulse response function(Impulse at the bed rock level)
Motion to be applied at the shake table top
0 1 2 3 4 5 6 7 8 9 10-15
-10
-5
0
5
10
15
t s
h(-t)
µ-s
train
Measured impulse response for strain(shown reversed in time)
0 1 2 3 4 5 6 7 8 9 10-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
t s
h(-t)
mm
Measured impulse response for inter storey drift(shown reversed in time)
0 1 2 3 4 5 6 7 8 9 10-6
-4
-2
0
2
4
6
t s
u(t)
m/s2
Control force; performance function defined with respect to strain
* 325 -strain; 2.5 s.mh µ τ= =
0 1 2 3 4 5 6 7 8 9 10-8
-6
-4
-2
0
2
4
6
8
10
t s
u(t)
m/s2
Control force; performance function defined with respect to inter-storey drift
* 1.3 mm; 2.5 s.mh τ= =
0 1 2 3 4 5 6 7 8 9 10-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
t s
Bas
e ac
cele
ratio
n m
/s2
UnbiasedBiased, h* = 325 µ-strain
Sample excitation
0 1 2 3 4 5 6 7 8 9 10-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
t s
Bas
e ac
cele
ratio
n m
/s2
UnbiasedBiased, h* = 1.3 mm
Sample excitation
0 0.5 1 1.5 2 2.5 310-5
10-4
10-3
10-2
10-1
100
101
t s
Γ(t)
Realization 1Realization 2Realization 3
Radon-Nikodym derivative
50 100 150 200 250 300 350 40010-3
10-2
10-1
100
h* µ-strain
P F
Method 1Method 2
Linear frame, uniaxial ground motions
Method 1Brute force : 3000 samples
Method 2 250 samples
0.2 0.4 0.6 0.8 1 1.2 1.4 1.610-3
10-2
10-1
100
h* mm
P F
Method 1, case (i)Method 1, case (ii)Method 2, case (i)Method 2, case (ii)
Nonlinear frame, uniaxial ground motions
Method 1Brute force : 1800 samples
Method 2 250 samples
0 50 100 150 200 250 30010-6
10-5
10-4
10-3
10-2
10-1
100
h* µ-strain
P F
Method 1Method 2
Nonlinear frame, biaxial ground motions
Method 1Brute force : 1800 samples
Method 2 250 samples
Extension to automotive applications
BiSS (Private) Limited
Current developments
• How to derive controls/importance sampling strategies that explicitly takes into account nonlinearity?
• How to deal with material nonlinearity?
Way out: particle splitting methods?
Reliability model updating in instrumented dynamical systems
Two steps procedure
(a) Structural system identification(b) Reliability model updating
StrategyStructural system identification• Maximum likelihood estimation
Reliability model updating• Modify the scope of the Bayesian filtering tools
( ) ( ) ( ) ( )( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
0
Process equation
0Measurement equation
, 0Quantitites to be determinedˆ ,0
ˆ ˆ ,0t
d t t dt t dt d t
t t t t T
t t T
t t t t t T
τ τ
τ τ
= + +
=
= + ≤ ≤
= ≤ ≤
= − − ≤ ≤
Linear - Gaussian state space model
A
H
P
X X f B
X X
Z X
X X Z
X X X X Z
µ
( ) ( ) ( ) ( ) ( ) ( )
( )( )
( )
( ) ( ) ( )
( ) ( )
0
0
0
1
1
1
Kalman's filterˆ
ˆ ˆ
ˆ ˆ0
0
0 , 0
t t
t
t t
d tt t t t t
dt
d tdt
t t t
−
−
−
= + − +
=
= + − +
=
=
=
−
= =
g
g
A K H
PAP PA PH R HP Q
P P
K PH R
P
= A + Q
= H RH AP I
XX Z X f
X X
Ω Ψ
Ω Ω Ψ
Ψ Ω Ψ
Ω Ψ
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
0
0
1 1
: future excitationsWhat is the reliability against future loading actions?Idea: Interpret Kalman filter equations as SDE-s.
ˆ ˆ ˆ
ˆ ˆ0
0 ,
t
t t
f t
d t t dt t t t t dt d t− − = + − +
=
−
=
A H R H
= A + Q
= H RH AP
X X Z X B
X X
Ω Ψ
Ω Ω Ψ
Ψ Ω Ψ
Ω Ψ( )
( ) [ ] ( ) ( ) [ ]
( )
*|
*
*
0
0
P 0, ,0
ˆP 0,
ˆP max
S
t T
t
t
t
P t h t T Z T
t h t T
t h
τ τ
≤ ≤
=
= ≤ ∀ ∈ ≤ ≤
= ≤ ∀ ∈
= ≤
I
Z X
X
X
φ
φ
φ
( )
( ) ( )
( ) ( )
*| | 0
*| 0
1
| ||
ˆ1 max 0
1ˆ ˆmax 0
1ˆVar
Back to the same dilemma!How to reduce the sampling variance?
s
F S t T
Nl
F t Tls
F FF
s
t
t
P P I h t
P I h tN
P PP
N
φ
φ
≤ ≤
≤ ≤=
= − = − ≤
= − ≤
−=
∑
Z Z
Z
Z ZZ
X
X
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )
( ) ( )
( ) ( ) ( )
( )
0 0
0
* *00 0
*| 0
1 1
Girsanavo's transformation
0 , 0
0 , 0
ˆmax 0 max 0
1 max
t T t T
Fg
t
t
t t
t t
d t t dt t t t t dt t t dt t d t
d t t t d t
I h t T I h t
P T I hN
φ φ≤ ≤ ≤ ≤
− − = + − + +
Γ = −Γ
= Γ = Γ
−
= =
− ≤ = Γ − ≤ Γ
= Γ −
A H R H
= A + Q
= H RH AP I
Z
X X Z X u B
u B
X X
X X
Ω Ψ σ σ
Ω Ω Ψ
Ψ Ω Ψ
Ω Ψ
( ) ( ) 01
0gN
l
t Tl
t tφ≤ ≤
=
≤ Γ ∑ X
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
2
1 0
0
Controls: Derive from the Kalman's filter equationTakes into account the measurements
; 0
0
mm
mj
j
u t dt
d t t dt t t t dt t t dt t T
τ
β τ=
⇒
=
= + − + ≤ ≤ =
∑ ∫
gA K HV V Z V u
V X
σ
( ) ( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( )
( ) [ ] ( )
0
*|
Process equation, ,
0
Measurement equation, 0
Quantity to be determined
P 0, ,0S
d t t dt t d t t d t
t t t t T
P h t h t T Tτ τ
= + +
=
= + ≤ ≤
= ≤ ∀ ∈ ≤ ≤
Nonlinear state space model
H
Z
X A X X B B
X X
Z X
X Z
σ σ
µ
• Girsanaov’s transformation• Distance minimizing controls• Modified state space model• Bootstrap based Monte Carlo filter
( ) ( ) ( )
( ) ( )
21 2
2 2
2
2 2s s s s s s
f f f f f f s s s s s
f
y y y e t w t w t
y y y y y
y t t
η ω ω
η ω ω η ω ω
+ + = +
+ + = +
+ + = − +M C K M
Y Y Y Ω ξ
( )1 1 1 5 5 5 1 2 15
Step-1System identification: 32 parameters
, , , , , , , , , , , ,ts a s a
x x x x s ak k k k k kθ θ ρ ρ η η η ψ =
0.2 0.4 0.6 0.8 1 1.2 1.4 1.610-3
10-2
10-1
100
h* mm
P F
ExperiementalAnalytical
0 1 2 3 4 5 6 7 8 9 10-0.015
-0.01
-0.005
0
0.005
0.01
0.015
t s
u(t)
m/s2
*
Control force1.6 mm; 4.0 s.mh τ= =
0.8 1 1.2 1.4 1.6 1.8 210-6
10-5
10-4
10-3
10-2
10-1
100
h* mm
P F
Method 1Method 3 Method 1
Proposed method500 samples
Method-3Brute force10E+05 samples
Closure
• Combining computing and experimental hardware in structural dynamic testing.
• Instrumented structures and ageing infrastructure.
Strong-Floor Reaction-wall
Net height : 5m Net Length: 4m Net Width: 3mFloor Thickness: 1m Wall Thickness : 1m