Surrogate models for uncertain dynamical systems: applications to earthquake engineering Bruno Sudret Chair of Risk, Safety and Uncertainty Quantification ETH Zurich Symposium on Uncertainty Quantification in Computational Geosciences BRGM (Orl´ eans, France) – January 16th, 2018
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Surrogate models for uncertain dynamicalsystems: applications to earthquake engineering
Bruno Sudret
Chair of Risk, Safety and Uncertainty Quantification
ETH Zurich
Symposium on Uncertainty Quantification inComputational Geosciences
BRGM (Orleans, France) – January 16th, 2018
Chair of Risk, Safety and Uncertainty quantification
The Chair carries out research projects in the field of uncertainty quantification forengineering problems with applications in structural reliability, sensitivity analysis,
model calibration and reliability-based design optimization
Research topics• Uncertainty modelling for engineering systems• Structural reliability analysis• Surrogate models (polynomial chaos expansions,
Kriging, support vector machines)• Bayesian model calibration and stochastic inverse
problems• Global sensitivity analysis• Reliability-based design optimization http://www.rsuq.ethz.ch
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 2 / 39
Excitationx(t) is generated by a probabilistic ground motion model Rezaeian & Der Kiureghian (2010)
x(t) = q(t,α)n∑i=1
si (t,λ(ti)) Ui
0 5 10 15 20 25 30−4
−3
−2
−1
0
1
2
3
4
t (s)
Acce
lera
tio
n (
m/s
2)
0 5 10 15 20 25 30−4
−3
−2
−1
0
1
2
3
4
t (s)
Acce
lera
tio
n (
m/s
2)
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 23 / 39
PC-NARX expansions Application to Bouc Wen model
Bouc-Wen model: prediction
0 5 10 15 20 25 30−4
−3
−2
−1
0
1
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3
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t (s)
Acce
lera
tio
n (
m/s
2)
0 5 10 15 20 25 30−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
t (s)
y(t
)
ReferencePC−NARX
0 5 10 15 20 25 30−4
−3
−2
−1
0
1
2
3
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t (s)
Acce
lera
tio
n (
m/s
2)
0 5 10 15 20 25 30−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
t (s)
y(t
)
ReferencePC−NARX
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 24 / 39
PC-NARX expansions Application to Bouc Wen model
Bouc-Wen model: prediction
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
Numerical model
PC
E
PC−NARX, εval,max
= 0.019
Maximal displacements0 0.05 0.1 0.15 0.2 0.25
0
10
20
30
40
50
max|y(t)|P
robabili
ty d
ensity function
ReferencePC−NARX
PDF of maximal displacements
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 25 / 39
Fragility curves
Outline
1 Introduction
2 Polynomial chaos expansions
3 PC-NARX expansions
4 Fragility curvesTheoryApplication: steel frame
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 25 / 39
Fragility curves Theory
Introduction to fragility curves• Earthquake engineering aims at assessing the
performance of structures and infrastructures w.r.trecorded or potential quakes
• Due to uncertainties in the localization, magnitude,structural behaviour and resistance, etc. probabilisticapproaches are commonly used
Fragility curvesFor a given performance criterion g ≤ gadm, the fragility curve represents theconditional probability of failure given an intensity measure IM :
Frag(IM ; gadm) = P (g ≥ gadm | IM)
Example• g = max
kmaxti∈[0,T ]
|δkti | (k-th interstorey drift)
• IM : peak ground acceleration (PGA), pseudo-spectral acceleration (PSa),cumulative absolute velocity (CAV), etc.
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 26 / 39
Fragility curves Theory
Fragility curvesL
HHH
a(t)^
L L
1 1
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
PGA (g)
Pro
babili
ty o
f fa
ilure
δo =0.01
δo =0.03
δo =0.05
Classical approach• Select a set of ground motions (recorded / synthetic)
• Compute the transient structural response (finiteelement analysis)
• Assume a parametric shape for the fragility curve,e.g. a lognormal shape:
Frag(IM ; δo) = P (∆ ≥ δo | IM) = Φ( log IM − α
β
)• Fit the parameters (α, β) form data
Limitations• Predefined shape of the curve• Subject to epistemic uncertainties when the number of ground motions is small
New proposal• Use non parametric statistics for the fragility curves• Use surrogate models of the transient analysis based on polynomial chaos
expansionsB. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 27 / 39
• Probabilistic demand model:log ∆ = A log IM +B + ζ Z Z ∼ N (0, 1)
• A and B determined by ordinary least squares estimation in a log-log scale• Results in a lognormal-like fragility curve:
Frag(IM ; δo) = P [log ∆ ≥ log δo] = 1− P [log ∆ ≤ log δo]
= Φ(
log IM − (log δo −B) /Aζ/A
).
Maximum likelihood estimation (ML) Shinozuka et al. (2000)
• Lognormal shape:
Frag(IM ; δo) = Φ(
log IM − logαβ
)• Estimation of α and β by maximum likelihood for each δo:
L(α, β, {IMi}Ni=1
)=
∏IMi: ∆i≥δo
[Frag(IMi; δo)
] ∏IMi: ∆i<δo
[1− Frag(IMi; δo)
]B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 28 / 39
Fragility curves Theory
Non parametric methods
Binned Monte Carlo estimate Mai, Konakli & Sudret, Frontiers Struct. Civ. Eng., (2017)
• Suppose Ns analyses are available for IM = IMo, with Nf such that ∆ ≥ δo.The fragility curve in this point could be estimated by Monte Carlo simulation:
Frag(IMo) = Nf (IMo)Ns (IMo)
• From the data cloud, a bin centered on IMo is considered, and points withinthe beam are “projected” onto the vertical line IM = IMo by linearization∆j(IMo) = ∆j
IMo
IMj.
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
PGA (g)
∆
PGAo
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
PGA (g)
∆
PGAo
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 29 / 39
Fragility curves Theory
Kernel density estimation
Fragility curves as a conditional CCDF Mai et al. , Frontiers Struct. Civ. Eng., (2017)
Frag(a; δo) = P (∆ ≥ δo|IM = a) =+∞∫δo
f∆(δ|IM = a) dδ
where:f∆(δ|IM = a) = f∆,IM (δ, a)
fIM (a)
Kernel density estimation• The joint- and the marginal PDFs are estimated by:
fX (x) = 1Nh
N∑i=1
K(x− xih
)fX (x) = 1
N |H|1/2
N∑i=1
K(H−1/2(x− xi)
)NB: Use of a constant bandwidth in the logarithmic scale
Mai, C., Polynomial chaos expansions for uncertain dynamical systems – Applications in earthquake engineering, PhD Thesis, ETH
Zurich, 2016
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 30 / 39
Fragility curves Application: steel frame
Outline
1 Introduction
2 Polynomial chaos expansions
3 PC-NARX expansions
4 Fragility curvesTheoryApplication: steel frame
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 30 / 39
Parameter Distribution Mean Standard deviation C.o.Vfy (MPa) Lognormal 264.2878 18.5 0.07E0 (MPa) Lognormal 210000 630 0.03
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 31 / 39
Fragility curves Application: steel frame
Stochastic ground motion
Stochastic excitation• Obtained by a modulated filtered white noise process Rezaeian & Der Kiureghian (2010)
x(t) = q(t,α)n∑i=1
si (t,λ(ti)) · ξi ξi ∼ N (0, 1)
• Parameters of the filterλ = (ωmid, ω′, ζf )T are calibratedon recorded signals
• Global parameters (Arias intensityIa, duration D5−95, strong phasepeak tmid) are transformed into theparameters α of the modulationfunction q(t,α) (e.g. gammadistribution)
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 32 / 39
Fragility curves Application: steel frame
Stochastic ground motion
Parameters of the excitation
Parameter Distribution Support Mean Standard deviationIa (s.g) Lognormal (0, +∞) 0.0468 0.164D5−95 (s) Beta [5, 45] 17.3 9.31tmid (s) Beta [0.5, 40] 12.4 7.44
M Shinozuka, M Feng, J Lee, and T Naganuma.Statistical analysis of fragility curves.J. Eng. Mech., 126(12):1224–1231, 2000.
B. Ellingwood.Earthquake risk assessment of building structures.Reliab. Eng. Sys. Safety, 74(3):251–262, 2001.
S. Rezaeian and A. Der Kiureghian.Simulation of synthetic ground motions for specified earthquake and site characteristics.Earthq. Eng. Struct. Dyn., 39(10):1155–1180, 2010.
C. V. Mai.Polynomial chaos expansions for uncertain dynamical systems – Applications in earthquake engineering.PhD thesis, ETH Zurich, Switzerland, 2016.
C.-V. Mai, M. D. Spiridonakos, E.N. Chatzi, and B. Sudret.Surrogate modeling for stochastic dynamical systems by combining nonlinear autoregressive with exogeneous input models andpolynomial chaos expansions.Int. J. Uncer. Quant., 6(4):313–339, 2016.
C.-V. Mai, K. Konakli, and B. Sudret.Seismic fragility curves for structures using non-parametric representations.Frontiers Struct. Civ. Eng., 11(2):169–186, 2017.
B. Sudret (Chair of Risk, Safety & UQ) Surrogates for dynamical systems BRGM – January 16th, 2018 39 / 39