Motivation Modeling and Parametric Uncertainty Summary
Dynamics of Soils and Structures underUncertainty
Boris JeremicKallol Sett (UB), Konstantinos Karapiperis and José Abell
University of California, DavisLawrence Berkeley National Laboratory, Berkeley
CompDyn,Crete, Greece, May 2015
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Outline
Motivation
Modeling and Parametric UncertaintyModeling UncertaintyParametric Uncertainty
Summary
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Introduction
Motivation
I Improve seismic design of soil structure systems
I Earthquake Soil Structure Interaction (ESSI) in time andspace, plays a major role in successes and failures
I Accurate following and directing (!) the flow of seismicenergy in ESSI system to optimize for
I Safety andI Economy
I Development of high fidelity numerical modeling andsimulation tools to analyze realistic ESSI behavior:Real ESSI simulator
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Introduction
Predictive CapabilitiesI Verification provides evidence that the model is solved
correctly. Mathematics issue.
I Validation provides evidence that the correct model issolved. Physics issue.
I Prediction under Uncertainty (!): use of computationalmodel to foretell the state of a physical system underconsideration under conditions for which the computationalmodel has not been validated.
I Modeling and Parametric Uncertainties
I Predictive capabilities with low Kolmogorov Complexity
I Modeling and simulation goal is to inform, not fit
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Uncertainties
Modeling Uncertainty
I Simplified modeling: Features (important ?) are neglected(6D ground motions, inelasticity)
I Modeling Uncertainty: unrealistic and unnecessarymodeling simplifications
I Modeling simplifications are justifiable if one or two levelhigher sophistication model shows that features beingsimplified out are not important
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Uncertainties
Parametric Uncertainty: Material Stiffness
5 10 15 20 25 30 35
5000
10000
15000
20000
25000
30000
SPT N Value
You
ng’s
Mod
ulus
, E (
kPa)
E = (101.125*19.3) N 0.63
−10000 0 10000
0.00002
0.00004
0.00006
0.00008
Residual (w.r.t Mean) Young’s Modulus (kPa)
Nor
mal
ized
Fre
quen
cy
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Uncertainties
Parametric Uncertainty: Material Properties
20 25 30 35 40 45 50 55 60Friction Angle [ ]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Prob
abili
ty D
ensi
ty fu
nctio
n
Min COVMax COV
10 20 30 40 50 60 70 80Friction Angle [ ]
0.0
0.2
0.4
0.6
0.8
1.0
Cum
ulat
ive
Prob
abili
ty D
ensi
ty fu
nctio
n
Min COVMax COV
0 200 400 600 800 1000 1200 1400Undrained Shear Strength [kPa]
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
Prob
abili
ty D
ensi
ty fu
nctio
n
Min COVMax COV
0 200 400 600 800 1000 1200 1400Undrained Shear Strength [kPa]
0.0
0.2
0.4
0.6
0.8
1.0
Cum
ulat
ive
Prob
abili
ty D
ensi
ty fu
nctio
n
Min COVMax COV
Field φ Field cu
20 25 30 35 40 45 50 55 60Friction Angle [ ]
0.00
0.05
0.10
0.15
0.20
0.25
Prob
abili
ty D
ensi
ty fu
nctio
n
Min COVMax COV
10 20 30 40 50 60 70 80Friction Angle [ ]
0.0
0.2
0.4
0.6
0.8
1.0
Cum
ulat
ive
Prob
abili
ty D
ensi
ty fu
nctio
n
Min COVMax COV
0 50 100 150 200 250 300Undrained Shear Strength [kPa]
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
Prob
abili
ty D
ensi
ty fu
nctio
n
Min COVMax COV
0 50 100 150 200 250 300 350 400Undrained Shear Strength [kPa]
0.0
0.2
0.4
0.6
0.8
1.0
Cum
ulat
ive
Prob
abili
ty D
ensi
ty fu
nctio
n
Min COVMax COV
Lab φ Lab cu
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Uncertainties
Realistic ESSI Modeling Uncertainties
I Seismic Motions: 6D, inclined, body and surface waves(translations, rotations); Incoherency
I Inelastic material: soil, rock, concrete, steel; Contacts,foundation–soil, dry, saturated slip–gap; Nonlinear buoyantforces; Isolators, Dissipators
I Uncertain loading and material
I Verification and Validation⇒ Predictions
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
Outline
Motivation
Modeling and Parametric UncertaintyModeling UncertaintyParametric Uncertainty
Summary
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
Real ESSI Models
I Full seismic motion input (body and surface waves) usingthe Domain Reduction Method (Bielak et al.)
I Inelastic (saturated or dry) soil/rock
I Inelastic (saturated or dry) contact (foundation – soil/rock)
I Buoyant (nonlinear) forces
I Inelastic structural modeling (elastic requested?!)
I Verification (extensive) and Validation (in progress)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
6D Free Field Motions
(MP4)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
6D Free Field Motions (closeup)
(MP4)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
6D Free Field at Location
(MP4)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
6D Earthquake Soil Structure Interaction
(MP4)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
From 6D to 1D?I Assume that a full 6D (3D) motions at the surface are only
recorded in one horizontal directionI From such recorded motions one can develop a vertically
propagating shear wave in 1DI Apply such vertically propagating shear wave to the same
soil-structure system
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
1D Free Field at Location
(MP4)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
1D ESSI of NPP
(MP4)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
6D vs 1D NPP ESSI Response Comparison
(MP4)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Modeling Uncertainty
6D vs 1D: Containment Acceleration Response
0.0 0.5 1.0 1.5 2.00.30.20.10.00.10.20.3
Acce
l-X [g
]
Containment building bottom
0.0 0.5 1.0 1.5 2.00.30.20.10.00.10.20.3
Acce
l-Y [g
]
3-D1-D
0.0 0.5 1.0 1.5 2.0Time [s]
0.30.20.10.00.10.20.3
Acce
l-Z [g
]
0.0 0.5 1.0 1.5 2.00.30.20.10.00.10.20.3
Acce
l-X [g
]
Containment building top
0.0 0.5 1.0 1.5 2.00.30.20.10.00.10.20.3
Acce
l-Y [g
]
3-D1-D
0.0 0.5 1.0 1.5 2.0Time [s]
0.30.20.10.00.10.20.3
Acce
l-Z [g
]
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Outline
Motivation
Modeling and Parametric UncertaintyModeling UncertaintyParametric Uncertainty
Summary
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Uncertain Material Parameters and Loads
I Decide on modeling complexity
I Determine model/material parameters
I Model/material parameters are uncertain!I MeasurementsI TransformationI Spatial variability
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Uncertainty Propagation through Inelastic System
I Incremental el–pl constitutive equation
∆σij = EEPijkl =
[Eel
ijkl −Eel
ijmnmmnnpqEelpqkl
nrsEelrstumtu − ξ∗h∗
]∆εkl
I Dynamic Finite Elements
Mu + Cu + Kepu = F
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Critique of our Previous Work, PEP and SEPFEMI Constitutive weighted coefficients N1 and N2 do not work
well for stress solution!
I We suggested that σ(t) be considered a δ-correlated, andbased on that simplified stiffness equations. Both theassumption and the resulting equation were not right.
I On a SEPFEM level, stiffness needs update basisfunctions and KL coefficients in each step. We updated theeigenvalues λi and kept the same structure(Karhunen-Loeve) in the approximation of the stiffness,which is not physical
I Implicitly assumed that the stiffness remains Gaussian,which is not the case
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Gradient Flow Theory of Probabilistic Elasto-Plasticity
Decomposition of an elastoplastic random process:(∂
∂t− Lrev
)P(σ, t) = 0 if σ ∈ Ωel
(∂
∂t− Lirr
)P(σ, t) = 0 if σ ∈ Ωel ∪ Ωpl
I Reversible (Lrev ) and Irreversible (Lirr ) operators
I Yield PDF is an attractor, similar to plastic corrector
I Ergodicity of the elastic-plastic process can be proven!
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Gradient Flow Theory of Probabilistic Elasto-PlasticityI Elastic, reversible process, Fokker-Planck (forward
Kolmogorov) equation
∂P(σ, t)∂t
= −∇ · (〈E ε〉P(σ, t)) + t Var[E ε]∆P(σ, t)
Lrev = ∇ · (t Var[Cε]∇− 〈Cε〉)I Plastic, irreversible process, Fokker-Planck (forward
Kolmogorov) equation
∂P(σ, t)∂t
= ∇ · (∇Ψ(σ)P(σ, t)) + D∆P(σ, t)
Lirr = D∇ ·(∇−
∇Py (σ)
Py (σ)
)Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Gradient Flow Theory of Probabilistic Elasto-Plasticity
I Limiting (final) distribution is considered to be known
I Underlying potential leading to this distribution is sought
I Transition from uncertain elastic to uncertain plasticresponse
I Only in a 1D elastoplastic problem does one end up with astationary distribution
I In higher dimensional problems this yield stress distributionis only "marginally" stationary along one or a combinationof the stress components.
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Probabilistic Elasto-Plasticity: von Mises Surface
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Gradient Theory of Probabilistic Elasto-Plasticity:Numerical Solution
I Using radial basis functions (a meshless method) forsolving Fokker-Planck equations for uncertainelastic-plastic response
I Details in a talk by Mr. Karapiperis later this afternoon(room 2, MS-6, 17:00-19:00, last talk)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Gradient Theory of Probabilistic Elasto-Plasticity:Verification, Elastic-Perfectly Plastic
0.03 0.02 0.01 0.00 0.01 0.02 0.03εx
1.0
0.5
0.0
0.5
1.0
σx [M
pa]
RBFMCS
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Stochastic Elastic-Plastic FEM (SEPFEM)
I KL-PC expansion of material random fields (stiffness, etc)D(x, θ) =
∑Mi=0 ri(x)Φi [ξr (θ)]
I PC expansion of displacement field
u(x, θ) =
p∑i=0
di(x)ψi [ξr (θ)]
I Stochastic Galerkin
N∑n=1
K ′mndni +N∑
n=1
P∑j=0
dnj
M∑k=1
bijkK ′′mnk = Φm〈ψi [ξr]〉
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
SEPFEM Statistical linearizationUpdate the FE stiffness in the elastoplastic regime:
I Solve elastoplastic FPE for each integration point∂Pnl(σ, t)
∂t=
∂
∂σ
(⟨Dk (1− P[Σy ≤ σ])
∆ε
∆t
⟩P)
+∂2
∂σ2
(tVar
[Dk (1− P[Σy ≤ σ])
∆ε
∆t
]P)
I Consider an equivalent linear FPE∂P lin(σ, t)
∂t= Neq
(1)∂P∂σ
+ Neq(2)∂2P∂σ2
I Linearization of the PC coeff. as an optimization problem
∂P lin(σ, t)∂t
=∂Pnl(σ, t)
∂t
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Dynamic, Time Domain, SEPFEM
I Gaussian formulation inadequate due to occurrence of"probabilistic softening" modes⇒ Need for positive definitekernel
I Stochastic forcing (e.g. uncertain earthquake)
I Stability of time marching algorithm (Newmark,Rosenbrock, Cubic Hermitian ) analyzed usingamplification matrix
I Long-integration error and higher order statistics phaseshift
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Wave Propagation Through Uncertain Soil
0 100 200 300 400 500Shear modulus G [MPa]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
min COVmax COV
30 20 10 0 10 20 30 40 50 60Shear strength τ [kPa]
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
min COVmax COV
0.0 0.1 0.2 0.3 0.4 0.5Time [s]
0.03
0.02
0.01
0.00
0.01
0.02
0.03
Disp
lace
men
t [m
]
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Uncertain Elastic Response at the Surface (COV = 120%)
0.0 0.1 0.2 0.3 0.4 0.5Time [s]
0.08
0.06
0.04
0.02
0.00
0.02
0.04
0.06
0.08
0.10
Disp
lace
men
t [m
]
MM - SD
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Displacement PDFs at the Surface (COV = 120%)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Displacement CDFs (Fragilities) at the Surface (COV = 120%)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Parametric Uncertainty
Probability of Exceedance, disp = 0.1m (COV = 120%)
0.0 0.1 0.2 0.3 0.4 0.5Time [s]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Prob
. of e
xcee
danc
e (d
= 0
.1m
)
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Summary
Concluding Remarks
I Uncertainty influences results of numerical predictions
I Uncertainty (modeling and parametric) must be taken intoaccount
I Goal is to predict and inform, not fit
I Philosophy of modeling and simulation systemReal ESSI simulator(aka: Vrlo Prosto, Muy Fácil, Molto Facile, ,
, Πραγματικά Εύκολο, , , TrèsFacile, Vistinski Lesno, Wirklich Einfach )
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty
Motivation Modeling and Parametric Uncertainty Summary
Summary
Acknowledgement
I Funding from and collaboration with the US-NRC,US-DOE, US-NSF, CNSC, AREVA NP GmbH, and ShimizuCorp. is greatly appreciated,
I Collaborators: Prof. Kavvas (UCD), Prof. Pisanò (TUDelft), Mr. Watanabe (Shimizu), Mr. Vlaski (AREVA NPGmbH), Mr. Orbovic (CNSC) and UCD students: Mr. Abell,Mr. Karapiperis, Mr. Feng, Mr. Sinha, Mr. Luo
Jeremic et al.
Dynamics of Soils and Structures under Uncertainty