dynamics of soils and structures under uncertaintysokocalo.engr.ucdavis.edu/~jeremic/ · motivation...

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


Outline

Motivation

Modeling and Parametric UncertaintyModeling UncertaintyParametric Uncertainty

Summary

Jeremic et al.



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.



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.



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.



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.



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.



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.




Outline

Motivation


Summary

Jeremic et al.




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.




6D Free Field Motions

(MP4)

Jeremic et al.


http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_01AApr2015/movie_input.mp4



6D Free Field Motions (closeup)

(MP4)

Jeremic et al.


http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_01AApr2015/movie_input_closeup.mp4



6D Free Field at Location

(MP4)

Jeremic et al.


http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4



6D Earthquake Soil Structure Interaction

(MP4)

Jeremic et al.


http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_npp_3d.mp4



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.




1D Free Field at Location

(MP4)

Jeremic et al.


http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4



1D ESSI of NPP

(MP4)

Jeremic et al.


http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_npp_1d.mp4



6D vs 1D NPP ESSI Response Comparison

(MP4)

Jeremic et al.


http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4



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.



Parametric Uncertainty

Outline

Motivation


Summary

Jeremic et al.




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.




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.




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.




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.




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.




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.




Probabilistic Elasto-Plasticity: von Mises Surface

Jeremic et al.




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.




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.




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.




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.




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.




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

PDF

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

PDF

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.




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.




Displacement PDFs at the Surface (COV = 120%)

Jeremic et al.




Displacement CDFs (Fragilities) at the Surface (COV = 120%)

Jeremic et al.




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.



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.



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.


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