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Motivation SEPFEM An Application Summary Stochastic Elastic-Plastic Finite Element Method Boris Jeremi´ c and Kallol Sett Department of Civil and Environmental Engineering University of California, Davis SEECCM 2009 Jeremi´ c and Sett Computational Geomechanics Group Stochastic Elastic-Plastic Finite Element Method

Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

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Page 1: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Stochastic Elastic-Plastic Finite ElementMethod

Boris Jeremic and Kallol Sett

Department of Civil and Environmental EngineeringUniversity of California, Davis

SEECCM 2009

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 2: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Outline

MotivationMotivation and Overview

SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations

An ApplicationSeismic Wave Propagation Through Uncertain Soils

Summary

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 3: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Motivation and Overview

Outline

MotivationMotivation and Overview

SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations

An ApplicationSeismic Wave Propagation Through Uncertain Soils

Summary

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 4: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Motivation and Overview

(Geo–) Materials are Inherently Uncertain

Spatial Variation of Friction Angle(After Mayne et al. (2000))

Typical COVs of Different Soil Properties(After Lacasse and Nadim 1996)

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 5: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Motivation and Overview

On UncertaintiesI Epistemic uncertainty - due to lack of knowledge

I Can be reduced bycollecting more data

I Mathematical tools not welldeveloped, trade-off withaleatory uncertainty

Hei

sen

ber

gp

rin

cip

le

un

cert

ain

tyA

leat

ory

un

cert

ain

tyE

pis

tem

ic

Det

erm

inis

tic

I Aleatory uncertainty - inherent variation of physical systemI Can not be reducedI Has highly developed mathematical tools

I Ergodicity – exchange ensemble average for time average?I Applicable to soils, biomaterials, up for discussion for

concrete, rock

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 6: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Motivation and Overview

Historical Overview

I Brownian motion, Langevin equation → PDF governed bysimple diffusion Eq. (Einstein 1905)

I With external forces → Fokker-Planck-Kolmogorov (FPK)for the PDF (Kolmogorov 1941)

I Approach for random forcing → relationship between theautocorrelation function and spectral density function(Wiener 1930)

I Approach for random coefficient → Functional integrationapproach (Hopf 1952), Averaged equation approach(Bharrucha-Reid 1968), Monte Carlo method

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 7: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Motivation and Overview

Soil Uncertainties and Quantification

I Natural,spatial variability of soil deposit (Fenton 1999)I Function of soil formation process

I Testing (point–wise) error (Stokoe et al. 2004)I Imperfection of instrumentsI Error in methods to register quantities

I Transformation (point–wise) error (Phoon and Kulhawy1999)

I Correlation by empirical data fitting (e.g. CPT data →friction angle etc.)

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 8: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Motivation and Overview

Recent State-of-the-ArtI Governing equation

I Dynamic problems → Mu + Cu + Ku = φI Static problems → Ku = φ

I Existing solution methodsI Random r.h.s (external force random)

I FPK equation approachI Use of fragility curves with deterministic FEM (DFEM)

I Random l.h.s (material properties random)I Monte Carlo approach with DFEM→ CPU expensiveI Perturbation method→ a linearized expansion! Error

increases as a function of COVI Spectral method→ developed for elastic materials so far

I Newly developed: Stochastic Elastic–Plastic FiniteElement Method

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 9: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Probabilistic Elasto–Plasticity

Outline

MotivationMotivation and Overview

SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations

An ApplicationSeismic Wave Propagation Through Uncertain Soils

Summary

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 10: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Probabilistic Elasto–Plasticity

Constitutive Problem Setup

I 3D incremental elasto–plasticity:

dσij/dt =

{Del

ijkl −Del

ijmnmmnnpqDelpqkl

nrsDelrstumtu − ξ∗r∗

}dεkl/dt

I Phase density ρ of σ(x , t) varies in time according to acontinuity Liouville equation (Kubo 1963)

I Continuity equation written in ensemble average form (eg.cumulant expansion method (Kavvas and Karakas 1996))

I van Kampen’s lemma (van Kampen 1976) →< ρ(σ, t) >= P(σ, t), ensemble average of phase density isthe probability density

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 11: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Probabilistic Elasto–Plasticity

Eulerian–Lagrangian FPK Equation

∂P(σ(xt , t), t)∂t

= − ∂

∂σ

»fiη(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t))

fl+

Z t

0dτCov0

»∂η(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t))

∂σ;

η(σ(xt−τ , t − τ), Del(xt−τ ), q(xt−τ ), r(xt−τ ), ε(xt−τ , t − τ)

–ffP(σ(xt , t), t)

–+

∂2

∂σ2

»Z t

0dτCov0

»η(σ(xt , t), Del(xt), q(xt), r(xt), ε(xt , t));

η(σ(xt−τ , t − τ), Del(xt−τ ), q(xt−τ ), r(xt−τ ), ε(xt−τ , t − τ))

– ffP(σ(xt , t), t)

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 12: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Probabilistic Elasto–Plasticity

Euler–Lagrange FPK EquationI Advection-diffusion equation

∂P(σ, t)∂t

= − ∂

∂σ

[N(1)P(σ, t)− ∂

∂σ

{N(2)P(σ, t)

}]I Complete probabilistic description of responseI Solution PDF is second-order exact to covariance of time

(exact mean and variance)I It is deterministic equation in probability density spaceI It is linear PDE in probability density space → Simplifies

the numerical solution processI Template FPK diffusion–advection equation is applicable to

any material model → only the coefficients N(1) and N(2)

are different for different material modelsJeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 13: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Probabilistic Elasto–Plasticity

Transformation of a Bi–Linear (von Mises) Response

0 0.0108 0.0216 0.0324 0.0432 0.0540

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003St

ress

(M

Pa)

Strain (%)

Mode

DeterministicSolutionStd. Deviations

Mean

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 14: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

SEPFEM Formulations

Outline

MotivationMotivation and Overview

SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations

An ApplicationSeismic Wave Propagation Through Uncertain Soils

Summary

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 15: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

SEPFEM Formulations

Stochastic Finite Element FormulationI Governing equations:

Aσ = φ(t); Bu = ε; σ = Dε

I Spatial and stochastic discretization

I Deterministic spatial differential operators (A & B) →Regular shape function method with Galerkin scheme

I Input random field material properties (D) →Karhunen–Loève (KL) expansion, optimal expansion, errorminimizing property

I Unknown solution random field (u) → Polynomial Chaos(PC) expansion

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 16: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

SEPFEM Formulations

Spectral Stochastic Elastic–Plastic FEM

I Minimizing norm of error of finite representation usingGalerkin technique (Ghanem and Spanos 2003):

N∑n=1

K epmndni +

N∑n=1

P∑j=0

dnj

M∑k=1

CijkK′epmnk = 〈Fmψi [{ξr}]〉

K epmn =

∫D

BnDepBmdV K′epmnk =

∫D

Bn√λkhkBmdV

Cijk =⟨ξk (θ)ψi [{ξr}]ψj [{ξr}]

⟩Fm =

∫DφNmdV

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 17: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

SEPFEM Formulations

Inside SEPFEM

I Explicit stochastic elastic–plastic finite elementcomputations

I FPK probabilistic constitutive integration at Gaussintegration points

I Increase in (stochastic) dimensions (KL and PC) of theproblem (parallelism)

I Development of the probabilistic elastic–plastic stiffnesstensor

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 18: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

SEPFEM Formulations

1–D Static Pushover Test Example

I Elastic–plastic material model,von Mises, linear hardening,< G >= 2.5 kPa,Var [G] = 0.15 kPa2,correlation length for G = 0.3 m,Cu = 5 kPa,C

′u = 2 kPa.

����������������������

L

F

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 19: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

SEPFEM Formulations

SEPFEM verification

1 2 3 4 5 6 7

0.02

0.04

0.06

0.08

0.1L

oad

(N)

Displacement at Top Node (mm)

Standard Deviations (SFEM)

Standard Deviations (MC)

Mean (SFEM)

Mean (MC)

Mean and standard deviations of displacement at the top node,von Mises elastic-plastic linear hardening material model,KL-dimension=2, order of PC=2.

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 20: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Outline

MotivationMotivation and Overview

SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations

An ApplicationSeismic Wave Propagation Through Uncertain Soils

Summary

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 21: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Seismic Wave Propagation Through Random Soil

Soil is 12.5 m deep 1–D soil column with random von Misesmaterial:

I Shear modulus:〈G〉 = 11.57MPa;Var [G] = 142.32MPa2;Cor. Length [G] = 0.61m

I Shear strength:〈qT 〉 = 4.99 MPa;Var [qT ] = 25.67 MPa2;Cor. Length [qT ] = 0.61m;Testing Error = 2.78 MPa2

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 22: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

"Uniform" Soil Site

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 23: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Three Approaches to Modeling

I Do nothing about site characterization (rely onexperience): conservative guess of soil data,COV = 225%, correlation length = 12m.

I Do better than standard site characterization:COV = 103%, correlation length = 0.61m)

I Improve site (material) characterization if probabilities ofexceedance are unacceptable!

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 24: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Evolution of Mean ± SD for Guess Case

5

1015

20 25

−400

−200

200

400

600mean ± standard deviation

mean

Dis

plac

emen

t (m

m)

Time (sec)

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 25: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Evolution of Mean ± SD for Real Data Case

5 1015

20 25

−200

−100

100

200

300

400

Time (sec)

Dis

plac

emen

t (m

m) mean ± standard deviation

mean

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 26: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Full PDFs for Real Data Case

I PDF at the finiteelement nodes can beobtained using, e.g.,Edgeworth expansion(Ghanem and Spanos2003)

I Numerous applications,especially where extremestatistics are critical

−400

0

200400

0

0.02

0.04

0.06

5

10

15

20

Tim

e (s

ec)

PDF

Displacement (mm)

−200

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 27: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Example: PDF at 6 s

−1000 −500 500 1000

0.0005

0.001

0.0015

0.002

0.0025

Displacement (mm)

PDF

Real Soil Data

Conservative Guess

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 28: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Example: CDF at 6 s

−1000 −500 500 1000

0.2

0.4

0.6

0.8

1

CD

F

Displacement (mm)

Real Soil DataConservative Guess

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 29: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Seismic Wave Propagation Through Uncertain Soils

Probability of Exceedance of 50cm

5 10 15 20 25

5

10

15

20

25

30

35

Displacement (cm)

Prob

abili

ty o

f E

xcee

danc

e of

50

cm (

%)

With Conservative Guess

With Real Soil Data

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method

Page 30: Stochastic Elastic-Plastic Finite Element Methodsokocalo.engr.ucdavis.edu/~jeremic/...Stochastic Elastic-Plastic Finite Element Method Motivation SEPFEM An Application Summary Seismic

Motivation SEPFEM An Application Summary

Summary

I Behavior of materials is probably probabilistic, and oneprobably has to deal with it (in a probabilistic way)!

I Probabilistic Elasto–Plasticity and StochasticElastic–Plastic Finite Element Methodology has beendeveloped and is being refined

I Human nature: how much do you want to know aboutpotential problem?

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method