the case for probabilistic elasto--plasticitysokocalo.engr.ucdavis.edu/~jeremic/ · (after lacasse...

Motivation Probabilistic Elasto–Plasticity SEPFEM Applications Summary

The Case for Probabilistic Elasto–Plasticity

Boris JeremicKallol Sett (UA) and Lev Kavvas

Department of Civil and Environmental EngineeringUniversity of California, Davis

GheoMatMasseria Salamina

Italy, June 2009

Jeremic Computational Geomechanics Group



OutlineMotivation

Stochastic Systems: Historical PerspectivesUncertainties in Material

Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response

Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example

ApplicationsSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making

Summary




Stochastic Systems: Historical Perspectives

OutlineMotivation





Summary





Failure Mechanisms for Geomaterials

Soil: Inside Failure of "Uniform" MGM Specimen(After Swanson et al. 1998)Jeremic Computational Geomechanics Group




Personal Motivation

I Probabilistic fish counting

I Williams’ DEM simulations, differential displacementvortices

I SFEM round table

I Kavvas’ probabilistic hydrology





Types of UncertaintiesI Epistemic uncertainty - due to lack of knowledge

I Can be reduced by collecting more dataI Mathematical tools are not well developedI trade-off with aleatory uncertainty

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

Hei

sen

ber

gp

rin

cip

le

un

cert

ain

tyA

leat

ory

un

cert

ain

tyE

pis

tem

ic

Det

erm

inis

tic





ErgodicityI Exchange ensemble averages for time averages

I Is soil elasto-plasticity ergodic?I Can soil elastic–plastic statistical properties be obtained by

temporal averaging?I Will soil elastic–plastic statistical properties "renew" at each

occurrence?I Are soil elastic–plastic statistical properties statistically

independent?

I Claim in literature that structural nonlinear behavior isnon–ergodic while earthquake characteristics are (?!)

I However, earthquake characteristics is representingmechanics (fault slip) on a different scale...





Historical OverviewI Brownian motion, Langevin equation → PDF governed by

simple 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), Numerical approaches, MonteCarlo method




Uncertainties in Material

OutlineMotivation





Summary





Material Behavior Inherently Uncertain

I Spatialvariability

I Point-wiseuncertainty,testingerror,transformationerror

(Mayne et al. (2000)





Motivation

Typical Coefficients of Variation of Different Soil Properties

(After Lacasse and Nadim 1996)Jeremic Computational Geomechanics Group




Soil Uncertainties and Quantification

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

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

I Transformation error (Phoon and Kulhawy 1999)I Correlation by empirical data fitting (e.g. CPT data →

friction angle etc.)





Probabilistic Material (Soil Site) Characterization

I Ideal: complete probabilistic site characterizationI Large (physically large but not statistically) amount of data

I Site specific mean and coefficient of variation (COV)I Covariance structure from similar sites (e.g. Fenton 1999)

I Moderate amount of data → Bayesian updating (e.g.Phoon and Kulhawy 1999, Baecher and Christian 2003)

I Minimal data: general guidelines for typical sites and testmethods (Phoon and Kulhawy (1999))

I COVs and covariance structures of inherent variabilityI COVs of testing errors and transformation uncertainties.





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 New developments for elasto–plastic applications




PEP Formulations

OutlineMotivation





Summary




PEP Formulations

Uncertainty Propagation through Constitutive Eq.

I Incremental el–pl constitutive equationdσij

dt= Dijkl

dεkl

dt

Dijkl =

Del

ijkl for elastic

Delijkl −

DelijmnmmnnpqDel

pqkl

nrsDelrstumtu − ξ∗r∗

for elastic–plastic




PEP Formulations

Previous WorkI Linear algebraic or differential equations → Analytical

solution:I Variable Transf. Method (Montgomery and Runger 2003)I Cumulant Expansion Method (Gardiner 2004)

I Nonlinear differential equations(elasto-plastic/viscoelastic-viscoplastic):

I Monte Carlo Simulation (Schueller 1997, De Lima et al2001, Mellah et al. 2000, Griffiths et al. 2005...)→ accurate, very costly

I Perturbation Method (Anders and Hori 2000, Kleiber andHien 1992, Matthies et al. 1997)→ first and second order Taylor series expansion aboutmean - limited to problems with small C.O.V. and inherits"closure problem"




PEP Formulations

Problem Statement

I Incremental 3D elastic-plastic stress–strain:

dσij

dt=

{Del

ijkl −Del

ijmnmmnnpqDelpqkl

nrsDelrstumtu − ξ∗r∗

}dεkl

dt

I Focus on 1D → a nonlinear ODE with random coefficient(material) and random forcing (ε)

dσ(x , t)dt

= β(σ(x , t),Del(x),q(x), r(x); x , t)dε(x , t)

dt= η(σ,Del ,q, r , ε; x , t)

with initial condition σ(0) = σ0




PEP Formulations

Evolution of the Density ρ(σ, t)

I From each initial point inσ-space a trajectorystarts out describingthe corresponding solutionof the stochastic process

I Movement of a cloud of initialpoints described by densityρ(σ,0) in σ-space,is governed by theconstitutive equation,

t

P( )

σσ




PEP Formulations

Stochastic Continuity (Liouville) Equation

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

∂ρ(σ(x , t), t)∂t

=

−∂η(σ(x , t),Del(x),q(x), r(x), ε(x , t))∂σ

ρ[σ(x , t), t ]

I with initial conditions ρ(σ,0) = δ(σ − σ0)




PEP Formulations

Ensemble Average form of Liouville EquationContinuity equation written in ensemble average form (eg.cumulant expansion method (Kavvas and Karakas 1996)):

∂ 〈ρ(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)

–ff〈ρ(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))

– ff〈ρ(σ(xt , t), t)〉

–




PEP Formulations

Eulerian–Lagrangian FPK Equationvan Kampen’s Lemma (van Kampen 1976) → < ρ(σ, t) >= P(σ, t),ensemble average of phase density is the probability density;

∂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)

–




PEP Formulations

E–L FPK Equation

I 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 process




PEP Formulations

Template Solution of FPK EquationI FPK diffusion–advection equation is applicable to any

material model → only the coefficients N(1) and N(2) aredifferent for different material models

∂P(σ, t)∂t

= − ∂

∂σ

[N(1)P(σ, t)− ∂

∂σ

{N(2)P(σ, t)

}]= −∂ζ

∂σ

I Initial conditionI Deterministic → Dirac delta function → P(σ,0) = δ(σ)I Random → Any given distribution

I Boundary condition: Reflecting BC → conservesprobability mass ζ(σ, t)|At Boundaries = 0

I Finite Differences used for solution (among many others)




PEP Formulations

Application of FPK equation to Material Models

I FPK equation is applicable to any incrementalelastic–plastic material model

I Solution in terms of PDF, not a single value of stress

I Influence of initial condition on the PDF of stress

I Mean stress yielding or

I Probabilistic yielding




Probabilistic Elastic–Plastic Response

OutlineMotivation





Summary





Elastic Response with Random GI General form of elastic constitutive rate equation

dσ12

dt= 2G

dε12

dt= η(G, ε12; t)

I Advection and diffusion coefficients of FPK equation

N(1) = 2dε12

dt< G >

N(2) = 4t(

dε12

dt

)2

Var [G]





Elastic Response with Random G

0.002

0.004

0.006

0

0.001

0.002

0.003

0.0040

2500

5000

7500

10000

0.00789

Strain (%)

0.0027

0.0426

Stress (MPa)Time (Sec)

Prob Density

0.002

0.004

0.006

< G > = 2.5 MPa; Std. Deviation[G] = 0.5 MPaJeremic Computational Geomechanics Group




Verification – Variable Transformation Method

0.002 0.004 0.006 0.008

0.0005

0.001

0.0015

0.002

0.0025

Strain (%)0 0.0426

Std. Deviation Lines


(Variable Transformation)

(Fokker−Planck)

(Fokker−Planck)

(Variable Transformation)Mean Line

Mean Line

Time (Sec)

Stre

ss (M

Pa)





Modified Cam Clay Constitutive Modeldσ12

dt= Gep dε12

dt= η(σ12,G,M,e0,p0, λ, κ, ε12; t)

η =

2G −

(36

G2

M4

)σ2

12

(1 + e0)p(2p − p0)2

κ+

(18

GM4

)σ2

12 +1 + e0

λ− κpp0(2p − p0)

Advection and diffusion coefficients of FPK equation

N(i)(1) =

⟨η(i)(t)

⟩+

∫ t

0dτcov

[∂η(i)(t)∂t

; η(i)(t − τ)

]

N(i)(2) =

∫ t

0dτcov

[η(i)(t); η(i)(t − τ)

]Jeremic Computational Geomechanics Group




Low OCR Cam Clay with Random G, M and p0

I Non-symmetry inprobabilitydistribution

I Differencebetweenmean, mode anddeterministic

I Divergence atcritical statebecause M isuncertain

0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.01

0.02

0.03

0.04

Time (s)

Strain (%)0 1.62


Mean LineDeterministic Line

Mode Line

Stre

ss (M

Pa)

50

30

10





Comparison of Low OCR Cam Clay at ε = 1.62 %

0.01 0.02 0.03 0.04

100

200

300

400

500

Det

. Val

ue =

0.0

2906

MPa

Shear Stress (MPa)

PDF

of S

hear

Str

ess

Random G, M, p0Random G, M

Random G, p0

Random G

I None coincides with deterministicI Some very uncertain, some very certainI Either on safe or unsafe side





High OCR Cam Clay with Random G and M

I Large non-symmetryin probabilitydistribution

I Significantdifferences inmean, mode,and deterministic

I Divergence atcritical state,M is uncertain 0 0.05 0.1 0.15 0.2 0.25 0.3 0.37

0

0.005

0.01

0.015

0.02

0.025

0.03

Strain (%)

Mean Line


Time (s)

0 2.0

Stre

ss (M

Pa)

3050

100

Deterministic LineMode Line





Probabilistic Yielding

I Weighted elastic and elastic–plastic Solution∂P(σ, t)/∂t = −∂

(Nw

(1)P(σ, t)− ∂(

Nw(2)P(σ, t)

)/∂σ

)/∂σ

I Weighted advection and diffusion coefficients are thenNw

(1,2)(σ) = (1− P[Σy ≤ σ])Nel(1) + P[Σy ≤ σ]Nel−pl

(1)

I Cumulative Probability Density function (CDF) of the yieldfunction

0.00005 0.0001 0.00015Σy HMPaL

0.2

0.4

0.6

0.8

1F@SyD = P@Sy£ΣyD

(a)(b)





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

0.0003

Stre

ss (

MPa

)

Strain (%)

Mode

DeterministicSolutionStd. Deviations

Mean

linear elastic – linear hardening plastic von MisesJeremic Computational Geomechanics Group




SPT Based Determination of Shear Strength

10 20 30 40

100

200

300

400

500

SPT N value

Su = (101.125*0.29) N 0.72

Und

rain

ed S

hear

Str

engt

h (k

Pa)

−300 −200 −100 0 100 200

0.002

0.004

0.006

0.008

0.01

Nor

mal

ized

Fre

quen

cy

Residual (w.r.t Mean) Undrained Shear Strength (kPa)

Transformation relationship between SPT N-value andundrained shear strength, su (cf. Phoon and Kulhawy (1999B)Histogram of the residual (w.r.t the deterministic transformationequation) undrained strength, along with fitted probabilitydensity function





SPT Based Determination of Young’s Modulus

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

Transformation relationship between SPT N-value andpressure-meter Young’s modulus, E (cf. Phoon and Kulhawy(1999B))Histogram of the residual (w.r.t the deterministic transformationequation) Young’s modulus, along with fitted probability densityfunction





Cyclic Response of Such Uncertain Material

−1 −0.5 0.5 1

−0.2

−0.1

0.1

0.2

Deterministic

Mean

She

ar S

tres

s (M

Pa)

Shear Strain (%)

−1 −0.5 0.5 1

0.05

0.1

0.15

0.2

Shear Strain (%)

She

ar S

tres

s (M

Pa)

Standard Deviation





G/Gmax Response

0.01 0.02 0.05 0.1 0.2 0.5 1

0.2

0.4

0.6

0.8

1

1.2

Shear Strain (%)

PI=200% (Vucetic and Dobry 1991)


PI=100% (Stokoe et al. 2004)Deterministic

Mean

G/G

max

Mean±Standard Deviation





Damping Response

0.01 0.02 0.05 0.1 0.2 0.5 1

5

10

15

20

Dam

ping

Rat

io (

%)

Shear Strain (%)

Corresponding to hysteresisloop of Mean of shear stress

Corresponding to hysteresis loop ofMean±Standard Deviation of shear stress

Deterministic

PI=100% (Stokoe et al. 2004)






SEPFEM Formulations

OutlineMotivation





Summary




SEPFEM Formulations

Governing Equations & Discretization Scheme

I Governing equations in geomechanics:

Aσ = φ(t); Bu = ε; σ = Dε

I Discretization (spatial and stochastic) schemesI Input random field material properties (D) →

Karhunen–Loève (KL) expansion, optimal expansion, errorminimizing property

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

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




SEPFEM Formulations

Spectral Stochastic Elastic–Plastic FEM

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

N∑n=1

Kmndni +N∑

n=1

P∑j=0

dnj

M∑k=1

CijkK ′mnk = 〈Fmψi [{ξr}]〉

Kmn =

∫D

BnDBmdV K ′mnk =

∫D

Bn√λkhkBmdV

Cijk =⟨ξk (θ)ψi [{ξr}]ψj [{ξr}]

⟩Fm =

∫DφNmdV




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

I Development of the probabilistic elastic–plastic stiffnesstensor




SEPFEM Verification Example

OutlineMotivation





Summary





1–D Static Pushover Test Example

I Linear elastic model:< G >= 2.5 kPa,Var [G] = 0.15 kPa2,correlation length for G = 0.3 m.

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





Linear Elastic FEM Verification

1 2 3 4 5

0.02

0.04

0.06

0.08

0.1

Loa

d (N

)

Displacement at Top Node (mm)

Mean (SFEM)

Standard Deviations (MC)

Mean (MC)

Standard Deviations (SFEM)

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





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.




Seismic Wave Propagation Through Uncertain Soils

OutlineMotivation





Summary





Applications

I Stochastic elastic–plastic simulations of soils andstructures

I Probabilistic inverse problems

I Geotechnical site characterization design

I Optimal material design





Seismic Wave Propagation through Stochastic Soil

I Soil as 12.5 m deep 1–D soil column (von Mises Material)I Properties (including testing uncertainty) obtained through

random field modeling of CPT qT〈qT 〉 = 4.99 MPa; Var [qT ] = 25.67 MPa2;Cor. Length [qT ] = 0.61 m; Testing Error = 2.78 MPa2

I qT was transformed to obtain G: G/(1− ν) = 2.9qT

I Assumed transformation uncertainty = 5%〈G〉 = 11.57MPa; Var [G] = 142.32MPa2

Cor. Length [G] = 0.61m

I Input motions: modified 1938 Imperial Valley





Random Field Parameters from Site DataI Maximum likelihood estimates

Typical CPT qT

Finite Scale

Fractal





"Uniform" CPT Site Data





Seismic Wave Propagation through Stochastic Soil

0.5 1 1.5 2Time HsL

-30

-20

-10

10

20

DisplacementHmmL

Mean± Standard DeviationJeremic Computational Geomechanics Group



Probabilistic Analysis for Decision Making

OutlineMotivation





Summary





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 characterization if probabilities of exceedanceare unacceptable!





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)





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





Full PDFs for Real Data Case

−400

0

200400

0

0.02

0.04

0.06

5

10

15

20

Tim

e (s

ec)

PDF

Displacement (mm)

−200





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





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





Probability of Exceedance of 20cm

5 10 15 20 25

10

20

30

40

50

60

70

Prob

abili

ty o

f E

xcee

danc

e of

20

cm (

%)

Displacement (cm)

With Conservative Guess

With Real Soil Data





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





Probabilities of Exceedance vs. Displacements

10 20 30 40 50

20

40

60

80

Prob

abili

ty o

fE

xcee

danc

e (%

)

Displacemnent (cm)

Real Soil DataConservative Guess




Summary

I Behavior of materials is probably probabilistic!

I Technical developments are available and are being refined

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



the case for probabilistic elasto--plasticitysokocalo.engr.ucdavis.edu/~jeremic/ · (after lacasse...

Documents