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Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary

Stochastic Elastic-PlasticFinite Element Method

Boris Jeremic

In collaboration withKallol Sett, You-Chen Chao and Levent Kavvas

CEE Dept., University of California, Davisand

ESD, Lawrence Berkeley National Laboratory

Intel Corporation Webinar Series,October 2010

Jeremic Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method


OutlineMotivation

Stochastic Systems: Historical PerspectivesUncertainties in Material

Probabilistic Elasto–PlasticityPEP FormulationsProbabilistic Elastic–Plastic Response

Stochastic Elastic–Plastic Finite Element MethodSEPFEM FormulationsSEPFEM Verification Example

Uncertain Response of SolidsProbabilistic Shear ResponseSeismic Wave Propagation Through Uncertain Soils

Summary




Stochastic Systems: Historical Perspectives

OutlineMotivation





Summary





History

I Probabilistic fish counting

I Williams’ DEM simulations, differential displacementvortexes

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

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ory

un

cert

ain

tyE

pis

tem

ic

Det

erm

inis

tic





Ergodicity

I Exchange ensemble averages for time averages

I Is elasto-plastic-damage response ergodic?I Can material elastic-plastic-damage statistical properties

be obtained by temporal averaging?I Will elastic-plastic-damage statistical properties "renew" at

each occurrence?I Are material elastic-plastic-damage statistical properties

statistically independent?

I Important assumptions must be made and justified





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 Uncertainties

I Spatialvariability

I Point-wiseuncertainty,testingerror,transformationerror

(Mayne et al. (2000)





Soil Uncertainties and Quantification

I Natural variability of material (for soil, Fenton 1999)I Function of material formation process

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

I Transformation error (for soil, Phoon and Kulhawy 1999)I Correlation by empirical data fitting (for soil, 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-damage 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-damage 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

Solution to Probabilistic Elastic-Plastic Problem

I Use of stochastic continuity (Liouiville) equation (Kubo1963)

I With cumulant expansion method (Kavvas and Karakas1996)

I To obtain ensemble average form of Liouville EquationI Which, with van Kampen’s Lemma (van Kampen 1976):

ensemble average of phase density is the probabilitydensity

I Yields Eulerian-Lagrangian form of theFokker-Planck-Kolmogorov equation




PEP Formulations

Eulerian–Lagrangian FPK Equation

∂P(σij(xt , t), t)∂t

=∂

∂σmn

»fiηmn(σmn(xt , t),Dmnrs(xt), εrs(xt , t))

fl+

Z t

0dτCov0

»∂ηmn(σmn(xt , t),Dmnrs(xt), εrs(xt , t))

∂σab;

ηab(σab(xt−τ , t − τ),Dabcd(xt−τ ), εcd(xt−τ , t − τ)–ff

P(σij(xt , t), t)–

+∂2

∂σmn∂σab

»Z t

0dτCov0

»ηmn(σmn(xt , t),Dmnrs(xt), εrs(xt , t));

ηab(σab(xt−τ , t − τ),Dabcd(xt−τ ), εcd(xt−τ , t − τ))–ff





PEP Formulations

Eulerian–Lagrangian FPK EquationI Advection-diffusion equation

∂P(σij , t)∂t

= − ∂

∂σmn

[N(1)P(σij , t)−

∂

∂σab

{N(2)P(σij , t)

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

(exact mean and variance)I Deterministic equation in probability density spaceI Linear PDE in probability density space→ simplifies the

numerical solution processI Applicable to any elastic-plastic-damage material model

(only coefficients N(1) and N(2) differ)




PEP Formulations

3D FPK Equation

∂P(σij(xt , t), t)∂t

=∂

∂σmn

»fiηmn(σmn(xt , t),Dmnrs(xt), εrs(xt , t))

fl+

Z t

0dτCov0

»∂ηmn(σmn(xt , t),Dmnrs(xt), εrs(xt , t))

∂σab;

ηab(σab(xt−τ , t − τ),Dabcd(xt−τ ), εcd(xt−τ , t − τ)–ff


+∂2

∂σmn∂σab

»Z t

0dτCov0

»ηmn(σmn(xt , t),Dmnrs(xt), εrs(xt , t));

ηab(σab(xt−τ , t − τ),Dabcd(xt−τ ), εcd(xt−τ , t − τ))–ff


I 6 equations, 3D probabilistic stress-strain responseI 3 equations, 3D in principal stress-strain spaceI 2 equations, 2D in principal stress-strain space




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)





Drucker-Prager Linear Hardening with Random G

dσ12

dt= Gep dε12

dt= η(σ12,G,K , α, α′, ε12; t)

Advection and diffusion coefficients of FPK equation

N(1) =dε12

dt

⟨2G − G2

G + 9Kα2 +1√3

I1α′

⟩

N(2) = t(

dε12

dt

)2

Var

2G − G2

G + 9Kα2 +1√3

I1α′





Verification of D–P E–P Response - Monte Carlo

0.004 0.008 0.012 0.01489

0.001

0.002

0.003

Time (Sec)0

Strain (%) 0.08040

Stre

ss (M

Pa)

(Actual)

(Actual)


Mean Line

Mean Line

Std. Deviation Lines(Fokker−Planck)

(Fokker−Planck)





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

0.0003St

ress

(M

Pa)

Strain (%)

Mode

DeterministicSolutionStd. Deviations

Mean




SEPFEM Formulations

OutlineMotivation





Summary




SEPFEM Formulations

Governing Equations & Discretization Scheme

I Governing equations of mechanics:

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

Truncated Karhunen–Loève (KL) expansion

I Representation of input random fields in eigen-modes ofcovariance kernel

qT (x , θ) = qT (x) +∑M

n=1√λnξn(θ)fn(x)∫

D C(x1, x2)f (x2)dx2 = λf (x1)

ξi(θ) =1√λi

∫D

[qT (x , θ)− qT (x)] fi(x)dx

I Error minimizing propertyI Optimal expansion→ minimization of number of stochastic

dimensions




SEPFEM Formulations

Polynomial Chaos (PC) ExpansionI Covariance kernel is not known a priori

u(x , θ) =L∑

j=1

ejχj(θ)bj(x)

I Can be expressed as functional of known random variablesand unknown deterministic function

u(x , θ) = ζ[ξi(θ), x ]

I Need a basis of known random variables→ PC expansion

χj(θ) =P∑

i=0

γ(j)i ψi [{ξr}]

u(x , θ) =L∑

j=1

P∑i=0

γ(j)i ψi [{ξr}]ejbj(x) =

P∑i=0

ψi [{ξr}]di(x)




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-damage 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-damagestiffness tensor




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.




Probabilistic Shear Response

OutlineMotivation





Summary





Probabilistic Material Response

0 0.135 0.27 0.405 0.54 0.675 0.81 0.945

0.1

0.0

0.2

0.3

0.4

Mean

Mode

She

ar S

tres

s (M

Pa)

Shear Strain (%)

400

200

150

100

25

300

Mean ± Standard Deviation

500





Elasticity and Strength, Mean and Std.Dev.

8040

12

10

8

6

4

2

00

Dep

th (

m)

Shear Modulus (MPa)Mean of

0 40 80

12

10

8

6

4

2

0

Dep

th (

m)

Standard Deviation of Shear Modulus (MPa)

0.20.0 0.4

0.412

10

8

6

4

2

0

Dep

th (

m)

Shear Strength (MPa)Mean of

0.0 0.2 0.4

12

10

8

6

4

2

0

Dep

th (

m)

Standard Deviation ofShear Strength (MPa)





Displacement Results at the Top

0.000 0.005 0.010 0.015 0.0200.00

0.10

0.20

0.30

Displacement (m)

Loa

d (k

N)

Mean

Mean ± Standard Deviation





Statistics of the Tangent Stiffness:Mean and λ for the 1st and 2nd KL-spaces

0.00 0.05 0.10 0.15 0.20 0.25 0.300

10

20

30

40

50

60

Load (kN)

Mea

n of

Tan

gent

Mod

ulus

(M

Pa)

0.00 0.05 0.10 0.15 0.20 0.25 0.300

200

400

600

800

λ

at 1st KL−space

Load (kN)

at 2nd KL−space





Mean Stiffness Evolution, Cor.Len. 0.1m, 1.0m

0 10 20 30 40 50 6012

10

8

6

4

2

0

Dep

th (

m)

Mean of Tangent Modulus (MPa)

Load = 0 kN0.10.3 0.2 0.15

0 10 20 30 40 50 6012

10

8

6

4

2

0

Dep

th (

m)

Load =0 kN0.10.150.20.3

Mean of Tangent Modulus (MPa)





Mean Prob. Stiffness (√λ1 f1) Evolution,

Cor.Len. 0.1m, 1.0m

0 10 20 30 4012

10

8

6

4

2

0

Dep

th (

m)

√λ

Load = 0 kN0.10.150.2

0.3

f at 1st KL−space

0 10 20 30 4012

10

8

6

4

2

0

Dep

th (

m)

f at 1st KL−space√λ

Load = 0 kN0.10.150.20.3





Mean Prob. Stiffness (√λ2 f2) Evolution,

Cor.Len. 0.1m, 1.0m

−20 0 20 4012

10

8

6

4

2

0

Dep

th (

m)

−40f at 2nd KL−space√λ

0 kNLoad =

0.10.3 0.2

−40 −20 0 20 4012

10

8

6

4

2

0

Dep

th (

m)

√λ f at 2nd KL−space

Load = 0 kN 0.1 0.30.2





Std.Dev Stiffness Evolution,Cor.Len. 0.1m, 1.0m

0 10 20 30 40 50 6012

10

8

6

4

2

0

Dep

th (

m)

Standard Deviation of Tangent Modulus (MPa)

Load = 0 kN0.10.20.3

0 10 20 30 40 50 6012

10

8

6

4

2

0

Dep

th (

m)


Load = 0 kN0.10.150.20.3





Probability for Softening (?), Cor.Len. 1.0m

0 10 20 30 40 50 6012

10

8

6

4

2

0

Dep

th (

m)

Load =0 kN0.10.150.20.3

Mean of Tangent Modulus (MPa)0 10 20 30 40 50 60

12

10

8

6

4

2

0

Dep

th (

m)


Load = 0 kN0.10.150.20.3




Seismic Wave Propagation Through Uncertain Soils

OutlineMotivation





Summary





Uncertain Dynamics

I Stochastic elastic–plastic simulations of soils andstructures

I Probabilistic inverse problems

I Geotechnical site characterization design

I Optimal material design





Decision About Site (Material) Characterization

I Do an inadequate site characterization (rely onexperience): conservative guess for soil data,COV = 225%, large correlation length (length of a model).

I Do a good site characterization: COV = 103%, correlationlength calculated (= 0.61m)

I Do an excellent (much improved) site characterization ifprobabilities of exceedance are unacceptable!





Random Field Parameters from Site DataI Maximum likelihood estimates

Typical CPT qT

Finite Scale

Fractal





"Uniform" CPT Site Data (Courtesy of USGS)





Statistics of Stochastic Soil Profile(s)

I Soil as 12.5m 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





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 Unacceptable Deformation (50cm)P

robab

ilit

y o

f E

xce

edan

ceof

50 c

m (

%)

Time (s)

Conservative Guess

Real Site

Characterized SiteExcellently

80

60

40

20

05 10 15 20 250





Risk Informed Decision Process

10 20 30 40 50

20

40

60

80

Prob

abili

ty o

f E

xcee

danc

e (%

)

Displacement (cm)

ConservativeGuessReal Site

Excellently Characterized Site




Summary

I Second-order (mean and variance) exact, method forsimulations of solids with probabilisticelastic-plastic-damage material and stochastic loading

I Input:I elastic-plastic-damage point-wise uncertainty (PDFs of

material paramaters, LHS),I spatial uncertainty (KL eigen modes, LHS) andI uncertain loading (RHS)

I Output: accurate PDFs and CDFs of displacements,velocities, accelerations, stress, strain...

I Probably numerous applications

