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Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary

Stochastic Elastic-Plastic Finite ElementMethod for Performance Risk Simulations

Boris Jeremic1 Kallol Sett2

1University of California, Davis

2University of Akron, Ohio

ICASPZürich, Switzerland

August 2011

Jeremic and Sett Computational Geomechanics Group

Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations


Outline

Motivation

Probabilistic Elasto–PlasticityPEP FormulationsStochastic Elastic–Plastic Finite Element Method

Applications to Risk AnalysisSeismic Wave Propagation Through Uncertain SoilsProbabilistic Analysis for Decision Making

Summary




Determining Risk for Civil Engineering Object Behavior

I Risk: inherent, intrinsic, constitutive part of civilengineering

I Uncertain loads (!)

I Uncertain materials (!!)

I Uncertain human factor (!)

Mu + Cu + Ku = F




Material Behavior Inherently Uncertain

I Spatialvariability

I Point-wiseuncertainty

I testingerror

I transformationerror

(Mayne et al. (2000)




PEP Formulations

Outline

Motivation



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

I What if all (any) material parameters are uncertainI Since material is inherently spatially variable and uncertain

at the point, PEP and SEPFEM methods were developed




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 the ForwardKolmogorov (Fokker-Planck-Kolmogorov) equation




PEP Formulations

Eulerian–Lagrangian FPK Equation

∂P(σij(xt , t), t)∂t

=∂

∂σmn

[{⟨ηmn(σmn(xt , t),Dmnrs(xt), εrs(xt , t))

⟩+

∫ 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)

]}P(σij(xt , t), t)

]+

∂2

∂σmn∂σab

[{∫ 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))

]}P(σij(xt , t), t)

]




PEP Formulations

Eulerian–Lagrangian FPK EquationI Advection-diffusion equation

∂P(σij , t)∂t

= − ∂

∂σab

[N(1)

ab P(σij , t)−∂

∂σcd

{N(2)

abcdP(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)ab and N(2)

abcd differ)




PEP Formulations

Probabilistic Elastic-Plastic Response

0.25

0.5

0.75

P[Shear Stress]

Shear Stress (MPa)

Shear Strain (%)




SEPFEM

Outline

Motivation



Summary




SEPFEM

Governing Equations & Discretization Scheme

I Governing equations:

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

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

Inside SEPFEM

I Stochastic elastic–plastic (explicit) 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




Seismic Wave Propagation Through Uncertain Soils

Outline

Motivation



Summary





Risk Assessment Applications

I Any problem (Mu + Cu + Ku = F) with knownI PDFs of material parameters,I PDFs of loading

can be analyzed using PEP and SEPFEM to obtain PDFsof DOFs, stress, strain...

I PEP solution is second order accurate (exact mean andstandard deviation)

I SEPFEM solution (PDFs) can be made as accurate asneed be

I Tails of PDFs can than be used to develop accurate riskI Application to a realistic case of seismic wave propagation





"Uniform" CPT Site Data





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





Seismic Wave Propagation through Stochastic Soil

0.5 1 1.5 2Time HsL

-30

-20

-10

10

20

DisplacementHmmL

Mean± Standard DeviationJeremic and Sett Computational Geomechanics Group




Full PDFs All Results

−400

0

200400

0

0.02

0.04

0.06

5

10

15

20

Tim

e (s

ec)

PDF

Displacement (mm)

−200




Probabilistic Analysis for Decision Making

Outline

Motivation



Summary





Example: Three Approaches to Modeling

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

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

I Do the best site (material) characterization to reduceprobabilities of exceedance





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 (Non-Exceedance) 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

Conservative Guess

Displacement (cm)

Real Site

Prob

abili

ty o

f E

xcee

danc

e

ExcellentlyCharacterized Site

of 2

0 cm

(%

)

0 5 10 15 20 25

80

60

40

20

0





Probability of Exceedance of 50cmPr

obab

ility

of

Exc

eeda

nce

of 5

0 cm

(%

)

Displacement (cm)

Conservative Guess

Real Site

Characterized SiteExcellently

80

60

40

20

05 10 15 20 250





Risk of Unacceptable Deformation

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 Behavior of all civil engineering objects (structures, soils...)is probably probabilistic!

I Presented methodology (PEP and SEPFEM) allows for(very) accurate numerical simulation of PDFs of DOFs(and stress, strain) from known (given) PDFs of materialproperties and PDFs of loads.

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



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