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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
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis 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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Material Behavior Inherently Uncertain
I Spatialvariability
I Point-wiseuncertainty
I testingerror
I transformationerror
(Mayne et al. (2000)
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
PEP Formulations
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis 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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
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)
]
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
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)
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
PEP Formulations
Probabilistic Elastic-Plastic Response
0.25
0.5
0.75
P[Shear Stress]
Shear Stress (MPa)
Shear Strain (%)
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
SEPFEM
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis 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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Seismic Wave Propagation Through Uncertain Soils
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Seismic Wave Propagation Through Uncertain Soils
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Seismic Wave Propagation Through Uncertain Soils
"Uniform" CPT Site Data
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Seismic Wave Propagation Through Uncertain Soils
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Seismic Wave Propagation Through Uncertain Soils
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
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Seismic Wave Propagation Through Uncertain Soils
Full PDFs All Results
−400
0
200400
0
0.02
0.04
0.06
5
10
15
20
Tim
e (s
ec)
Displacement (mm)
−200
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Probabilistic Analysis for Decision Making
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Probabilistic Analysis for Decision Making
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Probabilistic Analysis for Decision Making
Example: PDF at 6 s
−1000 −500 500 1000
0.0005
0.001
0.0015
0.002
0.0025
Displacement (mm)
Real Soil Data
Conservative Guess
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Probabilistic Analysis for Decision Making
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Probabilistic Analysis for Decision Making
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Probabilistic Analysis for Decision Making
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
Probabilistic Analysis for Decision Making
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations
Motivation Probabilistic Elasto–Plasticity Applications to Risk Analysis Summary
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?
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method for Performance Risk Simulations