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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
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Stochastic Systems: Historical Perspectives
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Stochastic Systems: Historical Perspectives
History
I Probabilistic fish counting
I Williams’ DEM simulations, differential displacementvortexes
I SFEM round table
I Kavvas’ probabilistic hydrology
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Stochastic Systems: Historical Perspectives
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
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Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Stochastic Systems: Historical Perspectives
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Stochastic Systems: Historical Perspectives
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Uncertainties in Material
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Uncertainties in Material
Material Behavior Inherently Uncertainties
I Spatialvariability
I Point-wiseuncertainty,testingerror,transformationerror
(Mayne et al. (2000)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Uncertainties in Material
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.)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Uncertainties in Material
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.
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Uncertainties in Material
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
PEP Formulations
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids 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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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"
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids 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 theFokker-Planck-Kolmogorov equation
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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
P(σij(xt , t), t)–
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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
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
P(σij(xt , t), t)–
I 6 equations, 3D probabilistic stress-strain responseI 3 equations, 3D in principal stress-strain spaceI 2 equations, 2D in principal stress-strain space
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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]
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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
Std. Deviation Lines
(Variable Transformation)
(Fokker−Planck)
(Fokker−Planck)
(Variable Transformation)Mean Line
Mean Line
Time (Sec)
Stre
ss (M
Pa)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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′
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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)
Std. Deviation Lines
Mean Line
Mean Line
Std. Deviation Lines(Fokker−Planck)
(Fokker−Planck)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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
Std. Deviation Lines
Mean LineDeterministic Line
Mode Line
Stre
ss (M
Pa)
50
30
10
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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)
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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
Std. Deviation Lines
Time (s)
0 2.0
Stre
ss (M
Pa)
3050
100
Deterministic LineMode Line
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Elastic–Plastic Response
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 Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
SEPFEM Formulations
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids 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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids 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
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 Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
SEPFEM Verification Example
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
SEPFEM Verification Example
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.
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L
F
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
SEPFEM Verification Example
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.
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
SEPFEM Verification Example
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 Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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)
Standard Deviation of Tangent Modulus (MPa)
Load = 0 kN0.10.150.20.3
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Probabilistic Shear Response
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)
Standard Deviation of Tangent Modulus (MPa)
Load = 0 kN0.10.150.20.3
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
Uncertain Dynamics
I Stochastic elastic–plastic simulations of soils andstructures
I Probabilistic inverse problems
I Geotechnical site characterization design
I Optimal material design
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
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!
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
Random Field Parameters from Site DataI Maximum likelihood estimates
Typical CPT qT
Finite Scale
Fractal
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
"Uniform" CPT Site Data (Courtesy of USGS)
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids 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 Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids 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 Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
Full PDFs for Real Data Case
−400
0
200400
0
0.02
0.04
0.06
5
10
15
20
Tim
e (s
ec)
Displacement (mm)
−200
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids 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)
Real Soil Data
Conservative Guess
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids 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 Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
Seismic Wave Propagation Through Uncertain Soils
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation Probabilistic Elasto–Plasticity SEPFEM Uncertain Response of Solids Summary
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
Jeremic Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method