Motivation SEPFEM An Application Summary
Stochastic Elastic-Plastic Finite ElementMethod
Boris Jeremic and Kallol Sett
Department of Civil and Environmental EngineeringUniversity of California, Davis
SEECCM 2009
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Outline
MotivationMotivation and Overview
SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations
An ApplicationSeismic Wave Propagation Through Uncertain Soils
Summary
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Motivation and Overview
Outline
MotivationMotivation and Overview
SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations
An ApplicationSeismic Wave Propagation Through Uncertain Soils
Summary
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Motivation and Overview
(Geo–) Materials are Inherently Uncertain
Spatial Variation of Friction Angle(After Mayne et al. (2000))
Typical COVs of Different Soil Properties(After Lacasse and Nadim 1996)
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Motivation and Overview
On UncertaintiesI Epistemic uncertainty - due to lack of knowledge
I Can be reduced bycollecting more data
I Mathematical tools not welldeveloped, trade-off withaleatory uncertainty
Hei
sen
ber
gp
rin
cip
le
un
cert
ain
tyA
leat
ory
un
cert
ain
tyE
pis
tem
ic
Det
erm
inis
tic
I Aleatory uncertainty - inherent variation of physical systemI Can not be reducedI Has highly developed mathematical tools
I Ergodicity – exchange ensemble average for time average?I Applicable to soils, biomaterials, up for discussion for
concrete, rock
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Motivation and Overview
Historical Overview
I Brownian motion, Langevin equation → PDF governed bysimple 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), Monte Carlo method
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Motivation and Overview
Soil Uncertainties and Quantification
I Natural,spatial variability of soil deposit (Fenton 1999)I Function of soil formation process
I Testing (point–wise) error (Stokoe et al. 2004)I Imperfection of instrumentsI Error in methods to register quantities
I Transformation (point–wise) error (Phoon and Kulhawy1999)
I Correlation by empirical data fitting (e.g. CPT data →friction angle etc.)
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Motivation and Overview
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 Newly developed: Stochastic Elastic–Plastic FiniteElement Method
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Probabilistic Elasto–Plasticity
Outline
MotivationMotivation and Overview
SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations
An ApplicationSeismic Wave Propagation Through Uncertain Soils
Summary
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Probabilistic Elasto–Plasticity
Constitutive Problem Setup
I 3D incremental elasto–plasticity:
dσij/dt =
{Del
ijkl −Del
ijmnmmnnpqDelpqkl
nrsDelrstumtu − ξ∗r∗
}dεkl/dt
I Phase density ρ of σ(x , t) varies in time according to acontinuity Liouville equation (Kubo 1963)
I Continuity equation written in ensemble average form (eg.cumulant expansion method (Kavvas and Karakas 1996))
I van Kampen’s lemma (van Kampen 1976) →< ρ(σ, t) >= P(σ, t), ensemble average of phase density isthe probability density
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Probabilistic Elasto–Plasticity
Eulerian–Lagrangian FPK Equation
∂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)
–
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Probabilistic Elasto–Plasticity
Euler–Lagrange FPK EquationI 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 processI Template FPK diffusion–advection equation is applicable to
any material model → only the coefficients N(1) and N(2)
are different for different material modelsJeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Probabilistic Elasto–Plasticity
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 and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
SEPFEM Formulations
Outline
MotivationMotivation and Overview
SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations
An ApplicationSeismic Wave Propagation Through Uncertain Soils
Summary
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
SEPFEM Formulations
Stochastic Finite Element FormulationI Governing equations:
Aσ = φ(t); Bu = ε; σ = Dε
I Spatial and stochastic discretization
I Deterministic spatial differential operators (A & B) →Regular shape function method with Galerkin scheme
I Input random field material properties (D) →Karhunen–Loève (KL) expansion, optimal expansion, errorminimizing property
I Unknown solution random field (u) → Polynomial Chaos(PC) expansion
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application 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
K epmndni +
N∑n=1
P∑j=0
dnj
M∑k=1
CijkK′epmnk = 〈Fmψi [{ξr}]〉
K epmn =
∫D
BnDepBmdV K′epmnk =
∫D
Bn√λkhkBmdV
Cijk =⟨ξk (θ)ψi [{ξr}]ψj [{ξr}]
⟩Fm =
∫DφNmdV
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
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 (parallelism)
I Development of the probabilistic elastic–plastic stiffnesstensor
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
SEPFEM Formulations
1–D Static Pushover Test Example
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
SEPFEM Formulations
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 and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Seismic Wave Propagation Through Uncertain Soils
Outline
MotivationMotivation and Overview
SEPFEMProbabilistic Elasto–PlasticitySEPFEM Formulations
An ApplicationSeismic Wave Propagation Through Uncertain Soils
Summary
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Seismic Wave Propagation Through Uncertain Soils
Seismic Wave Propagation Through Random Soil
Soil is 12.5 m deep 1–D soil column with random von Misesmaterial:
I Shear modulus:〈G〉 = 11.57MPa;Var [G] = 142.32MPa2;Cor. Length [G] = 0.61m
I Shear strength:〈qT 〉 = 4.99 MPa;Var [qT ] = 25.67 MPa2;Cor. Length [qT ] = 0.61m;Testing Error = 2.78 MPa2
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Seismic Wave Propagation Through Uncertain Soils
"Uniform" Soil Site
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Seismic Wave Propagation Through Uncertain Soils
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 (material) characterization if probabilities ofexceedance are unacceptable!
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application 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 and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application 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 and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Seismic Wave Propagation Through Uncertain Soils
Full PDFs for Real Data Case
I PDF at the finiteelement nodes can beobtained using, e.g.,Edgeworth expansion(Ghanem and Spanos2003)
I Numerous applications,especially where extremestatistics are critical
−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
Motivation SEPFEM An Application 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 and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application 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 and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Seismic Wave Propagation Through Uncertain Soils
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
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method
Motivation SEPFEM An Application Summary
Summary
I Behavior of materials is probably probabilistic, and oneprobably has to deal with it (in a probabilistic way)!
I Probabilistic Elasto–Plasticity and StochasticElastic–Plastic Finite Element Methodology has beendeveloped and is being refined
I Human nature: how much do you want to know aboutpotential problem?
Jeremic and Sett Computational Geomechanics Group
Stochastic Elastic-Plastic Finite Element Method