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Introduction FC-KL-exp. Numerical Studies Summary
Discretization of Random FieldsBased on the Karhunen-Loeve Expansion
Using the Finite Cell Method
Wolfgang BetzEngineering Risk Analysis Group
TU Munchen
Presentation of the Master’s thesis at SOFiSTiK
2012-08-03
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 1/21
Introduction FC-KL-exp. Numerical Studies Summary
1 IntroductionRandom fieldsEOLEKL-expansion
2 FC-KL-exp.pFEM-KL-exp.FC-KL-exp.Integration
3 Numerical Studies2D-Example
4 Summary
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 2/21
Introduction FC-KL-exp. Numerical Studies Summary Random fields EOLE KL-expansion
Motivation
Application of random fields - Examples:
soil properties in geotechnical engineering
groundwater heights
rainfall
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 3/21
Introduction FC-KL-exp. Numerical Studies Summary Random fields EOLE KL-expansion
Notation
random field (RF): H(x)
Gaussian random field - completely described by:mean function µ(x)covariance function Cov(x,x′) = σ(x) · σ(x′) · ρ(x,x′)
σ(x): standard deviation functionρ(x,x′): correlation coefficient function
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 4/21
Introduction FC-KL-exp. Numerical Studies Summary Random fields EOLE KL-expansion
Random field discretization
Number of random variables (RVs) in a random field
theoretically: infinite number of RVs (∞)
for each x ∈ Ω, H(x) represents a RV
discretized RF: finite number of RVs (M)
H(x)discretization−−−−−−−→ H(x) (1)
Categories of RF-discretization methods
point discretization methods
averaging discretization methods
series expansion methodsKarhunen-Loeve (KL) expansionEOLE method
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 5/21
Introduction FC-KL-exp. Numerical Studies Summary Random fields EOLE KL-expansion
EOLE method - basic idea
model a RV χi at each xi(Σχχ)nm = Cov(xn,xm)
solve eigenvalue problem:
ΣχχΦi = θiΦi (for M largest θi)
H(x) = µ(x) +∑M
i=1
√θihi(x)ξi
hi(x) = ΦTi b(x)
find bT (x) such thatminimize Var[εH(x)] subjected to E[εH ] = 0
EOLE-expansion
H(x) = µ(x) +
M∑i=1
ΦTi Σχx(x)√
θiξi (2)
with (Σχx(x))j = Cov(xj ,x)
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 6/21
Introduction FC-KL-exp. Numerical Studies Summary Random fields EOLE KL-expansion
EOLE method
EOLE-expansion
H(x) = µ(x) +
M∑i=1
ΦTi Σχx(x)√
θiξi (3)
(Σχx(x))j = Cov(xj ,x)
solve ΣχχΦi = θiΦi with (Σχχ)nm = Cov(xn,xm)
Note: geometry of Ω appears only indirectly
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 7/21
Introduction FC-KL-exp. Numerical Studies Summary Random fields EOLE KL-expansion
Karhunen-Loeve expansion
KL-expansion
H(x) = µ(x) +
∞∑i=1
√λi ϕi(x) ξi (4)
λi: eigenvalues of the covariance kernel
ϕi: eigenfunctions of the covariance kernel
orthonormal:∫Ωϕi(x)ϕj(x) dx = δij
ξi: uncorrelated standard normal RVs
orthonormal: E[ξiξj ] = δij
Integral eigenvalue problem∫x′∈Ω
ϕi(x′)Cov(x,x′) dx′ = λiϕi(x) (5)
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 8/21
Introduction FC-KL-exp. Numerical Studies Summary Random fields EOLE KL-expansion
Truncated KL-expansion
KL-expansion (exact representation)
H(x) = µ(x) +
∞∑i=1
√λi ϕi(x) ξi (6)
Truncated KL-expansion (approximation)
H(x) = µ(x) +
M∑i=1
√λi ϕi(x) ξi (7)
λi: M largest eigenvalues (in descending order)
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 9/21
Introduction FC-KL-exp. Numerical Studies Summary pFEM-KL-exp. FC-KL-exp. Integration
Approximation of the KL-eigenfunctions
Integral eigenvalue problem (KL-expansion)∫x′∈Ω
ϕi(x′)Cov(x,x′) dx′ = λiϕi(x) (8)
Approximation of the eigenfunctions
ϕi(x) =
N∑n=1
dinNn(x) = dTi N(x) (9)
with Nn(x) ∈ L2(Ω)
Remember - EOLE: hi(x) = ΦTi b(x); Φi known a priori.
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 10/21
Introduction FC-KL-exp. Numerical Studies Summary pFEM-KL-exp. FC-KL-exp. Integration
Minimization of the resulting error
Approximated integral eigenvalue problem∫x′∈Ω
ϕi(x′)Cov(x,x′) dx′ − λiϕi(x) = εiN (x) (10)
Minimization of the resulting error (Galerkin)∫ΩεiN (x)Nk(x) dx = 0 (11)
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 11/21
Introduction FC-KL-exp. Numerical Studies Summary pFEM-KL-exp. FC-KL-exp. Integration
Matrix eigenvalue problem
Matrix eigenvalue problem
Bdi = λiMdi (12)
where
Bkn =
∫x∈Ω
Nk(x)
∫x′∈Ω
Nn(x′)Cov(x,x′) dx′ dx (13)
Mij =
∫x∈Ω
Ni(x)Nj(x) dx (14)
Approximated truncated KL-expansion
H(x) = µ(x) +
M∑i=1
√λi ϕi(x) ξi (15)
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 12/21
Introduction FC-KL-exp. Numerical Studies Summary pFEM-KL-exp. FC-KL-exp. Integration
Finite cell - basic idea
1 2
3 4
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 13/21
Introduction FC-KL-exp. Numerical Studies Summary pFEM-KL-exp. FC-KL-exp. Integration
Finite cell - notation
global shape functions: Ni ∈ L2(Ω∗)
α(x) =
1 ∀x ∈ Ω
0 ∀x ∈ Ω∗ \ Ω(16)
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 14/21
Introduction FC-KL-exp. Numerical Studies Summary pFEM-KL-exp. FC-KL-exp. Integration
Finite cell approach of the pFEM-KL-expansion
Matrix eigenvalue problem
Bdi = λiMdi (17)
where
Bkn =
∫x∈Ω∗
α(x)Nk(x)
∫x′∈Ω∗
α(x′)Nn(x′)Cov(x,x′) dx′ dx (18)
Mij =
∫x∈Ω∗
α(x)Ni(x)Nj(x) dx (19)
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 15/21
Introduction FC-KL-exp. Numerical Studies Summary pFEM-KL-exp. FC-KL-exp. Integration
Staggered Gaussian integration
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 16/21
Introduction FC-KL-exp. Numerical Studies Summary 2D-Example
Error variance
Error variance
εσ(x) =Var
[H(x)− H(x)
]σ2(x)
(20)
Mean error variance
ε =
∫Ω εσ(x) dx
|Ω|(21)
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 17/21
Introduction FC-KL-exp. Numerical Studies Summary 2D-Example
Example of a plate with a hole
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 18/21
Introduction FC-KL-exp. Numerical Studies Summary 2D-Example
M = 100: relative error
0.0001
0.001
0.01
0.1
1
100 1000 10000
|ε N−ε r
ef|
ε ref
Ndof
FC (trunk)hFEMEOLE
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 19/21
Introduction FC-KL-exp. Numerical Studies Summary 2D-Example
M = 100: time
0.01
0.1
1
10
100
1000
10000
0.00010.0010.010.11
tim
e[s
]
|εN−εref |εref
FC (trunk)hFEMEOLE
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 20/21
Introduction FC-KL-exp. Numerical Studies Summary
Summary and Conclusion
FC-KL - Cons
Computationally very expensive to solve (especially in 3D)
FC-KL: double integral over covariance functionEOLE: just N ×N covariance function (and Lanczos methods)
Quite difficult to implement (compared to EOLE)
pFEM(double) integration of non-continuous non-smooth functions
Numerical stability of eigenvalue problem
FC-KL - Pros
Fast convergence against optimal representation
Realization computationally cheap to evaluateEfficient assembly of FE stiffness matrices
Simple mesh
Wolfgang Betz (ERA group) FC Random Field Discretization Based on the KL-Exp. 21/21