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