The Stochastic Finite Element Method_Theory and Applications

4/6/2009 IFMA Seminar, France 1

The Stochastic Finite Element Method: Theory and Applications

George Stefanou

Institute of Structural Analysis and Seismic ResearchNational Technical University of Athens, Greece

Clermont-FerrandJune 2009


Outline of the presentation

• Stochastic processes and fields

• Simulation of Gaussian stochastic processes/fields

• Simulation of non-Gaussian stochastic processes/fields

• The stochastic finite element method (SFEM)

• SFE (static) analysis of shells with random material and geometric properties -response variability calculation

• SFE-based stability analysis of shells with random imperfections

• Nonlinear dynamic analysis of frames with random material properties under seismic loading


Simulation of Gaussian stochastic fields

The spectral representation method (Rice 1954, Shinozuka & Deodatis 1991)

Fundamental theorem:

Spectral representation of field :

is asymptotically a Gaussian stochastic field as due to the central limit theorem

[ ]0

( ) cos( ) ( ) sin( ) ( )f x x du x dvκ κ κ κ∞

= +∫

( )f x1

0

ˆ ( ) cos( )N

n n nn

f x A xκ−

=

= +Φ∑

2 ( )n ff nA S κ κ= Δ n nκ κ= Δ

0,1, 2,..., 1n N= −

u

N

κκΔ =

ˆ ( )f x N →∞

Truncated form (finite number of terms)


Its mean value and autocorrelation function are identical to the corresponding targets as

is periodic with period Τ0:

The Fast Fourier Transform (FFT):

M is the number of points on which

f is generated:

FFT-based sample functions are always generated on a domain equal to one period Τ0

The reduction of step Δx (stochastic mesh refinement) leads to a larger number of terms M in the FFT series

N →∞

0

2T

πκ

=Δ

( )21

( )

0

ˆ RenpM i

i Mn

n

f p x B eπ⎛ ⎞− ⎜ ⎟

⎝ ⎠

=

⎡ ⎤Δ = ⎢ ⎥

⎢ ⎥⎣ ⎦∑

( )

2i

nin nB A e φ=

0,1,..., 1n M= −

1,...,1,0 −= Mpμ2=M

ˆ ( )f x


The Karhunen-Loève expansion method (Loève 1977, Ghanem & Spanos1991)

Exact form (infinite terms) Truncated form (finite number of terms)

eigenpairs of the covariance function obtained from the solution of the following eigenvalue problem:

(Fredholm integral equation of the second kind)

Exact solution feasible only for simple geometries and special forms of

Numerical solution with the conventional Galerkin approach Dense matrices very costly to compute and invert

∑∞

=

+=1

)()()(),(n

nnn xxxf θξφλμθ ∑=

+=N

nnnn xxxf

1

)()()(),(ˆ θξφλμθ

, ( )n n xλ φ 1 2( , )ffC x x

∫ =D

nnnff xdxxxxC )()(),( 21121 φλφ

1 2( , )ffC x x


• Some remarks on K-L expansion

Few K-L terms are needed for the simulation of strongly correlated (narrow-banded) stochastic fields

Few K-L terms are needed for the simulation of stochastic fields with smooth (differentiable) target covariance function

More efficient simulation when the exact (analytical) solution of the eigenvalueproblem is feasible (Huang et al. 2001)

The variance of the stochastic field is underestimated and the simulated field is in general not homogeneous since its variance is a function of x (Field & Grigoriu2004):

)],([])([])([)],(ˆ[ 2

1

2

1

θφλφλθ xfVarxxxfVarn

nn

N

nnn =≤= ∑∑

∞

==

ˆ ( , )f x θ


• Non-ergodic characteristics of K-L sample functions (Stefanou & Papadra-kakis 2007)

- Ergodicity in the mean:

Proof:

Without any loss of generality, we consider that

[ ]{ }Pr lim ( ) ( ) 1LL

f x E f x→∞

= =

[ ]( ) ( ) 0E f x f x= =

( ) ( ) ( ) ( )

0 0 01 1

1 1 1ˆ ˆ( ) ( ) ( ) ( )N NL L Li i i i

n n n n n nL

n n

f x f x dx x dx x dxL L L

λ ξ φ λ ξ φ= =

= = =∑ ∑∫ ∫ ∫

( ) ( )

01

1ˆlim ( ) lim ( )N Li i

n n nL LLn

f x x dxL

λ ξ φ→∞ →∞

=

⎡ ⎤= ⎢ ⎥⎣ ⎦∑ ∫

( )

01

?

1lim ( )

N Lin n nL

n

x dxL

λ ξ φ→∞

=

=

⎡ ⎤= ⎢ ⎥⎣ ⎦∑ ∫

Existence of limitRight hand side of the expression:

random variableK-L sample functions: in general

not ergodic in the mean


- Ergodicity in autocorrelation:

Proof:

( )ˆ ( ) ( )i Tff ffR Rξ ξ= ( )ˆ ( )if x∀

( ) ( ) ( ) ( ) ( )

0

1ˆ ˆ ˆ ˆˆ ( ) ( ) ( ) ( ) ( )Li i i i i

ffL

R f x f x f x f x dxL

ξ ξ ξ= + = +∫( ) ( ) ( ) ( )

0 01 1 1 1

1 1( ) ( ) ( ) ( )

N N N NL Li i i in m n m n m n m n m n m

n m n m

x x dx x x dxL L

λ λ ξ ξ φ ξ φ λ λ ξ ξ φ ξ φ= = = =

= + = +∑∑ ∑∑∫ ∫

( ) ( ) ( ) ( )

01 1

1ˆ ˆlim ( ) ( ) lim ( ) ( )N N Li i i i

n m n m n mL LL

n m

f x f x x x dxL

ξ λ λ ξ ξ φ ξ φ→∞ →∞

= =

⎡ ⎤+ = +⎢ ⎥⎣ ⎦∑∑ ∫

( ) ( )

01 1

1lim ( ) ( )

N N Li in m n m n m

Ln m

x x dxL

λ λ ξ ξ φ ξ φ→∞

= =

⎡ ⎤= +⎢ ⎥⎣ ⎦∑∑ ∫

1

( ) ( ) ( )N

Tff n n n

n

R x xξ λ φ ξ φ=

≠ = +∑ even in the case n=m

K-L sample functions: in general not ergodic in autocorrelation


• Solution of Fredholm integral equation

Exact solution feasible only for simple geometries (line, square, circle) and special forms of

Two main categories of numerical methods are available: integration formulae-based methods (e.g. quadrature method) and expansion methods (e.g. Galerkin)

Numerical solution with the conventional Galerkin approach Densematrices very costly to compute and invert

An efficient numerical solution of the Fredholm integral equation is indispensable especially when higher order eigenpairs are needed for an accurate representation of the stochastic field

1 2( , )ffC x x


The wavelet-Galerkin method

Wavelet basis functions enhance the performance of the Galerkin method in the solution of integral equations (Phoon et al. 2002)

If the kernel of the integral equation is a rapidly decreasing function, the application of the wavelet-Galerkin method leads to sparse matrices (Beylkin, Coifman & Rokhlin 1991)

Basic steps of the wavelet-Galerkin approach:

1. Selection of a set of M wavelet basis functions

Usually Haar wavelets are used (the simplest form of Daubechies wavelets)

Haar mother wavelet function:1

( ) 1

0

xψ⎧⎪= −⎨⎪⎩

0 0.5

0.5 1

x

x

otherwise

≤ <≤ <

1 2( ), ( ),..., ( )Mx x xψ ψ ψ

1 2( , )ffC x x


2. Approximation of each eigenfunction of the covariance kernel by a linear combi-nation of Haar wavelet basis functions:

di(n): wavelet coefficients

M = 2m, m: wavelet level

Remark 1. Number of terms in the truncated K-L expansion:

1( ) ( )

0

( ) ( ) ( )M

n nn i i

i

x d x x Dφ ψ−

Τ

=

= =Ψ∑

3. Solution of the generalized eigenvalue problem: ( ) ( )x D x AHDΤ ΤΨ Λ = Ψ

Remark 2. A crucial difference of this approach with the conventional Galerkin method: the computation of matrices A and H does not require numerical integration

Matrix A is the 2D wavelet transform of and H a diagonal MxM matrix

Remark 3. Matrix A is sparse by nature and can be made further sparse byignoring elements below a threshold value This may lead to computational instabilities in the numerical procedure

1 2( , )ffC x x

N M≤


• Numerical examples (Stefanou & Papadrakakis 2007)

1. First order Markov stochastic field with exponential covariance function

2/2

2 2Case 1: ( ) ( ) ( )

(1 )b

ff ff ff

bC R e S

bξ σξ ξ σ κ

π κ−= = ⇒ =

+- zero mean, unit variance- selected threshold values in step 3: 10-3, 10-7

2. Stochastic field with square exponential covariance function

- zero mean, unit variance- selected threshold value in step 3: 10-12

A smaller threshold value is selected for case 2 because the square exponential kernel is sparser by its nature

Parameters: Wavelet level m=7 (M=27=128 eigenpairs can be computed at most), number of K-L terms-points in the discretization of the wave number domain N=16, number of terms in the FFT series M=128

2 22 2 2

2 /Case 2: ( ) ( ) ( ) exp42

bff ff ff

b bC R e Sξ σ κξ ξ σ κ

π− ⎛ ⎞

= = ⇒ = −⎜ ⎟⎝ ⎠


Ensemble variance of truncated K-L expansion:

- Wavelet level m = 7- Threshold values in matrix A: 10-3, 10-7

Mean variance: 0.912 Good approximation for 5 K-L terms

Threshold value 10-3: substantial oscillations observed throughout the range [-1,1] due to the numerical instabilities in the calculation of eigenpairs

Threshold value 10-7: a stable solution is obtained

0.8

0.85

0.9

0.95

1

0 32 64 96 128

x [-1,1]

Va

ria

nc

e

thres.10^-3

thres.10^-7

2

1

ˆ[ ( )] ( )N

n nn

Var f x xλ φ=

=∑

The variance is fluctuating w.r.t. xThe error at the boundaries is larger compared

to the middle region


• Eigenvalue decay for θ=0.2, 0.4, 1.0 και 2.0 (θ: Vanmarcke’s scale of fluctuation)

The most rapid eigenvalue decay is observed in case 2 for θ=2.0 (strongly correlated stochastic field with smooth autocovariance function)

Few K-L terms are needed for an accurate simulation of random fields of this kind

Case 2: bθ π=Case 1: 2bθ =

0

0.2

0.4

0.6

0.8

1

1.2

0 8 16 24 32

index i

eig

enva

lue

θ=0.2

θ=0.4

θ=1.0

θ=2.0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 8 16 24 32

index i

eig

enva

lue

θ=0.2

θ=0.4

θ=1.0

θ=2.0


Case 1: θ=0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 4 8 12 16 20 24

Number of terms N

En

se

mb

le v

ari

an

ce

K-L

Spectral

0

0.2

0.4

0.6

0.8

1

1.2

0 4 8 12 16 20 24

Number of terms N

En

se

mb

le v

ari

an

ce

K-L

Spectral

Case 1: θ=2.0

0

0.2

0.4

0.6

0.8

1

1.2

0 4 8 12 16 20 24

Number of terms N

En

sem

ble

var

ian

ce

K-L

Spectral

0

0.2

0.4

0.6

0.8

1

1.2

0 4 8 12 16 20 24

Number of terms N

En

se

mb

le v

ari

an

ce

K-L

Spectral

Case 2: θ=0.2

Case 2: θ=2.0Convergence of each approach to the target

variance


Case 1 Case 2

Spectral representation produces ergodic sample functions in a sample-by-sample sense i.e. every sample function has the target mean and SDF (see below SDF plots)

0.0000.0711-0.01031.6556Skewness

1.0000.78480.78480.7848Variance

0.0000.00000.00000.0000Mean

2.02261.86843.2233Max. value

-1.9070-1.9034-1.3331Min. value

TargetAverageBestWorst

Samples generated with the spectral representation (N=16, θ=0.2)

0.000-0.07610.0153-1.1962Skewness

1.0000.67910.70320.8954Variance

0.000-0.0115-0.1370-0.3434Mean

1.59811.77061.1902Max. value

-1.7178-2.0820-3.3076Min. value


Samples generated with the K-L expansion(N=16, θ=0.2)

0.0000.0678-0.00041.7822Skewness

1.0000.90690.90690.9069Variance

0.0000.00000.00000.0000Mean

2.03282.11473.4019Max. value

-1.9791-2.3388-1.2252Min. value


Samples generated with the spectral representation (N=16, θ=0.2)

0.0000.05950.0166-0.5614Skewness

1.0000.75920.88030.5237Variance

0.0000.08720.08130.5846Mean

1.84551.89321.9598Max. value

-1.5988-1.7148-1.5411Min. value


Samples generated with the K-L expansion(N=16, θ=0.2)


0

0.2

0.4

0.6

0.8

1

1.2

-3 -2 -1 0 1 2 3

f

CD

F(f

)

Sample

Target

0

0.2

0.4

0.6

0.8

1

1.2

-3 -2 -1 0 1 2 3

f

CD

F(f

)

Sample

Target

K-L expansion: ensemble average

Spectral representation: ensemble average

Case 2θ=0.2

K-L expansion: Case 1, θ=0.2, m=9

Spectral representation: Case 1, θ=0.2, m=9

0

0.2

0.4

0.6

0.8

1

1.2

-3 -2 -1 0 1 2 3

f

En

se

mb

le C

DF

(f)

Sample

Target

0

0.2

0.4

0.6

0.8

1

1.2

-3 -2 -1 0 1 2 3

f

En

se

mb

le C

DF

(f)

Sample

Target


0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 10 20 30 40 50 60

wave number

SD

F

Target

Sample

10-33x10-4Spectral representation (1 sample)

60.02.5K-L expansion (1 sample)

97Wavelet level, power in FFT series (m)

Computational performance of K-L expansion and spectral representation

Ergodicity of spectral representation-based samples with regard to SDF: a perfect matching of the target SDF is observed (cases 1-2, θ=0.2)

Case 1

Case 2

02

( ) ( )

0 0

1 ˆ( ) ( ) exp( )2

Ti i

ffS f x i x dxT

κ κπ

= −∫

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 10 20 30 40 50 60

wave number

SD

F

Target

Sample


Simulation of non-Gaussian stochastic fields

Theoretically all the joint multi-dimensional density functions are needed to fully characterize a non-Gaussian stochastic field

Much of the existing research has focused on a more realistic way of defining a non-Gaussian sample function e.g. as a simple transformation of some under-lying Gaussian field with known second-order statistics:

Translation field theory (Grigoriu 1984, 1998):

- Spectral distortion:

- Compatibility of F - :

)]([)( 1 xgFxf Φ⋅= −

( ) ( )Tff ffS Sκ κ≠

1 11 2 1 2 1 2( ) [ ( )] [ ( )] [ , ; ( )]T

ff ggR F g F g g g R dg dgξ φ ξ∞ ∞

− −

−∞ −∞

= Φ Φ ⋅∫ ∫

)(ξTffR [ ]

1 1

( ) ( ) ( ) [ ( )] [ ( )]F F

TffR E f x f x E h g x h g xξ ξ ξ

− −⋅Φ ⋅Φ↓ ↓

⎧ ⎫⎪ ⎪= + = + ⇒⎨ ⎬⎪ ⎪⎩ ⎭


and the joint density of

• Characteristics of translation fields

The relationship between the two autocorrelation functions can have a closed form only in few cases

Strictly speaking, if the target F και are proven to be incompatible, there is no translation field with the prescribed characteristics.

The problem of incompatibility becomes even greater for highly skewed narrow-banded stochastic fields (Grigoriu 1998)

Analytical expressions of crossing rates are available for translation fields

1 ( ),g g x= )(2 ξ+= xgg

)](;,[ 21 ξφ ggRgg },{ 21 gg

)(ξTffR

where


- In order to address the problem of spectral distortion Iterative methods (e.g. Yamazaki-Shinozuka 1988, Deodatis-Micaletti 2001)

Repeated updating of in every iteration:

Unwanted correlations between the terms of the spectral representation series

After the first iteration, the underlying Gaussian field is no more Gaussian and homogeneous. Finally, the generated non-Gaussian sample functions will not have the prescribed marginal PDF

Use of extended non-Gaussian to non-Gaussian mapping:

- To address the issue of possible incompatibility between F and : Spectralpreconditioning Generated non-Gaussian field: approximately translationfield

)(κggS )()(

)()( )(

)(

)1( κκ

κκ

α

jggj

ff

Tffj

gg SS

SS

⎥⎥⎦

⎤

⎢⎢⎣

⎡=+

)]([)( 1 xgFFxf ∗− ⋅=

)(ξTffR


- Correlations between the terms of the spectral representation series (proof):

1( 1) ( 1)0

0

( ) 2 ( ) cos( )N

j jgg n n n

n

g x S xκ κ κ φ−

+ +

=

= Δ +∑2 2

( ) ( ) 1 ( )0 0

0 0

1 1( ) ( ) exp( ) [ ( )]exp( )

2 2

T Tj j j

ffS f x i x dx F g x i x dxT T

κ κ κπ π

−= − = Φ −∫ ∫

21

1 ( )

00

12 ( ) cos( ) exp( )

2

T Nj

gg n n nn

F S x i x dxT

κ κ κ φ κπ

−−

=

⎡ ⎤= Φ Δ + −⎢ ⎥⎣ ⎦∑∫

in iteration j+1

depends on the random phase angles φn

)()(

)()( )(

)(

)1( κκ

κκ

α

jggj

ff

Tffj

gg SS

SS

⎥⎥⎦

⎤

⎢⎢⎣

⎡=+

( ) ( )jffS κ

also depends on the random phase angles φn alteration of themarginal PDF of due to c.l.t.

( 1) ( )jggS κ+

( 1)0 ( )jg x+


Proposed enhanced hybrid method (Lagaros, Stefanou & Papadrakakis 2005)

Basic idea: replace the updating scheme of the Gaussian spectrum (source of all important difficulties in the simulation) by a NN-based regression model

The unwanted correlations between the terms of the spectral representation series become negligible

Use of the extended mapping:

Spectral preconditioning is not required an algorithm covering a widerrange of non-Gaussian fields is obtained (not only translation fields).

A very small number (<50) of iterations is required until convergence drastic reduction of the computational effort required for simulation

)(κggS

)]([)( 1 xgFFxf ∗− ⋅=


0

0.2

0.4

0.6

0.8

-4 -2 0 2 4

f

Ga

us

sia

n P

DF

(f) Target PDF

Sample PDF

0

0.2

0.4

0.6

-4 -2 0 2

f

PD

F(f

)

Target PDF

Sample PDF

Exact and sample Gaussian PDF at first iteration

Exact and sample beta PDF at final iteration (translation field case)

2

1

1(w) [ ( ) ( )]

2

NT

ff j ff jj

S κ S κ=

= −∑E


• Numerical example: a highly skewed narrow-banded stochastic field

Characteristics of target non-Gaussian field:- Lognormal distribution defined in the range [-1.30, 10.0]skewness γ = 2.763, kurtosis δ = 19.085

- Target correlation structure: σ = 1 και b = 5]exp[4

1)( 232 κκσκ bbS T

ff −=

-2

0

2

4

6

8

10

0 64 128 192 256

x (m)

Sa

mp

le f

un

cti

on

-2

0

2

4

6

8

10

0 64 128 192 256

x (m)

Sa

mp

le f

un

cti

on

Sample function generated using the D-M algorithm: [-1.23, 7.87]

Sample function generated using the proposed EHM: [-1.25, 8.89]


0

0.2

0.4

0.6

0.8

-2 0 2 4 6 8 10

f

PD

F(f)

Target PDF

Sample PDF

0

0.2

0.4

0.6

0.8

-2 0 2 4 6 8 10

f

PD

F(f)

Target PDF

Sample PDF

Marginal PDF of sample function generated using the D-M algorithm

versus target lognormal PDF

Marginal PDF of sample function generated using the proposed EHM

versus target lognormal PDF

0

0.2

0.4

0.6

0.8

0 2 4 6 8

wave number (rad/m)

Sp

ec

tra

l de

ns

ity

Starget

Sng

SDF of sample function generated using the D-M algorithm versus target SDF

0

0.2

0.4

0.6

0.8

0 2 4 6 8

wave number (rad/m)

Sp

ec

tra

l de

ns

ity Starget

Sng

SDF of sample function generated using the proposed EHM versus target SDF


Perfect matching of the target PDF due to the (exact) extended mappingPerfect matching of the target SDF achieved by the proposed ΕΗΜ (use of Rprop

training, Riedmiller & Brown 1993)

0.832EHM-Rprop

1.245EHM-Quickprop

0.625EHM-CG

2.082EHM-SD

14629279D-M*

Time (sec)IterationsMethod

* : without spectral preconditioning

Computational performance of D-M and proposed algorithm (EHM)

Substantially smaller cost of proposed EHM (~ 2 orders ofmagnitude)

Reduction of cost for the stochastic analysis of realistic structures with uncertain non-Gaussian properties


Statistical comparison of D-M and EHM algorithms

2951582

2861618

2910628

3353295

3594494

315935

2924335

284414

3846843

3229279

EHM-Rprop (Iterations)D-M* (Iterations)

* : without spectral preconditioning

D-M*

Mean value: 38242St. dev.: 28701Max: 94494Min: 4414

EHM-RpropMean value: 31.2St. dev.: 3.5Max: 38Min: 28


Identification of random shapes from images based on polynomial chaos expansion (Stefanou, Nouy & Clement 2009)

Physical problems with random geometry: disordered systems and random media, fluctuating domain boundaries, shells with cut-outs, heterogeneous materials with random distribution of inclusions…

Usually a limited number of samples of the geometry is available in practice: its complete probabilistic characterization (in terms of joint densities) is infeasible

Shape recovery from simple images allows obtaining many samples at a low cost + it is relatively precise

Material layout of a quadratic microstructure

Cylindrical shell with rectangular cut-out


• Short description of the procedure

- Shape recovery with the level-set technique: construction of a collection of discreti-zed level-set functions corresponding to a collection of images - Reduction of information through empirical Karhunen-Loève expansion:

( ) ( )

1

mk k

i ii

Xφφ μ=

≈ +∑U

- Probabilistic identification of random vector from samplesdecomposition on a polynomial chaos basis of degree p in dimension m(Wiener 1938):

- Identification of chaos coefficients:1. Without independence hypothesis: maximum likelihood estimation for random vector X2. With independence hypothesis: maximum likelihood estimation for each random variable or projection method based on empirical CDF

1( ,..., )mX X=X ( )kX

,

( ( ))m p

Hα αα

θ∈ℑ

≈ ∑X X ξ


The stochastic finite element method

Solution of stochastic elliptic boundary value problems

• Well posed problem (existence and uniqueness of solution):

και

• Weak form:

• Finite element approximation selection of suitable function spacese.g. and

[ ]( , ) ( , ) ( , ),

( , ) 0,

u f

u

κ θ θ θθ

−∇ ⋅ ∇ =

=

x x x

x

D∈x

D∈∂x

1( )C Dκ ∈

min max0 κ κ< <min maxPr ( , ) [ , ], 1x x Dκ θ κ κ⎡ ⎤∈ ∀ ∈ =⎣ ⎦

:u D×Θ→

( , , )PΘ ℑ

( , ) ( )A u v B v= 10 ( )v X H D∀ ∈ =

( , ) ,D

A u v u vdκ= ∇ ⋅∇∫ x ( )D

B v fvd= ∫ x ,u v X∈

deterministic

( , ) ( )A u v B v= 2 ( )Pv X L∀ ∈ ⊗ Θ stochastic

1 2{ , ,..., }x

hNX span Xφ φ φ= ⊂

1 2{ ( ), ( ),..., ( )}hNW span Wξ

ψ ψ ψ= ⊂ξ ξ ξ

1( ,..., )Mξ ξ=ξ


Formulation of the stochastic stiffness matrix

- Case of material randomness:

- Discretization of the stochastic fields

• Midpoint method

• Local average method:

• Weighted integral method:

[ ]),,(1),,( 0 zyxfDzyxD +=

( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( )

( )0 0 , ,

e e

e e e e e e e e e eT T

V V

k B D B dV B D B f x y z dV= +∫ ∫ ( ) ( ) ( )eee kkk Δ+= 0

( ) ( ) ( )0 (1 )e e ek k a= +

( ) ( ) ( ) ( )0

1

WNe e e e

l ll

k k X k=

= + Δ∑

3D elasticity problem:


Response variability calculation

• Direct Monte Carlo Simulation (e.g. Rubinstein 1981)

Solution of ΝSIM deterministic problems substantial computational cost especially for high dimensional problems and for large number of simulations

It must be combined with efficient discretization methods (e.g. local average method)

Particularly suitable for parallel computing environment (“embarrassingly parallel”)

Assisted by the spectacular growth of computing power, the only available method for the solution of large-scale realistic problems

∑=

=NSIM

jii ju

NSIMuE

1

)(1

)( ⎥⎦

⎤⎢⎣

⎡⋅−

−= ∑

=

NSIM

jiii uENSIMju

NSIMu

1

222 )()(1

1)(σ


• The perturbation method (Kleiber & Hien 1992)

Taylor series expansion of the stochastic finite element matrix and of the resulting response vector:

Calculation of the stochastic displacement vector:

Satisfactory results only for small coefficients of variation of the uncertain input parameters

Improvement in accuracy obtained using higher order approximations: small compared to the disproportional increase of computational effort

I II0

1 1 1

1...

2

N N N

i i ij i ji i j

K K K a K a a= = =

= + + +∑ ∑∑ I ,0

ii

KK

a a

∂∂

==

2II

0ij

i j

KK

a a a

∂∂ ∂

==

I II0

1 1 1

1...

2

N N N

i i ij i ji i j

u u u a u a a= = =

= + + +∑ ∑∑ 10 0 0u K P−=

( )I 1 I I0 0i i iu K P K u−= −

( )II 1 II I I I I II0 0ij ij i j j i iju K P K u K u K u−= − − −


• The spectral stochastic finite element method (Ghanem & Spanos 1991)

Karhunen-Loève expansion of the stiffness matrix + polynomial chaos expansion (PCE) of the displacement vector:

Application (mainly) to linear problems with small variability + Gaussian assumption for the uncertain input parameters (Sudret & der Kiureghian 2002)

Prohibitive computational cost for the solution of problems with large stochasticdimension (increase of the order of PCE)

1

0

( ) ( ),P

j jj

θ θ−

=

= Ψ∑U U

( ) ( ) ( )0

1

( ) ( ),M

e e el i

l

θ ξ θ=

= +∑k k ke

Tii

ei d

e

Ω= ∫Ω

BDBxk 0)( )(φλ

( )!1

! !

M pP

M p

++ =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−

−

−

1

1

0

1

1

0

1,10,1

1,110

1,000

PPPPP

P

P

F

F

F

U

U

U

KK

KK

KK

…

……

NPNP ×


- Spectral SFEM: recent advances and extensions

Cases of non-smooth solutions (non-linearities, discontinuities): use of other basis functions e.g. wavelets (Le Maitre et al. 2004) or adaptive sparse generalized PCE (Wan & Karniadakis 2005, Blatman & Sudret 2008).

Stochastic reduced basis methods (Nair et al. 2002-2008): limited to random linear systems

Non-intrusive, stochastic response surface approaches (Baroth et al. 2006, Berveilleret al. 2006): can take advantage of powerful deterministic FE codes

Spectral SFEM in a multi-scale setting (Xu 2007): stochastic variational approach + scale-bridging multi-scale shape functions

X-SFEM (Nouy et al. 2007-8): implicit representation of complex geometries using random level-set functions


Stochastic finite element analysis of shells

The multi-layered triangular shell element TRIC (Argyris et al. 1997)

TRIC is a triangular, shear-deformable facet shell element suitable for the analysis of thin and moderately thick isotropic as well as composite plate and shell structures

The element formulation is based on the natural mode finite element method (Argyris et al. 1979)

The treatment of the element kinematics eliminates automatically shear locking phenomena (Argyris et al. 2000)

No need to perform numerical integration for the computation of the deterministic stiffness matrix which is carried out in closed form derivation of a cost-effective stochastic stiffness matrix

Computational efficiency in large-scale stochastic finite element computations


kqc : axial and symmetric bending stiffness terms

kqh : anti-symmetric bending and shear stiffness terms

kaz : stiffness terms due to in-plane rotations (azimuth stiffness terms)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ΟΟ

ΟΟ

ΟΟ

=

×

×

×

×Ν

)33(

)33(

)66(

)1212(

az

qh

qc

k

k

k

k

The natural stiffness of the TRIC element is based only on the 12 independent natural straining modes:


• Derivation of the stochastic stiffness matrix

- Random variation of Young modulus (Argyris, Papadrakakis & Stefanou 2002)

( ) [ ]),(1, 0 yxfEyxE +=

∫=× V

tcctTtcqc dVBBk κ

)66(

( ) 0)66(

)(1 qcqc kak +=×

( )a f x y d= ∫1

ΩΩ

Ω

,

It is proved that the axial and symmetric bending stiffness terms have a local average form

( )[ ] [ ]{ } 111

33

−−−

×

+= sqh

bqhqh kkk

∫=V

thctTth

bqh dVBBk κ ( )

6

01

b bqh qh i i

i

k k X k=

= + Δ∑

dVBXBkV

shsTsh

sqh ∫= ( ) 0)(1 s

qhsqh kak +=

6 weighted integrals Χi

Local average form

1.

2.

2a.

2b.


the natural (triangular) coordinates

e.g. weighted integrals Χ1, Χ4 : ( )X f da a12= ∫ζ ζ ζ ζβ γ, , Ω

Ω

( )X f da a4 = ∫ζ ζ ζ ζ ζβ β γ, , ΩΩ

ζ ζ ζβ γa , ,

Matrix Δk1 of the fluctuating part of the anti-symmetric bending stiffness terms:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

++

−−−−++=Δ

γγγ

γγ

βγγβ

γγ

ββ

β

γγβββ

ββ

β

kzl

kzl

kzl

lsymm

kzll

kzll

lkz

ll

lkz

l

llkz

lkz

lkz

l

lk

a

a

aa

a

a

a

a

a

aa

a

a

a

aa

a

2

2

2

2

2

4

2

22

2

2

2

2

4

2

2

2

2

2

4

2

1

96.

93396000

( )/ 2

2 2 3 31

1/2

1

3

h Nk

ij ij ij k kkh

z k z dz z zκ κ +=−

= = −∑∫ γβα ,,, =ji

- κij the entries of the constitutivematrix κct

- k = 1,2,…,N the number of layers


( ) ( ) ( ) ( ) ( ) ( ){ }k a k a k a kz za

z z= + + +max , ,1 1 1β γ

The azimuth stiffness terms have also an inherent local average form

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=×

15.05.0

5.015.0

5.05.01

.)33(

z

z

zz

az

az

aaz

az k

ksymm

kk

kkk

kγγ

βγββ

γβ

3.

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⋅Ω= ∫∫∫ −−−

2/

2/

22

2/

2/

22

2/

2/

22

1,

1,

1max

h

h

h

h

h

ha

z dzzl

dzzl

dzzl

k γγγ

βββ

αα κκκ

kz the maximum of the three edge bending stiffness values:

( ) ∫−Ω

=2/

2/

22

h

h iii

iz dzz

lk κ

( )a f x y d= ∫1

ΩΩ

Ω

,


- Combined random variation of Young modulus and Poisson ratio (Stefanou & Papadrakakis 2004)

The entries of the elasticity matrix are nonlinear functions of Poisson ratio consideration of random variation of Lamé constants λ and μ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

+=

μμλλ

λμλκ

200

02

02

1221 ν

νλ−

=E

( )νμ+

==12

EG

( ) [ ]),(1, 0 yxfyx λλλ += ( ) [ ]),(1, 0 yxfyx μμμ +=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

=ν

νν

νκ

100

01

01

1 212

E

+

Axial and symmetric bending stiffness terms: local average formAnti-symmetric bending stiffness terms: 12 weighted integralsAnti-symmetric shear stiffness terms: local average formAzimuth stiffness terms: local average form


- Random variation of the thickness

The thickness h appears into the integrals expressing the stiffness matrix of the shell element

The application of the weighted integral method is not straightforward The local average approach is applied leading to greater computational efficiency

( ) )1(, 0 hahyxh += ∫Ω ΩΩ

= dyxfa hh ),(1


2. Scordelis-Lo shell

C

ν = 0.3

Α

rigid diaphragm

rigid diaphragm R = 300 mm L = 600 mm h = 3.0 mm E = 3000 N/mm2 ν = 0.3

L

P

P

C

h

x

y

z

R

1. Pinched cylinder

• Numerical examples(Stefanou & Papadrakakis 2004)


3. Hyperboloid shell

R1 = 4800 mm

R2 = 8000 mm

L = 20000 mm

h = 40 mm

E = 21000 N/mm2

ν = 0.25

L

y

xz

R1

R2

A

C


Use of 2D homogeneous Gaussian stochastic fields for the representation of the uncertain material and geometric properties

Sample functions of the stochastic fields generated by the spectral representation method

Selection of two different correlation structures:

[ ])(exp)(),( 22

22

21

21

22

22

21

2121

2

21 κκκκπσ

κκ bbbbbbS fff +−+=

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛−=

2

22

2

1121221 22

exp4

,κκ

πσκκ

bbbbS fff

Calculation of COV of displacement at the characteristic node C:

Investigation of the effect of various parameters of the stochastic fields (σf, Sff, b1, b2) on the response variability

)(

)()(

i

ii uE

uuCOV

σ=

1

2


1

2

σf = 0.2 b1 = b2 = 2.6


0

0,05

0,1

0,15

0,2

0,25

0,3

0,2 2 4 8 80 200

Correlation length parameter b1

CO

V o

f w

at

no

de

C E

h

E,h

0

0.05

0.1

0.15

0.2

0.25

0.065 0.65 1.3 2.6 26 65 130

Correlation length parameter b

CO

V o

f w

at

no

de

C

Eh

E,h

• Scordelis-Lo shell (σf = 0.1):

Random variation of Young modulus and thickness (local average method)

Important effect of thickness variation

• Hyperboloid shell (σf = 0.1):


Important effect of thickness variation


0

0.05

0.1

0.15

0.065 0.65 1.3 2.6 26 65 130

Correlation length parameter b

CO

V o

f w

at

no

de

C

E E,v (Local average) E,v (Weighted integral)

0

0,05

0,1

0,2 2 4 8 80 200


CO

V o

f w

at

no

de

C

E E (Weighted integral)

E,v (Local average) E,v (Weighted integral)

• Scordelis-Lo shell (σf = 0.1):

Random variation of Young modulus and Poisson ratio

Negligible effect of Poisson ratio variation


Random variation of Young modulus and Poisson ratio

Negligible effect of Poisson ratio variation


0

0,05

0,1

0,15

0,2

0,25

0,3

0,2 2 4 8 80 200


CO

V o

f w

at

no

de

C

hh(P.s.2)

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,2 2 4 8 80 200


CO

V o

f w

at

no

de

C E,h

E,h(P.s.2)



Investigation of the effect of correlation structure (power spectrum)

( ) ( )C ffVar w S κ∼


• Conclusions

The stochastic stiffness matrix of the TRIC shell element is formulated in terms of a minimum number of random variables

Good agreement between weighted integral and local average methods

Significant effect of correlation length parameter b on the response variability

Slight effect of correlation structure (expressed via the form of the power spectrum) on the response variability

Random variation in the shell thickness has significant effect on the response variability compared to the effect of random Young modulus

Small effect of random variation of Poisson ratio on the results


Stability analysis of shells with random imperfections

Large scatter in the measured buckling loads

Big discrepancy between the deterministic predictions of buckling loads and the corresponding experimental results

Presence of initial imperfections which occur during the manufacturing and construction stages

Realistic description of imperfections in the framework of a robust stochastic finite element formulation

Importance of available data banks for the realistic simulation of imperfections (estimation of probability distribution, correlation structure), e.g. Elishakoff & Arbocz (1982, 1985)


Methods based on random variables

Use of 2D Fourier series with random Fourier coefficients (e.g. Chryssanthopoulos& Poggi 1995, Noirfalise et al. 2007) for the representation of geometric imperfect-ions

This idea is related to the fact that an analytical buckling analysis of cylindrical shells yields a 2D Fourier series representation of the critical modes

Methods based on stochastic fields

Simulation of the imperfections using stochastic fields combined with the finite element method for the solution of the stability problem

Use of direct Monte Carlo simulation for the computation of buckling load variability


Stability analysis of cylindrical shells with random geometric imperfections(Schenk & Schueller 2003)

Use of the data bank by Arbocz & Abramovich (1979) for the estimation of the probability distribution and correlation structure of the imperfections

Selection of 2D non-homogeneous Gaussian stochastic fields for the description of the imperfections

250 direct Monte Carlo simulations for the computation of buckling load variability (finite element model with geometric non-linearity)

The experimentally determined scatter in the limit load can be predicted numerically

A more accurate prediction requires the incorporation of other kinds of imperfections e.g. material, thickness, boundary conditions


Stability analysis of cylindrical shells with random geometric, material, thickness and boundary imperfections (Papadopoulos, Stefanou & Papadrakakis 2009)

Use of TRIC shell element with geometric and material non-linearitySelection of 2D non-homogeneous Gaussian stochastic fields for the description

of geometric imperfections (data bank by Arbocz & Abramovich, 1979)Effect of non-Gaussian assumption for E, t on buckling load variability

0 1( , ) [1 ( , )]E x y E f x y= +

0 2( , ) [1 ( , )]t x y t f x y= +

Randomly varying quantities:

X

Z

Y

Loading P

θ

L

R

E=104410N/mm2

ν=0.3 L=202.3mm R=101.6mm t=0.11597mm

f1, f2: homogeneous non-Gaussian translation fields

( ) 0 1r , R a ( , ) g ( , )x y x y x y= + +

( ) [ ]0 2P P 1 g ( )x x= +


• Initial geometric imperfections: ( ) 0 1r , R a ( , ) g ( , )x y x y x y= + +

3442.9108940196.850.1110101.6A-14

3108.8104110196.850.1128101.6A-13

3853.0104800209.550.1204101.6A-12

3196.9102730203.200.1204101.6A-10

3724.8101350203.200.1153101.6A-9

3673.8104800203.200.1179101.6A-8

3036.4104110203.200.1140101.6A-7

P (N)E (N/mm2)L (mm)t (mm)R(mm)Shell

Geometry, material properties and experimental buckling loads of A-shells (Arbocz & Abramovich, 1979)

Measured initial unfolded shape of shell A7

a0(x,y)

The ensemble average a0(x,y) as well as the evolutionary power spectrum of stochastic field g1(x,y) are derived from a statistical analysis of experimentally measured imperfections of A-shells


• Material and thickness imperfections: use of non-Gaussian translation fields

q=1.6p=0.80.26-0.13L-beta - Case 3

q=0.8p=0.80.16-0.16U-beta - Case 2

q=12p=120.5-0.5Beta - Case 1

--+-1Lognormal

Shape parametersUpper boundLower bound

∞

( )2 2

2 1 2 1 1 2 21 2, exp

4 2 2gg f

b b b bS

κ κκ κ σπ

⎡ ⎤⎛ ⎞ ⎛ ⎞= − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

Correlation structure of the imperfections

• In-plane edge (boundary) imperfections: non-uniform axial loading

It is assumed that boundary imperfections are produced by a non-uniform random axial load distribution on the upper edge of the cylinder, modeled as a 1D homoge-neous Gaussian stochastic field

0 1( , ) [1 ( , )]E x y E f x y= +

0 2( , ) [1 ( , )]t x y t f x y= +

( ) [ ]0 2P P 1 g ( )x x= +

22 21

( ) exp ( )4 4

ggg g gS b b

σκ κ

π⎡ ⎤= −⎢ ⎥⎣ ⎦

loading surface

cylinder


• Numerical results

bg = 100 mm with σg = 5% are selected for the description of g2

b1 = b2 = 50 mm with σf = 10% are selected for the description of Young modulus and thickness these values are responsible for the minimum mean(Pu) and it is likely to lead to “worst case” scenarios w.r.t. lowest Pu

The choice of PDF has a significant effect on first and second order properties of Pudistribution

Max Cov(=0.16) for a Gaussian PDF, Min Cov(=0.06) for an L-beta PDF

NSIM=100


Gaussian E, t Lognormal E, t

Beta E, t U-beta E, t

(perfect)uP 5350N=


L-beta E, t

Experimental

• Conclusions

The choice of marginal PDF of E, t is crucial: it affects significantly the shape as well as the extreme values of Pu distribution

A large magnification of uncertainty has been observed in the Gaussian case: Cov(Pu) ~ 1.6σf

The lognormal and beta non-Gaussian assumptions led to estimates of the scatter of Pucloser to the experimental measurements

The tri-modal shape of buckling loads observed in the experiments has been reproduced by the corresponding numerical simulations

The lowest Pu has been found to represent only the 28-60% of the Pu of the perfect shell

Towards a robust design of imperfect shell structures


Nonlinear stochastic dynamic analysis of frames (Stefanou & Fragiadakis 2009)

• A 3-storey steel moment-resisting frame designed for a Los Angeles site (SAC/FEMA program)

• Fundamental mode period: T1=1.02 sec

• 5 integration sections defined in every beam-column element

• Geometrical nonlinearities not considered in the analysis

• Material law: bilinear with pure kinematic hardening• Loading: 3 sets of 5 strong ground motion records corresponding to 3 levels of

increasing hazard: low, medium and high + spectrum-compatible artificial accelerograms


0.638–,D16.96.9090WAHOLoma Prieta, 198915 (5/3)

0.370–,D16.96.9000WAHOLoma Prieta, 198914 (4/3)

0.371–,D28.86.9000Hollister South & PineLoma Prieta, 198913 (3/3)

0.358C,D25.56.7360LA, Hollywood Storage FFNorthridge, 199412 (2/3)

0.2C,D24.46.7360Wildlife Liquefaction ArraySuperstition Hills, 198711 (1/3)

0.209C,D28.86.9360Sunnyvale Colton AveLoma Prieta, 198910 (5/2)

0.18C,D24.46.7090Wildlife Liquefaction ArraySuperstition Hills, 19879 (4/2)

0.207C,D28.86.9270Sunnyvale Colton AveLoma Prieta, 19898 (3/2)

0.057C,D31.76.5090Plaster CityImperial Valley, 19797 (2/2)

0.239B,B31.36.7090LA, Baldwin HillsNorthridge, 19946 (1/2)

0.147C,D32.66.5285CompuertasImperial Valley, 19795 (5/1)

0.074C,D15.16.5090Westmoreland Fire StationImperial Valley, 19794 (4/1)

0.11C,D15.16.5180Westmoreland Fire StationImperial Valley, 19793 (3/1)



PGASoil4R3Mw2φο1StationEarthquakeID

( level)

The ground motion records used

1 Component 2 Moment magnitude 3 Closest distance to fault rupture 4 USGS, Geomatrix soil class


• 1D stochastic variation of Young modulus and yield stress described by zero-mean lognormal and L-betatranslation fields with σf =10%

• The variability of the maximum inter-storey drift θmax is examined using 1000 Monte Carlo simulations:

Lower bound Upper bound Shape parameters

Lognormal -1 +∞ - - L-beta -0.13 0.26 p=0.8 q=1.6

maxmax

max

( )( )

( )

σCOV

E

θθ

θ=

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-1 -0.5 0 0.5 1

normalised length (m)

Sam

ple

func

tion

b=0.2

b=1

b=2

b=10

b=20

b=100

• Investigation of the sensitivity of θmax w.r.t. the correlation length parameter b

A set of sample functions of lognormal stochastic fields characterizing the spatial variation of Young modulus E in a beam


Mean value, COV of θmax for different values of correlation length parameter b and the ground motion records of slide 61 (lognormal distribution of E, σy)

The different characteristics of the seismic records are transferred to the response statistics significant record-to-record variability

The effect of b on COV is important in many cases

A large magnification of uncertainty is observed in some cases max COV=18% (~1.8σf) for record 4/2

The mean value is practically not affected by b

1/1 2/1 3/1 4/1) 5/1 1/2 2/2 3/2 4/2 5/2 1/3 2/3 3/3 4/3 5/30

0.005

0.01

0.015

0.02

0.025

record

mea

n of

θm

ax

b=0.2b=1.0b=2.0b=10b=20b=100

1/1 2/1 3/1 4/1) 5/1 1/2 2/2 3/2 4/2 5/2 1/3 2/3 3/3 4/3 5/30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

record

CO

V( θ

max

) =

σ /

μ

b=0.2b=1.0b=2.0b=10b=20b=100


Mean value, COV of θmax for different values of correlation length parameter b and artificial accelerograms compatible with records 1/1, 4/2, 1/3 (lognormal distribution of E, σy)

The different characteristics of the seismic records are transferred to the response statistics significant record-to-record variability

The effect of b on COV is important in some cases

The magnification of uncertainty is even more pronounced in this case: max COV=23% (~2.3σf) for record 1/1

The mean value is practically not affected by b

1/1 4/2 1/30

0.005

0.01

0.015

record

mea

n of

θm

ax

b=0.2b=1.0b=2.0b=10b=20b=100

1/1 4/2 1/30

0.05

0.1

0.15

0.2

0.25

record

CO

V( θ

max

) =

σ /

μ

b=0.2b=1.0b=2.0b=10b=20b=100


Skewness of θmax for different values of correlation length parameter b and the ground motion records of slide 61 (lognormal, L-beta distribution of E, σy)

Important record-to-record variability

Significant influence of b in many cases

Values of skewness substantially different from those of the system properties due to the strong non-linearity of the problem

1/1 2/1 3/1 4/1) 5/1 1/2 2/2 3/2 4/2 5/2 1/3 2/3 3/3 4/3 5/3−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

record

Ske

wne

ss

b=0.2b=1.0b=2.0b=10b=20b=100

1/1 5/1 1/2 2/2 2/3 5/3−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

record

Ske

wne

ss

b=0.2b=2.0b=100

Input skewness values:- Lognormal distribution: 0.3- L-beta distribution: 0.59


• Conclusions

A stochastic response history analysis of a steel frame having uncertain non-Gaussian material properties and subjected to seismic loading has been performed.

The Young modulus and the yield stress were described by uncorrelated homogeneous non-Gaussian translation fields.

The effect of the probability distribution of the input parameters on the response variability was negligible due to the small input COV(=10%).

The significant influence of the scale of correlation of the stochastic fields and of the different seismic records on the response variability has been revealed.

A large magnification of uncertainty has been observed in some cases.

Importance of a realistic uncertainty quantification and propagation in nonlinear dynamic analysis of engineering systems.


Thank you for your attention!

Documents

The Stochastic Finite Element Method_Theory and Applications