Transcript CwU 2004 - Johns Hopkins University

The need for linking micromechanics of materials
with stochastic finite elements:
a challenge for materials science
Dimos C. Charmpis and Gerhart I. Schuëller
Institute of Engineering Mechanics
Leopold-Franzens University
Innsbruck, Austria, EU
Probability & Materials: From Nano- to Macro-Scale
Johns Hopkins University Homewood Campus, Baltimore, MD, USA
January 5-7, 2005
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Outline of the presentation

Scope

Modeling of material uncertainties in structural mechanics

Random fields

Stochastic finite element analysis

The need for micromechanically derived information

Options and assumptions for random fields
(using numerical example)

Conclusions
2
Spectrum of uncertainties
Mechanical
Model
Physical
Entire Spectrum
3
Evolution of Structural Analysis
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Deterministic
Structural
Analysis
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p2
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Non-deterministic
Structural
Analysis
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p2
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r2
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Modeling of material uncertainties in structural mechanics
Quantification of Randomness
Random Variables
Random Fields
(stochastic analysis)
f(x1 ,x2 )
x2
x1
FX1 X 2 ( x1 , x2 ) 
x1 x2
  f x , x  dx dx
1

2
1
2
Autocorrelation Function:
CHH x, y   CHH x   , y   
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Random fields (1)

Each uncertain variable is represented by a stochastic field
Variation of uncertain property P:
P0: mean value of the uncertain property P
f (x): 1D-1V zero-mean stochastic field
Px   P0 1  f ( x)
f (x,y): 2D-1V zero-mean stochastic field
Px, y   P0 1  f ( x, y)
1D
2D
Sample functions (1D):
Karhunen-Loeve Expansion
f x   f
( 0)
m
x     j f ( j ) x 
j 1
Spectral Representation Method
N 1
f x   2 [ Αn cos( n x  n )]
…
n 0
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Random fields (2)
Modeling the spatial variation of material properties
240
1D
2D
230
E (GPa)
220
210
200
190
180
0
1
2
3
4
5
6
7
8
9
10
x (m)
Autocorrelation function:
According to this function correlation decays as the distance of the
locations of two random field variables increases
Correlation length b:
Distance between two stochastic field locations, over which correlation
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approaches zero
Random fields (3)
explicit discretization of random fields into random variables
230
continuous sample
E (GPa)
220
discretized sample
210
200
190
0
1
2
3
4
5
6
7
8
9
10
x (m)
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Stochastic finite element analysis
E x   E0 1  f ( x)
e.g. stochastic modulus of elasticity
Stochastic elemental stiffness matrix:
k
e 

B
e 
e T
e  e 
e 
D0 B d 
B
e 
e T
D0e  B e  f e   x d e 
k e  k0e  k e
D0: mean value of constitutive matrix D
B: strain-displacement matrix
ne
Stochastic global stiffness matrix:
K   k e   K0  K
e 1
Linear static stochastic system of equations:
K0  K u  f
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The need for micromechanically derived information (1)
Research efforts in SFEM:

Intense from the mathematical/numerical point of view
 sophisticated SFEM approaches developed

Very little space left for progress in collecting and studying
measured data for spatial variation characteristics
Consequences:

Data are even today very scarce

Assumptions made for qualitative and quantitative random field
characteristics

Criticism on the correctness and usefulness of SFEM results
Implicit circumvention of this situation:

Parametric studies examining several options for the stochastic
parameters

Bounds on the response variability of structural quantities
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The need for micromechanically derived information (2)
A promising approach:
Probabilistic information for macroscopic properties
from micromechanical material characteristics

Acquire data on the micromechanical characteristics of various
material types

Detect basic rules connecting the random micro- and macroscales

Determine the particular material and microstructure types
available each time  obtain macroscopic stochastic information
with only few and inexpensive measurements
Therefore:

Systematic effort to establish a formal and complete link between
microstructural material models and stochastic material properties
at the macroscale

The contribution of materials scientists in this effort is crucial
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Options and assumptions for random fields
representing stochastic material properties
Issues affecting the type, the shape and the number of random fields
involved in a SFEM application:

statistical homogeneity and isotropy

probability distribution functions and corresponding parameter
values (mean, variance or coefficient of variation)

autocorrelation function

correlation length

stochastic modeling of one or multiple material properties

cross-correlations between different stochastic material
properties

cross-correlations of the same material property at different
structural components
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Numerical test example:
Stochastic FE analysis of wing-shaped structure
A
4.0m
2.0m
y
x
16.0m

94 triangular shell elements

Variable thickness (19cm  4cm)

Subjected to static & turbulence loads

Direct MCS (nsim=100,000),
COSSAN software

Calculation of exceedance
probabilities P(wA>wA,cr)

Independent
stochastic material
properties:
Ex, y   E0 1  f E ( x, y)
vx, y   v0 1  f v ( x, y)
E0=2.06∙1011N/m2, v0=0.3
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The effect of probability distribution functions (1)
1.E+00
Exceedance probability P(w A>wA,cr)
Lognormal
Normal
1.E-01
Uniform
Exponential
Rayleigh
1.E-02
Large_I
Small_I
1.E-03
1.E-04
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Critical displacement wA,cr (m)
E is a random field with μ=E0, σ=10%, exponential autocorrelation and b=10m
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The effect of probability distribution functions (2)
1.E+00
Exceedance probability P(w A>wA,cr)
Lognormal
Normal
1.E-01
Uniform
Exponential
Rayleigh
1.E-02
Large_I
Small_I
1.E-03
1.E-04
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Critical displacement wA,cr (m)
E is a random field with μ=E0, σ=10%, exponential autocorrelation and b=1m
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The effect of probability distribution functions (3)

Choice of probability distribution  significant effect

The effect becomes more pronounced for large correlation length
values

The effect tends to diminish (until it even practically disappears)
for smaller correlation length values
Generally:

No experimental data  arbitrary selections are typically made

Usually Gaussian (normal) random fields,
other distribution functions utilized more rarely (e.g. lognormal)

Insignificant effect on the first two statistical moments (mean,
variance) of a structural response quantity,
but strong influence on tails of a response distribution (e.g.
exceedance probabilities or probabilities of failure/reliability)
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The effect of random field mean μ
1.E+00
μ=(1-20%)Εο
Exceedance probability P(w A>wA,cr)
μ=(1-15%)Εο
μ=(1-10%)Εο
1.E-01
μ=(1-5%)Εο
μ=Εο
μ=(1+5%)Εο
1.E-02
μ=(1+10%)Εο
μ=(1+15%)Εο
μ=(1+20%)Εο
1.E-03
1.E-04
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Critical displacement wA,cr (m)
E is a lognormally distributed random field with σ=10%, exponential autocorrelation
and b=10m
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The effect of random field coefficient of variation σ
1.E+00
b=1.0m, σ=5%
Exceedance probability P(w A>wA,cr)
b=1.0m, σ=10%
b=1.0m, σ=15%
1.E-01
b=1.0m, σ=20%
b=10.0m, σ=5%
b=10.0m, σ=10%
1.E-02
b=10.0m, σ=15%
b=10.0m, σ=20%
1.E-03
1.E-04
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Critical displacement wA,cr (m)
E is a lognormally distributed random field with μ=E0, exponential autocorrelation and
b=1m, b=10m
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The effect of random field mean μ and coefficient of
variation σ

Choice of μ, σ  significant effect

The effect is stronger for large correlation length values

The effect diminishes for smaller correlation length values
Generally:

Values of μ, σ are typically assumed

σ often involved in parametric SFEM studies

Common values used for σ are in the order of 10%
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The effect of correlation lengths b (1)
1.E+00
Exceedance probability P(w A>wA,cr)
b=0.5m
b=1.0m
1.E-01
b=5.0m
b=10.0m
1.E-02
b=20.0m
1.E-03
1.E-04
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Critical displacement wA,cr (m)
E is a lognormally distributed random field with μ=E0, σ=10% and exponential
autocorrelation
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The effect of correlation lengths b (2)

Choice of correlation length  decisive effect

Influence on simulation results and

Effect on the behavior of other random field properties
Generally:

Most often involved in parametric SFEM studies

The number of random variables required to adequately represent
a field is practically a function of the correlation length

Directly affects the computing effort needed to generate random
field samples during SFEM simulations
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The effect of autocorrelation function types (1)
1.E+00
b=1.0m, exponential autocorrelation
1.E-01
b=1.0m, convex autocorrelation
b=10.0m, exponential autocorrelation
b=10.0m, triangular autocorrelation
b=10.0m, convex autocorrelation
1.E-02
1
1.E-03
1.E-04
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
Critical displacement wA,cr (m)
autocorrelation function value
Exceedance probability P(w A>wA,cr)
b=1.0m, triangular autocorrelation
0.9
exponential autocorrelation
0.8
triangular autocorrelation
0.7
convex autocorrelation
0.6
0.5
0.4
0.3
0.75
0.80
0.85
0.90
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
dimensionless distance
E is a lognormally distributed random field with μ=E0, σ=10% and b=1m, b=10m
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The effect of autocorrelation function types (2)



Exponential autocorrelation R ff σ ff2 exp  d  d  x 2  y 2  z 2


 bf 

Triangular autocorrelation

Convex autocorrelation
R ff

σ ff2


0

 d
d
 exp 
 2  2m
 bf
bf


R ff
 2
σ
  ff
0


1  m d

bf


 b 
 if d  0, f 

 m

else


 if 2  2m d  exp  d

 bf
bf


else

0


m

Choice of autocorrelation function  insignificant effect

Preference shown in exponential autocorrelation (physically
reasonable)

No complete experimental validation for any function
e 1
e
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The effect of multiple stochastic material properties (1)
1.E+00
Exceedance probability P(w A>wA,cr)
b=1.0m, σ(Ε)=10%, σ(ν)=0
b=1.0m, σ(Ε)=10%, σ(ν)=10%
1.E-01
b=10.0m, σ(Ε)=10%, σ(ν)=0
b=10.0m, σ(Ε)=10%, σ(ν)=10%
1.E-02
1.E-03
1.E-04
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
Critical displacement wA,cr (m)
E and v are lognormally distributed random fields with exponential autocorrelation
and b=1m, b=10m (for E: μ=E0, σ=10% – for v: μ=v0, σ=0 or σ=10%)
24
The effect of multiple stochastic material properties (2)

Stochastic modeling of E and v  negligible effect compared to
E-only stochastic variation

This may not be the case with other stochastic material properties
(e.g. E & ρ)
Generally:

Typically only one stochastic material property assumed (E)

The same physical reasoning leading to the assumption that a
particular material property varies randomly in space is very likely
to apply also to other material properties of the same structure
 multiple stochastic material properties
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Other random field properties
Statistical homogeneity and isotropy

Random fields usually assumed to be homogeneous, isotropic

Homogeneous field: distributional and autocorrelation
characteristics are invariant under translations in the spatial
domain

Isotropic field: distributional and autocorrelation characteristics
are invariant under rotations in the spatial domain
Cross-correlations between different stochastic material properties

Typically ignored, although they may have significant effect
Cross-correlations of the same material property at different
structural components

Typically ignored
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Conclusions (1)
Effect of random field assumptions based on numerical investigation:

Decisive:
correlation length

Significant:
probability distribution functions and related parameters (μ, σ)

Insignificant:
autocorrelation function types
simultaneous stochastic E & v
Effect of random field assumptions based on engineering reasoning:

Significant:
statistical homogeneity and isotropy
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Conclusions (2)

The need for experimental data to realistically quantify material
randomness is evident

This work identifies the most critical of the random field properties
needing closer and more urgent attention

The contribution of materials scientists in this effort is crucial

A challenge for material scientists:
they can make a valuable contribution in the understanding of
unexplored issues affecting the application of sophisticated
stochastic formulations that are being developed for decades
28