Epistemic Uncertainty Quantification of
Product-Material Systems
Grant No. 826547
CMMI, Engineering Design and Innovation
Shahabedin Salehghaffari
PhD Student, Computational Engineering
Masoud Rais-Rohani (PI, Research Advisor)
Prof. of Aerospace Engineering
Masoud@ae.msstate.edu
Douglas J. Bammann (Co-PI)
Prof. of Mechanical Engineering
Bammann@me.msstate.edu
Esteban B. Marin (Co-PI)
Research Associate Prof.
Ebmarin@cavs.msstate.edu
Tomasz A. Haupt (Co-PI)
Research Associate Prof.
Haupt@cavs.msstate.edu
Center for Advanced Vehicular Systems
Bagley College of Engineering
Abstract
Principles of evidence theory are used to develop a methodology for
quantifying epistemic uncertainty in constitutive models that are often
used in nonlinear finite element analysis involving large plastic
deformation. The developed methodology is used for modeling
epistemic uncertainty in Johnson-Cook plasticity model. All sources of
uncertainty emanating from experimental stress-strain data at different
temperatures and strain rates, as well as expert opinions for method of
fitting the model constants and the representation of homologous
temperature are considered. The five Johnson-Cook model constants
are determined in interval form and the presented methodology is used
to find the basic belief assignment (BBA) for them. The represented
uncertainty in intervals with assigned BBA are propagated through the
non-linear crushing simulation of an Aluminum 6061-T6 circular tube.
Comparing the propagated uncertainty with belief structure of the
crushing response—constructed by collection of all available
experimental, numerical and analytical sources—the amount of
epistemic uncertainty in Johnson-Cook model is estimated.
2
Sources of Uncertainties in Plasticity Models
Uncertainties in
Simulation of Large
Deformation Process
Model Selection
Uncertainty caused by
different choices of
Plasticity Models
(Johnson-Cook, EMMI,
BCJ, …)
Uncertain Material
Parameters reflecting
incomplete knowledge
of the defamation
mechanism of metals
Model Form
Uncertainty caused by
making simplifications
in mathematical
representation of
deformation process
Different Expert
Opinions for fitting
method of material
constants
Different Choices of
Experimental Data
(stress-strain curves):
Types, Strain Rates,
Temperatures
Uncertainties in
Experimental Data
method of
Experimentation,
Measuring stress
3
Uncertainty Modeling
1. Uncertainty Representation:
–
–
–
Establishment of an informative methodology for construction of Basic Belief
Assignment (BBA) using available sources of experimental data as well as
different expert opinions.
Using a proper aggregation rule to combine evidence from different sources with
conflicting BBA.
Uncertainty representation of Johnson-Cook models in intervals with assigned
BBA using the established methodology by collection of evidence from different
experimental sources and fitting approaches of material constants.
2. Uncertainty Propagation:
–
–
Propagation of the represented uncertainty through the non-linear crushing
simulation of an Aluminum 6061-T6 circular tube.
Obtaining bounds of simulation responses due to the variation of material
constants in intervals using Design and Analysis of Computer Experiments to
determine propagated belief structure.
3. Modeling Model Selection Uncertainty:
–
Using Yager’s aggregation rule to combine the propagated belief structure
obtained from different formulations of Johnson-Cook models.
4. Uncertainty Quantification:
–
Constructing belief structure of the simulation response through consideration of
available experimental, numerical and analytical sources of evidence.
4
From Evidence Collection to Evidence Propagation
I1
I2
0.05 Values 0.3
of
0.1
0.25
Constant 0.3
C2
Joint
Belief
(BBA)
I4
0. 0036
I5
I1
I2
0.2
I3
I1
Data,
Opinion
0.7
Values
0.1
of
Constant
I3
C1
I3
I1
I2
0. 0009
Propagated
BBA
[I1(C1), I5(C2),…, I3(Cn)]
I2
0. 10. 5
Values
0.4
of
Constant
Cn
m{[I1(C1), I5(C2),…, I3(Cn)]}
=m{[I1(C1)}×m{[I5(C5)}× …
×m {[I3(Cn)}
I3
I4
I5
In
5
Mathematical Tools of Evidence Theory
•
Consider Θ = {θ1, θ2, ..., θn} as exhaustive set of mutually exclusive events. Frame of
Discernment is defined as
– 2Θ = {f, {θ1}, …, {θn}, {θ1, θ2}, …, {θ1, θ2, ... θn} }
•
The basic belief assignment (BBA), represented as m, assigns a belief number [0,1] to
every member of 2Θ such that the numbers sum to 1.
•
The probability of event A lies within the following interval
– Bel(A) ≤ p(A) ≤ Pl(A)
•
Belief (Bel) represents the total belief committed to event A
•
Plausibility (Pl) represents the total belief that Intersects event A
Epistemic
Uncertainty
0
Bel(A)
1
Bel(Ā)
Pl(A)
6
Relationship Types Between Uncertainty Intervals
•
Ignorance Relationship
BBA: m({I1})=A / (A+B), m({I2})= 0, m({I1,I2})=B / (A+B)
Bel: Bel({I1})=A / (A+B), Bel({I2})= 0, Bel({I1,I2})=1
A
Data Points in interval 1 (I1) = A
Data Points in interval 2 (I2) = B
Total Data points = A+B
A
Pl: Pl({I1})=1, Pl({I2})= B / (A+B), Pl({I1,I2})=1
•
B
A
B
Agreement Relationship
Since two disjoint intervals are combined into a single
interval, BBA structure construction is meaningless
•
Conflict Relationship
BBA: m({I1})=A / (A+B), m({I2})= B / (A+B), m({I1,I2})= 0
Bel: Bel({I1})=A / (A+B), Bel({I2})= B / (A+B), Bel({I1,I2})=1
Pl: pl({I1})= A / (A+B), Pl({I2})= B / (A+B), Pl({I1,I2})=1
B
I1 I2
Ignorance
(B/A < 0.5)
I1 I2
Agreement
(B/A > 0.8)
I1 I2
Conflict
(0.5 ≤ B/A ≤ 0.8)
BBA Structure
7
Different Types of BBA
•
Bayesian: all intervals of uncertainty are
disjointed and treated as having conflict.
•
Consonant: Similar to the case of
ignorance, all intervals of uncertainty in
consonant BBA structure are in ignorance.
•
General: Intervals of uncertainty can be in
both forms of ignorance and conflict. It is
more
prevalent
in
uncertainty
quantification of physical systems.
8
Methodology for BBA Construction in Intervals
•
Step 1: Collect all possible values of uncertain data and
determine the interval of uncertainty that represents the universal
set.
•
Step 2: Plot a histogram (bar chart) of the collected data.
•
Step 3: Identify adjacent intervals of uncertainty that are in
agreement and combine them.
•
Step 4: Identify the interval with highest number of data points
(Im) and recognize its relationship with each of the adjacent
intervals to its immediate left and right (Ia),and construct the
associating BBA
•
Step 5: Consider the adjacent interval (Ic) to interval (Ia)
–
Ia and Im are in ignorance relationship: recognize
relationship type between intervals Ic and Im and construct
the associating BBA.
–
Ia and Im are in conflict relationship: recognize relationship
type between intervals Ia and Ic and construct the
associating BBA.
9
Aggregation of Evidence
Yager’s rule
q (C k ) 
m
1
( Ai ) m 2 ( B j )
B j  Ai  C k
m c (C k )  q (C k )
m ( Ai  B j ) 
m ( Ai ) m ( B j )
m c ( )  0
m c  X   q  X   q ( )
BBA of conflict between Information
from Multiple Sources is assigned to
the Universal Set (X) and interpreted as
degree of Ignorance
10
Uncertainty Representation of Johnson-Cook Models
 Expert Opinion 1: Johnson-Cook Model form

−
−
−
−
A -> yield stress
B and n -> strain hardening
C -> strain rate
m -> temperature
Unknown Constants
to be determined
by fitting methods
 Strain Rate Term Opinions
–
–
–
–
Log-Linear Jonson-Cook, 1983
Log-Quadratic Huh-Kang, 2002
Exponential Allen-Rule-Jones, 1997
Exponential Cowper-Symonds, 1985
 Temperature Term Opinions
 Expert Opinion 2: Fitting Methods


Method 1: Fit constants simultaneously
Method 2: Fit in three separate stages
 Expert Opinion 3:
Choice of experimental test
system
 Expert Opinion 4:
Choice of stress-strain curve
sets to fit constants
11
Uncertainty Representation of Johnson-Cook Models
Test Data for Aluminum Alloy 6061-T6
 Testing Requirements
Experimental Source 1
Curve #
−
−
−
Produce the required
dynamic loads
Determine the stress
state at a desired point
of a specimen
Measure the stress and
strain rates at the above
Type
Strain Rate
(s-1)
Temperature
(K)
1
Tension
634
605
2
Tension
627
3
Tension
4
Curve #
Type
Strain Rate
(s-1)
Temperature
(K)
11
Torsion
11
293
505
12
Torsion
1
293
624
472
13
Torsion
0.001
293
Tension
622
293
14
Torsion
0.1
293
5
Torsion
99
293
15
Compression
800
293
6
Torsion
48
293
16
Compression
0.008
293
7
Torsion
39
293
17
Compression
40
293
8
Torsion
239
293
18
Compression
2
293
9
Torsion
130
293
19
Compression
0.1
293
10
Torsion
126
293
-
-
-
-
Experimental Source 2
point
Resulting test data by different
approaches always subject to
epistemic uncertainty
Experimental Source 1
Experimental Source 3
1
Tension
4.8e-5
297
1
Compression
1000
298
2
Tension
28
297
2
Compression
2000
298
3
Tension
65
297
3
Compression
3000
298
4
Tension
1e-05
533
4
Compression
4000
298
5
Tension
18
533
5
Tension
5.7E-04
373
6
Tension
130
533
6
Tension
1500
373
7
Tension
1e-05
644
7
Tension
5.7E-04
473
8
Tension
23
644
8
Tension
1500
473
9
Tension
54
644
-
-
-
-
12
Uncertainty Representation Procedure
A
Experimental
Source 1
BBA Construction
for Constant A
Model 1 Method 1
Agreement
A1
m ([200.74, 274.29])=
(1330+1395)/4220=0.646
Conflict
311.07])=
A2 m([274.29,
920/4220=0.218
Ignorance
A3
m([90.4, 274.29])=
(210+120)/4220=0.078
Ignorance
A4
m([163.96, 274.29])=
245/4220=0.058
Histograms for
Model 1, Source 1,
Fitting Method 1
B
n
C
m
Experimental
Source 3
Experimental
Source 2
Experimental
Source 1
Histograms
Histograms
Histograms
BBA
for
M2
BBA
for
M1
BBA
for
M2
Combinations
BBA
Source 3
BBA
for
M1
Combinations
BBA
for
M2
BBA
for
M1
Combinations
BBA
Source 2
BBA
Source 1
Combinations
Intervals of Uncertainty
With Assigned BBA
for Each Type Johnson-Cook Model
13
Uncertainty Propagation
BBA Structure for Johnson-Cook Model 1
m({A1})
m({A2})
m({A3})
m({B1})
m({B2})
m({n1})
m({C1}) m({C2}) m({C3})
Generate Random Samples for
each Set of Uncertain Variables
m({n2})
m({n3})
Perform Crush Simulations to
Obtain Output of Interest (Mean
and Maximum Crush Force)
m({m1})
m({A1,B1,C1,n1,m1})
m({A1,B1,C1,n2,m1})
Establish metamodels Between
Uncertain Variables and output of
interest for each set
Perform global optimization analysis
using the established metamodel
To obtain intervals for output of interests
Consider All Sets of Uncertain Variables
m({A3,B2,C3,n3,m1})
m({A(i),B(j),C(k),n(l),m(o)})=
m ({A(i)})×m ({B(j)}) ×m ({C(k)})× m ({n(l)})× m ({m(o)})
Assign a BBA to each obtained interval
for output of interests
Aggregate Propagated BBA from
different sources
14
Uncertainty Propagation
Finite Element Model
Modeling Model Selection Uncertainty of
Johnson-Cook (JC) based Material Models
BBA Structure of
output of interest
using JC Type#1
Random Samples
•Variables: Material Constants
•Outputs: Time Duration &
Crush Length
Simulation Description
•Tube Length: 76.2 mm
•Tube Thickness: 2.4mm
•Tube Mean Radius: 11.5 mm
•Attached Mass: 127 g
•Mass Velocity: 101.3 m/s
•Element Number: 1500
BBA Structure of
output of interest
using JC Type#2
BBA Structure of
output of interest
using JC Type#3
BBA
Aggregation
Final representation of
uncertainty for outputs of
interest (final BBA structure for
Mean or Maximum Crush Load)
15
Uncertainty Propagation
 Metamodeling Technique
Collapsed shapes
of some samples
–Radial Basis Functions (RBF) with
Multi-quadric Formulation
n

f ( X )    if ( X  X i )
i 1
f (r ) 
r c
2
2
c  0 . 001
r = normalized X
 Design Variables: Material Constants
 Simulation Response: Crush Length
16
Construction of Belief Structure for Crush Length
•Available Sources of Evidence for Crush Length:
• Experimental (E): 13.9
• Analytical: 13.1
• Numerical: 12.03
BBA
Aggregation
0.22
0.0359
12
12.5
0.2985
0.2178 0.2278
13
13.5
14
17
Uncertainty Quantification
Belief:
Epistemic Uncertainty:
Belief Complement:
Universal set:
Element of Belief Structure for
Crush Length:
Propagated Belief
Structure
for Crush Length
0.22
Belief Structure
for Crush Length
0.0359
12
12.5
0.2985
0.2178 0.2278
13
13.5
14
18
Developed Approach for Uncertainty Modeling
Experimental
Stress-Strain
Curves
Propagated
Intervals of
Uncertainty
with Assigned
BBA
Uncertainty
Representation
Uncertainty
Propagation
Intervals of
Uncertainty
with Assigned
BBA
FE Simulation
of Crush Tubes
Using Material
Models
•Fully Covered: Increase Belief
•Not Covered: Decrease Belief
•Partially Covered: Increase
Plausibility and Ignorance
Propagated
Belief
Structure
Comparison
Comparison
Intervals of
Uncertainty
with Assigned
BBA
Uncertainty
Representation
of Output of Interests
Available
Evidences for
Crush Length
Belief
Structure for
Crush Length
19
References
• Salehghaffari, S., Rais-Rohani, M., “Epistemic Uncertainty Modeling of Johnson-Cook Plasticity Model, Part 1: Evidence
Collection and Basic Belief Assignment Construction ”, International Journal of Reliability Engineering & System Safety
(under review), 2010.
• Salehghaffari, S., Rais-Rohani, M.,“Epistemic Uncertainty Modeling of Johnson-Cook Plasticity Model, Part 2: Propagation
and quantification of uncertainty”, International Journal of Reliability Engineering & System Safety (under review), 2010.
• Johnson, G.R., Cook W.H., “A constitutive model and data for metals subjected to large strains, high strain rates and high
temperatures”,In: Proceedings of 7th international symposium on Ballistics, The Hague, The Netherlands 1983;. 4, 1999, pp.
557–564.
• Hoge, K.G., “ Influence of strain rate on mechanical properties of 6061-T6 aluminum under uniaxial and biaxial states of
stress”, Experimental Mechanics, 1966; 6: 204-211.
• Nicholas, T., “ Material behavior at high strain rates”, In: Zukas, J.A. et al., 1982. Impact Dynamics, John Wiley, New York,
27–40.
• Helton, J.C., Johnson, J.D., Oberkampf, W.L., “An exploration of alternative approaches to the representation of uncertainty in
model predictions”, International Journal of Reliability Engineering & System Safety ,2004; 85: 39–71.
• Shafer, G., “A mathematical theory of evidence”, Princeton, NJ: Princeton University Press; 1976.
• Yager, R., “On the Dempster-Shafer Framework and New Combination Rules”, Information Sciences, 1987; 41: 93-137.
• Bae, H., Grandhi, R.V., Canfield, R.A., “Epistemic uncertainty quantification techniques including evidence theory for largescale structures”, Computers & Structures, 2004; 82: 1101–1112.
• Bae, H., Grandhi, R.V., Canfield, R.A., “An approximation approach for uncertainty quantification using evidence theory”,
International Journal of Reliability Engineering & System Safety, 2004; 86: 215–225.
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