Epistemic Uncertainty Quantification of Product-Material Systems

Epistemic Uncertainty Quantification of Product-Material Systems

Grant No. 826547CMMI, Engineering Design and Innovation

Shahabedin SalehghaffariPhD Student, Computational Engineering

Masoud Rais-Rohani (PI, Research Advisor) Douglas J. Bammann (Co-PI)Prof. of Aerospace Engineering Prof. of Mechanical [email protected] [email protected]

Esteban B. Marin (Co-PI) Tomasz A. Haupt (Co-PI)Research Associate Prof. Research Associate Prof. [email protected] [email protected]

Center for Advanced Vehicular SystemsBagley College of Engineering

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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.

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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

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

Model Form Uncertainty caused by making simplifications

in mathematical representation of

deformation process

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Uncertainty Modeling1. 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.

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Data,Opinion

Values of

ConstantC2

Values of

ConstantCn

Values of

Constant C1

0.1

0.70.2

0.4

0. 50. 1

0.050.25 0.1

0.3

0.3

I1

I2

I3 I3

I2

I1

I1

I2

I3

I4

I5

Joint Belief (BBA)

[I1(C1), I5(C2),…, I3(Cn)]

0. 0036

m{[I1(C1), I5(C2),…, I3(Cn)]}=m{[I1(C1)}×m{[I5(C5)}× …×m {[I3(Cn)}

Propagated BBA

I1

I2

I3

I4

In

I5

From Evidence Collection to Evidence Propagation

0. 0009

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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

Bel(A) Bel(Ā)

Pl(A)

0 1Epistemic

Uncertainty

Relationship Types Between Uncertainty Intervals

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• 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 Pl: Pl({I1})=1, Pl({I2})= B / (A+B),

Pl({I1,I2})=1

• 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

Data Points in interval 1 (I1) = A Data Points in interval 2 (I2) = B Total Data points = A+B

(B/A < 0.5) (B/A > 0.8) (0.5 ≤ B/A ≤ 0.8)Ignorance Agreement Conflict

BBA Structure

A A A

B

BB

I1 I1 I1I2 I2 I2

Different Types of BBA

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• 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.

Methodology for BBA Construction in Intervals

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• 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.

Aggregation of Evidence

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Yager’s rule

kij CAB

jik BmAmCq )()()( 21

)()( kkc CqCm

0)( cm

)( qXqXmc

BBA of conflict between Information from Multiple Sources is assigned to

the Universal Set (X) and interpreted as degree of Ignorance

)()(

)(

ji

ji

BmAm

BAm

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 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

Unknown Constants to be determined

by fitting methods

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Uncertainty Representation of Johnson-Cook Models

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Curve #

Experimental Source 1 Curve #

Experimental Source 1

Type Strain Rate (s-1)

Temperature(K) Type Strain Rate

(s-1)Temperature

(K)

1 Tension 634 605 11 Torsion 11 293

2 Tension 627 505 12 Torsion 1 293

3 Tension 624 472 13 Torsion 0.001 293

4 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



9 Torsion 130 293 19 Compression 0.1 293

10 Torsion 126 293 - - - -

Experimental Source 2 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 - - - -

Testing Requirements

− 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

point Resulting test data by different

approaches always subject to epistemic uncertainty

Test Data for Aluminum Alloy 6061-T6

Uncertainty Representation Procedure

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Histograms forModel 1, Source 1, Fitting Method 1

Experimental Source 1

Combinations

ExperimentalSource 3



BBA for M2

BBA for M1

BBA for M2

BBA for M2

BBA for M1

BBA for M1

BBASource 1

BBASource 2

BBASource 3

Intervals of UncertaintyWith Assigned BBA

for Each Type Johnson-Cook Model

Combinations

Combinations Combinations

Histograms Histograms Histograms

m ([200.74, 274.29])=(1330+1395)/4220=0.646

m([274.29, 311.07])=920/4220=0.218

m([163.96, 274.29])=245/4220=0.058

m([90.4, 274.29])=(210+120)/4220=0.078

Agreement

Conflict

Ignorance

Ignorance

BBA Constructionfor Constant A

Model 1 Method 1

A1

A2

A4

A3

A

B n

C m

Uncertainty Propagation

m({A1,B1,C1,n1,m1})

m({A1,B1,C1,n2,m1})

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)})

Consider All Sets of Uncertain Variables

BBA Structure for Johnson-Cook Model 1

m({A1})m({A2})

m({A3})m({B2})m({B1})

m({C2})m({C1}) m({C3})m({n1}) m({n2})

m({n3})

m({m1})

Generate Random Samples for each Set of Uncertain

Variables

Perform Crush Simulations to Obtain Output of Interest

(Mean and Maximum Crush Force)

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

Assign a BBA to each obtained interval for output of interests

Aggregate Propagated BBA from different sources

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Finite Element Model

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

Final representation of uncertainty for outputs of interest (final BBA structure for Mean or Maximum Crush

Load)

Modeling Model Selection Uncertainty of Johnson-Cook (JC) based Material Models

BBA Structure of output of

interest using JC Type#1





BBA Aggregation


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Metamodeling Technique

– Radial Basis Functions (RBF) with Multi-quadric Formulation

r = normalized X

Design Variables: Material Constants Simulation Response: Crush Length

001.0)( 22

ccrrf

n

iiif

1

)()( XXX f

Collapsed shapes of some samples

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Construction of Belief Structure for Crush Length

BBA Aggregation

• Available Sources of Evidence for Crush Length: • Experimental (E): 13.9• Analytical: 13.1• Numerical: 12.03

12 12.5 13 13.5 14

0.0359 0.22780.21780.2985

0.22

Belief:Epistemic Uncertainty:Belief Complement:Universal set:Element of Belief Structure for Crush Length:

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Uncertainty Quantification

12 12.5 13 13.5 14

0.0359 0.22780.21780.2985

0.22

Belief Structure for Crush Length

Propagated Belief Structure

for Crush Length

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Developed Approach for Uncertainty Modeling

ExperimentalStress-Strain

Curves

Intervals ofUncertainty

with AssignedBBA

FE Simulationof Crush Tubes Using Material

Models

PropagatedIntervals ofUncertainty

with Assigned BBA

Available Evidences forCrush Length

Intervals ofUncertainty

with AssignedBBA

UncertaintyRepresentation

UncertaintyPropagation

• Fully Covered: Increase Belief

• Not Covered: Decrease Belief

• Partially Covered: Increase Plausibility and Ignorance

UncertaintyRepresentation

of Output of Interests

Comparison

Propagated Belief

Structure

Belief Structure for Crush Length

Comparison

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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 large-scale 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.

Epistemic Uncertainty Quantification of Product-Material Systems

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