Epistemic Uncertainty Quantification of Product-Material Systems

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					        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)                        Douglas J. Bammann (Co-PI)
Prof. of Aerospace Engineering                                   Prof. of Mechanical Engineering
Masoud@ae.msstate.edu                                            Bammann@me.msstate.edu

Esteban B. Marin (Co-PI)                                         Tomasz A. Haupt (Co-PI)
Research Associate Prof.                                         Research Associate Prof.
Ebmarin@cavs.msstate.edu                                         Haupt@cavs.msstate.edu

                             Center for Advanced Vehicular Systems
                                 Bagley College of Engineering
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.
Sources of Uncertainties in Plasticity Models

                           Uncertainties in
                         Simulation of Large
                         Deformation Process

   Model Selection                                        Model Form
                           Uncertain Material
Uncertainty caused by                                Uncertainty caused by
                          Parameters reflecting
  different choices of                               making simplifications
                          incomplete knowledge
   Plasticity Models                                    in mathematical
                             of the defamation
(Johnson-Cook, EMMI,                                   representation of
                           mechanism of metals
        BCJ, …)                                       deformation process

                           Different Choices of        Uncertainties in
  Different Expert
                            Experimental Data         Experimental Data
 Opinions for fitting
                           (stress-strain curves):        method of
 method of material
                            Types, Strain Rates,       Experimentation,
                               Temperatures            Measuring stress

                          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.
      From Evidence Collection to Evidence Propagation

      I1                                 I3
                  0.05Values 0.3                                    Joint                I1
                 0.25 of      0.1             I4                   Belief                          I3
                     Constant 0.3
       I2              C2                                          (BBA)                          I2
                                                                    0. 0036
                                    I5                                                  0. 0009
                                                           [I1(C1), I5(C2),…, I3(Cn)]     Propagated 
            I2                                     I1
                      Data,         I2
  0.2           Opinion             0. 1 5
      0.7                               0.              m{[I1(C1), I5(C2),…,
 Values                             Values                                               I4             I5
         0.1                   0.4                      I3(Cn)]}
   of                                  of
Constant                           Constant             =m{[I1(C1)}×m{[I5(C5)}× …
             I3       I3                                ×m {[I3(Cn)}
  C1                                  Cn

            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

                          0            Uncertainty              1
                              Bel(A)                    Bel(Ā

     Relationship Types Between Uncertainty Intervals

                                                                         §Data Points in interval 1 (I1) = A
•   Ignorance Relationship                                               §Data Points in interval 2 (I2) = B
    BBA: m({I1})=A / (A+B), m({I2})= 0, m({I1,I2})=B / (A+B)             §Total Data points = A+B

    Bel: Bel({I1})=A / (A+B), Bel({I2})= 0, Bel({I1,I2})=1
                                                                     A               A                A
    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
                                                                       I1 I2          I1    I2         I1 I2
                                                                     Ignorance      Agreement       Conflict
    Bel: Bel({I1})=A / (A+B), Bel({I2})= B / (A+B), Bel({I1,I2})=1                  (B/A > 0.8) (0.5 ≤ B/A ≤ 0.8)
                                                                     (B/A < 0.5)
    Pl: pl({I1})= A / (A+B), Pl({I2})= B / (A+B), Pl({I1,I2})=1
                                                                             BBA Structure
                      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.

      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

•   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

              §Yager’s rule

               BBA of conflict between Information
                from Multiple Sources is assigned to
              the Universal Set (X) and interpreted as
                       degree of Ignorance

 Uncertainty Representation of Johnson-Cook Models
§ Expert Opinion 1: Johnson-Cook  Model 
                                                     Unknown Constants 
           −   A  -> yield stress                                         
                                                        to be determined 
           −   B and n -> strain hardening
                                                        by fitting methods 
           −   C -> strain rate
           −   m -> temperature
           Ø Strain Rate Term Opinions
           –   Log-Linear Jonson-Cook, 1983
           –   Log-Quadratic Huh-Kang, 2002
           –   Exponential Allen-Rule-Jones, 1997
           –   Exponential Cowper-Symonds, 1985
           Ø Temperature Term Opinions

§ Expert Opinion 2: Fitting Methods
    Ø     Method 1: Fit constants simultaneously
    Ø     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                            Experimental Source 1
                                        Curve #                                          Curve #
                                                             Strain Rate   Temperature                           Strain Rate   Temperature
                                                    Type                                              Type
    −    Produce  the  required                                  (s-1)        (K)                                    (s-1)        (K)
                                          1        Tension      634           605          11        Torsion         11           293
         dynamic loads
                                          2        Tension      627           505          12        Torsion         1            293
                                          3        Tension      624           472          13        Torsion       0.001          293
    −    Determine  the  stress           4        Tension      622           293          14        Torsion        0.1           293
         state  at  a  desired            5        Torsion       99           293          15      Compression      800           293

         point of a specimen              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
    −    Measure  the  stress             9        Torsion      130           293          19      Compression      0.1           293
         and strain rates at the          10       Torsion      126           293           -           -             -             -

         above point                           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

Resulting  test  data  by  different      4        Tension     1e-05          533          4       Compression      4000          298
                                          5        Tension       18           533          5         Tension      5.7E-04         373
approaches  always  subject  to           6        Tension      130           533          6         Tension        1500          373
epistemic uncertainty                     7        Tension     1e-05          644          7         Tension      5.7E-04         473
                                          8        Tension       23           644          8         Tension        1500          473
                                          9        Tension       54           644           -           -             -             -

                      Uncertainty Representation Procedure
                                                                                                                            Histograms for
                                               Experimental                             A                                  Model 1, Source 1,
                                                 Source 1                                                                   Fitting Method 1

                                                                                        B                                     n

                                                                                        C                                     m

                                                                                 Experimental                Experimental              Experimental
BBA Construction                                                                   Source 3                    Source 2                  Source 1
 for Constant A                                                                 Histograms                  Histograms                Histograms
Model 1 Method 1                       Agreement
                                                                                 BBA        BBA              BBA       BBA            BBA         BBA
                                    m ([200.74, 274.29])=
                               A1 (1330+1395)/4220=0.646                          for        for              for       for            for         for
                                                                                 M2         M1               M2        M1             M2          M1

                                                      Conflict                 Combinations               Combinations             Combinations
                                                                                      BBA                         BBA                      BBA
                      Ignorance                       A2m([274.29, 311.07])=
                                                         920/4220=0.218             Source 3                    Source 2                 Source 1
     m([90.4, 274.29])=
A3 (210+120)/4220=0.078                                                                                   Combinations
                                                                                                         Intervals of Uncertainty
                               m([163.96, 274.29])=                                                        With Assigned BBA
                          A4     245/4220=0.058
                                                                                                   for Each Type Johnson-Cook Model

                                           Uncertainty Propagation

      BBA Structure for Johnson-Cook Model 1
                                                                           Generate Random Samples for
                                                                           each Set of Uncertain Variables

    m({A1})           m({A3})
                                          m({B1})       m({B2})

                                          m({n1})             m({n2})       Perform Crush Simulations to
    m({C1}) m({C2})   m({C3})                       m({n3})                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
                  m({A1,B1,C1,n2,m1})                                    using the established metamodel
                                                                          To obtain intervals for output of
  Consider All Sets of Uncertain Variables
                                                                        Assign a BBA to each obtained interval
                                                                                for output of interests
                  m({A(i),B(j),C(k),n(l),m(o)})=                           Aggregate Propagated BBA from
                                                                                  different sources
m ({A(i)})×m ({B(j)}) ×m ({C(k)})× m ({n(l)})× m ({m(o)})
             Uncertainty Propagation
Finite Element Model                      Modeling Model Selection Uncertainty of
                                            Johnson-Cook (JC) based Material

                                                            BBA Structure of
                                                            output of interest
                                                            using JC Type#1

                                           BBA Structure of                   BBA Structure of
         Random Samples                    output of interest                 output of interest
                                           using JC Type#2                    using JC Type#3
         •Variables: Material Constants
         •Outputs: Time Duration &
         Crush Length
         Simulation                                       Aggregation
         •Tube Length: 76.2 mm
         •Tube Thickness: 2.4mm
                                                        Final representation of
         •Tube Mean Radius: 11.5 mm                    uncertainty for outputs of
         •Attached Mass: 127 g                       interest (final BBA structure for
                                                      Mean or Maximum Crush Load)
         •Mass Velocity: 101.3 m/s
         •Element Number: 1500
                   Uncertainty Propagation
                                         Collapsed shapes
§ Metamodeling Technique                 of some samples
   –Radial Basis Functions (RBF) with
   Multi-quadric Formulation

      r = normalized X

§ DesignVariables: Material Constants
§ Simulation Response: Crush Length
   Construction of Belief Structure for Crush Length
•Available Sources of Evidence for Crush Length:
    • Experimental (E): 13.9
    • Analytical: 13.1
    • Numerical: 12.03


                                                    0.0359       0.2178 0.2278

                                                   12   12.5    13    13.5       14

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

  Belief Structure
                      0.0359           0.2178 0.2278
 for Crush Length
                     12    12.5      13      13.5       14
Developed Approach for Uncertainty Modeling
                 Uncertainty                             •Fully Covered: Increase Belief
                Representation     Intervals of          •Not Covered: Decrease Belief
                                   Uncertainty           •Partially Covered: Increase
                                  with Assigned          Plausibility and Ignorance
 Propagated     Uncertainty
                                 FE Simulation
 Intervals of   Propagation
                                 of Crush Tubes
 Uncertainty                     Using Material
with Assigned                        Models
 Intervals of
                of Output of Interests     Available               Belief
                                         Evidences for          Structure for
with Assigned
                                         Crush Length
    BBA                                                         Crush Length

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  (under review), 2010.

• Salehghaffari, S., Rais-Rohani, M.,“Epistemic Uncertainty Modeling of Johnson-Cook Plasticity Model, Part 2: Propagation
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