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## 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 ShahabedinSalehghaffari 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.eduBammann@me.msstate.edu Esteban B. Marin (Co-PI) Tomasz A. Haupt(Co-PI) Research Associate Prof. Research Associate Prof. Ebmarin@cavs.msstate.eduHaupt@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.**Uncertainty Modeling**• 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. • 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**• Joint Belief (BBA) • I3 • I1 • 0.3 • 0.05 • I1 • 0.1 • I4 • 0.25 • I3 • 0.3 • I2 • I2 • 0. 0036 • 0. 0009 • I5 • I1 Propagated BBA [I1(C1), I5(C2),…, I3(Cn)] • I1 • I2 • I2 • 0.2 • 0. 1 • 0. 5 • 0.7 m{[I1(C1), I5(C2),…, I3(Cn)]} =m{[I1(C1)}×m{[I5(C5)}× … ×m {[I3(Cn)} • I5 • I4 • 0.4 • 0.1 • I3 • I3 • In**0**1 Bel(A) Bel(Ā) Pl(A) 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 2Θsuch 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**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 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})=1Pl: 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 A A A B B B I1 I2 I1 I2 I1 I2 Ignorance Agreement Conflict (0.5 ≤ B/A ≤ 0.8) (B/A > 0.8) (B/A < 0.5) 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 IcandImand 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 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 • 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 11**Uncertainty Representation of Johnson-Cook Models**Test Data for Aluminum Alloy 6061-T6 • 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 12**Uncertainty Representation Procedure**Experimental Source 3 Experimental Source 2 Experimental Source 1 BBA Construction for Constant A Model 1 Method 1 Histograms Histograms Histograms Agreement BBA for M2 BBA for M1 BBA for M2 BBA for M1 BBA for M2 BBA for M1 A1 m ([200.74, 274.29])= (1330+1395)/4220=0.646 Combinations Combinations Conflict A2 m([274.29, 311.07])= 920/4220=0.218 BBA Source 3 BBA Source 2 BBA Source 1 Ignorance m([90.4, 274.29])= (210+120)/4220=0.078 Combinations A3 Ignorance Intervals of Uncertainty With Assigned BBA for Each Type Johnson-Cook Model m([163.96, 274.29])= 245/4220=0.058 A4 Histograms for Model 1, Source 1, Fitting Method 1 A • Experimental Source 1 B n m C Combinations**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({A2}) Perform Crush Simulations to Obtain Output of Interest (Mean and Maximum Crush Force) m({n1}) m({n2}) m({C1}) m({C2}) m({C3}) m({n3}) m({m1}) Establish metamodels Between Uncertain Variables and output of interest for each set m({A1,B1,C1,n1,m1}) m({A1,B1,C1,n2,m1}) Perform global optimization analysis using the established metamodel To obtain intervals for output of interests Consider All Sets of Uncertain Variables Assign a BBA to each obtained interval for output of interests 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)}) Aggregate Propagated BBA from different sources 14**Uncertainty Propagation**Modeling Model Selection Uncertainty of Johnson-Cook (JC) based Material Models Finite Element Model BBA Structure of output of interest using JC Type#1 BBA Structure of output of interest using JC Type#2 BBA Structure of output of interest using JC Type#3 • Random Samples • Variables: Material Constants • Outputs: Time Duration & Crush Length BBA Aggregation • 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)**Uncertainty Propagation**Collapsed shapes of some samples • Metamodeling Technique • Radial Basis Functions (RBF) with Multi-quadric Formulation 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 0.2985 0.2278 0.0359 0.2178 12 12.5 13 13.5 14**Uncertainty Quantification**Belief: Epistemic Uncertainty: Belief Complement: Universal set: Element of Belief Structure for Crush Length: Propagated Belief Structure for Crush Length 0.22 0.2985 Belief Structure for Crush Length 0.2278 0.0359 0.2178 12 12.5 13 13.5 14**Developed Approach for Uncertainty Modeling**Uncertainty Representation • Fully Covered: Increase Belief • Not Covered: Decrease Belief • Partially Covered: Increase Plausibility and Ignorance Experimental Stress-Strain Curves Intervals of Uncertainty with Assigned BBA Propagated Belief Structure FE Simulation of Crush Tubes Using Material Models Uncertainty Propagation Propagated Intervals of Uncertainty with Assigned BBA Comparison Comparison Uncertainty Representation of Output of Interests Intervals of Uncertainty with Assigned BBA Available Evidences for Crush Length Belief Structure for Crush Length**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.