Report

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 [email protected] Douglas J. Bammann (Co-PI) Prof. of Mechanical Engineering [email protected] Esteban B. Marin (Co-PI) Research Associate Prof. [email protected] Tomasz A. Haupt (Co-PI) Research Associate Prof. [email protected] 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 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