REC 2008; Zissimos P. Mourelatos Design under Uncertainty using Evidence Theory and a Bayesian Approach Jun Zhou Zissimos P. Mourelatos Mechanical Engineering.

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Presentation transcript:

REC 2008; Zissimos P. Mourelatos Design under Uncertainty using Evidence Theory and a Bayesian Approach Jun Zhou Zissimos P. Mourelatos Mechanical Engineering Department Oakland University Rochester, MI 48309, USA

REC 2008; Zissimos P. Mourelatos 2 Overview  Introduction  Design under uncertainty  Uncertainty theories  Evidence – Based Design Optimization (EBDO)  Fundamentals of Bayesian Approach  Bayesian Approach to Design Optimization (BADO)  A Combined Bayesian and EBDO Approach  Example  General conclusions

REC 2008; Zissimos P. Mourelatos Design Under Uncertainty Analysis / Simulation Input Output Uncertainty (Quantified) Uncertainty (Calculated) Propagation Design 1. Quantification 2. Propagation 3. Design

REC 2008; Zissimos P. Mourelatos  Aleatory Uncertainty (Irreducible, Stochastic)  Probabilistic distributions  Bayesian updating  Epistemic Uncertainty (Reducible, Subjective, Ignorance, Lack of Information)  Fuzzy Sets; Possibility methods (non-conflicting information)  Evidence theory (conflicting information) Uncertainty Types 25%30%45% Probability distribution Expert OpinionFinite samplesInterval Decreasing level of information

REC 2008; Zissimos P. Mourelatos 5 Evidence Theory Possibility Theory Probability Theory Uncertainty Theories

REC 2008; Zissimos P. Mourelatos Evidence-Based Design Optimization (EBDO)

REC 2008; Zissimos P. Mourelatos 7 Set Notation Universe (X) Power Set (All sets) Element A BC A B Evidence Theory

REC 2008; Zissimos P. Mourelatos 8 Evidence-Based Design Optimization (EBDO) Basic Probability Assignment (BPA): m(A) If m(A)>0 for then A is a focal element Y “Expert” A Y “Expert” B Y 0.x x2 0.x3 0.x6 0.x5 0.x4 5 Combining Rule (Dempster – Shafer)

REC 2008; Zissimos P. Mourelatos 9 Evidence-Based Design Optimization (EBDO) BPA structure for a two-input problem

REC 2008; Zissimos P. Mourelatos 10 Evidence-Based Design Optimization (EBDO) If we define, then where and

REC 2008; Zissimos P. Mourelatos 11 Evidence-Based Design Optimization (EBDO) Position of a focal element w.r.t. limit state Contributes to Belief Contributes to Plausibility

REC 2008; Zissimos P. Mourelatos 12 Evidence-Based Design Optimization (EBDO) Design Principle Therefore, is satisfied if OR If non-negative null form is used for feasibility, failure

REC 2008; Zissimos P. Mourelatos 13, Evidence-Based Design Optimization (EBDO) Formulation

REC 2008; Zissimos P. Mourelatos 14 Evidence-Based Design Optimization (EBDO) Calculation of

REC 2008; Zissimos P. Mourelatos 15 Feasible Region x2x2 x1x1 g 1 (x 1,x 2 )=0 g 2 (x 1,x 2 )=0 Objective Reduces initial design point frame of discernment EBDO optimum deterministic optimum Evidence-Based Design Optimization (EBDO) Implementation

REC 2008; Zissimos P. Mourelatos 16 25%30%45% Probability distribution Expert OpinionFinite samplesInterval Decreasing level of information Available Information

REC 2008; Zissimos P. Mourelatos Fundamentals of Bayesian Approach

REC 2008; Zissimos P. Mourelatos 18 Example : Success : Failure : Bayesian uncertain variable

REC 2008; Zissimos P. Mourelatos 19 Example Want to calculate : Number of sample points

REC 2008; Zissimos P. Mourelatos 20 Probabilistic Constraint : Determ. Random Bayesian R=0.78 Confidence Percentile = 85% Probability = 0.78

REC 2008; Zissimos P. Mourelatos 21 Probabilistic Constraint : Confidence Percentile Confidence Percentile = 85% Probability = 0.78 R = 0.78

REC 2008; Zissimos P. Mourelatos 22 Confidence Percentile Using Extreme Distribution of Smallest Value R=0.465 Confidence Percentile using Beta Distribution = 99.87% Confidence Percentile using Extreme Distribution = 57.3% Beta Extreme Value

REC 2008; Zissimos P. Mourelatos Bayesian Approach to Design Optimization (BADO)

REC 2008; Zissimos P. Mourelatos 24 BADO Formulation s.t. Target Confidence Percentile Confidence Percentile

REC 2008; Zissimos P. Mourelatos 25 Thin-Walled Pressure Vessel Example s.t. yielding

REC 2008; Zissimos P. Mourelatos 26 Thin-Walled Pressure Vessel Example

REC 2008; Zissimos P. Mourelatos A Combined Bayesian and EBDO Approach

REC 2008; Zissimos P. Mourelatos 28 Illustration with the Thin-Walled Pressure Vessel Example 100 Sample Points P=3/100=0.03 ??

REC 2008; Zissimos P. Mourelatos 29 Illustration with the Thin-Walled Pressure Vessel Example Define random variable X where, X is # of sample points insegment. X ~ Beta(3+1, ) = Beta(4, 98) Then: Using extreme distribution of smallest value with :

REC 2008; Zissimos P. Mourelatos 30 Illustration with the Thin-Walled Pressure Vessel Example

REC 2008; Zissimos P. Mourelatos 31 Illustration with the Thin-Walled Pressure Vessel Example

REC 2008; Zissimos P. Mourelatos 32 General Conclusions  There are formal design optimization tools depending on amount and form of available information; e.g. RBDO, EBDO, PBDO, BADO, Bayesian + Evidence.  There is a trade-off between conservative designs (loss of optimality) and available information. Deterministic RBDO EBDO BADO Less Information More Conservative Design PBDO Bayesian + Evidence

REC 2008; Zissimos P. Mourelatos 33 Q & A

REC 2008; Zissimos P. Mourelatos 34 Evidence-Based Design Optimization (EBDO) Implementation Feasible Region x2x2 x1x1 g 1 (x 1,x 2 )=0 g 2 (x 1,x 2 )=0 Objective Reduces hyper-ellipse initial design point frame of discernment B MPP for g 1 =0 deterministic optimum

REC 2008; Zissimos P. Mourelatos 35 Evidence-Based Design Optimization (EBDO) Implementation Feasible Region x2x2 x1x1 g 1 (x 1,x 2 )=0 g 2 (x 1,x 2 )=0 Objective Reduces hyper-ellipse initial design point frame of discernment EBDO optimum B MPP for g 1 =0 deterministic optimum

REC 2008; Zissimos P. Mourelatos Bayesian Approach Design Optimization (BADO) 1. Uncertainty is in the form of sample points. 2. The probability distribution is Beta distribution. Prior and posterior distributionsConfidence percentile Extreme value and Beta distribution s.t.,, 3. BADO formulation