NonProbabilistic Design Optimization with Insufficient Data using Possibility
Non-Probabilistic Design Optimization with Insufficient Data using Possibility and Evidence Theories Zissimos P. Mourelatos Jun Zhou Mechanical Engineering Department Oakland University Rochester, MI 48309, USA mourelat@oakland. edu REC 2006; Zissimos P. Mourelatos
Overview § Introduction Ø Design under uncertainty Ø Uncertainty theories § Possibility – Based Design Optimization (PBDO) Ø Uncertainty quantification and propagation Ø Design algorithms § Evidence – Based Design Optimization (EBDO) § Examples § Summary and conclusions REC 2006; Zissimos P. Mourelatos 2
Design Under Uncertainty Input Uncertainty (Quantified) Analysis / Simulation Propagation Design 1. Quantification 2. Propagation 3. Design REC 2006; Zissimos P. Mourelatos Output Uncertainty (Calculated)
Uncertainty Types Ø Aleatory Uncertainty (Irreducible, Stochastic) § Probabilistic distributions § Bayesian updating Ø Epistemic Uncertainty (Reducible, Subjective, Ignorance, Lack of Information) § Fuzzy Sets; Possibility methods (non-conflicting information) § Evidence theory (conflicting information) REC 2006; Zissimos P. Mourelatos
Uncertainty Theories Evidence Theory Possibility Theory Probability Theory REC 2006; Zissimos P. Mourelatos 5
Non-Probabilistic Design Optimization: Set Notation Universe A B Power Set (All sets) (X ) C B Element A Evidence Theory REC 2006; Zissimos P. Mourelatos 6
Possibility-Based Design Optimization (PBDO) REC 2006; Zissimos P. Mourelatos
Possibility-Based Design Optimization (PBDO) Evidence Theory No Conflicting Evidence (Possibility Theory) REC 2006; Zissimos P. Mourelatos 8
Quantification of a Fuzzy Variable: Membership Function - cut provides confidence level At each confidence level, or -cut, a set is defined as convex normal set REC 2006; Zissimos P. Mourelatos 9
Propagation of Epistemic Uncertainty Extension Principle The “extension principle” calculates the membership function (possibility distribution) of the fuzzy response from the membership functions of the fuzzy input variables. If where then Practical Approximations of Extension Principle • Vertex Method • Discretization Method • Hybrid (Global-Local) Optimization Method REC 2006; Zissimos P. Mourelatos 10
Optimization Method 1. 0 a a 0. 0 1. 0 Global a Global 0. 0 where : s. t. and s. t. REC 2006; Zissimos P. Mourelatos 11
Possibility-Based Design Optimization (PBDO) (Possibility Theory) § What is possible may not be probable § What is impossible is also improbable If feasibility is expressed with positive null form then, constraint g is ALWAYS satisfied if or for REC 2006; Zissimos P. Mourelatos 12
Possibility-Based Design Optimization (PBDO) Considering that , we have REC 2006; Zissimos P. Mourelatos 13
Possibility-Based Design Optimization (PBDO) s. t. ; OR s. t. ; REC 2006; Zissimos P. Mourelatos Double Loop 14
PBDO with both Random and Possibilistic Variables s. t. ; with Triple Loop , and s. t. , s. t. REC 2006; Zissimos P. Mourelatos 15
Evidence-Based Design Optimization (EBDO) REC 2006; Zissimos P. Mourelatos
Evidence-Based Design Optimization (EBDO) Basic Probability Assignment (BPA): m(A) If m(A)>0 for 5 then A is a focal element 0. 2 9. 5 8 6. 2 0. 1 11 0. 3 0. 4 Y “Expert” A 8. 7 7 5 11 0. 4 0. 3 Y “Expert” B Combining Rule (Dempster – Shafer) 6. 2 5 0. x 1 7 0. x 2 8. 7 8 0. x 3 0. x 4 9. 5 0. x 5 REC 2006; Zissimos P. Mourelatos 11 0. x 6 Y 17
Evidence-Based Design Optimization (EBDO) For define: where Assuming independence, where REC 2006; Zissimos P. Mourelatos 18
Evidence-Based Design Optimization (EBDO) BPA structure for a two-input problem REC 2006; Zissimos P. Mourelatos 19
Evidence-Based Design Optimization (EBDO) Uncertainty Propagation If we define, then where and REC 2006; Zissimos P. Mourelatos 20
Evidence-Based Design Optimization (EBDO) Position of a focal element w. r. t. limit state Contributes to Belief Contributes to Plausibility REC 2006; Zissimos P. Mourelatos 21
Evidence-Based Design Optimization (EBDO) Design Principle If non-negative null form is used for feasibility, feasible infeasible failure Therefore, is satisfied if OR is satisfied if REC 2006; Zissimos P. Mourelatos 22
Evidence-Based Design Optimization (EBDO) Formulation , REC 2006; Zissimos P. Mourelatos 23
Evidence-Based Design Optimization (EBDO) Calculation of REC 2006; Zissimos P. Mourelatos 24
Geometric Interpretation of PBDO and EBDO REC 2006; Zissimos P. Mourelatos
Possibility-Based Design Optimization (PBDO) Implementation x 1 initial design point g 1(x 1, x 2)=0 Feasible Region frame of discernment g 2(x 1, x 2)=0 Objective Reduces PBDO optimum deterministic optimum x 2 REC 2006; Zissimos P. Mourelatos 26
Evidence-Based Design Optimization (EBDO) Implementation x 1 initial design point g 1(x 1, x 2)=0 Feasible Region frame of discernment g 2(x 1, x 2)=0 Objective Reduces EBDO optimum deterministic optimum x 2 REC 2006; Zissimos P. Mourelatos 27
Evidence-Based Design Optimization (EBDO) Implementation hyper-ellipse x 1 initial design point g 1(x 1, x 2)=0 Feasible Region frame of discernment g 2(x 1, x 2)=0 Objective Reduces B MPP for g 1=0 deterministic optimum x 2 REC 2006; Zissimos P. Mourelatos 28
Evidence-Based Design Optimization (EBDO) Implementation hyper-ellipse x 1 initial design point g 1(x 1, x 2)=0 Feasible Region frame of discernment Objective Reduces B MPP for g 1=0 g 2(x 1, x 2)=0 EBDO optimum deterministic optimum REC 2006; Zissimos P. Mourelatos x 2 29
Cantilever Beam Example: RBDO Formulation s. t. where : REC 2006; Zissimos P. Mourelatos 30
Cantilever Beam Example: PBDO Formulation RBDO PBDO s. t. REC 2006; Zissimos P. Mourelatos 31
Cantilever Beam Example: EBDO Formulation EBDO s. t. BPA Structure REC 2006; Zissimos P. Mourelatos 32
Cantilever Beam Example: EBDO Formulation BPA structure for y, Y, Z, E REC 2006; Zissimos P. Mourelatos 33
Cantilever Beam Example: Comparison of Results REC 2006; Zissimos P. Mourelatos 34
Thin-walled Pressure Vessel Example yielding s. t. REC 2006; Zissimos P. Mourelatos 35
Thin-walled Pressure Vessel Example BPA structure for R, L, t, P and Y REC 2006; Zissimos P. Mourelatos 36
Thin-walled Pressure Vessel Example REC 2006; Zissimos P. Mourelatos 37
Summary and Conclusions § Possibility and evidence theories were used to quantify and propagate uncertainty. § PBDO and EBDO algorithms were presented for design with incomplete information. § EBDO design is more conservative than the RBDO design but less conservative than PBDO design. Deterministic RBDO PBDO Less Information More Conservative Design EBDO REC 2006; Zissimos P. Mourelatos 38
Q&A REC 2006; Zissimos P. Mourelatos 39
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