Mechanical Engineering Department 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

Overview Introduction Design under uncertainty Uncertainty theories Possibility – Based Design Optimization (PBDO) Uncertainty quantification and propagation Design algorithms Evidence – Based Design Optimization (EBDO) Examples Summary and conclusions

Design Under Uncertainty Analysis / Simulation Input Output Uncertainty (Quantified) Uncertainty (Calculated) Propagation Design Quantification 2. Propagation 3. Design

Uncertainty Types 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 Theories Evidence Theory Probability Theory Possibility Theory Probability Theory

Non-Probabilistic Design Optimization: Set Notation Universe (X) Power Set (All sets) Element A B C Evidence Theory

Possibility-Based Design Optimization (PBDO)

Possibility-Based Design Optimization (PBDO) Evidence Theory No Conflicting Evidence (Possibility Theory)

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

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

Optimization Method Global where : s.t. and s.t. 1.0 1.0 a a 0.0 0.0

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 for or

Possibility-Based Design Optimization (PBDO) Considering that , we have

Possibility-Based Design Optimization (PBDO) s.t. ; ; s.t. OR Double Loop

PBDO with both Random and Possibilistic Variables , with Triple Loop ; , s.t.

Evidence-Based Design Optimization (EBDO)

Evidence-Based Design Optimization (EBDO) Basic Probability Assignment (BPA): m(A) If m(A)>0 for then A is a focal element 0.2 0.1 0.4 0.3 Y 5 6.2 8 9.5 11 “Expert” A 5 0.3 0.4 Y 7 8.7 11 “Expert” B Y 0.x1 7 8.7 11 9.5 8 6.2 0.x2 0.x3 0.x6 0.x5 0.x4 5 Combining Rule (Dempster – Shafer)

Evidence-Based Design Optimization (EBDO) Assuming independence, For where define: Assuming independence,

Evidence-Based Design Optimization (EBDO) BPA structure for a two-input problem

Evidence-Based Design Optimization (EBDO) Uncertainty Propagation If we define, then where and

Evidence-Based Design Optimization (EBDO) Position of a focal element w.r.t. limit state Contributes to Belief Contributes to Plausibility

Evidence-Based Design Optimization (EBDO) If non-negative null form Design Principle If non-negative null form is used for feasibility, feasible infeasible failure Therefore, is satisfied if OR

Evidence-Based Design Optimization (EBDO) Formulation ,

Evidence-Based Design Optimization (EBDO) Calculation of

Geometric Interpretation of PBDO and EBDO

Possibility-Based Design Optimization (PBDO) Implementation x1 initial design point Feasible Region g1(x1,x2)=0 frame of discernment g2(x1,x2)=0 Objective Reduces PBDO optimum deterministic optimum x2

Evidence-Based Design Optimization (EBDO) deterministic optimum Implementation Feasible Region x2 x1 g1(x1,x2)=0 g2(x1,x2)=0 Objective Reduces initial design point frame of discernment EBDO optimum deterministic optimum

Evidence-Based Design Optimization (EBDO) deterministic optimum Implementation Feasible Region x2 x1 g1(x1,x2)=0 g2(x1,x2)=0 Objective Reduces hyper-ellipse initial design point frame of discernment B MPP for g1=0 deterministic optimum

Evidence-Based Design Optimization (EBDO) deterministic optimum Implementation Feasible Region x2 x1 g1(x1,x2)=0 g2(x1,x2)=0 Objective Reduces hyper-ellipse initial design point frame of discernment EBDO optimum B MPP for g1=0 deterministic optimum

Cantilever Beam Example: RBDO Formulation s.t. where :

Cantilever Beam Example: PBDO Formulation s.t. RBDO s.t. PBDO

Cantilever Beam Example: EBDO Formulation s.t. EBDO BPA Structure

Cantilever Beam Example: EBDO Formulation BPA structure for y, Y, Z, E

Cantilever Beam Example: Comparison of Results

Thin-walled Pressure Vessel Example yielding s.t.

Thin-walled Pressure Vessel Example BPA structure for R, L, t, P and Y

Thin-walled Pressure Vessel Example

More Conservative Design 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 EBDO PBDO Less Information More Conservative Design

Q & A