Non-Probabilistic Design Optimization with Insufficient Data using Possibility and Evidence Theories

1
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_at_oakland.edu

2
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

3
Design Under Uncertainty
4
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)

5
Uncertainty Theories
6
Non-Probabilistic Design Optimization Set
Notation
7
Possibility-Based Design Optimization (PBDO)
8
Possibility-Based Design Optimization (PBDO)
9
Quantification of a Fuzzy Variable Membership
Function
10
Propagation of Epistemic Uncertainty
Extension Principle
11
Optimization Method
where
and
12
Possibility-Based Design Optimization (PBDO)
13
Possibility-Based Design Optimization (PBDO)
14
Possibility-Based Design Optimization (PBDO)
s.t.



15
PBDO with both Random and Possibilistic Variables



16
Evidence-Based Design Optimization (EBDO)
17
Evidence-Based Design Optimization (EBDO)
Basic Probability Assignment (BPA) m(A)
18
(No Transcript)
19
Evidence-Based Design Optimization (EBDO)
BPA structure for a two-input problem
20
(No Transcript)
21
Evidence-Based Design Optimization (EBDO)
Position of a focal element w.r.t. limit state
22
(No Transcript)
23




24
Evidence-Based Design Optimization (EBDO)
25
Geometric Interpretation of PBDO and EBDO
26
x1
initial design point
g1(x1,x2)0
frame of discernment
g2(x1,x2)0
PBDO optimum
deterministic optimum
x2
27
(No Transcript)
28
(No Transcript)
29




30
Cantilever Beam Example RBDO Formulation
s.t.
31
Cantilever Beam Example PBDO Formulation
32
Cantilever Beam Example EBDO Formulation
33
Cantilever Beam Example EBDO Formulation
BPA structure for y, Y, Z, E
34
Cantilever Beam Example Comparison of Results
35
Thin-walled Pressure Vessel Example
yielding
36
Thin-walled Pressure Vessel Example
BPA structure for R, L, t, P and Y
37
Thin-walled Pressure Vessel Example
38
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.

39
Q A