1. Erdem Acar Sunil Kumar Richard J. Pippy Nam Ho Kim Raphael T. Haftka
Approximate Probabilistic Optimization Using Exact-Capacity-Approximate-Response-Distribution (ECARD)
Erdem Acar
Sunil Kumar
Richard J. Pippy
Nam Ho Kim
Raphael T. Haftka

2. Outline Introduction & Motivation
Introduce characteristic stress and correction factor
Details of Exact Capacity Approximate Response Distribution (ECARD) optimization method
Demonstration on two Examples:
Cantilever beam problem
Ten bar truss problem
Conclusion

3. Introduction: Design Optimization
Deterministic
Design governed by safety factor for loads, and knockdown factors for allowable stress and displacement.
Suboptimal Risk allocation because of uniform safety factor
Probabilistic
Optimum risk allocation by probabilistic analysis
Light weight components usually should have higher safety factors than heavy elements because, for them, weight for reducing risk is very small compared to heavier elements
Computational expense involved in reliability assessment

4. Dealing with the Computational Cost
Double loop optimization: Outer loop for design optimization, inner loop for reliability assessment by Lee and Kwak in 1987
Single loop methods: sequential deterministic optimizations by Du and Chen in 2004 known as Sequential Optimization and Reliability Assessment (SORA) method.
ECARD Optimization
Uses sequence of approximate inexpensive probabilistic optimizations
It reduces computational cost by approximate treatment of expensive response distribution

5. Introduction to ECARD Model
Limit State function can be expressed as
F (response, capacity) = Capacity - Response
CDF of capacity is usually easy to obtain from failure records : Required by Regulations
ECARD uses Exact CDF of capacity
It approximates the Response (e.g. stress ) Distribution (PDF) using Characteristic Response (* ) to estimate probability of failure for any given design
Characteristic stress is an equivalent deterministic stress having the same failure probability for random capacity (e.g. failure stress)

6. Exact Capacity Approximate Response Distribution (ECARD) Model

7. Exact Capacity Approximate Response Distribution (ECARD) Model

8. Exact Capacity Approximate Response Distribution (ECARD) Model

9. Exact Capacity Approximate Response Distribution (ECARD) Model

10. Exact Capacity Approximate Response Distribution (ECARD) Model

11. Exact Capacity Approximate Response Distribution (ECARD) Model

12. Correction factor Correction factor, k, is defined as ratio of * & 
It replaces derivatives of probability of failures in full probabilistic optimization and provides an approximate direction for optimizing objective function.
Simplifying assumption: ‘k’ is constant

13. Linearity assumption between * & 
If distribution shape does not change k can be approximated easily by shifting Nominal MCS values
For lognormally distributed failure stress and normally distributed stress, the linearity assumption is quite accurate over the range -10%  10%.

14. Initial Steps of ECARD Method
Calculate Characteristic stress,σp*, of the previous or given design using
Calculate deterministic stresses σ0 for the initial design using the mean values of all input variables
Calculate correction factor ‘k’ using finite differences. For instance:

15. ECARD approximate Optimization
To calculate Pfapprox :
‘k’ is estimated before start of the ECARD optimization procedure
As design changes in optimization procedure the changes in probability of failure are reflected by changes in Characteristic responses

16. Example 1: Cantilever Beam Problem
Random variable
Mean
Coefficient of variation
FX (lb)
500
20%
FY (lb)
1,000
10%
Young's Modulus, E (psi)
2.9107 5%
Failure Stress,σf (psi)
40,000

17. Cantilever Beam Problem: Deterministic optimization
where
SFL(=1.5) is safety factor for loads,
kc,1(=1) and kc,2 (=1) are knockdown factors for allowable stress and displacement.
Width
(in)
Thickness
Area
(in2)
2.27
4.41
10.04
Optimum design :

18. Cantilever Beam Problem: Probabilistic optimization
Ditlevsen’s First Order upper Bound
Leads to 6% reduction in Area over Deterministic Optimum Design by reallocating risk between different failure modes
Deterministic Design allocates Most of the risk to Displacement criteria but its cheaper to guard against Displacement constraint violation
Width
(in)
Thickness
Area
(in2)
PF(stress)
PF(Displacement)
PTotal
Deterministic optimum
2.27
4.41
10.04
9.8 x 10-5
2.67x 10-3
2.7x 10-3
Probabilistic optimum
2.65
3.56
9.44
2.410-3
3.310-4
2.710-3

19. Cantilever Beam Problem: ECARD Optimization
Only 5 Iterations of ECARD optimization needed
Leads to 0.2% heavier Design than Probabilistic Optimum Design which was 6% lighter than deterministic Design by proper risk allocation.
Probability of failure due to stress and displacement criteria have changed in opposite directions. Similar to full Probabilistic optimization.
Width
(in)
Thickness
Area
(in2)
PF(stress)
PF(Displacement)
PTotal
# Response PDF
Assessments
Deterministic
optimum
2.27
4.41
10.04
9.8x 10-5
2.67x 10-3
2.7x 10-3
Probabilistic
2.65
3.56
9.44
2.310-3
3.3110-4
2.710-3
455
ECARD 5th
Iteration
2.50
3.80
9.50
1.7710-3
9.810-4 10

20. Cantilever beam Problem: Convergence
Convergence of ECARD optimization technique to the full probabilistic optimum is not achieved exactly because of approximations in correction factor ‘k’.

21. Example 2: Ten-bar Truss Problem
Aluminum Truss:
Density = 0.1 lb/in³
Elasticity Modulus:
E = 10,000 ksi
Length: b = 360 in
P1 = P2 = 100,000 lbs
(includes a SF of 1.5)

22. Ten-bar Truss Problem: Deterministic Optimization
where,
W = Total Weight of Truss,
 = Density,
L = Length,
A = Cross-sectional Area,
N = Axial force in an element
Constraints:
Minimum Area = 0.1 in²
Maximum Stress in all elements = 25 ksi ,
Except in Element 9,it is 75 ksi

23. Ten-bar Truss Problem: Deterministic Optimization Results
Element
Area (in2)
Weight (lb)
Stress (ksi)
Pfd 1
7.9
284 25
2.1E-03 2
0.1 4
1.1E-02 3
8.1
292
-25
4.80E-04
3.9
140
2.19E-03 5
4.04E-04 6
1.07E-02 7
5.8
295
1.69E-03 8
5.5
281
1.89E-03 9
3.6
187
37.5
5.47E-13 10
Total
---
1498 --
4.08E-02
Light weight elements account for 50% of total failure probability

24. Ten Bar Truss Problem: Probabilistic Optimization Results
Errors in loads, cross sectional area, stress calculations and failure predictions leads to uncertainty
Results of full probabilistic optimization using 10,000 samples of Separable MCS
Element
Deterministic
Areas
Probabilistic Pf 1
7.9
7.2
2.1E-03
5.9E-03 2
0.1
0.3
1.0E-02
3.1E-03
--- --
Totals:
Ibs
Ibs
4.10E-02

25. Ten Bar Truss Problem: ECARD Optimization Results
Element
Determ. Des.
iter_01
iter_02
iter_03
iter_04
AREAS (in2) 1
7.9
7.45
7.48 2
0.1
ACTUAL PF
2.1E-03
5.5E-03
5.3E-03
5.2E-03
1.1E-02
3.1E-03
2.2E-03
Risk of failure of elements have changed in opposite direction
Element
Deterministic
Areas
Probabilistic Pf 1
7.9
7.2
2.1E-03
5.9E-03 2
0.1
0.3
1.0E-02
3.1E-03
Compare it with Full probabilistic optimization
Computational costs
Probabilistic Optimization
ECARD Optimization
# Expensive Response PDF Assessments
728 8
Cost
Comparison

26. Conclusions
A failure characteristic stress * is used to approximate changes in probability of failure with changes in design
Using this, ECARD dispenses with most of the expensive structural response calculations.
Cantilever beam: 455 to 10 expensive reliability assessments
Ten bar truss: 728 to 8 expensive reliability assessments
ECARD converges to near optima of allocated risk between failure modes much more efficiently than the deterministic optima

27. Any Questions or Comments?
Thank you