structural & multidisciplinary optimization lab mechanical and aerospace engineering 1...

1Structural & Multidisciplinary Optimization Lab

Mechanical and Aerospace Engineering

Approximate Probabilistic Optimization Using Exact-Capacity-Approximate-Response-Distribution (ECARD)Erdem Acar

Sunil KumarRichard J. PippyNam Ho KimRaphael T. Haftka



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



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



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



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)



Exact Capacity Approximate Response Distribution (ECARD) Model



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

*k



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%.



Initial Steps of ECARD Method1. Calculate Characteristic stress,σp*, of the previous

or given design using

2. Calculate deterministic stresses σ0 for the initial design using the mean values of all input variables

3. Calculate correction factor ‘k’ using finite differences. For instance:

* 1( )p fF Ps -=

** 1

*p

*k



ECARD approximate Optimization

x

min x

s.t. xapproxf fd

W

P P1

p

* k * 1 pfF P

* *1approxfP F

‘k’ is estimated before start of the ECARD optimization procedure

To calculate Pfapprox :

As design changes in optimization procedure the changes in probability of failure are reflected by changes in Characteristic responses



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) 2.9107 5%

Failure Stress,σf (psi) 40,000 5%



Cantilever Beam Problem: Deterministic optimization

,

,1 2 2

2 20

,2 3 2 2

min

600 600s.t. 0

10

4

w t

c f FL Y FL X

FL Y FL Xc

A wt

k S F S Fwt w t

D E S F S Fk

wtL t w

Width (in)

Thickness (in)

Area (in2)

2.27 4.41 10.04

Optimum design :

where

SFL(=1.5) is safety factor for loads,

kc,1(=1) and kc,2 (=1) are knockdown factors for allowable stress and displacement.



Cantilever Beam Problem:Probabilistic optimization

,

.

min

s.t. ~ 0.0027

w t

Stress DispF f f

A wt

P P P

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

Ditlevsen’s First Order upper Bound

Width(in)

Thickness(in)

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 2.410-3 3.310-4 2.710-3



Cantilever Beam Problem:ECARD Optimization

,

1 2

min

s.t. ~ 0.0027

w t

approx approx approxFS f f

A wt

P P P

Width(in)

Thickness(in)

Area(in2)

PF(stress) PF(Displacement) PTotal

# Response PDF

Assessments

Deterministicoptimum

2.27 4.41 10.04 9.8x 10-5 2.67x 10-3 2.7x 10-30

Probabilisticoptimum

2.65 3.56 9.44 2.310-3 3.3110-4 2.710-3455

ECARD 5thIteration

2.50 3.80 9.50 1.7710-3 9.810-4 2.710-310

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.

2 2

2 20

3 2 2

600 6000

10

4

f Y X

Y X

F Fwt w t

D E F F

wtL t w



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’.



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)



Ten-bar Truss Problem:Deterministic Optimization

10

1

i 1 2

min

, ,s.t.

ii i

A i

i allow ii

W L A

N P P

A

A

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



Ten-bar Truss Problem:Deterministic Optimization ResultsElement Area (in2) Weight (lb) Stress (ksi) Pfd

1 7.9 284 25 2.1E-03

2 0.1 4 25 1.1E-02

3 8.1 292 -25 4.80E-04

4 3.9 140 -25 2.19E-03

5 0.1 4 0 4.04E-04

6 0.1 4 25 1.07E-02

7 5.8 295 25 1.69E-03

8 5.5 281 -25 1.89E-03

9 3.6 187 37.5 5.47E-13

10 0.1 7 -25 1.07E-02

Total --- 1498 -- 4.08E-02

Light weight

elements account for

50% of total failure probability



Ten Bar Truss Problem:Probabilistic Optimization Results

10

1

10 10

1 1

min

s.t.

i iAi i

f fdi ii i

W L A

P P

ElementDeterministic

AreasProbabilistic

AreasDeterministic

PfProbabilistic

Pf

1 7.9 7.2 2.1E-03 5.9E-03

2 0.1 0.3 1.0E-02 3.1E-03

--- --- --- -- --

Totals: 1497.6 Ibs 1407.13 Ibs 4.10E-02 4.10E-02

Results of full probabilistic optimization using 10,000 samples of Separable MCS

Errors in loads, cross sectional area, stress

calculations and failure predictions

leads to uncertainty



Ten Bar Truss Problem:ECARD Optimization Results

ElementDeterm.

Des. iter_01 iter_02 iter_03 iter_04

AREAS (in2)

1 7.9 7.45 7.48 7.48 7.48

2 0.1 0.1 0.1 0.1 0.1

ACTUAL PF

1 2.1E-03 5.5E-03 5.3E-03 5.2E-03 5.2E-03

2 1.1E-02 3.1E-03 2.2E-03 2.1E-03 2.1E-03

ElementDeterministic

AreasProbabilistic

AreasDeterministic

PfProbabilistic

Pf

1 7.9 7.2 2.1E-03 5.9E-03

2 0.1 0.3 1.0E-02 3.1E-03

Risk of failure of elements have changed in opposite direction

Compare it with Full probabilistic optimization

Computational costs Probabilistic Optimization

ECARD Optimization

# Expensive Response PDF Assessments 728 8

Cost

Comparison



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



Any Questions or Comments?

Thank you

structural & multidisciplinary optimization lab mechanical and aerospace engineering 1...

Documents

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sequential optimization

characteristic response

exact cdf of capacity

failure stress slide

f response

random capacity

haftka slide