Upload
reginald-rich
View
219
Download
0
Tags:
Embed Size (px)
Citation preview
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
2Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
3Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
4Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
5Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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)
6Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
Exact Capacity Approximate Response Distribution (ECARD) Model
7Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
Exact Capacity Approximate Response Distribution (ECARD) Model
8Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
Exact Capacity Approximate Response Distribution (ECARD) Model
9Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
Exact Capacity Approximate Response Distribution (ECARD) Model
10Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
Exact Capacity Approximate Response Distribution (ECARD) Model
11Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
Exact Capacity Approximate Response Distribution (ECARD) Model
12Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
13Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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%.
14Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
15Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
16Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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%
17Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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.
18Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
19Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
20Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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’.
21Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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)
22Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
23Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
24Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
25Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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
26Structural & Multidisciplinary Optimization Lab
Mechanical and Aerospace Engineering
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