1 Structural & Multidisciplinary Optimization Lab Mechanical and Aerospace Engineering Approximate Probabilistic Optimization Using Exact- Capacity-Approximate- Response-Distribution (ECARD) Erdem Acar Sunil Kumar Richard J. Pippy Nam Ho Kim Raphael T. Haftka
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Structural & Multidisciplinary Optimization Lab Mechanical and Aerospace Engineering 1 Approximate Probabilistic Optimization Using Exact-Capacity- Approximate-Response-Distribution.
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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
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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
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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
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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)
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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
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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
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
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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
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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’.
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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
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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