development of reliability analysis and multidisciplinary design optimization (ramdo) software
TRANSCRIPT
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2015 Americas Altair Technology Conference
K.K. Choi, Nicholas Gaul, Hyeongjin Song and Hyunkyoo Cho
RAMDO Solutions, LLCIowa City, IA 52240
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Contents
● Multidisciplinary Simulation with Input Variability● Deterministic Design Optimization (DDO) vs.
Reliability-Based Design Optimizations (RBDO)● Capabilities in RAMDO Software Modeling Input Distributions Sensitivity-Based RBDO Sampling-Based RBDO
● Multidisciplinary Applications of RAMDO● Commercialization of RAMDO
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Input Variables X=[X1, X2,…, Xn]
OutputPerformances G(X)=[G1,.., Gnc]
Multidisciplinary Simulation with Input Variability
OutputPerformances G(X)
Output Variability of Performance G1(X)
Output Variability of Performance Gnc(X)
Input Variables X
Load Variability
ManufacturingVariability
Surrogate ModelingVariability
Material Property
Variability
Other Input Variable
Variabilities
CastingProcess
Variability
RAMDO will stimulate collaborationamong Design, Manufacturing &Testing Engineers.
FEA
Multibody Dynamics
CFD
Casting
Electromagnetics
Reliability
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Safety factor approach can be considered. Right safety factor? Over design or under design? How about multidisciplinary design optimization problem?
There are two approaches for reliability analysis:- FORM or SORM with Sensitivity Analysis to Find MPP - Use Surrogate Models with DoE Samples and MCS
DDO vs. RBDO
Minimize CostSubject to
: deterministic variables
( )1,) , ,( 0
L Uj j ncG =
≤ ≤
≤
x
xx
x x x
DDO Formulation
X2
Failure SurfaceG1(X)=0
Failure SurfaceG2(X)=0
Initial DesignX10
DDO Design is only ~ 50% Reliable
Minimize Cost
Subject to
( )1, ,
( ): mean of random
( ( ) 0)
variables
,j
L U
Tarj F j nP G P c=
≤
≤
≤
>
=
d
d d dμ X
X
d
RBDO Formulation
RBDO Design with 95% TargetReliability
95% Target ReliabilityLevel Set
Variability of Input Variables
95% Target ReliabilityLevel
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Modeling Input Distributions
● Two-step Weight-based Bayesian method is implemented in RAMDO using 7 marginal PDFs and 9 copulas to best fitthe data.
Example: Highly Correlated Fatigue Data of SAE 950X (HSLA Steel)
b
'fσ
c
'fε
Joint PDF isFrank Copula Correlation τ = − 0.906
Joint PDF is Gaussian Copula
Correlation τ = − 0.683
Marginal PDFsσf′ is Lognormalb is Normal
Marginal PDFsεf′ is Lognormalc is Normal
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Sensitivity-Based RBDO
Failure SurfaceG1(X)=0
Failure SurfaceG2(X)=0
95% Target ReliabilityLevel Set( )
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Maximize
Subject toj
j
t
G
β≤
U
U
Inverse Reliability Analysisto Search MPP
Failure ContourG2(X)=5 > 0
Failure ContourG1(X) =7 > 0
MPP2MPP1
MPP
Minimize Cost
Subject to
( )
( ( )) 0,
1, ,j
L U
G
j nc
≤
=
≤ ≤
d
X d
d d d
Performance Measure Approach (PMA) Using MPP
Also developed DRM-based PMA for highly nonlinear problems.
DDO Design
Feasible RegionRBDO Design
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Sensitivity-Based RBDO Case Studies for Durability
crack initiation point
crack initiation point
• 2-σ Design (2.275% target probability of failure).
• Weight reduced to 42.62 lbs from 53.0 lbs (20%).
• Improving fatigue life 10.8 times.
• 2-σ Design.• Used 16 Parallel
Processors.• Fatigue life
improved by 2084 times.
Stryker Left-Front A-Arm RBDO
HMMWV Left-Front A-Arm RBDO
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HMMWV Left-Front A-Arm Using Sensitivity-Based RBDO
Initial Design
RBDO Results Uncorrel. Fatigue Prop. (Incorrect)
Correl. Fatigue Prop. (Correct)
d1 0.1200 0.2926 0.2423 d2 0.1200 0.2858 0.1278 d3 0.1800 0.3418 0.2143 d4 0.1350 0.3208 0.2584 d5 0.2500 0.5852 0.4827 d6 0.1800 0.5000 0.5000 d7 0.1350 0.3278 0.2437 d8 0.1800 0.3886 0.1000
Volume 106.9 in3 227.55 in3 157.52 in3
Using correct correlated fatigue material property model, more than 45% weight is saved!
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● Surrogate models are used for Sampling-based RBDO.
● To mitigate curse-of-dimension, variable screening methodis developed for reduction dimension of RBDO problem.
● The variables that induce larger output variances are selected as important variables.
● Successfully selected 14 DVs out of 44-D Ford vehicle MDO problem, and obtained RBDO design that is quite close to RBDO design of the full 44-D model.
Variable Screening Method for Sampling-Based RBDO
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Dynamic Kriging (DKG) Surrogate Models
● In standard Kriging model, the responses are represented by
where is the regression coefficient, is polynomial basis function and is a model of Gaussian random process with zero mean and covariance .
● Select best mean structure from 0th, 1st, and 2nd order polynomials using cross validation (CV) error.
● Select best correlation model from 7 candidates using maximum likelihood estimation.
● Provides 7×3 = 21 options for surrogate models on each local window.
( , , )i jR θ x x
( )kf x
, =[ ( ), 1,..., , 1,..., ]n Kikf k K i n ×= =y = Fβ+Z F x
2( , ) ( , , )i j i jC Rσ=x x θ x x
1K×β1n×Z
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● For correlated input variables, DoE samplesare properly selected using copulas.
Local Windows for Surrogating Modeling
● Use Local Window (LW) for reliability analysis to mitigate curse-of-dimension. For 2 DVs – Global Domain has 25 LWs.
For 10 DVs – Global Domain has 9,765,625 LWs!
βt
u1
u2
1.2βt
● Hyper-Spherical LW is used for efficient utilization of DoE samples. For 2-DVs, useless gray area is 21.3%
For 8-DVs, useless gray area is 98.4%!
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Sampling-Based RBDO with Solvers as Black-BoxesInitial
Design
No
Yes
Optimization Converged?
Update Design
No
Optimum Design
Yes
Yes
Sequential DoE Sampling
No
Scan Local Window for Existing Samples
No. of Existing DoE Sample > Required
No. of DoE Samples?
Generate Initial DoE Samples: Transformation Gibbs Sampling
Computer Simulations at DoE Samples
Surrogate Model by Dynamic-Kriging
Is Surrogate Model Accurate?
Probabilistic Sensitivity Using Score Function
MCS for Reliability & Sensitivity Analyses
RBDO Optimizer Using Matlab
Make Local Window for New Design
SOLVERS:CAD & CAE Tools (FEA, CFD, MBD,
Casting, Stamping, Durability,
Electromagnetics, Etc.)
Input
Output
Safety Optimization And RobustnessResearch & Advanced Engineering
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Safety Optimization And RobustnessResearch & Advanced Engineering
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RBDO of FORD Vehicle MDO ProblemRBDO Formulation
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2
Min: Subject to: : ( _ ) 90% : ( _ ) 90%
Full Frontal Constraints:
40% Offset Constraint
:(
s
Weight
G P Chest G BaselineG P Crush dis Baseline
P BrakePeda
≤ ≥≤ ≥
) 90% ( ) 90% ( ) 90% ( ) 90% ( ) 90%
l BaselineP Footrest BaselineP LeftToepan BaselineP CntrToepan BaselineP RightToepan Baseline
≤ ≥≤ ≥
≤ ≥≤ ≥≤ ≥
NVH Constrai
( ) 90% ( ) 90% ( ) 90% ( ) 90
nts
%
:
P LeftIP BaselineP RightIP Baseline
P TorsionMode BaselineP VertBenMode Baseline
≤ ≥≤ ≥
≥ ≥≥ ≥
Design Variables:All 44 body thickness design variables are treated as random with normal distribution.
• Validated 2 Key Capabilities of RAMDO Software:
(Case I) Effectiveness of RAMDO RBDO Algorithm
(Case II) Effectiveness of Variable Screening Method
Safety Optimization And RobustnessResearch & Advanced Engineering
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Safety Optimization And RobustnessResearch & Advanced Engineering
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Case I: Effectiveness of RAMDO RBDO Algorithm(Using 44-D Surrogate Models)
Objective, Constraints, etc.Initial Designs RBDO Using RAMDO
BaselineDesign
RAMDODDO
NSGA-IIDesign
Starting from Baseline
Starting from RAMDO DDO
Starting from NSGA-II Design
Optimum Weight 269.47kg 222.91kg 240.12kg 225.68kg 225.66kg 225.67kg
G1 48.22% 49.61% 32.95% 10.05% 10.07% 9.96%G2 51.48% 51.44% 49.57% 10.09% 10.18% 10.11%G3 54.15% 57.18% 0.01% 0.00% 0.00% 0.00%G4 55.57% 37.65% 0.01% 0.12% 0.09% 0.10%G5 58.96% 4.38% 0.59% 1.91% 1.82% 1.84%G6 59.71% 24.55% 2.52% 10.00% 10.03% 9.92%G7 59.92% 61.26% 19.05% 10.06% 9.99% 9.89%G8 53.19% 0.12% 13.79% 9.14% 9.91% 10.04%G9 51.17% 51.90% 38.44% 9.96% 9.91% 9.95%G10 49.05% 0.00% 0.00% 0.00% 0.00% 0.00%G11 52.46% 52.24% 43.80% 10.05% 9.94% 10.08%
Terminal Cond. 1.00E-03 1.00E-03 1.00E-03Computation Time (h) 6 54 3# of Design Iterations 29 43 19
Safety Optimization And RobustnessResearch & Advanced Engineering
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Safety Optimization And RobustnessResearch & Advanced Engineering
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Case II: Effectiveness of Variable Screening Method
At each RBDO design, reliability analysis is carried out using the 44-D benchmark surrogate model.
Variables selected using RAMDO variable screening method disagrees only 1.26% or 1.17%, which are very close to the target value of 10%.
PerformanceMeasure
BaselineDesign
RAMDO VariableScreening (14-D)
Variable Screening+ Cost Function (18-D)
Optimum Weight 269.47kg 259.83kg 244.17kgG1 48.25% 9.93% 10.00%G2 51.34% 9.88% 10.04%G3 54.14% 0.00% 0.00%G4 55.57% 0.12% 0.09%G5 58.94% 1.83% 1.98%G6 59.70% 9.80% 10.05%G7 59.86% 10.03% 9.91%G8 53.23% 10.36% 9.97%G9 51.15% 10.16% 9.96%G10 49.10% 0.00% 0.00%G11 52.46% 11.26% 11.17%
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RAMDO provides Sensitivity-Based & Sampling-BasedRBDO of Simulation-Based Designs in● Fatigue Analysis & Durability● Stamping Process Design● Explosion Analysis & Survivability● Vehicle and Machine Dynamics● Noise, Vibration & Harshness (NVH)● Crashworthiness● Casting Process Design (Manufacturing)● Advanced & Hybrid Powertrain● Wind Power Systems● Human Centered Design● MEMS & Nano/Micromechanics Based
Materials Design● Robotic Systems● Electromagnetics● Fluid-Structure Interaction
Multidisciplinary Applications of RAMDO
And a lot more …..
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Multidisciplinary Applications of RAMDO 1. Hardin, R.A., Choi, K.K., Gaul, N.J. and Beckermann, C., “Reliability-Based Casting Process Design
Optimisation,” International Journal of Cast Metals Research, to appear, 2015.2. Jang, H-R., Cho, S., and Choi, K.K., “Sampling-based RBDO of Fluid-Solid Interaction (FSI)
Problems,” IMechE-C; Journal of Mechanical Engineering Science, Vol. 228 (10), 2014, pp. 1724-1742.
3. Choi, M., Cho, S., Choi, K.K., and Cho, H., “Sampling-based RBDO of Ship Hull Structures Considering Thermo-elasto-plastic Residual Deformation,” Mechanics Based Design of Structures and Machines, Vol. 43 (2), 2015, pp. 183–208 (Reduce Residual Deformation in Welding Process)
4. Kim, D-W., Choi, N-S., Choi, K.K., Kim, H-G., and Kim, D-H., “Optimization of a SMES Magnet in the Presence of Uncertainty Utilizing Sampling-based Reliability Analysis,” Journal of Magnetics (SCIE), Vol. 19(1), 2014, pp. 78-83 (2014). (Superconducting Magnetic Energy Storage System)
5. Kim, D-W., Choi, N-S., Choi, K.K. and Kim, D-H., “Sequential Design Method for Geometric Optimization of an Electro-Thermal Microactuator based on Dynamic Kriging Models,” CEFC 2014, Annecy, France, May 25-28, 2014. (Electro-Thermal Polysilicon Microactuator)
6. Volpi, S., Diez, M., Gaul N.J., Song, H., Iemma, U., Choi, K.K., Campana, E.F., Stern, F., “Development and Validation of a Dynamic Metamodel Based on Stochastic Radial Basis Functions and Uncertainty Quantification,” Structural and Multidisciplinary Optimization, DOI 10.1007/s00158-014-1128-5, 2014. (High-Fidelity CFD Outputs)
7. Li, H., Sugiyama, H., Gaul, N., and Choi, K.K., “Analysis of Wind Turbine Drivetrain Dynamicsunder Wind Load and Axial Misalignment Uncertainties,” The 3rd Joint International Conf. on Multibody System Dynamics, Busan, Korea, June 30-July 3, 2014.
8. Sen, O., Davis, S., Jacobs, G., Udaykumar, H.S., “Evaluation of Convergence Behavior of Metamodeling Techniques for Bridging Scales in Multi-scale Multimaterial Simulation,” Journal of Computational Physics, DOI: http://dx.doi.org/10.1016/j.jcp.2015.03.043. (Concluded Accuracy of DKG Method is the Best Among Tested Methods)
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Commercialization of RAMDO● Start-up Company – RAMDO Solutions, LLC in Fall 2013 Grants/Equity in: $3.6M in Basic Research Recruited Dr. Nicholas J. Gaul as the Chief Operating Officer.
● 2013 Iowa Center for Enterprise Elevator Pitch Competition Award - $2K (December 2013-12)
● Awarded GAP Funding - $75K (January 2014)● Obtained Iowa State LAUNCH Loan - $100K (February 2014)● Obtained PETTT Project on Army HPC DSP - $120K (April 2014)● TARDEC Matching Grant - $100K (August 2014)● Awarded SBIR Phase I Grant from U.S. Department of Defense
(U.S. Army TARDEC) - $150K (June 2014-April, 2015)● SBIR Phase II Grant - $1M for Two Years (June 2015)● Since RAMDO is a computational software for Multidisciplinary
RBDO, the company will work with PIDO (Process Integration & Design Optimization) software company(s) for partnership.
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http://www.ramdosolutions.com/