Development of Reliability Analysis and Multidisciplinary Design Optimization (RAMDO) Software

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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


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



95% Target ReliabilityLevel Set( )

2

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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RBDO of FORD Vehicle MDO ProblemRBDO Formulation

1

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


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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


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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/

Development of Reliability Analysis and Multidisciplinary Design Optimization (RAMDO) Software

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