lsopt probabilistic class - LS-OPT Support

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®

Probabilistic AnalysisWillem Roux (LSTC)

Probabilistic Analysis

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Content

• Objectives• Statistical Background• Structural Mechanics• Methodologies• Examples• Hands-on Tutorial

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Objectives

Modeling of Variability• Repeatability of Response

Design Criteria• Probability of failure• Robustness (Variance)

Redesign• Source of variability


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BASIC STATISTICS

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Response Variability

Response distributionMean Standard deviation

Probability of Failure


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Basic Computations

• Mean

• Variance

• Standard Deviation

1

1 n

ii

yn

μ=

= ∑

2 2

1

1 ( )1

n

ii

yn

σ μ=

= −− ∑

2σ σ=

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Basic Computations

Coefficient of variation

. .c o v σμ

=


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Covariance and Correlation

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Covariance and CorrelationCovariance

Coefficient of correlation

Always between –1 and 1Viewer shows confidence bounds on correlation.

1 2 1 1 2 2( , ) [( )( )]Cov y y E y yμ μ= − −

1 2

1 2

( , )Cov y yρσ σ

=


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Covariance and Correlation

Correlation plots are the X-ray machines of the stochastic world. It shows the interdependence of results.

Correlation is not causation though.

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Sum of Normal Variables

21 μμμ +=Y

21 XXY +=

22

21

2 σσσ +=Y


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Statistical DistributionsBetaBinomialLognormalNormalUniformUser definedWeibull

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Beta DistributionLower and Upper ValueTwo Shape parametersVersatile


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Binomial DistributionDiscrete DistributionProbability of Event and Number of Trials

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Lognormal DistributionMean and Standard Deviation


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Normal DistributionMean and Standard Deviation

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Truncated NormalMean, standard, deviation, lower and upper bound


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Uniform DistributionLower and Upper Bound

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User Defined


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WeibullShape and Scale

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Response Surfaces• Approximations to structural behavior• Build from responses from selected set of designs

• Experimental design

• Multi-dimensional Least Squares Fit


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Outliers/Residuals

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METHODOLOGIES I


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• Monte Carlo• Monte Carlo using Metamodels• Reliability Based Design Optimization• Robust Parameter Design• Outlier/bifurcation investigation• Metal Forming

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VariablesControl Variables (Design Parameters)

Nominal value controlled by designer• Gauge• Shape

Noise Variables (Environment)Values not controlled by designer but can vary• Load• Yield stress


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Probabilistic Variables

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Monte Carlo Analysis

• Random process• Large number of FE runs (100+)• Sampling

• Random• Latin Hypercube (Structured Monte Carlo)


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Latin Hypercube Sampling

• Each partition has equal probability.

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Monte Carlo Using Metamodels

• Response Surface / Neural Network• Medium number of FE runs (10 – 30+)• Monte Carlo analysis conducted using

metamodel and a large number of points (106)

• Identify design variable contributions


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2.Determσ

DσDσ

VarσVarσ

Design Variable

Res

pons

e

Metamodel-based Monte Carlo

Rσ

2Residualσ

Rσ

Stochastic Contributions

2Totalσ

TσTσ

Meta-Model(Least-Squares Fit)

Finite Element Simulations

Noisy Physical Response

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Contribution AnalysisContribution of variation of variable to variation of response

Variance of response

, ,g i x ii

Gx

σ σ∂=∂

2 2,

1

n

g x ii

σ σ=

= ∑


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EXAMPLES I

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Head Impact Problem


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Head Impact ProblemMonitor: HIC-d(Head Injury Criterion)Variables:

• Horizontal Angle of impact

15 degrees10% standard deviation

• Rib height12.5mm5% standard deviation

Height = 12.5

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VariationVary one variable at a time to investigate curvature.Linear response with some scatter (noise).

360

365

370

375

380

385

390

4 8 12 16 20Rib Height

HIC

-d

HIC-dLinear (HIC-d)

355360365370375380385390395

10 12 14 16 18 20

Impact Angle

HIC

-d

HIC-dLinear (HIC-d)


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Variation

Reponses nearly linearQuadratic Surface should fit accuratelyRange of response surface is 2σ

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Response VariationBaseline Design: HIC-d = 374.4Monte Carlo Analysis: 150 FE analysesQuadratic Response Surface: 60 FE analyses.Outliers have standard deviation of 2.35

4.21 (87%)4.82

4.85Standard deviation• Structural• Structural + Outlier

373.9373.9Mean

Metamodel(60 simulations)

Monte Carlo (150 simulations)


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Probability of Value

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

363 367 370 374 377 380 384 387 390

HIC-d

Prob

abili

ty

0.00

0.02

0.04

0.06

0.08

0.10

350

355

360

364

369

373

378

383

387

HIC-d

Prob

abili

tyMetamodel

Direct Monte Carlo Analysis

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Probability of Exceeding Bound

00.10.20.30.40.50.60.70.80.9

1

HIC-d

P[H

IC-d

>]

Monte CarloMetamodel

362 386370 378


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Contribution Analysis

0

5

10

15

20

25

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HIC-d

Var

ianc

e Hor AngleRib HeightResiduals

2σ

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Tutorial I

Reliability Analysis TutorialRELIABILITY/MONTE_CARLORELIABILITY/METAMODEL


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METHODOLOGIES II

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RBDO

Reliability Based Design Optimization includes uncertainty of variables and responses into optimization

Requires:• Statistical distribution of variable• Probabilistic bound on constraint.

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RBDO: Probabilistic boundUnderlying implementation shifts constraint bound with number

of standard deviations corresponding to probability.


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Robust Parameter Design

Robust parameter design creates designs insensitive to the variation of specific inputs.

Two sets of variables:• Noise variables• Control (normal) variables

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Search for a flat part of design space


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Control variables are adjusted to minimize the effect of the noise variables.

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Robust Parameter DesignLS-OPT Command

composite 'var x11' noise 'x11‘

Experimental designInteraction terms must be included.

Use multi-criteria design to described bigger-is-better etc.


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Tutorial II

RBDO TutorialRBDO

Robustness TutorialROBUST_PARAMETER_DESIGN

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STRUCTURAL MECHANICS


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Deterministic Variation

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Process Variation

Process is seeded by stochastic fields


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Mechanics of stochastic processes• Physical

• Instability• On-Off Impact

• Numerical• Discretization• Convergence criteria• Different computers• Different compilers

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Process Variation

Discretization (modeling) effects• Smoothness of FE mesh

• Postprocessing: time step size and filter selection

• Selection of node/element to monitor


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Stochastic Processes

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Stochastic Process Example


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Statistics of Stochastic Process

One event after another.)()()( 1 nXEXEXE ++= L

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2nσσσ ++= L

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Stochastic Fields

Our stochastic capability consists of:• Stochastic fields and• Stochastic variables.Random variables are part of LS-OPT ®

while stochastic fields are part of LS-DYNA ®.


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Stochastic FieldsStochastic process is seeded by a stochastic field.A stochastic field allows a property (e.g. shell

thickness) to vary over a part.LS-DYNA keywords:

*PERTURBATION_NODE*PERTURBATION_SHELL_THICKNESS

Methods• Harmonic Fields• Import DYNA displacements or eigen mode

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Random fields vs. variablesRandom Fields

*PERTURBATION in LS-DYNA ®, valid for geometry, shell thickness, and some material parameters.Correlation function defined as part of the keyword.

Random Variables

*PARAMETER, valid for any number in the input file.Statistical distribution assigned in LS-OPT ®


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Stochastic Field: Geometric imperfection

*PERTURBATION_NODE

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Stochastic Field: Blank ThicknessVariation due to rolling of steel.*PERTURBATION_SHELL_THICKNESS

.

See also Reliability Based Design Optimization with LS-OPT for a Metal Forming Application by Müllerschön, Lorenz & Roll.

Mill chatter Non-uniform roll contact

=+


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*PERTURBATION_MATThis allows the spatial variation of material

properties such as the yield stress. It is being developed. It should be available for

MAT024 in LS980.

Yield Strength Variation Resulting Plastic Strain just before rupture.

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Correlation functionA random field (or process) is characterized by its correlation function.

A description of the experimentally measured correlation function allows us to create many random fields in LS-DYNA with the same properties. We have predefined functions for which the user must define the parameters.The methodology works for very large problems (millions of nodes).

)]()([)( tXtXER ττ +=


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Effect of correlation functions type

Different Functions Types

Fields

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Effect of parameter values

Gaussian functions with different values of the function parameter

Fields


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Alternative methods of specifying stochastic fields

User defined fields: A file containing the stochastic field can be specified. This allows, for example, scaled values of the eigen modes to be used as perturbations in a buckling analysis.

Sinusoidal perturbations: A sine expansion describing the perturbation can be specified.

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Test vs. Analysis


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METHODOLOGIES II

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Unexpected events:• Buckling • Modeling

Investigate:• Accuracy Plots in Viewer• Visualize on FE model in LS-PrePost

Outlier Analysis

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Metal Forming

Visualization of metal forming results requires:

• Consider adaptivity (meshes differ for designs),

• Map to geometric location (not node),• Map to base design (run iteration.1).Finer meshes give better accuracy.


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Only the work piece is considered.For each node in the base mesh:

Find closest element in the mapped mesh.Find the closest point in the element.Use the element nodal results.Interpolated linearly between the nodes.

Metal Forming Results Interpolation

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Element results are transformed to nodal averaged as in LS-PREPOST.


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Sudden jumps, especially over a single element, are filtered. Prediction of failure should be preserved in the mapped results.

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Statistic of plastic strain

Mean Standard deviation

Red = 0.5


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Maximum plastic strain: Sources of variation

Draw bead variation associated with failure

Coefficient of correlation plot

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EXAMPLES II


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Head Impact Problem

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Outlier AnalysisSome displacements may be:

• Unrelated to a design variable change• Not repeatable


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Outliers Variation

Not related toDesign

Variables

Repeatable Displacement

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CorrelationHIC-d and displacements correlated


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Investigate OutliersDifferent buckling modes

Max Outlier Min Outlier

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Vehicle CrashVary angle of impact25 FE Runs


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History Variation

Outlier Standard Deviation

Smoking Gun?

Average Acceleration

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Displacement Variation


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Impact

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Tutorial III

Bifurcation/Outlier AnalysisOUTLIER

Metal FormingMETALFORMING_RELIABILITY


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PROBABILITY

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Basic Probability

Probability of subset of events

Count: P(A=B) = 3/9 = 1/3Need probabilistic rules for general case


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AND Rule

Independent:Probability of event A AND event B

( ) ( ) ( | )P A B P A P B A∩ =( ) ( ) ( )P A B P A P B∩ =

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=×

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OR Rule

Mutually exclusive:Probability of event A OR event B

( ) ( ) ( ) ( )P A B P A P B P A B∪ = + − ∩

( ) ( ) ( )P A B P A P B∪ = +

1 1 1 53 3 9 9+ − =


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Probability ExampleP[Failure due to A]=0.6P[Failure due to B]=0.4

P[Failure] == 0.6 + 0.4 - 0.6*0.4 = 0.76

( ) ( ) ( ) ( )P A B P A P B P A B∪ = + − ∩

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lsopt probabilistic class - LS-OPT Support

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