random fields: theory and applications · 2016. 2. 20. · • important random field variables can...

Random Fields: Theory and Applications

Dr.-Ing. Veit Bayer (dynardo GmbH)Dr.-Ing. Johannes Einzinger (ANSYS Germany)Dr.-Ing. Dirk Roos (dynardo GmbH)

Weimarer Optimierungs- und Stochastik- TageWOST 6.0, November 2009

2 Veit Bayer, WOST 6.0 © dynardo GmbH, Weimar

What is a Random Field?• Random variation of any measureable property

(e.g., geometry, material, load, ...) over space. • The spatial domain is defined by the observed structure.• A property takes random values at each point on the structure.

Values at different locations may be correlated.

(Example: random realizations of a cylinder)


Application ExamplesAxial Turbomachine: Robustness evaluation after optimization• Random geometry, material and operation parameters• Imperfect surface of turbine blade modeled as random field• Robustness analysis controlled by optiSLang,

fluid-structure simulation by ANSYS Workbench

(FE model of axial turbine) (Realization of random surface)


Application ExamplesKnuckle with manufacturing tolerances: Robustness evaluation for brake squeal

(By courtesy of Daimler AG)


Application ExamplesRobustness Evaluation with Random Fields

Statistics of measurements:means, covariance matrix

X1

X2

Simulate random parameters

Generate set of random specimen,replace in FE assembly to compute

Post-process results

Model preparation: - map measurement points to structure- define variable surface nodes


Theoretical BasicsWhat is a random field?• A random function, defined on a structure, which takes random

values at any location. One outcome is called realization, the set of all realizations is called ensemble.

• Stochastic properties at each point are defined by stochastic moments (mean, standard deviation …) and distribution type.

• Dependency between different locations is defined by the correlation function.


Theoretical Basics• Assuming Normal distribution throughout the structure, the

random field is mathematically described completely by• mean function

• correlation function

< or, covariance function

> < or, correlation coefficient function

>


Theoretical BasicsThe correlation coefficient function is a function of the distance

between two points. It is characterized by the correlation length

The correlation function must be positive semi-definite. Examples:Exponential, triangular.


Theoretical BasicsSpecial cases:• Normal distribution type: the random field is characterized

completely by mean and covariance function• Homogeneity: same stochastic properties at any point throughout

the structure• Isotropy: correlation depends on the distance between two points,

not the direction• Zero means: correlation function and covariance function are

identical

x1 x1

x2x2


SimulationDiscretization of a random field• At the nodes of the structure model,

e.g. for geometry imperfections, random load distribution, …• At element mid points or integration points,

e.g. for material properties, …

• Discretization yields a finite set of random variables Xi (ri), the mean function reduces to the mean vector, the covariance function becomes the covariance matrix


SimulationFor simulation, transform dependent random variables X to

independent r.v. Y• Eigenvalue decomposition of CXX

• Simulate

• Transformation between simulated variables and "real-world"

X = Y1 + Y2 + Y3 + …


SimulationSample mode shapes from a metal forming example

Thickness reduction

Mode shape #1

Mode shape #2

Mode shape #3


Data ReductionSources of data• Random field model,

discretized at nodes or elements

• Data obtained by simulation of a manufacturing process with random parameters

• Data obtained by measurements


Data ReductionMesh coarsening:• Number of nodes or elements is reduced,• giving a uniform spread of support points on the structure,• or a mesh which retains mesh topology (relative refinement)

• Data are mapped from original to fine mesh by local averaging• Simulated random field is interpolated by Moving Least Squares


Data ReductionSmoothing effect by mesh coarsening

Original datathickness

Mapped to coarse mesh

Back-transformed


Data ReductionTruncation of random field expansion• Eigenvalues are sorted in (usu. strongly) decreasing order• Highest eigenvalues contribute most to total variance

Neglect variables with minor contribution


Data Reduction• Quality of truncated series: variability fraction

• After mesh reduction: normalize variability to number of data


Data ReductionEffects of subspace projection

Thickness:original data

Modal base plastic strain:coincident with thickness

Self-projected with 15 mode shapes

Modal base elastic stress: no coincidence with thickness


Procedure Overview

FE model

Interpolatemoments

Additionalsupports

Correlationmodel

Simulateddata

Coarsemesh

Mapdata

Computestatistics

Definerandomvariables

Assumedmoments

Measureddata

Measurementpoints

Project tostructure

Simulate

Analyse

Post-process

Map tostructure


Example: Turbine BladeScope of analysis:• Given is the FE model of an

axial turbine blade• Imperfect geometry of the

blade surface is scanned• A random field is modelled by

the statistics of measurements• Random realizations of the

turbine blade are generated by optiSLang and put into

• Fluid-structure simulation with ANSYS Workbench, then

• Robustness post-processing by optiSLang


Example: Turbine BladeMeasurement of manufacturing tolerances:• Geometry scan at 1500 points on the surface• Measurement points need not be FE model nodes• Realizations of simulated random fields are mapped

on the FE structure


Example: Turbine Blade• Measurements are taken in the y- and z-direction• Standard deviations and correlations are evaluated• Means of imperfections are assumed zero

x

y

z


Example: Turbine BladeRandom field model• After decomposition of the

covariance matrix, 18 (9 y-, 9 z-) random amplitudes represent 99 % of the random field variability

Robustness evaluation• 15 random parameters

(material, geometry, operation) are included in the analysis

Which variables have significant influence on the performance of the turbine?

Object: AMPLITUDES_YObject info: 2 2 9 1 0

0.382764141388120.0074965679902850.00519877522136250.0033236367891740.00276139640235760.00199919771897930.00187636535663480.0016475523697930.0013368316745761

Object: AMPLITUDES_ZObject info: 2 2 9 1 0

0.287083430609080.00541863112801510.00369360067819380.0022287850603360.00198742676047010.00152944615712680.00142139404707410.00104114064233110.00092089226898308


Example: Turbine BladeLatin Hypercube simulation (50 samples), only random field variables:

Mode Y_5 Mode Z_3 Mode Y_8


Latin Hypercube (50 samples) including random parameters

Example: Turbine Blade


Example: Turbine BladeLatin Hypercube (50 samples) including random parameters


Final Remarks• Manufacturing tolerances may have significant influence on the

performance of structures.• Imperfections can be measured.• Important random field variables can be identified by methods of

robustness analysis, even in comparison to other (e.g. CAD, material, operation, …) parameters.

• The expansion by random amplitudes and shape functions allows for interpretation of imperfection shapes.

• Software is available to • analyze data,• model random fields,• simulate random fields,• control the analysis,• post-process results.


Final RemarksFurther developments• Refine mesh coarsening algorithms• Better support 3D solid structures• Estimate covariances from few data• Handling of big models, big data sets• Implementation into dynardo's software environment

Data,FE-model

Random FieldModeling

Random FieldPostprocessing

random fields: theory and applications · 2016. 2. 20. · • important random field variables can...

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