random fields in statistics on structure · 2016-02-20 · random fields in statistics on structure...

Random Fields in Statistics on Structure

Dr.-Ing. Veit BayerWeimar Optimization and Stochastic DaysWOST 7.0, Oktober 2010

2 Veit Bayer, WOST 7.0 © dynardo GmbH, Weimar

Why SoS and what is SoS ?Why: Engineers need to evaluate statistical data on the

structure to locate „hot spots“ of variation as well as investigate correlations

What: A post processor for Statistics on finite element Structures

• Visualization of descriptive statistics on the structure• Visualization of correlations and CoD

between random input and structural results• Identification of spatial dependencies using Random Fields

Key features:• Locate „hot spots“ of variation• Data reduction and smoothing by mesh coarsening and

random field projection• Identification of relevant scatter shapes• Visualization statistics of eroded (failed) elements


Random Field Methods

Measurementson Structure

Process Simulationè Random data

on Structure

Reference (FE-)Structure

Random FieldAnalysis and

Synthesis

Assess robustnessof structures to

spatial randomness

Analyse influence of random input

on spatial scatter

Simulation

Assumedrandom field

model





on Structure



Synthesis


spatial randomness


on spatial scatter

Simulation

Assumedrandom field

model



• Structural model reduction by mesh coarsening• Data reduction: mapping of data from input model to coarse mesh

by local averaging• Reduction of number of random variables

• Eigenvalue analysis of covariance matrix• Choose random variables of highest variance

given by the eigenvalues• Karhunen – Loève expansion of random field data

(series of deterministic shape functions – eigenvectors –scaled by random amplitudes)

• Mapping of coarse mesh data to original model by interpolation (moving least squares)

• Postprocessing of basic statistics, input – output correlation etc.

Random Field Analysis and Synthesis:


Random Field MethodsSmoothing effect by mesh coarsening

Original datathickness

Mapped to coarse mesh

Back-transformed


• Original data: (possibly dependent) random vector X• Eigenvalue decomposition of CXX

• Uncorrelated random variables Y

• Transformation between basic random variables and “real-world”,Analysis: Simulation:


X = Y1 + Y2 + Y3 + …


Random Field Methods• Truncation error: variability fraction

• … may be used also as measure of the contribution of one single mode shape

11 © dynardo GmbH, Weimar

SoS Post-processing on FE-meshes

SoSVisualize variation, correlation,identify random field shapes

Export local statisticsand imperfection amplitudes to optiSLang for furtherpost-processing

Process:Import data (multiple simulation runs or measurements)

12 © dynardo GmbH, Weimar

SoS Post-processing on FE-meshesPost-processing modes• Structure and imperfection shapes• Descriptive statistics:

Means,Standard deviations,Ranges,Quantiles,Eroded elements,Single designs and Design differencesetc.

• Correlations & CoDs• Quality Capability Statistics

Export options:• Samples of data at selected

node / element regions• Samples of modal amplitudes


Application Example• Stringer in a car body subject to crash simulation• 55 random inputs: sheet thickness and material parameters (also

of other parts), load parameters (velocity, barrier position etc.)

• Observed result: remaining effective plastic strain

[Will, J.; Frank, T.: Robustness analysis of structural crash load cases at Daimler AG, WOST 5.0, Weimar 2008]


• Effective plastic strain: Distribution of standard deviation corresponds to observed buckling

• Causes of buckling shall be found:• Which input scatter is responsible for peaks of plastic strain?

Application Example


Application Example• CoD of effective plastic strain w.r.t. all input variables

• … does not reveal the buckling shape, due to strong non-linearity and multiple inputs


Application ExamplePost-processing of mode shapes

• Mode 1: contribution ofQi = 96% of total variability

• Mode 2: 1.2%

• Mode 3: 0.8%

• All three: 98%


Application Example

Further analysis of random amplitude #1 in optiSLang:

Evaluate Metamodel of optimal prognosis

13 variables (out of 55) are found significant

CoP is 86%


Comparison to New ProcedureFirst mode shape after back-transformation

• First shape covers larger amount of variability

• Higher CoP

• Most significant inputs are similar

• More variables in themetamodel


Conclusions and Outlook• Analysis of random data distributed on a structure

using random field parametric helps toè Locate critical points on the structureè Find the causes of scatter by statistical means.

• SoS 2.3.0 released in summer 2010 offers post-processing with data-based mode shapes.

è Hence you can directly see the spatial influence of scattering inputs on the structure.

è Further analysis in optiSLang (CoI, MoP, CoP) offers further insight for causal analyses.

• Future developments aim at• improving efficiency of the software,• introducing new, more effective parametric for better handling

of small data sets on large structures,• generating imperfect structures by random field sampling,• ... meet customer requests.





on Structure



Synthesis


spatial randomness


on spatial scatter

Simulation

Assumedrandom field

model


Conclusions and Outlook• Analysis of random data distributed on a structure

using random field parametric helps toè Locate critical points on the structureè Find the causes of scatter by statistical means.

• SoS 2.3.0 released in summer 2010 offers post-processing with data-based mode shapes.

è Hence you can directly see the spatial influence of scattering inputs on the structure.

è Further analysis in optiSLang (CoI, MoP, CoP) offers further insight for causal analyses.

• Future developments aim at• improving efficiency of the software,• introducing new, more effective parametric for better handling

of small data sets on large structures,• generating imperfect structures by random field sampling,• ... meet customer requests.

random fields in statistics on structure · 2016-02-20 · random fields in statistics on structure...

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