Simulation of Random Fields for SensitivityAnalysis

1st Workshopon Nonlinear Analysis of Shell Structures

INTALES GmbH Engineering Solutions

University of Innsbruck, Faculty of Civil Engineering

University of Innsbruck, Faculty of Mathematics, Informatics and Physics

Natters/Tyrol, 15/06/2010

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 1 / 11

Random Fields

Definition

Random fields are simulating

the stochastic fluctuations of material properties (e.g.thickness, E-modulus, ...) subject to the

spatial domain of the considered structures.

⇒ (spatial) correlation of several values

Stochastic fluctuations are deviations of the nominal values forcertain material properties or irregularities of the geometry, causedby

(slightly) change of manufacturing terms

changing quality of raw material

stochastic influence, ...

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 2 / 11

Random fields - Examples

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 3 / 11

Random Fields - Simulation

Parameters

Before it is possible to generate a random field, a few parametershave to be defined first:

µ ... mean value or nominal value of a material propertyoften: simulating the random field with µ′=0 and afterwards µis added to the field values

σ ... the standard deviation of the random field

d (s, t) ... distance function to calculate the space betweenthe two locations s, t on the surface/FE-grid

c ... correlation length: the domain of influence between twopoints on a surfacec depends on the material in use, and the FE-grid

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 4 / 11

Random Fields - Simulation

c ≈ meshsize of FE-grid:independent field values in eachelementc � meshsize: stochastic mo-del not valid

C (s, t) ... covariance function to measure the relationbetween two points s, t on the structureC (s, t) only depends on the distance between the observedlocations, e.g.

C (s, t) = σ2 exp(− 1

c · d (s, t))

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 5 / 11

Random Fields - Simulation Method

Karhunen-Loeve expansion

orthogonal decomposition method

a series expansion method to simulate random fieldsthe series components are the following:

ξn random variablesλn constantsφn (t) a set of orthonormal functions

in the finite case: λn and φn are the eigenvalues andeigenfunctions of Cφn = λnφn

the ξn are normally distributed random variables for aGaussian random field

for numerical calculations, discretisation of the structure isoften necessary for using KL ⇒ the FE-grid can be useddirectly

problem: computation time for larger models

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 6 / 11

Random Fields - Simulation Method

Karhunen-Loeve expansion

orthogonal decomposition method

a series expansion method to simulate random fieldsthe series components are the following:

ξn random variablesλn constantsφn (t) a set of orthonormal functions

in the finite case: λn and φn are the eigenvalues andeigenfunctions of Cφn = λnφn

the ξn are normally distributed random variables for aGaussian random field

for numerical calculations, discretisation of the structure isoften necessary for using KL ⇒ the FE-grid can be useddirectly

problem: computation time for larger models

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 6 / 11

Random Fields - Simulation Method

Karhunen-Loeve expansion

orthogonal decomposition method

a series expansion method to simulate random fieldsthe series components are the following:

ξn random variablesλn constantsφn (t) a set of orthonormal functions

in the finite case: λn and φn are the eigenvalues andeigenfunctions of Cφn = λnφn

the ξn are normally distributed random variables for aGaussian random field

for numerical calculations, discretisation of the structure isoften necessary for using KL ⇒ the FE-grid can be useddirectly

problem: computation time for larger models

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 6 / 11

Random Fields - Simulation Method

Karhunen-Loeve expansion

orthogonal decomposition method

a series expansion method to simulate random fieldsthe series components are the following:

ξn random variablesλn constantsφn (t) a set of orthonormal functions

in the finite case: λn and φn are the eigenvalues andeigenfunctions of Cφn = λnφn

the ξn are normally distributed random variables for aGaussian random field

for numerical calculations, discretisation of the structure isoften necessary for using KL ⇒ the FE-grid can be useddirectly

problem: computation time for larger models

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 6 / 11

Random Fields - Simulation Method

Karhunen-Loeve expansion

orthogonal decomposition method

a series expansion method to simulate random fieldsthe series components are the following:

ξn random variablesλn constantsφn (t) a set of orthonormal functions

in the finite case: λn and φn are the eigenvalues andeigenfunctions of Cφn = λnφn

the ξn are normally distributed random variables for aGaussian random field

for numerical calculations, discretisation of the structure isoften necessary for using KL ⇒ the FE-grid can be useddirectly

problem: computation time for larger models

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 6 / 11

Random Fields and Sensitivity Analysis

(How) Do random fields effect the behavior (of astructure)?⇒ statistics, sensitivity analysis, comparisions

Loading Proportional Factor - LPF

measures, with which percentage of an intented load thestructure could be stressed, so that the model still converges.

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 7 / 11

Variation of LPF

Question: Variation of output LPF

Considered combinations:1 nominal material parameter (no RF)

randomly varying loads2 random field for material parameter

nominal loads3 random field for material parameter

randomly varying loads

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 8 / 11

Comparing the Distributions

no RF, varying loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

RF, varying loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

RF, nominal loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

Deterministic LPF = 3.52all input: nominal values

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 9 / 11

Comparing the Distributions

no RF, varying loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1F

Distribution of LPF

RF, varying loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

RF, nominal loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

Deterministic LPF = 3.52all input: nominal values

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 9 / 11

Comparing the Distributions

no RF, varying loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1F

Distribution of LPF

RF, varying loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

RF, nominal loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

Deterministic LPF = 3.52all input: nominal values

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 9 / 11

Comparing the Distributions

no RF, varying loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1F

Distribution of LPF

RF, varying loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

RF, nominal loads

3.2 3.4 3.6 3.8 40

0.2

0.4

0.6

0.8

1

F

Distribution of LPF

Deterministic LPF = 3.52all input: nominal values

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 9 / 11

Comparing the Boxplots

no RF, varying loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

RF, varying loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

RF, nominal loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

Effect of RF

larger scatter

higher mean LPF

higher median LPF

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 10 / 11

Comparing the Boxplots

no RF, varying loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

RF, varying loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

RF, nominal loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

Effect of RF

larger scatter

higher mean LPF

higher median LPF

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 10 / 11

Comparing the Boxplots

no RF, varying loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

RF, varying loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

RF, nominal loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

Effect of RF

larger scatter

higher mean LPF

higher median LPF

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 10 / 11

Comparing the Boxplots

no RF, varying loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

RF, varying loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

RF, nominal loads

3.2

3.4

3.6

3.8

4Boxplot of LPF

Effect of RF

larger scatter

higher mean LPF

higher median LPF

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 10 / 11

Outlook

Sensitivity Analysis

Sensitivity of output LPF with respect to input loads,given random field material parameters

Further outputs

Beside the LPF, also other paramters can be considered:

displacements

elastic/plastic strain energy density

von Mises stress

eigenvalues,...

Sensitivity w.r.t. random field parameters

analysis with varying random field parameters (σ, correlationlength)

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 11 / 11

Outlook

Sensitivity Analysis

Sensitivity of output LPF with respect to input loads,given random field material parameters

Further outputs

Beside the LPF, also other paramters can be considered:

displacements

elastic/plastic strain energy density

von Mises stress

eigenvalues,...

Sensitivity w.r.t. random field parameters

analysis with varying random field parameters (σ, correlationlength)

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 11 / 11

Outlook

Sensitivity Analysis

Sensitivity of output LPF with respect to input loads,given random field material parameters

Further outputs

Beside the LPF, also other paramters can be considered:

displacements

elastic/plastic strain energy density

von Mises stress

eigenvalues,...

Sensitivity w.r.t. random field parameters

analysis with varying random field parameters (σ, correlationlength)

K. Riedinger (University of Innsbruck) ACOSTA-Workshop Natters/Tyrol, 15/06/2010 11 / 11