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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