Transcript Document

Part 5
Parameter Identification
(Model Calibration/Updating)
Calibration using optiSLang
1) Define the Design space using
continuous or discrete optimization
variables
2) Scan the Design Space
- Check the variation
- Identify sensible parameters
and responses
- Check parameter bounds
- extract start value
Simulation
Test
Best Fit
optiSLang
3) Find the best possible fit
- choose an optimizer depending on the sensitive
optimization parameter dimension/type
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Part 5: Parameter Identifikation
Model Updating using optiSLang
Validation of Airbag Modeling via Identification
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Validation of numerical models with test
results (7 test configurations)
Modelling with Madymo
Sensitivity study to identify sensitive
parameters and responses and to verify
the design space
Definition of the objective function
Δamax
Zeit
=α
3
acceleration
integral
Zeit
+β
Part 5: Parameter Identifikation
acceleration peak
Zeit
+γ
pressure
integral
Model Updating using optiSLang
Validation of Airbag Modeling via Identification
• optiSLang’s genetic
algorithm for global search
• 15 generation
*10 individuals
*7 test configuration
• (Total:11 h CPU)
Simulation
Test
Best Fit
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optiSLang
Part 5: Parameter Identifikation
System Identification
Fitting of Experiments to Numerical Models
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Mechanical properties
of historical masonry
are unknown
Identification of
system parameters
via model updating
for dynamic
measurements
(system
identification)
Ringing the bell is
the critical load case
Sankt Michael church Jena
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Part 5: Parameter Identifikation
Application Identification of failure strain
1. Set up of an parametric simulation process, FE model of tensile
test in LSDYNA to identify Gurson Damage Material Parameter
2. Integrate the process in optiSLang
3. Run Sensitivity study to identify
sensitive parameters and responses
and verify the design space
4. Definition of the objective function for
Identification & optimize
Failure strain
from simulation
Target failure
strain
obj_func = |FAIL_STRAIN – TARGET_STRAIN|  0
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Part 5: Parameter Identifikation
Stress-Strain curve
Identification of one experiment
2 mm
• Identify one set of Gurson
material values (FC,FF,EN)
for mean experimental
value
4 mm
10 mm
• 7 Parameter using ARSM
algorithm for global search
1 start design from
sensitivity (best design)
• 4 min/design
(Total:8 h 1 CPU)
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Best Fit
Part 5: Parameter Identifikation
Simulation
3 calculations
per design
Identify min, mean and max experimental value
Identify Gurson material (FC,FF,EN)
values for mean, min, max
representing the scatter range of
experiments
12 Parameter using ARSM algorithm
for global search is used
1 start design from sensitivity
(best design)
2 mm
4 mm
10 mm
Simulation
• 4 min/design
(Approx. total:23 h 1 CPU)
Best Fit
FF0, FC,
Lo curve
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Part 5: Parameter Identifikation
3 *3 calculations
per design
Calibration of seismic fracturing
Sensitivity evaluation of 200 rock parameter and the
hydraulic fracture design Parameter due to seismic
hydraulic fracture measurements
Blue:Stimulated rock volume
Red: seismic frac measurement
With the knowledge about the most
important parameter the update was
significantly improved.
Non-linear coupled fluidmechanical analysis
Solver: ANSYS/multiPlas
Design evaluations: 160
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Part 5: Parameter Identifikation
Least Squares Minimization
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The likelihood of the parameters is proportional to the conditional
probability of measurements y* from a given parameter set p
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Assuming normally distributed measurement errors
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Maximizing the likelihood (minimizing the log-likelihood)
leads to the optimal parameter set
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If the errors are independent with constant standard deviation
we obtain the well-known least squares formulation
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Part 5: Parameter Identifikation
Example: Calibration of a damped oscillator
• Mass m, damping c, stiffness k
and initial kinetic energy
• Equation of motion:
• Undamped eigen-frequency:
• Lehr's damping ratio D
• Damped eigen-frequency
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Part 5: Parameter Identifikation
Example: Calibration of a damped oscillator
• Time-dependent displacement
function
• Identification of the input parameters
m, k, D and Ekin to optimally fit a
reference displacement function
• Objective function is the sum of squared errors between the
reference and the calculated displacement function values
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Part 5: Parameter Identifikation
Parameterization of signals
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Repeated block marker
Vector objects with variable length
Part 5: Parameter Identifikation
Definition of signal objects and functions
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Signal object consists of abscissa vector and several channels
Signal functions to extract value from a single signal or to
compare channels or different signals
Definition of constant reference signals for model calibration
Part 5: Parameter Identifikation
Definition of signal functions
1. Min/Max functions
SIG_MIN_Y
Extract the minimum ordinate of the channel
SIG_MIN_X
Extract the abscissa of the minimum ordinate of the channel
SIG_MAX_Y
Extract the maximum ordinate of the channel
SIG_MAX_X
Extract the abscissa of the maximum ordinate of the channel
2. Global functions
SIG_Y_RANGE Extract
SIG_MEAN
Extract
SIG_STDDEV
Extract
SIG_RMS
Extract
SIG_SUM
Extract
SIG_EUCLID
Extract
SIG_NORM
Extract
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the
the
the
the
the
the
the
range of ordinate values of the channel
mean of the channel
standard deviation of the channel
root mean square of the channel
sum of values of the channel
Euclidean norm of the channel
norm of specified order of the channel
Part 5: Parameter Identifikation
Definition of signal functions
3. Difference between two channels
SIG_DIFF_EUCLID
Extract the Euclidean norm of the difference
between two channels
SIG_DIFF_NORM
Extract the norm of specified order of the difference
between two channels
4. Functions in slots
SIG_***_SLOT
Extract the function parameter (functions in 1.-3.)
within the specified abscissa bounds
5. Global functions in steps
SIG_MEAN_STEPS
Extract the mean values within a specified number
of equally spaced intervals
SIG_STDDEV_STEPS
Extract the standard deviation within a number of
equally spaced intervals
SIG_RMS_STEPS
Extract the root mean square values within a
number of equally spaced intervals
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Part 5: Parameter Identifikation
Example: Sensitivity analysis using MOP
100 samples
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CoP of sum of squared errors is very low (45% CoP) and only m and
k are found to be significant
CoP of maximum values in time slot are much better (95% - 99%
CoD) and all inputs are indicated to be significant
Part 5: Parameter Identifikation
Example: Sensitivity analysis using MOP
100 samples
500 samples
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2000 samples
CoP of sum of squared errors increases if number of samples is
increased (from 45% to 84%) and one additional parameter
becomes significant
 Sensitivity study of objective function itself may require many
samples due to a certain complexity
 Analysis of single values may be more efficient
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Part 5: Parameter Identifikation
Example: Sensitivity analysis using MOP
100 samples
Full
m
k
19% 44%
500 samples
D
Ekin
Full
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72%
m
k
Ekin
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21%
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56%
8%
43%
RMSE
45%
Max0
99%
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42%
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57%
99%
Max2
97%
7%
41%
9%
47%
99%
10% 45%
Max4
97%
15% 42% 18% 29%
98%
15% 44% 16% 30%
Max6
98%
23% 36% 23% 23%
99%
20% 41% 22% 24%
Max8
95%
23% 28% 35% 16%
96%
19% 36% 26% 20%
• All inputs are significant for
at least some of the output
values
 Identification of all input
parameters is generally
possible
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Part 5: Parameter Identifikation
27% 53%
D
-
46%
Example: EA with global search
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Global optimization converges to small
difference between output and reference
Part 5: Parameter Identifikation
Signal post-processing
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optiSLang provides signal plots of each design in DOE or
optimization flow with best design and specified reference signal
Part 5: Parameter Identifikation
Example: Dependent parameters
Run 1: RMSE=0.183
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Run 2: RMSE=0.434
Different optimization runs lead
to different parameter sets with
similar differences
Part 5: Parameter Identifikation
Example: Dependent parameters
Reason for non-unique solution:
• The parameters Ekin and m as well as k and m appear only
pair-wisely in the displacement function
 Only the ratio between Ekin and m as well as k and m can be
identified
 We keep the value of m as constant
General procedure:
• Check designs from DOE with almost equal objective values
• Or perform multiple global optimization runs
• Sensitivity indices quantify the global influence of each input,
But: the dependency between input parameters with respect to
the minimum objective values can not be identified
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Part 5: Parameter Identifikation
Example: EA with reduced parameter set
Run 1: RMSE=1.587
Run 2: RMSE=0.287
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Different optimization runs
lead to similar parameter sets
with similar differences
 No parameter dependencies
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Part 5: Parameter Identifikation
Run 3: RMSE=0.769
Example: Gradient-based optimization
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Local gradient-based optimization
gives exact reference values for inputs
Fitting is perfect (almost zero rmse)
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Part 5: Parameter Identifikation
Example: Identification with noisy reference
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Measurements are more or less precise
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Reference displacement function is disturbed by Gaussian noise
with zero mean and standard deviation of 0.1 m
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Again global + local optimization with reduced input parameter
set k, D and Ekin
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Part 5: Parameter Identifikation
Example: Identification with noisy reference
Evolutionary Algorithm
(global search)
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Part 5: Parameter Identifikation
Example: Identification with noisy reference
Gradient based
(local search)
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Measurements errors may reduce the
identification quality
The accuracy of the identified parameters
depends on the number of measurements
and the sensitivity of the parameters
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Part 5: Parameter Identifikation
Estimation of model representation quality
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Assuming, that the model can reproduce the reality, the
measurement error can be defined as the deviation
of the fitted model from the reference solution
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Estimated error variance by assuming independent measurement
errors with constant variance
(p is the number of identified parameters, n the number of
measurement points and yi* are the measurement values)
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The quality of the model representation may be estimated by the
explained variance
Part 5: Parameter Identifikation
Estimation of model representation quality
Oscillator with exact measurements:
Oscillator with noisy measurements:
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But: this measure can not distinguish between errors in the fit
caused by inexact measurements or by inadequate models
Part 5: Parameter Identifikation