part 5 parameter identification (model calibration/updating)

Part 5Parameter Identification

(Model Calibration/Updating)

2 Part 5: Parameter Identifikation

1) Define the Design space using continuous or discrete optimization variables

Calibration using optiSLang

3) Find the best possible fit - choose an optimizer depending on the sensitive

optimization parameter dimension/type

Test

Best Fit

Simulation

optiSLang

2) Scan the Design Space - Check the variation- Identify sensible parameters

and responses- Check parameter bounds- extract start value


• 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

Model Updating using optiSLang

Δamax

pressure integral

acceleration integral

acceleration peak= α + β + γ

ZeitZeit Zeit

Validation of Airbag Modeling via Identification


Model Updating using optiSLang

Test

Best Fit

Simulation

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)


System Identification

Sankt Michael church Jena

Fitting of Experiments to Numerical Models

• 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

6 Part 5: Parameter Identifikation6

3. Run Sensitivity study to identify sensitive parameters and responses and verify the design space

4. Definition of the objective function for Identification & optimize

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

Stress-Strain curve

obj_func = |FAIL_STRAIN – TARGET_STRAIN| 0

Target failure strain

Failure strain from simulation

Application Identification of failure strain


Identification of one experiment

Best Fit

Simulation

• Identify one set of Gurson material values (FC,FF,EN) for mean experimental value

• 7 Parameter using ARSM algorithm for global search1 start design from sensitivity (best design)

• 4 min/design (Total:8 h 1 CPU)

10 mm

4 mm

2 mm

3 calculations per design


Best Fit

Simulation

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 used1 start design from sensitivity (best design)

• 4 min/design (Approx. total:23 h 1 CPU)

10 mm

4 mm

2 mm

3 *3 calculations per design

FF0, FC, Lo curve

Identify min, mean and max experimental value


Calibration of seismic fracturingSensitivity evaluation of 200 rock parameter and the hydraulic fracture design Parameter due to seismic hydraulic fracture measurements

Blue:Stimulated rock volumeRed: seismic frac measurement

With the knowledge about the most important parameter the update was significantly improved.

Non-linear coupled fluid-mechanical analysis

Solver: ANSYS/multiPlas

Design evaluations: 160


Least Squares Minimization• The likelihood of the parameters is proportional to the conditional

probability of measurements y* from a given parameter set p

• Assuming normally distributed measurement errors

• Maximizing the likelihood (minimizing the log-likelihood) leads to the optimal parameter set

• If the errors are independent with constant standard deviation we obtain the well-known least squares formulation


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


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


Parameterization of signals

• Repeated block marker • Vector objects with variable length


Definition of signal objects and functions

• 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


Definition of signal functions

1. Min/Max functionsSIG_MIN_Y Extract the minimum ordinate of the channelSIG_MIN_X Extract the abscissa of the minimum ordinate of the channelSIG_MAX_Y Extract the maximum ordinate of the channelSIG_MAX_X Extract the abscissa of the maximum ordinate of the

channel

2. Global functionsSIG_Y_RANGE Extract the range of ordinate values of the channelSIG_MEAN Extract the mean of the channelSIG_STDDEV Extract the standard deviation of the channelSIG_RMS Extract the root mean square of the channelSIG_SUM Extract the sum of values of the channelSIG_EUCLID Extract the Euclidean norm of the channelSIG_NORM Extract the norm of specified order of the channel


Definition of signal functions

3. Difference between two channelsSIG_DIFF_EUCLID Extract the Euclidean norm of the difference

between two channelsSIG_DIFF_NORM Extract the norm of specified order of the difference

between two channels

4. Functions in slotsSIG_***_SLOT Extract the function parameter (functions in 1.-

3.) within the specified abscissa bounds

5. Global functions in stepsSIG_MEAN_STEPS Extract the mean values within a specified number

of equally spaced intervalsSIG_STDDEV_STEPS Extract the standard deviation within a number

of equally spaced intervalsSIG_RMS_STEPS Extract the root mean square values within a

number of equally spaced intervals


Example: Sensitivity analysis using MOP

• 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

100 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

100 samples 500 samples 2000 samples



• All inputs are significant for at least some of the output values

Identification of all input parameters is generally possible

100 samples 500 samples

Full m k D Ekin Full m k D Ekin

RMSE 45% 19% 44% - - 72% 27% 53% - 21%

Max0 99% - 42% - 57% 99% - 46% - 56%

Max2 97% 7% 41% 9% 47% 99% 10% 45% 8% 43%

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%


Example: EA with global search

• Global optimization converges to small difference between output and reference


Signal post-processing

• optiSLang provides signal plots of each design in DOE or optimization flow with best design and specified reference signal


Example: Dependent parameters

• Different optimization runs lead to different parameter sets with similar differences

Run 1: RMSE=0.183 Run 2: RMSE=0.434


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


Example: EA with reduced parameter set

• Different optimization runs lead to similar parameter sets with similar differences

No parameter dependencies

Run 1: RMSE=1.587 Run 2: RMSE=0.287 Run 3: RMSE=0.769


Example: Gradient-based optimization

• Local gradient-based optimization gives exact reference values for inputs

• Fitting is perfect (almost zero rmse)


Example: Identification with noisy reference

• Measurements are more or less precise

• Reference displacement function is disturbed by Gaussian noise with zero mean and standard deviation of 0.1 m

• Again global + local optimization with reduced input parameter set k, D and Ekin


Example: Identification with noisy referenceEvolutionary Algorithm(global search)


Example: Identification with noisy reference

Gradient based (local search)

• 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


Estimation of model representation quality

• 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

• 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)

• The quality of the model representation may be estimated by the explained variance


Oscillator with exact measurements:

Oscillator with noisy measurements:

• But: this measure can not distinguish between errors in the fit caused by inexact measurements or by inadequate models

Estimation of model representation quality

part 5 parameter identification (model calibration/updating)

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