1. Part 5 Parameter Identification (Model Calibration/Updating)

2. 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

3. 3 Part 5: Parameter Identifikation 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 Δa max pressure integral acceleration integral acceleration peak = α+ β+ γ Zeit Validation of Airbag Modeling via Identification

4. 4 Part 5: Parameter Identifikation 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)

5. 5 Part 5: Parameter Identifikation 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. 6 Part 5: Parameter Identifikation 6 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

7. 7 Part 5: Parameter Identifikation 7 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 search 1 start design from sensitivity (best design) 4 min/design (Total:8 h 1 CPU) 10 mm 4 mm 2 mm 3 calculations per design

8. 8 Part 5: Parameter Identifikation 8 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 used 1 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

9. 9 Part 5: Parameter Identifikation 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 fluid- mechanical analysis Solver: ANSYS/multiPlas Design evaluations: 160

10. 10 Part 5: Parameter Identifikation 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

11. 11 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

12. 12 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

13. 13 Part 5: Parameter Identifikation Parameterization of signals Repeated block marker Vector objects with variable length

14. 14 Part 5: Parameter Identifikation 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

15. 15 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 the range of ordinate values of the channel SIG_MEAN Extract the mean of the channel SIG_STDDEV Extract the standard deviation of the channel SIG_RMS Extract the root mean square of the channel SIG_SUM Extract the sum of values of the channel SIG_EUCLID Extract the Euclidean norm of the channel SIG_NORM Extract the norm of specified order of the channel

16. 16 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

17. 17 Part 5: Parameter Identifikation 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

18. 18 Part 5: Parameter Identifikation Example: Sensitivity analysis using MOP 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 samples500 samples2000 samples

19. 19 Part 5: Parameter Identifikation Example: Sensitivity analysis using MOP All inputs are significant for at least some of the output values  Identification of all input parameters is generally possible 100 samples500 samples FullmkDEkinFullmkDEkin RMSE45%19%44%--72%27%53%-21% Max099%-42%-57%99%-46%-56% Max297%7%41%9%47%99%10%45%8%43% Max497%15%42%18%29%98%15%44%16%30% Max698%23%36%23% 99%20%41%22%24% Max895%23%28%35%16%96%19%36%26%20%

20. 20 Part 5: Parameter Identifikation Example: EA with global search Global optimization converges to small difference between output and reference

21. 21 Part 5: Parameter Identifikation Signal post-processing optiSLang provides signal plots of each design in DOE or optimization flow with best design and specified reference signal

22. 22 Part 5: Parameter Identifikation Example: Dependent parameters Different optimization runs lead to different parameter sets with similar differences Run 1: RMSE=0.183Run 2: RMSE=0.434

23. 23 Part 5: Parameter Identifikation Example: Dependent parameters Reason for non-unique solution: The parameters E kin and m as well as k and m appear only pair-wisely in the displacement function  Only the ratio between E kin 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

24. 24 Part 5: Parameter Identifikation Example: EA with reduced parameter set Different optimization runs lead to similar parameter sets with similar differences  No parameter dependencies Run 1: RMSE=1.587Run 2: RMSE=0.287Run 3: RMSE=0.769

25. 25 Part 5: Parameter Identifikation Example: Gradient-based optimization Local gradient-based optimization gives exact reference values for inputs Fitting is perfect (almost zero rmse)

26. 26 Part 5: Parameter Identifikation 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 E kin

27. 27 Part 5: Parameter Identifikation Example: Identification with noisy reference Evolutionary Algorithm (global search)

28. 28 Part 5: Parameter Identifikation 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

29. 29 Part 5: Parameter Identifikation 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 y i * are the measurement values) The quality of the model representation may be estimated by the explained variance

30. 30 Part 5: Parameter Identifikation 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