Part 5 Parameter Identification (Model Calibration/Updating)

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

2. 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 optiSLang Best Fit 3) Find the best possible fit - choose an optimizer depending on the sensitive optimization parameter dimension/type Part 5: Parameter Identifikation

3. Model Updating using optiSLang • 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 Validation of Airbag Modeling via Identification Δamax Zeit Zeit Zeit acceleration integral acceleration peak pressure integral = α + β + γ Part 5: Parameter Identifikation

4. 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 optiSLang Best Fit Part 5: Parameter Identifikation

5. System Identification • 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 Fitting of Experiments to Numerical Models Sankt Michael church Jena 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 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 Stress-Straincurve Failure strain from simulation Target failure strain obj_func = |FAIL_STRAIN – TARGET_STRAIN|  0 6 Part 5: Parameter Identifikation

7. Identification of one experiment 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) 2 mm 4 mm 10 mm Simulation 3 calculations per design Best Fit 7 Part 5: Parameter Identifikation

8. 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) Identify min, mean and max experimental value 2 mm 4 mm 10 mm Simulation 3 *3 calculations per design Best Fit FF0, FC, Lo curve 8 Part 5: Parameter Identifikation

9. 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 Part 5: Parameter Identifikation

10. 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 Part 5: Parameter Identifikation

11. 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 Part 5: Parameter Identifikation

12. 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 Part 5: Parameter Identifikation

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

14. 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 Part 5: Parameter Identifikation

15. Definition of signal functions • 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 • 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 Part 5: Parameter Identifikation

16. Definition of signal functions • 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 • Functions in slots SIG_***_SLOT Extract the function parameter (functions in 1.-3.) within the specified abscissa bounds • 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 Part 5: Parameter Identifikation

17. Example: Sensitivity analysis using MOP 100 samples • 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

18. Example: Sensitivity analysis using MOP 100 samples 500 samples 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 Part 5: Parameter Identifikation

19. 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 Part 5: Parameter Identifikation

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

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

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

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

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

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

26. 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 Part 5: Parameter Identifikation

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

28. 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 Part 5: Parameter Identifikation

29. 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 Part 5: Parameter Identifikation

30. Estimation of model representation quality 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 Part 5: Parameter Identifikation