Webinar Parameter Identification with optislang. Dynardo GmbH

Transcription

1 Webinar Parameter Identification with optislang Dynardo GmbH 1

2 Outline Theoretical background Process Integration Sensitivity analysis Least squares minimization Example: Identification of material parameters of spring steel 2

3 Theoretical Background 3

4 Inverse Identification of Model Parameters Identification of unknown model parameters by the calibration of the model with respect to given measurements Direct relation between measurements and model parameters is known only inversely as forward simulation model 4

5 The Forward Simulation Model parameters Simulation model Model responses For given set of model parameters p the model responses y can be calculated with a given simulation model Deviation of model responses and measurements y * can be evaluated For which parameter set p opt model responses and measurement agree sufficiently well? Measurements 5

6 Least Squares Minimization The likelihood of the parameters is proportional to the conditional probability of measurements y* from a given parameter set p For correct model (y* - y) is caused only by measurement errors Assuming normally distributed measurement errors: Maximizing the likelihood (minimizing the log-likelihood) leads to the least squares objective function 6

7 Least Squares Minimization If the errors are independent we obtain With constant standard deviation the objective simplifies 7

8 Least Squares Minimization Standard error of fit (root mean squared error) n number of discretization Weighting of different experiments in one objective function by normalizing the RMSE by the response ranges (or standard deviations) y max Response-Maximum y min Response-Minimum 8

9 Requirements to the Identification Procedure The simulation model needs to represent the main physical behavior (systematic model errors are not considered) Since the least squares minimization may lead to a local optimum a global optimization strategy is necessary Only sensitive parameters can be identified Different parameter combinations may lead to a similar objective Uniqueness of identified parameters has to be assessed 9

10 Calibration using optislang 1) Define the Design space using continuous or discrete optimization variables 2) Scan the Design Space - Check the variation - Identify sensitive 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 10

11 Process Integration 11

12 Process Integration Parametric model as base for User defined optimization (design) space Naturally given robustness (random) space Design variables Entities that define the design space Scattering variables Entities that define the robustness space The CAE process Generates the results according to the inputs Response variables Outputs from the system 12

13 optislang Integrations & Interfaces Direct integrations ANSYS Workbench MATLAB Excel Python AMESim SimulationX Supported connections ANSYS APDL Abaqus Adams AMESim Arbitary connection of ASCII file based solvers Signals can be directly imported from MATLAB, Excel, Python, AMESim, SimulationX & ASCII 13

14 Signals in optislang Signals are vector outputs having an abscissa (e.g. time axis) and several output channels (e.g. displacements, velocities) Comprehensive library of signal functions enables the user to extract local and statistical quantities and to analyze differences between several signal channels e.g. for calibration tasks Automatic mapping of non-consistent abscissa discretizations for the signals of each design and of the reference curves Direct access to signal plots in the optislang postprocessing and interactive connection to the statistic/optimization postprocessing 14

15 Signal Processing with Extraction Tool Kit (ETK) Now available in optislang inside ANSYS Reads many CAE binary output formats and text files Can read signals, vectors and matrices from solver files in text and binary format Can perform arbitrary mathematical operations from extracted objects Questions? 15

16 Sensitivity Analysis 16

17 Scanning the Design Space Inputs Design of Experiments Solver evaluation Outputs Uniform distribution of inputs is represented by Latin Hypercube Sampling Minimum number of samples should represent statistical properties, cover the input space optimally and avoid clustering For each design all responses are calculated 17

18 Metamodel of Optimal Prognosis (MOP) Approximation of solver output by fast surrogate model Reduction of input space to get best compromise between available information (samples) and model representation (number of inputs) Determination of optimal approximation model Assessment of approximation quality Evaluation of variable sensitivities 18

19 Least Squares Minimization 19

20 optislang Optimization Algorithms Gradient-based Methods Most efficient method if gradients are accurate enough Consider its restrictions like local optima, only continuous variables and noise Start Adaptive Response Surface Method Attractive method for a small set of continuous variables (<20) Adaptive RSM with default settings is the method of choice Nature inspired Optimization GA/EA/PSO imitate mechanisms of nature to improve individuals Method of choice if gradient or ARSM fails Very robust against numerical noise, nonlinearity, number of variables, 20

21 Decision Tree for Optimizer Selection optislang automatically suggests an optimizer depending on the parameter properties, the defined criteria and user specified settings Questions? 21

22 Examples 22

23 Tension Test of Spring Steel Finite element model in ANSYS Workbench Nonlinear material behavior Tensile bar is deformed by a predefined displacement Reaction forces at deformed tensile bar end (1) are monitored depending on deformation between named selection u1 (2) and u2 (3) and saved into the result file file.rst

24 Problem Definition Simulation with initial materials parameters vs. reference (measurements) 24

25 Problem Definition Identification of the material parameters to optimally fit the force-displacement curve to the measurements Unknown material parameters for nonlinear isotropic hardening (nliso): Young s modulus Yield stress σ 0 σ = σ 0 + R 0 ε pl + R (1-e -b εpl ) Linear hardening coefficient R 0 Exponential hardening coefficient R Exponential saturation parameter b Objective function is the sum of squared errors between the reference and the calculated force-displacement function values 25

26 Task Description Generation of a solver chain using ANSYS Workbench and Signal Processing Definition of the input parameters Definition of output and reference signals Sensitivity analysis of signal extraction terms using the given parameter bounds Single objective, unconstrained optimization by minimizing the sum of squared errors 26

27 Tension Test of Spring Steel Unknown parameters defined in ASCII input file 27

28 Definition of the Reference Signal Displacements and forces of measurements are parameterized as signal 28

29 Definition of the Output Signal Displacements and forces of simulation are parameterized as signal from a binary format (file.rst) 29

30 Definition of the Output Signal With Instant Visualization (1) it is possible to compare both signals Both signals do not have the same discretization (2) and length (3) To get the same length and discretization it is necessary to extract the abscissa from the Signal_Ref and than interpolate the Signal_raw to this abscissa

31 Definition of Signal Functions The displacement is divided in 7 equally spaced steps (1-7) to get more detailed information about the influence of the 5 material parameters At these steps the forces will be extracted

32 Definition of the Design Variables 1. Adjust lower and upper bounds for all parameters 2. Press Next 32

33 Results of the Sensitivity Analysis The reference is covered sufficiently by the simulations Parameter bounds seem to be adequate for the calibration 33

34 Results of the Sensitivity Analysis The CoP value of the signal difference indicates a good explainability of this function Linear hardening coefficient R 0 are not detected as important Check also single force_steps values 34

35 Results of the Sensitivity Analysis 35

36 Results of the Sensitivity Analysis Why is the linear hardening coefficient (1) R 0 not important? The linear hardening coefficient R 0 describes the slope of the asymptotic curve (2), which is zero in our example (3) force_steps[0] force_steps[6] 36

37 Results of the Sensitivity Analysis This localization will lead to a negative gradient (1) of the force-displacement-curve. To avoid this situation, a positive gradient is required as constraint: min_gradient_signal

38 Direct Optimization 38

39 Direct Optimization 39

40 Results of the Optimization Very good agreement between simulation and reference is achieved 40

41 optislang Training Program 41

42 Training ( optislang Basics 3 day introduction to process integration, sensitivity analysis, single- and multiobjective optimization, calibration and basics of robustness evaluation optislang and ANSYS Workbench 1 day introduction to the integration of ANSYS Workbench projects in a optislang solver chain and the parameterization of signals via APDL output Parameter Identification 1 day advanced seminar on model calibration using sensitivity analysis and least squares minimization Robust Design and Reliability Analysis 1 day advanced seminar on robustness & reliability analysis and robust design optimization Statistics on Structures 1 day introduction to our newly released software for postprocessing statistical data, evaluation and simulation of random effects on finite element structures 42

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