We implement a Bayesian Inference approach to calibrate the material parameters of a model for the superelastic deformation of NiTi shape memory alloy. We specify a diamond-shaped specimen geometry that is suited to calibrate both tensile and compressive material parameters from a single test. We adopt the Bayesian Inference calibration scheme to take full-field strain measurements obtained using digital image correlation together with global load data as an input for calibration. The calibration itself is performed by comparing the experimentally measured quantities of interest -- strain data at select locations and global load -- with the corresponding results from a simulation library. We present a machine learning based approach to enrich the simulation library and improve the calibration accuracy. This approach is versatile and can be used to calibrate other models of superelastic deformation from data obtained using various modalities. This probabilistic calibration approach can become an integral part of a framework to assess and communicate the risk-informed credibility of simulations performed in the design of superelastic NiTi articles such as medical devices.
In the specific example presented, following six material parameters in the superelastic material model implemented in Abaqus finite element modeling framework are calibrated.
: Austenite stiffness.
: Martensite stiffness.
: Maximum transformation strain.
: Upper plateau stress.
: Lower plateau stress.
: Compression plateau stress.
Matlab code to execute the full calibration pipeline is presented here. For details see the research publication describing the method 👇.
Paranjape, Harshad M., Kenneth I. Aycock, Craig Bonsignore, Jason D. Weaver, Brent A. Craven, and Thomas W. Duerig. 2021. “A Probabilistic Approach with Built-in Uncertainty Quantification for the Calibration of a Superelastic Constitutive Model from Full-Field Strain Data.” Computational Materials Science 192 (May): 110357. https://doi.org/10.1016/j.commatsci.2021.110357.
Below are pre-requisite tools to perform the material parameter calibration described here.
- Matlab with Image Processing Toolbox and Statistics and Machine Learning Toolbox.
- Matlab source code in this repository.
- GWMCMC sampler by Aslak Grinsted. Add this to the Matlab path.
- A diamond specimen with the geometry specified in the engineering drawing
data/NDC-56-03018_rev3_25102018_00.pdf. This specimen should be laser-cut and heat treated from the material to which the model is to be calibrated. - Instron or similar load frame to perform tension experiments on the specimen.
- A 2D digital image correlation (DIC) setup including tools for speckle pattern application on the diamond, digital camera, lighting, and other supporting equipment.
- NCORR DIC data processing software for Matlab.
The purpose of this step is to obtain full-field surface strain data during a tensile test. Experimental quantities of interest (QoI) will be extracted from this data and calibration will be subsequently performed against these QoI.
Cover a diamond specimen with speckle pattern as shown in the figure. Mount the speckled specimen on an Instron load frame at appropriate temperature and perform a tension test with two steps: Start from zero and load to 0.8 mm, then unload to 0.4 mm. Take images with a DIC setup. Process the DIC images using NCORR. Process images corresponding to those displacement increments at which you will be obtaining simulation output in Step 2 below. Save the results as data/QoI_expt_DIC_strains.mat. Copy the Instron load-displacement file to QoI_expt_load.csv.
Set process_mode = 1; in NitinolBayesCal.m. Specify the input values in the first process_mode block. Comments related to the inputs are provided in the program file.
The goal of this step is to create a library of simulations with material property inputs spanning a reasonable parameter space.
Set process_mode = 2; in NitinolBayesCal.m and run the script. It will create a number of Abaqus simulation input files in directory private/SimLib using the template private/SimLib_ftgdia-0p8_0p4.template. The simulation is with the identical specimen geometry and boundary conditions as in the experiment.
Copy the simulation input files to a suitable location and run them using Abaqus. Post-process each resultant odb file using two post-processing scripts in the SimLib folder: fem-get_s_e.py and fem-get_rf_u.py. This should create several output files with the extension data for each simulation. Copy the data files to SimLib directory. There is no need to copy the odb files.
In this step, the script will batch-process all simulation outputs (i.e., the data files) and store simulated QoIs. These will be later compared with the experimental QoI to perform the material property calibration.
Set process_mode = 3; in NitinolBayesCal.m and run the script.
This step in the analysis will provide the results of a least-squares calibration. An objective function is defined that depends on the experimental QoI and simulated QoI from each simulation in the library. The material parameters in the simulation library that minimize the objective function are reported.
Set process_mode = 4; in NitinolBayesCal.m and run the script. The calibration results will be printed and a plot of objective function vs. each of the material parameter value as shown above will be printed.
The purpose of this step is to perform a Markov Chain Monte Carlo (MCMC) sampling of the posterior probability distribution (i.e., the probability distribution of fitted material parameters) and obtain median, MAP, credible interval values.
Set process_mode = 5; in NitinolBayesCal.m. Make sure that the ranges of the variables E_a, E_m etc. are the same as those in process_mode = 2 block are a subset of the values in process_mode = 2. Run the script. The sampler will show a progress bar and final results will be printed in a few minutes. A corner plot of the marginal posterior distributions of the fitted parameters will also be displayed. While calibrated material parameters can be obtained using this procedure, a more credible values may be obtained by performing the following two steps.
In this step, the Matlab script fits an array of support vector machine models that map the six material parameters to QoIs. Thus, we develop a surrogate model that can be used to efficiently predict the QoIs for any input material parameters (even those that are not present in the simulation library).
Set process_mode = 6; in NitinolBayesCal.m and execute the script. The fitting and optimization procedure will take several minutes.
The goal of this final step is to perform MCMC sampling using the surrogate model fitted the step above.
Set process_mode = 5; in NitinolBayesCal.m and run the script. A progress bar will be displayed and the results will be printed after a few minutes.
This application uses Open Source components. You can find the source code of their open source projects along with license information below. We acknowledge and are grateful to these developers for their contributions to open source.
- The NCORR digital image correlation package.
- GWMCMC sampler implementation in Matlab by Aslak Grinsted.
Copyright 2020 Harshad M. Paranjape
Licensed under the Apache License, Version 2.0 (the "License");
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