Example for Arruda Boyce benchmark models
In the following, we illustrate the use of the Bayesian-EUCLID to discover the model of a known benchmark material (Arruda-Boyce model) using displacement field data obtained from FEM simulations.
Input format
The benchmark FEM data folders- dyn-euclid-master-data (dynamic data) and euclid-master-data (quasistatic data) are structured as shown in the figure below
Additional benchmark materials can be added in the format shown above. The only difference between dynamic and quasistatic data is that the former also includes nodal acceleration data in the output_nodes.csv file.
Parameters and code execution
The figure below shows a snippet of the config.py file, which contains the initialized parameters.
lambda_r is the regularization parameter that is used in process_raw_data(...) function in unsupervised_hyperelasticity.py file.
filter_value is the number of degrees of freedom subsampled from the available data at all quadrature points. In this case, at total of 100 degrees of freedom are subsampled from a total of 2752 data points.
a_v, b_v, a_p, b_p, a_sigma and b_sigma are the hierarchical spike slab hyperparameters shown in Fig. 2 in the Bayesian-EUCLID paper.
chain_length is the length of each individual Markov chain used to sample the posterior probability distribution. In this case a chain length of 1000 was used.
burn is the number of initial elements of each formed chain that are discarded. In this case the first 250 elements of each chain were discarded.
parallel_chains is the total number of chains used to sample the posterior probability distribution. A total of 4 chains were used in this work, which were later merged into a single chain of length 3000 elements, after the first 250 elements of each chain was discarded.
theta_fiber and theta_fiber2 are the assumed orientation angles of the two fibers considered in the feature library. Two fibers suffice for the Holzapfel benchmark considered in this work. Also, as per the work by Dong and Sun, (2021), any symmetrical distribution of arbitrary number fibers can be equivalently resolved into a two-fiber system along the directions of symmetry. Appendix C of the the Bayesian-EUCLID paper discusses the effect of assuming incorrect directions (\(\pm45^{\circ}\)) for the fibers.
The figures below show snippets of code execution in Anaconda Powershell Prompt. The main_hss.py file is run with the shown arguments. In the first case, the quasistatic Arruda-Boyce benchmark data is selected, while in the second the dynamic Arruda-Boyce benchmark data is selected.
The figure below shows a snippet of code progress as the chains are constructed and the energy plots are made.
Code output
The main output of the code is the chain object whose theta attribute contains a distribution of the discovered feature coefficients. These discovered feature coefficients are used to make the violin plots and energy evolution plots as shown below.
