PeriFlakes

Alejandro Francisco Queiruga
Lawrence Berkeley National Lab
2015-2018

Introduction

PeriFlakes is an open source solver of state-based Peridynamics. It is built off of cornflakes/popcorn and can autogenerate every possible formulation found in the literature, and then some more. Notably, the system uses symbolic differentiation to obtain the expressions for tangent matrices to use implicit time steppers or solve static equilibrium. This is an updated version of the code used to perform the study in

Queiruga, A. F. and G. J. Moridis, "Numerical experiments on the convergence properties of state-based peridynamic laws and influence functions in two-dimensional problems." Computer Methods in Applied Mechanics and Engineering 322 (2017): 97-122.

in which the algorithms and formulations are described in detail.

Running

The easiest way to run this code yourself is to pull the cornflakes image:

docker pull afqu/cornflakes

and then find this code in /opt_cornflakes/PeriFlakes. The Docker image is not automatically rebuilt with pushes to PeriFlakes, so do a git pull to get the latest version.

Datasets

The databases that the scripts will generate are published on Zenodo at:

Alejandro Francisco Queiruga. (2018). 2D Peridynamic Displacement Fields [Data set]. Zenodo. http://doi.org/10.5281/zenodo.1284634

They are simple sqlite3 databases, and the Jupyter notebook analysis.ipynb illustrates their structure and how to query them.

Conclusions

The purpose of this repository is to illustrate the limitations of the Peridynamics family of numerical methods. The popcorn file PeriFlakes/peri_kernels_pop.py performs extensive Ahead-Of-Time code generation to allow us to exhaustively search the hyperparameter space of possible Peridynamics programs. The simulation object was only designed to quickly graph the unit square domain and sweep through different configurations of the Peridynamics approximation.

The analysis of the results can be viewed and run in the Jupyter notebook analysis.ipynb. The fundamental limitation is the inability of the most widely used state-based to represent all basic modes of deformation of mechanics. The derfomation-gradient based correspondence model is able to converge linearly to all explored problems with certain hyperparameter choices, and its instability can be dealt with. The recommendation for using perdiynamics is thus:

Use the deformation gradient based correspondence model of Silling,2007 using either the stabilizer of Silling,2017 or the post-smoothing technique of Queiruga, 2017.
Use a cubic influence function with as small of an influence radius as possible, no shorter than $1.5\times$ the grid size.

Some arguments on the computational complexity of Peridynamics are made in the Jupyter notebook computation.ipynb.

The ficticious-nodes boundary correction method is explored in ficticious_nodes.ipynb.

Acknowledgements

This work was supported primarily by Laboratory Directed Research and Development (LDRD) funding from Berkeley Lab, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

License

PeriFlakes is released under version 3 of the GNU General Public License, as per LICENSE.txt.