Using Neural Networks to Obtain the Kernel for Peridynamic Thermal Diffusion Models

Using Neural Networks to Obtain the Kernel for Peridynamic Thermal Diffusion Models
Longzhen Wang a b, Jiangming Zhao a b, Florin Bobaru a
a University of Nebraska-Lincoln, Lincoln, NE 68588, United States
* Email of Corresponding Author: fbobaru2@unl.edu

In peridynamics, the ‘kernel’ in the peridynamic equation determines the nonlocal interaction between points inside the horizon region. Selecting the “right” kernel is important because its influence on results and converge behavior under specific discretization [1][2]. In previous studies [3-6], the kernels are obtained by simple fitting/matching the heat-flux under constant thermal/concentration gradient conditions (for thermal/mass diffusion, for example). The particular “shape” for the kernel functions can be assumed to be, for example, constant, linearly decreasing towards the edge of the horizon region, Gaussian, etc. It has been noticed that the discretized version of the peridynamic diffusion equation [4] shares a similar structure to the mathematical backbone of convolutional neural networks (CNNs). In this presentation, we use a convolutional neural network to train the PD kernel from data (temperature/concentration). For a 1D transient heat transfer problem, we trained a one-layer CNNs using TensorFlow based on the initial temperature and the temperature at 0.1s. Using the resulting kernel, we accurately predicted the temperature at the next step (t = 0.2 s). The error compared with the analytical solution for the classical problem was around 10^(-12) in the L2-norm. We also discuss future steps in utilizing neural networks in peridynamic modeling of other engineering problems.

References
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[5]    Z. Chen and F. Bobaru, “Peridynamic modeling of pitting corrosion damage,” J. Mech. Phys. Solids, vol. 78, pp. 352–381, 2015.
[6]    J. Zhao, Z. Chen, J. Mehrmashhadi, and F. Bobaru, “Construction of a peridynamic model for transient advection-diffusion problems,” Int. J. Heat Mass Transf., vol. 126, pp. 1253–1266, 2018.