Session: 12-03-01: Minisymposium on Peridynamic Modeling of Materials’ Behavior

Paper Number: 77494

Start Time: Monday, 12:15 PM

77494 - Data-Driven Learning of Nonlocal Models: From High-Fidelity Simulations to Constitutive Laws 

Nonlocal models use integral operators acting on a lengthscale, known as horizon. This feature allows nonlocal models to capture long-range forces at small scales and multiscale behavior, and to reduce regularity requirements on the solutions, which are allowed to be discontinuous or even singular. In recent decades, nonlocal equations have been successfully used to model several engineering and scientific applications, including fracture mechanics, subsurface transport, image processing, multiscale and multiphysics systems, finance, and stochastic processes.
However, it is often the case that nonlocal kernels defining nonlocal operators are justified a posteriori and it is not clear how to define such kernels to faithfully describe a physical system. The problem of learning an appropriate kernel for a specific application is one of the most challenging open problems in nonlocal modeling.
The literature on techniques for learning kernel parameters for a given functional form is vast and it is based on either control-based approaches or machine-learning approaches. However, the use of machine learning to learn the functional form of the kernel is still in its infancy.
In this work we show that machine learning can improve the accuracy of simulations of stress waves in one-dimensional composite materials.
In this work on nonlocal modeling of wave propagation through heterogeneous materials. By using an operator regression technique, we learn nonlocal kernels whose associated nonlocal wave equation is well posed by construction and can be used as an accurate surrogate for more detailed, high-fidelity wave propagation models. In particular, we present an application to wave propagation at the microscale in a heterogeneous solid. In this context, the machine-learned nonlocal kernel embeds the material constitutive behavior so that the material interfaces do not have to be treated explicitly and, more importantly, the material microstructure can be unknown. Furthermore, the corresponding nonlocal models allow for accurate simulations at scales that are much larger than the microstructure.
The method is an optimization-based technique in which the nonlocal kernel function is approximated via Bernstein polynomials. The kernel, including both its functional form and parameters, is derived so that when used in a nonlocal solver, it generates solutions that closely match high-fidelity data. The optimal kernel therefore acts as a homogenized nonlocal continuum model that accurately reproduces wave motion in a smaller-scale, more detailed model that can include multiple materials. Several one-dimensional numerical tests illustrate the accuracy of our algorithm. The optimal kernel is demonstrated to reproduce high-fidelity data for a composite material in applications that are substantially different from the problems used as training data.

Presenting Author: Marta D'Elia Sandia National Laboratories

Authors:

Marta D'Elia Sandia National Laboratories
Yue Yu Lehigh University
Huaiqian You Lehigh Univeristy
Stewart Silling Sandia National Laboratories