Session: 12-16-03: Drucker Medal Symposium

Paper Number: 99654

99654 - An Assessment of Neural Nets as a Discretization for Large-Deformation Elastoplasticity 

Data-driven approaches in applied mechanics and materials science applications have received significant attention in recent years. The communities are trying to bring data science methods, such as neural networks, to bear on some of key challenges, including the design of better materials and architected structures. Neural net approaches were also deployed as part of the so-called Physics Informed Neural Networks (PINNs) framework recently developed to bring more physics into predictions. PINNs accomplish this by defining an objective function that simultaneously minimizes the errors in the observed data, boundary conditions, and some form of the energy or PDE residual governing the problem at hand. A distinguishing feature of PINNs is that the dicretization of a PDE does not make use of traditional FEM, but rather the neural net itself. As a result, the accuracy of the discrete solution relies on that of the approximation power of the neural net, which is not currently very well understood. However, it is becoming clear that the choices made in the design of the neural nets (number of layers, type of activation functions) profoundly affect their approximation power. In addition, it appears that using a mixed form of an elliptic PDE results in a more efficient and accurate formulation, which is usually not the case in traditional FEM for solid mechanics.

In the present contribution we focus on the ability of the neural nets to model large-deformation elastoplastic behavior. We start with the appropriate formulation of the neural net approach to discretize the governing equation of large-deformation elastoplasticity on arbitrary domains and identify some key ingredients that maximize the accuracy and efficiency of the proposed formulation. We assess the neural-net-based solution accuracy and convergence under mesh refinement. We focus on some key error indicators, namely, global and local norms of the displacement, stress, and plastic strain errors, and compare the results with the traditional low-order FEM. The comparison will focus on the robustness of the neural net approaches (i.e., the ability of neural nets to generate discrete solutions under all input conditions and without significant intrusion from the analyst), their accuracy (i.e., how quickly the errors converge under mesh refinement), and the timings involved in generating the discrete solutions. We also provide further guidance on making neural-network-based methods more competitive with traditional modeling and discretization methods.

This presentation is dedicated to Prof. Horacio Espinosa on the occasion of him receiving the Drucker Medal of the ASME.

Presenting Author: Yuri Bazilevs Brown University

Presenting Author Biography: Yuri Bazilevs is the E. Paul Sorensen Professor of the Mechanics of Solids and Structures in the School of Engineering at Brown University. He was previously a Professor and Vice Chair in the Structural Engineering Department at the University of California, San Diego. Yuri is the original developer of Isogeometric Analysis (IGA), a new computational methodology that aims to integrate engineering design (CAD) and simulation (FEM). For his research contributions Yuri received a number of awards and honors, including the 2018 Walter E. Huber Research Prize from the ASCE, the 2020 Gustus L. Larson Award from the ASME, and the inaugural 2021 Centennial Mid-Career Award from the Materials Division of the ASME. He is included in the lists of Highly Cited Researchers, both in the Engineering (2015-2018) and Computer Science (2014-2019) categories. Yuri currently serves as the President of the US Association for Computational Mechanics (USACM) and as a member of the US National Committee for Theoretical and Applied Mechanics (USNCTAM). He recently completed his service as the Chairman of the Applied Mechanics Division of the ASME .

Authors:

Sijun Niu Brown University
Enrui Zhang Brown University
Vikas Srivastava Brown University
Yuri Bazilevs Brown University