Title: SO(3)-invariance of informed-graph-based deep neural network for anisotropic elastoplastic materials

Abstract

This work examines the frame-invariance (and the lack thereof) exhibited in simulated anisotropic elasto-plastic responses generated from supervised machine learning of classical multi-layer and informed-graph-based neural networks, and proposes different remedies to fix this drawback. The inherent hierarchical relations among physical quantities and state variables in an elasto-plasticity model are first represented as informed, directed graphs, where three variations of the graph are tested. While feed-forward neural networks are used to train path-independent constitutive relations (e.g., elasticity), recurrent neural networks are used to replicate responses that depends on the deformation history, i.e. or path dependent. In dealing with the objectivity deficiency, we use the spectral form to represent tensors and, subsequently, three metrics, the Euclidean distance between the Euler Angles, the distance from the identity matrix, and geodesic on the unit sphere in Lie algebra, can be employed to constitute objective functions for the supervised machine learning. In this, the aim is to minimize the measured distance between the true and the predicted 3D rotation entities. Following this, we conduct numerical experiments on how these metrics, which are theoretically equivalent, may lead to differences in the efficiency of the supervised machine learning as well as the accuracy and robustness ofmore » the resultant models. Neural network models trained with tensors represented in component form for a given Cartesian coordinate system are used as a benchmark. Our numerical tests show that, even given the same amount of information and data, the quality of the anisotropic elasto-plasticity model is highly sensitive to the way tensors are represented and measured. The results reveal that using a loss function based on geodesic on the unit sphere in Lie algebra together with an informed, directed graph yield significantly more accurate rotation prediction than the other tested approaches.« less

Authors:

Heider, Yousef ^[1]; Wang, Kun ^[2]; Sun, WaiChing  ^[2]

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+ Show Author Affiliations

1. Columbia Univ., New York, NY (United States); RWTH Aachen Univ. (Germany)
2. Columbia Univ., New York, NY (United States)

Publication Date:

Thu Feb 13 00:00:00 EST 2020

Research Org.:

Columbia Univ., New York, NY (United States)

Sponsoring Org.:

USDOE Office of Nuclear Energy (NE); National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR)

OSTI Identifier:

1801241

Alternate Identifier(s):

OSTI ID: 1599468

Grant/Contract Number:  

NE0008534; CMMI-1846875; FA9550-17-1-0169; FA9550-19-1-0318

Resource Type:

Accepted Manuscript

Journal Name:

Computer Methods in Applied Mechanics and Engineering

Additional Journal Information:

Journal Volume: 363; Journal Issue: C; Journal ID: ISSN 0045-7825

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

42 ENGINEERING; engineering; mathematics; mechanics; recurrent neural networks; lie algebra; anisotropic materials; machine learning; crystal plasticity; special orthogonal group