Published December 1, 2023 | Published
Journal Article

Data-Driven games in computational mechanics

  • 1. ROR icon University of Siegen
  • 2. ROR icon Institut de Recherche en Génie Civil et Mécanique
  • 3. ROR icon University of Bonn
  • 4. ROR icon California Institute of Technology

Abstract

We resort to game theory in order to formulate Data-Driven methods for solid mechanics in which stress and strain players pursue different objectives. The objective of the stress player is to minimize the discrepancy to a material data set, whereas the objective of the strain player is to ensure the admissibility of the mechanical state, in the sense of compatibility and equilibrium. We show that, unlike the cooperative Data-Driven games proposed in the past, the new non-cooperative Data-Driven games identify an effective material law from the data and reduce to conventional displacement boundary-value problems, which facilitates their practical implementation. However, unlike supervised machine learning methods, the proposed non-cooperative Data-Driven games are unsupervised, ansatz–free and parameter–free. In particular, the effective material law is learned from the data directly, without recourse to regression to a parameterized class of functions such as neural networks. We present analysis that elucidates sufficient conditions for convergence of the Data-Driven solutions with respect to the data. We also present selected examples of implementation and application that demonstrate the range and versatility of the approach.

Copyright and License

© 2023 Elsevier B.V. All rights reserved.

Acknowledgement

KW gratefully acknowledges support from the DFG through project WE2525/14-1 as part of the priority program SPP 2256 (no. 422730790). MO gratefully acknowledges additional support from the DFG through project 390685813–GZ 2047/1–HCM. LS gratefully acknowledges the financial support of the French Agence Nationale de la Recherche (ANR) through project ANR-19-CE46-0012 within the French-German Collaboration for Joint Projects in Natural, Life and Engineering (NLE) Sciences , as well as NExT ISite program of Nantes Université through International Research Project (IRP) iDDrEAM.

Conflict of Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.
We understand that the Corresponding Author Kerstin Weinberg is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). She is responsible for communicating with the other authors about progress submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author
Kerstin Weinberg reports financial support was provided by German Research Foundation. Laurent Stainier reports support from the French Agence Nationale de la Recherche (ANR). Michael Ortiz reports financial support was provided by German Research Foundation (DFG).

Data Availability

No data was used for the research described in the article.

Additional details

Created:
December 17, 2024
Modified:
December 17, 2024