The multiscale finite element method for nonlinear continuum localization problems at full fine-scale fidelity, illustrated through phase-field fracture and plasticity

Lam H. Nguyen, Dominik Schillinger

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

The residual-driven iterative corrector scheme recently presented by the authors for linear problems has opened a pathway to achieve the best possible fine-mesh accuracy in the multiscale finite element method (MsFEM). In this article, we focus on a series of algorithmic and variational extensions that enable efficient residual-driven correction for nonlinear localization problems. These include a synergistic combination of Newton and corrector iterations to reduce the algorithmic complexity, the use of corrector degrees of freedom in the Galerkin projection to eliminate the repeated recomputation of multiscale basis functions during Newton iterations, and a natural residual-based strategy for fully automatic fine-mesh adaptivity. We illustrate through numerical examples from phase-field fracture and plasticity that the MsFEM with residual-driven adaptive correction achieves full fine-scale fidelity while also being computationally more efficient than the pristine MsFEM. We also show that for localization problems, it significantly increases accuracy and robustness over standard oversampling.

Original languageEnglish (US)
Pages (from-to)129-160
Number of pages32
JournalJournal of Computational Physics
Volume396
DOIs
StatePublished - Nov 1 2019

Bibliographical note

Funding Information:
The authors gratefully acknowledge support from the National Science Foundation via the NSF grant CISE-1565997 . The first author (L.H. Nguyen) was partially supported by a Doctoral Dissertation Fellowship ( DDF ) awarded by the University of Minnesota for the academic year 2017-2018. The second author (D. Schillinger) gratefully acknowledges support from the National Science Foundation through the NSF CAREER Award No. 1651577 . The authors also acknowledge the Minnesota Supercomputing Institute (MSI) of the University of Minnesota for providing computing resources that have contributed to the research results reported within this article ( https://www.msi.umn.edu/ ).

Funding Information:
The authors gratefully acknowledge support from the National Science Foundation via the NSF grant CISE-1565997. The first author (L.H. Nguyen) was partially supported by a Doctoral Dissertation Fellowship (DDF) awarded by the University of Minnesota for the academic year 2017-2018. The second author (D. Schillinger) gratefully acknowledges support from the National Science Foundation through the NSF CAREER Award No. 1651577. The authors also acknowledge the Minnesota Supercomputing Institute (MSI) of the University of Minnesota for providing computing resources that have contributed to the research results reported within this article (https://www.msi.umn.edu/).

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

  • Multiscale finite element method
  • Nonlinear localization
  • Phase-field fracture
  • Plasticity
  • Residual-driven iterative correction

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