The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

Lam H. Nguyen, Stein K.F. Stoter, Martin Ruess, Manuel A. Sanchez Uribe, Dominik Schillinger

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging-based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase-field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.

Original languageEnglish (US)
Pages (from-to)601-633
Number of pages33
JournalInternational Journal for Numerical Methods in Engineering
Issue number4
StatePublished - Jan 27 2018

Bibliographical note

Funding Information:
National Science Foundation, Grant/Award Number: CISE-1565997; NSF CAREER Award, Grant/Award Number: CAREER-1651577

Funding Information:
Dominik Schillinger and the Minnesota group gratefully acknowledge support from the National Science Foundation through the NSF grant CISE-1565997 and the NSF CAREER AwardCAREER-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 paper ( The authors are grateful to Thomas Baum and Jan S. Kirschke (Dept. of Neuroradiology, Technische Universität München, Germany) for providing access to the medical imaging data of the vertebra and to Zohar Yosibash (Ben-Gurion-University of the Negev, Beer Sheva, Israel) for helpful comments.

Publisher Copyright:
Copyright © 2017 John Wiley & Sons, Ltd.

Copyright 2018 Elsevier B.V., All rights reserved.


  • diffuse domain methods
  • phase-fields
  • the diffuse Nitsche method
  • weak Dirichlet boundary conditions

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