The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Dominik Schillinger; Isaac Harari; Ming Chen Hsu; David Kamensky; Stein K.F. Stoter; Yue Yu; Ying Zhao

doi:10.1016/j.cma.2016.06.026

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Dominik Schillinger, Isaac Harari, Ming Chen Hsu, David Kamensky, Stein K.F. Stoter, Yue Yu, Ying Zhao

Civil, Environmental, and Geo- Engineering

Research output: Contribution to journal › Article › peer-review

71 Scopus citations

Abstract

We explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L² error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest.

Original language	English (US)
Pages (from-to)	625-652
Number of pages	28
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	309
DOIs	https://doi.org/10.1016/j.cma.2016.06.026
State	Published - Sep 1 2016

Bibliographical note

Funding Information:
This study is based on discussions that took place during the workshop “Advanced Computational Methods for Design, Analysis and Optimization” at Iowa State University in December 2015. We thank the organizers Ming-Chen Hsu and Adarsh Krishnamurthy and acknowledge the support of the Department of Mechanical Engineering at Iowa State University . The first author gratefully acknowledges support from the National Science Foundation ( ACI-1565997 ). The second author was supported by the Israel Science Foundation (Grant No. 1008/13 ) and the Diane and Arthur B. Belfer Chair in Mechanics and Biomechanics. The last author was supported by the “Excellence Initiative” of the German Federal and State Governments and the Graduate School of Computational Engineering at the Technical University of Darmstadt . We 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 ( https://www.msi.umn.edu/ ).

Publisher Copyright:
© 2016 Elsevier B.V.

Keywords

Immersed finite element methods
Non-symmetric Nitsche method
Weak boundary and coupling conditions

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title = "The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements",

abstract = "We explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L2 error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest.",

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AU - Stoter, Stein K.F.

AU - Yu, Yue

AU - Zhao, Ying

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N2 - We explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L2 error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest.

AB - We explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L2 error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest.

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The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Abstract

Bibliographical note

Keywords

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