A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method

By Eric Béchet, Nicolas Moës, Barbara I. Wohlmuth. 2009

184 citations recorded by the source corpus.

This paper introduces a new algorithm to define a stable Lagrange multiplier space to impose stiff interface conditions within the context of the extended finite element method. In contrast to earlier approaches, we do not work with an interior penalty formulation as, e.g. for Nitsche techniques, but impose the constraints weakly in terms of Lagrange multipliers. Roughly speaking a stable and optimal discrete Lagrange multiplier space has to satisfy two criteria: a best approximation property and a uniform inf–sup condition. Owing to the fact that the interface does not match the edges of the mesh, the choice of a good discrete Lagrange multiplier space is not trivial. Here we propose a new algorithm for the local construction of the Lagrange multiplier space and show that a uniform inf–sup condition is satisfied. A counterexample is also presented, i.e. the inf–sup constant depends on the mesh‐size and degenerates as it tends to zero. Numerical results in two‐dimensional confirm the theoretical ones. Copyright © 2008 John Wiley & Sons, Ltd.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/7e2cc8cc3a80852263fe43aa760b38cb85134dde

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In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

Robust imposition of Dirichlet boundary conditions on embedded surfaces (2012, methodology): “The method introduced in [16] thus computes the shape function of the Lagrange multiplier at vital vertex v,N v , as a weighted sum of traces onto the embedded surface, T , of bulk shape functions:”
Stable imposition of stiff constraints in explicit dynamics for embedded finite element methods (2012, methodology): “[15] introduced a more general algorithm to discretize the interfacial mesh based on identifying a minimum set of vertices essential to impose the constraint efficiently while satisfying the LBB condition.”
Fictitious domain finite element methods using cut elements (2012, methodology): “In [11] the authors propose a Lagrange multiplier method with an inconsistent penalty term and in [2] using inf-sup stable finite element spaces.”
Stabilized low-order finite elements for frictional contact with the extended finite element method (2010, methodology): “In [2,31], a stabilized Lagrange space is designed to satisfy the Ladyzhenskaya–Babuska–Brezzi (LBB) condition [1,14], the basic idea being to reduce the number of Lagrange multipliers by certain rules.”