Imposing Dirichlet boundary conditions in the extended finite element method

By Nicolas Moës, Eric Béchet, Matthieu Tourbier. 2006

137 citations recorded by the source corpus.

This paper is devoted to the imposition of Dirichlet‐type conditions within the extended finite element method (X‐FEM). This method allows one to easily model surfaces of discontinuity or domain boundaries on a mesh not necessarily conforming to these surfaces. Imposing Neumann boundary conditions on boundaries running through the elements is straightforward and does preserve the optimal rate of convergence of the background mesh (observed numerically in earlier papers). On the contrary, much less work has been devoted to Dirichlet boundary conditions for the X‐FEM (or the limiting case of stiff boundary conditions). In this paper, we introduce a strategy to impose Dirichlet boundary conditions while preserving the optimal rate of convergence. The key aspect is the construction of the correct Lagrange multiplier space on the boundary. As an application, we suggest to use this new approach to impose precisely zero pressure on the moving resin front in resin transfer moulding (RTM) process while avoiding remeshing. The case of inner conditions is also discussed as well as two important practical cases: material interfaces and phase‐transformation front capturing. Copyright © 2006 John Wiley & Sons, Ltd.

Subfields: Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/0844b99c7e4bd4137e08e4311c81808c9cfc1bf3

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In Computational Mechanics

In Mechanics of Materials

Citations in the corpus

A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method (2009, methodology): “One natural possibility is to use standard hat functions associated with the one-dimensional mesh on Γ as in (Moës, Béchet, and Tourbier 2006).”
An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction (2008, background): “A recent discussion of applying Dirichlet boundary conditions in the context of XFEM can be found in [31].”
The extended finite element method in thermoelastic fracture mechanics (2008, methodology): “Two papers dealing with the imposition of boundary conditions on the primary field (the displacement in their case) are worth mentioning: Reference [24] uses a penalty method in the frame of the finite element method with discontinuities, Reference [ 25 ] uses a method with Lagrangian multipliers an”
Implicit boundary method for finite element analysis using non‐conforming mesh or grid (2008, methodology): “[20] have extended the XFEM for imposition of essential boundary conditions using carefully selected Lagrange multiplier space on the boundaries.”
Strict and effective bounds in goal‐oriented error estimation applied to fracture mechanics problems solved with XFEM (2010, methodology): “Construction of admissible fields in the XFEM framework It was shown in [43] that when the XFEM approach is used to obtain an approximate solution of the reference problem it is still possible to get an approximate displacement field uh which verifies the prescribed Dirichlet boundary conditions.”