On methods for stabilizing constraints over enriched interfaces in elasticity

Enriched finite element approaches such as the extended finite element method provide a framework for constructing approximations to solutions of non-smooth problems. Internal features, such as boundaries, are represented in such methods by using discontinuous enrichment of the standard finite element basis. Within such frameworks, however, imposition of interface constraints and/or constitutive relations can cause unexpected difficulties, depending upon how relevant fields are interpolated on un-gridded interfaces. This work address the stabilized treatment of constraints in an enriched finite element context. Both the Lagrange multiplier and penalty enforcement of tied constraints for an arbitrary boundary represented in an enriched finite element context can lead to instabilities and artificial oscillations in the traction fields. We demonstrate two alternative variational methods that can be used to enforce the constraints in a stable manner. In a ‘bubble-stabilized approach,’ fine-scale degrees of freedom are added over elements supporting the interface. The variational form can be shown to have a similar form to a second approach we consider, Nitsche’s method, with the exception that the stabilization terms follow directly from the bubble functions. In this work, we examine alternative variational methods for enforcing a tied constraint on an enriched interface in the context of two-dimensional elasticity. We examine several benchmark problems in elasticity, and show that only Nitsche’s method and the bubble-stabilization approach produce stable traction fields over internal boundaries. We also demonstrate a novel difference between the penalty method and Nitsche’s method in that the latter passes the patch test exactly, regardless of the stabilization parameter’s magnitude. Results for more complicated geometries and triple interface junctions are also presented. Copyright 2008 John Wiley & Sons, Ltd.