Search

SUMMARY A new technique for treating the mechanical interactions of overlapping finite element meshes is presented. Such methods can be useful for numerous applications, for example, fluid–solid interaction with a superposed meshed solid on an Eulerian background fluid grid. In this work, we consider the interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background mesh and is often a structured grid. Many of the previously proposed methods employ surface defined Lagrange multipliers or penalties to enforce the boundary constraints. It has become apparent that these methods will cause mesh locking under certain conditions. Appropriately applied, the Nitsche method can overcome this locking, but, in its canonical form, is generally not applicable to non-linear materials such as hyperelastics. The relationship between interior point penalty, discontinuous Galerkin and Nitsche's method is well known. Based on this relationship, a nonlinear theory analogous to the Nitsche method is proposed to treat nonlinear materials in an embedded mesh. Here, a discontinuous Galerkin derivative based on a lifting of the interface surface integrals provides a consistent treatment for non-linear materials and demonstrates good behavior in example problems. Published 2012. This article is a US Government work and is in the public domain in the USA.

The past 10–15 years have seen important extensions of the mortar method, a technique for joining dissimilar grids popularized by the domain decomposition community, to the more general problem of contact and impact interactions in finite element analysis. This development has taken place largely in response to several long-standing problems in computational contact mechanics: lack of robustness in solution of the nonlinear and nonsmooth equations of evolution; degradation of spatial convergence rates in problems involving nonconforming meshes on interfaces; lack of a variationally consistent technique for stress recovery on interfaces; and so on. This survey paper summarizes some of the major steps in development of mortar contact formulations. It begins with a basic summary of the mortaring idea in the context of tied contact, it discusses key concepts required for the extension of these methods to large deformation, large sliding formulations of contact-impact, and it previews new results where lessons learned from mortar contact formulations can be extended to a much broader class of interface mechanics applications, considering in particular enriched interface formulations and embedded interface approaches to fluid–structure interaction.

A new technique for treating the mechanical interactions of overlapping finite element meshes is proposed. Numerous names have been applied to related approaches, here we refer to such techniques as embedded mesh methods. Such methods are useful for numerous applications e.g., fluid-solid interaction with a superposed meshed solid on an Eulerian background fluid grid or solid-solid interaction with a superposed meshed particle on a matrix background mesh etc. In this work we consider the interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background mesh and is often a grid. We first employ a classical mortar type approach [see Baaijens (Int J Numer Methods Eng 35:743---761, 2001)] to impose constraints on the interface. It turns out that this approach will work well except in special cases. In fact, many related approaches can exhibit mesh locking under certain conditions. This motivates the proposed version of Nitsche's method which is shown to eliminate the locking phenomenon in example problems.

SUMMARY 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.

The past decade or so has shown the mortar method to be a helpful foundation for formulation of new methods for contact/impact analysis. As originally formulated, the advantage of the mortar method is that it preserves inf-sup conditions associated with interfacial constraints, such that stability and convergence are more or less guaranteed, at least when some regularity of the solution is expected. In more recent work, interest has been shown by a number of researchers in extending many of these ideas to treatment of interfaces in extended finite element frameworks, and also to embedded surface techniques. In these cases, use of the mortar method with naive multiplier space choices can readily lead to instability. This paper highlights some of these issues, and contemplates stabilization methods which seem to be effective in such settings.

This paper serves as an investigation into the response of the NASA space shuttle wing in the event of a debris impact during launch. The explicit finite element method is used to model the leading edge of the shuttle wing and the event of impact with a piece of a material from the shuttle's external tank. The results will be utilised by the Impact Penetration Sensor System (IPSS) to help characterise the signal received by the system in the event of an actual impact. Numerical results are validated by experiments done at the Southwest Research Institute (SwRI).