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Abstract: The eXtended Finite Element Method (X-FEM) has been successfully used in two-phase flow problems involving a moving interface. In order to simulate problems involving more than two phases, the X-FEM has to be further eXtended. The proposed approach is presented in the case of a quasi-static Stokes n-phase flow and it is based on using an ordered collection of level set functions to describe the location of the phases. A level set hierarchy allows describing triple junctions avoiding overlapping or “voids” between materials. Moreover, an enriched solution accounting for several simultaneous phases inside one element is proposed. The interpolation functions corresponding to the enriched degrees of freedom require redefining the associated ridge function accounting for all the level sets. The computational implementation of this scheme involves calculating integrals in elements having several materials inside. An adaptive quadrature accounting for the interfaces locations is proposed to accurately compute these integrals. Examples of the hierarchical X-FEM approach are given for a n-phase Stokes problem in 2 and 3 dimensions. [Copyright &y& Elsevier]

Abstract: This paper presents a general mass lumping technique for explicit dynamics simulations using the eXtended Finite Element Method with arbitrary enrichment functions. The proposed mass lumping technique is a generalization of previously published results for cracks and holes. Time step estimates are studied for crack singular enrichment functions and for hole enrichment. In both cases, we show that the critical time step does not tend to zero and is of the same order as that of the same unenriched element. The performance of the method is demonstrated on several numerical examples that compare well with classical finite elements and previously published X-FEM results. [Copyright &y& Elsevier]

Abstract: This paper presents a general explicit time integration scheme for dynamics simulations using the eXtended Finite Element Method with standard critical time step. We use the generalized mass lumping technique proposed in Part I of this paper. This technique allows us to consider arbitrary enrichment functions in the X-FEM for explicit dynamics simulations. In this second part, the proposed approach allows the use of standard finite elements critical time step estimates. For that purpose, we develop a classical element-by-element strategy that couples the standard central difference scheme with the unconditionally stable-explicit scheme proposed by Chang [S.Y. Chang, An explicit method with improved stability property, Int. J. Numer. Methods Engrg. 77 (8) (2008) 1100–1120]. This scheme coupled with X-FEM allows us to recover FE critical time steps independently of the enrichment functions considered. Furthermore, a study of the stability property of this new explicit scheme is proposed in the X-FEM framework, both with fixed enrichments and evolving enrichments with time. Some examples illustrate the good properties of the new explicit numerical time scheme and some applications to dynamics crack growth are given. [Copyright &y& Elsevier]