Cao Rial, María TeresaMoreno, C.Quintela Estévez, Peregrina2024-03-082024-03-082024Mathematics and Computers in Simulation Volume 218, April 2024, Pages 266-2910378-4754http://hdl.handle.net/10347/33079Models involving interfaces with discontinuities or even singularities of some fields across them are very frequent in real life problems modelling. In the last decades, the use of the eXtended Finite Element Method (XFEM) instead of the traditional FEM has become more and more popular, mainly because of two advantages: the mesh of the domain can be independent of the interface position, therefore avoiding remeshing, and it allows to enrich an area with specific shape functions fitted to the particular properties (singularities, discontinuities) of the expected solution, obtaining more accurate results with less computational efforts. Nevertheless, a critical point of XFEM is its implementation since it varies from one problem to another, due to the different kind (and number) of degrees of freedom on each node. A diligent organization of nodes, degrees of freedom and enrichment functions is fundamental to achieve an efficient implementation. Our aim in this paper is to provide a common reference framework for the implementation of XFEM from a mathematical point of view, providing the readers with a set of tools that will allow them to apply it to any kind of problem. To this aim, we present a detailed description of XFEM implementation, with special emphasis on the terms that involve integration over interfaces. The proposed tools are presented in a general context, and as an example, we will apply them to a problem of solids mechanics. In particular, we will contextualize the procedure on a Rayleigh waves propagation problem in a cracked structure considering a Signorini contact condition on the crack sides.eng© 2023 The Authors. Published by Elsevier B.V. on behalf of International Association for Mathematics and Computers in Simulation (IMACS). This is an open access article under the CC BY-NC-ND licenseAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Extended finite element methodMathematical tools for numerical implementationIntegration over interfacesRayleigh wavesjournal article10.1016/j.matcom.2023.11.012open access