Embedded solids of any dimension in the X-FEM. Part I - Building a dedicated P1 function space

This paper focuses on the design of a dedicated P1 function space to model elliptic boundary value problem on a manifold embedded in a space of higher dimension. Using the traces of the linear P1 shape functions, it introduces an algorithm to reduce the function space into an equivalent space having the same properties than a P1 Lagrange approximation. Convergence studies involving problems of codimension one or two embedded in 2D or 3D show good accuracy with regard to classical finite element and analytical solutions. The effects of the relative position of the domain with respect to the mesh are studied in a sensitivity analysis; it illustrates how the proposed solution allows to keep the condition number bounded. A comparative study is performed with the method introduced by Olshanskii et al. (2009) [35] on a closed surface to validate our approach. The robustness of the proposed approach is investigated with regard to their method and that of Burman et al. (2016) [42]. This paper is the first in a series of two, on the topic of embedded solids of any dimension within the context of the extended finite element method. It investigates problems involving borderless domains or domains with boundary subject to Dirichlet constraint defined only on the boundaries of the bulk mesh, while the forthcoming paper overcomes this limitation by introducing a new stable Lagrange multiplier space for Dirichlet boundary condition (and more generally stiff condition), that is valid for every combination of the background mesh and manifold dimensions. The combination of both algorithms allows to handle any embedding, i.e., 1D, 2D and 3D problems embedded in 2D or 3D background meshes.