Embedded solids of any dimension in the X-FEM defined on higher-order approximations

Abstract

Embedded interface methods bring a significant simplification of the modelling process before analysis. Complex geometries and moving boundaries may be described with great flexibility, reducing the meshing step to that of a simple bounding box. Following this idea – to dissociate the field approximation from the geometric description – manifolds of different dimensions may be embedded in the same bulk mesh. However, special attention should be given to the difference of dimensions between that of problem domain and that of bulk mesh (the codimension). Whereas the direct use of the shape functions of the bulk mesh is possible for a problem domain of codimension zero, this approach is no longer possible in other configurations, for instance a beam in a 3D mesh. Unlike approaches introducing independent overlapping meshes for each subdomain, function spaces may be built from the traces of higher dimensional spaces built upon the bulk mesh. For closed curves in 2D and closed surfaces in 3D, the resulting discrete method based on P1 FE have already been studied in the literature. To avoid badly conditioned linear systems, specific treatments are required, e.g. preconditioning approaches or stabilization techniques. Here, we propose to deplete wisely the trace space. We investigate higher-order function spaces to solve the diffusion equation in embedded solids of any codimension. A new space-reducer algorithm is introduced to design the dedicated spaces that avoids ill-conditionning while treating boundary conditions. We present the results of several numerical experiments with convergence analyses. To conclude, applications of this technique to embedded beams or shells is discussed