Embedded solids of any dimension in the X-FEM. Part II - Imposing Dirichlet boundary conditions

- We propose a methodology to impose Dirichlet boundary conditions with non-matching meshes for any embedding.
- A new algorithm is introduced to design dedicated stable Lagrange multiplier spaces on boundaries of arbitrary dimension.
- The accuracy and the stability of the method are investigated with regard to the FEM and the inf-sup condition.
- Comparisons with Nitsche's method and another stable Lagrange multiplier method are performed.
- This work is the first investigation of Dirichlet boundary conditions defined on 1D submanifolds embedded in a 3D non-conforming mesh.

This paper focuses on the design of a stable Lagrange multiplier space dedicated to enforce Dirichlet boundary conditions on embedded boundaries of any dimension. It follows a previous paper in a series of two, on the topic of embedded solids of any dimension within the context of the extended finite element method. While the first paper is devoted to the design of a dedicated P1 function space to solve elliptic equations defined on manifolds of codimension one or two (curves in 2D and surfaces in 3D, or curves in 3D), the general treatment of Dirichlet boundary conditions, in such a setting, remains to be addressed. This is the aim of this second paper. A new algorithm is introduced to build a stable Lagrange multiplier space from the traces of the shape functions defined on the background mesh. It is general enough to cover: (i) boundary value problems investigated in the first paper (with, for instance, Dirichlet boundary conditions defined along a line in a 3D mismatching mesh); but also (ii) those posed on manifolds of codimension zero (a domain embedded in a mesh of the same dimension) and already considered in Béchet et al. (2009) [48]. In both cases, the compatibility between the Lagrange multiplier space and that of the bulk approximation (the dedicated P1 function space used in (i), or classical shape functions used in (ii))-resulting in the inf-sup condition-is investigate through the numerical Chapelle-Bath test. Numerical validations are performed against analytical and finite element solutions on problems involving 1D or 2D boundaries embedded in a 2D or 3D background mesh. Comparisons with Nitsche's method and the stable Lagrange multiplier space proposed in Hautefeuille et al. (2012) [44], when they are feasible, highlight good performance of the approach.