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

Abstract

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 (

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

“…To overcome this limitation, we explore -in a Part II [6] following this paper -a new way of building stable Lagrange multipliers with the objective of addressing all the range of 51 problems with embedded solids.…”

“…The general treatment of boundary conditions is actually a problem in its own right, we will therefore propose some solutions in a subsequent paper [6]. This paper is focused on the definition of an optimal way of representing fields on an embedded manifold of codimension one to two.…”