Abstract :
[en] 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. 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, when they are feasible, highlight good performance of the approach.
Commentary :
• We propose a methodology to impose Dirichlet boundary conditions for any embedding, i.e. 1D, 2D and 3D problems embedded in 2D or 3D (non-matching) background meshes.
• A new stable Lagrange multiplier approach is presented to enforce Dirichlet boundary conditions involved in boundary value problems defined on embedded surfaces.
• A single algorithm is introduced and allows to design dedicated Lagrange multiplier spaces on boundaries of arbitrary dimension.
• Convergence analyses showed the accuracy, the optimality, and the stability of the method, with regard to the FEM and the inf–sup condition, in all configurations of boundary dimension.
• Comparisons with Nitsche’s method and another stable Lagrange multiplier method are performed wherever possible.
• This work is the first investigation of Dirichlet boundary conditions defined on submanifolds of codimension two in a 3D non-conforming mesh.
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