Abstract :
[en] 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 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. 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.
Commentary :
Highlights:
• We propose an approach to solve 1D, 2D and 3D elliptic problems embedded in 2D or
3D non-conforming meshes within a single framework, neither modifying the variational
formulation nor requiring a dedicated solver.
• A priori well-adapted function spaces are designed on manifolds of codimension one or
two embedded in 2D or 3D background meshes.
• A new algorithm is developed to reduce linear P1 shape functions into a suitable space.
• The optimality and the efficiency of the method is shown through convergence analyses
covering all codimension configurations, using the classical FEM as reference.
• Specific treatments are investigated in a sensitivity analysis to take care of the condi-
tioning of the linear system.
• A comparative study is performed with the method introduced by Olshanskii et al. 2009
on a closed surface to validate our approach.
• The robustness of the method is shown through detailed eigenvalue analyses with
regard to the method of Olshanskii et al. 2009 and that of Burman et al. 2016.
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