Shifted boundary method for poisson problems in libMesh

Abstract

The non-conformal finite elements formulations applicable to situations in which the computation of conformal meshes are significantly expensive, up to a point where a problem might be redered unfisible. The embedded finite element method is one approach to diminish the mesh generation burden in finite element analysis. It consists of dealing with a description of a boundary that does not necessarily match the problem’s physical boundary. It can potentially shrink the workflow giving the opportunity of immediately inputting a CAD geometry or tomographic image into a simulation, without necessarily using isogeometric elements or performing substantial preprocessing. This work presents an implementation of the recently proposed embedded formulation for Poisson problems in the general purpose library libMesh. In the formulation, the boundary condition is shifted and enforced weakly by a Nitsche approach, and is referred as Surrogated Boundary. This is accomplished provided the surrogate boundary is close enough to the physical boundary so a Taylor expansion can be used to describe the chopped off region. This approach provides a significant computational relief compared to the alternative of adaptative point integration selection, especially when dealing with complex domains where the total point-locating operations’ cost can be significantly high. The reported convergence rate is also examined.