On Several Methods for Imposing Dirichlet Boundary Conditions in Embedded Grids

Abstract eng:
We analyze several possibilities to prescribe essential boundary conditions in the context of embedded grid methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. We first focus on imposing Dirichlet boundary conditions strongly [1]. As starting variational approach we consider Nitsche’s method, which we revisit. This allows us to propose two options that yield non-symmetric problems but that turn out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition. In the first method proposed, the degrees of freedom are those corresponding to nodes exterior to the computational domain, whereas in the second method the nodes are adjacent to the boundary from the interior. The first approach turns out to be more accurate, and the second more robust. In both cases, though, the approximation of the boundary conditions is of second order in the mesh size when using linear finite elements. Secondly, we consider a weak imposition of boundary conditions. The starting point of the strategy we propose is the method presented in [2], for which we develop a symmetric version. This method imposes Dirichlet boundary conditions weakly but does not require of any user defined stabilization or penalty parameter. In order to do so, a hybrid-formulation which introduces an additional elementwise discontinuous stress field is used. However, this additional stress field is only required in the elements which are cut by the immersed boundary, and since it is discontinuous across inter-element boundaries, it can be condensed prior to solving the resulting system of equations. The method shows optimal order of convergence and satisfies the design condition of not needing additional degrees of freedom in order to impose boundary conditions. Additional terms are required in order to guarantee stability in the convection-diffusion equation and the Stokes problem. The proposed method is then easily extended to the transient Navier-Stokes equations. The proposed methods are applied to the scalar convection-diffusion-reaction equation and to the transient incompressible Navier-Stokes problem. Stabilized finite element methods are used to deal with convection dominated flows and to allow the use of equal velocity-pressure interpolations. Examples are shown to illustrate the good behavior of the approaches described. References [1] R. Codina, J. Baiges, Approximate boundary conditions in immersed boundary methods, International Journal for Numerical Methods in Engineering, 80, 1379-1405,2009. [2] A. Gerstenberger, W.A. Wall, An embedded Dirichlet formulation for 3D continua. International Journal for Numerical Methods in Engineering, 82, 537-563, 2010.

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