# The Shifted Boundary Method and adaptive mesh: Numerical tools for high order embedded computations

In this presentation, two approaches aiming at improving accuracy within immersed/embedded boundary (IB/EB) methods are discussed. We will start with an introduction on IB and EB methods before focusing on a recent approach of EB family in the context of Finite Element (FE) formulations: the Shifted Boundary Method (SBM). The SBM intends to tackle some well-known flaws of EB methods such as the reduced order of accuracy, or a poor algebraic conditioning. This technique relies on two main ingredients: weak Nitsche-type Boundary Conditions (BC), and one-sided Taylor expansions. First, we will provide an improved formulation of the original SBM [1] for elliptic PDEs solved in mixed formulation. The FE spaces can be enriched to obtain a quadratic approximation of the solution, while remaining within a P1 FE method. Such a strategy, based on previous work [2], allows to provide in the embedded context an overall second order accuracy on both the primary solution and its derivatives, and a third order accuracy on the solution when only Dirichlet BC are considered. Secondly, to illustrate the capability of the SBM, even in the context of moving interfaces, free surface flow resolution will be presented.

The second part of the presentation is devoted to the use of mesh adaptation to further improve the solutions. The idea is to reduce the error resulting from the embedding of the BC by adapting the mesh to the immersed geometry. Two techniques are discussed. The first one, usually referred as “h-adaptation”, is a metric based remeshing [3]. The second one, often called “r-adaptation”, is a constant connectivity adaptive deformation of the mesh, being particularly appealing for unsteady simulations for its easy combination with the Arbitrary Lagrangian Eulerian (ALE) framework.

[1] A. Main, G. Scovazzi, The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems. Journal of Computational Physics, 372, 2018.

[2] A. Mazaheri, H. Nishikawa, Improved second-order hyperbolic residual-distribution scheme and its extension to third-order on arbitrary triangular grids. Journal of Computational Physics, 300, 2015.

[3] F. Alauzet, A. Loseille, A decade of progress on anisotropic adaptation for computational fluid dynamics. Comput Aided Design, 72, 2016. 23rd International Meshing Roundtable Issue : Advances in Mesh Generation.