Anisotropic variational mesh adaptation for embedded finite element methods

Embedded or immersed boundary methods (IBM) are powerful mesh-based techniques that permit to solve partial differential equations (PDEs) in complex geometries circumventing the need of generating a mesh that fits the domain boundary, which is indeed very difficult and has been the main bottleneck of the simulation pipeline for decades. Embedded methods exploit a simple background mesh that covers the domain and the difficulties are (1) the imposition of boundary conditions, (2) the ill-conditioning generated by poorly intersected elements and (3) the lack of resolution required in boundary layers. Whereas several methods are available in the literature to address the first two difficulties, the third one still deserves attention. Meshless methods, Chimera grids or adaptive h or p-refinement strategies have been proposed but none of them include alignment techniques. In this work we introduce an adaptive refinement/alignment strategy for the embedded finite element method (FEM). This is done exploiting the variational moving-adaptive mesh method (VMAM), a well-known technique in mesh generation, in which a mapping from a simple computational domain to the physical one is obtained by minimizing a functional. This method is very difficult to apply to complex geometries when used to generate body fitted meshes because the requirement of mapping the boundary of computational domain to the boundary of the physical one is simply too severe and it fails to converge or provides a nonbijective map. Applying this technique combined with the embedded FEM the computational domain is mapped to a bounding box of the physical domain and therefore the mapping of the boundaries is trivial. The resulting mesh is aligned to and refined near the physical boundary through the definition of the variational problem that determines the mapping. To define the domain boundary within the embedded mesh strategy, we employ an analytical level set function for simple geometries and an explicit boundary representation created by computer aided design (CAD) software. In both cases, the distance and the normal to the boundary are computed to define the metric or monitor matrix used in the VMAM, which permits to control the refinement and alignment of the mesh. Additionally, a stabilization strategy to overcome oscillations caused by poorly cut elements based on a ghost penalty approach is introduced. We test our method through several two-dimensional (2D) and three-dimensional (3D) examples, including complex geometries, demonstrating significant improvements in accuracy with respect to the standard embedded FEM on Cartesian meshes.