Coupled <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo linebreak="goodbreak">−</mml:mo> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> mesh adaptation in embedded finite element framework

• An efficient r -method is developed to align the mesh on the domain boundary. • An h -method strategy refines the mesh in critical regions of the computational domain. • The r - and h -methods are coupled to make use of the advantages of both techniques. • The coupled r − h method is integrated with an embedded finite element solver. • The proposed method improves the accuracy of embedded strategies in complex domains. Among mesh-based methods, embedded methods are widely used in the literature for solving partial differential equationss (PDEss) in complex domains without the challenge of creating body-fitted (or, more generally, domain-fitted) meshes. In these methods, the physical domain is embedded in a simple background domain that can be easily meshed. However, the uniform accuracy of the background mesh in these methods is often inadequate in critical regions, such as boundary layers and vortex zones. While h -refinement methods for mesh adaptation are commonly employed to enhance resolution in regions with complex local phenomena like vortices, they have limitations in areas with steep gradients in one direction but little to no gradients in the other(s), such as boundary layers. These limitations include increased computational costs due to the lack of mesh alignment strategies. Mesh alignment can be achieved through r -methods, which deform the mesh without adding new elements, avoiding overhead computational costs and mesh repartitioning. However, while r -methods improve mesh alignment in boundary layers, they can lead to insufficient element density in other areas where local phenomena, such as vortices, occur. In this study, we introduce a variational moving adaptive mesh method (VMAM), a subset of r -methods, combined with h -refinement ( h -method), designed to operate on distributed memory parallel systems. This r − h method, coupled with embedded finite element method (EFEM), addresses the challenge of inadequate resolution in key regions. The r -method aligns the mesh in boundary layers, while the h -method refines it in local critical areas, such as regions where vortices occur. The alignment in VMAM is achieved through a mapping strategy that transfers the simple computational domain to the physical domain. This mapping, which is challenging in domain-fitted approaches, is applied easily in the embedded method due to the simplicity of the background domain. In the present method, a level-set approach is used to define the physical domain. For simple cases, this is determined using analytical expressions, whereas in more complex cases, explicit boundary representations from computer aided design (CAD) are employed. The method is examined on several two-dimensional (2D) and three-dimensional (3D) cases, demonstrating its effectiveness and improvements in the simulation results compared to those obtained by non-adaptive EFEM on Cartesian grids.