A novel technique for local mesh refinement in the finite element method based on the alternative shape functions

Unlike triangular/tetrahedral elements used in finite element methods for two/three-dimensional problems, local refinement of meshes composed of quadrilateral/hexahedral elements while maintaining compatibility is challenging and often results in severe distortion of elements. A well-known and widely used approach to address this issue is local mesh refinement based on transitional elements with hanging nodes. The key point in this method is enforcing displacement continuity at the transitional element boundaries in the presence of hanging nodes. Transitional elements introduced in the literature employ varied formulations depending on their placement within the mesh and are also constrained by a maximum number of hanging nodes along the element boundary. Therefore, implementing them for a general case is quite complicated.
This paper presents a novel transitional element based on alternative shape functions, which offers a unified formulation for different placements of transitional elements in the mesh and it applies to any number of hanging nodes. Additionally, an analytical proof is provided to demonstrate the continuity and partition of unity properties in the proposed method used in local mesh refinement. Finally, numerical examples in two and three dimensions are simulated to compare the accuracy and convergence of the proposed method against the existing methods in the field.