Construction of H1-conforming hierarchical shape functions for elements of all shapes and transfinite interpolation

The paper reviews the construction of hierarchical H1-conforming shape functions for 1D (edge), 2D (triangle, quad) and 3D (hexahedron, tetrahedron, prism, pyramid) elements. The logic of implementation is based on identifying a set of core (kernel, bubble) 1D and 2D functions, and the use of specific edge-to-element and face-to-element extensions. The resulting implementation allows for an instantaneous modification (or redefinition) of core functions without any changes in the rest of the FE code. The 3D shape functions conform automatically to 1D edge and 2D face shape functions defined in local edge and face coordinates. In other words, the dependence of 3D shape functions on different edge and face orientations is taken into account at the level of element shape functions routine. This eliminates the use of sign factors during the assembly and simplifies dramatically the implementation of constrained approximation allowing for the presence of hanging nodes. The construction of shape functions is directly related to transfinite interpolation techniques used in mesh based geometry (MBG) descriptions.