Revisiting Transfinite Elements: Unifying Element Formulations for IGA, SEM, NEFEM, p-FEM and h-FEM

Abstract The article at hand reviews the state-of-the-art in finite element methods based on transfinite function spaces, which were originally introduced by Coons and subsequently refined by Gordon and co-workers. The main objective is to present the benefits of transfinite interpolation techniques, its connection and its use in popular approaches that include isogeometric elements, NURBS-enhanced finite elements, and advanced transition elements following the p N h - or x N y -frameworks. Note that a secondary objective is to revitalize interest in this element-type, which, despite its remarkable advantages, has not gained widespread adoption. To shed light on this aspect, our review addresses the historical reluctance to embrace transfinite interpolation, attributing it to the perceived mathematical complexity of shape functions and the limited disclosure of element formulation details in initial publications due to industrial confidentiality. By elucidating these features, we aspire to rekindle appreciation for one of the most powerful element formulations in the finite element method, leading to renewed interest in and exploration of its application. Finally, we summarize the unique key features of transfinite elements , which are: (i) Conventional finite elements of Lagrange, Serendipity, and isogeometric types are included as special cases in the formulation of transfinite elements; (ii) Polytope elements, enhancing mesh generation flexibility (e.g., image-based analysis), are generated in a straightforward manner; (iii) Transition elements capable of coupling different element families and/or meshes with varying element sizes can be easily derived; and (iv) Exact geometry representations using the blending function method, which is a direct application of the transfinite interpolation concept to the geometry mapping, are achieved.