Unstructured Transfinite Elements Including T-Splines

This paper presents a unified framework for constructing partially unstructured B-spline transfinite finite elements with arbitrary nodal distributions. Four novel distinct classes of elements are investigated. The first consists of classical transfinite elements reformulated using B-spline basis functions. The second includes elements defined by arbitrary control point networks arranged in parallel layers along one direction. The third features arbitrarily placed boundary nodes combined with a tensor-product structure in the interior. The fourth class comprises classical T-spline elements, characterized by selectively omitted internal nodes that produce sparsely populated, incomplete grids. For all four classes, novel macro-element formulations are introduced, enabling flexible and customizable nodal configurations while preserving the partition of unity property. The key innovation lies in reinterpreting the generalized coefficients as discrete samples of an underlying continuous univariate function, which is independently approximated at each station in the T-index space. This perspective generalizes the classical transfinite interpolation by allowing both the blending functions and the univariate trial functions to be defined using non-cardinal bases such as Bernstein polynomials or B-splines, offering enhanced adaptability for complex geometries and nonuniform node layouts.