From Lagrange to Bernstein: Generalized Transfinite Elements with Arbitrary Nodes

This paper presents a unified framework for constructing transfinite finite elements with arbitrary node distributions, using either Lagrange or Bernstein polynomial bases. Three distinct classes of elements are considered. The first includes elements with structured internal node layouts and arbitrarily positioned boundary nodes. The second comprises elements with internal nodes arranged to allow smooth transitions in a single direction. The third class consists of elements defined on structured T-meshes with selectively omitted internal nodes, resulting in sparsely populated or incomplete grids. For all three classes, new macro-element (global interpolation) formulations are introduced, enabling flexible node configurations. Each formulation supports representations based on either Lagrange or Bernstein polynomials. In the latter case, two alternative Bernstein-based models are developed: one that is numerically equivalent to its Lagrange counterpart, and another that offers modest improvements in numerical performance.