Polyhedral control-net splines for analysis

Abstract

Generalizing tensor-product splines to smooth functions whose control nets can outline more general topological polyhedra, bi-cubic polyhedral-net splines form a first-order differentiable piecewise polynomial space. Each polyhedral control net node is associated with one bi-cubic function. A polyhedral control net admits grid-, star-, n-gon-, polar- and three types of T-junction configurations. Analogous to tensor-product splines, polyhedral-net splines can both model curved geometry and represent higher-order functions on the geometry. This paper explores the use of polyhedral-net splines for solving elliptic partial differential equations on curved smooth free-form surfaces without additional meshing.