Manifold-based B-splines on unstructured meshes

TL;DR

This paper presents new manifold-based spline constructions that exactly reproduce B-splines on unstructured meshes, enabling improved isogeometric analysis of complex surface geometries with arbitrary topology.

Contribution

Introduction of univariate manifold-based splines capable of exactly reproducing B-splines on unstructured meshes for isogeometric analysis.

Findings

Splines automatically reproduce B-splines in regular mesh regions.

Numerical analysis confirms convergence properties.

Results extend to tensor-product spline constructions.

Abstract

We introduce new manifold-based splines that are able to exactly reproduce B-splines on unstructured surface meshes. Such splines can be used in isogeometric analysis (IGA) to represent smooth surfaces of arbitrary topology. Since prevalent computer-aided design (CAD) models are composed of tensor-product B-spline patches, any IGA suitable construction should be able to reproduce B-splines. To achieve this goal, we focus on univariate manifold-based constructions that can reproduce B-splines. The manifold-based splines are constructed by smoothly blending together polynomial interpolants defined on overlapping charts. The proposed constructions automatically reproduce B-splines in regular parts of the mesh, with no extraordinary vertices, and polynomial basis functions in the remaining parts of the mesh. We study and compare analytically and numerically the finite element convergence of several univariate constructions. The obtained results directly carry over to the tensor-product case.