Multiply Periodic Splines

TL;DR

This paper introduces multiply periodic splines on hyperbolic discs that can simplify CAD models and enable integrated CAD and FEA, with a theoretical framework and initial examples.

Contribution

It presents a new class of splines on hyperbolic discs that reduce the number of pieces needed for complex CAD models and facilitate multiresolution analysis.

Findings

Single-piece splines can model complex CAD geometries.

Framework for constructing multiply periodic splines is proposed.

Future work will include rigorous derivation of B-splines.

Abstract

Spline functions have long been used in numerical solution of differential equations. Recently it revives as isogeometric analysis, which offers integration of finite element analysis and NURBS based CAD into a single unified process. Usually many NURBS pieces are needed to build geometrically continuous CAD models. In this paper, we introduce some multiply periodic splines defined on hyperbolic disc. A single piece of such splines is enough to build complex CAD models. Multiresolution analysis on surfaces of high genus built from such splines can be carried out naturally. CAD and FEA are integrated directly on such models. It is difficult to derive such splines, only a theoretical framework is presented, together with some simple examples. Rigorous derivation and construction of B-splines will be given in future papers.