# Branched splines

## TL;DR

This paper introduces branched splines, a new class of spline functions on branched surfaces, enabling complex CAD models with a single piece and facilitating integrated CAD and FEA on high-genus surfaces.

## Contribution

It presents a theoretical framework for branched splines on complex surfaces, reducing the number of spline pieces needed for detailed CAD models and enabling direct multiresolution analysis.

## Key findings

- Single-piece branched splines can model complex geometries.
- Facilitates integration of CAD and FEA on high-genus surfaces.
- Provides a theoretical basis with simple illustrative examples.

## 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 splines defined on branched covering of sphere, torus or general domains of $R^2$, which are called branched splines in this paper. A single piece of such splines is enough to build some 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. A theoretical framework is presented in this paper, together with some simple examples.