# Manifold-based isogeometric analysis basis functions with prescribed sharp features

## TL;DR

This paper develops manifold-based basis functions for isogeometric analysis that can handle surfaces with sharp features, boundaries, and creases, enabling accurate analysis of complex shell geometries.

## Contribution

It introduces a novel manifold-based basis function construction combining differential geometry and conformal mappings, accommodating sharp creases and boundaries in isogeometric analysis.

## Key findings

- Basis functions handle arbitrary sharp features and boundaries.
- Suitable for Kirchhoff-Love thin shell analysis with kinks.
- No normalization needed for partition of unity functions.

## Abstract

We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed $C^0$ continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in $\mathbb R^3$ is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in $\mathbb R^2$, so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to earlier constructions no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish local polynomial approximants that are always $C^0$ continuous across tagged edges. The derived non-rational manifold-based basis functions are particularly well suited for isogeometric analysis of Kirchhoff-Love thin shells with kinks.