Analysis-suitable $G^1$ multi-patch parametrizations for $C^1$ isogeometric spaces

TL;DR

This paper investigates the approximation properties of $C^1$ isogeometric spaces on multi-patch geometries with $C^0$ interfaces, introducing analysis-suitable $G^1$ parametrizations to achieve optimal convergence.

Contribution

The paper defines analysis-suitable $G^1$ geometries and analyzes their impact on the approximation capabilities of $C^1$ isogeometric spaces, including theoretical and numerical insights.

Findings

Analysis-suitable $G^1$ geometries enable optimal convergence.

Non-analysis-suitable geometries prevent optimal approximation.

Numerical tests confirm theoretical predictions.

Abstract

One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from $p$-degree splines (and extensions, such as NURBS), they enjoy up to $C^{p-1}$ continuity within each patch. However, global continuity beyond $C^0$ on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only $C^0$ at the patch interface. On such domains we study the $h$-refinement of $C^1$-continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the $C^1$-continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recent studies by Kapl et al. have given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) $C^1$ splines. This is the starting point of our study. We introduce the class of analysis-suitable $G^1$ geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of $C^1$ isogeometric spaces over analysis-suitable $G^1$ parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of $C^1$ isogeometric spaces is prevented.