# An IGA Framework for PDE-Based Planar Parameterization on Convex Multipatch Domains

## TL;DR

This paper introduces a PDE-based elliptic grid generation algorithm for planar parameterization on convex multipatch domains, supporting arbitrary spline continuity levels, facilitating isogeometric analysis applications.

## Contribution

It presents a novel PDE-based parameterization method that supports arbitrary spline continuity and multi-patch domains, expanding the applicability of elliptic grid generation in isogeometric analysis.

## Key findings

- Supports $C^{0}$ and higher spline continuities
- Effective for multi-patch domain parameterization
- Demonstrated through three numerical experiments

## Abstract

The first step towards applying isogeometric analysis techniques to solve PDE problems on a given domain consists in generating an analysis-suitable mapping operator between parametric and physical domains with one or several patches from no more than a description of the boundary contours of the physical domain. A subclass of the multitude of the available parameterization algorithms are those based on the principles of Elliptic Grid Generation (EGG) which, in their most basic form, attempt to approximate a mapping operator whose inverse is composed of harmonic functions. The main challenge lies in finding a formulation of the problem that is suitable for a computational approach and a common strategy is to approximate the mapping operator by means of solving a PDE-problem. PDE-based EGG is well-established in classical meshing and first generalization attempts to spline-based descriptions (as is mandatory in IgA) have been made. Unfortunately, all of the practically viable PDE-based approaches impose certain requirements on the employed spline-basis, in particular global $C^{\geq 1}$-continuity. This paper discusses a PDE-based EGG-algorithm for the generation of planar parameterizations with arbitrary continuity properties (where arbitrary stands for spline bases with global $C^{\geq 0}$-continuity). A major use case of the proposed algorithm is that of multi-patch parameterization, made possible by the support of $C^{\geq 0}$-continuities. This paper proposes a specially-taylored solution algorithm that exploits many characteristics of the PDE-problem and is suitable for large-scale applications. It is discussed for the single-patch case before generalizing its concepts to multipatch settings. This paper is concluded with three numerical experiments and a discussion of the results.