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

Abstract

The first step towards applying isogeometric analysis techniques to solve PDE problems on a given domain consists in generating an analysis-suitable map** 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 map** 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 map** 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 ^≥1-continuity.

This paper discusses an EGG-algorithm for the generation of planar parameterizations with locally reduced smoothness (i.e., with support for locally only C ⁰-continuous bases). A major use case of the proposed algorithm is that of multipatch parameterizations, made possible by the support of C 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.