PDE-Based Parameterisation Techniques for Planar Multipatch Domains

TL;DR

This paper introduces a PDE-based framework for parameterizing complex planar multipatch domains, facilitating isogeometric analysis by ensuring valid interior domain descriptions from boundary contours.

Contribution

It proposes novel PDE formulations and numerical algorithms for harmonic map-based domain parameterization tailored for multipatch isogeometric analysis applications.

Findings

Developed PDE-based algorithms compatible with IGA software

Enabled control over parametric properties like cell size and boundary layers

Provided a framework for valid interior domain mapping from boundary contours

Abstract

This paper presents a PDE-based parameterisation framework for addressing the planar surface-to-volume (StV) problem of finding a valid description of the domain's interior given no more than a spline-based description of its boundary contours. The framework is geared towards isogeometric analysis (IGA) applications wherein the physical domain is comprised of more than four sides, hence requiring more than one patch. We adopt the concept of harmonic maps and propose several PDE-based problem formulations capable of finding a valid map between a convex parametric multipatch domain and the piecewise-smooth physical domain with an equal number of sides. In line with the isoparametric paradigm of IGA, we treat the StV problem using techniques that are characteristic for the analysis step. As such, this study proposes several IGA-based numerical algorithms for the problem's governing equations that can be effortlessly integrated into a well-developed IGA software suite. We augment the framework with mechanisms that enable controlling the parametric properties of the outcome. Parametric control is accomplished by, among other techniques, the introduction of a curvilinear coordinate system in the convex parametric domain that, depending on the application, builds desired features into the computed harmonic map, such as homogeneous cell sizes or boundary layers.