Scaled Boundary Parametrizations in Isogeometric Analysis

TL;DR

This paper introduces a new class of parametrizations for Isogeometric Analysis based on scaled boundary methods, providing a flexible framework for complex domains and demonstrating their effectiveness through computational examples.

Contribution

It develops a general framework for scaled boundary parametrizations in IGA, including smoothness, regularity, and handling non-star-shaped domains, and relates them to standard IGA methods.

Findings

Framework for scaled boundary parametrizations in IGA

Analysis of singularities at the scaling center

Comparison of parametrization strategies on a screw compressor geometry

Abstract

This paper deals with a special class of parametrizations for Isogeometric Analysis (IGA). The so-called scaled boundary parametrizations are easy to construct and particularly attractive if only a boundary description of the computational domain is available. The idea goes back to the Scaled Boundary Finite Element Method (SB-FEM), which has recently been extended to IGA. We take here a different viewpoint and study these parametrizations as bivariate or trivariate B-spline functions that are directly suitable for standard Galerkin-based IGA. Our main results are first a general framework for this class of parametrizations, including aspects such as smoothness and regularity as well as generalizations to domains that are not star-shaped. Second, using the Poisson equation as example, we explain the relation between standard Galerkin-based IGA and the Scaled Boundary IGA by means of the Laplace-Beltrami operator. Further results concern the separation of integrals in both approaches and an analysis of the singularity in the scaling center. Among the computational examples we present a planar rotor geometry that stems from a screw compressor machine and compare different parametrization strategies.