Robust Trimmed Multipatch IGA with Singular Maps

TL;DR

This paper introduces a robust isogeometric analysis method for elliptic problems with singular geometries, using regularized metrics and weak enforcement of conditions to maintain accuracy and stability.

Contribution

It develops a novel weak formulation for multipatch IGA that handles singular maps without specialized spaces, improving robustness in challenging geometries.

Findings

Method remains stable with aggressive singular parameterizations

Achieves optimal approximation order through regularization scaling

Numerical results confirm robustness and accuracy

Abstract

We consider elliptic problems in multipatch isogeometric analysis (IGA) where the patch parameterizations may be singular. Specifically, we address cases where certain dimensions of the parametric geometry diminish as the singularity is approached - for example, a curve collapsing into a point (in 2D), or a surface collapsing into a point or a curve (in 3D). To deal with this issue, we develop a robust weak formulation for the second-order Laplace equation that allows trimmed (cut) elements, enforces interface and Dirichlet conditions weakly, and does not depend on specially constructed approximation spaces. Our technique for dealing with the singular maps is based on the regularization of the Riemannian metric tensor, and we detail how to implement this robustly. We investigate the method's behavior when applied to a square-to-cusp parameterization that allows us to vary the singular behavior's aggressiveness in how quickly the measure tends to zero when the singularity is approached. We propose a scaling of the regularization parameter to obtain optimal order approximation. Our numerical experiments indicate that the method is robust also for quite aggressive singular parameterizations.