Overlapping Multi-Patch Isogeometric Method with Minimal Stabilization

TL;DR

This paper introduces a new isogeometric analysis method for complex geometries created by Boolean operations, using minimal stabilization and Nitsche's method to ensure stability and accuracy.

Contribution

It proposes a novel approach for IGA on union geometries with non-conforming patches, incorporating minimal stabilization and theoretical guarantees.

Findings

Method recovers stability on trimmed interfaces

Achieves optimal error estimates

Successfully solves Poisson's equation on complex geometries

Abstract

We present a novel method for isogeometric analysis (IGA) to directly work on geometries constructed by Boolean operations including difference (i.e., trimming), union and intersection. Particularly, this work focuses on the union operation, which involves multiple independent, generally non-conforming and trimmed spline patches. Given a series of patches, we overlay one on top of another in a certain order. While the invisible part of each patch is trimmed away, the visible parts of all the patches constitute the entire computational domain. We employ the Nitsche's method to weakly couple independent patches through visible interfaces. Moreover, we propose a minimal stabilization method to address the instability issue that arises on the interfaces shared by small trimmed elements. We show in theory that our proposed method recovers stability and guarantees well-posedness of the problem as well as optimal error estimates. In the end, we numerically verify the theory by solving the Poisson's equation on various geometries that are obtained by the union operation.