Local h-, p-, and k-Refinement Strategies for the Isogeometric Shifted Boundary Method Using THB-Splines

TL;DR

This paper explores local refinement strategies for the isogeometric shifted boundary method using THB-splines, focusing on accuracy, stability, and efficiency in handling complex geometries with trimmed domains.

Contribution

It introduces novel local p- and k-refinement schemes for THB-splines within SBM and proposes an enhanced shift operator with mixed partial derivatives.

Findings

Degree elevation improves convergence for Neumann boundary conditions.

Refinement strategies significantly influence accuracy and stability.

Enhanced shift operator with mixed derivatives improves boundary condition enforcement.

Abstract

The concept of trimming, embedding, or immersing geometries into a computational background mesh has gained considerable attention in recent years, particularly in isogeometric analysis (IGA). In this approach, the physical domain is represented independently from the computational mesh, allowing the latter to be generated more easily compared with body-fitted meshes. While this facilitates the treatment of complex geometries, it also introduces challenges, such as ill-conditioning of the stiffness matrix caused by small cut elements and difficulties in accurately enforcing boundary conditions. A recently proposed technique to address these issues is the Shifted Boundary Method (SBM), which represents the computational domain solely through uncut elements and enforces boundary conditions via a Taylor expansion from a surrogate boundary to the true boundary. Previous studies have shown that, for Neumann boundary conditions, the flux evaluation requires additional derivatives in the Taylor expansion, effectively reducing the order of convergence by one. In this work, we investigate for the first time the performance of SBM combined with Truncated Hierarchical B-splines (THB-splines) under various local refinement strategies. In particular, we propose local p- and k-refinement schemes for THB-splines and compare them with local h-refinement and the unmodified SBM. Furthermore, we propose an enhanced shift operator that incorporates mixed partial derivatives, in contrast to the standard operator. The study assesses accuracy, stability, and computational efficiency for benchmark problems on trimmed domains. The results highlight how different refinement strategies affect convergence behavior in trimmed IGA formulations using SBM and demonstrate that targeted degree elevation can mitigate the Neumann boundary limitations of the standard method.