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Description
Trimming and immersion techniques have become standard tools for handling complex geometries in isogeometric analysis (IGA). By decoupling the physical domain from the computational background mesh, mesh generation is simplified compared to body-fitted approaches. However, this embedding strategy introduces numerical challenges, including ill-conditioning due to small cut elements.
The Shifted Boundary Method (SBM) addresses these issues by restricting the computational domain to uncut elements and imposing boundary conditions through a Taylor expansion from a surrogate boundary to the true boundary. For Neumann conditions, the flux evaluation requires higher-order derivatives in the Taylor expansion, which effectively reduces the achievable convergence rate by one order.
This contribution investigates, for the first time, the combination of the SBM with Truncated Hierarchical B-splines (THB-splines). We systematically study local h-, p-, and k-refinement strategies and analyze their influence on accuracy, stability, and computational efficiency in trimmed domains. In addition to the standard shift operator, we introduce an enhanced formulation that incorporates mixed partial derivatives within the Taylor expansion.
Benchmark problems are used to evaluate the convergence behavior under different refinement strategies. The results indicate that local degree elevation can compensate for the loss of optimal convergence rates associated with Neumann boundary conditions in the classical SBM.
The study clarifies the role of refinement strategies in trimmed IGA formulations and provides guidance for the robust application of SBM in locally refined spline discretizations.