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arxiv: 2512.17666 · v2 · submitted 2025-12-19 · 🧮 math.NA · cs.NA

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

Christoph Hollweck , Andrea Gorgi , Nicolo Antonelli , Marcus Wagner , Roland W\"uchner This is my paper

Pith reviewed 2026-05-16 20:47 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords isogeometric analysisshifted boundary methodTHB-splineslocal refinementtrimmed domainsNeumann boundary conditionsconvergence ratesh-p-k refinement
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The pith

Targeted local degree elevation in THB-splines mitigates the convergence reduction for Neumann boundaries in the Shifted Boundary Method.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines how local h-, p-, and k-refinement strategies using Truncated Hierarchical B-splines interact with the Shifted Boundary Method on trimmed geometries in isogeometric analysis. The standard SBM represents the domain through uncut elements and enforces boundary conditions with a Taylor expansion from a surrogate boundary, but Neumann conditions require extra derivatives that drop the convergence order by one. The authors introduce local p- and k-refinement schemes for THB-splines plus an enhanced shift operator that includes mixed partial derivatives, then compare their effects on accuracy, stability, and efficiency against plain h-refinement and unmodified SBM. A sympathetic reader would care because the approach avoids ill-conditioning from small cut elements and body-fitted mesh generation while offering a practical route to higher accuracy on complex immersed domains.

Core claim

The central claim is that local p- and k-refinement schemes for THB-splines, paired with an enhanced shift operator that incorporates mixed partial derivatives, change convergence behavior in SBM-based trimmed IGA formulations and that targeted degree elevation can mitigate the order reduction that standard SBM exhibits for Neumann boundary conditions, while stability is maintained across benchmark trimmed problems.

What carries the argument

The enhanced mixed-derivative shift operator applied within the Shifted Boundary Method to locally refined THB-splines.

Load-bearing premise

The local p- and k-refinement schemes together with the mixed-derivative shift operator preserve stability and optimal convergence rates across trimmed configurations without introducing new ill-conditioning.

What would settle it

A numerical test on a trimmed domain with Neumann boundary conditions in which p-refinement and the mixed-derivative operator still produce the same one-order drop in convergence rate as unmodified SBM.

Figures

Figures reproduced from arXiv: 2512.17666 by Andrea Gorgi, Christoph Hollweck, Marcus Wagner, Nicolo Antonelli, Roland W\"uchner.

Figure 1
Figure 1. Figure 1: Two-scale relation for different p-refined basis functions. The first row shows basis functions on level 0. The second row shows their corresponding children after p-refinement. The third row illustrates the scaled children whose weighted sum reproduces the parent basis function. With the two-scale relation at hand, the conceptual idea of THB-splines is identical for all refinement types. In contrast to cl… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of refinement strategies: h-, p-, and k-refinement. The first row shows the initial basis with p “ 2 and respective control points for all strategies. The second row illustrates the basis functions after global refinement for each method. The third row displays the final THB-spline basis after local refinement of the leftmost three basis functions (grey), which are replaced by finer-level child … view at source ↗
Figure 3
Figure 3. Figure 3: Refinement domains and final mesh Mˆ for local k-refinement [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of the SBM for B-splines of order [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Surrogate domain and boundary for an immersed circle with standard B-splines of degree [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Surrogate domain for an immersed circle with THB-splines of degree [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Manufactured solution upx, yq “ xy p1 ´ xqp1 ´ yq rsinh p10p1 ´ xqp1 ´ yqq ` sinh p10xyqs ` 10 “ p1 ´ xq 2 p1 ´ yq 2 ` x 2 y 2 ‰ . 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.6 0.8 1 y |u−uh| kukL2 0 0.5 1 1.5 ×10−1 (a) no refinement, 144 DOFs, 100 Elements 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.6 0.8 1 y |u−uh| kukL2 0 0.5 1 1.5 2 ×10−2 (b) h-refinement (1 step), 294 DOFs, 250 Elements 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.… view at source ↗
Figure 9
Figure 9. Figure 9: Error distribution for various local refinement strategies. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Effect of the enhanced shift operator rS p d on the convergence for the manufactured solution in Eq. 30 with Dirichlet boundary conditions applied to the geometry shown in [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Effect of the enhanced shift operator Srp d on convergence for the manufactured solution in Eq. 30, with Neumann conditions on the exterior boundary and Dirichlet conditions on the interior boundary of the geometry shown in [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 14
Figure 14. Figure 14: The color legend is the same as in the previous example. [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 12
Figure 12. Figure 12: Convergence study for the geometry in Fig. 6 with di [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Head-to-head convergence comparison, from 20 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Convergence study for the geometry 6 with di [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Head-to-head convergence comparison, from 20 [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Convergence study for the geometry 5 with immersed exterior Neumann boundary and immersed interior Dirichlet boundary, di [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Head-to-head convergence comparison, from 20 [PITH_FULL_IMAGE:figures/full_fig_p024_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Example visualizations of the geometries with 30 [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Step-by-step convergence study for the geometry in Fig. 18a, using di [PITH_FULL_IMAGE:figures/full_fig_p025_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Step-by-step convergence study for the geometry in Fig. 18b, using di [PITH_FULL_IMAGE:figures/full_fig_p026_20.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to investigate local h-, p-, and k-refinement strategies for the Isogeometric Shifted Boundary Method (SBM) using THB-splines on trimmed domains. It proposes an enhanced shift operator incorporating mixed partial derivatives to overcome the order reduction in Neumann boundary conditions associated with the standard SBM. Through numerical benchmarks on trimmed domains, it demonstrates the impact of different refinement strategies on convergence behavior and shows that targeted degree elevation can mitigate the Neumann limitations.

Significance. If the results hold, this work advances the field by providing practical refinement strategies for immersed IGA methods, improving accuracy and efficiency for complex geometries. The enhanced operator and THB-spline refinements could lead to more stable and optimal convergent methods for problems involving Neumann conditions, building on prior SBM and THB literature.

major comments (2)
  1. [Enhanced shift operator description] The central claim that the mixed-derivative shift operator preserves consistency and stability when paired with local THB-spline refinements lacks supporting a-priori error estimates. The Taylor expansion accuracy under variable knot multiplicities and hierarchical truncation is not analyzed, placing the full burden on numerical benchmarks which may be configuration-specific.
  2. [Numerical benchmarks] The abstract asserts that benchmark results support the mitigation of Neumann order reduction via targeted p-refinement, but no quantitative error tables, convergence rates, or stability metrics are provided in the available text. This undermines the ability to assess the strength of the claims regarding generality across trimmed configurations.
minor comments (2)
  1. [Abstract] The abstract could more clearly specify the benchmark problems used and the observed convergence orders to strengthen the summary of results.
  2. [Refinement strategies] Clarify the distinction between p- and k-refinement in the context of THB-splines, as the notation may be non-standard for some readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to clarify and strengthen our manuscript. We address each major point below and indicate the planned revisions.

read point-by-point responses

  1. Referee: The central claim that the mixed-derivative shift operator preserves consistency and stability when paired with local THB-spline refinements lacks supporting a-priori error estimates. The Taylor expansion accuracy under variable knot multiplicities and hierarchical truncation is not analyzed, placing the full burden on numerical benchmarks which may be configuration-specific.

    Authors: We acknowledge that a full a-priori error analysis for the enhanced shift operator under THB-spline hierarchies would provide stronger theoretical backing. The manuscript focuses on numerical investigation of local refinement strategies; the mixed-derivative operator is derived directly from the Taylor expansion to restore consistency for Neumann data. We will add a dedicated subsection deriving the consistency of the operator, explicitly addressing the effect of knot multiplicity changes and truncation on the expansion remainder, and include additional numerical tests on configurations with varying hierarchical levels to support generality. A complete a-priori proof is beyond the present scope. revision: partial

  2. Referee: The abstract asserts that benchmark results support the mitigation of Neumann order reduction via targeted p-refinement, but no quantitative error tables, convergence rates, or stability metrics are provided in the available text. This undermines the ability to assess the strength of the claims regarding generality across trimmed configurations.

    Authors: The full manuscript contains detailed numerical results, including L2 and H1 error tables, observed convergence rates, and condition-number metrics for several trimmed geometries. We will revise the text to ensure these quantitative data are explicitly referenced from the abstract onward, add a summary table of convergence rates across all refinement strategies, and include two additional trimmed-domain examples to further illustrate generality. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation or claims.

full rationale

The paper proposes local h-, p-, and k-refinement strategies for THB-splines in the context of the Shifted Boundary Method and introduces an enhanced shift operator incorporating mixed partial derivatives. These are evaluated via numerical benchmarks on trimmed domains, with convergence behavior compared to the unmodified SBM. No equations, parameters, or central claims reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the work builds on prior SBM and THB literature as external foundations while presenting independent numerical assessments of accuracy, stability, and efficiency.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable. The approach inherits standard assumptions from SBM (Taylor expansion validity near the boundary) and THB-splines (truncation properties) without additional postulates stated.

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