Constructing $C^1$ limit surfaces from unstructured splines via averaging and refinement

Abstract: In this paper we present a construction for unstructured splines over quadrilateral meshes by iterative averaging and refinement. We represent the spline as a multi-patch B-spline, where the degrees of freedom are those B-spline coefficients on the quadrilateral patches that are not associated with interior edges and vertices of the mesh, i.e., their corresponding Greville points lie inside the patches. In every averaging step, we replace the remaining B-spline coefficients associated with interior edges and vertices by suitable averages of neighboring degrees of freedom. In the refinement step we apply regular splits to all patches by knot insertion. This process results in a subdivision scheme that, for degree $p=2$, is similar to the almost-$C<sup>1$ spline construction from (Takacs, Toshniwal. CMAME, 2023) and behaves similar to Doo-Sabin subdivision, cf. (Doo, Sabin. CAD, 1978), and that can be defined for arbitrary degrees and regularities inside the patches. We derive two families of spline constructions, based on simple and coplanar averaging, respectively, and analyze their spectral properties when interpreted as subdivision schemes. Using this interpretation, we show that they are $C<sup>1$ in the limit. Moreover, the coplanar averaging scheme produces splines that are $C<sup>1$ at all vertices for every level of refinement, whereas the simple averaging is $C<sup>1$ only in the limit. For both constructions, we have control over the subdominant eigenvalue, which has multiplicity two and can range between $\frac{1}{4}$ and $1$, with $\frac{1}{2}$ often being the desired option. The resulting basis functions form a partition of unity. Moreover, they form a non-negative partition of unity for suitably selected averaging parameters.

• The paper introduces a systematic spline construction ensuring C¹ continuity for unstructured quadrilateral meshes via iterative averaging and refinement.
• It provides tunable spectral properties and explicit eigenvalue control to optimize smoothness near extraordinary vertices.
• Numerical experiments validate optimal L∞ and L² convergence rates, making the approach effective for isogeometric analysis and surface modeling.

$C^1$ Limit Surface Construction from Unstructured Splines via Averaging and Refinement

Introduction and Motivation

The paper "Constructing $C^1$ limit surfaces from unstructured splines via averaging and refinement" (2606.07149) presents a systematic spline construction for quadrilateral meshes with arbitrary topology, targeting robust $C^1$-continuity for isogeometric analysis and geometric modeling. The approach generalizes classical Doo–Sabin and almost-$C^1$ spline subdivision [takacs2023almost] toward arbitrary polynomial degrees, leveraging iterative averaging combined with refinement. Unlike conventional subdivision schemes, which emphasize infinite ring generation around extraordinary vertices, this method offers direct, tunable spectral properties, including explicit subdominant eigenvalue control, ensuring smoothness and optimal approximation near singularities.

Construction Overview and Methodology

The method is grounded in hierarchical multi-patch tensor-product B-splines, associating DOFs with interior patch locations, boundary edges, and corner vertices, but not with interior edges or vertices. Degree $p$ and regularity $r$ are chosen per patch, with degrees $r<p$ allowed, enabling flexible adaptation to local mesh complexity.

The main construction loop comprises three steps per refinement level:

1. DOF Assignment and Extraction: Select only face interior, boundary edge, and corner DOFs, filtering the complete set of B-spline coefficients via a sparse extraction operator.
2. Iterative Averaging: Edge and vertex-associated B-spline coefficients—omitted from the DOF set—are recomputed as convex or linear combinations (depending on averaging strategy) of surround DOFs. This enforces $C^1$-conditions along patch interfaces and at mesh vertices, including extraordinary cases.
3. Refinement: All patches undergo uniform knot insertion (tensor-product B-spline refinement), expanding the mesh with increased resolution but maintaining the consistency of the multi-patch parametrization.

Figure 1: Initial DOFs sampled from a quadrilateral mesh.

Figure 2: B-spline coefficients at mesh level $\ell$, with selected DOFs (blue), new coefficients by averaging (red/black), and refined coefficients at level $\ell+1$.

Averaging differentiates between two variants:

• Simple Averaging: Edge and vertex coefficients are simple weighted combinations of local neighbors. Weights can be chosen for convexity and spectral placement, with the only restriction being symmetry and affine partitioning (sum to one).
• Coplanar Averaging: Weights ensure that edge coefficients are coplanar for all extraordinary vertex valences, yielding $C^1$0-equivalence at extraordinary vertices on all refinement levels—not just in the limit.

Figure 3: (a) Smooth interior edge coefficients from face DOFs. (b) Regular interior vertex coefficients. (c) Smooth boundary vertex coefficients via edge DOFs.

This hierarchical, refinement-based process can be interpreted as a subdivision scheme, with the key property that, for regular mesh regions, the process exactly reproduces classical B-spline refinement.

Spectral Properties and Smoothness Analysis

The local subdivision matrix around an extraordinary vertex encodes the geometry and smoothness. The paper performs a detailed spectral analysis of this operator, characterizing the (sub-)dominant eigenvalues as a function of the averaging weights and mesh valence $C^1$1. Key outcomes include:

• There always is a dominant eigenvalue $C^1$2, corresponding to the identity partition of unity.
• The subdominant eigenvalue $C^1$3 (with multiplicity two) can be explicitly tuned via the parameter $C^1$4 in the averaging weights, ensuring $C^1$5 continuity in the limit for $C^1$6.
• The remaining eigenvalues are strictly less than $C^1$7 in magnitude, ensuring contractive behavior and control-point locality.

Figure 4: Characteristic rings for degrees $C^1$8 and valences $C^1$9 with $C^1$0 and parameter $C^1$1 for $C^1$2.

Coplanar averaging (in contrast to simple) always enforces $C^1$3 continuity at the extraordinary vertex at any finite refinement level, not just asymptotically, and the construction provides a concrete formula for feasible weights. The explicit connection between weights, subdominant eigenvalue, and continuity ensures that tuning $C^1$4 enables trade-offs between local shape quality and approximation power.

Numerical Experiments

Surface Quality Around EVs

The paper validates the construction through extensive tests on neighborhoods of extraordinary vertices with varying valence (e.g., 3, 5, 6) and polynomial degree. An example with valence 5 demonstrates that both simple and coplanar averaging yield $C^1$5 surfaces in the limit, but only coplanar averaging guarantees $C^1$6 smoothness at the EV on all refinement levels.

Figure 5: Initial DOFs for valence-$C^1$7 configuration.

Figure 6: Refined surface for $C^1$8 (coplanar averaging, valence-5, $C^1$9).

Figure 7: Surface with $C^1$0, $C^1$1 (coplanar averaging).

A parametric study shows that decreasing $C^1$2 causes elements near extraordinary vertices to shrink faster, increasing flattening and potentially reducing $C^1$3 jumps, which is reflected in convergence rates of maximum normal deviations.

Convergence Rates: Interpolation & $C^1$4-Approximation

On regular domains (unstructured multi-patch square or sphere), the paper interpolates or fits smooth functions and the unit sphere, examining $C^1$5 and $C^1$6-errors. Results indicate:

• For $C^1$7 with optimal choice $C^1$8, $C^1$9 errors exhibit cubic convergence ($p$0), the best attainable rate given the smoothness barrier.
• For $p$1 and higher, optimal convergence in $p$2 is retained when $p$3 matches mesh-dependent constraints ($p$4 for $p$5), aligning with theoretical predictions [takacs2025approximation].
• For non-optimal $p$6 (e.g., $p$7), errors still decrease, but convergence is suboptimal.
• Basis functions always form a partition of unity, and for admitted weight ranges, are pointwise non-negative, guaranteeing convex hull property.

Figure 8: Surfaces for $p$8 after refinement to level $p$9 with DOFs, surfaces for $r$0, and for $r$1.

Figure 9: Surface for $r$2.

Practical and Theoretical Implications

This construction establishes a practical toolkit for $r$3 isogeometric analysis and surface modeling on unstructured quadrilateral meshes, facilitating high-regularity bases without the artifacts associated with standard subdivision surfaces near EVs. Direct insight into the spectral properties of the refinement operator enables explicit control over surface quality and approximation power.

Key theoretical insights include:

• Locality: All basis functions have compact support. Support size is proportional to $r$4, controlled by refinement level and EV tuning.
• Partition of Unity & Non-negativity: Satisfies affine invariance and, under parameter restriction, the convex hull property.

The approach seamlessly integrates into IGA pipelines, making it suitable for high-order PDE solvers, especially those requiring globally $r$5-continuous spaces for thin-shell or Kirchhoff–Love models.

Future Directions

Further directions mentioned in the paper include:

• Extension to almost-$r$6 ($r$7) splines and bivariate generalized continuity.
• Localized parameter adjustment for improved smoothness at high-valence EVs.
• Further reduction in $r$8 jump (e.g., normal variation) near EVs without sacrificing support locality or the convex hull property.
• In-depth comparison with other smooth spline constructions, especially for high-order PDE applications [verhelst2024comparison, hughes2021smooth].

Conclusion

This work offers a principled, extensible framework for constructing globally $r$9 spline spaces via averaging and refinement on unstructured quadrilateral meshes. The localized, tunable averaging procedures—supported by meticulous smoothness and convergence analysis—enable flexible adaptation to arbitrary degree, valence, and regularity. Applications in geometric design and isogeometric analysis are immediate, with robust numerical evidence for optimal and near-optimal approximation in both $r<p$0 and $r<p$1 senses for suitable eigenvalue scaling. The results lay substantial groundwork for future developments in high-order continuity and adaptive spline analysis.