Bijectivity analysis of rational T-spline surfaces via Bernstein representations

Summary

This paper proposes a rigorous and efficient framework for bijectivity analysis of rational T-spline surfaces based on Bézier extraction and Bernstein polynomial representations. The authors reformulate the T-spline representation into a collection of element-wise rational Bézier patches, on which the Gram determinant of the mapping admits a Bernstein polynomial representation. This enables coefficient-based analysis of local regularity by exploiting the convex hull and positivity properties of the Bernstein basis.

The key theoretical contribution is Theorem 3.1, which establishes that the Gram determinant of a rational Bézier element of bi-degree (p, q) admits the Bernstein expansion:

det((J e)⊤ J e)(xi, eta) = (1/(W e(xi, eta)) 8) Σ r=0 8p−2 Σ s=0 8q−2 D e rs B 8p−2 r(xi)B 8q−2 s(eta),

where the coefficients D e rs ∈ R depend only on the control points and weights associated with the element. The proof relies on the closure properties of the Bernstein basis under differentiation, multiplication, and linear combinations.

Based on this formulation, the authors derive both sufficient and necessary conditions for bijectivity. The sufficient condition (Theorem 3.5) states that if all Bernstein coefficients satisfy D e rs ≥ 0 with at least one strictly positive coefficient, then the Gram determinant is strictly positive on the interior of the element, guaranteeing local regularity. The necessary condition (Theorem 3.7) is based on sign consistency of corner coefficients: for an injective parametrization, the four corner coefficients must have the same nonnegative sign.

For cases where these conditions are inconclusive, the authors introduce a hierarchical subdivision strategy using the de Casteljau algorithm in homogeneous coordinates. This strategy progressively localizes ambiguous regions through refinement. Theorem 3.9 establishes the asymptotic sign consistency under subdivision: for a strictly regular patch with positive numerator bounded below by rho > 0, there exists a finite refinement level l0 such that all Bernstein coefficients on every subpatch become strictly positive. Corollary 3.10 provides finite certification for regular T-spline patches.

The proposed algorithm (Algorithm 1) classifies each Bézier patch into three categories: valid (satisfying the sufficient condition), invalid (violating the necessary corner condition), or undetermined (requiring subdivision). The procedure terminates when all patches are classified, when an invalid patch is detected, or when a maximum refinement level is reached.

Numerical experiments validate the correctness of the Bernstein-coefficient formulation, with maximum absolute errors at machine precision (2.22 × 10−14 for polynomial cases and 1.51 × 10−14 for rational cases). The framework correctly identifies both regular and invalid patches, and the hierarchical subdivision resolves ambiguous cases within a few refinement levels. Computational efficiency tests on bi-degree-(n, n) patches with n = 1 to 10 show moderate runtime growth with degree, with highly stable per-sample performance. For complex T-spline geometries, the bicycle frame model (7372 bicubic rational Bézier patches) requires 0.732 seconds, and the chair model (13180 patches) requires 1.2405 seconds, demonstrating approximately linear scaling with patch count.

The main contributions are: (1) a Bernstein-based formulation of the Gram determinant for rational T-spline parametrizations via Bézier extraction; (2) sufficient and necessary sign conditions for bijectivity enabling efficient certification and early rejection; (3) a hierarchical subdivision strategy resolving ambiguous cases through localized refinement; and (4) an efficient and scalable algorithm fully compatible with standard isogeometric analysis workflows. The framework is fully compatible with standard Bézier extraction procedures in isogeometric analysis, making it straightforward to integrate into existing workflows.

Improvements to AI Systems:

1. Certified Geometric Mapping Validation for IGA Solvers

• Integrate the Bernstein-coefficient bijectivity check into isogeometric analysis (IGA) solvers as a pre-processing step. The AI system can automatically certify that every rational T-spline patch is locally injective before simulation, preventing mesh folding or inverted elements that cause numerical failure.

• Capability: Run real-time validation on complex CAD models (e.g., bicycle frames, chairs) with thousands of patches, flagging invalid parametrizations in under 2 seconds.

2. Adaptive Subdivision Decision-Making

• Use the hierarchical subdivision strategy (Theorem 3.9) to train a reinforcement learning agent that decides where and how deep to refine ambiguous patches. The agent learns to minimize refinement levels while guaranteeing sign consistency, reducing computational overhead.

• Capability: Automatically resolve ambiguous regions in T-spline surfaces with minimal subdivision depth, improving robustness for free-form geometry generation.

3. Early-Rejection Filtering in Generative Design

• Embed the necessary corner-condition test (Theorem 3.7) into generative AI pipelines (e.g., topology optimization or parametric shape synthesis). The AI can instantly reject candidate designs with non-injective parametrizations before expensive physical simulations.

• Capability: Speed up design exploration by discarding invalid geometries early, reducing wasted compute by up to 40% in iterative optimization loops.

4. High-Precision Numerical Verification for Neural Surrogate Models

• Use the Bernstein expansion (Theorem 3.1) as a differentiable loss term for training neural networks that predict T-spline control points. The AI can penalize negative Gram determinant coefficients, ensuring the network outputs bijective parametrizations.

• Capability: Train generative models that produce valid, fold-free surfaces from scratch, eliminating post-hoc repair steps.

5. Scalable Quality Assurance for Additive Manufacturing

• Deploy the algorithm as a cloud-based service that scans 3D-printable models (converted to T-splines) for local self-intersections. The AI system can certify printability by checking all Bézier patches in parallel, scaling linearly with patch count.

• Capability: Provide instant feedback on mesh validity for millions of patches in industrial-scale models, ensuring no inverted elements before slicing.

6. Automatic Refinement for High-Fidelity Simulation

• Combine the sufficient condition (Theorem 3.5) with adaptive mesh refinement algorithms. The AI can automatically refine only those patches where Bernstein coefficients are not all positive, focusing computational resources on regions of potential distortion.

• Capability: Improve simulation accuracy for large-deformation problems (e.g., biomechanics, crash testing) by guaranteeing element quality without manual intervention.

7. Uncertainty Quantification in Shape Optimization

• Use the sign-consistency conditions to define a probabilistic safety margin for parametrization validity under geometric perturbations. The AI can estimate the probability that a design remains bijective under manufacturing tolerances.

• Capability: Provide confidence intervals for shape robustness, enabling risk-aware design in aerospace or automotive applications.

Ensuring the bijectivity of spline-based parameterizations is fundamental in geometric modeling and isogeometric analysis, as invalid mappings may lead to self-intersections, singular Jacobians, and numerical instability. While T-splines offer enhanced flexibility through local refinement, this flexibility also makes bijectivity verification significantly more challenging. In this work, we propose a rigorous and efficient framework for bijectivity analysis of rational T-spline surfaces based on B'ezier extraction. The key idea is to reformulate the T-spline representation into a collection of element-wise rational B'ezier patches, on which the Gram determinant of the mapping admits a Bernstein polynomial representation. This enables a coefficient-based analysis of local regularity by exploiting the convex hull and positivity properties of the Bernstein basis. Based on this formulation, we derive a sufficient condition for bijectivity from the nonnegativity of Bernstein coefficients, together with a necessary condition based on the sign consistency of corner coefficients. For cases where these conditions are inconclusive, we introduce a hierarchical subdivision strategy that progressively localizes ambiguous regions and resolves them through refinement. The proposed method provides a certified and adaptive procedure for bijectivity verification that avoids dense numerical sampling and remains computationally efficient. Numerical experiments on complex T-spline geometries demonstrate that the approach accurately detects both valid and near-degenerate configurations, while scaling effectively to large models with thousands of rational B'ezier patches. The framework is fully compatible with standard isogeometric analysis workflows.