Regularity analysis and verification of Coons volume mappings

TL;DR

This paper presents a systematic framework for analyzing and verifying the regularity of Coons volume mappings, ensuring their suitability for stable and high-quality volumetric modeling in computational design.

Contribution

It introduces a general sufficient condition and a Bézier coefficient-based criterion for regularity verification, along with an efficient algorithm applicable to multi-patch B-spline volumes.

Findings

Verification algorithm completes in milliseconds

Regularity relates to geometric similarity of boundary surfaces

Method validated through numerical experiments

Abstract

The Coons volume provides a classical approach for constructing three-dimensional parametric mappings via boundary surface interpolation and is widely employed in volumetric mesh generation, computer-aided geometric design, and isogeometric analysis. However, due to curvature variations and continuity limitations of the boundary surfaces, the Jacobian determinant of a Coons volume may locally vanish or become negative, resulting in a non-regular mapping. This undermines mesh quality and compromises the stability of subsequent numerical computations. Ensuring the regularity of Coons volumes is therefore critical for robust parametric modeling. This paper develops a systematic framework for analyzing and verifying the regularity of Coons volumes. We first derive a general sufficient condition applicable to arbitrary boundary parameterizations, independent of specific analytical forms. For B\'ezier-form Coons volumes, we introduce a criterion based on the B\'ezier coefficients of the Jacobian determinant, transforming the verification problem into checking the positivity of control coefficients. Furthermore, we construct a necessary condition by applying a subdivision strategy combined with the B\'ezier blossoming technique, ensuring that regularity is preserved in all subdomains. By integrating these conditions, we design an efficient verification algorithm whose correctness and computational performance are validated through numerical experiments. We observe that the regularity of a Coons volume is closely related to the geometric similarity of its opposite boundary surfaces. Moreover, through B\'ezier extraction, the algorithm is extended to multi-patch B-spline volumes of arbitrary topology. Numerical tests show that the method completes regularity verification in milliseconds, enabling real-time application.