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 Bézier-form Coons volumes, we introduce a criterion based on the Bézier 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 Bézier 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 Bézier 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. This work contributes both theoretical and computational tools for quality assurance in volumetric parametric modeling.

Isogeometric analysis has brought a paradigm shift in integrating computational simulations with geometric designs across engineering disciplines. This technique necessitates analysis-suitable parameterization of physical domains to fully harness the synergy between Computer-Aided Design and Computer-Aided Engineering analyses. Existing methods often fix boundary parameters, leading to challenges in elongated geometries such as fluid channels and tubular reactors. This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations. We employ a sophisticated Schwarz–Christoffel mapping technique, which is instrumental in computing boundary correspondences. A refined boundary curve reparameterization process complements this. Our dual-strategy approach maintains the geometric exactness and continuity of input physical domains, overcoming limitations often encountered with the existing reparameterization techniques. By employing our proposed boundary parameter matching method, we show that even a simple linear interpolation approach can effectively construct a satisfactory analysis-suitable parameterization. Our methodology offers significant improvements over traditional practices, enabling the generation of analysis-suitable and geometrically precise models, which is crucial for ensuring accurate simulation results. Numerical experiments show the capacity of the proposed method to enhance the quality and reliability of isogeometric analysis workflows.