Spline-based Solid Subdivision Schemes over Arbitrary Tetrahedral Meshes

The basis function of the subdivision algorithm on the regular mesh and its derivative.

The basis function of the subdivision algorithm on the regular mesh and its derivative.

Yu-Sung Chang and Hong Qin


Technical Report: Submitted to Computer-Aided Geometric Design

In this paper, we propose novel trivariate spline-based solid subdivision schemes over arbitrary tetrahedral meshes in 3D space. The motivation of the proposed solid subdivision schemes are to represent solid objects with complex topology and heterogeneous material properties. The subdivision schemes are based on trivariate double-directional box splines which serve as their basis functions. By devising a special quasi-regular structure comprising tetrahedra and octahedra in 3D, we derive the subdivision rules for the regular cases. The subdivision algorithm uses a certain directional choice of the diagonals in octahedra to precisely evaluate the basis functions in the limit. We extend the subdivision rules to cope with arbitrary tetrahedral meshes. To avoid the asymmetry of the original scheme, an additional subdivision scheme based on averaging is presented. We prove the C<sup>1</sup> continuity of our subdivision scheme using existing mathematical techniques, such as the spectral analysis of subdivision matrices and the characteristic map method. Both theoretical and numerical results are presented in detail. Moreover  we support our hypothetical assumptions with empirical data which cover most of the practical cases. With the outline of the implementation of the algorithm, we present some experimental modeling results using our subdivision scheme, including solid models with non-trivial topology, volumetric objects with heterogeneous materials, and simple numerical simulation on solid objects.

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