Refinable tri-variate C ¹ splines for box-complexes including irregular points and irregular edges

Area of Science:

• Computer-Aided Design
• Geometric Modeling
• Numerical Analysis

Background:

• Traditional tensor-product splines are limited to regular grids.
• Unstructured hexahedral meshes present challenges for spline representations.
• Existing methods struggle with irregular features in complex geometries.

Purpose of the Study:

• To generalize C1 cubic tensor-product splines to unstructured hexahedral meshes (box-complexes).
• To develop a spline space that accommodates irregular mesh features.
• To ensure continuity and handle singularities in complex geometric domains.

Main Methods:

• Defined C1 splines over box-complexes, generalizing tensor-product splines.
• Utilized binary splitting of polynomial pieces to isolate mesh irregularities.
• Ensured derivative continuity across element boundaries.
• Addressed singularities at irregular features via local variable changes.

Main Results:

• The proposed splines generalize C1 cubic tensor-product splines.
• Polynomials are split in irregular regions, isolating singularities.
• Singularities are removable, allowing for a well-behaved spline space.
• The resulting spline space is refinable and contains 2^3 linearly independent functions per box.

Conclusions:

• C1 splines over box-complexes provide a robust method for geometric modeling on unstructured hexahedral meshes.
• This approach effectively handles mesh irregularities while maintaining spline continuity.
• The method offers a refinable and flexible framework for complex 3D modeling applications.