Hodge decomposition of vector fields in Cartesian grids.

Area of Science:

• Computational Geometry
• Geometric Modeling
• Scientific Simulation

Background:

• Explicit shape representations (e.g., meshes) are common, but implicit representations (e.g., level-set functions) are widely used in geometric modeling and simulations.
• The L²-orthogonal Hodge decomposition is a crucial computational tool for scalar and vector fields, but its application to implicit representations with boundary conditions poses challenges.
• Existing mesh-based frameworks do not directly translate to implicit representations, particularly regarding domain projections.

Purpose of the Study:

• To develop a comprehensive Hodge decomposition method suitable for implicit shape representations.
• To unify normal and tangential components within a Cartesian representation for enhanced computational analysis.
• To overcome the difficulties associated with projections in implicit domains for accurate field decomposition.

Main Methods:

• Introduction of a novel 5-component Hodge decomposition tailored for implicit representations.
• Unification of normal and tangential field components within a Cartesian coordinate system.
• Application of the decomposition under standard Dirichlet/Neumann boundary conditions, respecting topological properties.

Main Results:

• The proposed method successfully performs L²-orthogonal Hodge decomposition on implicit representations.
• Numerical experiments demonstrate the effectiveness of the 5-component decomposition across various objects.
• Validation confirms rigorous L²-orthogonality and accurate cohomology calculations, as evidenced by single-cell RNA velocity analysis.

Conclusions:

• The developed 5-component Hodge decomposition provides a robust computational tool for analyzing fields on implicit representations.
• This approach overcomes limitations of previous methods, enabling accurate decomposition and cohomology analysis.
• The unified treatment of normal and tangential components enhances the applicability of Hodge decomposition in geometric modeling and simulations.