Metric-Driven 3D Frame Field Generation.

Area of Science:

• Computational geometry
• Finite element analysis
• Computer-aided design

Background:

• Controlling element size and shear is vital for high-quality hexahedral (hex) or hex-dominant meshes.
• Non-orthonormal frame fields are essential for this control but difficult to generate under general Riemannian metric fields.
• Existing methods struggle with generating integrable frame fields with precise metric control.

Purpose of the Study:

• To propose a method for generating integrable frame fields for Riemannian metric fields that are locally flat.
• To develop a metric-aware smoothness measure for assessing frame field integrability.
• To demonstrate improved mesh parameterization and metric field consistency on various shapes.

Main Methods:

• Solving for frame fields on Riemannian metrics that are flat away from singularities, admitting local isometry to Euclidean space.
• Applying Cartan's first structural equation to the rotation field to derive conditions for zero covariant derivatives.
• Introducing a metric-aware smoothness measure based on local integrability.

Main Results:

• Demonstrated that the rotation field must have zero covariant derivatives under the induced connection for locally flat metrics.
• Developed a method to generate smooth metric fields satisfying intuitive constraints.
• Achieved singularity-aware parameterization on cuboid shapes and improved integrability/consistency on generic shapes.

Conclusions:

• The proposed approach effectively generates integrable frame fields for locally flat Riemannian metrics.
• The metric-aware smoothness measure provides a robust criterion for integrability on such meshes.
• The method enhances the quality and consistency of hex-dominant meshes by better aligning parameterization with input metric fields.