Integrable symmetric fields in computational design : from stable woven shells to seamless parameterizations

Many problems in science and specifically in computer graphics, particularly in areas like geometry processing and computational fabrication, leverage a general toolkit that can be conceptualized as leveraging energy driven pattern formation to solve inverse problems. A large class of such problems ultimately involve minimizing an energy with (field) degrees of freedom discretized over a geometric domain. In medical imaging, Diffusion Tensor Imaging (DTI) seeks to reconstruct interwoven neural pathways. In structural engineering, truss networks are designed to align with internal stress fields. In geometry processing, there is a longstanding grand challenge of developing a robust hexahedral mesh generation package, and state of the art approaches employ a pipeline where this mesh generation process is guided by an orientation field. In each case, a powerful algorithmic pipeline involves first designing a continuous direction field, and then rationalizing it into a discrete, coherent structure. A powerful algorithmic approach for problems of this form is to solve an optimization problem that attempts to "integrate" the direction field, e.g. by recovering a scalar potential whose gradients match the directions. The central limitation is that, in general, the guiding field is not integrable. As a result the rationalization step, which attempts to integrate the field, must introduce error. Controlling this error to reduce it's impact on the quality and physical validity of the ultimate rationalized designs is a central question in numerous research studies in this area. This thesis presents the first formulation of an exact formulation of discrete integrability which extends to symmetric frame fields in volumes. The key insight of our work was the introduction of a new "mixed-moment" representation for symmetric frames to the geometry processing literature. We prove that this representation is injective for the larger symmetric space GL(3)/O, which was unsupported by previous formulations, but necessary in order to exactly discritize integrability. Additionally, we develop algorithms and release reference implementations of solvers for designing smooth frame fields in the presence of non-linear constraints. These constraints pose a challenge for off-the-shelf optimization packages. We validate the algorithms we develop on the fabrication task of weaving geometrically complex shell structures out of straight ribbons, and the geometry processing task of volumetric parameterization.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Computer Science

Grantor

The University of Texas at Austin

Year dc:date.issued

2025

Author dc:creator

• Vekhter, Joshua Samuel

Advisor dc:contributor.advisor

• Vouga, Paul Etienne

• Computational design
• Integrability
• Hexahedral meshing
• Symmetric integrability
• Basket weaving
• Fabrication
• Discrete differential geometry

Language dc:language.iso

English

Identifier URI

https://doi.org/10.26153/tsw/63119

OAI identifier oai:identifier

oai:repositories.lib.utexas.edu:2152/135802