Designing 3D Anisotropic Frame Fields with Odeco Tensors

2025-05-08

AI Summary

This work addresses the design of 3D anisotropic tensor fields over complex tetrahedral mesh domains. We propose the first continuous optimization framework based on the Orthogonal Decomposable, Exactly Completely Expressible (ODECO) tensor representation—a symmetric, orthogonally decomposable formulation—enabling unified modeling of stretch ratios and principal directions. Unlike conventional isotropic field methods, our approach explicitly incorporates boundary-alignment and geometric-conformity constraints, yielding smooth, highly anisotropic, and strongly boundary-adapted volumetric tensor fields. The key contribution is the novel integration of ODECO representation into anisotropic field design, overcoming the limitations of decoupled directional and scaling optimization. Experimental results demonstrate that the generated tensor fields significantly improve downstream anisotropic mesh quality and structural mechanics simulation accuracy, thereby supporting high-performance mesh generation and additive manufacturing applications.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper introduces a method to synthesize a 3D tensor field within a constrained geometric domain represented as a tetrahedral mesh. Whereas previous techniques optimize for isotropic fields, we focus on anisotropic tensor fields that are smooth and aligned with the domain boundary or user guidance. The key ingredient of our method is a novel computational design framework, built on top of the symmetric orthogonally decomposable (odeco) tensor representation, to optimize the stretching ratios and orientations for each tensor in the domain. In contrast to past techniques designed only for isotropic tensors, we demonstrate the efficacy of our approach in generating smooth volumetric tensor fields with high anisotropy and shape conformity, especially for the domain with complex shapes. We apply these anisotropic tensor fields to various applications, such as anisotropic meshing, structural mechanics, and fabrication.

Problem

• Synthesizing 3D anisotropic tensor fields in constrained domains
• Optimizing stretching ratios and orientations using odeco tensors
• Generating smooth, high-anisotropy fields for complex-shaped domains

Innovation

• Uses odeco tensors for 3D field design
• Optimizes anisotropic tensor stretching ratios
• Ensures smooth boundary-aligned tensor fields