Designing 3D Anisotropic Frame Fields with Odeco Tensors

TL;DR

This work tackles the problem of designing smooth anisotropic volumetric tensor fields in constrained tetrahedral domains, going beyond isotropic approaches. It introduces AOTF, a framework based on symmetric orthogonally decomposable (odeco) tensors that decouples orientation $\boldsymbol{\Theta}$ from stretching $\boldsymbol{\Lambda}$, and optimizes a Dirichlet energy $E_s$ together with a soft input term $E_{\boldsymbol{\Lambda}}$, while enforcing boundary alignment. The authors provide a detailed odeco representation (interior and boundary forms), a warm-start optimization strategy, and theoretical analysis of shape conformity, including curvature alignment and feature preservation. They demonstrate the approach through applications in anisotropic meshing, fabrication of anisotropic microstructures, and elastic material designs, achieving improved smoothness and boundary conformity with flexible user guidance. The method promises significant impact on anisotropic hex/tet meshing and material-design workflows where directional anisotropy is essential.

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.

Figures (17)

• Figure 1: An optimal 3D anisotropic odeco tensor field (AOTF) is demonstrated through integral curves and clipped views. It automatically balances field smoothness, normal alignment, curvature alignment, feature preservation, and user-specified interior constraint (i.e., a gray curve shown in the middle image), leveraging the estimated surface curvature magnitudes as the user's input guidance. $E_{s}^i$ is the vertex-wise smoothness energy in this work.
• Figure 2: We represent each odeco tensor $f$ (right) as a rotation of the canonical odeco tensor $\hat{f}$ (middle). This canonical tensor $\hat{f}$ can be expressed as a linear combination of spherical harmonics bases and stretching ratios $\boldsymbol{\lambda}_i$ (left).
• Figure 3: Comparison of shape-conforming anisotropic tensor (frame) field designs. (a) Anisotropic tensor field designed via the method of palmer2020algebraic, guided by both estimated principal curvature magnitudes and directions. (b) Anisotropic tensor field designed by our AOTF method guided only from curvature magnitudes. Our optimal AOTF demonstrates automatic and better shape conformity, a highly desirable characteristic for one important downstream application -- anisotropic meshing.
• Figure 4: The pipeline of our warm start-based joint optimization of tensor orientations and stretching ratios. (a) Given $\boldsymbol{\Lambda^{In}}$ (user's sparse input) denoted by different colors; (b) Warm start stretching ratios $\boldsymbol{\Lambda}_{warm}$ in volume domain by diffusion; (c) Warm start orientations $\boldsymbol{\Theta}_{warm}$ with coordinate descent method; (d) Joint optimization on orientations and stretching ratios with $\boldsymbol{\Lambda}_{warm}$ and $\boldsymbol{\Theta}_{warm}$ as initial guesses. Step (c) accelerates the procedure of searching better local minima as shown in (e). $E_{T}$ is the total energy of AOTF in Eq. (\ref{['eq:problem']}).
• Figure 5: Dirichlet energy of a normal-aligned odeco tensor. In this example, we set the stretching ratios as $\lambda^x_i=2,\lambda^y_i=1,\lambda^z_i=1$. The computed curvature-aligned term of $||\nabla f(\boldsymbol{\theta}_i,\boldsymbol{\lambda}_i)||_2^2$ coincides with Proposition \ref{['prop_curvaalign']}.

Theorems & Definitions (3)

• Proposition 5.1
• Proposition 5.2
• Proposition 5.3