Morphology-Preserving Remeshing Approach to Particulate Microstructures via Harmonic Decomposition

TL;DR

The paper addresses the challenge of generating high-quality, morphology-preserving meshes for particulate microstructures analyzed via harmonic decomposition. It introduces a diffusion-based remeshing framework on spheroidal analysis domains to equalize curvilinear coordinates while preserving Fourier weights and global area/volume, employing isotropic and anisotropic Laplace-Beltrami operators and a pullback correction for stable reconstructions. Key contributions include demonstrating substantial improvements in area-density uniformity and mesh quality on both closed and open genus-0 surfaces, as well as realistic 2D and 3D microstructures (concrete and stone masonry), along with a hierarchical diffusion strategy to reduce computation and open-source release for reproducibility. The approach enables more accurate, morphology-faithful simulations (finite/discrete element methods) of complex microstructures and offers avenues for broader applications such as area-preserving sampling and boundary-conformal meshing.

Abstract

Harmonic decomposition of surfaces, such as spherical and spheroidal harmonics, is used to analyze morphology, reconstruct, and generate surface inclusions of particulate microstructures. However, obtaining high-quality meshes of engineering microstructures using these approaches remains an open question. In harmonic approaches, we usually reconstruct surfaces by evaluating the harmonic bases on equidistantly sampled simplicial complexes of the base domains (e.g., triangular spheroids and disks). However, this traditional sampling does not account for local changes in the Jacobian of the basis functions, resulting in nonuniform discretization after reconstruction or generation. As it impacts the accuracy and time step, high-quality discretization of microstructures is crucial for efficient numerical simulations (e.g., finite element and discrete element methods). To circumvent this issue, we propose an efficient hierarchical diffusion-based approach for resampling the surface-i.e., performing a reparameterization-to yield an equalized mesh triangulation. Analogous to heat problems, we use nonlinear diffusion to resample the curvilinear coordinates of the analysis domain, thereby enlarging small triangles at the expense of large triangles on surfaces. We tested isotropic and anisotropic diffusion schemes on the recent spheroidal and hemispheroidal harmonics methods. The results show a substantial improvement in the quality metrics for surface triangulation. Unlike traditional surface reconstruction and meshing techniques, this approach preserves surface morphology, along with the areas and volumes of surfaces. We discuss the results and the associated computational costs for large 2D and 3D microstructures, such as digital twins of concrete and stone masonry, and their future applications.

Figures (11)

• Figure 1: Reconstructing and meshing 2D contours via the elliptic harmonics approach proposed in Shaqfa2024_SOH. (A) Uniform sampling of the hyperbolic curvilinear coordinates $\eta$ (left) and the corresponding reconstruction (right). Below is the radial distribution of the spacing between two consecutive points on the reconstructed contour, where the red dashed line represents the mean of the spacings. (B) The herein proposed diffusion-based sampling of the $\eta$ coordinates (left), where the resulting reconstruction (right) corresponds to uniformly sampled points on the contour as it converges to the mean spacings of (A). (C) The corresponding finite element (FEM) mesh of the reconstruction bulk in case (A). (D) The FEM mesh of the reconstructed contour in case (B).
• Figure 2: The difference between the reconstruction $\mathcal{H}^{-1}: \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ of surfaces via uniform and nonuniform sampling of the spheroidal coordinates $(\eta, \phi)$. (A) The input surface $\mathcal{S} \subset \mathbb{R}^3$. (B) SOH reconstruction using $\eta\phi$--coordinates of a scaled icosahedron, resulting in nonuniform surface reconstruction. (C) Same as (B), but with a nonuniform sampling of $\eta\phi$--coordinates, obtained via isotropic diffusion, that results in uniform surface reconstruction. Color maps represent normalized local areas of the triangulated surfaces; blue is for small and red is for large areas relative to the mean triangular area. To visualize the duality of sampling, we introduced the gray-scale insets (below) that show the texture deformation on the spheroidal coordinates and the corresponding reconstruction.
• Figure 3: Schematics for the anisotropic diffusion kernel. (A) The principal distortion directors (PDD) of equilateral and non-equilateral triangles, and (B) the corresponding isotropic and anisotropic diffusibility of distorted triangles.
• Figure 4: Isotropic diffusion for remeshing David's head bust via SOH. (B) We used five refinement cycles of an icosphere for the reconstruction process with $N_{max} = 50$ and $I_{max} = 30$. The color maps in (B) reflect the area density on the surface (red for high errors, and blue for relatively small errors). (C) Shows the PDF of the area density before (in red) and after (in blue) the diffusion.
• Figure 5: Computational time comparisons of different diffusion and reconstruction configurations. (A) Compares the CPU time versus the surface resolution (icosahedral refinements: $\{2, 3, 4, 5\}$) with fixed $N_{max} = 50$. (B) Compares the CPU time versus $N_{max}$ used throughout the diffusion iterations $N_{max} \in \{5, 10, 20, 30, 40, 50\}$. The same inset shows the error in the converged mean area density as a function of $N_{max}$.