📝 SummarXiv

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.