Particle-Based Shape Modeling

• Particle-based shape modeling is a computational framework that represents complex 3D structures using pseudo-landmarks to capture and analyze volume, surface area, and shape characteristics.
• It integrates geometric, statistical, and physical modeling to express key properties as power-law functions of the specific surface, enabling clear visualization of shape-size distributions.
• PSM supports advanced simulation workflows in granular materials and anatomical analysis, enhancing predictive modeling in DEM, fluid-structure coupling, and statistical shape optimization.

Particle-based shape modeling (PSM) is a computational paradigm that parameterizes complex, typically three-dimensional geometries using ensembles of particles (pseudo-landmarks) distributed across or within each shape instance. PSM provides a unified framework for quantifying, comparing, synthesizing, and statistically analyzing the morphology of granular entities and anatomical structures. The approach integrates geometric, statistical, and physical modeling, enabling precise control over the multi-attribute representation of size, surface area, volume, and shape descriptors within both material and biological contexts.

1. Core Principles and Motivation

Traditionally, granular material and morphological analyses have separated particle size and particle shape, which obscures the intrinsic covariation between size-dependent shape properties such as sphericity, specific surface, and surface area-to-volume ratio. PSM addresses this limitation by unifying volume ($V$), surface area ($S$), equivalent-sphere diameter ($D$), and shape indexes (e.g., Wadell’s sphericity $S_p$ or its inverse $B$) in a single analytical framework through the specific surface ($s = S/V$). This integration enables the representation of all 3D particle geometry attributes as power-law functions of $s$, so that correlations and trends in shape-size distribution are immediately visible in particle geometry space (PGS), a log–log geometric domain where each geometric attribute traces a straight line of distinct slope (Tripathi et al., 24 Feb 2025).

In the PGS framework,

• $V(s) = 10^c\,s^a$, where $a<0$ captures the empirical relationship between shape and size;
• $S(s) = sV(s)$ (surface area as a function of specific surface);
• $S$0 (shape index as a function of specific surface).

Embedding a conventional particle size distribution (PSD) $S$1 into PGS upgrades the univariate distribution to a multidimensional geometric density consistent for all derived measures.

2. Mathematical Formulations and Statistical Models

The central mathematical insight of PSM is the expressibility of all key geometric quantities via $S$2, with the following defining relationships for a particle:

• $S$3
• $S$4
• $S$5
• $S$6

This structure yields, in PGS ($S$7 vs.\ $S$8),

• Volume: horizontal line ($S$9),
• Surface area: slope $D$0 ($D$1),
• Shape-factored size: slope $D$2 ($D$3),
• Shape index: slope $D$4 ($D$5).

A PSD $D$6 is mapped to $D$7 via empirical shape-size regressions, transforming size-only data into complete multidimensional characterizations. For example, in the crushed Virginia granite dataset, the regression $D$8 captures increasing sphericity with particle size ($D$9). Given any percentile $S_p$0, the associated $S_p$1 can be computed, supporting full characterization and simulation (Tripathi et al., 24 Feb 2025).

3. Methodological Implementations and Computational Workflows

In practical PSM-enabled workflows, several distinct technical strategies are deployed:

• Shape Approximation Algorithms: For the generation of discrete particle geometries, techniques such as the multi-sphere method (MSS) produce an optimal set of overlapping spheres whose union approximates a target voxelized shape $S_p$2 by minimizing the symmetric difference $S_p$3 (Buchele et al., 6 Mar 2026). Each iteration proceeds via efficient Euclidean distance transforms and local maximum search, yielding an adaptive sphere filling that scales linearly in both shape voxel number and sphere count, outperforming prior heuristics in fidelity and computational cost.
• Statistical Geometric Shape Models: For materials with faceted or irregular morphologies, statistical 'Poisson plane field' models construct particle ensembles by intersecting a parent domain with randomly oriented planes. The resulting convex polyhedra encode realistic, experimentally-observed shape and size distributions for granular solids without parameter fitting (Smith et al., 2012).
• Stochastic Star-shaped Models: For star-shaped particle classes, the radial boundary function $S_p$4 is modeled as an (possibly anisotropic) Gaussian random field on the unit sphere. Covariance decays, parameterized via 'adaptive Schoenberg expansion,' directly determine local fractal or Hausdorff dimension, enabling multifractal particle morphologies with spatially variable roughness (Alegría, 2019).

4. Integrating Physics, Simulation, and Applications

PSM frameworks are embedded within multiphysics and simulation environments to enable predictive modeling of collective and individual particle behaviors.

• Discrete Element Methods (DEM): Particles generated via PSM are used in quasi-static packing and compaction simulation, often leveraging shape models ranging from convex polyhedra to multi-sphere or implicit metaball representations. Rigorous energy-based optimization or contact-resolved dynamics produce jammed packings at densities close to theoretical maxima (e.g., $S_p$5 for random polyhedra (Smith et al., 2012)).
• Fluid–structure coupling: In fluid–particle systems, analytic implicit surfaces (e.g., metaball representations) bridge DEM and Lattice Boltzmann Method (LBM) solvers via sharp-interface coupling and local refilling schemes, enabling accurate and robust simulation of sedimentation, segregation, and transport with arbitrary particle morphologies (Zhang et al., 2022, Zhao et al., 2022).
• Thermal and transport modeling: Direct simulation of effective transport properties, such as thermal conductivity, leverages the true PSM-generated microstructure. Empirical and theoretical results show non-percolating behavior and saturation of effective transport for high contrast ratios, necessitating proper representation of the granular geometry for physical prediction (Smith et al., 2012).
• Soft and Deformable Particles: Continuum frameworks managing shape via in-plane mass-point shells (with elasto-plastic force laws and core pressure) accurately resolve deformation, pore-filling, and fracture phenomena in soft particle or cellular assemblies, with outcomes spanning highly elongated, polygonal, or irregular morphologies under external loading (Trivino et al., 28 Mar 2025).

5. Statistical Shape Analysis and Optimization

Beyond materials, PSM techniques support anatomical surface modeling and statistical shape analysis.

• Particle Correspondence and Statistical Models: Shape distribution in a population is modeled by fixing pseudo-landmarks (particles) across anatomical surfaces, simultaneously optimizing for geometric fidelity (surface sampling), statistical compactness (low-dimensionality via PCA entropy), and correspondence consistency (minimum Frobenius or neighborhood-matching loss) (Xu et al., 10 Jul 2025, Xu et al., 2024).
• Implicit Surface and RBF Representations: Particle placements may be constrained by implicit radial basis function (RBF) reconstructions of the signed distance field, which enables self-supervised adaptation to geometric complexity while maintaining high-fidelity point distribution models (Xu et al., 2024, Xu et al., 10 Jul 2025).
• Local Adaptivity and Correspondence: Geodesic neighborhood regularization and localized correspondence objectives facilitate high adaptivity while ensuring anatomical consistency, with high evaluation performance on metrics such as mean two-way surface distances and PCA compactness in various biomedical datasets (Xu et al., 10 Jul 2025).

6. Extensions: Constraints, Spatiotemporal Modeling, and Future Directions

PSM has been extended to tackle modeling over arbitrary regions-of-interest (ROI) by means of free-form and cutting-plane constraints, expressed via differentiable scalar mesh fields and quadratic penalty optimization. This allows complex subset selection for region-specific statistical analyses, supporting efficient enforcement of both geodesic and planar boundaries (Xu et al., 2023).

Spatiotemporal extensions generalize the core entropy-based PSM scheme to the temporal domain: joint optimization of inter-subject (cross-sectional) and intra-subject (longitudinal) correspondences supports compact, data-driven 4D shape models as demonstrated in cardiac applications. Optimization alternates between static particle updates and trajectory smoothing, and achieved improved generalization and specificity for dynamic organ shape representations over image-based methods (Adams et al., 2022).

A plausible implication is that ongoing research will further unify geometric, statistical, and physical components in PSM, expand to higher-order features (curvature, topology), and refine domain-specific constraints (e.g., differentiable geodesic losses or multiscale adaptivity), thereby consolidating PSM as a central modeling paradigm across granular materials, soft matter, and biomedical shape analysis.