Quaternion-based dynamic algorithm for random generation of solid 4D cylindrical curves in RVE modeling

Abstract

A quaternion‐based dynamic algorithm is developed to populate Representative Volume Elements (RVEs) with solid 4D cylindrical fibers, combining spatial centerline coordinates ($x,y,z$) and quaternion‐encoded orientation. Fiber geometries are generated via parametric equations incorporating variable wavenumbers, phase angles, and amplitude functions, then discretized into equal‐arc segments to compute true curved lengths and volumes. Collision detection employs a closed‐form root‐finding solution for the overlap potential and Bézier‐clipping to guarantee $d_{\min }\ge 2r$, enabling non-overlapping placement at volume fractions up to 50 %. Random Sequential Adsorption (RSA) is integrated with quaternion mathematics to achieve dense, yet random, fiber packing. Scalability follows an empirical complexity of $O\left(N^{1.5}\right)$ for $N$ fibers. Micro-scale RVEs were successfully generated across fiber volume fractions up to 50 %, validated through second‐order orientation tensor analysis and statistical evaluations using Probability Density Functions (PDFs) and Cumulative Distribution Functions (CDFs). High-aspect-ratio tests (L/D = 10, 50, 100) confirm collision-free packings with normalized nearest-neighbor spacings ${\min }_{\text{NN}}/D>1$, mean spacing decreasing from 252 to 82.8 diameters, and relative dispersion ${\sigma }_{\text{NN}}/{\mu }_{\text{NN}}<0.5$. This singularity-free, computationally efficient method advances realistic composite microstructure modeling under extreme curvature and slenderness.