# Modelling and simulation of multifractal star-shaped particles

## TL;DR

This paper introduces a flexible stochastic modeling framework for 3D star-shaped particles with variable fractal dimensions, using anisotropic Gaussian fields and a novel simulation algorithm, demonstrated through multifractal Earth topography examples.

## Contribution

It presents a new approach to model and simulate multifractal star-shaped particles with variable Hausdorff dimensions using anisotropic Gaussian fields and a spectral turning bands-based algorithm.

## Key findings

- Successful simulation of multifractal Earth topography.
- Demonstration of the model's ability to capture shape variability.
- Introduction of a new simulation algorithm for complex fractal particles.

## Abstract

The problem of constructing flexible stochastic models to describe the variability in shape of solid particles is challenging. Natural objects often exhibit mono- or multi-fractal features, i.e. irregular shapes and self-similar patterns. This paper presents a general framework for modelling three-dimensional star-shaped particles with a locally variable Hausdorff (or fractal) dimension. In our approach, the radial function of the particle is represented by an anisotropic Gaussian random field on the sphere. We additionally derive a simulation algorithm being parenthetical to the spectral turning bands method proposed in Euclidean spaces. We illustrate the use of our proposal through numerical examples, including a multifractal simulated version of the Earth topography.