Fractal structures are characterized by the self-similarity they exhibit at different length scales. This self-similarity can be characterized using a measure called the fractal dimension, which quantifies the complexity by the ratio of the change in geometry by the change in scale. This measure is used to describe these geometries mathematically by employing fractal order differential operators related to the structure’s fractal dimension. Fractal structures arise remarkably frequently in nature; some examples include snowflakes, lightning and blood vessels. In field-coupled intelligent materials, complex multi-scale material structures exhibit the same self-similar properties of fractals and therefore it is useful to model these structures in a fractal mathematical framework to better understand their behavior. Diffusion-limited aggregation (DLA) is an iterative process that produces a two-dimensional fractal structure, whereby particles undergo a random walk and cluster together to form larger structures of particles. The structures produced using this process can be observed in many natural systems such as electrodeposition and dielectric breakdown. A finite difference model was developed to simulate heat diffusion on different DLA structures with varying fractal dimension. The effect of the structure’s fractal dimension on the rate of heat transfer across the structure was analyzed.

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Mario A. Carvajal, Basanta R. Pahari, and William Oates, "Finite difference modeling of heat diffusion on diffusion-limited aggregation generated fractal structures," Proc. SPIE 12484, Behavior and Mechanics of Multifunctional Materials XVII, 1248407 (Presented at SPIE Smart Structures + Nondestructive Evaluation: March 13, 2023; Published: 18 April 2023); https://doi.org/10.1117/12.2658500.