CHAPTER 004
Growth Systems
106 DIFFUSION LIMITED AGGREGATION
LAPLACIAN GROWTH
THE PHENOMENON DESCRIPTION
THE PHENOMENON DESCRIPTION
Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, 1 is applicable to aggregation in any system where diffusion is the primary means of transport in the system. DLA can be observed in many systems such as electrodeposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown.
Laplacian instability occurs when a smooth interface evolving under a Laplacian field develops rapidly growing spikes and branches. Many fields are Laplacian, including the steadystate heat equation, electric potential, and an incompressible fluid pressure field. The instability has been connected to many disparate phenomena, such as dendrites on snowflakes, forks on lightning, quasi-steady-state fracture, lobes onlichen, coral, riverbeds, vasculature, and urban sprawl patterns. 2
1 DLA is a simple computer simulation of the formation of clusters by particles diffusing through a medium that jostles the particles as they move. T. Witten, L. Sandler 1981
2 Theodore Kim, Jason Sewall, Avneesh Sud and Ming C. Lin, Fast Simulation of Laplacian Growth, University of North Carolina at Chapel Hill, USA,
MACROSCALE GROWTH We are interested in the Diffusion Limited Aggregation growth system in terms of a macroscale deployment of our network structure. Firstly, the DLA growth system acts in a diffusion logic, which characterizes our overall investigations. Also, it offers the possibility of acquiring several levels of control, specifically concerning the direction of the growth, through the specification of the position of the seeds of the growth. Nevertheless, this extension also relies on other parameters, which result in unexpected, interesting geometrical and structural formations. In general, the DLA growth logic is chosen for the simulation of the networks growth in an urban level, where several input parameters and obstacles will exist, but the deployment of the system will have to occur through processing feedback from the urban environment.
MICROSCALE GROWTH We are interested in the Laplacian growth system in terms of a microscale organisation of our network structure. More specifically, we are including the logic of the Laplacian growth in most of our digital simulation studies, either these concern surface formations or the description of the feedback loop of the system. An essential element of this logic is the reconfiguration of the system's boundaries according to external forces and environmental conditions. A use of the latter in our system's microscale growth is being done in the case of surface formation through the consideration of the external pressure conditions (Hele Shaw simulation-feedback from external constraints), as well as in the case of the feedback loop through heating the network locally, where the heat diffusion informs the system and leads to decision-making.