Voronoi diagram: An adaptive spatial tessellation for processes simulation

Modelling and simulation of spatial processes is increasingly used for a wide variety of applications including water resources protection and management, meteorological prevision and forest fire monitoring. As an example, an accurate spatial modelling of a hydrological system can assist hydrologists to answer questions such as: "where does ground water come from?", “how does it travel through a complex geological system?" and “how is water pollution behaviour in an aquifer?” It also allows users and decision-makers to better understand, analyze and predict the groundwater behaviour. In this chapter, we briefly present an overview of spatial modelling and simulating of a dynamic continuous process such as a fluid flow. Most of the research in this area is based on the numerical modelling and approximation of the dynamic behaviour of a fluid flow. Dynamic continuous process is typically described by a set of partial differential equations (PDE) and their numerical solution is carried out using a spatial tessellation that covers the domain of interest. An efficient solution of the PDE requires methods that are adaptive in both space and time. The existing numerical methods are applied in either a static manner from the Eulerian point of view, where the equations are solved using a fixed tessellation during a simulation process, or in a dynamic manner from the Lagrangian point of view, where the tessellation moves. Some methods are also based on the mixed EulerianLagrangian point of view. However, our literature review reveals that, unfortunately, these methods are unable to efficiently handle the spatial-dynamic behaviour of phenomenon. Therefore, in this chapter, we investigate spatial tessellation based on Voronoi diagram (VD) and its dual Delaunay tessellation (DT) which is good candidate to deal with dynamics behaviour of fluid flow. Voronoi diagram is a topological data structure that discretizes the dynamic phenomenon to a tessellation adaptive in space and time.