Voronoi diagram: An adaptive spatial tessellation for processes simulation

In computational fluid dynamics, there are two fundamental approaches to simulate fluid flow: Eulerian and Lagrangian flow formulations The former is based on a fixed tessellation while the latter uses a moving tessellation. Eulerian methods offer the advantage of a fixed tessellation that is easier to generate, and they can efficiently handle 3 www.intechopen.com Modeling, Simulation and Optimization -Tolerance and Optimal Control 42 dispersion dominated transport problems. Eulerian methods are therefore used for a majority of numerical modelling of fluids flow. However, in these methods, the time step size and the tessellation size have to be selected to ensure realistic solutions and avoid overshoot and undershoot of concentrations. However, since a uniformly fine tessellation is computationally costly, these methods are generally not well suited to handle of moving concentration fronts and advection-dominated and tracking problems. Lagrangian methods model the processes by tracking the changing location, shape and values of particles over space and provide an accurate and efficient solution to advectiondominated problems with steep concentration gradients. For Lagrangian methods, the tessellation movement softens the solution behaviour in time, such that larger time steps can be taken compared to a fixed spatial tessellation However, the connectivity between tessellation elements remains unchanged during simulation, which may cause difficulties such as tessellation tangling and deformation that become especially acute in non-uniform media with multiple sources and complex boundary conditions In addition, due to the relative nodal motion, the tessellation used for Lagrangian methods becomes distorted over time and complete re-tessellating is frequently required Development of mixed Eulerian-Lagrangian methods has led to the class of arbitrary Lagrangian-Eulerian (ALE) codes These codes reduce tessellation distortion by continuous "remapping" or "reconnecting" of the mesh. Tessellation remapping can be regarded as an Eulerian process, because mass is transported across tessellation cell boundaries. The principle of continual remapping led to the Free-Lagrange method. The difference between the Free-Lagrange and the classical Lagrange methods is that the latter attempt to maintain the initial tessellation connectivity during simulation. The Free-Lagrange method allows updating the tessellation connectivity as part of the problem to be solved. Simulation of free-surface flow and variable-density flow and transport are two examples that are well-suited for dynamic modelling. For example, for a hydrogeological system, the free surface or water table is an imaginary surface below ground where the absolute groundwater pressure is atmospheric. The water table moves and dynamic modelling could be used to track its temporal and spatial evolution by using a moving tessellation that conforms to the motion of the free surface. One complexity of using a moving tessellation is to maintain the tessellation alignment with stratigraphic layers or with geological formations having different hydraulic properties. The tessellation alignment can be maintained by continuously updating physical parameters, such as hydraulic conductivity, porosity, storage coefficient, as the tessellation moves. Density-variable flow and transport is another application well-suited for a moving tessellation. A classical example is given by salt-rock formations, where groundwater may become very rich in salt. They used a dynamic Lagrangian approach to track the sharp fresh-salt water interface and state that, when high concentrations prevail, fixed tessellation methods are inefficient. Moving tessellations have not been efficiently used in 3D since three-dimensionality adds several complexities for simulations. To minimize the problems, the Lagrangian algorithm presented by

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