Keywords: mesh generation, mesh smoothing, mesh untangling, adaptive refinement, mass consistent wind model, 3D finite element method.

In the finite element simulation of environmental processes that occur in a three-dimensional domain defined over complex terrains, a mesh generator capable of adapting itself to the topographic characteristics and to the numerical solution is essential. The objective of this work is to present a review of our recent results in these topics [1,2,3,4,5,6].

A tetrahedral mesh of a region bounded in its lower part by the terrain and in its upper part by a horizontal plane will be created. To do this we a 3D Delaunay triangulation of a previously established distribution of points is made, whose density increases with the complexity of the orography. The point generation in the domain is done attending to a vertical spacing function over different layers defined from the terrain to the upper part of the domain. The adaptive position of nodes in the terrain surface is automatically determined by applying a 2D refinement/derefinement algorithm of nested meshes. To avoid conforming problems between mesh and orography, the tetrahedral mesh will be designed with the help of an auxiliary parallelepiped, in such a way that every terrain node is projected on its lower plane. Once the 3D Delaunay triangulation of the set of points has been constructed on the parallelepiped, points are replaced on their real positions keeping the mesh topology. In this last stage there can be occasional low quality elements, or even inverted elements, thus making it necessary to apply any untangling and smoothing procedures.

For this reason, we have developed a simultaneous untangling and smoothing procedure to optimise the resulting mesh. The quality improvement mesh optimisation techniques that preserve its connectivity are obtained by an iterative process in which each node of the mesh is moved to a new position that minimises a certain objective function. In general, objective functions are derived from some quality measure of the local submesh, that is, the set of tetrahedra connected to the adjustable or free node. Although these objective functions are suitable to improve the quality of a mesh in which there are non inverted elements, they are not when the mesh is tangled. The substitution of objective functions having barriers by modified versions that are defined and regular on all R³ is proposed. With these modifications, the optimisation process is also directly applicable to meshes with inverted elements, making a previous untangling procedure unnecessary.

Once the adapted mesh in accordance with the geometrical characteristics of our domain is constructed, an adaptive local refinement of tetrahedral meshes is implemented in C++ in order to improve the numerical solution obtained by the finite element method. The refinement technique, based on the eight-subtetrahedron subdivision, allows a higher discretization of the selected regions of the domain. This process may be repeated until the numerical solution is accurately approximated.

Air pollution models usually start from the computation of the velocity field of the fluid. In this paper, a model for computing such a field based on the contribution of the observed wind flow and the vertical buoyancy or momentum plume rise defined by a Gaussian plume model is presented. This initial velocity field is adjusted to verify incompressibility and impermeability conditions by using a mass consistent model. The problem is solved with 3D adaptive finite element method. All these techniques will be apply to realistic and test problems. An example of a 3D discretization can be seen in Figure 1.

1

J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero, J.M. González-Yuste, "Simultaneous untangling and smoothing of tetrahedral meshes", Computer Methods in Applied Mechanics and Engineering, 192, 2775-2787, 2003. doi:10.1016/S0045-7825(03)00299-8

2

J.M. González-Yuste, R. Montenegro, J.M. Escobar, G. Montero, E. Rodríguez, "Local refinement of 3-D triangulations using object-oriented methods", Advances in Engineering Software, 35, 693-702, 2004. doi:10.1016/j.advengsoft.2003.07.003

3

R. Montenegro, G. Montero, J.M. Escobar, E. Rodríguez, J.M. González-Yuste, "Tetrahedral mesh generation for environmental problems over complex terrains", Lecture Notes in Computer Science, 2329, 335-344, 2002. doi:10.1007/3-540-46043-8_33

4

G. Montero, R. Montenegro, J.M. Escobar, "A 3-D model for wind field adjustment", Journal of Wind Engineering and Industrial Aerodynamics, 74-76, 249-261, 1998. doi:10.1016/S0167-6105(98)00022-1

5

G. Montero, E. Rodríguez, R. Montenegro, J.M. Escobar, J.M. González-Yuste, "Genetic algorithm for an improved parameter estimation with local refinement of tetrahedral meshes in a wind model, Advances in Engineering Software", 36, 3-10, 2005. doi:10.1016/j.advengsoft.2004.03.011

6

G. Montero, R. Montenegro, J.M. Escobar, E. Rodríguez, J.M. González-Yuste, "Velocity field modelling for pollutant plume using 3-D adaptive finite element method", Lecture Notes in Computer Science, 3037, 642-645, 2004. doi:10.1007/b97988

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