Keywords: 3D triangulations, finite element, adaptive meshes, object oriented method, adaptive refinement/derefinement, data structures, wind field models.

In previous works [2], we have presented a refinement algorithm for tetrahedral meshes. This algorithm is based on eight-subtetrahedral subdivision, according to an error indicator or an estimation of the numerical solution. If is the initial mesh, we can build a sequence of nested meshes , where is obtained from a local refinement of the previous level . The error indicator will be associated to the element , and it will be refined if , being the maximum value of the error indicator in , and the refinement parameter ( ).

In [3] the implementation for the derefinement algorithm is presented, where mesh nodes are removed if the difference between the numerical and the interpolated solution is lower than . The value of the derefinement parameter depends on the required precision.

Usually, it is difficult to obtain reliable error indicators that specify the elements that should be refined in the mesh. In the wind field model presented in [1], the error indicator is the gradient of the solution in each element. But not always is it possible to dispose of any indicator.

We now present a different method. The mesh will be globally refined, so error indicators or estimations are not required. After that, derefinement process is carried out according to the parameter. Each iteration of this method implies higher computational cost than local refinement attending to an error indicator, as all tetrahedrals in the mesh must be refined. But the total number of iterations of the proposed method could be much less if we do not chose a optimal refinement strategy.

Theoretically, this method is fully automatic. The sequence of meshes might be convergent to , where , being the number of nodes of . That is, new elements added by refinement of are not required to improve the numerical solution according to the parameter . But in practice, we have dealt with different cases. In one of them, we have obtained two groups of meshes which and , so the convergence will be never reached. The applied solution was to compare the number of nodes of with any previous mesh (). If the difference is less than a given percentage, the process can stop as both meshes are similar: .

Another case treated is related to the wind field model [1]. The difference in the numerical solution between terrain elements and their adjacent elements is not improved by the refinement process. New elements introduced would keep the difference with new terrain elements, and they would be refined time and again. Therefore, the parameter is not enough to ensure the mentioned convergence. We have forced the derefinement process in those elements that have edges with length less than a parameter .

An implementation of this method is presented with both, real and test problems. The test problem is a 3D Gaussian and a real problem is the wind model applied to the south of La Palma island.

1

G. Montero, E. Rodríguez, R. Montenegro, J.M. Escobar and 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, 2005; 36:3-10. doi:10.1016/j.advengsoft.2004.03.011

2

J.M. González-Yuste, R. Montenegro, J.M. Escobar, G. Montero and E. Rodríguez, "Local Refinement of 3-D Triangulations Using Object-Oriented Methods", Advances in Engineering Software, 2004; 35:639-702. doi:10.1016/j.advengsoft.2003.07.003

3

J.M. González-Yuste, R. Montenegro, J.M. Escobar, G. Montero and E. Rodríguez, "Implementation of a Refinement/Derefinement Algorithm for Tetrahedral Meshes", B.H.V. Topping, (Editor), Proceedings of the The Fourth International Conference on Engineering Computational Technology, Civil-Comp Press, Stirling UK, 2004. doi:10.4203/ccp.80.13

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