Efficiency and error in the finite element method with application to structures

Alexandra D. Danciu

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Abstract

The sixth chapter presents the evolutionary procedure proposed through this thesis. In the first part an overview of the use of evolutionary methods in meshing is done.
The purpose of the present chapter is to show that, by using evolutionary procedures, after a few generations, one can obtain acceptable triangular meshes that could be used in analysis after a minimum correction. The triangulations obtained are optimized after two parameters: the quality of the triangles obtained and the refinement of the mesh. Through the implemented operators I tried to keep the quality of the newly generated triangles and, where possible, to improve the quality of the initial triangles.
All the steps of the procedure are presented in detail. The operators presented in the literature have been implemented and their performance evaluated for the new procedure proposed. Two new cross-over operators have been proposed, explained in detail and implemented, as well as a new mutation operator responsible for the mesh refinement. A situation appearing during mutation was named as fan mutation and ways to resolve it have been presented. The fitness of the individual is evaluated through a cumulative function, that takes into account the quality of the individual though the deviation fitness and the refinement obtained through the refinement fitness .
For a plane octagonal domain the influence of the number of generations, the division rate, the mutation rate, the cross-over rate and the initial triangulation are investigated.
Next the issue of solving a domain with holes is addressed. The procedure starts with a set of initial triangles that take into account the geometry of the body to be meshed and the operators proposed are not modifying the geometry; therefore, any type of geometry can be meshed.
The strong and the weak points of the procedure are investigated and ways of improving it are proposed at the end of the chapter. As well, a short overview of the unstructured meshing techniques is done.
Is is shown in this chapter that the proposed procedure can improve the fitness of the initial triangulation with 25 to 56.5%. The number of generations for which the results were obtained is relatively small, 25 to 100 generations. It was shown that, for the operators implemented, the initial triangulation can affect the final result, the evolution of those individuals having a common vertices for all the triangles is much weaker than the evolution of the other individuals. It was shown that the increase of the number of individuals in the population and by increasing the number of generations, the results can be improved. Therefore one of the recommendations is the investigation of the performances of the procedure in grid or cloud computing.

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