Keywords: mesh smoothing, mesh untangling, mesh generation, adaptive refinement, nested meshes, 3-D finite element method.

There are two basic ways to improve the quality of a pre-existing mesh. The first, usually named mesh optimization, consists in moving each node to a new position that improves the quality of the surrounding elements. This technique preserves the topology of the mesh, that is, it does not modify the connectivity of the nodes. The second one involves some changes in the node connections. For example, edge swapping is a well known technique included in this category. In this work we propose an hybrid method that combines both approaches.

Firstly, we focus in tetrahedral mesh optimization. The quality improvement in mesh optimization may be obtained by an iterative process in which each node of the mesh is moved to a new position that minimises a certain objective function. The objective function is derived from some quality measure [1] of the local submesh, that is, the set of tetrahedra connected to the adjustable or free node. Among the many objective functions described in the literature [2] those having barriers are specially indicated to improve the quality of a valid mesh in which there are non inverted elements. In this case, the barrier avoids the possible appearance of inverted elements in the optimization process. Nevertheless, the existence of barriers prevents these objective functions from working properly when the mesh is tangled. For example, if the free node is out of the feasible region (the positions where the free node must be located to get a valid submesh) the barrier avoids reaching the appropriate minimun. It can even happen that the feasible region does not exist, for example, when the fixed boundary of the local submesh is tangled. In all these situations these objective functions are not well defined on all R³ and, therefore, they are not suitable to improve the quality of the mesh. To overcome this problem we can proceed as Freitag et al in [3], where an optimization method consisting of two stages is proposed. In the first one, the inverted elements are untangled by an algorithm that maximises their negative Jacobian determinants; in the second, the resulting mesh from the first stage is smoothed using a objective function with barrier, based on the element condition number.

In this paper we propose an alternative to these procedures, such that the untangling and smoothing are carried out in the same stage. In order to do this, we substitute the objective functions by modified versions that are defined and regular on all R³. With these modifications, the optimization process is also directly applicable to meshes with inverted elements, making a previous untangling procedure unnecessary [4]. This simultaneous procedure allows the number of iterations for reaching a prescribed quality to be reduced. Nevertheless, the improvement in the mesh quality without changing its connectivity is bounded. This bound is associated with the topology of the mesh and with the constraints imposed by the boundary of the domain. In practice, we observe that both average and minimum quality tends to become steady to its respective bounds as the number of iteration increases. As result, once a sufficient number of iterations has been done, the mesh quality will not improve significatively and the process must then stop.

In this work we propose to combine the above optimization techniques with the mesh refinement algorithm presented in [5]. The main idea consists in increasing the node number, and thus, the degrees of freedom, in the neighbourhood of the regions where the elements have poor quality. Then, we refine all the elements whose quality are below to a certain threshold. Once it is done, we initiate another stage of optimization until the quality of the mesh reaches a limit. The overall process can be repeated several times until the required quality is obtained or no additional improvement is got. To illustrate the effectiveness of our approach, we present several applications where it can be seen the validity of the proposed strategies.

1

P.M. Knupp, "Algebraic Mesh Quality Metrics", SIAM. J. Sci. Comput., 23, 193-218, 2001. doi:10.1137/S1064827500371499

2

P.M. Knupp, "Achieving Finite Element Mesh Quality Via Optimization of the Jacobian Matrix Norm and Associated Quantities. Part II-A Frame Work for Volume Mesh Optimization and the Condition Number of the Jacobian Matrix", Int. J. Numer. Meth. Engng., 48, 1165-1185, 2000. doi:10.1002/(SICI)1097-0207(20000720)48:8<1165::AID-NME940>3.0.CO;2-Y

3

L.A. Freitag, P.M. Knupp, "Tetrahedral Mesh Improvement Via Optimization of the Element Condition Number", Int. J. Numer. Meth. Engng., 53, 1377-1391, 2002. doi:10.1002/nme.341

4

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, Springer Verlag, Berlin Heidelberg New York, 335-344, 2002. doi:10.1007/3-540-46043-8_33

5

J.M. González-Yuste, R. Montenegro, J.M. Escobar, G. Montero, E. Rodríguez, "An Object Oriented Method for Tetrahedral Mesh Refinement", Third International Conference on Engineering Computational Technology, Civil-Comp Press, in CD-ROM, Paper 72, 1-18, 2002. doi:10.4203/ccp.76.10

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