Keywords: mesh smoothing, untangling, three dimensional surface meshes.

Mesh generation algorithms are used to generate valid meshes, however in some cases the mesh changes and the same conditions should be ensured during the analysis. To ensure that the mesh stays valid, mesh modification, smoothing or optimization algorithms must be applied to the mesh.

The mesh optimization algorithm must be robust and should always produce a valid mesh. This paper focuses on the adaptive smoothness functional optimization method published by Tinoco-Ruiz and Barrera-Sanchez [1,2,3]. They were interested in meshes for finite difference schemes and for their requirement that the meshes must be "convex or unfolded". This method is a discrete variational approach where "a function of the inner points is designed and it is expected that its minimum be attained in a grid with good geometrical properties" [2].

This paper investigates and demonstrates how this method can be extended to unstructured quadrilateral meshes and for three dimensional surface meshes with quadrilateral elements. The paper introduces a new objective function for the three dimensional case and a new procedure to calculate this objective function.

1

J.G. Tinoco-Ruiz, P. Barrera-Sanchez, "Smooth and Convex Grid Generation over General Plane Regions", Mathematics and Computers in Simulation, 46(2), 87-102, 1998. doi:10.1016/S0378-4754(98)00074-3

2

J.G. Tinoco-Ruiz, P. Barrera-Sanchez, "Area control in Generating Smooth and Convex Grid over General Plane Regions", Journal of Computational and Applied Mathematics, 103(1), 19-32, 1999. doi:10.1016/S0377-0427(99)00237-X

3

P. Barrera, G. Tinoco, "Area Functional in Plane Grid Generation", Proceedings of the 6th International Conference on Numerical Grid Generation in Computational Field Simulation, Avery Village, Londres, 1998

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