Keywords: mesh generation, mesh smoothing, surface meshes, surface mesh optimization, adaptive meshes.

This paper presents a new procedure to improve the quality of triangular meshes defined on surfaces. The improvement is obtained by an iterative process in which each node of the mesh is moved to a new position that minimizes a certain objective function. This objective function is derived from an algebraic quality measures of the local mesh, , [1] (the set of triangles connected to the adjustable or free node). The optimization is done in the parametric mesh, where the presence of barriers in the objective function maintains the free node inside the feasible region. In our case, the parametric space is a plane, chosen in terms of the local mesh, in such a way that this mesh can be optimally projected performing a valid mesh, that is, without inverted elements. In this way, the original problem on the surface is transformed into a two-dimensional one on the parametric space. The barrier has an important role because it avoids the optimization algorithm to create a tangled mesh when it starts with a valid one. Nevertheless, objective functions constructed by algebraic quality measures are only directly applicable to inner nodes of 2-D or 3-D meshes [2], but not to its boundary nodes.

To overcome this problem, the local mesh sited on a surface , is orthogonally projected on a plane in such a way that it performs a valid local mesh . Therefore, it can be said that is geometrically conforming with respect to . Here is the free node on and is its projection on . The optimization of is got by the appropriated optimization of . To do this we try to get ideal triangles in that become equilateral in . In general, when the local mesh is on a surface, each triangle is placed on a different plane and it is not possible to define a feasible region on . Nevertheless, this region is perfectly defined in .

To construct the objective function in , it is first necessary to define the objective function in and, afterward, to establish the connection between them. A crucial aspect for this construction is to keep the barrier of the 2-D objective function. This is done with a suitable approximation in the process that transforms the original problem on into an entirely two-dimensional one on .

The optimization of becomes a two-dimensional iterative process. The optimal solutions of each two-dimensional problem form a sequence of points belonging to . We have checked in many numerical tests that is always a convergent sequence. It is important to underline that this iterative process only takes into account the position of the free node in a discrete set of points, the points on corresponding to and, therefore, it is not necessary that the surface is smooth. Indeed, the surface determined by the piecewise linear interpolation of the initial mesh is used as a reference to define the geometry of the domain.

If the node movement only responds to an improvement of the quality of the mesh, it can happen that the optimized mesh loses details of the original surface. To avoid this problem, every time the free node is moved on , the optimization process only allows a small distance between the centroid of the triangles of and the underlaying surface (the true surface, if it is known, or the piece-wise linear interpolation, if it is not).

1

P.M. Knupp, "Algebraic mesh quality metrics", SIAM J. Sci. Comp., 23, 193-218, 2001. doi:10.1137/S1064827500371499

2

L.A. Freitag, P.M. Knupp, "Tetrahedral mesh improvement via optimization of the element condition number", Int. J. Num. Meth. Eng., 53, 1377-1391, 2002. doi:10.1002/nme.341

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