A robust direct variational approach for generation of quadrangular and triangular grids on planar domains
Introduction
Grid generation techniques are largely used in several different fields to obtain the numerical solution of partial differential equations. The properties required for the grid used in these approximations usually depend on the problem considered and on the discretization scheme employed. We consider the generation of triangular grids, whose elements are all triangles, and of quadrangular grids, whose elements are all quadrilaterals.
The grid quality is an important concept that needs to be considered in the generation step. Sometimes this concept depends on the particular problem that must be solved and several different grid quality measures are proposed in the scientific literature, see [12], [2], [15] for details. In a general context poorly shaped triangles or quadrilaterals can cause serious difficulties in the corresponding numerical application. For example, in triangular grids, triangles with a large internal angle increase the discretization error in the corresponding finite element solution [1], on the contrary, triangles with an internal angle too small increase the condition number of the element matrix [11]; note that in a low quality triangular grid these two cases usually occur in the same time. Thus, some desirable geometric properties are requested, in particular, triangular grids usually need to be made of triangles as much as possible equilaterals; quadrangular grids usually need to be made of convex quadrilaterals as much as possible near to rectangles.
We consider the direct variational method for quadrangular grid generation. In this method the geometric properties of the desired grid are expressed as a function of the inner vertices of the grid, and the grid results from the minimum of such function. This approach has been presented by Castillo and Steinberg [3], [4], [5], and several different modifications have been introduced in order to improve such a method, see [13], [16], [6], [7], [8] for some examples.
We propose a new direct variational method for the generation of quadrangular grids. This method can be seen as an improvement of the method proposed in [8], and it is based on a modification of the usual length and area functionals, taking into account the geometric properties of a triangular grid related to the quadrangular grid under consideration. As a consequence a high quality triangular grid can be obtained in a straightforward way from this quadrangular grid.
Some numerical experiments are reported to show the stability properties of the proposed method. In particular, for each domain taken into account in the numerical experiments, we compare the grid computed by using proposed method with the best possible one computed by using the usual direct variational method. This comparison is based on some classical grid quality measures and some classical grid generation examples.
In Section 2 we briefly describe the grid generation problem and the direct variational formulation. In Section 3 we present the suggested modification of the functional. In Section 4 we give some numerical results. In Section 5 we give some conclusions and future developments of the work.
Section snippets
The direct variational approach
Let be the real Euclidean space. Let be a compact and connected domain, be the unit square. Let M and N be two positive integer numbers, we define the following set of indices: , , , , . Let , where the superscript T means transposed, we denote with , that is the squared length of , and ,
The use of triangular grid in direct variational approach
We modify the length and area functional in order to lower the dependence of with respect to the geometry of domain . Given a rectangular grid over R and given two positive integers , each rectangle of is split in rectangles along the horizontal direction and rectangles along the vertical direction. Clearly in the corresponding quadrangular grid each quadrilateral , , of is split in subquadrilaterals , and . In
The numerical experiments
Let be a quadrangular grid over , we denote with the triangular grid associated to , that is obtained in the following way: each quadrilateral of is split in the four triangles obtained by joining the vertices of the quadrilateral with its barycenter. We note that for grid having quadrilaterals the associated triangular grid has triangles, moreover, if grid has convex quadrilaterals, then is unfolded. In Fig. 4 we have an example of grid .
Let be a triangular grid,
Conclusions
We proposed a direct variational method for the quadrangular grid generation problem. This method is a refinement of the one proposed in [10] and it arises from a modification of the usual length and area functionals, where a proper triangular grid is considered. The main contribution of the paper is given by a convincing numerical experience, where the robustness of the method is tested.
Several different developments of this method can be considered. In particular, we suggest further
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