A robust direct variational approach for generation of quadrangular and triangular grids on planar domains

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Abstract

We consider the problem of the quadrangular and triangular grid generation. The algorithm proposed for the solution of this problem is based on a generalization of the so called variational method. In particular, this generalization takes into account a combination of length and area functionals for a given triangular grid related to the quadrangular grid under consideration. As a consequence of this generalization high quality triangular grids of a given domain are obtained in a straightforward way from the quadrangular grids computed by this algorithm. We test the proposed algorithm on standard numerical examples.

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 R be the real Euclidean space. Let ΩR2 be a compact and connected domain, RR2 be the unit square. Let M and N be two positive integer numbers, we define the following set of indices: I={(i,j),i=0,1,,M;j=0,1,,N}, I1={(i,j),i=0,1,,M1;j=0,1,,N}, I2={(i,j),i=0,1,,M;j=0,1,,N1}, I0={(i,j),i=1,2,,M1;j=1,2,,N1}, I=II0. Let v_1=(x1,y1)T,v_2=(x2,y2)TR2, where the superscript T means transposed, we denote with v_12=x12+y12, that is the squared length of v_1, and v_1×v_2=x1y2y1x2,

The use of triangular grid in direct variational approach

We modify the length and area functional in order to lower the dependence of w with respect to the geometry of domain Ω. Given a rectangular grid R over R and given two positive integers nh,nv, each rectangle of R is split in nh rectangles along the horizontal direction and nv rectangles along the vertical direction. Clearly in the corresponding quadrangular grid Q=u_(R) each quadrilateral Qi,j, (i,j)I1I2, of Q is split in nhnv subquadrilaterals Qi,jh,k, h=0,1,,nh1 and k=0,1,,nv1. In

The numerical experiments

Let Q be a quadrangular grid over Ω, we denote with TQ the triangular grid associated to Q, that is obtained in the following way: each quadrilateral of Q is split in the four triangles obtained by joining the vertices of the quadrilateral with its barycenter. We note that for grid Q having NM quadrilaterals the associated triangular grid TQ has 4NM triangles, moreover, if grid Q has convex quadrilaterals, then TQ is unfolded. In Fig. 4 we have an example of grid TQ.

Let T 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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