Algorithm

A popular method of graded triangulation is by use of a background mesh for the entire domain. The background mesh is a triangulated mesh generated using only the boundary points. Such a mesh can be generated by the use of initial Delaunay triangulation. Now the graded triangulation differs from the AFM only in that the function for calculating the trial point is modified to accommodate the grading. This function now uses the information of the background mesh such that the length of the triangle's edges increase from the minimum value to the maximum. At any point in the domain the edge length of the sides of the triangle under construction are calculated to ensure gradual variation by finding the background triangle that encloses the point. The barycentric coordinates of the point with respect to the three nodes of the triangle are calculated. The mesh density at any node is defined as the mean of the lengths of the edges meeting at that point. If the mesh density at the nodes of the triangle containing the point are L₁, L₂ and L₃ and the respective barycentric coordinates of the point are $\alpha$, $\beta$ and $\gamma$ (note that $\alpha + \beta + \gamma = 1$), then the required lengths of the new edges are given by

$$L = \alpha L_1 + \beta L_2 + \gamma L_3$$

The disadvantage of this method is that the Delaunay triangulation for initial triangulation should be done before starting the actual triangulation, but it results in a far better mesh than direct graded triangulation which does not use any background mesh.