The generation of the initial mixed mesh and the effect of the following mesh optimization process is demonstrated on a quadrilateral domain with a square and a circular opening (Fig. 10). The first figure depicts the domain of interest. In the next three figures, the propagation of the front is illustrated. The completed initial mesh after one cycle of smoothing is shown in the fifth figure. The final optimized mesh is displayed in the last figure. In all figures, the triangular elements are shaded and the edges on the current front are drawn in bold. The actual performance of the proposed algorithm is presented on several examples. In the first example, a locally graded mesh is generated over a planar complex domain (Fig. 11). The second example presents a uniform mesh over a torus (Fig. 12). In the last example, a mesh of a chair subjected to curvature based mesh size control is demonstrated (Fig. 13). The numbers of generated triangular and quadrilateral elements, in the initial and final mesh, together with the average and worst quality and minimum and maximum dihedral angle are summarized in Table 1 for individual examples. Note that the quality of a triangular element is evaluated according to the following formula
where is the area and , , and are the side lengths of the triangle. The quality of a convex quadrilateral element is determined as
where , , , and are the qualities of four different triangles obtained by two possible diagonal swappings. Both qualities are calibrated to range from 0 to 1.