One problem on generating 2D regular grids based on mappings - Azarenok, B. N.; Charakhch’yan, A. A. (Mathematical models and computer simulations, 2015)
To study the known problem on generating a regular grid by means of the Winslow method in a rectangular domain with a boundary kink known as a backstep, a high-accuracy numerical method for the inverse harmonic mapping of the unit square onto the domain with a certain mapping of the domain boundaries is developed. The behavior of mapping level lines that enter the point of the boundary kink is studied. Near the kink, the angle between the boundary and the straight line connecting a point on the level line with a point of the boundary kink is found as a function of the coordinate of the point on the level line in the unit square. It is shown that the level line of the mapping is tangent to the boundary at the kink point. Near the kink point the mapping is not quasi-isometric. The regular grid generated according to the intersection points between the level lines connected by straight lines contains a self-intersecting cell that remains when the grid step along the boundary decreases. Based on the universal elliptical equations reproducing any nondegenerate mapping of the parametric rectangle onto a given domain, a simple two-parametric control of the grid nodes in the backstep that makes it possible to effectively control the slope angle of the grid line entering the point of the kink, thereby removing the escape of the grid lines from the domain boundary, is suggested. In the case of a small number of grid points 31 × 31, the nondegenerate grid is generated by selecting a suitable value of one parameter. If the number of grid nodes is increased by a factor of 8 along both directions (grid size 241 × 241), the nonconvex cells appear within the domain, which are easily removed by using the variational barrier method. Another possibility to avoid nonconvex cells is to decrease the grid dimension along the second direction (grid size 241 × 121).