Single QIAC

Consider the initial reference grid and the embedded quadrilateral shown in Fig. . We will use this example to illustrate the basic single QIAC algorithm:

1.

Assign logical (grid) coordinates for the vertices and center of the quadrilateral and identify the closest reference point for each vertex:

• Identify the grid point closest to each vertex of the quadrilateral and assign its logical coordinates to the vertex. In Fig. if the grid lines are numbered from 0 to 10, then the grid index closest to the vertex V₀ at (0.2,0.6) is (2,6), V₁ is at (5,2), V₂ is at (8,4), and V₃ is at (5,8). Here we have ordered the four vertices in counterclockwise in the logical (grid) coordinate system.
• Determine a starting vertex for the algorithm in a consistent manner. We use an algorithm that identifies the vertex in the lower left corner for the logical grid system. The horizontal coordinate of this vertex is smaller than the next vertex and the sum of absolute value for the slopes of the two edges is less in the x-direction than in the y-direction. For this example, only V₀ and V₁ satisfy the first condition. For the sum of absolute value of the logical slopes between V₀ - V₁ and V₂ - V₃, is ((6-2)/(5-2) + (8-4)/(8-5) = 8/3). This sum between V₁ - V₂ and V₀ - V₃ is ((4-2)/(8-5) + (8-6)/(5-2) = 4/3). Therefore, we define V₁ to be the low left corner in logical coordinates.
• Define reference points in the logical coordinate system.
  • Define the logical center of the quadrilateral as the average of the logical coordinates of the four vertices. For this example the logical center is at (5, 5).
  • Define average logical length in the x-direction of the logical coordinate as (8-5+5-2)/2+1=4. The average logical length in y direction is (6-2+8-4)/2+1=5.
  • Define the logical coordinates for the four vertices based on the logical coordinates of the center and average length in each direction. The logical coordinates for the low left corner V₁ is (5-4/2,5-5/2)=(3,3). The logical coordinates for other three vertices is V₂=(7,3), V₃=(7,7), V₀=(3,7).

2.

Split the domain into nine parts by connecting the four vertices with the domain boundaries by moving the closest reference point to each vertex. as shown in Fig. . For this example, the closest reference point of each vertex and vertex itself are at the same location. For general grid, this may not be true. Some parts may be degenerated into a line if one line segment of the quadrilateral is on the domain boundary.

3.

Using the inverse interpolation, compute the grid locations for the line segments on the quadrilateral, along the lines connecting these segments to the domain boundaries and on the domain boundaries as shown in Fig. .

• First define the grid points on the lines connecting the vertices with the domain boundaries. Note that to prevent the grid from crossing each other on a boundary, the locations on the domain boundaries and the points nearest to the boundaries are not changed by the inverse interpolation.
• Using the inverse interpolation, define the grid positions on the domain boundary.
•   Define the grid points between the vertices on the QIAC.

4.

If the domain boundary has kink point, move the reference grid point closest to the kink point to the kink point, as we did for aligning the grid with an SIAC. (Note, that there are no kink points in the example shown in Fig. ). Because we modify the grid point locations only on the four lines, where the edges of quadrilateral lie, and the domain boundaries, the grid we generate after this step is very irregular.

5.

  Using the same smoothing operator as for the SIAC, generate the aligned grid while freezing the grid points on the QIAC and domain boundary.

     

Figure: The initial QIAC is embedded in a uniform 3#3 initial grid.

=3.75in quainput1.eps

Figure: The domain is split into nine subdomains by exending the vertices to the domain boundaries.

=3.0in quainput2.eps

Figure: The reference grid points are defined for the line segments on the QIAC, the lines connecting the vertices to the domain boundaries and along the domain boundaries.

=3.0in quainput5.eps

Figure: The initial grid points nearest the IACs are moved to the newly defined reference grid points.

=3.0in quainput6.eps

Figure: The grid is regularized by the smoothing iteration.

=3.0in quainput7.eps

To avoid a singularity, the algorithm requires the reference grid to be sufficiently fine so that in the first step different vertices have different logical coordinates. Even with this restriction, two vertices may have the same nearest reference point, which could lead to the singularity of the quadrilateral. This often happens for a thinner layer where two vertices are very close. This can be avoided by using a finer reference grid to find the reference point for each vertex. The minimum spacing for the finer grid should be larger than the minimum distance between any two vertices. Rather than have the algorithm fail when this occurs, we define the reference vertices sequentially by binding the vertex is closest to a reference point to that point and eliminating it for consideration as a reference point for any other vertices. This approach is robust, and can create a locally fine grid near the quadrilateral.