Grid Refinement

 

The necessity of grid refinement is always given when the simulation grid cannot resolve the attributes accurately. The initial simulation grid must be adapted to the initial attribute profiles prior to the discretization of the equation system. The grid elements are refined according to gradient and dose criteria deduced from the input quantities. During the refinement the original parent grid element is replaced by a configuration of child elements. The so-called ``red'' and ``green'' refinement elements are given by Figures 4.1-6a and b, respectively, which are used within the TRIGEN grid generator. The new generated mid-edge points are connected to the neighboring triangle by green refinement to ensure grid consistency.

  
Figure 4.1-6: Grid refinement technique for parent elements; a) red refinement at mid-edge points b) green refinement to connect mid-edge points and c) circumcenter refinement suitable for Delaunay grids.

Another possibility is to split up the parent element at the circumcenter into three child elements (see Fig. 4.1-6c). If the parent element is a Delaunay triangle, the child elements are loosing this feature. Additional algorithms like e.g. edge swapping have to be applied to ensure also Delaunay-criteria for the child elements. The advantage of this method is that only one new grid point is generated, where in the case of red refinement three new mid-edge points are inserted. Furthermore, as the new generated point is the circumcenter of the parent triangle, one can be sure that no other point is within the local vicinity, hence nasty point density checks are avoided.

Special precautions have to be taken during the red-green refinement procedure to avoid that green triangles are refined again, because these green triangles lead to obtuse angles which degenerate the quality of the mesh. In the case when a green triangle is marked for refinement the twin green triangles have to be replaced by one red triangle prior the next refinement step. The following algorithm presents details of a grid refinement algorithm used in an extend version of the TRIGEN grid generator.

 generate_geometry_grid; 
/* start refinement */ 
initialize_interpolation_service; 
red=0; 
newtriangles=oldtriangles; 
assemble_neighbor_info; 
for (i=0;i < maxlevels;i++) 
{ 
for (j=0;j < newTriangles;j++) 
{ 
if (! triangle_interpolated) 
interpolate_attributes_on_triangles; 
mark_triangles_as_interpolated; 
if (! triangle_checked) 
check_refinement_criteria_on_triangles; 

mark_bad_triangles_for_refinement(&red); 
mark_traingle_as_checked; 
} 
newTriangleNumber = oldTriangles + 4 * red; 
newPointNumber = oldPoints + 3 * red; 
reallocate_grid_structure(newTriangleNumber,newPointNumber); 
if (green_refine_detected > 0) 
{ 
red_refine_green_parent; 
} 
split_refine_grid_lines(&ptIndex); 
red_green_refinement(&triIndex); 
split_neighbor_info; 
newTriangles = triIndex; 
newPoints = ptIndex; 
} 
write_grid;

This algorithm ensures a sufficient refinement by a recursive marking function which avoids hierarchical refinement level differences of more than one between neighboring triangles. The refined mesh obtained by the above refinement algorithm is given in Figure 4.1-7a. Thereby the source/drain doping is resolved according to a local gradient and dose criterion. In some regions the relatively coarse initial parent triangles can be recognized. As the decision whether a triangle should be refined or not is performed at the mid-edge points of a triangle, some information can be lost if the profhierarchical refinement level differenceiles are extremely shallow compared to the triangle size.

The same doping profile was taken by the TRIANGLE grid generator program to refine the geometry grid. The internal refinement techniques are much more sophisticated than compared to other grid generators, including edge swapping algorithm to obtain better grid quality (see Fig. 4.1-7b). The refinement is based on minimum angle conditions, triangle area constraints and on local gradient and dose criteria. A flexible interface allows the implementation of nearly every suitable refinement criterion.

  
Figure 4.1-7: Triangular grid refinement according to source/drain doping profile: a) using TRIGEN and green-red refinement (4462 triangles generated) and b) using TRIANGLE and circumcenter refinement and several quality operations (3241 triangles generated).