A Stretched and Twisted Cube.

The Distorted Mesh.

We start with a mesh of a unit cube. It has 8,007 cells. The edges of the tetrahedra have lengths close to 0.1. (Each of the figures below show the modified condition number of the elements.)

  cp ../../../data/geom/mesh/33/cube_1_1_1_0.1.txt mesh.txt 

The initial mesh of the cube. The minimum modified condition number is 0.58; the mean is 0.86.

Then we distort the mesh by stretching it along the z axis by a factor of 2 and twisting it around the z axis by an angle of .

  python.exe ../../../data/geom/mesh/utilities/vertices_map.py stretch mesh.txt distorted.txt
  python.exe ../../../data/geom/mesh/utilities/vertices_map.py twist distorted.txt distorted.txt 

The distorted mesh. The minimum modified condition number is 0.16; the mean is 0.59.

Topological and Geometric Optimization.

First we apply topological optimization. We use a dihedral angle deviation of 0.5 to define edge features. Intersecting edge features are corner features.

  utility/boundary33.exe distorted.txt boundary.txt
  optimization/topologicalOptimize3.exe -manifold=boundary.txt -angle=0.5 r1.txt r1t1.txt 

The mesh after topological optimization. The minimum modified condition number is 0.19; the mean is 0.68.

Next we apply one sweep of geometric optimization.

  optimization/geometricOptimize3.exe -boundary=boundary.txt -dihedralAngle=0.5 r1t1.txt r1t1g1.txt 

The mesh after geometric optimization. The minimum modified condition number is 0.28; the mean is 0.73.

Finally, we apply an additional cycle of topological and geometric optimization.

  optimization/topologicalOptimize3.exe -manifold=boundary.txt -angle=0.5 r1t1g1.txt r1t2g1.txt
  optimization/geometricOptimize3.exe -boundary=boundary.txt -dihedralAngle=0.5 r1t2g1.txt r1t2g2.txt 

The mesh after two cycles of topological and geometric optimization. The minimum modified condition number is 0.41; the mean is 0.78.

We see that the optimization has significantly improved the quality of the mesh. However, because of the distortion, the cells are on average about twice as large as in the original mesh. The following sections use refinement to deal with this.

Refinement Based on Edge Length.

We refine the mesh to deal with the stretching. We split any edges that are longer than 0.2. This results in a mesh with 20,867 cells.

  optimization/refine33.exe -length=0.2 -manifold=boundary.txt -angle=0.5 distorted.txt r1.txt 

The mesh after refinement based on edge length. The minimum modified condition number is 0.18; the mean is 0.63.

Next, as we did above, we apply two cycles of topological and geometric optimization.

The mesh after refinement based on edge length and two cycles of topological and geometric optimization. The minimum modified condition number is 0.40; the mean is 0.80.

We see that the refinement has little affect on the final quality of the mesh. It only reduces the size of the cells.

Refinement Based on Cell Quality.

A different strategy is to refine the mesh based on cell quality. We refine cells that have have been significantly distorted. To do this, we select the cells with modified condition number less than 0.5. This approach results in fewer edge splits than in the above section. The refined mesh has 13,034 cells.

  utility/cellAttributes33.exe -mcn distorted.txt mcn.txt
  utility/selectCells.exe -upper=0.5 mcn.txt indices.txt
  optimization/refineCells33.exe indices.txt distorted.txt rq1.txt 

The mesh after refinement based on cell quality. The minimum modified condition number is 0.19; the mean is 0.60.

Next, as we did above, we apply two cycles of topological and geometric optimization.

The mesh after refinement based on cell quality and two cycles of topological and geometric optimization. The minimum modified condition number is 0.41; the mean is 0.80.

In terms of mesh quality, refinement based on cell quality is little different than no refinement or refinement based on edge length.