Gotcha!

The grid approximation output now looks as it should, thanks to some carefully planned and intelligent testing (aka calculated guessing). As I had expected after my previous attemptsfailed, the parameters used to scale the force between a grid and vertex were too low.

Actually, the parameters were far too low. I had been using the parameters used in Walshaw's work; -Cwk²/d, where C was 0.2 (or some value less than 1). For my implementation, the better results are using a value of 50. These are preliminary values, a more suitable value will be found in the future.

The higher this value, the more a grid will affect a vertex (so pushing the vertices out of its square and not keeping it on the edge). Below is a few images to show the differences. On the left is the force calculation, on the right, the output.

0.1 * weight * distance As can be seen the graph is sticking to the grid.
0.5 * weight * distance Same again, the graph is sticking to the grid, but the vertices within squares appear better spaced.
1.0 * weight * distance As above, the structure begins to look clearer at this point and the graph resembles its normal self (loosely)
5.0 * weight * distance As the force increases, the graph looks more and more as it should (with no grid) but there is still vertices sticking.
25.0 * weight * distance Skipping ahead a large value, the graph looks almost normal again (I know subjective), but there are still vertices clumping
50.0 * weight * distance Skip to another larger value and the graph now looks as it should (albeit "squashed"). This output resembles that given in Chris Walshaws paper.
1000.0 * weight * distance I wanted to see what would happen with an extremely large value, so I put in 1000. The graph takes an unusual shape, it looks compressed but also very smooth