topopt tutorial on 1d multilayer film optimization

[stevengj]

As discussed with @oskooi, it might be nice to have a tutorial with "1d" topology optimization, i.e. designing a multilayer film. This can be done even in 2d and 3d just by making the material grid a 1x1xN grid (which can then be "extruded" over a higher-dimensional block object).

1. For example, designing a multilayer film to minimize transmission should result in essentially a quarter-wave stack (possibly with some modification for the first and last layers). Just minimize transmission with the abovementioned 1d material grid, ramping up β slowly as usual to let it decide on the number of layers.
2. We should also be able to fix the number of layers. If you just set β=infinity (with the new subpixel-smoothing algorithm) then the optimization will tend not to change the topology (because the derivative is not correct for changes in topology) unless it happens to take a very large step. You can probably prevent it from doing this by simply counting the number of layers (from the projected material grid) on each step and adding a large penalty to the objective function if the number of layers changes, which will cause the optimization algorithm to backtrack.

(You could also use a level-set representation if you have a differentiable mapping function that maps a set of layer thickness to a density ρ, ala #2235.)

On a separate (but related) note, we should probably update the fill_factor calculation in the subpixel smoothing code here to always compute the fill factor for a 3d spherical smoothing kernel — i.e. take the perspective that even "2d" and "1d" structures are actually 3d structures that are invariant in 1 or 2 directions, respectively.

[erikshipton]

Isn't this kind of like the Needle Optimization for thin films? I apologize if thats obvious and been pointed out elsewhere but Needle optimization for thin films kind of feels like topological optimization in 1D.