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Argonne National Labs, which is 5th fastest I think, is used for weather models, material science, and more[1]

[1]http://www.alcf.anl.gov/

There are dual-use systems which run classified and non-classified codes (they can be partitioned.. They typically live at LANL or LLNL rather than ANL or Berkeley (Berkeley in particlar doesn't do any classified work). Note that most classified codes are actually physics/material/explosion/plasma physics simulations and you can't always tell they are running on your system.

I checked the list of projects running on ALCF and I see some pretty obvious nuclear weapon stockpile stewardship and weapon design projects, such as "Validation Simulations of Macroscopic Burning-Plasma Dynamics"

Nucleosynthesis of heavier materials in supernova explosions and neutron star mergers are two cases of problems that I guess are used for "publicly" validating dark codes.

(I know a guy that Los Alamos is trying to recruit to do some dark work for them, and he does nucleosynthesis in neutron star mergers.)

Those would be useful, but typically the validation codes use terms like "multiscale combustion physics" or "coupled neutron/radiation transport".

I think the folks simulating supernova and neutron stars have a lot of physics overlaps, but I don't think that data is used directly for stockpile stewardship.

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