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I have solved this problem, though I don't understand quite why...
I believe the problem was caused by using a "No Penetration" contact set between two touching faces. Possibly the solver was trying to find a way for them not to touch, and it sent everything flying 1e6 inches in every direction in its attempts to do so.
I changed the contact set to "Free (no interaction)" and used other means (Bonded Connection, Pin with key and retaining ring) of securing the component which previously had "No Penetration" to stop it from moving. This method brought more realistic results.
So, if you ever see your sheet metal melt, check your "No Penetration" contacts. If the two parts are touching, try changing the contact set to a rigid connection, a link, or bonded. You will probably see better results.
I haven't solved this problem, because I can create assemblies with no penetration contact sets where the sheet metal doesn't melt. It's frustrating to have to work-around a bug you can't understand. It's most annoying because the melted parts always displace way more than they should, which throws the rest of the results off.
It usually comes up because I want to use bolt connectors (and to do that, you need to define "no penetration" between the flange faces). So I tried making a smaller, simpler assembly to check and see what I might be doing that is making this happen. Well, I get nothing but good results from that small assembly! I can't re-create the melting problem.
Has anybody else run into this? Any further insight? Can anybody re-create the problem?
I have tried to re-create the problem and I may have come up with the cause. I am working with large assemblies so I have been doing small studies first, then copying the contact sets up to the next largest assembly. So the "no penetration" contact set was created in a different configuration than the one in which I see my sheet metal melting. Even though the faces are correct, I believe there is some error when copying to another configuration and using it in another study than where it was created. The shell elements melt when this error is present.
If you are copying contact sets, leave out the "no penetration" sets and re-create them in each study that you do. Then use bolted connectors as usual. If your sheet metal melts, delete the "no penetration" contact sets and re-make them. Hopefully you don't have too many.
Until Solidworks comes up with a more robust solver, this workaround will have to do.