How are you updating the calculations and how often?
You could definitely script the publishing of the extract from tabcmd: tabcmd Commands
But if you could tell us a bit more about your specific situation, we could better help.
In addition to what Diego suggested above, i was wondering if you've also looked at publishing empty extracts to Tableau Server.
It could be useful in big extracts cases where the published version doesn't contain the forced fail condition in the extract definition, so any refresh – immediate or on a schedule – will populate the extract in full.