Couple of questions, how are you processing it? What service is being killed - can you connect to SSAS or SQL Server after the cube has failed?

The error looks like the SQL database connection is being killed. That could be beacause the SQL is timing out but most likely it is due to another problem that isn't in the log you provided. I think SSAS is killing the SQL connection because it has failed to process (key errors or ran out of diskspace for example)

Right clicking on the database & processing via SSMS will often give you a better idea of why it is failing

Mack