direct train from barcelona to madrid international airport

No it does not. It stops at Madrid Atocha station. And then you transfer from there by train or bus.
https://www.welcomepickups.com/madrid/airport-transfer-to-atocha-train-station/

From Atocha, you can take the metro (subway) line 8 or the cercanias (suburban train) line C1 or C10. They both run frequently and take about 1/2 hour. The cercanias is the cheapest - it's free with your AVE ticket from Barcelona. You can get the cercanias ticket from machines in Atocha. You either scan your AVE ticket or input your ticket number - it's been a couple of years, so I don't remember exactly.

"You either scan your AVE ticket or input your ticket number - it's been a couple of years, so I don't remember exactly. "

At least in 2015, on your ticket was printed a CombinadoCercanias, which was a number you put into a machine to get the ticket for the cercanias train from Atocha to the airport. I had to ask someone in Barcelona Sants station for help with this, so don't be afraid to ask for assistance in Madrid Atocha station (it's huge and confusing, with multiple levels).

Chani and Harold: this is exactly what I am looking for. I just went thru Los Angeles from T1/A to T2 from inside the bldgs because airport employee said "just go straight"....that "straight" is 25 min with us dragging our suitcase, when we could have go out the bldg where our domestic plane landed...go outside and catch shuttle bus which is a mere 10 min and comfortably seated on a bus, lol. Thank you very much again.