I remember reading somewhere that the purpose of yoga's asanas was to prepare the body for a seated meditation practice...

I personally can't do half-lotus, I just bend the knees and place one foot in front of the other. I also do a lot of spine strengthening and flexibility exercise, and use a meditation bench. This has allowed me to sit for 40 minutes (I've been training for two months only), but I mostly only do several shorter sessions of 10-15 minutes each.