I discovered a simple solution to Alienware M18 R1 battery charging problem. Thought I would post about this in the event someone else is having a similar problem and other solutions didn't work or are just too convoluted or doesn't work every time.
Initially I found the solution of going through a series of steps to get it to work, but my gosh, what a pain. The steps are to unplug the power adapter from outlet, unplug the power adapter from computer, turn off the computer, remove the battery, do a power drain (hold down the start button for a while), and install the battery. Then you turn the computer back on, plug in the power adapter (only after the green light has gone completely off) and then plug the adapter into the wall. That worked...sometimes.
Sometimes I would just unplug the adapter from the wall outlet and the computer, wait for the green light to go out and then plug the adapter back into the computer and then plug the adapter into the wall outlet. That also worked...sometimes.
I discovered something by accident recently which is very simple and has worked each time.
To get my Alienware M18 R1's battery to charge, I unplug the power adapter from the computer, shut the lid, plug in the adapter, wait for all the pretty lights to go out, wait a few seconds and then open the lid. When I get to the desktop I will see the animated battery charging icon. I wait for the animation to stop, hover the cursor over the icon and see the glorious message "XX minutes to full charge".
I hope this works for you too. Good luck!