Villumsen, who missed out on bronze by 1.8sec at the London Olympics four years ago, said she felt flat right from the start.
"Not real good - from the beginning, I was a bit tensed up. Usually you're better after 10 or 15km but today it just didn't happen."
The world and Commonwealth champion, Villumsen recorded a time of 44 minutes 54.71 seconds and 28.29sec behind Armstrong who won her third consecutive gold medal in the event.