Pushdown automation where number of letter `a` is at least as twice as letter `b` in the word
General strategy:

What you want to do is every time you read an input symbol a, if the stack has a b on it delete the b, and if the stack is empty/has a's, add an a. Every time you read a b, if the stack is empty/has b's add bb, if the stack has one a, remove the a and add a b, and if the stack has two or more a's, delete two of them. You're in an accepting state as long as the stack has no b's on it.

You can implement this strategy if you use epsilon transitions and intermediate states in a fairly straightforward way.