[Raph] had a pair of foam mattresses from ... 50 V 20 A variable DC power supply. [Raph] determined the current experimentally: the wire failed at 20 A, and cutting speed was too low at 12 A.

What you really need is a hot wire for nice clean cuts. [Elite Worm] built a hot wire foam cutter ... also a circle-cutting jig that looks really handy. This design uses a 12 V power regulator ...