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Make surface geometry for the air mattress including some internal webs between the top and bottom faces of the air mattress. Simply apply pressure to all the external surfaces of the air mattress. Don’t apply any pressure to the internal webs.
You will want a surface to represent a floor, which can be a rigid body. You may want a solid body on the top of the mattress to press down on it. You would use Frictional Contact between the solid and the top of the mattress and another Frictional Contact between the floor and the bottom of the mattress. You would turn on Standard Earth gravity to pull the body down.