Smooth part of mesh geometry / subsurf only faces on one side This is very specific task as I can see here.
The picture: 
Need to make this selected part more smooth, better use subsurf with detached frame part, as you can see it. And mark upper and lower edges sharp to not be REALLY round. But the problem, it will be round anyway and also very much deformed quads all over the place. 

What directly required is to add more transverse edges at selected area and move it to form more smooth surface. It only will make more same tris as you can see, but less long. Btw this element also requires smoothing upper edges row at both sides. And what is the thing, no modifier can do that. Catmull-Clark is pure mess of different shaped surface everytime. 

And right here you gonna see wrong smoothing even with auxiliary edges.
Can you help me?
Actually found it myself

Just set corners of sharp edges to required crease level and they gonna be untouched.
So simply, experimenting gives surprises. Also somehow this time no random lines breaking, clearly new quads symmetrically follows the base shape.