Cleaning mesh topology

Hi everyone,

I am quite new to mesh operation, and trying to figure out how to subdivide mesh faces in the way that I can control the subdivision inset faces

At the moment I did it with a very complicated way and I feel that there should be an easier and practical way to do it. The second problem that I am encountering is that the face order is not clean and I believe it happened because of my messy subdividing method. What I aim for is a mesh that has faces sorted in radial layer by layer as the snapshot posted here

I attach the definition below. If someone could help me sort this out.

Thank you

Mesh Topology.gh (43.1 KB)

There is Mesh Loft component in Mesh Plus

It also has Catmull-Clark Subdivision component. Catmull is for quad mesh.