Conformal map onto a circle, from an identification space composed of five square regions.

I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at hand. I don't have a rigorous mathematical background (mech. engineer in computational fluid dynamics), so I would appreciate if someone here could advise me on the route that I should take in order to derive such a conformal mapping.

The end application is to generate a smooth computational mesh that looks like this. I have generated a mesh like this using other means, but the smoothness of the mesh vertices is not sufficient for extremely fine meshes. This results in spurious oscillations in the numerical problem I am trying to solve.

This question is a couple of years old, but so far no analytic solutions have been offered.

If you're willing to consider a computational solution, You might want to try this Boundary First Flattening app.

Boundary First Flattening (BFF) is a free and open source application for surface parameterization. Unlike other tools for UV mapping, BFF allows free-form editing of the flattened mesh, providing users direct control over the shape of the flattened domain—rather than being stuck with whatever the software provides. The initial flattening is fully automatic, with distortion mathematically guaranteed to be as low or lower than any other conformal mapping tool. The tool also provides some state-of-the art flattening techniques not available in standard UV mapping software such as cone singularities, which can dramatically reduce area distortion, and seamless maps, which help eliminate artifacts by ensuring identical texture resolution across all cuts. BFF is highly optimized, allowing interactive editing of meshes with millions of triangles.

The BFF application is based on the paper, “Boundary First Flattening” by Rohan Sawhney and Keenan Crane.

You can supply it with a mesh model of a cube that is missing one of its faces. This is a surface with a boundary. You should be able to use the app to create a mesh whose boundary is a circle corresponding to the boundary of your model and whose interior is a nearly conformal flattening of the remaining five faces of the cube.