I would like to ask how to model minimal surface within this box frame, so that the continuity is good and boundary edge is within the box?
On the left you see 6 surfaces, but it is not continuous and on the right side is a simple patch, which results in more or less good continuity but the edge is bad (not linear).
If you make the surface from 6 mesh quads, make extra copies around the periodic boundaries, convert to SubD, then convert that SubD to NURBS and select an interior patch which is not affected by the naked boundary, you get something pretty smooth, and continuous across the periodic boundaries.
Continuity is not absolutely perfect at the interior 6-valent vertex, but that’s what you get with the Catmull-Clark limit surface.
Positioning the control points with the standard parametric equations (as eg in Lunchbox) doesn’t actually solve the question of how to make a continuous and exact NURBS minimal surface either though - the resulting surface is only an approximation of a minimal one, since the equation is for the points on the minimal surface, and the NURBS surface does not interpolate these points.
Interestingly the quad mesh to NURBS converter in MoI does somehow manage to get what appears to be perfect G2 continuity even right at the star point. So it is possible. I haven’t figured out the magic recipe though except for in a few special cases.
Here’s a comparison - Rhino mesh-to-SubD-to-NURBS on the left, and MoI mesh-to-NURBS on the right