[bgoated01]

> Isn't IGA's shtick just replacing classical shape functions with the splines used to specify the geometry?

That's how it started, yes. The splines used to specify the geometry are trimmed surfaces, and IGA has expanded from there to the use of splines generally as the shape functions, as well as trimming of volumes, etc. This use of smooth splines as shape functions improves the accuracy per degree of freedom.

> If I recall correctly convergence rates are exactly the same

Okay, looks like I remembered wrong here. What we do definitely see is that in IGA you get the convergence rates of higher degrees without drastically increasing your degree of freedom, meaning that there is better accuracy per degree of freedom for any degree above 1. See for example Figures 16 and 18 in this paper: https://www.researchgate.net/profile/Laurens-Coox/publicatio...

> geometry and the fields of quantities of interest do not have the same spatial distributions.

Using the same shape functions doesn't automatically mean that they will have the same spatial distributions. In fact, with hierarchical refinement in splines you can refine the geometry and any single field of interest separately.

> What is left in terms of potential?

The biggest potential other than higher accuracy per degree of freedom is perhaps trimming. In FEM, trimming your shape functions makes the solution unusable. In IGA, you can immerse your model in a "brick" of smooth spline shape functions, trim off the region outside, and run the simulation while still getting optimal convergence properties. This effectively means little to no meshing required. For a company that is readying this for use in industry, take a look at https://coreform.com/ (disclosure, I used to be a software developer there).