Isogeometric analysis  has the advantage of integrating geometric and mesh models using NURBS or Spline. At the same time, I would like to ask a question to my friends: for traditional FEM and FVM, why can't we use mesh to represent geometry and use the mesh which represents the geometry to do the computation directly?
In detail, according to the discrete differential geometry, if we can find the equivalent definitions between continuous and discrete geometry, we can represent a continuous geometry with a mesh. We know that for CG software (for example Blender), the main way of modelling is to use the mesh to represent the continuous geometry, we modify the geometry by modifying the mesh. But for the scientific computing mesh generation software (for example Gmsh), we usually need a geometric model first and then set some parameters to do the meshing. There is a big difference between them. I know CG mainly uses the surface mesh in 3D space. And scientific calculation uses mainly solid mesh. But I think that CG's way of using mesh to represent geometry is simpler and suitable for shape change (optimization). So I wonder why can't we use mesh to represent geometry like CG in scientific computation. The geometry is directly represented by a mesh, and then you can do the computation directly on it. I think for the pioneers of FEM and FVM, the idea of combining the geometry and mesh must have been considered before. But now almost all the simulation software don't combine the geometry and mesh model. So I wonder what is the main difficulty of not doing that. So I would like to ask the question to my friends: for traditional FEM and FVM, why can't we use mesh to represent geometry and use the mesh which represents the geometry to do the computation directly?
This question is inspired by the answers of my previous question: Can the mesh generation methods in FVM and FEM be totally based on the knowledge of the mesh generation theory in computer graphics?
1. Hughes, T. J. R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng. 194, No. 39-41, 4135-4195 (2005). ZBL1151.74419.