Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement

The paper introduces the concept of isogeometric analysis, in which basis functions generated from NURBS (Non-Uniform Rational B-Splines) are used to construct an exact geometric model. For purposes of analysis the basis is refined and/or its order elevated without changing the geometry or its parameterization. The authors present analogues of finite element h- and p-refinement schemes and introduce a new, more efficient, higher-order concept called k-refinement; refinements are easily implemented and exact geometry is maintained at all levels without needing subsequent communication with a CAD description.

In structural mechanics, the authors establish that the basis functions are complete with respect to affine transformations, so all rigid body motions and constant strain states are represented exactly, and standard patch tests are satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions, while the k-refinement strategy converges toward monotone solutions for advection–diffusion processes with sharp internal and boundary layers—a result the authors call very surprising. They argue isogeometric analysis is a viable alternative to standard polynomial-based finite element analysis with several advantages.