We have used unstructured (simplex) splines to formulate an unstructured version of isogeometric analysis. The basis functions are built from an underlying point cloud, and not on a mesh, unlike standard finite elements (FE). The basis functions are smoother than the usual FE basis, and their regularity can be adapted by aligning some points in the cloud:
Numerically, these bases retain many good properties of standard isogeometric analysis, including a better simulation timestep in time-explicit schemes, and, for the Helmholtz problem, a better precision per degree of freedom and a convergence over a larger range of frequencies.
We have used these bases to simulate the propagation of waves for geophysics, acoustic and helioseismology applications.
Multiply-connected domains
Compared to the standard, tensor-product form of isogeometric analysis, this unstructured formulation allows to model domains with holes, which is convenient for example in acoustics, such as sound propagation in a church: Helioseismology also has to deal with complex topologies: This kind of domains would be much more difficult to simulate using standard isogeoetric analysis, which is based on tensor-product patches with the topology of a disk