Isogeometric analysis with Powell-Sabin splines

Basic idea of isogeometric analysis

Isogeometric Analysis (IgA) is a recent framework to improve the connection between numerical simulation and Computer Aided Design (CAD) systems. Splines are common representation tools in CAD systems, because they provide the concept of control points and control nets, allowing the design of complex smooth surfaces in a geometrically intuitive way. On the other hand, the numerical simulation and analysis phase is usually done in terms of (low order) polynomial-based finite element representations. The idea of IgA is to use the exact spline description of the geometry provided by the CAD systems, and by means of an isoparametric approach, to approximate the unknown solution of the differential equations using the same type of spline functions.

Tensor-product NURBS (Non-Uniform Rational B-Splines) are today standard tools in CAD systems, and thus the original choice for IgA. A drawback inherent to the tensor-product topology, is the restriction to rectangular regular meshes. Especially in applications with local difficulties, this can badly affect the global quality of the approximation, and often results in a too high spline dimension. Splines defined on irregular triangulations are an interesting alternative for IgA. They have the advantage that an adaptive local refinement strategy is very easy to implement because of the flexibility of triangulations, while they retain most of the useful properties of the tensor-product splines. In particular, we focus on the application of Powell-Sabin splines into the isogeometric context.