Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis

Isogeometric analysis is a numerical simulation method which uses the NURBS based representation of CAD models. NURBS stands for non-uniform rational B-splines and is a generalization of the concept of B-splines. The isogeometric method uses the tensor product structure of 2-or 3-dimensional NURBS functions to parameterize domains, which are structurally equivalent to a rectangle or a hexahedron. The special case of singularly parameterized NURBS surfaces and NURBS volumes is used to represent non-quadrangular or non-hexahedral domains without splitting, which leads to a very compact and convenient representation.If the parameterization of the physical domain is available, the test functions for the isogeometric analysis are obtained by composing the inverse of the domain parameterization with the NURBS basis functions. In the case of singular parameterizations, however, some of the resulting test functions are not well defined at the singular points and they do not necessarily satisfy the required integrability assumptions. Consequently, the stiffness matrix integrals which occur in the numerical discretizations may not exist.After summarizing the basics of the isogeometric method, we discuss the existence of the stiffness matrix integrals for 1-, 2- and 3-dimensional second order elliptic partial differential equations. We consider several types of singularities of NURBS parameterizations and derive conditions which guarantee the existence of the required integrals. In addition, we present cases with diverging integrals and show how to modify the test functions in these situations.

► We set up the stiffness matrix for a model problem in isogeometric analysis.
► If singularities are present some entries of the stiffness matrix may be unbounded.
► We analyze the underlying function space in 1D and for two different settings in 2D.
► In the 2D model cases all entries of the stiffness matrix are bounded.
► For 1D and some 2D cases this is not true and the function spaces have to be modified.