Computer Methods in Applied Mechanics and Engineering, 200(49–52):3568–3582, Elsevier, 2011.
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.
@Article{         Takacs_2011aa,
abstract      = {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.},
author        = {Takacs, Thomas and Jüttler, Bert},
doi           = {http://dx.doi.org/10.1016/j.cma.2011.08.023},
file          = {Takacs_2011aa.pdf},
issn          = {0045-7825},
journal       = {Computer Methods in Applied Mechanics and Engineering},
keywords      = {parameterization,isogeometric,NURBS,B-Splines},
number        = {49--52},
pages         = {3568--3582},
publisher     = {Elsevier},
title         = {Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis},
url           = {http://www.sciencedirect.com/science/article/pii/S0045782511002787},
volume        = {200},
year          = {2011},
shortjournal  = {Comput. Meth. Appl. Mech. Eng.}
}