Blending spline surfaces over polygon mesh and their application to isogeometric analysis

Abstract

Finite elements are allowed to be of a shape suitable for the specific problem. This choice defines thereafter the accuracy of the approximated solution. Moreover, flexible element shapes allow for the construction of an arbitrary domain topology. Polygon meshes are a common representation of the domain that cover any choice of the finite element shape. Being an alternative tool for modeling and analysis, blending spline surfaces support representation on polygon grids. The blending splines have a hierarchical structure, which is obtained by generating local surfaces that cover each node support and then blended with a special type of basis functions. This type of splines in their tensor product form is suitable for application to isogeometric analysis problems. A more general representation constructed on polygonal elements can be used on a wider range of domain topology in comparison with tensor product surfaces. In this paper we introduce a novel approach to constructing curvilinear polygon meshes in the blending spline representation in application to the isogeometric analysis context. The focus is on generating a novel special type of basis functions on a connected collection of polygons, with triangles and quadrilaterals as particular cases. The purpose of the proposed paper is to show applications of this construction to various numerical problems, as well as to generalize the approach to evaluating these basis functions on arbitrary planar domains.