B++ splines with applications to isogeometric analysis

A novel spline, named B++ Splines (Boundary Plus Plus Splines), is developed to address the expression of a trimmed NURBS patch in an analytic form, which is a central idea of surface representation. The presented method converts each trimmed NURBS patch into a B++ spline patch that incorporates of specific boundary points as the boundary presentation. Emphasis is placed on the construction of a new analytic formula of a trimmed NURBS patch defined by the boundary points at the trimming curves and a group of enriched control points. B++ spline basis functions are linearly independent, build a partition of unity and satisfy the Kronecker delta property. Each B++ spline basis function is a linear combination of the basis functions of the trimmed NURBS patch. These properties allow imposing the Dirichlet boundary conditions strongly at the boundary of the trimmed patch without the necessity of modifying the basis functions of the trimmed patch. Isogeometric analysis using B++ splines for two-dimensional elastic solids is also proposed. Several numerical examples are used to demonstrate the reliability of the presented method. The numerical example for the patch test illustrates that the B++ spline patch passes the standard patch test.