An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions

Abstract

The NURBS-based isogeometric analysis offers a seamless integration between the CAD and subsequent finite element analysis and has been shown to be very effective for wide classes of problems due to its accurate geometric description. However, similar to the moving least square or reproducing kernel meshfree shape functions, the NURBS basis functions generally are not interpolatory functions. In this work it is shown that the direct imposition of the inhomogeneous essential boundary conditions to the NURBS control points may lead to significant errors with deteriorated rates of convergence. Consequently an improved formulation for NURBS-based isogeometric analysis is proposed. This is fulfilled by employing a transformation method that relates the control variables to the collocated nodal values at the essential boundary. By using open knot vectors, the resulting NURBS basis functions associated with the interior control points vanish at the boundary. Thus unlike the meshfree approximation, the NURBS control points can be clearly partitioned into boundary and interior groups. Therefore the transformation method can be only applied to the boundary control points via invoking collocation of physical values at a set of boundary points. The NURBS approximation enhanced with transformed physical boundary variables is kinematically admissible and the essential boundary conditions can be enforced straightforwardly like the finite element method. Several potential and elasticity problems evince that much higher solution accuracy with optimal convergence rates can be achieved by the present improved formulation compared with the method with direct imposition of essential boundary conditions to control variables.