Isogeometric collocation method with intuitive derivative constraints for PDE-based analysis-suitable parameterizations

Abstract

This paper presents a general formulation of an isogeometric collocation method (IGA-C) for the parameterization of computational domains for the isogeometric analysis (IGA) using non-uniform rational B-splines (NURBS). The boundary information of desired computational domains for IGA is imposed as a Dirichlet boundary condition on a simple and smooth initial parameterization of an initial computational domain, and the final parameterization is produced based on the numerical solution of a partial differential equation (PDE) that is solved using the IGA-C method. In addition, we apply intuitive derivative constraints while solving the PDE to achieve desired properties of smoothness and uniformity of the resulting parameterization. While one may use any general PDE with any constraint, the PDEs and additional constraints selected in our case are such that the resulting solution can be efficiently solved through a system of linear equations with or without additional linear constraints. This approach is different from typical existing parameterization methods in IGA that are often solved through an expensive nonlinear optimization process. The results show that the proposed method can efficiently produce satisfactory analysis-suitable parameterizations.