An Isogeometric Collocation Method Based On Blue-Noise Sampling

Isogeometric analysis(IGA)is a novel numerical method for solving equations of physical fields,which is motivated by improving the integration between numerical simulations of physical phenomena and the computer-aided design tools,and provides a New way to associate product design with analysis and optimization via a common mathematical platform.Isogeometric analysis method is based on the NURBS function as a basis function to approximate the analytical solution of differential equations.If the order of the NURBS function used to approximate the analytical solution is high enough,the differential equations can be solved by isogeometric collocation method(IGA-C).In addition,the IGA-C method performs better than the Galerkin method in terms of accuracy and time ratio.However,the accuracy of the IGA-C method depends on the choice of collocation points,and the choice of ideal collocation point remains an open problem.Sampling is an essential technique in computer science.Sampling translates a continuous signal into its discrete counterpart or selects a subset from a discrete set of signals.In computer graphics,sampling plays an important role in many applications,such as rendering,stippling,texture synthesis,object distribution and simulation.Among all the sampling techniques,blue-noise sampling is the most popular method in recent papers.Because of the isogeometric collocation method can be regarded as a fitting problem for the imposing function on the right side of the differential equation and the fitting accuracy is closely related to the nature of this imposing function.If the feature points of the imposing function are used as the collocation points of the isogeometric collocation method,the fitting accuracy can be improved and the calculation accuracy of the isogeometric collocation method can be improved,too.Therefore,this paper presents an adaptive blue noise sampling collocation method that regards the feature points of the imposing function as the collocation points to isogeometric calculation.Numerical experiments show that the collocation points generated by the adaptive blue noise sampling collocation method significantly improve the calculation accuracy of the isogeometric collocation method.