Study On Knot Selection In Isogeometric Analysis And Spline Fitting

Isogeometric analysis(IGA)was introduced to bridge the gap between computer aided geometric design(CAGD)and finite element analysis(FEA).The core idea of IGA is to use the same smooth and higher-order basis functions for the representation of both the geometry in CAGD and the approximation of solutions fields in FEA.The primary goal of IGA is to simplify the high-cost mesh generation process in FEA and make the tools of CAGD and FEA interact more tightly.In addition,IGA has been proved to be an excellent computational mechanics technology,which on a per-degreeof-freedom basis shows higher accuracy and robustness than finite element methods.In contrast to IGA,Isogeometric collocation(IGA-C)method is based on the discretization of the strong form of the partial differential equations,which takes advantage of that spline functions can be easily adjusted to any order of polynomial degree and continuity required by the differential operators.It turns out that IGA-C is more competitive with respect to IGA on the basis of an accuracy-to-computational-cost ratio.In this paper,we analyse the convergence rate of collocation method in curve/surface fitting problems at first.Then,we verify that sampling feature points of exact solutions to select knots and construct more suitable approximation solution space can improve the precision or even the convergence rate.At last,we study the knot selection heuristically and raise a new knots insertion algorithm.By numerical examples,we show that suitable knot selection results in higher accuracy for problems with exact solution having sharp peaks.Curve fitting with splines is a fundamental problem in CAD and CAE.How to choose the number of knots and the position of knots in spline fitting is still a difficult issue.This paper presents a method based on feature points sampling and sparse optimization model.Fisrt,sampling feature points of the given data and generating an initial knot vector;Second,removing redundant knots by solving a sparse optimization problem under the condition that the fitting performance is good enough.Our experiments show that the approximatin spline curve obtained by our method has less number of knots with cheap time cost.