Vector Radial Basis Interpolation Based On Fictitious Points And Its Application In N-S Equations

The vector radial basis function(VRBF)is extended from the radial basis function(RBF),the divergence-free and curl-free VRBFs are obtained by using curl and divergence operators acting on RBFs.At the same time,these properties of RBF which are the isotropic,simple and meshless are preserved in VRBF.Therefore,it has important applications in solving numerical solutions of partial differential equations.In general,this paper mainly uses the divergence-free VRBF,combined with fictitious points to study the influence of points on the interpolation,and extends this method to the numerical solution of Navier-Stokes equations.First,briefly introduce the background,current situation and significance of the research.Next,describe the basic definitions and theoretical foundations used in this article.Then,utilize the generalized minimal residual method(GMRES)for RBF interpolation by using the fictitious points as the interpolation center points.It is necessary to observe the effect of fictitious points on interpolation results.The results show that the accuracy of interpolation obtained by using fictitious points is higher than that traditional points.Second,extend the method to VRBF interpolation.For the shape parameters of basis function,the best shape parameters are selected by the leave-one-out cross-validation method(LOOCV).The numerical results show that the effect of using fictitious points as the center points is better.Next,apply the above method to get the numerical solution of the two-dimensional incompressible Navier-Stokes(N-S)equation.Assuming that the fluid flows in a square cavity,time-independent and the pressure is stable.In other words,it's the dimensionless N-S equations.We can simplify the equation by the divergence property of divergence free VRBF is consistent with incompressibility of equations.Considering that the equation is nonlinear,so we adopt iterative method to solve and discuss the convergence of the iter-ation.The initial value is given for iteration,and compare the error between the velocity generated by iteration and the initial velocity.Then,the effectiveness of VRBF interpolation can be obtained by programming.What the conclusion we can get is when the Reynolds number is large,the interpolation effect of fictitious points as the center points is better.Finally,the advantages and disadvantages of the algorithm are analyzed,which lays the foundation for generalization to more fluid mechanics problems and the calculation of other partial differential equations.