A Review on Higher Order Spline Techniques for Solving Burgers Equation using B-Spline methods and Variation of B-Spline Techniques

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers' equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these method

Rapid evaluation of radial basis functions

Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail

A Cubic Spline Projection Method for Computing Stationary Density Functions of Frobenius-Perron Operator

Title from PDF of title page, viewed August 22, 2022Dissertation (Ph.D)--Department of Mathematics and Statistics, Department of Physics and Astronomy. University of Missouri--Kansas City, 2022Includes bibliographical references (pages 143-150)Stationary density functions of Frobenius-Perron operators have critical applications in many fields of science and engineering. Accordingly, approximating stationary density functions f* is important and the focus of this dissertation. Among the computational methods of approximating the smooth f*, the linear spline and quadratic spline projection methods have been proven effective. However, we intend to improve the convergence rate of the previous methods. We will fulfill this goal by using cubic spline functions since cubic spline functions are twice continuously differentiable on the whole domain. Theoretically, we prove the existence of a nonzero sequence of cubic spline functions {fₙ} that converges to the stationary density function f* of the Frobenius-Perron operator in L¹-norm. The numerical experimental results assure that the cubic spline projection method gives the fastest convergence rate so far. In addition, when the stationary density function f* lies in the cubic spline space, the cubic spline projection method computes f* exactly no matter what n may be.Introduction -- Preliminaries -- Spline Space -- Projection Method -- Convergence Analysis of Cubic Spline Projection Method -- Numerical Results -- Appendi

Splines and vector splines

The thrust of this report concerns spline theory and some of the background to spline theory and follows the development in (Wahba, 1991). We also review methods for determining hyper-parameters, such as the smoothing parameter, by Generalised Cross Validation. Splines have an advantage over Gaussian Process based procedures in that we can readily impose atmospherically sensible smoothness constraints and maintain computational efficiency. Vector splines enable us to penalise gradients of vorticity and divergence in wind fields. Two similar techniques are summarised and improvements based on robust error functions and restricted numbers of basis functions given. A final, brief discussion of the application of vector splines to the problem of scatterometer data assimilation highlights the problems of ambiguous solutions

Isogeometric regular discretization for the Stokes problem

The inf-sup stability and optimal convergence of an isogeometric C 1 discretization for the Stokes problem are shown. In this discretization the velocities are the pushforward through the geometrical map of cubic C1 non-uniform rational B-spline (NURBS) functions and the pressures are the pushforward of quadratic C1 NURBS. This paper follows the work in Bazilevs et al. (2006, Math. Models Methods Appl. Sci., 16, 1031-1090) where the authors showed the numerical result of this discretization and proved the inf-sup stability for C0 NURBS functions. The use of more regular functions is useful to decrease the degrees of freedom and thus the computational cost. The analysis is performed by means of the Verfürth trick, the macro-element technique, some approximation properties and the inf-sup condition for tensor products of B-spline spaces. © 2010 The author

Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization

In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries consisting of a set of B-spline curves. Instead of forming the computational domain by a simple boundary, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including B\'ezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior B\'ezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C1=G1-continuity constraints on the interface of neighboring B\'ezier patches with respect to each quad in the quadrangulation, the high-quality B\'ezier patch parameterization is obtained by a C1-constrained local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach