NUMERICAL AND GEOMETRIC PROPERTIES OF INTERVAL B-SPLINES

Trigonometric B-spline curves with shape parameters are equally important and useful for modeling in Computer-Aided Geometric Design (CAGD) like classical B-spline curves. This paper introduces the cubic polynomial and rational cubic B-spline curves using new cubic basis functions with shape parameter ξ∈[0,4]. All geometric characteristics of the proposed Trigonometric B-spline curves are similar to the classical B-spline, but the shape-adjustable is additional quality that the classical B-spline curves does not hold. The properties of these bases are similar to classical B-spline basis and have been delineated. Furthermore, uniform and non-uniform rational B-spline basis are also presented. C3 and C5 continuities for trigonometric B-spline basis and C3 continuities for rational basis are derived. In order to legitimize our proposed scheme for both basis, floating and periodic curves are constructed. 2D and 3D models are also constructed using proposed curves.

A reduced-order model based on cubic B-spline basis function and SSP Runge–Kutta procedure to investigate option pricing under jump-diffusion models

In this study, we present a collocation method based on cubic B-spline functions for solving systems of Volterra integro-differential equations involving both classical and fractional derivatives in the Caputo sense (LSVIDEs-CF). The approach begins by dividing the problem domain into a finite number of subintervals, followed by the construction of cubic B-spline basis functions within each segment. Control points are introduced as the unknowns in the approximate numerical solution, which is expressed as a cubic combination of these basis functions. The given system of VIFDEs-CF is then reduced to a system of algebraic equations, which are efficiently solved using the Jacobian matrix method. In practice, the integrals are approximated using the Clenshaw–Curtis quadrature rule. The implementation of this method is supported by Python software, ensuring effective computational processing. Numerical examples are provided to demonstrate the efficiency of the proposed method. An itemized version of the algorithm is also presented to facilitate its implementation.

Dynamic analysis of two-dimensional elasticity problems using the Isogeometric analysis method and its comparison with the finite element method is the subject of this research. For this purpose, formulation of the governing differential equation is obtained by using the B-spline basis functions. Some numerical examples employing consistent and lumped mass matrices are presented in order to compare the results of Isogeometric analysis and finite elements. The effect of increasing the number of control points and the degree of B-spline basis functions on the accuracy of the solution are also investigated. Furthermore, since the Newmark method is employed for transient dynamic analysis, the effects of parameters α and β on the solution accuracy are investigated. The obtained results demonstrate higher accuracy of the isogeometric analysis method in comparison with the FEM, despite employing fewer numbers of degrees of freedom and consequently reduction in the dimensions of system of equations and computational effort.

This paper describes a wavelet–based ray–tracing algorithm that carries out high quality images. Our appoach takes advantage of the multiresolution representation of the volume data and the self–similarity of the basis functions. The results are flexible high order approximations of the low albedo rendering equation. We propose accurate integration and evaluation schemes for shading and intensity computation. This paper also describes the calculation of error bounds of the intensity integration. The theory and its implementation are discussed and shaded images are presented.

Even degree B-spline curves and surfaces: A note on the paper “B-spline Curves and Surfaces Viewed as Digital Filters” by A. Goshtasby, F. Cheng, and B. Barsky

Conversion from NURBS to Bézier representation

Reconstruction of B-spline curves and surfaces by adaptive group testing

Optimal quadrature for univariate and tensor product splines

Numerical and stability analysis of linear B-spline and local radial basis functions for solving two-dimensional distributed-order time-fractional telegraph models

An open uniform knot vector ensures that the resulting B-spline curve starts at the first control point and ends at the last control points, providing a more intuitive and predictable shape at the boundaries. When dealing uncertainty in data, open uniform knot vector B-spline are particularly useful because they offer several advantages. In this paper, the picture fuzzy set approach was used to introduce the picture fuzzy B-spline curve interpolation model with open uniform knot vector. Firstly, picture fuzzy control point relation is introduced by using basic concepts of picture fuzzy set which is involved picture fuzzy number and picture fuzzy relation. The B-spline basis function is then blended with open uniform knot vector with order and the picture fuzzy control point relation. Afterwards, using interpolation technique, fuzzy B-spline curves interpolation are generated and visualized using open uniform knot vector. In conclusion, a few numerical examples for creating the desired curve are displayed.

Numerous fields, including the physical sciences, social sciences, and earth sciences, benefit greatly from the application of fractional calculus (FC). The fractional-order derivative is developed from the integer-order derivative, and in recent years, real-world modeling has performed better using the fractional-order derivative. Due to the flexibility of B-spline functions and their capability for very accurate estimation of fractional equations, they have been employed as a solution interpolating polynomials for the solution of fractional partial differential equations (FPDEs). In this study, cubic B-spline (CBS) basis functions with new approximations are utilized for numerical solution of third-order fractional differential equation. Initially, the CBS functions and finite difference scheme are applied to discretize the spatial and Caputo time fractional derivatives, respectively. The scheme is convergent numerically and theoretically as well as being unconditionally stable. On a variety of problems, the validity of the proposed technique is assessed, and the numerical results are contrasted with those reported in the literature.

We interpret a type of fuzzy controller as an interpolator of B-spline hypersurfaces. B-spline basis functions of different orders are regarded as a class of membership functions (MFs) with some special properties. These properties lead to several interesting conclusions about fuzzy controllers if such membership functions are employed to specify the linguistic terms of the input variables. We show that by appropriately designing the rule base, C/sup n/-continuity of the output can be achieved (n is the order of the B-spline basis functions). The issues of function approximation and heuristic control using such a fuzzy system are also discussed.

In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.

Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II)