Abstract-In this paper a new kind of splines, called quadratic non-uniform algebraic trigonometric B-splines (quadratic NUAT Bsplines) with multiple shape parameters are constructed over the space spanned by { } . As each piece of the curves is generated by three consecutive control points, they possess many properties of the quadratic B-spline curves and quadratic trigonometric B-spline curves. These curves have continuity with non-uniform knot vector. These curves are closer to the control polygon than the quadratic B-spline curves and quadratic trigonometric B-spline curves when choosing special shape parameters. The shape parameters serve to local control in the curves. The changes of one shape parameter will only affect two curve segments. Taking different values of the shape parameters, one can globally or locally adjust the shapes of the curves. The generation of tensor product surfaces by these new splines is straightforward. Keywords-Quadratic NUAT B-spline; shape parameter; continuity; spline surface. IntroductionThe trigonometric B-splines were presented in [12]. The recurrence relation for the trigonometric B-splines of arbitrary order was established in [8]. The construction of exponential tension B-splines of arbitrary order was given in [7]. It was further shown in [13] that the trigonometric Bsplines of odd order form a partition of a constant in the case of equidistant knots. In recent years, several bases in new spaces other than the polynomial space have been proposed for geometric modeling in CAGD. For instance, in [10] a basis is constructed for Cm=span { }. In [11] a basis for the space of trigonometric polynomials { } is constructed. Some bases are constructed in [9] for the spaces { }, { } and { }. In [1] C-Bézier curves are constructed in the space spanned by . Non-uniform algebraic trigonometric B-splines (NUAT B-splines), are generated in [14] over the space spanned by in which k is an arbitrary integer larger than or equal to 3. But all these curves do not have any shape parameter. In [15,16] C-B-splines are presented in the space spanned by { } with parameter .

In this paper a new kind of splines, called cubic trigonometric polynomial B-spline (cubic TP B-spline) curves with a shape parameter, are constructed over the space spanned by As each piece of the curve is generated by three consecutive control points, they posses many properties of the quadratic B-spline curves. These trigonometric curves with a non-uniform knot vector are C1 and G2 continuous. They are C2 continuous when choosing special shape parameter for non-uniform knot vector. These curves are closer to the control polygon than the quadratic B-spline curves when choosing special shape parameters. With the increase of the shape parameter, the trigonometric spline curves approximate to the control polygon. The given curves posses many properties of the quadratic B-spline curves. The generation of tensor product surfaces by these new splines is straightforward.

Analogous to the quartic B-splines curve, a piecewise quartic trigonometric polynomial B-spline curve with two shape parameters is presented in this paper. Each curve segment is generated by three consecutive control points. The given curve posses many properties of the B-spline curve. These curves are closer to the control polygon than the different other curves considered in this paper, for different values of shape parameters for each curve. With the increase of the value of shape parameters, the curve approach to the control polygon. For nonuniform and uniform knot vector the given curves have C0, G3; C1, G3; C1, G7; and C3 continuity for different choice of shape parameters. A quartic trigonometric Bézier curves are also introduced as a special case of the given trigonometric spline curves. A comparison of quartic trigonometric polynomial curve is made with different other curves. In the last, quartic trigonometric spline surfaces with two shape parameters are constructed. They have most properties of the corresponding curves.

Herein, we present the design and development of a ‘Non-uniform Rational B-spline (NURBS)’ based iso-geometric approach for the analysis of a number of ‘Boundary Value Problems (BVPs)’ relevant in hydrodynamics. We propose a ‘Potential Function’ based ‘Boundary Element Method (BEM)’ and show that it holds the advantage of being computationally efficient over the other known numerical methods for a wide range of external flow problems. The use of NURBS is consistent, as inspired by the ‘iso-geometric analysis’, from geometric formulation for the body surface to the potential function representation to interpolation. The control parameters of NURBS are utilised and they have been explored to arrive at some preferable values and parameters for parameterization and the knot vector selection. Also, the present paper investigates the variational strength panel method, and its computational performance is analyzed in comparison with the constant strength panel method. The two variations have been considered, e.g. linear and quadratic. Finally, to illustrate the effectiveness and efficiency of the proposed NURBS based iso-geometric approach for the analysis of boundary value problems, five different problems (i.e. flow over a sphere, effect of the knot vector selection on analysis, flow over a rectangular wing section of NACA 0012 aerofoil section, performance of DTMB 4119 propeller (un-skewed), performance of DTNSDRC 4382 propeller (skewed)) are considered. The results show that in the absence of predominant viscous effects, a ‘Potential Function’ based BEM with NURBS representation performs well with very good computational efficiency and with less complexity as compared to the results available from the existing approaches and commercial software programs, i.e. low maximum errors close to 110−3 , faster convergence with even up to 75 % reduction in the number of panels and improvements in the computational efficiency up to 32.5 % even with low number of panels.