NON-UNIFORM RATIONAL B-SPLINE BASED ISO-GEOMETRIC ANALYSIS FOR A CLASS OF HYDRODYNAMIC PROBLEMS

2021 ◽  
Vol 162 (A2) ◽  
Author(s):  
M Goel ◽  
R Sharma ◽  
S K Bhattacharyya ◽  
Tae-wan Kim
Keyword(s):  
Boundary Value Problems ◽  
Potential Function ◽  
Computational Efficiency ◽  
Boundary Value ◽  
Geometric Analysis ◽  
Geometric Approach ◽  
Panel Method ◽  
B Spline ◽  
Computational Performance ◽  
Wide Range

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 110−3 , 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.

Non-Uniform Rational B-Spline Based Iso-Geometric Analysis for a Class of Hydrodynamic Problems

2020 ◽  
Vol 162 (A2) ◽  
Author(s):  
M Goel ◽  
R Sharma ◽  
S K Bhattacharyya ◽  
Tae-wan Kim
Keyword(s):  
Boundary Value Problems ◽  
Potential Function ◽  
Computational Efficiency ◽  
Boundary Value ◽  
Geometric Analysis ◽  
Geometric Approach ◽  
Panel Method ◽  
B Spline ◽  
Computational Performance ◽  
Wide Range

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.

scholarly journals The Laplace equation in the exterior of the Hankel contour and novel identities for hypergeometric functions

2013 ◽  
Vol 469 (2157) ◽  
pp. 20130081 ◽  
Author(s):  
A. S. Fokas ◽  
M. L. Glasser
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Laplace Equation ◽  
Special Functions ◽  
Boundary Value ◽  
Zeta Functions ◽  
Neumann Boundary ◽  
Neumann Boundary Value Problem ◽  
Wide Range ◽  
Definition Of

By using conformal mappings, it is possible to express the solution of certain boundary-value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identity for special functions. A convenient tool for deriving this type of identity is the so-called global relation , which has appeared recently in a wide range of boundary-value problems. As a concrete application, we analyse the Neumann boundary-value problem for the Laplace equation in the exterior of the Hankel contour, which appears in the definition of both the gamma and the Riemann zeta functions. By using the explicit solution of this problem, we derive a number of novel identities involving the hypergeometric function. Also, we point out an interesting connection between the solution of the above Neumann boundary-value problem for a particular set of Neumann data and the Riemann hypothesis.

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