Communications in Numerical Analysis

Volume 2013 (2013), Article ID cna-00178, 12 Pages

doi: 10.5899/2013/cna-00178


Research Article


Numerical solution of the helmholtz equation for the superellipsoid via the galerkin method


Yajni Warnapala1 *, Hy Dinh1


1Roger Williams University, Department of Mathematics, One Old Ferry Road, Bristol, RI 02809


* Corresponding author. Email address: ywarnapala@rwu.edu; Tel. 401-254-3097.


Received: 19 November 2012; Accepted: 04 January 2013


Copyright © 2013 Yajni Warnapala and Hy Dinh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The objective of this work was to find the numerical solution of the Dirichlet problem for the Helmholtz equation for a smooth superellipsoid. The superellipsoid is a shape that is controlled by two parameters. There are some numerical issues in this type of an analysis; any integration method is affected by the wave number k, because of the oscillatory behavior of the fundamental solution. In this case we could only obtain good numerical results for super ellipsoids that were more shaped like super cones, which is a narrow range of super ellipsoids. The formula for these shapes was: $x=cos(x)sin(y)^{n},y=sin(x)sin(y)^{n},z=cos(y)$ where $n$ varied from 0.5 to 4. The Helmholtz equation, which is the modified wave equation, is used in many scattering problems. This project was funded by NASA RI Space Grant for testing of the Dirichlet boundary condition for the shape of the superellipsoid. One practical value of all these computations can be getting a shape for the engine nacelles in a ray tracing the space shuttle. We are researching the feasibility of obtaining good convergence results for the superellipsoid surface. It was our view that smaller and lighter wave numbers would reduce computational costs associated with obtaining Galerkin coefficients. In addition, we hoped to significantly reduce the number of terms in the infinite series needed to modify the original integral equation, all of which were achieved in the analysis of the superellipsoid in a finite range. We used the Green's theorem to solve the integral equation for the boundary of the surface. Previously, multiple surfaces were used to test this method, such as the sphere, ellipsoid, and perturbation of the sphere, pseudosphere and the oval of Cassini Lin and Warnapala [9], Warnapala and Morgan [10].


Keywords: Helmholtz Equation; Galerkin Method; Superellipsoid Mathematics Subject Classifications (2000); 45B05; 65R10

References

  1. K. E. Atkinson, The numerical solution of the Laplace's equation in three dimensions, SIAM J. Numer. Anal, 19 (1982) 263-274.


  2. K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, (1998).


  3. David Colton, Rainer Kress, Integral equation methods in scattering theory, (1983).


  4. C. Criado, N. Alamo, Thomas rotation and Foucault pendulum under a simple unifying geometrical point of view, Int. J. Non-Linear Mech, 44 (2009) 923-927.


  5. D. S. Jones, Integral equations for the exterior acoustic problem, Q. J. Mech. Appl. Math, 27 (1974) 129-142.


  6. G. Jost, Integral equations with Modified Fundamental Solution in Time-Harmonic Electromagnetic Scattering, IMA J. Appl. Math, 40 (1988) 129-143.


  7. R. E. Kleinman, G. F. Roach, On modified Green functions in exterior problems for the Helmholtz equation, Proc. R. Soc. Lond. A, 383 (1982) 313-333.


  8. T. C. Lin, The numerical solution of Helmholtz equation using integral equations, Ph.D. thesis, University of Iowa, Iowa City, Iowa, (1982),


  9. T. C. Lin, Y. Warnapala-Yehiya, The Numerical Solution of the Exterior Dirichlet Problem for Helmholtz's Equation via Modified Green's Functions Approach, Comput. Math. Appl, 44 (2002) 1229-1248.


  10. Y. Warnapala, E. Morgan, The Numerical Solution of the Exterior Dirichlet Problem for Helmholtz's Equation via Modified Green's Functions Approach for the Oval of Cassini, Far East J. Appl. Math, 34 (2009) 1-20.