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2009•Journal of Southwest China Normal UniversityRequires access

The Virtual Boundary Element Method for Exterior Problems of the 3-D Helmholtz Equation

Lijun Jia

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Abstract

A virtual boundary method for solving the Dirichlet and Neumann exterior problems of the 3-D Helmholtz equation,which is valid for all wave number is presented in this paper.When wave number is an eigenvalue of the interior Dirichlet or Neumann problem for the Laplacian,the solution of the boundary integral equation correspond to eigenvalue is not unique.Based on the extension of potential function,Virtual boundary integral equation is deduced.The uniqueness problem is granted by choosing different virtual boundary,with which the wave number do not coincide with the eigenvalue of the original interior Dirichlet or Neumann problem for the Laplacian.The results of numerical examples demonstrate that the scheme presented is practical and effective for the exterior problems of the 3-D Helmholtz equation with any wave numbers.

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What this paper is about

A virtual boundary method for solving the Dirichlet and Neumann exterior problems of the 3-D Helmholtz equation,which is valid for all wave number is presented in this paper.When wave number is an eigenvalue of the interior Dirichlet or Neumann problem for the Laplacian,the solution of the boundary integral equation correspond to eigenvalue is not unique.Based on the extension of potential function,Virtual boundary integral equation is deduced.The uniqueness problem is granted by choosing different virtual boundary,with which the wave number do not coincide with the eigenvalue of the original interior Dirichlet or Neumann problem for the Laplacian.The results of numerical examples demonstrate that the scheme presented is practical and effective for the exterior problems of the 3-D Helmholtz equation with any wave numbers.

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Available abstract

A virtual boundary method for solving the Dirichlet and Neumann exterior problems of the 3-D Helmholtz equation,which is valid for all wave number is presented in this paper.When wave number is an eigenvalue of the interior Dirichlet or Neumann problem for the Laplacian,the solution of the boundary integral equation correspond to eigenvalue is not unique.Based on the extension of potential function,Virtual boundary integral equation is deduced.The uniqueness problem is granted by choosing different virtual boundary,with which the wave number do not coincide with the eigenvalue of the original interior Dirichlet or Neumann problem for the Laplacian.The results of numerical examples demonstrate that the scheme presented is practical and effective for the exterior problems of the 3-D Helmholtz equation with any wave numbers.

Key concepts: Helmholtz equation, Mathematics, Neumann boundary condition, Mathematical analysis, Robin boundary condition, Mixed boundary condition, Dirichlet boundary condition, Dirichlet eigenvalue

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