Dual reciprocity boundary element analysis of Helmholtz equations with variable coefficients

Abstract

Practical engineering problems have complex domains and boundary conditions which make them difficult to solve using analytical methods. Hence, such engineering problems are often solved numerically to a desired level of accuracy. There are several approximation methods for making numeric solutions, of which the boundary element method (BEM) has been adopted in my thesis work where problem’s dimensionality is shortened by one. Three-dimensional elements can be modelled with two-dimensional elements and similarly two-dimensional domains can be modelled with line elements. BEM requires less processing time and storage space than domain approaches such as finite element or finite difference methods. For this reason, boundary element method has been preferred here. Initially, the formulation of boundary element is employed to Laplace type equation. Later, we attempted Helmholtz type equations. Since Helmholtz equation involves non-homogenous terms, therefore to solve Helmholtz equations we have to go for domain integration technique. However, domain integration technique lacks the flavor of a true “boundary-only” solution. By adopting dual reciprocity boundary element method (DRBEM) this lacuna can be overcome. In my thesis work by using dual reciprocity boundary element method Helmholtz type equation is solved. For practical engineering problem numerical problem on acoustics is solved. A FORTRAN 77 program is developed for the present purpose with constant elements. This eases computation while enhancing understanding. For numerical validation purpose we solve a potential flow problem which satisfies Laplace equation, then we solve two-dimensional acoustics type problem which satisfies Helmholtz equation. Finally, we take a Poisson’s equation problem as a case study, compare results with finite difference method and do some mesh convergence study.