The Construction And Analysis Of A Conjugate Operator In Acoustic Waveguides With Curved Interface

In this paper a conjugate operator M is constructed to compute coefficients in marching method. With numerical experiments we demonstrate that this method has improved the simulation process of Helmholtz equation.First of all,we use PML(Perfectly Matched Layer)technique to truncate Helmholtz equation defined in an unbounded domain into the bounded area.Mathematically a complex stretching transform is operated on z coordinate which changes original equation into a complex one.Secondly,to avoid "staircase"approximation of the internal interface,a local orthogonal transform is used to "flatten"the curved internal surface.Under new coordinate system,the interface conditions can be easily handled and the transformed equation is suitable for marching method without adding much computation cost.Thirdly,there are some direct methods of solving Helmholtz equation,such as finite element method and finite difference method.However,they are not suitable for Helmholtz equation in large range,considering the large linear system they produce.The computation cost and storage space are highly expensive.As a result,iterative method,such as One-way method and marching method are employed here to simulate the wave field.During the process of marching computation,the eigenvalues and eigenvectors of related operator L should be given out.Multiple unsymmetric rayleigh quotient iterative method is employed here to solve the problem.Because local eigenfunctions of L operator are not orthogonal,related expansion coefficients are computed by solving several large linear systems.To avoid solving equations we construct the conjugate operator M of L operator,with the eigenvectors of M operator orthogonal to corresponding eigenvectors of L operator the coefficients during eigenfunction expansion can be derived without solving linear systems.The improvement saves much computation cost and speeds the marching process.Numerical experiments demonstrate the orthogonality of eigenvectors between L and M operators, besides L and M operators share the same eigenvalue distribution under certain condition, which means construction of M operator is feasible.Further more we simulate the wave field with one-way method and marching method using and not using the M operator for comparison. The former results keep acceptable accuracy.This method can be further applied for unbounded optical waveguides.