Acoustic Eigenvalue Analysis And Near-Field Acoustics Analysis Based On Radial Integration Boundary Element Method And Adaptive Integration Method

For steady-state acoustic problems, the fundamental solution of the Helmholtz equation is dependent on the frequency. So, the matrices of influence coefficients by discretizing the boundary integral equation implicitly contain frequency term and the frequency term can not be factored out. Early attempts of this nonlinear eigenvalue problem were confined to using the determinant search method. Consequently, the traditional boundary element method has a well-known difficulty when calculating acoustic eigenvalue problems. Meanwhile, nearly singular integrals will be encountered in near-field acoustics analysis by using the traditional boundary element method. At this time the integrand varies sharply, so the conventional Gaussian quadrature becomes inefficient or even inaccurate. This paper focuses on these two points to start research work. The main contents in the dissertation and innovative achievements are summarized as follows:The integral equation and boundary integral equation of acoustics Helmholtz equation is obtained by using the fundamental solution of Laplace equation, which is not dependent on the frequency and results in domain integrals in the integral equation and boundary integral equation. The radial integration method is presented to transform domain integrals to boundary integrals. Consequently, the advantages of the boundary element method in that only the boundary of the problem needs to be discretized into elements are maintained. In this paper, the acoustics radial integration boundary element method is presented by extending the radial integration method to computational acoustics, which is calculated in the field of complex numbers. The proposed method eliminates the frequency dependency of the coefficient matrices in the traditional boundary element method and the dependence on particular solutions of the dual reciprocity method and the particular integral method, meanwhile, enhanced determinant search method in the multiple reciprocity method is avoided. Eventually, the acoustic eigenvalue analysis procedure based on the acoustics radial integration boundary element method resorts to a generalized eigenvalue problem.In the scope of boundary element method for eigenvalue analysis of acoustical cavity covered with porous materials, the following two factors led to the non-linear. One is that the matrices of influence coefficients by discretizing the boundary integral equation implicitly contain frequency term. The other comes from the frequency dependency of the acoustic impedance of sound absorbing materials. The contour integral method is used to solve nonlinear eigenvalue problems in the traditional acoustics boundary element method. While in this method, the contour integral method parameter is required. It is crucial for impedance boundary conditions as spurious eigenvalues could be produced near the damped solutions, therefore degrades the reliability of the contour integral method. In this paper, the frequency dependency derived from above two factors can be eliminated by using the acoustics radial integration boundary element method and polynomials approximating. Finally, with the help of the state-space form, a generalized eigenvalue problem is obtained. This method avoids nonlinear transcendental eigenvalue equations.When the radiator is planar noise source, a Green function can be chosen such that its normal derivative vanishes on the planar surface, so that the Helmholtz integral equation is degraded into the Rayleigh integral. The exact radiation acoustical analysis of simple sound radiators, which is controlled by Rayleigh integral, can be obtained. The obtained analytical solutions refer to far-field acoustics variables by approximating the distance from the radiator to observation point. Assuming that the velocity on the plane radiator is consistent, the Rayleigh integral is degraded into the oscillating piston, which is embedded in an infinite rigid baffle. In this case, the near-field solution is still not obtained. From a numerical analysis point of view, nearly singular integrals will be encountered in near-field acoustics boundary element analysis. The Gaussian integral could not effectively tackle this issue because the integrand varies sharply and special treatment is required in this case. In fact, the nearly singular are not singular in the sense of mathematics. In this paper, an adaptive integration technique is presented to tackle nearly singular integrals which arise in three-dimensional near-field acoustics boundary element analysis. This method involves subdividing the element into sub-elements to avoid using excessively high Gauss order according to Davies&Bu criterion or Gao&Davies criterion. By introducing the Jacobian of the sub-element, nearly singular integrals can be calculated numerically without the nodal values of sub-elements. The given numerical examples show that the proposed approach is robust.