Averaged velocity boundary element method for sound radiation from vibrating structures : on the use of element volume velocities for improving computational efficiency

Abstract

In the field of noise control engineering the numerical computation of the sound radiation from vibrating structures plays an important role in product design. This study is concernedwith the Boundary ElementMethod (BEM), which enables accurate modeling of sound radiation from structures with complex geometry and with complex patterns of vibration. Proper acoustical modeling usually requires large numbers of Degrees Of Freedom (DOF) and conventional BEM algorithms need to solve fully populated linear systems of equations. Therefore, the major practical obstacle for using BEM with conventional solvers is formed by tremendous computational costs and by hardware limitations. In this thesis Averaged Velocity BEM (AVBEM) is developed, which is attractive for its simplified implementation and its fast assembly. AVBEM is used to find mesh discretization rules that keep errors within a priori chosen value with a minimum number of elements. For AVBEM with a conventional iterative solver the CPU times on a modern desktop PC will remain within about one minute for model sizes up to 6000 DOF. This CPU time is for a singe frequency line. It is shown that when using the economic discretization rules derived in this study, AVBEM can handle a wide range of noise control engineering problems for such model sizes. AVBEM has an original combination of three distinctive features. First, the acoustic variables are, at the element level, represented by averaged quantities. For the boundary elements on the vibrating structure, averaged element velocities are prescribed. Second, sub-parametric elements are used, namely elements with a zero order approximation for the acoustic variables and a higher order approximation for the geometry. These higher order polynomial shape functions ensure a proper geometrical modeling without increasing the size of the models. Third, a highly accurate approximation of the input variable (averaged velocity) is used. This is done by modeling the boundary velocity with auxiliary nodes and high order polynomials and integrating it for each element. Most of other BEM variants use isoparametric elements, i.e. elements with the same order of polynomial shape functions for the approximation of acoustic variables and geometry. Such BEM variants are designated as Shape Function BEM or SFBEM. Numerical simulations are used to derive the rules for economic mesh discretization. The majority of the simulations are for plates with flexural vibration. These plates are placed in different baffles and sound power is computed. The ratio of acoustical and structural wavelengths is systematically varied and the models are analyzed both with AVBEM and SFBEM. A few calculations on other structures complete the simulations for sound power. Additional to the plate simulations for sound power, a few similar calculations are performed for sound pressure in the near field and far field of plates. The discretization errors in these simulations remain basically within the error range as for sound power. A basic summary of the discretization study is that for relatively thin-walled structures with flexural wavelengths smaller than the acoustical wavelength, AVBEM needs significantly less DOF (often 2 to 3 time less) than SFBEM for the same accuracy. On relatively thick-walled structures and on rigid boundaries, only a small DOF reduction is obtained. But for these structures, an already coarse discretization (approximately 3 DOF per acoustical wavelength) ensure an accuracy of 2 dB. These rules are further simplified for situations where the structural wavelength is not known: 3.5 DOF per acoustical wavelength should be used to ensure a 2 dB accuracy. Compared to the industrial practice of using 6 DOF per acoustical wavelength, this coarser discretization reduces the model size with a factor 3. However, this rule can not be applied for very thin-walled structures on which the structural wavelength is less than half the acoustical wavelength. For these structures, it is necessary to discretize with respect to the structural wavelength. Using discretizations with 3 DOF per structural wavelength is assumed to give errors less than 2 dB. This implies more than 6 DOF per acoustical wavelength. The combination of the fast assembly of AVBEM with an iterative solver and with the economic meshing of the models leads to large CPU time gains for a wide range of engineering applications. It is recommended to convert Averaged Velocity BEM into Averaged Variable BEM to make it versatile for a broad range of noise control engineering applications. For this purpose only the implementation of two extra Boundary Conditions is needed. Furthermore, application of AVBEM is recommended for so-called coupled problems.