Study On The Interior Acoustic Problem Based On The Radial Basis Collocation Meshless Method

Study on the interior acoustic problem is very important for the acoustic design of the equipments with cavity. As a very powerful tool, finite element method (FEM) has solved many acoustic problems in practical projects. However, the FEM solutions will becomeinaccurate as the frequency increasing, which is directly related to the dispersion.On the other hand, as a relatively new numerical method, meshless method is independence of the concept of the element which traditional element-type method depends on, thus it has some special features. And radial basis collocation method (RBCM) has the advantages of high accurate and easy programming. Furthermore, some researchers’study shows that, comparing with the FEM, meshless method has better performance in controlling the dispersion. Thus it has the potential value in solving the interior problem in a wider frequency rage.In this paper, RBCM is introduced into the solution of the interior acoustic problem, the following work is done:The traditional RBCM is modified in this paper. The radial basis function (RBF) is a function of distance, its great performance in the scattered data interpolation has attracted many researchers’attentions. And many efficient meshless methods have been formulated by employing the RBF, among these methods, the RBCM is a typical one. Considering the advantages of this meshless method, such as high accuracy, easy to implement and without integration, it has been successfully used to solve the differential equations, In terms of the size of the support domain, the RBCM can be divided into two kinds:the global RBCMand the local RBCM. The accuracy of the global one is high, but which will suffer from the ill-condition in solving the large scale problems. On the contrary, the local one can circumvent the ill-condition problem while its accuracy is worse than the global one. In order to overcome these problems, we try to improve the performance of these two kinds of methods by changing the shape parameters in the RBF, which leads to the variable shaped RBF.When solving the interior acoustic problem with meshless method, the treatments for the boundary conditions are study. The boundary conditions have a great influence on the accurate of the solution, and meshless method has its own specialty in dealing with the boundary conditionsWith respect to different boundary conditions, we present different treatments with meshless method. In order to handle the Dirichlet boundary, variable shaped parameter method, dense point method and high order additional polynomial method are presented. For the boundary conditions with derivative, we employ the Hermit collocation method, fictitious point method into the boundary treatment. In addition, the well developed model updating technology used in structure is introduced into the establishment of the impedance boundary condition.The contribution of the error in the solution of the interior acoustic problem is presented, the estimation of the dispersion is proposed and the advantage of the meshless method in solving the interior acoustic problems is validated. The composition of the error in solving the interior acoustic problem with FEM is analyzed theoretically. As the frequency increases, the dispersion is the key reason for the increment of the error.Comparison between the FEM solution and the RBCM solution validated the excellent performance of our method in controlling the dispersion, which makes the RBCM can solve the interior problem in a wider frequency range. In the end, combined with the boundary condition treatments, the numerical implementation scheme is given, and the effectiveness of our method is validated by numerical examples with different boundaries and the experiments.RBCM has the following advantages:it has high accurate,it does not need mesh and it is easy in implementation, and it is an independent spatial dimension which can be easily extended to high-dimensional problems. What is more, the RPCM can solve the interior acoustic problem in a wider frequency range while keeping its advantages.