Research Of 3D Fast Multipole Method Of Fundamental Solutions On Mechanical Noise Prediction And Its Application

The numerical methods for noise prediction in mechanical engineering mainly include the finite element method(FEM) and the boundary element method(BEM). FEM can solve many acoustic problems. However, it could not solve the exterior acoustic problems, for it cannot automatically identify boundaries. In addition, mesh division needs expensive computational resources and computational time for large scale problems. BEM can automatically satisfy the radiation condition of far field and only require mesh division on the surface, which improves the efficiency of mesh division. Therefore, it can be especially applied to the radiation condition of far field. However, the disadvantages of BEM are the singular integral and the uncertainty solutions at the eigenfrequency. In compared with BEM, the conventional method of fundamental solutions(CMFS) has all the advantage of BEM. The source points of CMFS are positioned outside of the domain, thus it avoids singular numerical integrals. With its application in acoustic field prediciton, the system matrix in CMFS is dense and non-symmetric matrix, it requires large memory storage and has low computational efficiency. In order to improve the computational efficiency of the CMFS, the fast multipole method(FMM) is combined with MFS, then the fast multipole method of fundamental solutions(FMMFS) is formed, which is used to predict mechanical noise.Combined FMM with CMFS, FMMFS is formed for two dimensions. The realization steps and the computational complexity of FMMFS are proposed. The acoustic radiation of the two-dimensional pulsating circular ring is taken as an example. The computational accuracy, computational efficiency and the impact of the distribution of source points on the computational accuracy are studied. The results show that the conformal distribution of source points for the computational accuracy of CMFS is better than that the annular distribution of source points for two-dimensional problems.To solve the problems for three-dimensional noise prediction, FMMFS is extended from two dimensions to three dimensions. The theoretical formulas of three-dimensional FMMFS are derived. FMMFS is used to compute the acoustical performance of the muffler, the computation results of FMMFS are in line with that of BEM, which verifies FMMFS. With the more degree of freedom of the practical problem, the computational efficiency of FMMFS becomes higher.The problems for noise prediction in practical engineering are mainly half-space acoustic problems. Thus, FMMFS is extended from full-space to half-space. The relative formulas of half-space FMMFS for the multipole expansion and the expansion transformation are derived. The computational precision of the fast multipole method of fundamental solutions mainly depends on the number and distribution of source points. The distance between the source points and collocation points is too less and computational errors are increased. Conversely, computational errors become small, meanwhile this enables source points become denser and induce the matrix’s ill-condition. In order to overcome the difficulty of determining the distribution of source points, the genetic algorithms are used to optimize the distribution of source points. Combined half-space FMMFS and the genetic algorithms, the genetic algorithms-based FMMFS is proposed. In the study on the complex surface of the engine, the genetic algorithms are applied to determine the distribution of source points of the model of the engine and then FMMFS is applied to predict the acoustic radiation field of engine surface. The results reveal that the optimal distance between the collocation points and source points is 34.5mm. When truncation terms are 15, it not only avoids uncertainty in low frequency by the too small truncation terms, but also guarantees the computational efficiency. With the optimized distribution of source points, the number of source points and collocation points can be reduced and meanwhile the computational precision is maintained, therefore the computational efficiency is improved. Genetic algorithms-based FMMFS is applied for the prediction of the acoustic radiation field of engine surface, the predicition accuracy of the radiated acoustic field of the engine is improved. Experimental results are in good agreement with computational results, which shows that Genetic algorithms-based FMMFS is validated. Genetic algorithms-based FMMFS may also allow its further application to the mechanical noise prediction.