Study On Computational Methods And Applications For Large Scale Acoustic Problems Based On The Fast Multipole Boundary Element Method

Boundary element method(BEM), a numerical method based on integral equation, can reduce the dimension of the model(from three and two dimension to two and one dimension, respectively), and only requires the boundary to be discretized. It is easy to handle the far field boundary condition with the BEM since the integral equation automatically satisfies the radiation condition. As one semi-analytical numerical method, BEM is more accurate than other numerical methods. Thus it is suitable to be applied in the analyses of acoustic problems. Although dimension is reduced, the coefficient matrix generated by the BEM is full, non-symmetrical and even singular, which results in huge memoery cost and computational complexity. Those features prevent BEM from the analyses of large scale acoustic problems.Currently, the analyses and studies for acoustic problems tend to large scale and multi-physics models, such as simulations for scattering and radiation of submarine, taking off and landing noise of airplane, acoustic characteristics of organs and tissues and the optimization of coupled structure-acoustics. Those models will generate DOFsat the level of hundreds of thousands or even several millions and thus urgently demand the BEM can solve such large scale acoustic problems within reasonable time. The fast multipole method(FMM) can reduce the momery and improve the computational efficiency dramatically. It presents a potential way to solve the large scale acoustic problems.To meet the demand for the computation of large scale acoustic problems, the present dissertation performs systemic study for free space, multi-domain and half space acoustic problems based on FMM. New theories and algorithms of the fast multipole boundary element method are proposed to solve the two bottlenecks, larege memoery and huge computational complexity, of BEM. The developed methods are applied to predict the acoustic radiation and structure-acoustic optimization for large scale structure. Those successful applications demonstrated the super computational capability for large scale acoustic problemsand the potential in the acoustic optimization of developed methods.The main contributions of the dissertation are as follows:An analytical evaluation method and its error analysis are proposed to compute the two left line integrals in the explicit hyper-singular expression for constant element discretization in BEM. It overcomes the difficulty in the implementation of Burton-Miller formulations for constant element.Analytical integration formulations are derived for moments with constant and linear element type in high frequency FMBEM(HF-FMBEM). It improves the efficiency and accuracy of moments and results in the speed up of the HF-FMBEM. The analytical moments in the HF-FMBEM are applied in upward pass in the low frequency FMBEM(LF-FMBEM). It results in improving the efficiency and accuracy of moments as well as keeping the advantage of low memory cost in LF-FMBEM. Combining the improved LF-FMBEM and HF-FMBEM, a wideband FMBEM is developed which can be used for large scale acoustic problems in a wide range of frequencies.The Burton-Miller based FMBEM is extended to solve multi-domain acoustic problems. A strategy for assigning a single tree structure for each domain is developed to handle the difficulty in the implementation of multi-domain FMBEM. A boundary block preconditioner is proposed to improve the condition number of the coefficient matrix which results in reduction of iteration in solution. The proposed FMBEM can improve the solution efficiency for simulations of practical multi-domain acoustic problems, such as porous material, under-water structure-acoustic and organs and tissues, and so on.A FMBEM for three-dimensional half-space acoustic wave problems over an impedance plane is proposed. A new algorithm is developed to divide the downward pass into real domain to real and imagine domain, respectively. A piecewise analytical expression of transfer function of moment to local(M2L) is derived which can compute the M2 L accurately and efficiently. Merging the piecewise analytical M2 L translators with the multi-level tree structure can further improve the efficiencies of M2 L and direct coefficients computation among adjacent cells. The developed half-space FMBEM makes the FMBEM can be applied in analysis for the three-dimensional half-space acoustic wave problems over an impedance plane.For baffled-planar structure, a new method based on the conjunction of FMBEM and iterative eigenvalue solver is proposed for computations of radiation modes. For three dimensional models, the concept of mapped acoustic radiation modes is proposed which arises from theequalent source method, multipole expansion of the Green’s function and the BIE. The spherical harmonic function is proved to be a set of mapped radiation modes. A simple method to compute the sound power of vibrating structure is developed based on the mapped radiation modes. The developed theory of mapped acoustic radiation modes facilitates the sound power evaluation of large scale vibrating structure, and makes optimization of the structure-acoustics acceptable.The developed theory and algorithm of FMBEM are applied to predict the radiation acoustic for a vibrating cylinder model. Aninterpolation method for the cylinder model based on Fourier series and Lagrange functions is proposed to obtain boundary values at arbitrarypositions on the body of the cylinder from the sampled singals. The developed FMBEM is used to compute the radiation field based on the interpolated boundary conditions. Comparisonsdemonstrate that predicted and measured results agree very well with each other. It also validates indirectly that the interpolation method is accurate.Combining the developed FMBEM and the theory of mapped acoustic radiation modes, optimizations are performed for the ribbion positions of compressor casing and the damping layout of the submarine model respectively with radiation sound power being the object function. To further improve the computational efficiency in the optimization of damping layout of the submarine model, commercial finite element methods and the developed FMBEM are integrated in the matlab development envirmoents and paralleled. Good optimization results are obtained within resonable time, which demonstrate the potential and significanceof the proposed optimization strategy for large scale structure-acoustic optimization.A software package based on the research code is developedwhich can server as a powerful numerical tool for the analysis of large scale acoustic problems.