A Study On Fast Boundary Element Methods Based On The Hierarchical Matrices For Acoustic Problems

The boundary element method only needs to discrete the boundary,has high accuracy and is ideal for infinite problems.Hence,it is widely applied to solve all kinds of acoustic problems in engineering.However,the coefficient matrices from BEM are in general dense and even fully populated.A coefficient matrix requires O(N~2)memory consumption with N being the number of degrees of freedom(DOFs).Typically,the direct solvers require O(N~3)operations while the iterative solvers need O(N~2)operations.These drawbacks prevent the conventional BEM(CBEM)from being applied to large-scale problems.On a personal computer,the boundary element method can only solve the problems whose DOFs are less than 10,000.The fast algorithms,such as the fast multipole method and the adaptive cross approximation algorithm,make the CBEM break through the bottleneck that the CBEM cannot solve the large-scale problems.Nevertheless,the multipole expansion process and local expansion process in the fast multipole BEM(FMBEM)are so complex that the implementation and parallelization of FMBEM are very complicated.The ACABEM is a pure algebraic method based on the H-matrices.It does not depend on specific physical problems.The condition for right choice of rows is very rigorous and not easy to be implemented for cubes and cuboids with many DOFs.The H-matrices format divides the coefficient matrix into a series of submatrices blocks according to the tree structure.The far-field submatrices blocks will be approximately expressed by the low rank approximation algorithm such as the adaptive cross approximation(ACA)algorithm and the singular value decomposition.The whole coefficient matrix is compressed and only needs a few data to express it.Compared with the FMBEM,the implementation and parallelization of the H-matrices are much easier.It is because that the H-matrices format has block structure characteristics.In addition,the construction of efficient preconditioners may be significantly easier for the H-matrices.Hence,it is imperative to develop fast boundary element methods for acoustic problems,which are simple and easy to be implemented and parallelized,based on the H-matrices.In this paper,theoretical research is combined with numerical simulation.The main content in the paper include the followed sections.Based on the CBEM,the ACABEM is implemented.The sound radiation problem of a pulsating sphere with different pulse frequencies is analyzed by the CBEM and ACABEM.The corresponding accuracy,efficiency and memory consumption are compared.The adaptivity of the ACABEM within a wide frequency range is analyzed.The study of the ACABEM sets up the basic program framework for subsequent methods.Based on the multipole expansion theory,the multipole expansion algorithm(MEA)is presented for the low rank matrices in this paper.Substituting the MEA algorithm for the ACA algorithm in ACABEM,a new method referred as ME-H-BEM is developed.Firstly,the results of the ME-H-BEM with different number of expansion terms are analyzed.It is indicated that the number of expansion terms is over-estimated by the analytical formulation in the multipole expansion theory.Secondly,the sound radiation problem of a pulsating sphere is analyzed by the CBEM,ACABEM,ME-H-BEM and FMBEM.The accuracy,efficiency and memory consumption of above numerical methods are compared in two cases.The results indicate that the ME-H-BEM can obtain the same accuracy as the CBEM,ACABEM and FMBEM.The computational expense of the ME-H-BEM is higher than that of the ACABEM but lower than that of the CBEM.The CPU time used for a single coefficient matrix-vector product in the ME-H-BEM is less than that in the FMBEM.In addition,the ME-H-BEM is much easier to be implemented compared with the FMBEM.Thirdly,the adaptivity of the ACABEM and ME-H-BEM are compared by analyzing a pulsating cube model and a pulsating drum model.It is indicated that the ME-H-BEM has better adaptivity and can achieve higher accuracy than the ACABEM.Fourthly,the applicable frequency range of the ME-H-BEM is analyzed by solving the sound radiation problem of the pulsating sphere.The results indicate that the ME-H-BEM is suitable for a wide range of frequencies.Lastly,the ME-H-BEM is applied to analyze the sound radiation problem of a steam turbine generator.As the number of expansion terms is appropriately set,the ME-H-BEM has an advantage in terms of efficiency as compared with the ACABEM.It is indicated that the ME-H-BEM can be applied to solve acoustic problems in engineering.To accelerate the solution of ME-H-BEM and reduce its memory consumption,hybrid approximation BEMs are developed taking advantage of the high efficiency and low memory consumption property of the ACABEM and the high accuracy advantage of the ME-H-BEM.Two combination ways are tried in this paper.Traversing all the cells in the tree structure,the first way combining the ACA algorithm with the MEA algorithm is described as followed.The submatrices corresponding to a cell and its neighbor cells are completely evaluated by the CBEM and stored entirely.The ACA algorithm is applied to approximate the far-field submatrices generating from the cell and its nearest interaction cells.The MEA algorithm is used to approximate the far-field submatrices corresponding to the cell and other interaction cells.This combined method is named as the adaptive cross approximation-multipole expansion approximation BEM(ACA-MEA-BEM).To verify whether the algorithm applied to approximate the submatrices generating from cells and their nearest interaction cells dominates the accuracy,efficiency and memory consumption of a combined BEM,the second way is also tried to combine the ACA and MEA.In contrast to the ACA-MEA-BEM,the MEA algorithm is used to approximate the far-field submatrices corresponding to the cell and its nearest interaction cells.The ACA algorithm is applied to decompose the far-field submatrices corresponding to the cell and other interaction cells.The second combined BEM is named as the multipole expansion approximation-adaptive cross approximation BEM(MEA-ACA-BEM).Numerical examples are elaborately setup to compare the accuracy,efficiency and memory consumption of the ACABEM,ME-H-BEM and hybrid methods.It is indicated that the hybrid BEMs can reach the same level of accuracy as the ACABEM and ME-H-BEM.The second combined way even can obtain more accurate results in some frequency range compared with the ACABEM.The computational expense of the hybrid BEMs are higher than that of the ACABEM but lower than that of the ME-H-BEM,especially for the ACA-MEA-BEM.The algorithm used to approximate the far-field submatrices corresponding to the cells and their nearest interactional cells determines the accuracy,efficiency and memory consumption of the hybrid BEMs.The applicability of the hybrid BEMs in engineering is illustrated by the sound radiation problem of a compressor model and above steam turbine generator.Lastly,the CPU time complexity and storage requirement complexity of the ACABEM,ME-H-BEM and hybrid BEMs are quantitatively analyzed by the least square method.Similar with the ACABEM and ME-H-BEM,the proposed hybrid BEMs have both operation and storage logarithmic-linear complexity.