大规模边界元模态分析的高效数值方法

随着大规模快速边界元计算技术的发展,在复杂结构的动态设计、振动与噪声分析中愈来愈多地采用边界元法,因此求解大规模边界元特征值问题、进行复杂结构和声场模态分析,成为工程应用中一个十分重要,但却极具挑战性的课题,目前国际上还没有十分有效的数值方法.本文针对边界元法中典型的非线性特征值问题,提出了一种通用、高效的数值解法,称为基于预解矩阵采样的Rayleigh-Ritz投影法,记为RSRR.首先,通过求解一系列频域边界元问题来构造特征向量搜索空间,进而可以采用Rayleigh-Ritz投影,将原问题转化为一个可以采用现有方法求解的小规模缩减特征值问题;其次,为了降低Rayleigh-Ritz投影过程的计算量,基于解析函数的Cauchy积分公式,构造了边界元系数矩阵的插值近似方法,以及缩减特征值问题系数矩阵的快速计算方法,给出了插值项数的估计策略;最后,将RSRR与声学快速边界元法结合,应用于大规模吸声结构的复模态分析.数值算例表明,RSRR方法能够可靠地求出给定频段内的全部特征值和特征向量,具有计算效率高、精度高、通用等优点.

AN EFFICIENT NUMERICAL METHOD FOR LARGE-SCALE MODAL ANALYSIS USING BOUNDARY ELEMENT METHOD

Thanks to the great advances in fast boundary element method (BEM) achieved in the last two decades, the BEM has been increasingly used in the dynamic design of engineering structures, the analysis of noise and vibration. Consequently, solving large-scale eigenvalue problems and performing modal analysis for complicated structures and acoustic fields using the BEM becomes an very important but challenging task; so far there are no effective numerical methods for this purpose. This paper aims to extend the application of the newly-developed resolvent sampling based Rayleigh-Ritz projection method (RSRR) to the solution of the general nonlinear eigenvalue problems (NEP) in BEM. First, in order to generate reliable eigenvector search spaces, a series of BEM linear systerms in frequency domain are solved. Then the original NEP can be transformed to a reduced NEP based the classical Rayleigh-Ritz procedure, and the reduced NEP could be solved by those exiting NEP solvers easily. Second, to reduce the prohibitive computational burden involved in solving the projected NEP by the Rayleigh-Ritz procedure, a BEM matrix interpolation technique and a fast computation method for reduced NEP systerm matrix are proposed based on the discretized Cauchy integral formula of analytic functions. Then a simple rule for estimating the number of terms in the interpolation is proposed as well. Finally, the RSRR method is used to solve large-scale practical acoustic modal analysis problems using fast BEM with complicated sound absorbing boundary conditions. Numerical results indicate that the method can robustly dig out all the interested eigenvalues and the corresponding eigenvectors with good accuracy and high computational efficiency.