多连通域双调和波特征值问题的边界积分法

Boundary Integral Method for the Biharmonic Wave Eigenvalue Problem in Multiply Connected Domains

摘要: 弹性薄板振动的数值分析对航空航天结构设计与机械减振至关重要。传统有限元法精度不足，边界积分方程法在多连通域中易产生虚假特征频率。为此，本文提出新型虚假特征频率识别策略，结合基于多层位势边界积分方程的Boyd方法，求解固支边界多连通域的双调和波特征值问题。首先，基于多层位势理论推导多连通域的第二类Fredholm边界积分方程，将特征值问题转化为行列式求零点问题；其次，采用广义高斯求积实现数值离散，并通过Boyd求根方法高效求解特征频率；进一步，依据虚假特征频率与内边界外散射问题相关的特性，构建多连通域虚假频率识别方案；最后，通过数值算例验证了Boyd方法的精度与效率，并证实所提识别策略的可行性。

Abstract: Numerical analysis of elastic thin plate vibration is crucial for aerospace structural design and mechanical vibration reduction. The conventional finite element method suffers from insufficient accuracy, while the boundary integral equation method is prone to spurious eigenfrequencies in multiply connected domains. To address these issues, this paper proposes a novel identification strategy for spurious eigenfrequencies, combined with the Boyd method based on the multi-layer potential boundary integral equation, to solve the biharmonic wave eigenvalue problem of clamped boundary multiply connected domains. First, based on the multi-layer potential theory, the second-kind Fredholm boundary integral equation for multiply connected domains is derived, transforming the eigenvalue problem into a problem of finding zeros of the determinant. Second, the generalized Gaussian quadrature is adopted for numerical discretization, and eigenfrequencies are solved efficiently via the Boyd root-finding method. Furthermore, exploiting the characteristic that spurious eigenfrequencies are associated with the exterior scattering problem at inner boundaries, a scheme for identifying spurious frequencies in multiply connected domains is constructed. Finally, numerical examples verify the accuracy and efficiency of the Boyd method and validate the feasibility of the proposed identification strategy.