Abstract Spectral element method (SEM) is a robust and efficient mathematical technique for dynamic analysis of structures in frequency domain. Unlike finite element method (FEM), in SEM, the dynamic stiffness matrix forms a nonlinear eigenvalue problem (NLEP) to compute the natural frequencies and vibration modes of the structure which cannot be solved using linear numerical eigen-solvers. In this paper, two distinct numerical methods, i.e. (1) a root finding method of rational polynomial functions and (2) a linearization of Lagrange matrix interpolating polynomials, have been used to compute the eigenvalues of a problem more efficiently employing SEM. These proposed methods can solve NLEP in a stable, efficient and accurate way even in the presence of singularities. The accuracy of these methods are numerically evaluated by comparing with the solutions from the modal analysis using FEM.

中文翻译：

---

谱元法的非线性特征值分析

摘要 谱元法（SEM）是一种稳健而高效的数学技术，可用于结构的频域动态分析。与有限元法 (FEM) 不同，在 SEM 中，动态刚度矩阵形成非线性特征值问题 (NLEP) 来计算结构的固有频率和振动模式，这是使用线性数值特征求解器无法解决的。在本文中，两种不同的数值方法，即（1）有理多项式函数的求根方法和（2）拉格朗日矩阵插值多项式的线性化，已被用于更有效地使用 SEM 计算问题的特征值。即使在存在奇点的情况下，这些提出的方法也能以稳定、有效和准确的方式求解 NLEP。