Nonlinear eigenvalue solver for spectral element of beam structures: An exponential matrix polynomial approximation with weighted residual method

The spectral element method (SEM) is a widely used frequency-domain technique for dynamic structural analysis. However, its frequency-dependent dynamic stiffness matrix leads to a nonlinear/transcendental eigenvalue problem (NLEP) for obtaining natural frequencies and mode shapes. Traditional numerical approaches, such as iterative root-finding and matrix polynomial linearization, are often used to solve NLEP. While linearization is robust, it becomes computationally expensive with increasing degrees of freedom and polynomial order. This study introduces a novel exponential matrix polynomial approximation of the dynamic stiffness matrix, combined with a weighted residual technique to compute polynomial coefficients without trigonometric or hyperbolic functions. The polynomial eigenvalue problem is then transformed into a generalized eigenvalue problem using a matrix pencil. The proposed method efficiently handles NLEP, even in cases with singularities and closely spaced modes. A lower-order matrix polynomial approximation improves computational efficiency while maintaining accuracy, outperforming Lagrange interpolating polynomials. The natural frequencies and mode shapes of thin-walled beams and frame structures with various boundary conditions are determined using the present NLEP solver, showing a close match with FEM results. However, in FEM the computational time increases exponentially with higher modes, whereas in SEM solved via NLEP, the increase is only marginal.