Chuanzeng Zhang

Jinyou Xiao, Shuangshuang Meng, Chuanzeng Zhang et al.

A new algorithm, denoted by RSRR, is presented for solving large-scale nonlinear eigenvalue problems (NEPs) with a focus on improving the robustness and reliability of the solution, which is a challenging task in computational science and engineering. The proposed algorithm utilizes the Rayleigh-Ritz procedure to compute all eigenvalues and the corresponding eigenvectors lying within a given contour in the complex plane. The main novelties are the following. First and foremost, the approximate eigenspace is constructed by using the values of the resolvent at a series of sampling points on the contour, which effectively circumvented the unreliability of previous schemes that using high-order contour moments of the resolvent. Secondly, an improved Sakurai-Sugiura algorithm is proposed to solve the projected NEPs with enhancements on reliability and accuracy. The user-defined probing matrix in the original algorithm is avoided and the number of eigenvalues is determined automatically by provided strategies. Finally, by approximating the projected matrices with the Chebyshev interpolation technique, RSRR is further extended to solve NEPs in the boundary element method, which is typically difficult due to the densely populated matrices and high computational costs. The good performance of RSRR is demonstrated by a variety of benchmark examples and large-scale practical applications, with the degrees of freedom ranging from several hundred up to around one million. The algorithm is suitable for parallelization and easy to implement in conjunction with other programs and software.

Jinyou Xiao, Junjie Rong, Wenjing Ye et al.

The frequency-domain approach (FDA) to transient analysis of the boundary element method, although is appealing for engineering applications, is computationally expensive. This paper proposes a novel adaptive frequency sampling (AFS) algorithm to reduce the computational time of the FDA by effectively reducing the number Nc of sampling frequencies. The AFS starts with a few initial frequencies and automatically determines the subsequent sampling frequencies. It can reduce Nc by more than 2 times while still preserving good accuracy. In a porous solid model with around 0.3 million unknowns, 4 times reduction of Nc and the total computational time is successfully achieved.

Jinyou Xiao, Chuanzeng Zhang, Tsung-Ming Huang et al.

Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by matrix structures, property of eigen-solutions, size of the problem, etc. This paper aims to break those limitations and to develop robust and universal NEP solvers for large-scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach (CIA), the RIA provides the possibility to select sampling points in more general regions and has advantages in improving accuracy and reducing computational cost. Second, a resolvent sampling scheme using the RIA is proposed for constructing reliable search spaces for the Rayleigh-Ritz procedure, based on which a robust eigen-solver, denoted by RSRR, is developed for solving general NEPs. RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of RSRR are demonstrated by a variety of benchmark and practical problems.