Numerical Algorithms For The Nonlinear Eigenvalue Problems In Boundary Element Methods

Boundary Element Method(BEM)demonstrates great advantages in solving acoustics problems due to its boundary-only discretization,high accuracy and easy of treating unbounded domains.However,Modal analysis for Acoustics in BEM is a nonlinear eigenvalue problem(NEP),which is still a challenge in large-scale engineering applications.Therefore,developing numerical algorithms for NEPs in BEM is of great importance.This thesis introduces a novel numerical algorithm(RSRR)for solving large-scale NEPs with the robustness and reliability of the solution.It is developed based on Rayleigh–Ritz procedure to be able to compute all eigenvalues and the corresponding eigenvectors lying within a given contour in the complex plane.It also has great potential in solving large-scale NEPs because the main idea of this method is to condense original NEPs by constructing approximate eigenspace.The main contribution of this thesis lies in the following four aspects:(1)The approximate eigenspace is constructed by using the values of the resolvent at a series of sampling points on the contour,which effectively circumvents the unreliability of previous schemes using high-order contour moments of the resolvent.(2)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 the provided strategies.(3)By approximating the projected matrices with the Chebyshev or Cauchy interpolation technique,RSRR is further extended to solve NEPs in BEM,which reduces the calculation times of BEM matrices and avoids high computational costs.(4)Combined Additive preconditioning technique with Woodbury equation,this thesis proposes a preconditioner for(near)singular equations derived from situations where resolvents come across the eigenfrequencies or the distances between them are too near.The proposed preconditioning technique shows effective improvement for slow convergence or even non-convergence in solving(near)singular equations.Adequate numerical examples are used to validate the advantages of the proposed algorithm(RSRR)compared with the existing SSRR in terms of accuracy and efficiency.The numerical method(RSRR)is also suitable for parallelization and easy to implement in conjunction with other programs and software,which has great advantages in solving large-scale NEPs.