Jinyou Xiao

Efficiency improvement of the polar coordinate transformation for evaluating BEM singular integrals on curved elements

The polar coordinate transformation (PCT) method has been e xt sively used to treat various singular integrals in the boundary element method (BEM). However, the resultant i ntegrands of the PCT tend to become nearly singular when (1) the aspect ratio of the element is large or (2) the fiel d point is closed to the element boundary; thus a large number of quadrature points are needed to achieve a relative ly high accuracy. In this paper, the first problem is circumvented by using a conformal transformation so that th e geometry of the curved physical element is preserved in the transformed domain. The second problem is alleviated by using a sigmoidal transformation, which makes the quadrature points more concentrated around the near singul arity. By combining the proposed two transformations with the Guig giani’s method in [M. Guiggiani,et al. A general algorithm for the numerical solution of hypersingular boundary integral equations. ASME Journal of Applied Mechanics, 59(1992), 604-614], one obtains an e fficient and robust numerical method for computing the weakly, stronglyand hyper-singular integrals in high-order BEM w ith curved elements. Numerical integration results show that, compared with the original PCT, the present method can reduce the number of quadrature points considerably, for given accuracy. For further verification, the method is i ncorporated into a 2-order Nyström BEM code for solving acoustic Burton-Miller boundary integral equation. It is s hown that the method can retain the convergence rate of the BEM with much less quadrature points than the existing PCT. T he method is implemented in C language and freely available.

Resolvent sampling based Rayleigh–Ritz method for large-scale nonlinear eigenvalue problems

Abstract 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 circumvents 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 the 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.

Efficiency improvement of the frequency-domain BEM for rapid transient elastodynamic analysis

The frequency-domain fast boundary element method (BEM) combined with the exponential window technique leads to an efficient yet simple method for elastodynamic analysis. In this paper, the efficiency of this method is further enhanced by three strategies. Firstly, we propose to use exponential window with large damping parameter to improve the conditioning of the BEM matrices. Secondly, the frequency domain windowing technique is introduced to alleviate the severe Gibbs oscillations in time-domain responses caused by large damping parameters. Thirdly, a solution extrapolation scheme is applied to obtain better initial guesses for solving the sequential linear systems in the frequency domain. Numerical results of three typical examples with the problem size up to 0.7 million unknowns clearly show that the first and third strategies can significantly reduce the computational time. The second strategy can effectively eliminate the Gibbs oscillations and result in accurate time-domain responses.

A kernel-independent fast multipole BEM for large-scale elastodynamic analysis

Purpose – The purpose of this paper is to develop an easy-to-implement and accurate fast boundary element method (BEM) for solving large-scale elastodynamic problems in frequency and time domains. Design/methodology/approach – A newly developed kernel-independent fast multipole method (KIFMM) is applied to accelerating the evaluation of displacements, strains and stresses in frequency domain elastodynamic BEM analysis, in which the far-field interactions are evaluated efficiently utilizing equivalent densities and check potentials. Although there are six boundary integrals with unique kernel functions, by using the elastic theory, the authors managed to accelerate these six boundary integrals by KIFMM with the same kind of equivalent densities and check potentials. The boundary integral equations are discretized by Nystrom method with curved quadratic elements. The method is further used to conduct the time-domain analysis by using the frequency-domain approach. Findings – Numerical results show that by t...

An efficient adaptive frequency sampling scheme for large-scale transient boundary element analysis

Aims at reducing the computational cost for BEM transient analysis using the frequency-domain approach.Proposes an adaptive algorithm to determine the optimal sampling frequencies.Achieves 2 to 4 times reduction of the total computational cost. 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 N c of sampling frequencies. The AFS starts with a few initial frequencies and automatically determines the subsequent sampling frequencies. It can reduce N c 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 N c and the total computational time is successfully achieved.

An improved r-adaptive Galerkin boundary element method based on unbalanced Haar wavelets

An r-adaptive boundary element method (BEM) based on unbalanced Haar wavelets (UBHWs) is developed for solving 2D Laplace equations in which the Galerkin method is used to discretize boundary integral equations. To accelerate the convergence of the adaptive process, the grading function and optimization iteration methods are successively employed. Numerical results of two representative examples clearly show that, first, the combined iteration method can accelerate the convergence; moreover, by using UBHWs, the memory usage for storing the system matrix of the r-adaptive BEM can be reduced by a factor of about 100 for problems with more than 15 thousand unknowns, while the error and convergence property of the original BEM can be retained.