# Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

Jeng-Tzong Chen (Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan and Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, Taiwan)
Shyh-Rong Kuo (Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan)
Yu-Lung Chang (Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan)
Shing-Kai Kao (Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan)

ISSN: 0264-4401

Article publication date: 3 July 2017

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## Abstract

### Purpose

The purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically.

### Design/methodology/approach

To avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered.

### Findings

Results using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function.

### Originality/value

The analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.

## Acknowledgements

Financial support from the Ministry of Science and Technology under Grant No. MOST 103-2221-E-019-012-MY3 and MOST 104-2221-E-019-007-MY3 for National Taiwan Ocean University is gratefully acknowledged.

## Citation

Chen, J.-T., Kuo, S.-R., Chang, Y.-L. and Kao, S.-K. (2017), "Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis", Engineering Computations, Vol. 34 No. 5, pp. 1527-1550. https://doi.org/10.1108/EC-06-2016-0187

## Publisher

:

Emerald Publishing Limited

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