Solution for the degenerate scale for a rigid curve in antiplane elasticity by using a weakly singular integral equation

This paper provides a numerical solution for the degenerate scale for a rigid curve in antiplane elasticity. The degenerate scale problem for the rigid curve is formulated on the usage of the logarithmic potential. After assuming the displacement to be a vanishing value along the rigid curve, the boundary integral equation (BIE) is formulated. The problem can be first formulated in the degenerate scale. After making a coordinate transform, we can obtain the relevant BIE in the ordinary scale. Finally, a numerical solution is achieved. Several numerical examples are provided. In addition, the degenerate scale problem for the multiple rigid curves is also solved.

On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale for a circular domain

ABSTRACT In this paper, we proposed two ways to understand the rank deficiency in the continuous system (boundary integral equation method, BIEM) and discrete system (boundary element method, BEM) for a circular case. The infinite-dimensional degree of freedom for the continuous system can be reduced to finite-dimensional space using the generalized Fourier coordinates. The property of the second-order tensor for the influence matrix under different observers is also examined. On the other hand, the discrete system in the BEM can be analytically studied, thanks to the spectral property of circulant matrix. We adopt the circulant matrix of odd dimension, (2N + 1) by (2N + 1), instead of the previous even one of 2N by 2N to connect the continuous system by using the Fourier bases. Finally, the linkage of influence matrix in the continuous system (BIE) and discrete system (BEM) is constructed. The equivalence of the influence matrix derived by using the generalized coordinates and the circulant matrix are proved by using the eigen systems (eigenvalue and eigenvector). The mechanism of degenerate scale for a circular domain can be analytically explained in the discrete system.

An indirect BIE free of degenerate scales

<p style='text-indent:20px;'>Thanks to the fundamental solution, both BIEs and BEM are effective approaches for solving boundary value problems. But it may result in rank deficiency of the influence matrix in some situations such as fictitious frequency, spurious eigenvalue and degenerate scale. First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system. The influence of contaminated boundary density on the field response is also discussed. It's well known that the CHIEF method and the Burton and Miller approach can solve the unique solution for exterior acoustics for any wave number. In this paper, we extend a similar idea to avoid the degenerate scale for the interior two-dimensional Laplace problem. One is the external source similar to the null-field BIE in the CHIEF method. The other is the Burton and Miller approach. Two analytical examples, circle and ellipse, were analytically studied. Numerical tests for general cases were also done. It is found that both two approaches can yield an unique solution for any size.</p>