An indirect BIE free of degenerate scales

Jeng-Tzong Chen; Shing-Kai Kao; Jeng-Hong Kao; Wei-Chen Tai

doi:10.3934/cpaa.2021114

Article Contents

2022, Volume 21, Issue 6: 1969-1985. Doi: 10.3934/cpaa.2021114

This issue Previous Article Ground state solutions for the fractional problems with dipole-type potential and critical exponent Next Article Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping

An indirect BIE free of degenerate scales

a.
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan
b.
Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan
c.
Department of Civil Engineering, National Cheng-Kung University, Tainan, 70101, Taiwan
d.
Bachelor Degree Program in Ocean Engineering and Technology, National Taiwan Ocean University, Keelung, 20224, Taiwan
e.
Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan

^* Corresponding author
^* Corresponding author

Received: November 2020

Revised: May 2021

Early access: August 2021

Published: June 2022

The financial support from the Ministry of Science and Technology, Taiwan under Grant No. MOST 106-2221-E-019-009-MY3 and No. MOST 107-2221-E-019 -003, and we also appreciate Mr. Kuen-Ting Lien to provide numerical results.

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Row space, column space, null space and range of $ [U] $ and $ [T] $ matrices in the indirect and direct BEMs when the degenerate scale occurs

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Figure 2. Numerical evidences for the degenerate scale of four cases in the BEM

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Table 1. Comparison of four BIEs

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Table 2. Study on the degenerate scale by using different approaches

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Table 3. Degenerate scales appearing in the conventional BEM, the approach of adding rigid body mode, the Burton and Miller approach and the method of introducing the fictitious source point

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Table 4. Comparison of the four kernel functions

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