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: October 31, 2020

Revised: April 30, 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

Abstract Full Text(HTML) Figure(2) / Table(4) Related Papers Cited by

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.

Keywords:

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

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

Table 1. Comparison of four BIEs

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

Table 4. Comparison of the four kernel functions

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. J. Burton and G. F. Miller, On the stability of time-domain integral equations for acoustic wave propagation, Discrete Contin. Dyn. Syst., 323 (2016), 4367-4382. doi: 10.2307/2152750.
[2]	R. Chapko and B. T. Johansson, On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging., 6 (2012), 25-38. doi: 10.2307/2152750.
[3]	G. Chen and J. X. Zhou, Boundary Elements Methods with Applications to Nonlinear Problems, Atlantis press, Paris, 2010. doi: 10.1007/978-1-4612-0873-0.
[4]	I. L. Chen, J. T. Chen, S. R. Kuo and M. T. Liang, A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation method, J. Acoust. Soc. Am., 109 (2001), 982-998. doi: 10.2307/2152750.
[5]	J. T. Chen, J. W. Lee, Y. L. Huang, C. H. Shao and C. H. Lu, On the linkage between influence matrices in the BIEM and BEM to explain the mechanism of degenerate scale, J. Mech., 37 (2021), 339-345. doi: 10.2307/2152750.
[6]	J. T. Chen, S. R. Lin and K. H. Chen, Degenerate scale for a torsion bar problem using BEM, Third International Conference on BeTeQ, (2002). doi: 10.2307/2152750.
[7]	J. T. Chen, L. W. Liu and H. K. Hong, Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply-connected problem, R. Soc. London Ser. A-Math. Phys. Eng. Sci., 459 (2003), 1891-1924. doi: 10.2307/2152750.
[8]	J. T. Chen, J. H. Lin, S. R. Kuo and Y. P. Chiu, Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants, Eng. Anal. Bound. Elem., 25 (2001), 819-828. doi: 10.2307/2152750.
[9]	J. T. Chen and W. C. Shen, Degenerate scale for multiply connected Laplace problems, Mech. Res. Commun., 34 (2007), 69-77. doi: 10.2307/2152750.
[10]	J. T. Chen, Y. T. Lee, S. R. Kuo and Y. W. Chen, Analytical derivation and numerical experiments of degenerate scale for an ellipse in BEM, Eng. Anal. Bound. Elem., 36 (2012), 1397-1405. doi: 10.2307/2152750.
[11]	J. T. Chen. S. R. Kuo, Y. L. Huang and S. K. Kao, Linkage of logarithmic capacity in potential theory and degenerate scale in the BEM for the two tangent discs, Appl. Math. Lett., 102 (2020), 106-135. doi: 10.2307/2152750.
[12]	J. T. Chen. S. R. Kuo. K. T. Lien and Y. L. Huang, On the degenerate scale of an infinite plane containing two unequal circles, Adv. Appl. Math. Mech., 12 (2020), 1280-1300. doi: 10.2307/2152750.
[13]	J. T. Chen. S. R. Kuo and Y. L. Huang, Revisit of logarithmic capacity of line segments and double-degeneracy of BEM/BIEM, Eng. Anal. Bound. Elem., 120 (2020), 238-245. doi: 10.2307/2152750.
[14]	J. T. Chen, C. F. Lee, I. L. Chen and J. H. Lin, An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation, Eng. Anal. Bound. Elem., 26 (2002), 559-569. doi: 10.2307/2152750.
[15]	J. T. Chen, S. R. Lin and K. H. Chen, Degenerate scale problem when solving Laplace equation by BEM and its treatment, Int. J. Numer. Meth. Eng., 62 (2002), 559-569. doi: 10.2307/2152750.
[16]	J. T. Chen, H. D. Han, S. R. Kuo and S. K. Kao, Regularized methods for ill-conditioned system of the integral equations of the first kind, Inverse Probl. Sci. Eng., 22 (2014), 1176-1195. doi: 10.2307/2152750.
[17]	J. T. Chen, S. R. Kuo, S. K. Kao and J. Jian, Revisit of a degenerate scale: A semi-circular disc, Comp. Appl. Math., 283 (2015), 182-200. doi: 10.2307/2152750.
[18]	J. T. Chen, Y. T. Lee, Y. L. Chang and J. Jian, A self-regularized approach for rank-deficiency systems in the BEM of 2D Laplace problems, Inverse Probl. Sci. Eng., 25 (2017), 89-113. doi: 10.2307/2152750.
[19]	J. T. Chen, S. K. Kao and J. W. Lee, Analytical derivation and numerical experiment of degenerate scale by using the degenerate kernel of the bipolar coordinates, Engng. Anal. Bound. Elem., 85 (2017), 70-86. doi: 10.2307/2152750.
[20]	C. Constanda, On the solution of the Dirichlet problem for the two dimensional Laplace equation, Proc. Am. Math. Soc., 119 (1993), 877-884. doi: 10.2307/2152750.
[21]	C. Cowan and A. Razani, Singular solutions of a Lane-Emden system, Discrete Contin. Dyn. Syst., 41 (2021), 621-656. doi: 10.2307/2152750.
[22]	C. L. Epstein, L. Greengard and T. Hagstrom, On the stability of time-domain integral equations for acoustic wave propagation, Discrete Contin. Dyn. Syst., 36 (2016), 4367-4382. doi: 10.2307/2152750.
[23]	G. Fichera, Boundary Problems in Differential Equations, The University of Wisconsin Press, Madison (WI), (1959). doi: 10.1007/978-1-4612-0873-0.
[24]	M. A. Golberg, Solution methods for integral equations, Plenum press, New York and London, (1978).
[25]	H. C. Hu, Necessary and sufficient boundary integral equations for plane harmonic functions, China Ser. A-Math. Phys. Astron., 35 (1992), 861-869. doi: 10.2307/2152750.
[26]	M. A. Jaswon and G. T. Symm, Integral equation method in potential theory and elastostatics, in Computational mathematics and applications, Academic Press, (1977).
[27]	S. R. Kuo, J. T. Chen, J. W. Lee and Y. W. Chen, Analytical derivation and numerical experiments of degenerate scale for regular N-gon domains in BEM, Appl. Math. Comput., 219 (2012), 1397-1405. doi: 10.2307/2152750.
[28]	S. R. Kuo and J. T. Chen, Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs, Appl. Math. Lett., 29 (2013), 929-938. doi: 10.2307/2152750.
[29]	S. R. Kuo, S. K. Kao, Y. L. Huang and J. T. Chen, Revisit of the degenerate scale for an infinite plane problem containing two circular holes using conformal mapping, Appl. Math. Lett., 92 (2019), 99-107. doi: 10.2307/2152750.
[30]	G. Li, F. Gu and F. Jiang, Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation, Commun. Pure Appl. Anal., 19 (2020), 1449-1462. doi: 10.2307/2152750.
[31]	C. K. Lin and K. C. Wu, Fundamental solution of the Laplace equation (dimensional analysis viewpoint), Proceedings of the 7th Taiwan-Philippine Symposium in Mathematics, (2007), 94-102. doi: 10.2307/2152750.
[32]	H. A. Schenck, Improved integral formulation for acoustic radiation problems, J. Acoust. Soc. Am., 44 (1976), 41-58. doi: 10.2307/2152750.
[33]	G. Strang, Introduction to Linear Algebra, 5$^{nd}$ edition, Wellesley Cambridge Press, Wellesley, 2016. doi: 10.1007/978-1-4612-0873-0.