• Previous Article
    Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping
  • CPAA Home
  • This Issue
  • Next Article
    Ground state solutions for the fractional problems with dipole-type potential and critical exponent
June  2022, 21(6): 1969-1985. doi: 10.3934/cpaa.2021114

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

Received  November 2020 Revised  May 2021 Published  June 2022 Early access  August 2021

Fund Project: 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

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.

Citation: Jeng-Tzong Chen, Shing-Kai Kao, Jeng-Hong Kao, Wei-Chen Tai. An indirect BIE free of degenerate scales. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1969-1985. doi: 10.3934/cpaa.2021114
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.

show all references

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.

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
Figure 2.  Numerical evidences for the degenerate scale of four cases in the BEM
Table 1.  Comparison of four BIEs
Table 2.  Study on the degenerate scale by using different approaches
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
Table 4.  Comparison of the four kernel functions
[1]

Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146

[2]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[3]

Georg Ostrovski, Sebastian van Strien. Payoff performance of fictitious play. Journal of Dynamics and Games, 2014, 1 (4) : 621-638. doi: 10.3934/jdg.2014.1.621

[4]

István Győri, László Horváth. On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1665-1702. doi: 10.3934/dcds.2020089

[5]

Reinhard Farwig, Ronald B. Guenther, Enrique A. Thomann, Šárka Nečasová. The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 511-529. doi: 10.3934/dcds.2014.34.511

[6]

P. Álvarez-Caudevilla, J. D. Evans, V. A. Galaktionov. The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 807-827. doi: 10.3934/dcds.2015.35.807

[7]

Joep H.M. Evers, Sander C. Hille, Adrian Muntean. Modelling with measures: Approximation of a mass-emitting object by a point source. Mathematical Biosciences & Engineering, 2015, 12 (2) : 357-373. doi: 10.3934/mbe.2015.12.357

[8]

Isaac Harris, Dinh-Liem Nguyen, Thi-Phong Nguyen. Direct sampling methods for isotropic and anisotropic scatterers with point source measurements. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022015

[9]

Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767

[10]

Alain Haraux. On the fast solution of evolution equations with a rapidly decaying source term. Mathematical Control and Related Fields, 2011, 1 (1) : 1-20. doi: 10.3934/mcrf.2011.1.1

[11]

Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091

[12]

Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055

[13]

Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial and Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487

[14]

Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075

[15]

H. M. Yin. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Communications on Pure and Applied Analysis, 2002, 1 (1) : 127-134. doi: 10.3934/cpaa.2002.1.127

[16]

Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022

[17]

João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321

[18]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic and Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707

[19]

Saeed Assani, Jianlin Jiang, Ahmad Assani, Feng Yang. Scale efficiency of China's regional R & D value chain: A double frontier network DEA approach. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1357-1382. doi: 10.3934/jimo.2020025

[20]

Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (347)
  • HTML views (349)
  • Cited by (0)

[Back to Top]