Treatment of singularities in the method of fundamental solutions for two-dimensional Helmholtz-type equations. (English) Zbl 1193.35223
Summary: We investigate a meshless method for the accurate and non-oscillatory solution of problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are approximated by the method of fundamental solutions (MFS). It is well known that the existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. The solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. This difficulty is overcome by subtracting from the original MFS solution the corresponding singular functions, without an appreciable increase in the computational effort and at the same time keeping the same MFS approximation. Four examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated and the numerical results presented show an excellent performance of the approach developed.
MSC:
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 74G70 | Stress concentrations, singularities in solid mechanics |
Keywords:
Helmholtz-type equations; singular problems; singularity subtraction technique (SST); method of fundamental solutions (MFS)References:
| [1] | Apel, T.; Sändig, A.-M.; Whiteman, J. R., Graded mesh refinement and error estimates for finite element solution of elliptic boundary value problems in non-smooth domains, Math. Methods Appl. Sci., 19, 63-85 (1996) · Zbl 0838.65109 |
| [2] | Apel, T.; Nicaise, S., The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Methods Appl. Sci., 21, 519-549 (1998) · Zbl 0911.65107 |
| [3] | Beskos, D. E., Boundary element method in dynamic analysis: Part II (1986-1996), ASME Appl. Mech. Rev., 50, 149-197 (1997) |
| [4] | Chen, J. T.; Liang, M. T.; Chen, I. L.; Chyuan, S. W.; Chen, K. H., Dual boundary element analysis of wave scattering from singularities, Wave Motion, 30, 367-381 (1999) · Zbl 1074.74644 |
| [5] | Chen, J. T.; Wong, F. C., Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition, J. Sound Vib., 217, 75-95 (1998) |
| [6] | Chen, J. T.; Lin, J. H.; Kuo, S. R.; Chyuan, W., Boundary element analysis for the Helmholtz eigenvalue problems with a multiply connected domain, Proc. R. Soc. London, 457, 2521-2546 (2001) · Zbl 0993.78021 |
| [7] | Huang, C.; Wu, Z.; Nevels, R. D., Edge diffraction in the vicinity of the tip of a composite wedge, IEEE Trans. Geosci. Remote Sensing, 31, 1044-1050 (1993) |
| [8] | Harari, I.; Barbone, P. E.; Slavutin, M.; Shalom, R., Boundary infinite elements for the Helmholtz equation in exterior domains, Int. J. Numer. Methods Eng., 41, 1105-1131 (1998) · Zbl 0911.76035 |
| [9] | Hall, W. S.; Mao, X. Q., A boundary element investigation of irregular frequencies in electromagnetic scattering, Eng. Anal. Bound. Elem., 16, 245-252 (1995) |
| [10] | Barbone, P. A.; Montgomery, J. M.; Michael, O.; Harari, I., Scattering by a hybrid asymptotic/finite element, Comput. Methods Appl. Mech. Eng., 164, 141-156 (1998) · Zbl 0962.76044 |
| [11] | Wood, A. S.; Tupholme, G. E.; Bhatti, M. I.H.; Heggs, P. J., Steady-state heat transfer through extended plane surfaces, Int. Commun. Heat Mass Transfer, 22, 99-109 (1995) |
| [12] | Kraus, A. D.; Aziz, A.; Welty, J., Extended Surface Heat Transfer (2001), Wiley: Wiley New York |
| [13] | Nowak, A. J.; Brebbia, C. A., Solving Helmholtz equation by boundary elements using multiple reciprocity method, (Calomagno, G. M.; Brebbia, C. A., Computer Experiment in Fluid Flow (1989), CMP/Springer Verlag: CMP/Springer Verlag Berlin), 265-270 |
| [14] | Agnantiaris, J. P.; Polyzer, D.; Beskos, D., Three-dimensional structural vibration analysis by the dual reciprocity BEM, Comput. Mech., 21, 372-381 (1998) · Zbl 0922.73074 |
| [15] | Chen, J. T.; Chen, K. H., Dual integral formulation for determining the acoustic modes of a two-dimensional cavity with a degenerate boundary, Eng. Anal. Bound. Elem., 21, 105-116 (1998) · Zbl 1062.76533 |
| [16] | Schiff, B., Eigenvalues for ridged and other waveguides containing corners of angle \(3 \pi / 2\) or \(2 \pi\) by the finite element method, IEEE Trans. Microwave Theory Tech., 39, 1034-1039 (1991) |
| [17] | Cai, W.; Lee, H. C.; Oh, H. S., Coupling of spectral methods and the p-version for the finite element method for elliptic boundary value problems containing singularities, J. Comput. Phys., 108, 314-326 (1993) · Zbl 0790.65093 |
| [18] | Lucas, T. R.; Oh, H. S., The method of auxiliary mapping for the finite element solutions of elliptic problems containing singularities, J. Comput. Phys., 108, 327-342 (1993) · Zbl 0797.65083 |
| [19] | Wu, X.; Han, H., A finite-element method for Laplace- and Helmholtz-type boundary value problems with singularities, SIAM J. Numer. Anal., 134, 1037-1050 (1997) · Zbl 0873.65100 |
| [20] | Xu, Y. S.; Chen, H. M., Higher-order discretised boundary conditions at edges for TE waves, IEEE Proc. Microw. Antennas Propag., 146, 342-348 (1999) |
| [21] | Mantič, V.; París, F.; Berger, J., Singularities in 2D anisotropic potential problems in multi-material corners, real variable approach, Int. J. Solids Struct., 40, 5197-5218 (2003) · Zbl 1060.74548 |
| [22] | Marin, L.; Lesnic, D.; Mantič, V., Treatment of singularities in Helmholtz-type equations using the bounadry element method, J. Sound Vib., 278, 39-62 (2004) · Zbl 1236.65150 |
| [23] | Li, Z.-C.; Lu, T. T., Singularities and treatments of elliptic boundary value problems, Math. Comput. Model., 31, 97-145 (2000) · Zbl 1042.35524 |
| [24] | Kupradze, V. D.; Aleksidze, M. A., The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput. Math. Math. Phys., 4, 82-126 (1964) · Zbl 0154.17604 |
| [25] | Mathon, R.; Johnston, R. L., The approximate solution of elliptic boundary value problems by fundamental solutions, SIAM J. Numer. Anal., 14, 638-650 (1977) · Zbl 0368.65058 |
| [26] | Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9, 69-95 (1998) · Zbl 0922.65074 |
| [27] | Golberg, M. A.; Chen, C. S.; potential, The method of fundamental solutions for, Helmholtz and diffusion problems, (Golberg, M. A., Boundary Integral Methods: Numerical and Mathematical Aspects (1999), WIT Press and Computational Mechanics Publications: WIT Press and Computational Mechanics Publications Boston), 105-176 · Zbl 0945.65130 |
| [28] | Fairweather, G.; Karageorghis, A.; Martin, P. A., The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem., 27, 759-769 (2003) · Zbl 1060.76649 |
| [29] | Cho, H. A.; Golberg, M. A.; Muleshkov, A. S.; Li, X., Trefftz methods for time dependent partial differential equations, CMC: Comput. Materials & Continua, 1, 1-37 (2004) |
| [30] | Karageorghis, A.; Fairweather, G., The method of fundamental solutions for the numerical solution of the biharmonic equation, J. Comput. Phys., 69, 434-459 (1987) · Zbl 0618.65108 |
| [31] | Karageorghis, A., Modified methods of fundamental solutions for harmonic and biharmonic problems with boundary singularities, Numer. Meth. Part. Diff. Equations, 8, 1-19 (1992) · Zbl 0760.65103 |
| [32] | Poullikkas, A.; Karageorghis, A.; Georgiou, G., Methods of fundamental solutions for harmonic and biharmonic boundary value problems, Comput. Mech., 21, 416-423 (1998) · Zbl 0913.65104 |
| [33] | Poullikkas, A.; Karageorghis, A.; Georgiou, G., The method of fundamental solutions for inhomogeneous elliptic problems, Comput. Mech., 22, 100-107 (1998) · Zbl 0913.65103 |
| [34] | Berger, J. R.; Karageorghis, A., The method of fundamental solutions for heat conduction in layered materials, Int. J. Numer. Methods Eng., 45, 1681-1694 (1999) · Zbl 0972.80014 |
| [35] | Berger, J. R.; Karageorghis, A., The method of fundamental solutions for layered elastic materials, Eng. Anal. Bound. Elem., 25, 877-886 (2001) · Zbl 1008.74081 |
| [36] | Poullikkas, A.; Karageorghis, A.; Georgiou, G., The numerical solution for three-dimensional elastostatics problems, Comput. Struct., 80, 365-370 (2002) |
| [37] | Balakrishnan, K.; Ramachandran, P. A., The method of fundamental solutions for linear diffusion-reaction equations, Math. Comput. Model., 31, 221-237 (2000) · Zbl 1042.35569 |
| [38] | Karageorghis, A.; Fairweather, G., The method of fundamental solutions for axisymmetric elasticity problems, Comput. Mech., 25, 524-532 (2000) · Zbl 1011.74005 |
| [39] | Poullikkas, A.; Karageorghis, A.; Georgiou, G., The numerical solution of three-dimensional Signorini problems with the method of fundamental solutions, Eng. Anal. Bound. Elem., 25, 221-227 (2001) · Zbl 0985.78004 |
| [40] | Alves, C. J.S.; Valtchev, S. S., Numerical comparison of two meshfree methods for acoustic wave scattering, Eng. Anal. Bound. Elem., 29, 371-382 (2005) · Zbl 1182.76924 |
| [41] | Young, D. L.; Jane, S. J.; Fan, C. M.; Murugesan, K.; Tsai, C. C., The method of fundamental solutions for 2D and 3D Stokes problems, J. Comput. Phys., 211, 1-8 (2006) · Zbl 1160.76332 |
| [42] | Tsai, C. C.; Young, D. L.; Fan, C. M.; Chen, C. W., MFS with time-dependent fundamental solutions for unsteady Stokes equations, Eng. Anal. Bound. Elem., 30, 897-908 (2006) · Zbl 1195.76324 |
| [43] | Johansson, B. T.; Lesnic, D., A method of fundamental solutions for transient heat conduction, Eng. Anal. Bound. Elem., 32, 697-703 (2008) · Zbl 1244.80021 |
| [44] | Katsurada, M.; Okamoto, H., A mathematical study of the charge simulation method, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math., 35, 507-518 (1988) · Zbl 0662.65100 |
| [45] | Katsurada, M., A mathematical study of the charge simulation method II, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math., 36, 135-162 (1989) · Zbl 0681.65081 |
| [46] | Katsurada, M., Asymptotic error analysis of the charge simulation method in Jordan region with an analytic boundary, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math., 37, 635-657 (1990) · Zbl 0723.65093 |
| [47] | Katsurada, M., Charge simulation method using exterior mapping functions, Jpn. J. Ind. Appl. Math., 11, 47-61 (1994) · Zbl 0816.35017 |
| [48] | Kondepalli, P. S.; Shippy, D. J.; Fairweather, G., Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions, J. Acoust. Soc. Am., 91, 1844-1854 (1992) |
| [50] | Tankelevich, R.; Fairweather, G.; Karageorghis, A., Potential field based geometric modeling using the method of fundamental solutions, Int. J. Numer. Methods Eng., 68, 1257-1280 (2006) · Zbl 1130.65035 |
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