Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.
@article{bwmeta1.element.doi-10_2478_s11533-009-0069-5,
author = {Yuri Melnikov},
title = {Efficient representations of Green's functions for some elliptic equations with piecewise-constant coefficients},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {53-72},
zbl = {1221.35112},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0069-5}
}
Yuri Melnikov. Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients. Open Mathematics, Tome 8 (2010) pp. 53-72. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0069-5/
[1] Ang W.T., Clements D.L., A boundary-integral equation method for the solution of a class of crack problems, J. Elasticity, 1987, 17, 9–21 http://dx.doi.org/10.1007/BF00042444 | Zbl 0602.73096
[2] Clements D.L., Haselgrove M.D., A boundary-integral equation method for a class of crack problems in anisotropic elasticity, Int. J. Comput. Math., 1983, 12, 267-278 | Zbl 0501.73099
[3] Courant R., Hilbert D., Methods of Mathematical Physics, vol.2, Interscience, New York, 1953 | Zbl 0051.28802
[4] Deutz J.W., Schober H.R., Boundary value problems using Green’s functions, Comput. Phys. Commun., 1983, 30, 87–91 http://dx.doi.org/10.1016/0010-4655(83)90125-X
[5] Dolgova I.M., Melnikov Yu.A., Construction of Green’s functions and matrices for equations and systems of elliptic type, Translation Russian PMM (J. Appl. Math. Mech.), 1978, 42, 740–746 http://dx.doi.org/10.1016/0021-8928(78)90017-5
[6] Duffy D., Green’s Functions with Applications, CRC Press, Boca Raton, 2001 | Zbl 0983.35003
[7] Embree M., Trefethen L.N., Green’s functions for multiply connected domains via conformal mapping, SIAM Rev., 1999, 41, 745–761 http://dx.doi.org/10.1137/S0036144598349277 | Zbl 0938.30003
[8] Gradstein I.S., Ryzhik I.M., Tables of Intergrals, Series and Products, Academic Press, New York, 1980
[9] Irschik H., Ziegler F., Application of the Green’s function method to thin elastic polygonal plates, Acta Mech., 1981, 39, 155–169 http://dx.doi.org/10.1007/BF01170339 | Zbl 0459.73047
[10] Marshall S.L., A rapidly convergent modified Green’s function for Laplace equation in a rectangular region, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 1999, 155, 1739–1766 http://dx.doi.org/10.1098/rspa.1999.0378
[11] Melnikov Yu.A., Some applications of the Green’s function method in mechanics, Internat. J. Solids Structures, 1977, 13, 1045–1058 http://dx.doi.org/10.1016/0020-7683(77)90075-0
[12] Melnikov Yu.A., Koshnarjova V.A., Green’s matrices and 2-D elasto-potentials for external boundary value problems, Appl. Math. Model., 1994, 18, 161–167 http://dx.doi.org/10.1016/0307-904X(94)90259-3 | Zbl 0796.73006
[13] Melnikov Yu.A., Shirley K.L., Matrices of Green’s type for the potential equation on a cylindrical surface joined to a hemisphere, Appl. Math. Comput., 1994, 65, 241–252 http://dx.doi.org/10.1016/0096-3003(94)90180-5 | Zbl 0815.65119
[14] Melnikov Yu.A., Green’s Functions in Applied Mechanics, Comput. Mech. Publications, Boston - Southampton, 1995 | Zbl 0898.73001
[15] Melnikov Yu.A., Influence functions for 2-D compound regions of complex configuration, Comput. Mech., 1996, 17, 297–305 http://dx.doi.org/10.1007/BF00368552 | Zbl 0845.35016
[16] Melnikov Yu.A., Green’s function formalism extended to systems of mechanical differential equations posed on graphs, J. Eng. Math., 1998, 34, 369–386 http://dx.doi.org/10.1023/A:1004396614908 | Zbl 0928.34024
[17] Melnikov Yu.A., Influence Functions and Matrices, Marcel Dekker, New York - Basel, 1999
[18] Melnikov Yu.A., Sheremet V.D., Some new results on the bending of a circular plate subject to point forces, Math. Mech. Solids, 2001, 6, 29–46 http://dx.doi.org/10.1177/108128650100600102 | Zbl 1024.74034
[19] Melnikov Yu.A., Matrices of Green’s type of steady-state heat conduction in multiply connected piecewise homogeneous regions, Eng. Anal. Bound. Elements, 2003, 27, 779–787 http://dx.doi.org/10.1016/S0955-7997(03)00044-4 | Zbl 1045.80001
[20] Melnikov Yu.A., Influence functions of a point source for perforated compound plates with facial convection, J. Eng. Math., 2004, 49, 253–270 http://dx.doi.org/10.1023/B:ENGI.0000031187.96637.ea | Zbl 1155.80002
[21] Morse P.M., Feshbach H., Methods of Theoretical Physics, vol.2, McGraw-Hill, New York - Toronto - London, 1953 | Zbl 0051.40603
[22] Pan E., Han F., Green’s functions for transversely isotropic piezoelectric multilayered half-spaces, J. Eng. Math., 2004, 49, 271–288 http://dx.doi.org/10.1023/B:ENGI.0000031183.83519.19 | Zbl 1068.74541
[23] Roach G.F., Green’s Functions, Cambridge University Press, New York, 1982
[24] Sheremet V.D., Handbook of Green’s Functions and Matrices, WITPress, Southampton - Boston, 2002
[25] Smirnov V.I., A Course of Higher Mathematics, Pergamon Press, Oxford - New York, 1964 | Zbl 0121.25904
[26] Stakgold I., Green’s functions and Boundary Value Problems, John Wiley, New York, 1980
[27] Tewary V.K., Wagoner R.H., Hirth J.P., Elastic Green’s functions for a composite solid with a planar interface, J. Mater. Res., 1989, 4, 113–123 http://dx.doi.org/10.1557/JMR.1989.0113
[28] Tewary V.K., Elastic Green’s function for a bimaterial composite solid containing a free surface normal to the interface, J. Mater. Res., 1991, 6, 2592–2608 http://dx.doi.org/10.1557/JMR.1991.2592
[29] Ting T.C.T., Green’s functions for a bimaterial consisting of two orthotropic quarter planes subjected to an antiplane force and a screw dislocation, Math. Mech. Solids, 2005, 10, 197–211 http://dx.doi.org/10.1177/1081286505036318 | Zbl 1074.74018
[30] Yang B., Tewary V.K., Efficient Green’s function method of line and surface defects in multilayered elestic and piezoelastic materials, Comput. Model. Eng. Sci., 2006, 15, 165–178
[31] Yang B., Wong S.-C., Qu S., A micromechanics analysis of nanoscale graphite platelet-reinforced epoxy using defect Green’s function, Comput. Model. Eng. Sci., 2008, 24, 81–94