On invertibility of elastic single-layer potential operator

Journal of Elasticity

Volumn 74, Issue 2, 2004, Pages 147-173

  Vodicka R ^a   Mantic V ^b  

^a TECHNICAL UNIVERSITY OF KO ICE   (Slovakia)

^b UNIVERSITY OF SEVILLE   (Spain)

Author keywords

(Symmetric galerkin) boundary element method; Boundary integral equation of the first kind; Condition number; Logarithmic capacity; Multiply connected domain; Plane elasticity; Single layer potential; Solution uniqueness

Indexed keywords

BOUNDARY CONDITIONS; BOUNDARY ELEMENT METHOD; FUNCTIONS; INTEGRAL EQUATIONS; MATHEMATICAL OPERATORS; MATHEMATICAL TRANSFORMATIONS;

BOUNDARY INTEGRAL EQUATIONS; CONDITION NUMBERS; PLANE ELASTICITY;

ELASTICITY;

EID: 3142689958     PISSN: 03743535     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:ELAS.0000033861.83767.ce     Document Type: Article

References (30)

1. A. Blázquez, V. Mantič, F. Paris and I. Cañas, On the removal of rigid body motions in the solution of elastostatic problems by direct BEM. Internat. J. Numer. Methods Engrg. 39 (1996) 4021-4038.
2. M. Bonnet, G. Maier and C. Polizzotto, Symmetric Galerkin boundary element method, Appl. Mech. Rev. 15 (1998) 669-704.
3. G. Chen and J. Zhou, Boundary Element Methods. Academic Press, London (1992).
4. J.T. Chen, S.R. Kuo and J.H. Lin, Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity. Internat. J. Numer. Methods Engrg. 54 (2002) 1669-1681.
5. J.T. Chen, C. Lee, I.L. Chen and J.H. Lin, An alternative method for degenerate scale problems in boundary element method for the two-dimensional Laplace equation. Engrg. Anal Boundary Element 26 (2002) 559-569.
6. S. Christiansen, On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel. Appl. Anal. 13 (1982) 1-18.
7. S. Christiansen, Modifications of some first kind integral equations with logarithmic kernel to improve numerical conditioning. Computing 34 (1985) 221-242.
8. S. Christiansen, The conditioning of some numerical methods for first kind boundary integral equations. J. Comput. Appl. Math. 67 (1996) 43-58.
9. S. Christiansen, Derivation and analytical investigation of three direct boundary integral equations for the fundamental biharmonic problem. J. Comput. Appl. Math. 91 (1998) 231-247.
10. S. Christiansen, Detecting non-uniqueness of solutions to biharmonic integral equations through SVD. J. Comput. Appl. Math. 134 (2001) 23-35.
11. C. Constanda, On non-unique solutions of weakly singular integral equations in plane elasticity. Quart. J. Mech. Appl. Math. 47 (1994) 261-268.
12. C. Constanda, The boundary integral equation method in plane elasticity. Proc. Amer. Math. Soc. 123(11) (1995) 3385-3396.
13. C. Constanda, Integral equations of the first kind in plane elasticity. Quart. Appl. Math. 53(4) (1995) 783-793.
14. M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal 19(3) (1988) 613-626.
15. M. Costabel and M. Dauge, Invertibility of the biharmonic single layer potential operator. Integral Equations Oper. Theory 24 (1996) 46-67.
16. R. Dautray and J.L. Lions, Physical Origins and Classical Methods, Mathematical Analysis and Numercial Methods for Science and Technology, Vol. 1. Springer, Berlin (2000).
17. M. Gurtin, The linear theory of elasticity. In: S. Flügge (ed.): Encyclopedia of Physics. Springer, Berlin (1972) pp. 1-295.
18. F. Hartmann, The physical nature of elastic layers. J. Elasticity 12(1) (1982) 19-29.
19. W. He, H. Ding and H. Hu, A necessary and sufficient boundary integral formulation for plane elasticity problems. Comm. Numer. Methods Engrg. 12 (1996) 413-424.
20. U. Heise, The spectra of some integral operators for plane elastostatical boundary value problems. J. Elasticity 8(1) (1978) 47-79.
21. U. Heise, Dependence of the round-off error in the solution of boundary integral equations on geometrical scale factor. Comput. Methods Appl. Mech. Engrg. 62 (1987) 115-126.
22. G. Hsiao and W. Wendland, On a boundary integral method for some exterior problems in elasticity. In: Proc. Tbiliss. Univ. Math. Mech. Astron. (1985) pp. 31-60.
23. M.A. Jaswon and G.T. Symm, Integral Equation Methods in Potential Theory and Elastostatics. Academic Press, London (1977).
24. G. Kuhn, G. Löbel and I. Potrč, Kritishes Lösungsverhalten der direkten Randelementmethode bei logarithmischen Kern. Z. Appl. Math. Mech. 67 (1987) 361-363.
25. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge Univ. Press, Cambridge (2000).
26. C. Müller and U. Heise, Numerical calculations of eigenvalues of integral operators for plane elastostatic boundary value problems. Comput. Methods Appl. Mech. Engrg. 21 (1980) 17-47.
27. N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Soviet Union Academy of Sciences Press, Moscow (1954) (in Russian).
28. O. Steinbach and W.L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl. 262 (2001) 733-748.
29. R. Vodička, V. Mantič and F. París, On the removal of non-uniqueness in the solution of elastostatic problems by Symmetric Galerkin BEM. Internat. J. Numer. Methods Engrg. (submitted).
30. Y. Yan and I.H. Sloan, Integral equations of the first kind with logarithmic kernels. J. Integral Equations Appl. 1(4) (1988) 549-579.