A boundary element procedure for contact problems in plane linear elastostatics. (English) Zbl 0773.73096

Summary: We present a new solution procedure for contact problems in plane linear elastostatics via boundary integral variational inequalities having as unknowns the trace of the displacement field and its boundary traction. We admit the case of only traction-contact boundary conditions without prescribing the displacements along some part of the boundary of the elastically deformed body. Without imposing any regularity assumption we establish norm convergence of piecewise polynomial boundary element approximations for mechanically definite problems. In detail we investigate piecewise quadratic and piecewise cubic approximations to the displacement field which lead to nonconform approximation schemes.


74S15 Boundary element methods applied to problems in solid mechanics
74A55 Theories of friction (tribology)
74M15 Contact in solid mechanics
74B05 Classical linear elasticity
Full Text: DOI EuDML


[1] I. BABUŠKA and A. K. Aziz, Survey lectures on the mathematicalformulation ofthe finite element method, in The Mathematical Foundation of the Finite Element Method (A. K. Aziz, ed.) Academic Press, New York, 1972, pp. 3-359. Zbl0268.65052 MR421106 · Zbl 0268.65052
[2] M. COSTABEL, Boundary integral operators on Lipschitz domains : Elementary results, SIAM J. Math. Anal. 19, 1988, pp. 613-626. Zbl0644.35037 MR937473 · Zbl 0644.35037
[3] R. DAUTRAY and J.L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 4, Integral Equations and Numerical Methods, Springer, Berlin, 1990. MR1081946 · Zbl 0766.47001
[4] G. DUVAUT and J. L. LIONS, Inequalities in Mechanics and Physics, Springer, Berlin, 1976. Zbl0331.35002 MR521262 · Zbl 0331.35002
[5] J. ELSCHNER, On spline approximation for a class of integral equations, I : Galerkin and collocation methods with piecewise polynomials, Math. Methods in the Applied Sciences 10, 1988, pp. 543-559. Zbl0662.65112 MR965421 · Zbl 0662.65112
[6] H. ENGELS, Numerical Quadrature and Cubature, Academic Press, New York, 1980. Zbl0435.65013 MR587486 · Zbl 0435.65013
[7] G. I. ÈSKIN, Boundary Value Problems for Elliptic Pseudodifferential Equations, Translations of Mathematical Monographs, Vol. 52, American Mathematical Society, Providence, 1981. Zbl0458.35002 MR623608 · Zbl 0458.35002
[8] G. FICHERA, Boundary value problems of elasticity with unilateral constraints, in Handbuch der Physik - Encyclopedia of Physics, Band VI a/2 Festkörper-mechanikll, Springer, Berlin, 1972, pp. 391-424.
[9] R. GLOWINSKI, Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984. Zbl0536.65054 MR737005 · Zbl 0536.65054
[10] R. GLOWINSKI, J. L. LIONS and R. TRÉMOLIÈRES, Numerical Analysis ofVariational Inequalities, North-Holland, Amsterdam, 1981. Zbl0463.65046 MR635927 · Zbl 0463.65046
[11] P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985. Zbl0695.35060 MR775683 · Zbl 0695.35060
[12] [12] J. GWINNER, Discretization of semicoercive variational inequalities, Aequationes Mathematicae 42, 1991, pp. 72-79. Zbl0739.65058 MR1112184 · Zbl 0739.65058
[13] G. HÄMMERLIN and K. H. HOFFMANN, Numerische Mathematik, Springer, 1989. Zbl0669.65001 MR1305149 · Zbl 0669.65001
[14] H. HAN, A direct boundary element method for Signorini problems, Math. Computation 55, 1990, pp.115-128. Zbl0705.65084 MR1023048 · Zbl 0705.65084
[15] I. HLAVAČEK, J. HASLINGER, J. NEČAS and J. LOVIŠEK, Solution of Variational Inequalities in Mechanics, Springer, Berlin, 1988. Zbl0654.73019 MR952855 · Zbl 0654.73019
[16] [16] I. HLAVAČEK and J. LOVIŠEK, A finite element analysis for the Signorini problem in plane elastostatics, Aplikace Mat. 22, 1977, pp. 215-228. Zbl0369.65031 MR446014 · Zbl 0369.65031
[17] G. C. HSIAO, E. P. STEPHAN, W. L. WENDLAND, On the Dirichlet problem in elasticity for a domain exterior to an arc, J. Computational Appl. Mathematics 34, 1991, pp. 1-19. Zbl0742.73026 MR1095192 · Zbl 0742.73026
[18] N. KIKUCHI and J. T. ODEN, Contact Problems in Elasticity : a Study ofariational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988. Zbl0685.73002 MR961258 · Zbl 0685.73002
[19] V.A. KONDRATIEV and O.A. OLEINIK, On Korn’s inequalities, C.R. Acad. Sci. Paris I 308, 1989, pp. 483-487. Zbl0698.35067 MR995908 · Zbl 0698.35067
[20] V. D. KUPRADZE, Potential Methods in the Theory of Elasticity, Israël Program for Scientific Translations, Jerusalem, 1965. Zbl0188.56901 MR223128 · Zbl 0188.56901
[21] V. D. KUPRADZE, T. G. GEGELIA, M. O. BASHELEISHVILI and T. V. URCHULADZE, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland, Amsterdam, 1979. Zbl0406.73001 MR530377 · Zbl 0406.73001
[22] N. I. MUSKHELISHVILI, Some Basic Problems of the Mathematical Theory of lasticity, Noordhoff, Groningen, 1963. Zbl0124.17404 MR176648 · Zbl 0124.17404
[23] J. NEČAS, Les méthodes directes en théorie des équations elliptiques, Academia, Masson, Prague, Paris, 1967. Zbl1225.35003 MR227584 · Zbl 1225.35003
[24] J. C. NEDELEC, Approximation des Équations Intégrales en Mécanique et en Physique, Lecture Notes, Centre Math. Appl., École polytechnique, Palaiseau, France 1977.
[25] J. C. NEDELEC, Integral equations with non integrable kernels, Integral Equations and Operator Theory 5, 1982, pp. 562-582. Zbl0479.65060 MR665149 · Zbl 0479.65060
[26] [26] J. A. NITSCHE, On Korn’s second inequality, R.A.I.R.O. Anal. Numér. 15, 1981, pp. 237-248. Zbl0467.35019 MR631678 · Zbl 0467.35019
[27] P. D. PANAGIOTOPOULOS, Inequality Problems in Mechanics and Applications, Birkhauser, Basel, 1985. Zbl0579.73014 MR896909 · Zbl 0579.73014
[28] P. D. PANAGIOTOPOULOS, Boundary integral < equation > methods for the Signorini-Fichera problem, in Boundary Elements 7, vol. 2, exp. No. 12, 1985, pp. 73-83. Zbl0599.73117 MR948223 · Zbl 0599.73117
[29] R. SAUER, Anfangswertprobleme bei partiellen Dijferentialgleichungen, Springer, Berlin, 1952. Zbl0046.31902 MR52009 · Zbl 0046.31902
[30] A. SiGNORlNI, Sopra alcune questioni di elastostatica, Atti délia Società Italiana per il Progresso della Scienze, 1933. JFM59.1413.02 · JFM 59.1413.02
[31] W. L. WENDLAND, On some mathematical aspects of boundary elementmethods for elliptic problems, in MAFELAP V (J. R. Whiteman, éd.), Academic Press, New York, 1985, pp. 193-227. Zbl0587.65079 MR811035 · Zbl 0587.65079
[32] W. L. WENDLAND and E. P. STEPHAN, A hypersingular boundary integral method for two-dimensional screen and crack problems, Arch. Rational Mech. Anal 112, 1990, pp. 363-390. Zbl0725.73091 MR1077265 · Zbl 0725.73091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.