Comparison of Lagrange Multiplier Method Implementation for Solving Two-Dimensional Contact Problems

Authors: Galanin M.P., Gliznutsina P.V., Lukin V.V., Rodin A.S. Published: 27.09.2017
Published in issue: #5(74)/2017  
DOI: 10.18698/1812-3368-2017-5-35-48

Category: Mechanics | Chapter: Mechanics of Deformable Solid Body  
Keywords: deformable solid, contact problem, finite element method, Lagrange multiplier method

We consider a two-dimensional contact problem involving two deformable solids. We used the finite element method based on quadrilateral bilinear elements to approximate our elastic problem. Three implementations of the Lagrange multiplier method account for contact conditions: node-to-surface, surface-to-surface and surface-to-surface employing sub-segments. We carried out test calculations, solving the Hertz problem and comparing our results to the analytical solution. A comparative analysis of these methods shows that the two surface-to-surface contact implementations are more accurate than the node-to-surface implementation. The surface-to-surface contact method that employs sub-segments makes it possible to smooth out stress field fluctuations, but this effect only works for a limited number of problems


[1] Johnson K.L. Contact mechanics. Cambrige University Press, 1985. 452 p.

[2] Burago N.G., Kukudzhanov V.N. A review of contact algorithms. Mech. Solids, 2005, vol. 40, no. 1, pp. 35–71.

[3] Wriggers P. Computational contact mechanics. Springer, 2006. 521 p.

[4] Tsvik L.B. Priority principle in conjugating and contact problems of deformable bodies. Prikladnaya mekhanika, 1980, vol. 16, no. 1, pp. 13–18 (in Russ.).

[5] Bogatyr’ S.M., Galanin M.P., Kuznetsov V.I., et al. Mathematical simulation of thermoelastic contact interaction of axisymmetric bodies. Inzhenernyy zhurnal: nauka i innovatsii [Engineering Journal: Science and Innovation], 2013, no. 4 (in Russ.). DOI: 10.18698/2308-6033-2013-4-667 Available at: http://engjournal.ru/eng/catalog/mathmodel/hidden/667.html

[6] Korobeynikov S.N. Nelineynoe deformirovanie tverdykh tel [Nonlinear deformation of solids]. Novosibirsk, SO RAN Publ., 2000. 262 p.

[7] Demidov S.P. Teoriya uprugosti [Elasticity theory]. Moscow, Vysshaya shkola Publ., 1979. 432 p.

[8] Zenkevich O. Metod konechnykh elementov v tekhnike [Finite elements method in physics]. Moscow, Mir Publ., 1975. 543 p.

[9] Sagdeeva Yu.A., Kopysov S.P., Novikov A.K. Vvedenie v metod konechnykh elementov [Introduction to finite elements method]. Izhevsk, Udmurt University Publ., 2011. 44 p.

[10] Galanin M.P., Gliznutsina P.V., Lukin V.V., Rodin A.S. Lagrange multiplier method implementations for two-dimensional contact problems. KIAM Preprint, 2015, no. 89, 27 p. (in Russ.). Available at: http://library.keldysh.ru/preprint.asp?id=2015-89

[11] Taylor Robert L. Finite element solution of contact problems from: 1974 to 2004. Available at: http://faculty.ce.berkeley.edu/rlt/presentations/hughes.pdf (accessed: 09.06.2015).