Space semidiscrete formulation of contact algorithm based on the Schwarz's decomposition method for deformable bodies

A. S. Karavaev   ; S. P. Kopysov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 3, pp. 396-413

Résumé

Implicit integration scheme for Schwarz alternating method for dynamic contact interaction problems of two interacting volumetric bodies without friction is considered. The paper presents the results of testing a contact algorithm of Schwarz domain decomposition using HTT-$\alpha$ scheme in consistent method redistribution of mass on the boundary of contact. To prevent artificial oscillations on the contact boundary together with common dissipative properties of the $\alpha$-scheme, the consistent mass redistribution method was used. The main advantage of this approach is the option to use multigrid methods to speed up the algorithm on subdomains, also there is no need for contact elements, contact parameters, Lagrange multipliers or regularization. Numerical examples including various contact zones, different materials of contact bodies and comparisons with measurements of other methods show the wide applicability of the derived algorithm.

Détail

Keywords: dynamic contact analysis, Schwarz alternating method, implicit schemes.
Mots-clés : mass redistribution

Bibliographie

[1] Bourago N. G., Kukudzhanov V. N., “A review of contact algorithms”, Mechanics of Solids, 40:1 (2005), 35–71

[2] Laursen T. A., Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis, Springer, New York, 2003, xv+454 pp. | DOI | MR

[3] Tsvik L. B., “Generalization of the Schwartz algorithm to the case of regions in contact without overlapping”, Dokl. Akad. Nauk SSSR, 224:2 (1975), 309–312 (in Russian) | Zbl

[4] Galanin M. P., Lukin V. V., Rodin A. S., Stankevich I. V., “Application of the Schwarz alternating method for simulating the contact interaction of a system of bodies”, Comput. Math. Math. Phys., 55:8 (2015), 1393–1406 | DOI | DOI | MR | Zbl

[5] Eck C., Wohlmuth B., “Convergence of a contact-Neumann iteration for the solution of two-body contact problems”, Math. Models Methods Appl. Sci., 13:8 (2003), 1103–1118 | DOI | MR | Zbl

[6] Bayada G., Sabil J., Sassi T., “A Neumann-Neumann domain decomposition algorithm for the Signorini problem”, Appl. Math. Lett., 17:10 (2004), 1153–1159 | DOI | MR | Zbl

[7] Sassi T., Ipopa M., Roux F. X., “Generalization of Lions' nonoverlapping domain decomposition method for contact problems”, Domain decomposition methods in science and engineering XVII, eds. Langer U., Discacciati M., Keyes D. E., Widlund O. B., Zulehner W., Springer, Berlin–Heidelberg, 2008, 623–630 | DOI | MR | Zbl

[8] Konyukhov A., Schweizerhof K., Computational contact mechanics: geometrically exact theory for arbitrary shaped bodies, Springer, 2013 | DOI | MR | Zbl

[9] Yastrebov V. A., Breitkopf P., Numerical methods in contact mechanics, John Wiley Sons, 2013, 416 pp. | DOI | Zbl

[10] Karavaev A. S., Kopysov S. P., “The method of unstructured hexahedral mesh generation from volumetric data”, Komp'yuternye issledovaniya i modelirovanie, 5:1 (2013), 11–24 (in Russian) | Zbl

[11] Karavaev A. S., Kopysov S. P., “The Schwarz method of domain decomposition in contact problems”, Problemy mekhaniki i materialovedeniya, Trudy Instituta mekhaniki Ural'skogo otdeleniya Rossiiskoi Akademii Nauk, Institute of Mechanics, Ural Branch of the Russian Academy of Sciences, Izhevsk, 2017, 6–20 (in Russian)

[12] Karavaev A. S., Kopysov S. P., Kuz'min I. M., “Conservative interpolation method between non-matching surface meshes”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, no. 4, 64–75 (in Russian) | DOI

[13] Shabrov N. N., The finite element method in calculating the details of thermal engines, Mashinostroenie, L., 1983, 212 pp.

[14] Hilber H. M., Hughes T. R. J., Taylor R. L., “Improved numerical dissipation for time integration algorithms in structural dynamics”, Earthquake Engineering and Structural Dynamics, 5:3 (1977), 283–292 | DOI

[15] Deuflhard P., Krause R., Ertel S., “A contact-stabilized Newmark method for dynamical contact problems”, Internat. J. Numer. Methods Engrg., 73:9 (2008), 1274–1290 | DOI | MR | Zbl

[16] Laursen T. A., Chawla V., “Design of energy conserving algorithms for frictionless dynamic contact problems”, Internat. J. Numer. Methods Engrg., 40:5 (1997), 863–886 | 3.0.CO;2-V class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[17] Khenous H. B., Laborde P., Renard Y., “Mass redistribution method for finite element contact problems in elastodynamics”, Eur. J. Mech. A Solids, 27:5 (2008), 918–932 | DOI | MR | Zbl

[18] Hager C., Hüeber S., Wohlmuth B. I., “A stable energy-conserving approach for frictional contact problems based on quadrature formulas”, Internat. J. Numer. Methods Engrg., 73:2 (2008), 205–225 | DOI | MR | Zbl

[19] Krause R., Walloth M., “Presentation and comparison of selected algorithms for dynamic contact based on the Newmark scheme”, Appl. Numer. Math., 62:10 (2012), 1393–1410 | DOI | MR | Zbl

[20] Belytschko T., Neal M. O., “Contact-impact by the pinball algorithm with penalty and Lagrangian methods”, Internat. J. Numer. Methods Engrg., 31:3 (1991), 547–572 | DOI | Zbl

[21] Sitzmann S., Robust algorithms for contact problems with constitutive contact laws, Zur Erlangung des Doktorgrades Dr.-Ing., Der Technischen Fakultat der Friedrich-Alexander-Universitat Erlangen-Nurnberg, 2016, 164 pp.

[22] Dostál Z., Kozubek T., Brzobohatý T., Markopoulos A., Vlach O., “Scalable TFETI with optional preconditioning by conjugate projector for transient frictionless contact problems of elasticity”, Comput. Methods Appl. Mech. Engrg., 247–248 (2012), 37–50 | DOI | MR | Zbl