Differential-difference iterative domain decomposition algorithms for unilateral multibody contact problems of elasticity

Fìz.-mat. model. ìnf. tehnol. 2017, 25:125-140

  • Ihor Prokopyshyn Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine
  • Stepan Shakhno Ivan Franko National University o f Lviv , 1 Universytetska St.,79000, Lviv, Ukraine
Keywords: problems with non-differentiating operator, theory of elasticity contact tasks, variational inequalities, nonlinear variational equations, fine method, differential-difference iterative methods, Newton semi-smooth method, methods of decomposition of the area, finite element method

Abstract

Implicit two-point differential-difference parallel iterative domain decomposition algorithms are proposed to solve the multibody contact problems of elasticity. A program implementation of these algorithms based on the finite element approximations is made for the case of plane contact problems. The influence of the iterative parameters on the convergence rate of presented algorithms is investigated. The numerical efficiency of different two-point and one-point iterative algorithms is compared.

References
  1. Bartish, M. Ya., Shcherbyna, Yu. M. (1972). Pro odyn riznytsevyi metod rozviazuvannia neliniinykh opera- tornykh rivnian. Dop. AN URSR. Ser. A., 7, 579-582.
  2. Shakhno, S. M. (2006). Pro riznytsevyi metod z kvadratychnoiu zbizhnistiu dlia rozviazuvannia neliniinykh operatornykh rivnian. Mat. studii., 26(1), 105-110.
  3. Shakhno, S. M. (2009). Pro dvokrokovyi iteratsiinyi protses v uzahalnenykh umovakh Lipshytsia dlia podilenykh rizyts pershoho poriadku. Mat. metody ta fiz.-mekh. polia., 52(1), 59-66.
  4. Shakhno, S. M. (2009). On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations. J. Comput. Appl. Math., 231, 222-235.
    DOI https://doi.org/10.1016/j.cam.2009.02.010
  5. Shakhno, S. M., Yarmola, H. P. (2011). Dvotochkovyi metod dlia rozviazuvannia neliniinykh rivnian z nedyferentsiiovnym operatorom. Mat. studii, 36(2), 213-220.
  6. Shakhno, S. M., Melnyk, I. V., Yarmola, H. P. (2013). Analiz zbizhnosti kombinovanoho metodu dlia rozviazuvannia neliniinykh rivnian. Mat. metody ta fiz.-mekh. polia, 56(1), 31-39.
  7. Hernandez, M. A., Rubio, M. J. (2002). The secant method for nondifferentiable operators. Appl. Math. Lett., 15(4), 395-399.
    DOI https://doi.org/10.1016/S0893-9659(01)00150-1
  8. Argyros, I. K. (2004). A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space. J. Math. Anal. Appl., 298(2), 374-397.
    DOI https://doi.org/10.1016/j.jmaa.2004.04.008
  9. Chen, X., Nashed, Z., Qi, L. (2000). Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal., 38, 1200-1216.
    DOI https://doi.org/10.1137/S0036142999356719
  10. Ulbrich, M. (2003). Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim., 13(3), 805-842.
    DOI https://doi.org/10.1137/S1052623400371569
  11. Hintermüller, M., Ito, K., Kunisch, K. (2003). The primal-dual active set strategy as semismooth Newton method. SIAM J. Optim., 13(3), 865-888.
    DOI https://doi.org/10.1137/S1052623401383558
  12. Kikuchi, N., Oden, J. T. (1988). Contact Problem in Elasticity: A Study of Variational Inequalities and Finite Element Methods. Philadelphia: SIAM.
    DOI https://doi.org/10.1137/1.9781611970845
  13. Kravchuk, A. S. (1978). Postanovka zadachi o kontakte neskolkih deformiruemyh tel kak zadachi nelinejnogo programmirovaniya. PMM, 42(3), 467-473.
  14. Lions, Zh.-L. (1972). Nekotorye metody resheniya nelinejnyh kraevyh zadach. Moskva: Mir.
  15. Kuzmenko, V. I. (1979). O variacionnom podhode k teorii kontaktnyh zadach dlya nelinejnouprugih sloistyh tel. PMM, 43(5), 893-901.
  16. Prokopyshyn, I. I. (2010). Skhemy dekompozytsii oblasti na osnovi metodu shtrafu dlia zadach kontaktu pruzhnykh til. (Dysertatsiia na zdobuttia naukovoho stupenia kandydata fiz.-mat. nauk). Lviv.
  17. Dyyak, I. I., Prokopyshyn, I. I., Prokopyshyn, I. A. (2012). Penalty Robin-Robin domain decomposition methods for unilateral multibody contact problems of elasticity: Convergence results.
    DOI https://doi.org/10.1007/978-3-642-35275-1_77
  18. Prokopyshyn, I. I. (2012). Metody dekompozytsii oblasti dlia zadach pro odnostoronnii kontakt neliniino pruzhnykh til. Fiz.-mat. modeliuvannia ta inform. tekhnolohii, 15, 75-87.
  19. Martyniak, R. M., Prokopyshyn, I. A., Prokopyshyn, I. I. (2013). Kontakt pruzhnykh til za naiavnosti neliniinykh vinklerivskykh poverkhnevykh shariv. Mat. metody ta fiz.-mekh. polia, 56(3), 43-56.
  20. Prokopyshyn, I. I., Martyniak, R. M. (2011). Chyslove doslidzhennia kontaktnoi vzaiemodii dvokh til z vyimkoiu metodom dekompozytsii oblasti. Problemy obchysliuvalnoi mekhaniky i mitsnosti konstruktsii, 16, 240-251.
  21. Prokopyshyn, I. I., Dyyak, I. I., Martynyak, R. M., Prokopyshyn, I. A. (2013). Penalty Robin-Robin domain decomposition schemes for contact problems of nonlinear elasticity. Lect. Notes Comput. Sci. Eng., 91, 647-654.
    DOI https://doi.org/10.1007/978-3-642-35275-1_77
  22. Prokopyshyn, I. I. (2015). Metody dekompozytsii oblasti dlia zadachi pro statychnu rivnovahu systemy pruzhnykh til, ziednanykh cherez tonki neliniini prosharky. Fiz.-mat. modeliuvannia ta inform. tekhnolohii, 21, 173-185.
  23. Shvets, R. M., Martynyak, R. M., Kryshtafovych, A. A. (1996). Discontinuous contact of an anisotropic halfplane and a rigid base with disturbed surface. Int. J. Engng. Sci., 34(2), 183-200.
    DOI https://doi.org/10.1016/0020-7225(95)00091-7
Published
2018-11-19
How to Cite
Prokopyshyn, I., & Shakhno, S. (2018). Differential-difference iterative domain decomposition algorithms for unilateral multibody contact problems of elasticity. PHYSICO-MATHEMATICAL MODELLING AND INFORMATIONAL TECHNOLOGIES, (25), 125-140. https://doi.org/10.15407/fmmit2017.25.125