OPTIMA 2018: Optimization and Applications pp 35-48

# A Modified Duality Method for Solving an Elasticity Problem with a Crack Extending to the Outer Boundary

• Robert Namm
• Georgiy Tsoy
• Ellina Vikhtenko
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)

## Abstract

A modified dual method for solving an elasticity problem with a crack extending to the outer boundary is considered. The method is based on a modified Lagrange functional. The convergence of the method is investigated in detail under a natural assumption of $$H^1$$-regularity of the solution to the crack problem. Basic duality relation for the primal and dual problems is proposed.

## Keywords

Non-penetration condition Crack Duality scheme Modified lagrange functional Generalized newton method

## Notes

### Acknowledgments

This study was supported by the Russian Foundation for Basic Research (Project 17-01-00682 A). Numerical experiments were performed on a computational cluster of the Shared Facility Center “Data Center of FEB RAS” [20].

## References

1. 1.
Morozov, N.F.: Mathematical Problems of Crack Theory. Nauka, Moscow (1982)Google Scholar
2. 2.
Khludnev, A.M.: Problems of Elasticity Theory in Non-Smooth Domains. Fizmatlit, Moscow (2010)Google Scholar
3. 3.
Kravchuk, A.S.: Variational and Quasi-Variational Inequalities in Mechanics. MGAPI, Moscow (1997)Google Scholar
4. 4.
Kovtunenko, V.A.: Numerical simulation of the non-linear crack problem with non-penetration. Math. Methods Appl. Sci. 27, 163–179 (2004).
5. 5.
Vtorushin, E.V.: A numerical investigation of a model problem for deforming an elasto-plastic body with a crack under non-penetration condition. Sib. Zh. Vych. Mat. 9(4), 335–344 (2006)
6. 6.
Rudoy, E.M.: Domain decomposition method for a model crack problem with a possible contact of crack edges. Comput. Math. Math. Phys. 55(2), 305–316 (2015).
7. 7.
Rudoy, E.M.: Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion. J. Appl. Ind. Math. 10(2), 264–276 (2016).
8. 8.
Namm, R.V., Tsoy, G.I.: A modified dual scheme for solving an elastic crack problem. Num. Anal. Appl. 10(1), 37–46 (2017).
9. 9.
Woo, G., Namm, R.V., Sachkov, S.A.: An iterative method based on a modified Lagrangian functional for finding a saddle point in the semicoercive signorini problem. Comput. Math. Math. Phys. 46(1), 23–33 (2006).
10. 10.
Vikhtenko, E.M., Namm, R.V.: Duality scheme for solving the semicoercive signorini problem with friction. Comput. Math. Math. Phys. 47(12), 2023–2036 (2007).
11. 11.
Vikhtenko, E.M., Maksimova, N.N., Namm, R.V.: Sensitivity functionals in variational inequalities of mechanics and their applications to duality schemes. Num. Anal. Appl. 7(1), 36–44 (2014).
12. 12.
Vikhtenko, E.M., Woo, G., Namm, R.V.: Sensitivity functionals in contact problems of elasticity theory. Comput. Math. Math. Phys. 54(7), 1218–1228 (2014).
13. 13.
Vikhtenko, E.M., Namm, R.V.: On duality method for solving model crack problem. Tr. IMM UrO RAN 22, 36–43 (2016)
14. 14.
Hlavačhek, I., Haslinger, Ya., Nechas, I., Lovišhek, Ya.: Numerical Solution of Variational Inequalities. Springer, New York (1988)Google Scholar
15. 15.
Kufner, A., Fuchik, S.: Nonlinear Differential Equations. Nauka, Moscokw (1988)Google Scholar
16. 16.
Vasiliev, F.P.: Methods for Solving Extremal Problems. Nauka, Moscow (1981)Google Scholar
17. 17.
Vikhtenko, E.M., Woo, G., Namm, R.V.: The methods for solution semi-coercive variational inequalities of mechanics on the basis of modified Lagrangian functionals. Dalnevos. Mat. Zh. 14, 6–17 (2014)
18. 18.
Vikhtenko, E.M., Woo, G., Namm, R.V.: Modified dual scheme for finite-dimensional and infinite-dimensional convex optimization problems. Dalnevos. Mat. Zh. 17, 158–169 (2017)
19. 19.
Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101–108 (2009).
20. 20.
Shared Facility Center “Data Center of FEB RAS” (Khabarovsk). http://lits.ccfebras.ru