Domain decomposition method for crack problems with nonpenetration condition
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 995-1009.

The work deals with an iteration method for numerical solving the equilibrium problem of two-dimensional elastic body with a crack under the nonpenetration condition. The method is based on the domain decomposition and Uzawa’s algorithm. To construct an algorithm, the domain is partitioned into two subdomains whose common boundary contains the crack. In each subdomain the linear problems are solved. We use Lagrangian multipliers to couple the solutions and provide the nonpenetration condition on the crack.

Reçu le :
DOI : 10.1051/m2an/2015064
Classification : 65N55, 65K15, 35Q74, 74R10
Mots clés : Crack, nonpenetration condition, domain decomposition method, Lagrange multipliers, Uzawa’s algorithm
Rudoy, Evgeny 1

1 Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk State University, 630090, Novosibirsk, Russia.
@article{M2AN_2016__50_4_995_0,
     author = {Rudoy, Evgeny},
     title = {Domain decomposition method for crack problems with nonpenetration condition},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {995--1009},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {4},
     year = {2016},
     doi = {10.1051/m2an/2015064},
     zbl = {1457.65242},
     mrnumber = {3521709},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2015064/}
}
TY  - JOUR
AU  - Rudoy, Evgeny
TI  - Domain decomposition method for crack problems with nonpenetration condition
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 995
EP  - 1009
VL  - 50
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2015064/
DO  - 10.1051/m2an/2015064
LA  - en
ID  - M2AN_2016__50_4_995_0
ER  - 
%0 Journal Article
%A Rudoy, Evgeny
%T Domain decomposition method for crack problems with nonpenetration condition
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 995-1009
%V 50
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2015064/
%R 10.1051/m2an/2015064
%G en
%F M2AN_2016__50_4_995_0
Rudoy, Evgeny. Domain decomposition method for crack problems with nonpenetration condition. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 995-1009. doi : 10.1051/m2an/2015064. http://archive.numdam.org/articles/10.1051/m2an/2015064/

G. Allaire, Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation. Oxford University Press (2007). | MR | Zbl

M. Bach, A.M. Khludnev and V.A. Kovtunenko, Derivatives of the energy functional for 2D-problems with a crack under Signorini and friction conditions. Math. Methods Appl. Sci. 23 (2000) 515–534. | DOI | MR | Zbl

G.P. Astrakhantsev, Domain decomposition method for the problem of bending heterogeneous plate. Comput. Math. Math. Phys. 38 (1998) 1686–1694. | MR | Zbl

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

G. Bayada, J. Sabil and T. Sassi, Convergence of a Neumann-Dirichlet algorithm for two-body contact problems with non local Coulomb’s friction law. ESAIM: M2AN 42 (2008) 243–262. | DOI | Numdam | MR | Zbl

J. Céa, Optimisation, Théorie et algorithmes. Dunod, Gauthier - Villars Paris (1971). | MR | Zbl

G.P. Cherepanov Mechanics of Brittle Fracture. New York, McGraw-Hill (1979). | Zbl

J. Daněk, I. Hlaváček and J. Nedomac, Domain decomposition for generalized unilateral semi-coercive contact problem with given friction in elasticity. Math. Comput. Simul. 68 (2005) 271–300. | DOI | MR | Zbl

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag - Berlin, Heidelberg, New York (1976). | MR | Zbl

I. Ekeland and R. Temam, Convex Analysis and Variational Problems. North-Holland Publishing Company, Amsterdam, Oxford (1976). | MR | Zbl

L.C. Evans, Partial Differential Equations. AMS Press (1998). | Zbl

P. Grisvard,Singularities in Boundary Value Problems. Masson, Springer, Paris (1991). | MR | Zbl

J. Haslinger, R. Kučera and T. Sassi, A domain decomposition algorithm for contact problems: analysis and implementation. Math. Model. Nat. Phenom. (2009) 4 123–146. | DOI | MR | Zbl

F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | DOI | MR | Zbl

M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. (2003) 13 865–888. | DOI | MR | Zbl

M. Hintermüller, V. Kovtunenko and K. Kunisch, The primal-dual active set method for a crack problem with non-penetration. IMA J. Appl. Math. (2004) 69 1–26. | DOI | MR | Zbl

K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications. SIAM, Philadelphia (2008). | MR | Zbl

M.S.D. Jacob, P.R. Arora and M. Saleem, M.A. Elsadig and S.M. Sapuan, Fretting fatigue crack initiation: An experimental and theoretical study. Int. J. Fatigue. 29 (2007) 1328–1338. | DOI

A.M. Khludnev and V.A. Kovtunenko, Analysis of cracks in solids. Southampton; WIT-Press, Boston (2000).

A.M. Khludnev and V.A. Kozlov, Asymptotics of solutions near crack tips for Poisson equation with inequality type boundary conditions. Z. Angew. Math. Phys. 59 (2008) 264–280. | DOI | MR | Zbl

A.M. Khludnev and G. Leugering, On elastic bodies with thin rigid inclusions and cracks. Math. Methods Appl. Sci. 33 (2010) 1955–1967. | MR | Zbl

A.M. Khludnev and A. Tani, Overlapping domain problems in the crack theory with possible contact between crack faces. Q. Appl. Math. 66 (2008) 423–435. | DOI | MR | Zbl

N. Kikuchi and J.T. Oden, Contact Problems in Elasticity. SIAM, Philadelphia (1988). | MR | Zbl

J. Koko, Uzawa conjugate gradient domain decomposition methods for coupled Stokes flows. J. Sci. Comput. 26 (2006) 195–215. | DOI | MR | Zbl

J. Koko, Uzawa bloc relaxation domain decomposition method for a two-body frictionless contact problem. App. Math. Lett. (2009) 22 1534–1538. | DOI | MR | Zbl

V.A. Kovtunenko, Numerical simulation of the non-linear crack problem with nonpenetration. Math. Meth. Appl. Sci. (2004) 27 163–179. | DOI | MR | Zbl

V.A. Kozlov, V.G. Mazya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Vol. 85 of Math. Surveys and Monographs, American Mathematical Society, Providence, RI (2001). | MR | Zbl

Yu.M. Laevsky, A.M. Matsokin, Decomposition methods for the solution to elliptic and parabolic boundary value problems Sib. Zh. Vychisl. Mat. 2 (1999) 361–372. | Zbl

E. Laitinen, A.V. Lapin and J. Pieskä, Splitting iterative methods and parallel solution of variational inequalities. Lobachevskii J. Math. 8 (2001) 167–184. | MR | Zbl

N.P. Lazarev and E.M. Rudoy, Shape sensitivity analysis of Timoshenko’s plate with a crack under the nonpenetration condition. Z. Angew. Math. Mech. 94 (2014) 730–739. | DOI | MR | Zbl

A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations. Clarendon Press (1999). | MR | Zbl

E.M. Rudoy, Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body. Z. Angew. Math. Phys. 66 (2015) 1923–1937. | DOI | MR | Zbl

E.V. Vtorushin, Numerical investigation of a model problem for deforming an elastoplastic body with a crack under non-penetration condition Sib. Zh. Vychisl. Mat. (2006) 9 335–344. (in Russian) | Zbl

K. Yosida, Functional Analysis. Springer-Verlag, Berlin Heidelberg GmbH (1968). | Zbl

Cité par Sources :