A Preconditioned Domain Decomposition Algorithm for Contact Problem

A Preconditioned Domain Decomposition Algorithm for Contact Problem HARRACHOV 2007 A. Lotfi A

Széchenyi István University Department of Mathmatics and Computational Sciences Abstract: • The purpose of this work is to study, the quasistatic two-body contact problem for small strains without friction. • To obtain an approximation of this problem, we use the finite element method • Non-overlapping DD method is used to solve the resulting finite element system • Preconditioner construction to solve the interface problem: 1. The schur complement operator is spectrally equivalent to H 1/2 seminorm [3] [4] 2. The circulant matrix representations of H 1/2 seminorm [3] [4] A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 2

Széchenyi István University Department of Mathmatics and Computational Sciences References: • [1] Kikuchi, N. and Oden, J. T. : Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988). • [2] B. Kiss, G. Molnárka: A preconditioned Domain Decomposition Algorithm for the Solution of the Elliptic Neumann Problem. In: W. Hackbush ed. , Parallel Algorithm for PDE, Proceeding of 6 th GAMM Seminar, Kiel, 119 -129, 1990. • [3] B. Kiss, A. Krebsz. : On the Schur Complement Preconditioners. Computers and Structures, Vol. 73, 537 --544, 1999. • [4] B. Kiss, A. Krebsz, K. Szalay. : On the Separability of the H 1/2 Seminorm on Convex Polyhedral Domains. HEJ (http: hej. szif. hu) vol(1) ANM-010227 -B, 20. A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 3

Széchenyi István University Department of Mathmatics and Computational Sciences OUTLINE • • Continuos problem Variational problem Finite element discretisation Domain Decomposition Method and Algorithm • The interface problem • Numerical results A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 4

Széchenyi István University Department of Mathmatics and Computational Sciences Continuos problem A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 5

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 6

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 7

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 8

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 9

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 10

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 11

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 12

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 13

Algorithm Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 14

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 15

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 16

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 17

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 18

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 19

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 20

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 21

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 22

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 23

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 24

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 25

Széchenyi István University Department of Mathmatics and Computational Sciences A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 26

Széchenyi István University Department of Mathmatics and Computational Sciences Thank you for your attention A Preconditioned DD Algorithm for Contact Problem / HARRACHOV 27