@article{1084.49029,
author="Hinterm\"uller, M. and Kovtunenko, V.A. and Kunisch, K.",
title="{The primal-dual active set method for a crack problem with
non-penetration.}",
language="English",
journal="IMA J. Appl. Math. ",
volume="69",
number="1",
pages="1-26",
year="2004",
doi={10.1093/imamat/69.1.1},
abstract="{The purpose of this very important and useful paper is to study the
Lam\'e problem in a 2D domain with a crack under a non-penetration condition
which is imposed at the crack faces. This condition is considered as a
variational inequality. From a mathematical point of view one principal
difficulty of crack problems lies in the non-regular character of boundaries
caused by the presence of a crack within the domain (leads to a reduced
method of the solution).
Main results: The primal-dual active set
algorithm for both symmetric and non-symmetric crack problems with
non-penetration is formulated and shown to be numerically efficient. In the
finite-dimensional case, superlinear local convergence is proved and global
convergence is obtained under the positiveness assumption made on some
matrix connecting the jump of the traces and stress at the crack by the
active set iteration. From this assumption, monotonicity of the estimates of
the active set follows. The authors propose the numerical tests which
indicate the following. In comparison with a regularization method, the
active set method requires a significantly smaller number of iterations. The
number of iterations required depends only moderately on the mesh size of
the discretization. The crucial point: The active sets converge
monotonically independently of the initialization. These assertions are
tested for the closed and opened crack faces, for the symmetric and
nonsymmetric loading, for the homogeneous and bonded isotropic materials,
and for various Lam\'e constants.}",
reviewer="{Jan Lov\'\i\v sek (Bratislava)}",
keywords="{linear elasticity; Lam\'e equations; non-penetration; variational
inequality; Dirichlet boundary conditions; Neumann-type boundary conditions;
stress-free crack faces; variational (weak) formulation; Lagrange
multiplier; active set algorithm; numerical simulation}",
classmath="{*49M25 (Finite difference methods)
74R10 (Brittle fracture)
35Q72 (Other PDE from mechanics)
90C51 (Interior-point methods)
}",
}