@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) }", }