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Unformatted text preview: Phase-Field Relaxation of Topology Optimization with Local Stress Constraints Martin Burger * Roman Stainko † January 14, 2005 Abstract We introduce a new relaxation scheme for structural topology opti- mization problems with local stress constraints based on a phase-field method. The starting point of the relaxation is a reformulation of the material problem involving linear and 0–1 constraints only. The 0–1 constraints are then relaxed and approximated by a Cahn-Hilliard type penalty in the objective functional, which yields convergence of minimizers to 0–1 designs as the penalty parameter decreases to zero. A major advantage of this kind of relaxation opposed to standard ap- proaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter. The relaxation scheme yields a large-scale optimization problem with a high number of linear inequality constraints. We discretize the problem by finite elements and solve the arising finite-dimensional pro- gramming problems by a primal-dual interior point method. Numeri- cal experiments for problems with stress constraints based on different criteria indicate the success and robustness of the new approach. Keywords: Topology Optimization, Local Stress Constraints, Phase-Field Methods, Interior-Point Methods. AMS Subject Classification: 74P05, 74P10, 74P15, 90C51, 74S05. 1 Introduction Topology optimization denotes problems of finding optimal material distri- butions in given design domains subject to certain criteria and, possibly, satisfying several additional constraints. In the last two decades, advances in homogenization, optimization theory, and numerical analysis, as well as * Institut f¨ur Industriemathematik, Johannes Kepler Universit¨ at, Altenbergerstr. 69, A 4040 Linz, Austria. e-mail: email@example.com. † Spezialforschungsbereich SFB F 013 Numerical and Symbolic Scientific Computing. Altenbergerstr. 69, A 4040 Linz, Austria. e-mail: firstname.lastname@example.org. 1 new engineering approaches caused topology optimization techniques to be- come a standard tool of engineering design (cf. [5, 17] for an overview), in particular in structural mechanics. In structural optimization there are two design-constraint combinations of particular importance, namely the maximization of material stiffness at given mass and the minimization of mass while keeping a certain stiffness. The formulation of the first combination as the so-called minimal compli- ance problem has become standard, and seems to be well-understood with respect to its mathematical properties (cf. e.g. [2, 8, 25]), and various suc- cessful numerical techniques have been proposed (cf. e.g. [3, 7, 19, 26, 31])....
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