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Unformatted text preview: PhaseField 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 phasefield 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 CahnHilliard 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 largescale optimization problem with a high number of linear inequality constraints. We discretize the problem by finite elements and solve the arising finitedimensional pro gramming problems by a primaldual 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, PhaseField Methods, InteriorPoint 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. email: martin.burger@jku.at. † Spezialforschungsbereich SFB F 013 Numerical and Symbolic Scientific Computing. Altenbergerstr. 69, A 4040 Linz, Austria. email: roman.stainko@jku.at. 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 designconstraint 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 socalled minimal compli ance problem has become standard, and seems to be wellunderstood 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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This note was uploaded on 04/12/2010 for the course ME master taught by Professor Mon during the Spring '09 term at Hanyang University.
 Spring '09
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 Strain, Stress

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