## Phase–field relaxation of topology optimization with local stress constraints (2005)

Venue: | Local Stress Constraints, SFB-Report 04-35 (SFB F013, University Linz, 2004), and submitted |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Burger05phase–fieldrelaxation,

author = {Martin Burger and Roman Stainko},

title = {Phase–field relaxation of topology optimization with local stress constraints},

booktitle = {Local Stress Constraints, SFB-Report 04-35 (SFB F013, University Linz, 2004), and submitted},

year = {2005}

}

### OpenURL

### Abstract

We introduce a new relaxation scheme for structural topology optimization 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 approaches 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 programming problems by a primal-dual interior point method. Numerical experiments for problems with stress constraints based on different criteria indicate the success and robustness of the new approach.

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Citation Context ...roblem has become standard, and seems to be well-understood with respect to its mathematical properties (cf. e.g. [2, 8, 25]), and various successful numerical techniques have been proposed (cf. e.g. =-=[3, 7, 19, 26, 31]-=-). The treatment of the second problem is by far less understood and until now there seems to be no approach that is capable of computing reliable (global) optima within reasonable computational effor... |

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Citation Context ...aints describing the elastic equilibrium and the local inequality constraints for stresses and displacements into a system of linear inequality constraints as recently proposed by Stolpe and Svanberg =-=[29, 30]-=-. This reformulation is approximate at the continuum level, but exact for finite element discretizations with suitable parameter choice. The main difficulty is that the arising problem also involves 0... |

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Citation Context ...he main source of difficulties in this problem is a lack of constraint qualification in the feasible set defined by the local stress constraints, which already appear for simple truss structures (cf. =-=[23, 28]-=-). Moreover, there are several complications for specific methods, e.g. convergence issues of homogenized stress criteria for material interpolation schemes (cf. [5, 16]). We start by describing the m... |

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Citation Context ...of these mixed linear programming problems grows fast with the number of degrees of freedom in the discretization, so that the problem could be solved only for very coarse discretizations so far (cf. =-=[27, 30]-=-). Instead of solving mixed linear programming problems, we propose to use a phase-field relaxation of the reformulated problem. The phase-field relaxation consists in using an interpolated material d... |

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Citation Context ...he main source of difficulties in this problem is a lack of constraint qualification in the feasible set defined by the local stress constraints, which already appear for simple truss structures (cf. =-=[23, 28]-=-). Moreover, there are several complications for specific methods, e.g. convergence issues of homogenized stress criteria for material interpolation schemes (cf. [5, 16]). We start by describing the m... |

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Citation Context ...pendent loads, another type of problem where standard material interpolation schemes encounter diffculties. The approach has recently been applied to minimal compliance type problems by Wang and Zhou =-=[33]-=-. The remainder of this paper is organized as follows: in Section 2 we review the constraint reformulation due to [30] and extend the approach to an approximate reformulation of the continuous problem... |

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Citation Context |

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Citation Context ...ce ∂{χ = 1} for d = 3. The boundedness of the perimeter regularizes the topology optimization problem, in particular it excludes checkerboard effects as the discretization size decreases to zero (cf. =-=[18, 22]-=-). In this paper we use a different approach to the relaxation of the local constraints. Starting point of our analysis is a reformulation of the equality constraints describing the elastic equilibriu... |

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