This article is a continuation of the paper Kočvara and Stingl (Struct Multidisc Optim 33(4–5):323–335, 2007). The aim is to describe numerical techniques for the solution of topology and material optimization problems with local stress constraints. In particular, we consider the topology optimization (variable thickness sheet or “free sizing”) and the free material optimization problems. We will present an efficient algorithm for solving large scale instances of these problems. Examples will demonstrate the efficiency of the algorithm and the importance of the local stress constraints. In particular, we will argue that in certain topology optimization problems, the addition of stress constraints must necessarily lead not only to the change of optimal topology but also optimal geometry. Contrary to that, in material optimization problems the stress singularity is treated by the change in the optimal material properties.
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The entire presentation is given for two-dimensional bodies, to keep the notation simple. Analogously, all this can be done for three-dimensional solids.
Not to be confused with the singularity of the stress function, e.g., in the corner of an L-shaped domain.
Of course, the strain still depends on ρ implicitly, through u. However, this dependence does not force the strain to vanish when ρ tends to zero.
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This research was supported by the EU Commission in the Sixth Framework Program, Project No. 30717 PLATO-N, by the Academy of Sciences of the Czech Republic through grant No. A100750802, and by DFG cluster of excellence 315.
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Kočvara, M., Stingl, M. Solving stress constrained problems in topology and material optimization. Struct Multidisc Optim 46, 1–15 (2012). https://doi.org/10.1007/s00158-012-0762-z
- Topology optimization
- Material optimization
- Stress based design
- Nonlinear semidefinite programming