Abstract
Stability is an important issue in topology optimization, since results of the optimization are often framework structures. If some trusses of these structures are subjected to compression, they maybe buckle and the structure fails.
A brief review about literature on topology optimization considering linear buckling is given. That includes the material interpolation for linear buckling analysis to avoid spurious modes, mode switching and duplicated eigenvalues.
In this contribution a continuously differentiable material interpolation scheme is explained to avoid spurious modes. It is shown how to cope with several modes (mode switching, duplicated eigenvalues) and how to use buckling safety as objective or constraint. The optimized structures are compared with results for compliance design. It is shown, that it is not useful to use buckling and mass as the only structure responses, because substructures subjected to tension can become very thin without a negative influence on the buckling safety. Thus tensile stresses are not limited. Buckling and mass have to be combined with other structure responses, like stress or compliance, to achieve a useful structure. Also the combination with manufacturing constraints for deep drawing is discussed in this contribution. Therefore 3D structures with up to a million design variables are presented.
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Notes
- 1.
Because \( {\mathbf{G}} \) and \( {\mathbf{K}} \) are symmetric band matrices and \( {\mathbf{K}} \) is positive definite, \( {\mathbf{K}} \) can be split efficiently to \( {\mathbf{K}} = {\mathbf{R}}^{\text{T}} {\mathbf{R}} \) using Cholesky decomposition. The eigenvalue problem can be written as \( \left( {{\mathbf{R}}^{{ - {\text{T}}}} {\mathbf{GR}}^{ - 1} + \psi_{j} {\mathbf{E}}} \right){\mathbf{R}}\varvec{\varphi }_{j} = 0 \), which is a standard eigenvalue problem \( \left( {{\bar{\mathbf{G}}} + \psi_{j} {\mathbf{E}}} \right)\bar{\varvec{\varphi }}_{j} = 0 \).
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Dienemann, R., Schumacher, A., Fiebig, S. (2019). Considering Linear Buckling for 3D Density Based Topology Optimization. In: Rodrigues, H., et al. EngOpt 2018 Proceedings of the 6th International Conference on Engineering Optimization. EngOpt 2018. Springer, Cham. https://doi.org/10.1007/978-3-319-97773-7_36
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DOI: https://doi.org/10.1007/978-3-319-97773-7_36
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