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Geometry and topology optimization of plane frames for compliance minimization using force density method for geometry model

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Abstract

A new method is proposed for simultaneous optimization of shape, topology and cross section of plane frames. Compliance against specified loads is minimized under constraint on structural volume. Difficulties caused by the melting nodes can be alleviated to some extent by introducing force density as design variables for defining the geometry, where the side constraints are assigned for force density to indirectly avoid the existence of extremely short members. Force density method is applied to an auxiliary cable-net model with different boundary and loading conditions so that the regularity of force density matrix is ensured by positive force densities. Sensitivity coefficients of the objective and constraint functions with respect to the design variables are also explicitly calculated. After the optimal geometry of the frame is obtained, the topology is further improved by removing the thin members and combining closely spaced nodes. It is demonstrated in the numerical examples of three types of frames that rational geometry and topology can be achieved using the proposed method, and the effect of bending moment on the optimal solution is also discussed.

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Acknowledgements

This research is partially supported by China Scholarship Council (File No. 201806050114).

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Authors and Affiliations

  1. Department of Architecture and Architectural Engineering, Kyoto University, Kyoto-Daigaku Katsura, Nishikyo, Kyoto, 615-8540, Japan

    Wei Shen & Makoto Ohsaki

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  1. Wei Shen
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  2. Makoto Ohsaki
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Correspondence to Wei Shen.

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Appendix

Appendix

The intermediate solutions in optimization procedure of Example 1 with \( \overline{V} \)= 1 are shown in Figs. 20 and 21 for Cases P and N, respectively. Note that both the results correspond to the best results of problem (20), and the red and blue color of members stand for positive and negative member forces, respectively. The figure at the top-right of Fig. 21 is trimmed to be consistent with the others, although some of the nodes and members are cut off.

Fig. 20
figure 20

Intermediate solutions of Example 1 with Case P

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Fig. 21
figure 21

Intermediate solutions of Example 1 with Case N

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As seen from Figs. 20 and 21, in Case N the structure undergoes drastic variation at the beginning mainly due to the change of signs of the force; on the other hand, iteration in optimization procedure with Case P has “smoother” shape variation leading to a monotonic convergence to the solution.

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Shen, W., Ohsaki, M. Geometry and topology optimization of plane frames for compliance minimization using force density method for geometry model. Engineering with Computers 37, 2029–2046 (2021). https://doi.org/10.1007/s00366-019-00923-w

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