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Robust truss topology optimization via semidefinite programming with complementarity constraints: a difference-of-convex programming approach

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Abstract

The robust truss topology optimization against the uncertain static external load can be formulated as mixed-integer semidefinite programming. Although a global optimal solution can be computed with a branch-and-bound method, it is very time-consuming. This paper presents an alternative formulation, semidefinite programming with complementarity constraints, and proposes an efficient heuristic. The proposed method is based upon the concave–convex procedure for difference-of-convex programming. It is shown that the method can often find a practically reasonable truss design within the computational cost of solving some dozen of convex optimization subproblems.

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Notes

  1. A truss is an assemblage of straight bars (called members) connected by pin-joints (called nodes) that do not transfer moment. See Sect. 2 for some concrete examples.

  2. The compliance of a truss, formally defined by (16), is equivalent to the twice strain energy of the truss at the equilibrium state under the prescribed boundary conditions (as far as the prescribed displacements are zeros [35]). It can be regarded as a global measure of the displacements, and hence by minimizing the compliance the global stiffness of the truss is maximized.

  3. The ground structure method is commonly used in truss topology optimization. It prepares an initial setting, called the ground structure, consisting of many members connected by nodes. The cross-sectional areas of the members are treated as design variables, while the locations of the nodes are specified. See Sect. 2 for more account.

  4. With reference to concrete examples, we will thoroughly discuss this issue in Sect. 2.

  5. A solution having a chain cannot be in equilibrium with uncertain loads applied at intermediate nodes of the chain. Therefore, the worst-case compliance of the solution is infinitely large.

  6. The concave–convex procedure is also known as the convex–concave procedure [39, 62].

  7. It should be clear that the worst-case compliance for the solution in Fig. 2c is smaller than that for the solution in Fig. 2a.

  8. In this example, overlapping longer members are not incorporated into the ground structure, because with overlapping members the global optimization method (YALMIP [40]) did not converge within realistic computational time.

  9. In fact, \(-\hat{g}_{i}(\cdot \,; \varvec{x}^{(k)})\) is a majorization function of \(-g_{i}\).

  10. Choice of an initial point in the numerical experiments is explained in Sect. 6

  11. The degrees of freedom of a truss are the possible components of the nodal displacements that define the configuration of the truss.

  12. Problem (17) is convex. Various reformulations are known in literature; see, e.g., Achtziger et al. [2] and Jarre et al. [31]. For example, replacing \(\mathop {\mathrm {diag}}\nolimits (\varvec{s})Q\) in (22) with \(\tilde{\varvec{p}}\), one can readily obtain SDP that minimizes w under constraint \( \begin{bmatrix} w&\tilde{\varvec{p}}^{\top } \\ \tilde{\varvec{p}}&K(\varvec{x}) \\ \end{bmatrix} \succeq 0 \), (17b), and (17c). This formulation was used in the numerical experiments. It should be clear that a ground structure with overlapping members is used for generating the initial point, \(\varvec{x}^{(0)}\).

  13. Ground structures without overlapping members are used to obtain the solutions in Figs. 1b, 3b and  4b.

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Acknowledgements

This work is partially supported by JSPS KAKENHI 26420545 and 17K06633.

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  1. Mathematics and Informatics Center, The University of Tokyo, Hongo 7-3-1, Tokyo, 113-8656, Japan

    Yoshihiro Kanno

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Kanno, Y. Robust truss topology optimization via semidefinite programming with complementarity constraints: a difference-of-convex programming approach. Comput Optim Appl 71, 403–433 (2018). https://doi.org/10.1007/s10589-018-0013-3

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