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The equivalent static loads method for structural optimization does not in general generate optimal designs

Abstract

The Equivalent Static Loads Method (ESLM) is an algorithm intended for dynamic response structural optimization. The algorithm attempts to solve a sequence of static response structural optimization problems approximating the original problem. It is argued in several published articles that if the ESLM converges, then it finds a KKT point of the considered dynamic structural response optimization problem. The theoretical convergence properties of the ESLM are however not as strong as previously reported. We propose and analyze easily reproducible counter examples based on a two-bar truss illustrating that the ESLM in general fails in finding optimal designs to the considered dynamic response problem.

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Notes

  1. We use the term KKT point to refer to a feasible point x for which there exists positive Lagrange multipliers satisfying the KKT conditions to the studied inequality constrained problem. The term primal-dual KKT point refers to the pair (x , λ ) satisfying the KKT conditions.

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Acknowledgements

We would like to express our sincere thanks to Professor Jakob Søndergaard Jensen from DTU Electrical Engineering and our colleague Kasper Sandal for providing constructive comments and suggestions that lead to improvements of the presentation. We would also like to thank the two reviewers for constructive comments and suggestions.

This research was financially supported by the strategic research project ABYSS: Advancing BeYond Shallow waterS - Optimal design of offshore wind turbine support structures (www.abyss.dk) which is funded by Innovation Fund Denmark.

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Correspondence to Mathias Stolpe.

Appendices

Appendix A: A worst-case two-bar truss example

An alternative time-discretized version of the truss example is the worst-case version of problem (\(\mathcal {P}_{D}\)) which reads

$$\begin{array}{@{}rcl@{}} \underset{\mathbf{x} \in \mathbb{R}^{n}}{\text{minimize}} && \underset{\ell_{1} \leq i \leq \ell_{2}}{\max} \{f_{i}(\mathbf{x})\}\\ \text{subject to} && \displaystyle \mathbf{x} \in X, \end{array} $$
(21)

where,

$$f_{i}(\mathbf{x}) = {\Delta} t \mathbf{u}_{i}(\mathbf{x})^{T} \mathbf{K}(\mathbf{x}) \mathbf{u}_{i}(\mathbf{x}). $$

Problem (21) is in general non differentiable and is therefore reformulated into

$$\begin{array}{@{}rcl@{}} \underset{\mathbf{x} \in \mathbb{R}^{n}, \tau \in \mathbb{R}}{\text{minimize}} && \tau \\ \text{subject to} && f_{i}(\mathbf{x})\leq \tau \,\,\, \text{ for }i = \ell_{1},\ell_{1}+ 1,\ldots,\ell_{2}\\ && \mathbf{x} \in X, \tau \geq 0. \end{array} $$
(22)

Problem (22) satisfies the format considered in e.g. Park and Kang (2003) in that the objective function is independent of the displacements and the nonlinear constraints only consider one time step per constraint. The static response sub-problem to be solved in the ESLM at iteration k corresponding to (22) becomes

$$\begin{array}{@{}rcl@{}} \underset{\mathbf{x} \in \mathbb{R}^{n},\tau \in \mathbb{R}}{\text{minimize}} && \tau\\ \text{subject to} && \tilde{f}_{i,\mathbf{x}^{k}}(\mathbf{x})\leq \tau, \text{ for } i = \ell_{1},\ell_{1}+ 1,\ldots,\ell_{2}\\ && \mathbf{x} \in X, \tau \geq 0, \end{array} $$
(23)

where

$$\tilde{f}_{i,\mathbf{x}^{k}}(\mathbf{x}) = {\Delta} t (\mathbf{r}_{i}^{e}(\mathbf{x}^{k}))^{T} \mathbf{K}^{-1}(\mathbf{x}) \mathbf{r}_{i}^{e}(\mathbf{x}^{k}). $$

Problem (23) enjoys similar properties as the static response sub-problem (\(\mathcal {S}_{D}\)). The objective function is linear and the constraint functions are convex resulting in a convex feasible set. The feasible set of (23) is non-empty since for any xX it is possible to find a sufficiently large τ > 0 such that the nonlinear constraints are satisfied. Additionally, the feasible set is convex and it is possible to find a feasible point (x,τ) with x ∈int(X) and τ > 0, i.e. the Slater constraint qualifications (Boyd and Vandenberghe 2004) are satisfied for (23). Again, uniqueness of the optimal solution is not guaranteed in the general case, but is ensured for the two-bar truss example.

For the two-bar truss the static response sub-problem for the worst-case situation becomes

$$\begin{array}{@{}rcl@{}} \underset{\mathbf{x} \in \mathbb{R}^{2}, \tau \in \mathbb{R}}{\text{minimize}} && \tau\\ \text{subject to} && \frac{\Delta t}{x_{1} + \frac{2}{3} x_{2}} ({r_{i}^{e}}(\mathbf{x}^{k}))^{2} \leq \tau \text{ for } i = \ell_{1},\ldots,\ell_{2}\\ && \mathbf{x} \in X, \tau \geq 0. \end{array} $$

Solving the static response sub-problem for the two-bar truss problem can again be done analytically and in the same manner as for the first version of the problem, i.e. the optimal solution is again unique and given by \(\hat {\mathbf {x}}\) independent of the equivalent static loads (under the condition that at least one is non zero). The equivalent static loads methods from Algorithm 1 thus converges in the same manner as before, i.e. at most two static response sub-problems are solved.

The point x is also found numerically using the optimization approaches as outlined in Section 5. The point is verified as a KKT point to the dynamic response problem (22) in the following way. At the point x the maximum in the objective function in (21) is attained in one time step i and the directional derivatives \(\nabla {f}_{i^{*}} (\mathbf {x}^{*})^{T}\mathbf {d}\) in the two feasible directions \({\mathbf {d}}_{1}^{*} = \hat {\mathbf {x}} - \mathbf {x}^{*} = \frac {1}{4}\)(3-2)T and \({\mathbf {d}}_{2}^{*} =\) (0-1)T are both strictly positive. Since these two vectors span the tangent cone at x and constraint qualifications are satisfied the point is as a KKT point. Our numerical experiments also show that the SQP method applied directly to (22) finds Lagrange multipliers such that the KKT conditions for (22) are accurately satisfied.

The point \(\hat {\mathbf {x}}\) does however not correspond to a KKT point to the dynamic response problem (22). Evaluating the dynamic response at for the design \(\hat {\mathbf {x}}\) shows that the maximum in (21) is attained at one time-point \(\hat {i}\). This implies that the objective function in (21) is differentiable at the point \(\hat {\mathbf {x}}\). The derivative with respect to the design variables is given by \(\nabla f_{\hat {i}}(\hat {\mathbf {x}})\). The directional derivatives \(\nabla f_{\hat {i}}(\hat {\mathbf {x}})^{T} \hat {\mathbf {d}}_{1} < 0\) and \(\nabla f_{\hat {i}}(\hat {\mathbf {x}})^{T} \hat {\mathbf {d}}_{2} > 0\). This shows that \(\hat {\mathbf {x}}\) does not correspond to a KKT point of (22).

Appendix B: Analytical design sensitivities

Here, we presents the analytical design sensitivities of the dynamic compliance and displacements, for both the original dynamic response problem and the ESLM static sub-problem. For the purpose of brevity we omit the function arguments. The design sensitivity of the displacement with respect to the j th design variable becomes

$$ \frac{\partial u}{\partial x_{j}} = \frac{\partial u_{0}}{\partial x_{j}} \sin(\omega t +\phi) + \frac{u_{0}}{1+\beta} \frac{\partial \beta}{\partial x_{j}}\cos(\omega t +\phi), $$
(24)

where

$$ \frac{\partial u_{0}}{\partial x_{j}} = -\frac{{u}_{0}^{3}}{{r}_{0}^{2}} \left( k - w^{2} m\right) \left( \frac{\partial k}{\partial x_{j}} - \omega^{2} \frac{\partial m}{\partial x_{j}} \right), $$
(25)

and

$$\frac{\partial \beta}{\partial x_{j}} = \frac{\beta^{2}\omega_{n}}{2k\zeta\omega} \left( \frac{\partial k}{\partial x_{j}} - \omega^{2} \frac{\partial m}{\partial x_{j}} \right). $$

The design sensitivity of the dynamic compliance can be written as

$$ \frac{\partial f_{c}}{\partial x_{j}} = \left( \frac{2}{u_{0}} \frac{\partial u_{0}}{\partial x_{j}} + \frac{1}{k} \frac{\partial k}{\partial x_{j}} \right)f_{c}, $$
(26)

which is valid for a time domain of duration T ω /2 for which the compliance is independent of the phase shift.

Inside the k th static response sub-problem, the design sensitivity of the displacement is

$$ \frac{\partial w}{\partial x_{j}} = -\frac{1}{k^{2}} \frac{\partial k}{\partial x_{j}} r^{e}(\mathbf{x}^{k}) $$
(27)

and the design sensitivity of the the dynamic compliance is

$$ \frac{\partial \tilde{f}_{c,\mathbf{x}^{k}}}{\partial x_{j}} =- \frac{1}{k} \frac{\partial k}{\partial x_{j}} \tilde{f}_{c,\mathbf{x}^{k}}. $$
(28)

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Stolpe, M., Verbart, A. & Rojas-Labanda, S. The equivalent static loads method for structural optimization does not in general generate optimal designs. Struct Multidisc Optim 58, 139–154 (2018). https://doi.org/10.1007/s00158-017-1884-0

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Keywords

  • Structural optimization
  • Equivalent static loads method
  • Convergence properties
  • Sensitivity analysis