Topology Optimization of Isolated Response Curves in 3D Geometrically-nonlinear Beam 159 Fig. 2 Example of topology with the 3D MMC framework Bifurcation detection and optimisation A computational framework based on the harmonic balance method is employed to compute the dynamic response. Hill’s stability analysis is performed to detect and locate the potential bifurcations. Similarly to [2], the objective function for the optimisation is based on a bifurcation measure, able to handle multiple bifurcations and to locate them at chosen frequency and/or amplitude levels. It is written as follows: f(T, P(θ))=|T−P|Ψ(θ)+ 1 |T| X τ∈T Yπθ∈P π(θ) −τ τ 1/|P| (3) with T the set of target bifurcation, P(θ) the set of bifurcations predicted for the model parameters θ. The notation |T| represents the number of elements in the set T. Geometry Parameterisation Traditional topology optimisation algorithms, such as SIMP or LSM, require the analytic definition of the gradient to ensure reasonable computational time. As this information is unavailable in our case, and as the problem is nonlinear, non-convex, with many local minima, the choice is made to go for global optimisation algorithms. As they are parametric, the Moving Morphable Component (MMC) framework is employed here to ensure the parametrisation of the beam. Coupled to the global optimisation algorithm, it was proven to be efficient in dealing with topology optimisation for nonlinear vibration mitigation [1]. A 3D individual component is characterised by 9 parameters, which define explicitly a Level-Set Function (LSF) [5]. The global geometry is defined as the union of the unitary LSF. By moving, rotating, and shrinking each component, complex topologies can be described. The LSF is then projected on the initial mesh to define the density field. An example of topology obtained with the MMC framework is given in Figure 2. Global Optimisation The optimisation problem is non-convex with many local minima. For this reason, a global optimisation algorithm is considered. The strategy proposed here couples two algorithms, namely the CMA-ES [9] and the PSO [10]. A preliminary study has been done on the case of linear vibrations, and it appears that the CMA-ES is very robust and efficient to reach approximately the global minima, but then requires many model calls to reach the precise solution. On the other hand, the PSO algorithm is very efficient when initialised near the optimal point. The solution proposed here couples the two algorithms. Acknowledgments Yichang Shen would like to thank HORIZON MSCA Postdoctoral Fellowship action and the UKRI Postdoc Guarantee Fellowship provided by the EPSRC, with Grant Ref: EP/X026027/1. References 1. Denimal, E., Renson, L., Wong, C., and Salles, L. “Topology optimisation of friction under-platform dampers using moving morphable components and the efficient global optimization algorithm”. Structural and Multidisciplinary Optimization, 65(2):56 (2022). 2. Me´lot, A., Denimal, E., and Renson, L. “Multi-parametric optimization for controlling bifurcation structures”. Proceedings of the Royal SocietyA, 480(2283):20230505 (2024). 3. Me´lot, A., Goy, E.D., and Renson, L. “Control of isolated response curves through optimization of codimension-1 singularities”. Computers & Structures, 299:107394 (2024).
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