Adrien Mélot

Selected Publications

1. CAS

   Control of isolated response curves through optimization of codimension-1 singularities

   A. Mélot, E. Denimal Goy, and L. Renson

   Computers & Structures 2024

   Abs Bib

   We introduce a computational framework for controlling the location of isolated response curves, i.e. responses that are not connected to the main solution branch and form a closed curve in parameter space. The methodology relies on bifurcation tracking to follow the evolution of fold bifurcations in a codimension-2 parameter space. Singularity theory is used to distinguish points of isola formation and merger from codimension-2 bifurcations and an optimization problem is formulated to delay or advance the onset or merger of isolated response curves or control their position in the state/parameter space. We illustrate the methodology on three examples: a finite element model of a cantilever beam with cubic nonlinearity at its tip, a two-degree-of-freedom oscillator with asymmetry and a two-degree-of-freedom base-excited oscillator exhibiting multiple isolas. Our results show that the location of points of isola formation and mergers can effectively be controlled through structural optimization.

   @article{melotOptiIRC,
     title = {{Control of isolated response curves through optimization of codimension-1 singularities}},
     author = {Mélot, A. and Denimal Goy, E. and Renson, L.},
     year = {2024},
     doi = {https://doi.org/10.1016/j.compstruc.2024.107394},
     journal = {Computers \& Structures}
   }

2. RSPA

   Multi-parametric optimization for controlling bifurcation structures

   A. Mélot, E. Denimal, and L. Renson

   Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 2024

   Abs Bib

   Bifurcations organise the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters (>10²). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite element model of a clamped-free Euler-Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.

   @article{melotOptiBif,
     title = {{Multi-parametric optimization for controlling bifurcation structures}},
     author = {Mélot, A. and Denimal, E. and Renson, L.},
     year = {2024},
     doi = {10.1098/rspa.2023.0505},
     journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}
   }

3. Int. J. Non-Linear Mech.

   Robust design of vibro-impacting geared systems with uncertain tooth profile modifications via bifurcation tracking

   A. Mélot, E. Rigaud, and J. Perret-Liaudet

   International Journal of Non-Linear Mechanics 2023

   Abs Bib

   The present work investigates the influence of uncertain tooth profile modifications on the nonlinear dynamic response of a spur gear pair induced by a backlash nonlinearity. To this end, an original approach based on bifurcation tracking is developed. The equations of motion are solved in the frequency domain with the harmonic balance method (HBM) coupled to an arc-length continuation algorithm and a bordering technique. The evolution of the bifurcation points with respect to the uncertain parameter is computed in a deterministic way. The study focuses on minimizing the amplitude-jump instabilities induced by the backlash nonlinearity around the primary resonance peak. The proposed methodology allows for a fast and reliable estimation of the tooth profile modification that minimizes the amplitude-jump instability by defining two criteria using the results of the bifurcation tracking algorithm. Probability density functions (PDF) of various indicators of the severity of vibro-impacts can be computed with Monte-Carlo (MC) simulation with minimal computational burden. Results show that the tooth profile modification that minimizes the amplitude-jump instabilities differs from the optimum obtained with static computations.

   @article{melotRobust,
     author = {M{\'e}lot, A. and Rigaud, E. and Perret-Liaudet, J.},
     title = {Robust design of vibro-impacting geared systems with uncertain tooth profile modifications via bifurcation tracking},
     journal = {International Journal of Non-Linear Mechanics},
     volume = {149},
     pages = {104336},
     year = {2023},
     issn = {0020-7462},
     doi = {https://doi.org/10.1016/j.ijnonlinmec.2022.104336},
   }

4. Nonlinear Dyn.

   Bifurcation tracking of geared systems with parameter-dependent internal excitation

   A. Mélot, E. Rigaud, and J. Perret-Liaudet

   Nonlinear Dynamics 2022

   Abs Bib

   We herein propose an algorithm for tracking smooth bifurcations of nonlinear systems with interdependent parameters. The approach is based on a complex formulation of the well-known harmonic balance method (HBM). Hill’s method is used to assess the stability of the computed forced response curves, and a minimally extended system is built to allow for the parametric continuation of the detected bifurcation points. The feasibility of coupling HBM-based minimally extended systems and arc-length continuation algorithms is established and demonstrated. The method offers an efficient way of determining the stability regions of the system. The methodology is applied on a spur gear pair model including the backlash nonlinearity and subjected to transmission error and mesh stiffness fluctuation whose harmonic contents depend on several parameters that do not appear explicitly in the equations of motion.

   @article{melot_bifurcation_2022,
     title = {Bifurcation tracking of geared systems with parameter-dependent internal excitation},
     volume = {107},
     issn = {1573-269X},
     doi = {https://doi.org/10.1007/s11071-021-07018-6},
     number = {1},
     journal = {Nonlinear Dynamics},
     author = {M{\'e}lot, A. and Rigaud, E. and Perret-Liaudet, J.},
     year = {2022},
     pages = {413--431},
   }