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

Adrien Mélot (1) , Enora Denimal (2) , Ludovic Renson (3)

1 I4S - Statistical Inference for Structural Health Monitoring
2 PLATON - Uncertainty Quantification in Scientific Computing and Engineering
3 Imperial College London

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.

Mots clés: Harmonic balance method; Nonlinear vibrations; Bifurcation tracking; Singularity; Stability; Isola

Domaines: Science non linéaire [physics]; Mathématiques [math]; Physique [physics]; Mécanique [physics]; Mécanique des structures [physics.class-ph]; Vibrations [physics.class-ph]

Adrien Mélot, Enora Denimal, Ludovic Renson. Control of isolated response curves through optimization of codimension-1 singularities. Computers & Structures, In press, pp.1-19. ⟨10.1016/j.compstruc.2024.107394⟩. ⟨hal-04555084⟩