Influence of damage on the global stability of nonlinear systems

Job description

Contexte et atouts du poste

Inria is the French National Research Institute for Digital Science and Technology.

As a center of scientific excellence, it oversees
the French Digital Programs Agency and stands at the forefront of Europe’s digital transformation.

The institute conducts world-
class research across various disciplines, fostering international and industrial collaborations, pioneering innovations, software
development, artificial intelligence, quantum and cybersecurity technologies, and deep-tech entrepreneurship.

Globally, Inria ranks
16th among top AI research organizations and holds the position of Europe’s leading institute for frontier research in digital sciences

Mission confiée

Nonlinear systems are ubiquitous in modern engineering, including turbomachinery, aerospace structures, robotic manipulators,
and devices designed to harvest or absorb energy.

These systems often exhibit complex behaviors that challenge our ability to
predict their response to real-world conditions, such as noise or external shocks.

This internship offers a unique opportunity to dive
into this exciting field, working on a cutting-edge project that investigates how damage influences the stability of such systems.

A key feature of nonlinear systems is multistability, where multiple stable states coexist.

For example, a mechanical structure
might settle into different operational states depending on its initial conditions or the disturbances it encounters.

Your role will
be to study basins of attraction, which describe the set of initial conditions that lead a system to a specific stable state over time.
You will numerically analyze how damage, such as cracks, material fatigue, etc, alters these basins and affects the system’s global
stability [1].

This work will involve exploring how damage reshapes the basins’ boundaries and assessing the system’s resilience to
external perturbations, providing insights beyond traditional analyses.

Your work will bridge applied mathematics and mechanical engineering, combining theoretical insights with computational tech-
niques.

You will combine computational tools and explore and develop metrics [2, 3, 4, 5] to tackle questions like: How does damage
shift a system’s behavior?

Can we predict when a damaged structure might fail to recover from a disturbance?

The project involves
numerical simulation and theoretical developments.

Depending on progress and interests, there will also be an opportunity to
contribute to algorithmic and code development aimed at improving the computation of basins of attraction.

Beyond the tech-
nical skills, you will gain experience in interdisciplinary research, working alongside experts to address challenges with real-world
implications, such as improving the safety and reliability of mechanical systems.

This project is ideal for students in applied mathematics or mechanical engineering, with interests in nonlinear dynamics, stability
theory, statistics and/or computational methods.

[1] M.

Stender and N.

Hoffmann, “bstab: An open-source software for computing the basin stability of multi-stable dynamical systems,”
Nonlinear Dynamics, vol.

107, pp.

1451–1468, 2022.

[Online].

Available: P.

Menck, J.

Heitzig, N.

Marwan, and J.

Kurths, “How basin stability complements the linear-stability paradigm,” Nature Physics, vol.

9, pp.
89–92, 2013.

[Online].

Available: M.

Soliman and J.

Thompson, “Integrity measures quantifying the erosion of smooth and fractal basins of attraction,” Journal of Sound and
Vibration, vol.

135, pp.

453–475, 1989.

[Online].

Available: G.

Habib, “Dynamical integrity assessment of stable equilibria: a new rapid iterative procedure,” Nonlinear Dynamics, vol.

106, pp.
2073–2096, November 2021.

[Online].

Available: O.

Avci, O.

Abdeljaber, S.

Kiranyaz, M.

Hussein, M.

Gabbouj, and D.

J.

Inman, “A review of vibration-based damage detection in civil
structures: From traditional methods to machine learning and deep learning applications,” Mechanical Systems and Signal Processing, vol.
147, p.

107077, Jan.

2021.

[Online].

Available:

Principales activités

• Implementing models of nonlinear systems that include damage evolution laws,
• Study multistability and basins of attraction in nonlinear mechanical systems,
• Analyze the evolution of the topology of these basins as a function of the damage level,
• Develop metrics to quantify global stability and system robustness,
• Contribute to research that bridges mathematical/statistical modeling with engineering applications.

Compétences

We are looking for a Master’s or final-year engineering student with a strong interest in applied mathematics, mechanics, and
numerical simulation.

A background in structural dynamics and nonlinear systems, along with programming skills (Julia, Matlab, or
equivalent), would be beneficial.

Scientific curiosity, autonomy, and rigor are essential qualities for this research-oriented project

Avantages

Required Skill Profession

Engineers