An abstract existence result for generalised Hughes’ model
Presented in Roma, Bangalore, Rouen and Tours
I am a PhD student at Institut Denis Poisson, Université de Tours since September 2021.
My research mostly focus on first order non-linear equations, you can find more about it in the research section.
These equations can model the pedestrian flow in evacuation contexts. You can find results of simulations in the simulation section.
Apart from the theoretical research, I also create synthesizers and scenography. These activities have their respective sections.
I teach mathematics and computer science in the university of Tours and the Supinfo school. Here is a list of the subject I taught:
In the one-dimensional setting, the Hughes' model simulate the evacuation of a corridor ( here we choose the corridor to be the domain \( [-1,1] \) ) with exits on the right and the left. It can be rewritten as:
\[\left\{ \begin{matrix} \partial_t \rho(t,x) + \partial_x \left[ \textrm{sign} (x - \xi(t)) f(\rho(t,x)) \right] = 0 \\
\int_{-1}^{\xi(t)} c(\rho(t,x)) \textrm{d} x = \int_{\xi(t)}^1 c(\rho(t,x)) \textrm{d} x. \end{matrix} \right.\]
Here \( f(\rho) = \rho v(\rho) \) is the flux function corresponding to Lighthill-Whitham-Richards (LWR) model and \(v\) corresponds to the speed of pedestrians.
Also, \( c(\rho) \) is the running cost paid by agents to travel in a location with density \( \rho \).
The unknown \( \xi \) named the "turning curve" corresponds to the point in space where pedestrian pay the same total cost to leave the corridor through the left or the right exit.
Below, two simulations of this model using a finite volume scheme. The turning curve \(\xi\) is in red and the density is plotted on \([-1.5,1.5]\) even if the dynamics of the turning curve only depends on what's inside \([-1,1]\).
In the two-dimensional setting, the Hughes' model takes the following form:
\[\left\{ \begin{matrix} \partial_t \rho(t,x) + \textrm{div} \left( \vec{V}(x) f(\rho(t,x)) \right) = 0 \\
\left|\nabla u(x) \right| = c(\rho(t,x)) \\
\vec{V}(x) = - \frac{\nabla u(x)}{|\nabla u (x)|}. \end{matrix}\right.\]
Here \( u \) is a solution of the Eikonal equation with a Dirichlet condition \(u=0\) at the exits.
Below is a simulation of the 2D Hughes model where the exits are represented in green and the density is represented by a gradient from red (high dentity) to black (low density).
The musical instruments I make are synthesizers inspired by wind instruments. The Polyclav is a numeric modular synth made to be compatible with the DM48 midi harmonica from Lekholm instruments. The Soubasynth is a synth inspired by the sousaphone. The Brassynth is an improved and portable version of the Soubasynth compatible with trombone and trumpet mouthpieces. The Soubasynth and the Brassynth both use a Bela board. This constructor made a post about my instruments on their blog.
You can contact me by e-mail at theo.girard /@/ univ-tours.fr
Bureau : E1350 bâtiment E2
Faculté Sciences et techniques, Campus Grandmont
40 avenue Monge
37200 Tours, France