Projects

Project 1 - Modelling the effects of an earthquake on buildings

Instructor: Sanja Ružičić - Faculty of Sciences University of Novi Sad

An earthquake is a sudden movement of the ground caused by the release of stress accumulated along geologic faults or by volcanic activity. Some of them have caused a great devastation and destruction of people's properties. It is known that building suffers the greatest damage if a frequency of the ground motion is close to its own frequency. In that case we say that the building and the ground motion are in resonance. Even though the resonance can cause a major damage, its effects can be reduced. So, nowadays it should be imperative to design the buildings that will resist the earthquakes forces and that are more likely to suffer reparable damage. It is natural to assume that the earthquake ground motion is characterized by its duration, displacement from the equilibrium point, velocity and acceleration which is a critical factor in determining how much damage a building will suffer.
The aim of this project is to give the mathematical model and investigate how an earthquake effects the buildings of different structures since different buildings respond in widely differing manners to the same earthquake ground motion. It is expected that some numerical simulations will be performed.

keywords: differential equations; numerical linear algebra; Matlab simulations

Project 2 - Drug delivery into the eye: therapeutic lenses versus intracameral implants

Intrsuctor: José Augusto Ferreira - Centre for Mahematics of University of Coimbra

Glaucoma is one of the most common diseases of the anterior segment of the eye that is characterized by the damage of the optic nerve due to the increase of the intraocular pressure (IOP). The anomalous behaviour of IOP can be a consequence of several physiological alterations of the actors in the aqueous humor dynamics: the ciliary body - where the aqueous humor is produced, and the trabecular meshwork - a porous tissue that is responsible for the aqueous humor drainage.

Topical drops and ointments are the conventional drug delivery formulations to treat eye diseases. However, due to several eye barriers only a small amount of drug reaches the anterior chamber. New drug delivery systems were proposed to avoid eye barriers, like therapeutic contact lens ([1]) that increases the residence time of the drug in the cornea, and biodegradable intracameral implants ([2]) that are placed in the anterior chamber. Both drug delivery systems are made of polymers that lead to a controlled drug release.

This project aims to analyse the efficacy of the delivery systems: a therapeutic lens and a biodegradable intracameral implant. We consider that the increase of IOP is due to the decrease of the porosity of the trabecular meshwork. From mathematical point of view, the efficacy is measured considering the drug mass that reaches the trabecular meshwork. The mathematical models that describe the drug release from these drug delivery systems and their transport until the target are defined by systems of partial differential equations (PDEs) that are completed with interface, boundary and initial conditions. It should be pointed out that the aqueous humor dynamics has an important role in the drug transport in the anterior chamber.

The project will be developed considering two subgroups that treat separately each problem. The students should be familiar with PDEs and numerical methods for PDEs.

Keywords: Drug delivery ; glaucoma ; partial differential equations ; numerical methods

Project 3 - Checking the "evenness" of a floor

Instructor: Thomas Goetz - University of Koblenz-Landau

The company CHEMICON is construction floor in building and the need to ensure that the final floor satisfies the criteria of the DIN standard. Since manual measurements (using the stick and measuring the gap) a extremely time consuming, the want to use a laser scanner taking images of the surface. Based on this ditialized surface image, it has to be checked whether the DIN standard is satiesfied or not. For the given project we do have some real data from actual laser scans of various buildings.

Keywords : Evenness of a surface ; digitalized data ; geometry

Project 4 - Epidemic models for infectious diseases

Instructor: Danijela Rajter-Ciric - Faculty of Sciences University of Novi Sad

The spread of infectious diseases have always been something that scientists have tried to prevent. Apart of medical and biological issues, epidemics and pandemics can have huge economic and social impacts on the society. In order to prevent the spread of a disease, one should first try to understand it. Mathematical models have been widely used in epidemiology. They are very useful for understanding the transmission dynamics of the disease. Many of these models can later be used for prediction of future epidemics (or pandemics) and for investigating the possible impact of medical interventions. The epidemic modelling will be investigated on the example of influenza (or some other disease(s) by students choice).

Project 5 - The zombie invasion - how an epidemic spreads depending on the population density?

Instructor: Jaroslaw Gruszka, Wroclaw University of Science and Technology, Wroclaw, Poland

Regardless of what the majority of people might think of it, there are still thousands who strongly believe that the ultimate end of the human civilisation will be a widespread rise of zombies - hostile to the humankind and willing to entirely destroy it. And - as we all know - wherever we find any indication of humas' uncertainty and doubtfulness, we should also be able to see science entering this area and dispel those doubts. Hence, the aim of this project is to develop a general model of how the zombie invasion could spread over a given piece of land. A very useful tool to do this accurately could be population density maps, easily available on the Internet. It is apparent that the more densely-populated a given area is, the easier it will be for the epidemic to spread. The scope of the project will therefore be to access and digitalise population density maps, devise the model of spreading the epidemic over the area pictured in these maps and create visualisations of the entire epidemic-spreading process. The devised model should be well-parametrised and general enough not only to describe the zombie attack, but also some other types of contagious diseases (epidemic of which can have a bit bigger change of occurring in the real world).

keywords: basics of differential equations and numerical methods of solving them; semi-advanced programming skills in Python; including libraries like numpy, matplotlib and pillow

Project 6 - Detection of resonances for the modelling of resonant tunneling devices.

Instructor: Clément Jourdana - University of Grenoble Alpes

Due to the constant miniaturization of electronic components, physical phenomena generated in such devices are extremely complex. To predict the electronic behavior of devices and design new configurations, numerical simulations can play an important role in complement of experimental tests. This project concerns the modelling of the electron transport through a resonant tunneling nanostructure, whose the mechanism is based on the quantum tunneling effect. We assume that electrons are in a mixed state. Each energy state is described by a one dimensional stationary Schr¨odinger equation. For resonant tunneling devices, the current-voltage characteristic, which is the relation between the electric current through the device and the voltage across its terminals, strongly depends on certain energy states called resonant states. After getting familiar with the model and discussing the discretization of equations, the aim of this project will be to propose an efficient way to detect the resonances analyzing the shape of the transmission coefficient. This detection is essential in view of computing accurate I-V curves without solving a non reasonable number of equations.

keywords: PDE/ODE, numerical analysis, Python/Matlab programming skills

Project 7 - Speaker diarization

Instructor: Jean-Baptiste Durand - Grenoble INP

The task consisting in determining who speaks and when in videos is referred to speaker diarization. In real-life situations, this task is complicated by the fact that speakers may move during the video, and even leave or enter the scene. The general version of the problem consists in detecting the number of people and their positions over time and simultaneously, detect if each of them is speaking or not. In this simplified project, video have been preprocessed using some (imperfect) algorithm, which detects the number of people and positions of their joints. Heatmaps of sound locations are also available. In a training set, labels are assigned to positions and speech occurrences to indicate both who is speaking and to implement tracking.

keywords: Statistical analysis, python programming. Notions in sequence analysis and signal processing are a plus.

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