Méthode de Newton

vignette|Une itération de la méthode de Newton. En analyse numérique, la méthode de Newton ou méthode de Newton-Raphson est, dans son application la plus simple, un algorithme efficace pour trouver numériquement une approximation précise d'un zéro (ou racine) d'une fonction réelle d'une variable réelle. Cette méthode doit son nom aux mathématiciens anglais Isaac Newton (1643-1727) et Joseph Raphson (peut-être 1648-1715), qui furent les premiers à la décrire pour la recherche des solutions d'une équation polynomiale. Thomas Simpson (1710-1761) élargit considérablement le domaine d'application de l'algorithme en montrant, grâce à la notion de dérivée, comment on pouvait l'utiliser pour calculer une solution d'une équation non linéaire, pouvant ne pas être un polynôme, et d'un système formé de telles équations. Présentation Sous sa forme moderne, l'algorithme peut être présenté brièvement comme suit : à chaque itération, la fonction dont on cherche un zéro est linéarisée en l'

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Hydrocontest – Computational Fluid Dynamics of Hydrofoils

Fabio Stradelli

HYDROcontest is a challenge open to students from 13 universities from all over the world. The aim of the competition is to design the motorboat that best fulfills the tradeoff between high speed and low energy consumption. In order to optimize the shape of the hydrofoils sustaining the boat, suitable Computational Fluid Dynamics models and parallel simulations were employed. The variational multiscale (VMS) model proposed by Bazilevs et al. (2009) was linearized by means of a Newton method, discretized in time through a fully-implicit scheme, transformed into a dimensionless form and implemented in C++, using the Finite Elements library LifeV.

2014

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Interactive, Real Time Structural Simulation

Julien Lebet

The aim of this thesis is to develop a model that provides a fast analysis of structures subjected to large displacements. This is done by transforming the structure in a set of nite particles and by applying a method called dynamic relaxation to it. The latter simulates a dynamic behaviour of the structure by applying Newton's second law to the set of particles until the equilibrium is reached. This can be accomplished using only explicit methods and it does not require to build a global stiness matrix nor to inverse it. The position and rotations of the elements are followed during the process and their forces are always computed locally by using their linear stiness matrix. This methodology is applied to multiple type of elements and the accuracy and speed of the model is nally tested through various examples.

2020

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Inverse problems in tumor modelling

Léo Aurélio Diserens

In this project, we deepen the analysis of a tumour growth model, recently proposed by Garcke et al. in [1]. This model describes tumour and healthy cells evolution as well as tumour cells’ nutrients, mixture velocity and pressure in the domain. Furthermore, it takes into account chemotaxis and apoptosis death of tumour cells, through a system of parabolic nonlinear PDE, that is a Cahn-Hilliard Darcy model, together with an advection-diffusion-reaction equation describing the evolution of nutrients. We perform a dimensional analysis and we build a numerical solver by use of the finite element method in space, a Backward Euler in time and a Newton method to tackle the nonlinearity. We perform several numerical simulations in order to recover results obtained in the article and to catch a general growth of the tumour depending on parameters of interest. Finally, a PDE-constrained optimization problem is formulated and solved, aiming at determining the shape of the tumur after a fixed time from an initial guess of its location. From the numerical simulations we obtained for the nutrients, we notice that the concentration of nutrients in an observable zone around the tumor region could possibly bring enough information to achieve this goal. Therefore, a previous numerical simulation of nutrients will be taken as a target, in order to recover the controlled tumor function, previously simulated numerically. In this respect, preliminary numerical results show that, to some extents, it is possible to identify the general shape of the tumor, even if the exact result of the numerical simulation could not be recovered.

2016

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ME-454: Modelling and optimization of energy systems

The goal of the lecture is to present and apply techniques for the modelling and the thermo-economic optimisation of industrial process and energy systems. The lecture covers the problem statement, the solving methods for the simulation and the single and multi-objective optimisation problems.

MATH-456: Numerical analysis and computational mathematics

The course provides an introduction to scientific computing. Several numerical methods are presented for the computer solution of mathematical problems arising in different applications. The software MATLAB is used to solve the problems and verify the theoretical properties of the numerical methods.

EE-470: Power systems dynamics

This course focuses on the dynamic behavior of a power system. It presents the basic definitions, concepts and models for angular stability analysis with reference to transient stability, steady state stability and long term stability.Fundamentals related to voltage stability are introduced as well.