Ing Ind - Inf (Mag.)(ord. 270) - BV (478) NUCLEAR ENGINEERING - INGEGNERIA NUCLEARE

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097725 - MATHEMATICAL AND PHYSICAL MODELING IN ENGINEERING [C.I.]

Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA

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097725 - MATHEMATICAL AND PHYSICAL MODELING IN ENGINEERING [C.I.]

Obiettivi dell'insegnamento

The main objective of the course is to acquire knowledge of mathematical and physical modeling of physical systems and to develop critical skills for their application in solving engineering problems.

The first goal of part [2] of the course is to make the student understand, both in a theoretical and an application oriented setting, the construction of perturbative approaches with a rigorous application of asymptotic series. A second objective is to analyse some microscopic models, with particular emphasis on phase transitions and Monte Carlo methods.

Risultati di apprendimento attesi

The student is expected:

- to master the mathematical develoment of all the mathematical and physical methods exposed during the course;

- to use dimensional analysis and variational principles to model the governing equations at the physical problem at hand;

- to develop the critical skills for selecting the most adapted method and for providing a rigorous derivation of the required mathematical solution;

- to use widespreadly adopted symbolic programming languages for object and modeling management (e.g. Mathematica);

- to become articulate and fluent in expressing and communicating all the key ideas and proofs of the basic theorems and concepts.

Argomenti trattati

Asymptotic Analysis. Classification of singular points in ordinary differential equations. Irregular singular points: dominant balance method. Introduction to regular and singular perturbation theory.

Asymptotic regimes. Dimensionless Analysis: Buckingham theorem. Examples of dimensionless number in physical system models. Boundary layer theory. Matching conditions: Van Dyke rule and intermediate variable method. Multiple-scale method: perturbation on unbounded domains, nonlinear oscillations.

Microscopic models. Equilibrium statistical mechanics in a nutshell, microcanonical and canonical ensemble, Maxwell-Boltzmann distribution. Ising model as a paradigmatic example. Paramagnetism, lattice gas, Lebwohl-Lasher model. Mean-field model, Landau theory, phase transitions. Monte Carlo method, importance sampling, Markov chain Monte Carlo, Metropolis algorithm. Simulation of the Ising model. Critical slowing down. Ehrenfest urn model for irreversibility, Vicsek model for active materials.

Prerequisiti

Basic ideas of calculus. Numerical series. Theory of ordinary differential equations and basic notions about partial differential equations. Elementary knowledege of classical Rational Mechanics and Continuum Mechanics (forces, balance equations, Cauchy stress tensor, kinematics of continua).

Modalità di valutazione

The exams wil be organized in agreement with the Academic Calendar and the Rules of the School of Engineering.

The verification of knowledge and capabilities acquired by the student is based, for each part, on a written test of about 2h, followed by an oral part.

The two written tests can be taken together during the same exam session, or separately in different exam sessions.

The final grade is based on an overall evaluation of the two parts of the exam.

The verification of knowledge and capabilities acquired by the student in part [2] of the course is based on a written test of about 2h, followed by an oral part or a presentation.

The written test consists in two exercises based on physical problems, in which the student is expected to provide answers to few itemized questions.

In the oral part the student is expected to answer theoretical questions on course topics, and to verify student's skills at making connections between course topics and previous courses.

The teachers might place lecture notes in the Beep area of the Course.

Bender, Carl M., Orszag, Steven A., Advanced mathematical methods for scientists and engineers , Editore: Springer, ISBN: 978-1-4757-3069-2
Madras, Neal, Lectures on Monte Carlo Methods, Editore: American Mathemataical Society, Anno edizione: 2002, ISBN: 978-0821829783

Forme didattiche

Tipo Forma Didattica

Ore di attività svolte in aula

(hh:mm)

Ore di studio autonome

(hh:mm)

Lezione

30:00

45:00

Esercitazione

20:00

30:00

Laboratorio Informatico

0:00

0:00

Laboratorio Sperimentale

0:00

0:00

Laboratorio Di Progetto

0:00

0:00

Totale

50:00

75:00

Informazioni in lingua inglese a supporto dell'internazionalizzazione

Insegnamento erogato in lingua
Inglese

Disponibilità di materiale didattico/slides in lingua inglese

Disponibilità di libri di testo/bibliografia in lingua inglese

Possibilità di sostenere l'esame in lingua inglese

Disponibilità di supporto didattico in lingua inglese