 Loading... Risorse bibliografiche Risorsa bibliografica obbligatoria Risorsa bibliografica facoltativa
 Scheda Riassuntiva
 Anno Accademico 2023/2024 Scuola Scuola di Ingegneria Industriale e dell'Informazione Insegnamento 096233 - MATHEMATICAL AND NUMERICAL METHODS IN ENGINEERING [I.C.] 096231 - MATHEMATICAL AND NUMERICAL METHODS IN ENGINEERING  Docente Vergara Christian Cfu 7.00 Tipo insegnamento Modulo Di Corso Strutturato

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing Ind - Inf (Mag.)(ord. 270) - BV (478) NUCLEAR ENGINEERING - INGEGNERIA NUCLEARE*AM096296 - NUMERICAL METHODS IN ENGINEERING
Ing Ind - Inf (Mag.)(ord. 270) - MI (471) BIOMEDICAL ENGINEERING - INGEGNERIA BIOMEDICA*AM096233 - MATHEMATICAL AND NUMERICAL METHODS IN ENGINEERING [I.C.]

 Obiettivi dell'insegnamento
 The goal of the course is to provide students with notions and tools about the understanding, the development and the analysis of numerical methods for the approximation of partial differential equations. In particular, the course will treat two families of discretization methods for steady and time dependent problems: finite differences and finite elements. Particular attention will be devoted to simple (initial and boundary value) problems deriving from continuum mechanics, which are prarticularly important for engineers. The learning process will involve lectures on the theory and computer laboratories about the application of simple examples of numerical methods.

 Risultati di apprendimento attesi
 The lectures and the laboratories will provide students with: i) knowledge and understanding (DD1) of  * finite difference approximation of time and space operators (of first and second order); * finite element approximation in space (of first and seond order operators); * the fundamental concepts of: consistency, stability and convergence; ii) the ability to apply the previous knowledge (DD2) to simple examples on the calculator. In particular the student is required to * be able to run and suitably modify a computer script provided by the instructor; * critically discuss the resuls of the numerical experiments above in prespective of the theory; The instructor expects a broad comprehension of the subjects which sould not be limited to the statement of theoretical results. Instead, the acquired knowledge should enable students to express critical judgment and make informed choices on numerical methods for partial differential equations (DD3). The students are expected to express their answers in a mathematically rigorous and clear way.

 Argomenti trattati
 First Part – Finite difference approximation of partial differential equations Review Finite difference formulae to approximate derivatives. Numerical approximation of ordinary differential equations, convergence, absolute stability. 2. First-order conservation laws Approximation with finite differences. Convergence, consistency, zero-stability and absolute stability. Forward Euler-centered scheme. Upwind, Lax-Friedrichs and Lax-Wendroff schemes. Analysis of the schemes, CFL condition and its meaning. Backward Euler-centered scheme. A quick description of systems and of non-linear problems. 3. Diffusion Discretization of the heat equation with finite differences. Implicit and explicit time marching schemes, the theta-method, stability analysis. 4. Laplace-Poisson equation Discretization with finite differences of a one-dimensional elliptic problem. Imposition of the Dirichlet and Neumann boundary conditions. Algebraic formulation and matrix properties. Diffusion-convection and diffusion-reaction problems. 5. Wave equation Discretization of the wave equation with finite difference explicit and implicit schemes. Leapfrog and Newmark schemes. Stability properties.   Second Part – Variational formulations and discretizations via finite element method. 6. Weak formulation and Finite Elements approximation of stationary problems Bilinear form, abstract variational problems and Lax-Milgram lemma. Variational formulation of elliptic problems and applications to transport-reaction-diffusion equations. Introduction to the Galerkin method for a one-dimensional elliptic problem. Consistency, stability and convergence. Cea' Lemma. The finite elements method. Linear and quadratic finite elements. Definition of Lagrangian basis functions, of composite interpolation and error estimates. Extension to the 2D case. Approximation of the diffusion-convection-reaction problem: comparison with the finite difference case and stability analysis. Stabilization with the upwind strategy and the mass lumping technique. 7. Evolution problems Approximation with the Galerkin method, the semi-discrete problem. Explicit and implicit time marching schemes, the theta-method. Stability properties. A quick description of finite elements for hyperbolic problems.

 Obiettivi di sviluppo sostenibile - SDGs
 Questo insegnamento contribuisce al raggiungimento dei seguenti Obiettivi di Sviluppo Sostenibile dell'Agenda ONU 2030: SDG9 - INDUSTRY, INNOVATION AND INFRASTRUCTURE

 Prerequisiti
 We recommend that students who attend this course have knowledge of linear algebra and numerical analysis, in particular: * numerical solution of linear systems (by direct and iterative methods); * polynomial interpolation; * basic numerical methods for the approximation of ordinary differential equations;

 Modalità di valutazione

 Bibliografia A. Quarteroni, Numerical Models for Differential Problems (III edition) , Editore: Springer, Anno edizione: 2017, ISBN: 978-3-319-49315-2 Salsa S., Vegni F., Zaretti A., Zunino P.,, A primer on PDEs, Models, Methods, Simulations, Editore: Springer Quarteroni A., Modellistica Numerica per Problemi Differenziali, Editore: Springer, Anno edizione: 2012

 Software utilizzato
 Software Info e download Virtual desktop Ambiente virtuale fruibile dal proprio portatile dove vengono messi a disposizione i software specifici per all¿attività didattica PC studente Indica se è possibile l'installazione su PC personale dello studente Aule Verifica se questo software è disponibile in aula informatizzata Altri corsi Verifica se questo software è utilizzato in altri corsi FreeFem++ Vedi sito produttore SI SI MATHWORKS Matlab SI SI

 Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lezione
42:00
63:00
Esercitazione
28:00
42:00
Laboratorio Informatico
0:00
0:00
Laboratorio Sperimentale
0:00
0:00
Laboratorio Di Progetto
0:00
0:00
Totale 70:00 105:00

 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
 schedaincarico v. 1.8.3 / 1.8.3 Area Servizi ICT 28/09/2023