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 Scheda Riassuntiva
 Anno Accademico 2018/2019 Scuola Scuola di Ingegneria Industriale e dell'Informazione Insegnamento 096233 - MATHEMATICAL AND NUMERICAL METHODS IN ENGINEERING [I.C.] Docente Cipriani Fabio Eugenio Giovanni , Vergara Christian Cfu 12.00 Tipo insegnamento Corso Integrato

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
096295 - MATHEMATICAL METHODS IN ENGINEERING
Ing Ind - Inf (Mag.)(ord. 270) - MI (471) BIOMEDICAL ENGINEERING - INGEGNERIA BIOMEDICA*AM096233 - MATHEMATICAL AND NUMERICAL METHODS IN ENGINEERING [I.C.]
Ing Ind - Inf (Mag.)(ord. 270) - MI (491) MATERIALS ENGINEERING AND NANOTECHNOLOGY - INGEGNERIA DEI MATERIALI E DELLE NANOTECNOLOGIE*AM096296 - NUMERICAL METHODS IN ENGINEERING

 Obiettivi dell'insegnamento
 The goal of the course is twofold. First, we present some classical differential models of Mathematical Physics, developing and analyzing analytical methods for the computation of the solutions and some finite difference schemes for their approximation. Second, we introduce the variational formulation of some boundary value problems, together with the finite element method for their numerical approximation. The course is characterized by a constant synergy between modeling, theoretical aspects and numerical simulation.

 Risultati di apprendimento attesi
 The lectures and the laboratories will provide students with: i) knowledge and understanding of* the main properties of the solutions of some classes of linear partial differential equations;* the basic tools used for solving linear partial differential equations;* modeling of some physical problems* 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 to simple examples. In particular the student is required to* solve linear partial differential equations analytically and numerically* discuss the properties of mathematical models* be able to run and suitably modify a computer script provided by the instructor;* critically discuss the resuls of the numerical experiments above in perspective of the theory. The instructors expect a broad comprehension of the subjects which should not be limited to the statement of theoretical results.Instead, the acquired knowledge should enable the students to express critical judgment and make informed choices on analytical and numerical methods for partial differential equations. The students are expected to express their answers in a mathematically rigorous and clear way.

 Argomenti trattati
 First Part – Differential modeling and finite difference approximation Review Mathematical Methods: Differential calculus for functions of several real variables. Series of functions. Numerical Methods: Finite difference formulae to approximate derivatives. Numerical approximation of ordinary differential equations, convergence, absolute stability. First-order conservation laws Mathematical Methods: Transport equation. Traffic flow models. Method of characteristics. Rankine-Hugoniot relation. Shock and rarefaction waves. Entropy condition. Numerical Methods: 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. Diffusion Mathematical Methods: Heat equation. Well-posed problems. Separation of variables. Maximum principles. Fundamental solution. Cauchy problem in a strip and in the half-space with several boundary conditions (isolation, transmission, mixed). Duhamel principle. Numerical Methods: Discretization of the heat equation with finite differences. Implicit and explicit time marching schemes, the theta-method, stability analysis. Laplace-Poisson equation Mathematical Methods: Harmonic functions. Mean value properties. Maximum principles. Well-posed problems. Poisson’s formula for the disk. Newtonian potentials. Numerical Methods: 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. Wave equation Mathematical Methods: String equations. Well-posed problems and separation of variables. D’Alembert formula. Numerical Methods: Discretization of the wave equation with finite difference explicit and implicit schemes. Leapfrog and Newmark schemes. Stability properties.   Second Part – Functional Analysis, variational formulations and discretizations via finite element method. Basics of Functional Analysis Mathematical Methods: Lebesgue integral. Projection theorem and Riesz representation theorem. Schwartz distributions. Sobolev spaces. Weak formulation and Finite Elements approximation of stationary problems Numerical Methods: 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. Evolution problems Numerical Methods: 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.

 Prerequisiti
 We recommend that students who attend this course have knowledge of linear algebra, calculus and numerical analysis, in particular: * differential calculus for functions of several real variables;* surfaces and integral surfaces, the divergence theorem;* series of functions, Fourier series;* linear second order ordinary differential equations;* 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
 Quarteroni A., Modellistica Numerica per Problemi Differenziali, Editore: Springer Salsa S., Vegni F., Zaretti A., Zunino P., A primer on PDEs, Models, Methods, Simulations, Editore: Springer

 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 Docenti FreeFem++ Vedi sito produttore SI SI Vergara Christian MATHWORKS Matlab SI SI Vergara Christian

 Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lezione
75:36
113:24
Esercitazione
44:24
66:36
Laboratorio Informatico
0:00
0:00
Laboratorio Sperimentale
0:00
0:00
Laboratorio Di Progetto
0:00
0:00
Totale 120:00 180:00

 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. 1.6.8 / 1.6.8 Area Servizi ICT 04/08/2021