Master of science-level of the Bologna process in Ingegneria Elettronica (Electronic Engineering) - Torino Master of science-level of the Bologna process in Ingegneria Energetica E Nucleare - Torino Master of science-level of the Bologna process in Nanotechnologies For Icts (Nanotecnologie Per Le Ict) - Torino/Grenoble/Losanna
The course is taught in English.
The main teaching goal is to guide the students to acquire the expertise needed to efficiently perform the numerical simulations that, in subsequent courses or in their Master thesis, they might have to deal with. To this end, the description and analysis of the finite element method will be presented, as well as its application to classical engineering problems represented by linear partial differential equations.
The necessary knowledge for setting out and solving engineering problems by means of Matlab computing programs will also be provided.
The course is taught in English.
The main teaching goal is to guide the students to acquire the expertise needed to efficiently perform the numerical simulations that, in subsequent courses or in their Master thesis, they might have to deal with. To this end, the description and analysis of the finite element method will be presented, as well as its application to classical engineering problems represented by linear partial differential equations.
The necessary knowledge for setting out and solving engineering problems by means of Matlab computing programs will also be provided.
Ability to recognize and classify a partial differential equation underlying a given mathematical model. Knowledge of the basic finite element methods for the solution of engineering problems described by differential equations. Skill to select or construct efficient numerical methods for solving PDE engineering problems, using the Matlab software.
Ability to recognize and classify a partial differential equation underlying a given mathematical model. Knowledge of the basic finite element methods for the solution of engineering problems described by differential equations. Skill to select or construct efficient numerical methods for solving PDE engineering problems, using the Matlab software.
Basic notions of linear algebra, calculus, numerical methods and Matlab programming.
Basic notions of linear algebra, calculus, numerical methods and Matlab programming.
INTRODUCTION (4 hours)
Lectures:
- Generalities on Partial Differential Equations (PDE) and on numerical methods.
Examples of classical PDE problems, in particular with reference to the electronic engineering and
nanotechnologies. Stationary problems, eigenvalue problems and evolution problems.
Classification of second order equations. Boundary conditions on the physical domain, initial conditions for
nonstationary problems. Strong and weak solutions.
- A short description of numerical methods alternative to Finite Elements.
STATIONARY PROBLEMS (41 hours)
Lectures and problem sessions (30.5 hours):
- The Finite Element Method (FEM) for second order linear elliptic problems defined on bounded intervals or
on bounded plane domains (25 hours).
Sobolev spaces and weak formulations of differential problems. Integration by parts and Green’s formulas.
Problem space domain discretization. Finite elements and finite element spaces. Definitions and examples.
Some main results on approximation theory.
Application of the FEM to linear elliptic problems, first defined on bounded intervals, and then on bounded
plane domains, with different types of boundary conditions (Dirichlet, Neumann, Robin, mixed-type, of
periodicity).
Some basic convergence results.
Eigenvalue problems.
- Construction of the FEM stiffness matrix and solution of the associated linear system (5.5 hours)
Numerical integration formulas. Conditioning of a linear system. Basic methods for the numerical solution of
linear systems.
Iterative methods for large sparse linear systems; in particular, preconditioned conjugate gradient and GMRes
methods.
Computer Lab sessions (10.5 hours):
- Numerical solution of engineering problems using the Matlab software.
FEM FOR LINEAR TIME-DEPENDENT PROBLEMS (15 hours)
Lectures and problem sessions (10.5 hours):
- Weak formulations in the physical domain of parabolic and hyperbolic problems. Application of the FEM.
Basic on initial value ordinary differential equation (ODE) problems. Stiff systems. Some basic numerical
methods.
Finite difference schemes for the time discretization of ODE systems generated after performing the (space)
FEM discretization.
Computer Lab sessions (4.5 hours):
- Numerical solution of engineering problems using the Matlab software.
INTRODUCTION (4 hours)
Lectures:
- Generalities on Partial Differential Equations (PDE) and on numerical methods.
Examples of classical PDE problems, in particular with reference to the electronic engineering and
nanotechnologies. Stationary problems, eigenvalue problems and evolution problems.
Classification of second order equations. Boundary conditions on the physical domain, initial conditions for
nonstationary problems. Strong and weak solutions.
- A short description of numerical methods alternative to Finite Elements.
STATIONARY PROBLEMS (41 hours)
Lectures and problem sessions (30.5 hours):
- The Finite Element Method (FEM) for second order linear elliptic problems defined on bounded intervals or
on bounded plane domains (25 hours).
Sobolev spaces and weak formulations of differential problems. Integration by parts and Green’s formulas.
Problem space domain discretization. Finite elements and finite element spaces. Definitions and examples.
Some main results on approximation theory.
Application of the FEM to linear elliptic problems, first defined on bounded intervals, and then on bounded
plane domains, with different types of boundary conditions (Dirichlet, Neumann, Robin, mixed-type, of
periodicity).
Some basic convergence results.
Eigenvalue problems.
- Construction of the FEM stiffness matrix and solution of the associated linear system (5.5 hours)
Numerical integration formulas. Conditioning of a linear system. Basic methods for the numerical solution of
linear systems.
Iterative methods for large sparse linear systems; in particular, preconditioned conjugate gradient and GMRes
methods.
Computer Lab sessions (10.5 hours):
- Numerical solution of engineering problems using the Matlab software.
FEM FOR LINEAR TIME-DEPENDENT PROBLEMS (15 hours)
Lectures and problem sessions (10.5 hours):
- Weak formulations in the physical domain of parabolic and hyperbolic problems. Application of the FEM.
Basic on initial value ordinary differential equation (ODE) problems. Stiff systems. Some basic numerical
methods.
Finite difference schemes for the time discretization of ODE systems generated after performing the (space)
FEM discretization.
Computer Lab sessions (4.5 hours):
- Numerical solution of engineering problems using the Matlab software.
The above lecture contents include also corresponding exercise sessions, for a total of 15 hours. Important aspects of the lecture topics as well as the solution of some problems will be discussed, to help the students to better understand the lectures.
An additional computer room activity (15 hours) is also included. Numerical properties of the numerical methods presented in the lectures will be tested, and some engineering problems will be solved by using the Matlab software.
The above lecture contents include also corresponding exercise sessions, for a total of 15 hours. Important aspects of the lecture topics as well as the solution of some problems will be discussed, to help the students to better understand the lectures.
An additional computer room activity (15 hours) is also included. Numerical properties of the numerical methods presented in the lectures will be tested, and some engineering problems will be solved by using the Matlab software.
Lecture notes, as well as notes on the computer lab sessions, can be downloaded by the students from the Portale della Didattica.
Further reading:
A. Quarteroni, Numerical Models for Differential Problems, Springer Verlag, 2009.
Lecture notes, as well as notes on the computer lab sessions, can be downloaded by the students from the Portale della Didattica.
Further reading:
A. Quarteroni, Numerical Models for Differential Problems, Springer Verlag, 2009.
Modalità di esame: Test informatizzato in laboratorio; Prova scritta (in aula);
Exam: Computer lab-based test; Written test;
...
At the end of the computer lab activities, students are asked to show that they have acquired the ability to solve, using the Matlab software, a given PDE problem. This test lasts 1h 20m and it is worth up to 2 points. The grade one obtains will remain valid for the three exam sessions of the current academic year. During the test, students are only allowed to consult their Lab session notes.
Then, in the regular exam sessions, students must take a written test which lasts for two hours, concerning the entire course programme. This test consists of one question and two exercises. The first question, worth up to 4 points, is devoted to definitions and mathematical properties of the finite element method or of related topics. The goal of this question is to verify if a student knows how to justify the steps required by the FEM, to apply the latter properly.
In each of the following two exercises (worth up to 12 points each), students are asked to apply the finite element method to a given problem, by performing all the required main steps, to show that they have understood how the method works and what kind of approximations it involves.
During this examination, it is forbidden to consult notes or books and electronic devices.
Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.
Exam: Computer lab-based test; Written test;
At the end of the computer lab activities, students are asked to show that they have acquired the ability to solve, using the Matlab software, a given PDE problem. This test lasts 1h 20m and it is worth up to 2 points. The grade one obtains will remain valid for the three exam sessions of the current academic year. During the test, students are only allowed to consult their Lab session notes.
Then, in the regular exam sessions, students must take a written test which lasts for two hours, concerning the entire course programme. This test consists of one question and two exercises. The first question, worth up to 4 points, is devoted to definitions and mathematical properties of the finite element method or of related topics. The goal of this question is to verify if a student knows how to justify the steps required by the FEM, to apply the latter properly.
In each of the following two exercises (worth up to 12 points each), students are asked to apply the finite element method to a given problem, by performing all the required main steps, to show that they have understood how the method works and what kind of approximations it involves.
During this examination, it is forbidden to consult notes or books and electronic devices.
In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.