Servizi per la didattica

PORTALE DELLA DIDATTICA

03RMEND

A.A. 2019/20

Course Language

Inglese

Course degree

Master of science-level of the Bologna process in Energy And Nuclear Engineering - Torino

Course structure

Teaching | Hours |
---|---|

Lezioni | 21 |

Esercitazioni in aula | 3 |

Esercitazioni in laboratorio | 36 |

Teachers

Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
---|---|---|---|---|---|---|---|

Subba Fabio | Professore Associato | ING-IND/19 | 21 | 3 | 0 | 0 | 1 |

Teaching assistant

Context

SSD | CFU | Activities | Area context |
---|---|---|---|

ING-IND/19 | 6 | D - A scelta dello studente | A scelta dello studente |

2019/20

The course focuses on the solution of 1D and 2D steady-state and transient heat conduction problems. These were chosen both as one of the fundamental classes of problems in energy applications and as a paradigm for the general case of elliptic and parabolic partial differential equations (PDE). Both problems are addressed using both the finite difference and the finite element approach. The fundamental tool for the solution of 1D heat conduction problems is MATLAB, to which the initial part of the lab classes is devoted. The chosen tool for the solution of 2D heat conduction problems is the Freefem++ freeware, to which the final part of the lab classes is devoted.

The course focuses on the solution of 1D and 2D steady-state and transient heat conduction problems. These were chosen both as one of the fundamental classes of problems in energy applications and as a paradigm for the general case of elliptic and parabolic partial differential equations (PDE). Both problems are addressed using both the finite difference and the finite element approach. The fundamental tool for the solution of 1D heat conduction problems is MATLAB, to which the initial part of the lab classes is devoted. The chosen tool for the solution of 2D heat conduction problems is the Freefem++ freeware, to which the final part of the lab classes is devoted.

Through this course the student is expected to acquire: - A feeling of the importance/relevance of the numerical, as opposed to analytical, solution of engineering problems such as that of heat conduction - A good knowledge of the finite difference and finite elements methods for solution of the above-mentioned problems, - The ability to implement and solve them using MATLAB and Freefem++, - The ability to critically and quantitatively assess the accuracy of the results obtained with the computer (quality assurance).

Through this course the student is expected to acquire: - A feeling of the importance/relevance of the numerical, as opposed to analytical, solution of engineering problems such as that of heat conduction - A good knowledge of the finite difference and finite elements methods for solution of the above-mentioned problems, - The ability to implement and solve them using MATLAB and Freefem++, - The ability to critically and quantitatively assess the accuracy of the results obtained with the computer (quality assurance).

The knowledge acquired in the following BSc courses (or equivalent ones) will be needed: Calculus (Analisi matematica I e II, Geometria), Computer science (Informatica), Applied thermodynamics and heat transfer (Termodinamica applicata e trasmissione del calore), with particular reference to vector and matrix algebra, to the solution of ordinary differential equations, to the basic elements of programming, numerical analysis, and to steady-state and transient problems of heat conduction.

The knowledge acquired in the following BSc courses (or equivalent ones) will be needed: Calculus (Analisi matematica I e II, Geometria), Computer science (Informatica), Applied thermodynamics and heat transfer (Termodinamica applicata e trasmissione del calore), with particular reference to vector and matrix algebra, to the solution of ordinary differential equations, to the basic elements of programming, numerical analysis, and to steady-state and transient problems of heat conduction.

The 1D steady-state heat conduction problem
• Finite difference approximation of derivatives
• Imposing boundary conditions
• Algebraic approximation of the original ordinary differential equation
• Concepts of accuracy and mesh independence
• Solution of 1D steady state problems in Cartesian and radial coordinates using MATLAB The 1D transient heat conduction problem
• Fundamental solution of the heat conduction problem
• The method of lines as a general approach to the solution of initial-boundary value PDEs
• Numerical schemes for time marching
• Solution of 1D transient problems in Cartesian and radial coordinates using MATLAB The 2D heat conduction problem
• Weak formulation
• Imposing boundary conditions
• Finite element vs. finite difference approximation
• Concepts of mesh generation/triangulation
• Quadrature formulae
• Solution of 2D steady state and transient problems in Cartesian and cylindrical coordinates using Freefem++.

The 1D steady-state heat conduction problem
• Finite difference approximation of derivatives
• Imposing boundary conditions
• Algebraic approximation of the original ordinary differential equation
• Concepts of accuracy and mesh independence
• Solution of 1D steady state problems in Cartesian and radial coordinates using MATLAB The 1D transient heat conduction problem
• Fundamental solution of the heat conduction problem
• The method of lines as a general approach to the solution of initial-boundary value PDEs
• Numerical schemes for time marching
• Solution of 1D transient problems in Cartesian and radial coordinates using MATLAB The 2D heat conduction problem
• Weak formulation
• Imposing boundary conditions
• Finite element vs. finite difference approximation
• Concepts of mesh generation/triangulation
• Quadrature formulae
• Solution of 2D steady state and transient problems in Cartesian and cylindrical coordinates using Freefem++.

24 h of standard lectures, combined with a total of 36 h of computational lab (3+ h per week). Under the guidance of professor and teaching assistants, the students will address the solution of problems of increasing complexity.

24 h of standard lectures, combined with a total of 36 h of computational lab (3+ h per week). Under the guidance of professor and teaching assistants, the students will address the solution of problems of increasing complexity.

- Notes by the teacher
- MATLAB and Freefem++ user manuals.
- Selected chapters from:
• J. Cooper, “Introduction to Partial Differential Equations with MATLAB” (Springer, 2008)
• C. Johnson, “Numerical solutions of PDEs by the finite element method” (Cambridge UP, 1987)

- Notes by the teacher
- MATLAB and Freefem++ user manuals.
- Selected chapters from:
• J. Cooper, “Introduction to Partial Differential Equations with MATLAB” (Springer, 2008)
• C. Johnson, “Numerical solutions of PDEs by the finite element method” (Cambridge UP, 1987)

Exam: written test; optional oral exam;
The assessment is made by written exam. Each student works on a PC in the lab and is asked to: 1) solve different numerical problems, using Matlab and/or Freefem++, and summarizing the results in the form of suitable plots; 2) justify the choice of the methods used for the solution; 3) discuss the quality/accuracy of the obtained numerical solution. These three items, collected by the student in a short report (doc file), contribute as follows to the final grade: 1) 70%; 2) 10%; 3) 20%. If the mark in the written exam is ≥ 28, an oral follow-up is foreseen

1) computer lab-based test (mandatory) +
2) oral discussion (access based on results of the mandatory part)
In 1) Each student works on a PC in the lab using vLAIB (virtual LAIB), integrated in the Exam platform with proctoring systems (Respondus), and is asked to:
a) Solve some numerical problems, using Matlab and/or Freefem++ scripts, and summarizing the results in the form of suitable plots (weight in the final score: 70%); the scripts must be uploaded and will be evaluated;
b) If required, justify the choice of the methods used for the solution as a comment in the script (weigth in the final score: 10%);
c) If required, discuss the quality/accuracy of the obtained numerical solution as a comment in the script (weight in the final score: 20%).
The maximum score allowed for the written test is 30, its duration is ~2 hours.
If the score obtained in part 1) is not higher than 27, this will be coincident with the final grade. Students scoring 28 or more will be required to participate to an oral discussion. The grade obtainable in the oral discussion can reach “30 cum laude”.

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Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY