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 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.

- 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: Computer lab-based test; Optional oral exam;

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”.

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.