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Anno Accademico

2017/2018

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 [2]

Cfu 
7.00

Tipo insegnamento

Modulo Di Corso Strutturato

Docenti: Titolare (Cotitolari)

Zunino Paolo,
Vergara Christian

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  *  A  M  096296  NUMERICAL METHODS IN ENGINEERING  Ing Ind  Inf (Mag.)(ord. 270)  MI (471) BIOMEDICAL ENGINEERING  INGEGNERIA BIOMEDICA  *  A  M  096233  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  *  A  M  096296  NUMERICAL METHODS IN ENGINEERING 
Programma dettagliato e risultati di apprendimento attesi 
First Part – Differential modeling and finite difference approximation
 Review
Finite difference formulae to approximate derivatives. Numerical approximation of ordinary differential equations, convergence, absolute stability.
 Firstorder conservation laws
Approximation with finite differences. Convergence, consistency, zerostability and absolute stability. Forward Eulercentered scheme. Upwind, LaxFriedrichs and LaxWendroff schemes. Analysis of the schemes, CFL condition and its meaning. Backward Eulercentered scheme. A quick description of systems and of nonlinear problems.
 Diffusion
Discretization of the heat equation with finite differences. Implicit and explicit time marching schemes, the thetamethod, stability analysis.
 LaplacePoisson equation
Discretization with finite differences of a onedimensional elliptic problem. Imposition of the Dirichlet and Neumann boundary conditions. Algebraic formulation and matrix properties. Diffusionconvection and diffusionreaction problems.
 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.
 Weak formulation and Finite Elements approximation of stationary problems
Bilinear form, abstract variational problems and LaxMilgram lemma. Variational formulation of elliptic problems and applications to transportreactiondiffusion equations. Introduction to the Galerkin method for a onedimensional 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 diffusionconvectionreaction problem: comparison with the finite difference case and stability analysis. Stabilization with the upwind strategy and the mass lumping technique.
6. Evolution problems
Approximation with the Galerkin method, the semidiscrete problem. Explicit and implicit time marching schemes, the thetamethod. Stability properties. A quick description of finite elements for hyperbolic problems.

Note Sulla Modalità di valutazione 
There are five examination dates (two in JanuaryFebruary, two in JuneJuly, one in September). The course consists in two moduli, one in Mathematical Methods (ref. G. Arioli and G. Grillo) and one in Numerical Methods (ref. C. Vergara and P. Zunino). The exam in Mathematical Methods is written and consists in both questions on the theory and exercises. The exam in Numerical Methods consists in a written part and an optional oral part. The written part of both moduli takes place in the same day. Students can take the oral exam for the Numerical Methods modulus only when the corresponding written grade is at least 15 out of 30.
The final grade is the (rounded up) arithmetic mean of the grades obtained in the two moduli. To get the grade “30 cum laude” one should obtain such grade in both the subparts. If this is not the case, a single "30 cum laude" in one of the parts will be considered as 30 in performing the arithmetic mean. It is possible to take the exam in one modulus (Mathematical or Numerical Methods) in one of the five examination dates and in the other modulus in another examination date, provided such dates are in the same academic year. It is mandatory to take the written and the oral part of the Numerical Methods modulus within the same examination date, i.e. it is not possible to give the written part in one examination date and the oral part in another one.
The participation tothe written exam of one of the two moduli, automatically discards any previous grade obtained for that modulus, even if the student chooses to withdraw.
NOTES
1) Mathematical Methods in Engineering and Numerical Methods in Engineering are also single courses which can be taken independently.
2) Browsing texts, notes, and electronic devices are not allowed during the tests. It is mandatory to bring an ID (e.g. identity card, driver’s licence,…) in order to be identified.
3) Registration to the exam is mandatory. Unregistered students will not be admitted.

Quarteroni A., Modellistica Numerica per Problemi Differenziali, Editore: Springer, Anno edizione: 2012

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Tipo Forma Didattica

Ore didattiche 
lezione

42.0

esercitazione

0.0

laboratorio informatico

26.0

laboratorio sperimentale

0.0

progetto

0.0

laboratorio di progetto

0.0

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Inglese

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