72764 - NUMERICAL METHODS

Conoscenze e abilità da conseguire

At the end of the course, the students know the basic notions of scientific numerical calculus. They possess the knowledge of some of the most important computational tools useful for offshore systems, processes, and structures, with particular reference to numerical methods for the solution of systems of linear equations and differential equations. They are able to use the MATLAB environment for testing methods and critically evaluate the results obtained.

Contenuti

The course is divided into two modules:

• Module 1: Numerical Computation with MATLAB (Prof. F. Zama)
• Module 2: Numerical Modelling (Prof. N. Fantuzzi)

Module 1:

• Matlab programming.
• Numerical derivation and integration:
  • Finite difference formulas
  • Basic Quadrature Formulas
• Numerical Interpolation:
  • Lagrange and Chebyshev Interpolation
  • Piecewise Polynomial Interpolation.
• Ordinary Differential Equations:
  • Numerical solution of Initial Value Problems.

Module 2:

   Some classes of differential equations:

• Elliptic equations
• Parabolic equations
• Hyperbolic equations

  Interpolation and collocation:

• Cardinal functions
• Gauss integral
• Transformation and derivation
• Solution of a boundary value problem
• Solution with basis recombination

   Time-step approximations:

• Parabolic equations
• Finite differences in time
• Finite differences in space
• Hyperbolic

Change of coordinates

• Chebyshev polynomials

Virtual work and energy principles in mechanics:

• Principle of virtual displacements
• Principle of minimum of total potential energy
• Hamilton’s principle for discrete systems (Lagrange method)

• Hamilton’s principle for continuum mechanics

Direct variational methods:

• Strong, weak, and variational formulations
• Ritz method
• Weighted-residual methods

Finite element method:

• 1D isoparametric elements
• 2D isoparametric elements

 

Testi/Bibliografia

• U. Ascher, C. Greif, A first course in Numerical Methods, SIAM, 2011.
• J. P. Boyd: Chebyshev and Fourier spectral methods. Dover, 2000.
• A. J. M. Ferreira, N. Fantuzzi: MATLAB codes for finite element Analysis. Springer, 2020.

Throughout the course, Course Notes and Matlab scripts will be provided.

 

Metodi didattici

Lectures, guided exercises in the laboratory, and group work.

Given the type of activity and teaching methods adopted, the attendance of this course requires the prior participation of all students in the training modules 1 and 2 on safety in the study places ( https://elearning-sicurezza.unibo. it/ ) in e-learning mode.

Modalità di verifica e valutazione dell'apprendimento

Exam:

• Part 1: Written test, with theory and Matlab programming
• Part 2: Homework and multiple choice quiz.

The final grade is obtained as the average of the two parts; part 1 is a prerequisite for part 2.

Strumenti a supporto della didattica

The teaching material will be available on the University of Bologna e-learning platform (https://virtuale.unibo.it).

Orario di ricevimento

Consulta il sito web di Fabiana Zama

Consulta il sito web di Nicholas Fantuzzi