THEORY: Introduction to some methods used for the numerical resolution of geodynamic problems (Finite Difference Method, Finite Volume Method, Spectral Method, Finite Element Method). Properties of Consistency, Stability, Conservation, Limits and Accuracy of a numerical method. Finite element method: Introduction to discrete systems. Discretization of a continuum into a set of finite elements. Reference to a simple elastic system to introduce the concepts of Nodal Forces, Nodal Displacement, Stiffness Matrix. Generalization of the Finite Element Method. Shape functions and their properties. Integral form equivalent to a differential equation. Weak Integral Form. Weighted residue method. Galerkin method. Overview of some methods of numerical integration (Quadrature 1D. Newton-Cotes Quadrature. Gaussian Quadrature).
LABORATORY: Elements of programming (Fortran language) aimed at writing a numerical algorithm. Galerkin formulation applied to the stationary Equation of Heat Conduction. Implementation of some of the subroutines needed to numerically integrate the stationary heat conduction equation. Graphical representation of the numerical solution and comparison with the analytical solution.
Prerequisites for admission
Basic knowledge of programming, integral calculus and linear systems.
Few traditional lessons on the blackboard. Frequent use of PowerPoint projections. For practical lessons, each student will have a computer at disposal to implement, step by step and with the support of the teacher, the numerical algorithm for solving a simple problem.
After each lesson on theoretical topics, a pdf file, which contains an exhaustive presentation of the topics covered during the lesson and which can be used as a text for the study, is made available on the teaching web page accessible through the ARIEL portal.
Texts for further information: Zienkiewich, The Finite Element Methods. Vol. I, any edition.
Some copies of the texts are available in the library of the Department of Earth Sciences "A. Desio".
Assessment methods and Criteria
The exam consists of a written test that aims to verify the knowledge of the arguments covered during the lessons. The test lasts 3 hours and consists in a) a series of questions on theoretical topics (an open answer) and b) the modification of the numerical algorithm implemented during the practical lessons on the basis of an assigned topic and in the discussion of the new results.