Numerical and computational methods

Recommended Prerequisites

Calculus I,II,III, Numerical analysis, Algebra, Statiscics, Mechanics I e II. Fortran, C++, Pascal.

Teaching Methods

Theoretical classes with detailed exposition, making resource to audiovisual media, of the fundamental concepts, principles and theories. Resolution of some practical examples in each chapter. The classes are concentrated in some weeks of the semester.

Learning Outcomes

This course involves the contact with mathematical and classical and advanced numerical methods. The principal aim is to provide skills for the exercise of theoretical and computational research.

Work Placement(s)

No

Syllabus

1. Matrices
- Matrices, tensors and vectors: fundamental concepts.
- Numerical solution of problems: inversion of matrices, applications.
2. Functions with vector values and integral calculation in Rn
- Derivatives and integrals, partial differentiation, gradient, extremes of multiple variable functions.
- Revision on integration, multiple and surface integrals.
- Methods for numerical integration.
3. Diferential equations
- Revisions, linear equation of order n., homogeneous and non-homogeneous; systems of linear equations, Laplace transform.
4. Calculus of variations
- Method of Lagrange, functionals, Lagrange multipliers, direct methods (Rayleigh-Ritz, Kantorovitch, Galerkin, finite diferences, Trefftz), problems with boundary conditions and initial value, aplication to mechanical problems.
 5. Continuation avanced methods
- Introdution, “predictor-corrector” method, “piecewise linear”, and Newton-Raphson, “arc-length” (normal and simplified) and Watson.

Assessment Methods

Contínua
Realização de trabalhos teóricos e práticos: 100.0%

Final
Exam: 100.0%

Bibliography

- Allgower, E. L., Georg, K. (1990), Numerical continuation methods – An introduction, Berlin-Heidelberg: Springer-Verlag.

 - Apostol, T., M. (1967), Calculus, Vol. I, 2nd Ed. John Wiley and Sons.

- Apostol, T., M. (1969), Calculus, Vol. II, 2nd Ed. John Wiley and Sons.

- Berge, C. (2001), Theory of graphs, Mineola, NY: Dover.

- Steven C. Chapra, Raymond P. Canale – Numerical Methods for Engineers, McGraw-Hill International, ISBN 0-07-079984-9, 1990.

- Heitor Pina – Métodos Numéricos, McGraw-Hill, ISBN 972-9298-04-8, 1995.

- O.C. Zienkiewicz, K. Morgan – Finite Elements and Approximation, John Wiley & Sons, 1983, USA.

- Oñate, E., - Structural Analysis with the Finite Element Method. Linear Statics. Volume 1. Basis and Solids,  CIMNE, Springer, ISBN: 978-1-4020-8732-5, 2009.

- K.-J. Bathe, E.L. Wilson – Numerical Methods in Finite Element Analysis, ISBN 0-13-627190-1, 1976.

- Bathe, K. - Finite Element Procedures, Prentice Hall, ISBN: 0-13-301458-4, 1996.