Numerical and Computational Methods

Recommended Prerequisites

Computer Programming: Computer program main structure, including data type, input-output instructions, decision-making tasks, repetitive tasks and functions.

Linear Algebra: Indicial notation. Matrices and systems of linear algebraic equations, including the Gauss elimination method. Matrix operations. Invertible matrices. Determining the inverse matrix. Vector spaces. Linear transformations. Matrix of a linear transformation. Determinants. Eigenvalues and eigenvectors. Norm, distance and orthogonal projection.

Teaching Methods

The theoretical lectures take the form of master classes where the problems are outlined, using examples, and the numerical methods are discussed. In theoretical-practical classes selected problems, which illustrate the issues discussed in the theoretical lectures, are analyzed and solved. In all theoretical-practical classes, the students are invited to solve some problems on the subject under study at home. In practical classes students develop and implement programs to test the application of the numerical methods to problem previously discussed in the theoretical-practical class.

Learning Outcomes

Provide skills in the numerical analysis filed to engineering students through a significant theoretical background and an applied component focusing on the introduction to Computational Mechanics.

Explore the numerical methods used by numerical simulation commercial programs through the development of simple numerical algorithms and their programming. The convenient exploitation of these programs allows the students to acquire the necessary awareness about the numerical difficulties that may arise and possible solutions that can be adopted to overcome those difficulties.

Work Placement(s)

No

Syllabus

1. Tensor analysis - Fundamentals of tensor analysis: notation, tensor fields and basic operations in Cartesian coordinates.

2. Nonlinear equations - general conditions for their solving; iterative methods: bisection, Newton, fixed point. Stopping criteria for iterative methods.

3. Systems of linear equations - Iterative methods: Jacobi, Gauss-Seidel.

4. Systems of nonlinear equations - Newton-Raphson iterative method.

5. Polynomial interpolation - Lagrange polynomial. Interpolation error.

6. Numerical integration - Newton-Cotes formulas (e.g. trapezius and Simpson); composed formulas, Gauss formulas; numerical integration error.

7. Differential equations of first order - Taylor methods. Euler method and Runge-Kutta of order 2 and order 4. Systems of differential equations.

8. Development of numerical methods: Development of algorithms and program implementation.

Head Lecturer(s)

Marta Cristina Cardoso de Oliveira

Assessment Methods

Assessment
Resolution Problems: 10.0%
Exam: 90.0%

Bibliography

S.C. Chapra, R.P. Canale, Métodos Numéricos para Engenharia, Mc Graw Hill, 2008.

L.F. Menezes; M.C. Oliveira, Textos de apoio à disciplina, 2007.

José Alberto Rodrigues, Métodos Numéricos - Introdução, Aplicação e Programação, Colecção Matemática 20, Ed. Sílabo, 2003.

F. Correia dos Santos, Fundamentos de Análise Numérica, Colecção Matemática 19, Ed. Sílabo, 2002.

Heitor Pina, Métodos Numéricos, Ed. McGraw-Hill, 1995.

A.J.C. Varandas, J. Brandão, A.A.C.C. Pais; Introdução à programação FORTRAN e cálculo científico, Minerva, Coimbra, 1994.

J.C. Vaissière, J.P. Nougier; Programmes et exercices sur les méthodes numériques, Masson, Paris, 1991.

Stephen J. Chapman, Introduction to fortran 90/95; McGraw-Hill, Boston, 1998. (Basic Engineering Séries and Tools).