Mathematical Modelling

Recommended Prerequisites

Mathematical Analysis I and II, Linear Algebra and Anlytical Geometry, Continuum Mechanics, Numerical Methods.

Teaching Methods

In the teaching of this course, it is expected to present theoretical concepts and knowledge related to the themes defined in the course syllabus. In addition to the contents transmitted in theoretical-practical classes, its application to practical cases with the teacher's support allows an effective learning oriented to the types of problems proposed. In the application to practical cases computational tools will be used.

Learning Outcomes

It is intended that this course allows the development and application of numerical modeling concepts focused on engineering problems. It is introduced in this unit the mathematical modeling of complex engineering problems, typically formulated based on partial differential equations and solved using advanced numerical methods. The mathematical principles of the most common methods used in engineering, namely Finite Differences, Finite Volumes and Finite Elements, are lectured. It is intended to provide students with knowledge on the the methods' formulation but also on its application, and so computational modeling tools are used. The software Matlab will be used, which students already learn and use in other courses.

Work Placement(s)

No

Syllabus

1.Introduction to the curricular unit

• Importance and relevance of numerical methods in Engineering

• General examples for motivation

2. Types of PDEs

• Elliptical, parabolic and hyperbolic PDEs, and relation to physical phenomena.

• Initial and boundary conditions.

3. 1D Problems

• FDM and FVM

-Formulation

-Simple application example

• FEM

-Formulation

-Simple application example

4.Numerical solution of time-independent PDEs in 2D and 3D

• Finite difference and finite volume methods applied to numerical resolution of PDEs.

-Formulation.

-Application to practical problem solving - computer simulation.

• Finite element method applied to numerical resolution of EDPs.

-Formulation, Introduction to variational principles and weigthed residues.

-Formulation of elements.

-Application to practical problem solving - computer simulation.

5.Time dependent problems

• Explicit and implicit schemes.

• Practical application.

Head Lecturer(s)

Luís Manuel Cortesão Godinho

Assessment Methods

Assessment
Resolution Problems: 30.0%
Exam: 70.0%

Bibliography

Chapra, S. C., Canale, R. P., Numerical methods for engineers, 7th Ed., NY: McGraw Hill, 2015

 

Bathe, K.-J., Finite Element Procedures, Upper Saddle River (NJ): Prentice Hall, 1996

 

Stoer, J, Bulirsch, R., Introduction to numerical analysis, 2nd Ed., NY: Springer, 1993

 

Ferziger, J. H., Peric, M., “Computational methods for fluid dynamics”, Springer, 1996

 

F. Teixeira-Dias, J. Pinho-da-Cruz, R.A. Fontes Valente, R.J. Alves de Sousa (2010), Método dos Elementos Finitos – Técnicas de Simulação Numérica em Engenharia, ETEP.