Computational Methods In Engineering – Instituto Politécnico de Coimbra

Base Knowledge

Programming knowledge in Matlab.

Teaching Methodologies

In the classes of this curricular unit, exposure, analysis, exercise resolution, application problems and discussion of computationally obtained results are used.

An e-learning platform (distance learning) is used to complement and extend the face-to-face classes, highlighting the use of thematic forums, as another activity for exposure, discussion and resolution of doubts and application problems.

Learning Results

The objectives of this curricular unit are to strengthen, increase and apply basic knowledge in mathematics and programming, which are indispensable for the understanding and scientific treatment of subjects taught and used in other course units of Mechanical Engineering, based on an analytical, algorithmic and computational treatment, thus contributing to obtain the following specific skills:

– Apply computational mathematical methods in the analysis and resolution of engineering problems;

– Use numerical modeling methods in the structural calculation of any mechanical component, using its own or commercial programs.

Program

THEORETICAL-PRACTICAL CLASSES

1. Laplace transform

Definition and properties. Laplace Transform Table. Heaviside function or unit step and Dirac Delta function or unit impulse. Decomposition of a rational fraction into a sum of simple elements (expansion into partial fractions). Inverse Laplace Transform. Convolution Theorem. Troubleshooting initial values and systems of differential equations. Practical application problems – Dynamic Systems. Computational treatment using Matlab and CAS programs.

2. Polynomial interpolation

Polynomial: definition, operations and properties. Taylor Formula and Polynomial. Interpolator polynomial: definition, graphical interpretation and Newton’s formula of divided differences.

3. Differentiation and numerical integration

Formulas for progressive, regressive and centered differences. Trapezoids rule and Simpson rule. Quad Matlab function.

4. Ordinary Differential Equations and Differential Equation Systems. Initial values problem (PVI)

Euler and Runge-Kutta methods (RK2 and RK4). Matlab functions: ODE23, ODE45 and others. Introduction to GUI’s (Graphical User Interface) in Matlab and its application for interface and output of a PVI solution.

5. Partial Derivative Equations (EDPs)

Definition and properties. Laplace, Poisson, Diffusion / Heat, Convex / Transport and Wave equations. Differential problems with initial and boundary conditions. Numerical methods of solving equations with partial derivatives. Introduction to the Finite Difference Method (MDF) and the Finite Element Method (MEF). Weak Formulation (FF) of a differential problem and application of the Ritz-Galerkin (R-G) method. Application problems, mathematical modeling by MDF and MEF, algorithms and respective programming in Matlab.

PRACTICAL CLASSES

1. Programming in Matlab

Matlab programming reviews (one-dimensional and multidimensional arrays, predefined functions, expressions, instructions, creating functions, importing and exporting data, 2D graphics). Cell Arrays. Structures. 3D graphics. Error handling. Image processing. App Designer. Live Script.

2. Application exercises

Profile of fluid velocities (Numerical integration). Heat transfer on a sphere (Solving differential equations). Bi-supported beam subject to loads (App Designer). 3D graphical representations (App Designer). Image processing (App Designer). Support subject to mechanical stress (FEM) with Live Script. Heat transfer in a ring (FEM) with Live Script.

3. Final work on applications to Mechanical Engineering

Final work with specific applications for students in the specialties of “Construction and Maintenance of Mechanical Equipment” and “Design, Installation and Maintenance of Thermal Systems”.

Curricular Unit Teachers

Raquel Almeida de Azevedo Faria

Internship(s)

NAO

Bibliography

Recommended Bibliography:

GRADE, A. (2020). Apresentações das Aulas Práticas de MCE. ISEC (available on the academic platform InforEstudante)
CORREIA, A. (2008). Apontamentos de AM2 e Matemática Aplicada. ISEC (available on the academic platform InforEstudante)
CHAPMAN S. (2005). Programação em MATLAB para engenheiros. Reimp (available from the ISEC Library: 1A-1-317)
HAHN, B., VALENTINE, D. (2010). Essential MATLAB for Engineers and Scientists (4th ed.). Academic Press (available from the ISEC Library: 3-7-80)
JALURIA, Y. (1988). Computer Methods for Engineering. Allyn and Bacon (available from the ISEC Library: 1A-7-5)
FAUSETT, L. V. (1999). Applied numerical analysis using MATLAB. Prentice Hall (available from the ISEC Library: 3-4-202)
HARMAN, T., DABNEY, J., RICHERT, N. (2000). Advanced engineering mathematics with MATLAB, Brooks/Cole (available from the ISEC Library: 3-7-58)
MOLER, C. B. (2004). Numerical computing with MATLAB, Siam (available from the ISEC Library: 3-4-23)
KREYSZIG, E. (1999). Advanced Engineering Mathematics (8th ed.). J. Wiley (available from the ISEC Library: 3-7-95)
MORAIS, V., VIEIRA, C. (2006). MATLAB 7 & 6 : Curso Completo, FCA (available from the ISEC Library: 1A-1-453)
ROSS, S. (1984). Differential Equations (3rd ed.). J. Wiley (available from the ISEC Library: 3-11-6)
BURDEN, R. L., FAIRES, J. D. (2001). Numerical Analysis, (7th ed.). Brooks/Cole (available from the ISEC Library: 3-4-67)
GLYN, J. (1996). Modern Engineering Mathematics (2nd ed.). Addison – Wesley (available from the ISEC Library: 3-2-193)