37261 - Numerical Analysis

Learning outcomes

The integrated course Numerical and Mathematical Methods for Engineering provides the student with advanced analytical and computational tools for the formulation and solution of engineering problems. The student develops an understanding of both the theoretical foundations of applied mathematics and the numerical techniques used to model and solve differential and applied problems, with particular reference to aerospace and mechanical engineering. In this module, the student learns the main numerical methods for solving differential problems of engineering relevance. The course covers the fundamental numerical–mathematical aspects and algorithmic methodologies for the discretization and computational solution of differential equations, emphasizing accuracy, stability, and efficiency in numerical simulations.

Course contents

Prerequisites:

General mathematical knowledge and computer skills from completed BSc study. More precisely, prerequisites of Geometry and Algebra, Mathematical Analysis and MATLAB programming are required. Moreover, a prior knowledge of fundamental concepts and basic topics of Numerical Analysis is also needed.

Program:

1. Iterative Methods for Solving Nonlinear Equations and Systems of Nonlinear Equations

Review of the main iterative methods for finding the roots of nonlinear equations in one variable: the bisection method, Newton's method, and the secant method. Iterative methods for the numerical solution of systems of nonlinear equations: Newton's method and quasi-Newton methods.

An application of nonlinear system solving in aerospace engineering.

2. Numerical Differentiation: Finite Difference Approximations of Derivatives

Numerical approximation of derivatives using finite difference formulas. The method of undetermined coefficients. Determination of the step size that minimizes the total approximation error.

Applications: solution of aerospace engineering problems using finite difference methods.

3. Numerical Integration Using Gaussian Quadrature

Brief overview of Newton-Cotes quadrature rules and their limitations. Open, closed, and semi-open Gaussian quadrature rules. Numerical algorithms for computing the nodes and weights of the Gauss-Legendre, Gauss-Lobatto, and Gauss-Radau quadrature rules.

An application of numerical integration in computational aerodynamics.

4. Numerical Methods for Ordinary Differential Equation Initial Value Problems (ODE-IVPs)

Ordinary differential equations (ODEs) and initial value problems (IVPs). Existence, uniqueness, and stability of solutions. One-step numerical methods for ODE-IVPs: explicit and implicit fixed-step Runge-Kutta methods, and adaptive Runge-Kutta methods. Accuracy, consistency, and stability of numerical methods. Stiff problems.

Application: simulation of the dynamic response of the Tacoma Narrows Bridge.

Readings/Bibliography

For exam preparation, students may refer to the course materials uploaded on the Virtuale platform.

Recommended reference books for further reading are:

[1] T. Sauer: Numerical Analysis, 3rd ed., Pearson, 2018.

[2] J. Kiusalaas: Numerical Methods in Engineering with MATLAB, 3rd ed., Cambridge University Press, 2015.

[3] K. Atkinson, W. Han, D. Stewart: Numerical Solution of Ordinary Differential Equations, John Wiley and Sons, 2009.

[4] A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics, 2nd ed., Springer, 2007.

[5] R.J. LeVeque: Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.

To fill any knowledge gaps related to the prerequisites of Numerical Analysis and/or to the fundamentals of MATLAB programming it is advisable to consult:

[1] E. E. Mikhailov: Programming with MATLAB for Scientists: A Beginner’s Introduction, CRC Press, Inc., 2017.

[2] A. Quarteroni, F. Saleri, P. Gervasio: Scientific Computing with MATLAB and Octave (4th edition), Springer, 2014.

[3] C. Moler: Numerical Computing with MATLAB, SIAM, 2004.

[4] J. Stoer, R. Bulirsch: Introduction to Numerical Analysis (3rd ed.), Springer, 2002.

Teaching methods

The course is structured in lectures and exercises in the computer laboratory. More precisely, the lectures are followed by laboratory exercises aimed at implementing the illustrated numerical methods in MATLAB and developing an adequate sensitivity and awareness of their use.

Lecture slides and other course materials, including practice exercises, application examples, and sample exams, will be made available on the Virtuale platform.

Although attendance is neither mandatory nor taken into account in the final assessment, it is strongly recommended, as it facilitates the learning process and contributes to a deeper understanding of the course topics.

In consideration of the type of activity and of the adopted teaching methods, the attendance of this training activity requires the prior participation of all students in Modules 1 and 2 of Health and Safety training courses in e-learning mode.

Assessment methods

The final examination is designed to assess the achievement of the following learning outcomes:

• knowledge of the numerical and mathematical foundations, as well as the main algorithmic methodologies, for solving nonlinear equations and systems of nonlinear equations, for numerical differentiation and integration, and for solving initial value problems for ordinary differential equations;

• the ability to design and implement numerical methods and MATLAB programs for the solution of engineering problems, and to critically evaluate their accuracy, efficiency, and reliability.

The examination will take place in the computer laboratory. Students will have 120 minutes to complete two exercises which require both the development of MATLAB code and written answers to theoretical questions covering the topics discussed during the lectures.

For assessment purposes, only the files uploaded to the Esami On Line (EOL) platform before the end of the examination will be considered.

During the examination, students are not permitted to consult textbooks, notes, or any computer-based support tools. Access to online course materials is also prohibited, as is the use of generative Artificial Intelligence (AI) tools.

The examination is considered passed if the student achieves at least 18 points out of a maximum of 32. A score above 30 results in the award of “30 cum laude”, in recognition of an outstanding level of achievement.

Instructions and deadlines for declining a grade will be provided by email upon publication of the examination results.

To participate in the exam the student must register (at least 2 days in advance) in the lists available on the AlmaEsami web platform. On the day of the exam the student will be able to access the laboratory and take the test only after showing an identification document.

The dates of the exams can be consulted on the AlmaEsami web platform and are visible several months in advance.

Clarifications on the recording of the exam grade for those who have the NUMERICAL ANALYSIS course (6 CFU) as a module of the integrated course of NUMERICAL AND MATHEMATICAL METHODS FOR ENGINEERING (12 CFU)

If the NUMERICAL ANALYSIS course (6 CFU) is one of the two modules that, together with MATHEMATICAL METHODS FOR ENGINEERING (6 CFU), constitutes the integrated course of NUMERICAL AND MATHEMATICAL METHODS FOR ENGINEERING (12 CFU), the grade that will be recorded will be calculated with the arithmetic average of the single grades that the student has obtained in the two modules. It should be noted that the result of the average will be rounded to the nearest integer. Only if the resulting average is exactly equidistant between two integers, the grade will be obtained by rounding up to the next highest integer. Moreover, in order to obtain the "30 cum laude" final evaluation, the student must be in one of the two following cases:

- receiving "30 cum laude" in both modules;

- obtaining "30 cum laude" in one module and 30 in the other one.

Lastly, it should be noted that the recording of the final evaluation requires the passing of both the exam of NUMERICAL ANALYSIS (6 CFU) and the exam of MATHEMATICAL METHODS FOR ENGINEERING (6 CFU) in a time interval not exceeding 12 months.

Students with special educational needs related to learning disorders and/or disabilities are invited to contact the responsible office (see https://site.unibo.it/studenti-con-disabilita-e-dsa/en) as soon as possible, so that they can propose suitable adaptations to be submitted to the lecturer at least 15 days before the exam date.

Teaching tools

LCD projector and PC are used in addition to the traditional blackboard.

The course includes also a laboratory activity in which the MATLAB software will be used.

PDF files of slideshows, examples and solved exercises will be made available on the online platform Virtuale.

Office hours

See the website of Lucia Romani