MECH4000 Advanced engineering mathematics Emneplan

This course provides a selection of advanced topics in Mathematics, essential to studying continuum mechanics, finite element method, computational fluid mechanics, and mechatronics. These include matrix analysis, with focus on eigenvalue problems and positive definiteness, multiple, line and surface integrations and applications of Gauss and Stokes theorems, classification, and the solution of certain types of PDEs using methods such as Laplace and Fourier transforms, discretization of domains and equations, and variational calculus with applications.

The knowledge, methods, and associated problem-solving techniques are applicable in fluid mechanics, solid mechanics, and mechatronics, as ample exemplification will elucidate. Matrix analyses, partial differential equation (PDE) solving methods, and variational calculus are covered topics. The students will learn about the formulation and implementation in code for the finite element and finite volume methods used in solid and fluid mechanics, respectively. It is expected that students be able to develop and implement solution algorithms in Python.

The course assumes basic knowledge of Python and programming in that environment as a quintessential element of problem-solving in engineering mathematics. It is also expected that students can solve problems arising in single variable calculus and are familiar with the theory of ODEs.

Knowledge:   

The candidate

• can classify matrixces according properties such as symmetry, bandwidth and eigenstructure.

•  can identify multiple integration and recognize how line and surface integrals may beevaluated directly or by using Gauss or Stokes theorem

•  can explain variational operator in contradistinction to the differential operator and establish when the use of the former is required.

• can recognize a PDE and outline what methods can be used to solve it including Fourierand Laplace transforms

•  can recognize the difference between a continuous and a discretized system and procedures to discretize the governing PDEs of the continuous system

Skills:   

The candidate

• can calculate the eigenstructures of a square matrix and determine diagonalizability and positive definiteness. 

• can apply the method of QR factorization to solve a system of simultaneous linear equations 

• can apply multiple integration, line, and surface integrals to work out domain and boundary integrals and discretize an equation to set up the algorithm for its numerical solution

•  can solve discretized equations through writing a code in Pyhton

• can solve parabolic, hyperbolic, and elliptic PDEs using the methods of Fourier or Laplace transforms where applicable

•  can apply variational calculus to derive PDEs based on the minimization of the integral form

•  can use the principle of least action to derive Euler-Lagrange equations of mechanics

General competence  

The candidate

•   can communicate extensive, independent work, mastering the language and terminology of engineering mathematics

• can discuss mechanical engineering problems with peers using the terminology of engineering mathematics

• can contribute to innovative thinking and novel solutions precipitated by thinking in different paradigms of mathematical modelling and the application of the classification methods learnt

• can use the methods learnt to model phenomena in solid mechanics, fluid mechanics and mechatronics using the language of mathematics, in particular PDEs, variational methods and linear transformations and solve these problems analytically and/or numerically.

ectures and tutorials (including problem solving and the use of Python coding to solving relevant mathematical problems).

None

Written 6-hour exam under supervision.

In the event of a resit or rescheduled exam, an oral examination may be used instead. If an oral exam is used, the examination results cannot be appealed.

None

Graded scale A-F

One internal examiner. External examiner is used periodically.