Course Overview

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Course Highlights

The breakdown of the course is:

1) Block 1 (Week 1 and Week 2):
In this block, the finite difference method for the Poisson equation on the interval (Week 1) and the square (Week 2) will be discussed. The differential equation and the boundary conditions will be discretized on a mesh by a finite difference scheme. This will result in linear equations for the grid unknowns. A stencil notation for these linear equations will be introduced. A linear system will be formed and solved. The discrete solution will be visualized on the mesh. Properties of the discretization method and the linear system will be discussed.

2) Block 2 (Week 3 and Week 4):
In this block, the Galerkin finite element method for the one-dimensional (Week 3) and two-dimensional (Week 4) Poisson problem will be introduced. The problem in so-called strong form will be transformed into the problem in so-called weak or variational form. The problem in weak form will then be discretized in space using Lagrangian basis or shape functions. A linear system represents the Poisson equation on a discrete level will be assembled and solved. The finite difference and finite element solution methods will be compared.

3) Block 3 (Week 5, Week 6 and Week 7):
In this block, the application of the finite element will be extended to treat more complex problems. Partial differential equations for the scalar and vector potential for the magnetic flux or field will be derived from the Maxwell equations. More complex geometries that represent transformers, actuators or rotating extrical machines will be meshed using unstructured triangular meshes. The Maxwell equations will be discretized on these meshes. The discrete linear system will be assembled by a element-by-element procedure and solved. The computed vector potential solution will be processed to compute the magnetic flux and the magnetic field.

MAIN GOAL

The main goal of the course is to apply the finite difference and finite element method to solve boundary value problems on a scale of problems of increasing complexity in electrical engineering applications.

Learning Outcomes

After completion of this course, you will be able to:

• + Formulate various boundary value problems for the one-dimensional and two-dimensional Poisson equation.
• + Derive these boundary value problems from the Maxwell equations for the magnetic field by introducing the scalar and vector potential.
• + Apply the finite difference and finite element method to solve these boundary value problems on a scale of problems of increasing complexity.
• + State properties of the finite difference and finite element method applied to Poisson boundary value problems.
• + Describe the implementation of the finite difference and the finite element method in software.
• + Give a physical interpretation of the numerical solution obtained for magnetic field problems.

Meet Your Instructor

Domenico Lahaye

Professor

Admissions

Entry Requirements

• + Linear algebra: concepts of vector, matrix, matrix-vector multiplication, solving linear systems, computing eigenvalues and eigenvectors of a matrix.
• + Calculus of functions of single variable: concepts of derivative of a function, finite difference approximation, integral of a function, quadrature, ordinary differential equations.
• + Calculus of functions of more variables: concepts of partial derivative of a function, surface and line integral, quadrature, partial differential equations.
• + Electric and magnetic fields: concepts of electric field, magnetic field, Maxwell equations, electrical machines and transformers.

Teaching and Assessment Methods

• + Weekly lectures
• + Weekly practical lab sessions
• + Homework assignments
• + Portfolio work
• + Oral examination

Application Deadline: TBC

Fees & Funding

Tuition Fees

TBC