 # Introduction to the Finite Element Method in optics

Duration: 35h allocated as follows : 17h (lectures) + 16h (experimental lab exercises)

Who should attend?

The course is aimed at

• Master level students or PhD students in Physics
• Master level students in Mathematics

who intend to use free or commercially available Finite Element Software package for applications in engineering with a sound theoretical experience on the theory of the FEM.

Prerequisites:

Matrix algebra, advanced analysis, basic concepts of Lebesgue Integral theory, at undergraduate level.

Course objectives

As a result of this course, you will:

• Understand the fundamental principles underlying the Finite Element Method
• Understand how to control modelling errors
• Gain insight into appropriate use of Finite Element Method
• Learn a Best Practice approach to Finite Element analysis
• Use finite element software packages for advanced problems in optics

Description

Very frequently, equations in physics are too complicated to find solutions in closed form or by purely analytical means (e.g. by Laplace and Fourier transform methods, or in the form of a power series). Numerical approximations to the unknown analytical solution are therefore necessary.

The Finite Element Method (FEM) represents a powerful and general method for the approximate solution of partial differential equations. Although more complex to formulate and to implement than the popular Finite Difference Method (FDM), the FEM offers several important advantages such as the possibility of accurately following material interfaces, of imposing boundary continuity requirements for the approximated electromagnetic vector fields, etc.

The main challenge in using FEM software is to ensure that the calculation corresponds to the "real world" problem and to understand and quantify the approximations made.

The aim of this course is to provide an introduction to both the mathematical theory and the numerical implementation of the FEM, with a special emphasis on applications in optics. Lectures will alternate with practical exercises using FEM software packages such as COMSOL Multi-Physics or Freefem++ and project work in groups of 3 or 4 students. These projects will involve "real world" problems in photonics. Complementary lectures on more specialized topics such as the Perfectly matched layer technique) will be given.

Course topics
• Variational (weak) formulation of Partial Differential Equation (PDE) problems
• Lagrange Finite element space
• Finite element discretization of elliptic PDE
• Edge/Vector Finite Elements for Maxwell equations Main lecturer : Eric Darrigrand, Ph.D from Université Bordeaux 1 and CEA/CESTA (France), is Assistant Professor in Numerical Analysis at the Mathematics Institute of Rennes. His research areas are related to Fast Multipole Methods in Electromagnetism. He is a contributor to the Finite Element Librairies Melina++ and Xlife++.

Lecturers : Stéphane Balac, Yvon Lafranche, Fabrice Mahé   