In this project, we wanted to familiarize ourselves with the boundary element method (BEM) for 2d coaxial conductors (the depth is supposed to be infinite) in a context of electromagnetism. To do so, we have solved a well-known problem in physics, the 2D Laplace potential with various geometries.
We used the simplest form of BEM : each element has a constant value over his domain and the element have a linear geometry.
The project was done in a context of a summer internship at Polytechnique Montréal University, with Pr. Frédiric Sirois as the supervisor.
The BEM only requires a discretization of the boundary. Also, only one type of condition is imposed on the boundary : either (exclusive) Dirichlet or Neumann.
- The interior or outer domain is not discretized with elements. The solution can be easily reconstructed in all space when the BEM has solved entirely the boundary.
- Less elements are required.
- Each boundary element are treated as a physical source. The interaction between the elements is treated with a Green-integral-source formalism. Thus, the problem becomes more complex with the physical potential. Plus, a considerable amount of mathematical work is required before implementing the BEM.
- Coaxial conductor 2D Laplace with Dirichlet condition of 0 for the outer circle and 1 for the inner circle.
- Hole conductor (outer square with inner circle) 2D Laplace with Dirichlet condition of 0 for the bottom and 1 for the top (opposite sides). Other sides are Neumann with 0.
- Slope 2D Laplace with Dirichlet condition of
Edward H-Hannan
Pr. Frédéric Sirois