# Séminaire POEMS sur les méthodes d'équations intégrales

#### 14h: T. Betcke "Software frameworks for computational boundary element methods"

In recent years Galerkin boundary element methods have become an important tool for large-scale simulations in bounded and unbounded homogeneous media. While the underlying mathematical theory is becoming more mature the development and implementation of fast and robust boundary element methods is still an active field of research. In this talk we present the BEM++ software framework for the solution of a range of boundary element problems from electrostatics, acoustics and computational electromagnetics. The guiding principle of BEM++ is to hide the implementational complexity by providing an interface that is as close to the mathematical description as possible.

We will present the underlying ideas and theory and present a range of interesting applications, in particular in acoustics and computational electromagnetics to demonstrate the design philosophy and capabilities of BEM++.

#### 15h30: A. Sellier "A new boundary approach to determine the MHD Stokes flow about a solid axisymmetric body translating in a conducting and quiescent liquid parallel to both its axis of revolution and a prescribed uniform ambient magnetic field"

For a solid body of revolution translating in a conducting quiescent liquid
parallel to its axis of revolution and a given uniform ambient
magnetic field it turns out that symmetries show that the magnetic field

remains uniform in the liquid whereas there is no electric field.
Morevover, the MHD liquid flow is axisymmetric with no swirl. Under the
low Reynolds number assumption, this flow is governed by the quasi-steady
Stokes equations driven by a non-uniform Lorentz body force acting in the
entire unbounded liquid domain. This work presents a new boundary approach
to accurately compute the liquid velocity and pressure fields. More precisely,
the talk will successively address the following points:

- Give the governing assumptions and equations
- Give and discuss the results obtained by Chester in 1957 for a sphere by expanding the flow as an infinite serie of special solutions.
- Give two fundamental axisymmetric MHD Stokes flows recently obtained and produced by distributing axial or radial forces on a ring.
- Deduce from 3. suitable integral representations for the required liquid velocity and pressure fields about the translating body. Establish coupled boundary-integral equations which govern the axial and radial components of the surface traction exerted by the flow on the body surface.
- Present the numerical strategy adopted to invert the previous integral equations and then obtain the liquid flow. Benchmark this approach against Chester's results for a sphere.
- Give and discuss new results for the flow patterns about different bodies of revolution: spheroid, torus and pear-shaped body. Those patterns will be given for different values of the Hartmann number (which is related to the magnetic field magnitude, the body typical length scale and the liquid uniform viscosity and permeability).