Computational mathematics: From wave-structure interaction to magnetic equilibrium in fusion reactors
The first half of the talk will be devoted to the problem of time-domain wave scattering by elastic obstacles. In this situation, part of the incident wave is scattered off the obstacle and part of it excites a perturbation that propagates throughout it according to its physical properties. From the mathematical point of view, this results in two equations coupled through transmission conditions at the interface of the solid. In order to deal efficiently with the scattered wave in the unbounded domain, I will propose a formulation that couples boundary integral equations at the interface with partial differential equations inside the scatterer. To exemplify the analysis I will consider the case of a linear elastic body and will show extensions to bodies with more complex physical properties.
The second part of the talk pertains an application coming from plasma physics. In axially symmetric magnetic confinement devices, the equilibrium between the magnetic and hydrostatic forces can be formulated in terms of a free boundary problem involving a semi-linear elliptic equation for a scalar potential posed in free space. Given a reactor configuration and the location and intensities of the external coils, the total magnetic field and the location of the plasma have to be determined.
In order to deal computationally with the unbounded domain, an artificial boundary containing the reactor is introduced and the exterior problem is reformulated as an integral equation. The resulting couple integro-differential system must be then discretized and solved. The location and properties of the plasma boundary are heavily dependent on the behavior of the external parameters, such as the location and current intensities in the coils, material properties of the reactor, etc. all of which are subject to variability. A satisfactory solution must therefore also address and quantify the uncertainty due to the stochasticity in the problem parameters.
This work has been done in collaboration with Antoine Cerfon (New York University), George Hsiao and Francisco-Javier Sayas (University of Delaware), Nestor Sanchez and Manuel Solano (Universidad de Concepción, Chile), and Howard Elman and Jiaxing Liang (University of Maryland, College Park).