Section: Application Domains
Computational electromagnetics and bioelectromagnetics
Electromagnetic devices are ubiquitous in present day technology. Indeed, electromagnetism has found and continues to find applications in a wide array of areas, encompassing both industrial and societal purposes. Applications of current interest include (among others) those related to communications (e.g transmission through optical fiber lines), to biomedical devices and health (e.g tomography, powerline safety, etc.), to circuit or magnetic storage design (electromagnetic compatibility, hard disc operation), to geophysical prospecting, and to nondestructive evaluation (e.g crack detection), to name but just a few. Equally notable and motivating are applications in defense which include the design of military hardware with decreased signatures, automatic target recognition (e.g bunkers, mines and buried ordnance, etc.) propagation effects on communication and radar systems, etc. Although the principles of electromagnetics are well understood, their application to practical configurations of current interest, such as those that arise in connection with the examples above, is significantly complicated and far beyond manual calculation in all but the simplest cases. These complications typically arise from the geometrical characteristics of the propagation medium (irregular shapes, geometrical singularities), the physical characteristics of the propagation medium (heterogeneity, physical dispersion and dissipation) and the characteristics of the sources (wires, etc.).
The significant advances in computer modeling of electromagnetic interactions that have taken place over the last two decades have been such that nowadays the design of electromagnetic devices heavily relies on computer simulation. Computational electromagnetics has thus taken on great technological importance and, largely due to this, it has become a central discipline in presentday computational science. The team currently considers two applications dealing with electromagnetic wave propagation that are particularly challenging for the proposed numerical methodologies.
Interaction of electromagnetic waves with biological tissues. Electromagnetic waves are increasingly present in our daily environment, finding their sources in domestic appliances and technological devices as well. With the multiplication of these sources, the question of potential adverse effects of the interaction of electromagnetic waves with humans has been raised in a number of concrete situations quite recently. It is clear that this question will be a major concern for our citizens in a near future, especially in view of the everrising adoption of wireless communication systems. Beside, electromagnetic waves also find applications in the medical domain for therapeutic and diagnostic purposes. Two main reasons motivate our commitment to consider this type of problem for the application of the numerical methodologies developed in the NACHOS projectteam:

first, from the numerical modeling point of view, the interaction between electromagnetic waves and biological tissues exhibit the three sources of complexity listed above and are thus particularly challenging for pushing one step forward the stateofthe art of numerical methods for computational electromagnetics. The propagation media is strongly heterogeneous and the electromagnetic characteristics of the tissues are frequency dependent. Interfaces between tissues have rather complicated shapes that cannot be accurately discretized using Cartesian meshes. Finally, the source of the signal often takes the form of a complicated device (e.g a mobile phone or an antenna array).

second, the study of the interaction between electromagnetic waves and living tissues finds applications of societal relevance such as the assessment of potential adverse effects of electromagnetic fields or the utilization of electromagnetic waves for therapeutic or diagnostic purposes. It is widely recognized nowadays that numerical modeling and computer simulation of electromagnetic wave propagation in biological tissues is a mandatory path for improving the scientific knowledge of the complex physical mechanisms that characterize these applications.
Despite the high complexity in terms of both heterogeneity and geometrical features of tissues, the great majority of numerical studies have been conducted using the widely known FDTD method. In this method, the whole computational domain is discretized using a structured (Cartesian) grid. Due to the possible straightforward implementation of the algorithm and the availability of computational power, FDTD is currently the leading method for numerical assessment of human exposure to electromagnetic waves. However, limitations are still seen, due to the rather difficult departure from the commonly used rectilinear grid and cell size limitations regarding very detailed structures of human tissues. In this context, the general objective of the works of the NACHOS projectteam is to demonstrate the benefits of high order unstructured mesh based Maxwell solvers for a realistic numerical modeling of the interaction of electromagnetic waves and living tissues.
Interaction of electromagnetic waves with charged particle beams. Physical phenomena involving charged particles take place in various physical and technological situations such as in plasmas, semiconductor devices, hyperfrequency devices, charged particle beams and more generally, in electromagnetic wave propagation problems including the interaction with charged particles by taking into account self consistent fields. The numerical simulation of the evolution of charged particles under their selfconsistent or applied electromagnetic fields can be modeled by the three dimensional VlasovMaxwell equations. The Vlasov equation describes the transport in phase space of charged particles submitted to external as well as selfconsistent electromagnetic fields. It is coupled nonlinearly to the Maxwell equations which describe the evolution of the selfconsistent electromagnetic fields. The numerical method which is mostly used for the solution of these equations is the ParticleInCell (PIC) method. Its basic idea is to discretize the distribution function f of the particles which is the solution of the Vlasov equation, by a particle method, which consists in representing f by a finite number of macroparticles and advancing those using the Lorentz equations of motion. On the other hand, Maxwell equations are solved on a computational mesh of the physical space. The coupling is done by gathering the charge and current densities from the particles on the mesh to get the sources for the Maxwell equations, and by interpolating the field data on the particles when advancing them. In summary the ParticleInCell algorithm, after the initialization phase, is based on a time loop which consists of the following steps: 1) particle advance, 2) charge and current density deposition on the mesh, 3) field solve, 4) field interpolation at particle positions. More physics, like particle injection or collisions can be added to these basic steps.
PIC codes have become a major research tool in different areas of physics involving selfconsistent interaction of charged particles, in particular in plasma and beam physics. Twodimensional simulations have now become very reliable and can be used as well for qualitative as for quantitative results that can be compared to experiments with good accuracy. As the power of supercomputers was increasing three dimensional codes have been developed in the recent years. However, even in order to just make qualitative 3D simulations, an enormous computing power is required. Today's and future massively parallel supercomputers allow to envision the simulation of realistic problems involving complex geometries and multiple scales. In order to achieve this efficiently, new numerical methods need to be designed. This includes the investigation of high order Maxwell solvers, the use of hybrid grids with several homogeneous zones having their own structured or unstructured mesh type and size, and a fine analysis of load balancing issues. These issues are studied in details in the team in the context of discontinuous Galerkin discretization methods on simplicial meshes. Indeed, the team is one of the few groups worldwide [40] considering the development of parallel unstructured mesh PIC solvers for the threedimensional VlasovMaxwell equations.