Advanced Methods for Computational Electromagnetics
Research Project
|
01.10.2012
- 30.09.2014
Computational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly oscillatory and wave-dominated ; moreover, they can develop singularities at material interfaces and boundaries. Second, the phenomena of interest typically involve complicated geometries, inhomogeneous media, and even nonlinear materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an artificial boundary to confine the region of interest to a finite computational domain. Thus, we seek efficient, reliable and inherently parallel computational methods which handle - complex geometry, corner singularities and material interfaces, - highly varying material properties, - wave-dominated behavior, - unbounded domains. To fully address the wide range of difficulties inherent to Maxwell's equations, great flexibility is needed from any computational approach. In continuation of our previous work, we shall pursue the development of novel explicit local time-stepping (LTS) methods and combine them with discon- tinuous Galerkin (DG) discretizations to achieve the required flexibility. These DG discretization techniques greatly facilitate the handling of material interfaces and non-matching grids; they also permit the coupling of different elements of arbitrary shapes and polynomial order. Starting from our new symmetric , interior penalty DG finite element discretization we have recently devised first explicit, energy conserving, high-order local time-stepping methods for electromagnetic wave propagation in non-conducting media. In our current work, we are studying new explicit LTS methods of arbitrarily high accuracy, which also handle lossy media or applied currents. While these LTS methods are based on classical multistep methods, we now wish to derive high-order explicit LTS methods starting from classical Runge-Kutta and low-storage Runge-Kutta methods for even greater flexibility and efficiency. Since wave phenomena typically occur in very large or truly unbounded regions, perfectly matched layers (PML) or nonreflecting boundary conditions (NBC) must be used to truncate the computational domain, yet without introducing spurious reflections. In our recent work we have derived a new PML for the wave equation in second-order form, which requires fewer auxiliary variables than previous fomulations and hence is more efficient. Currently we are extending that approach to Maxwell's equations and shall combine it with a DG discretization and our explicit local time-stepping methods. To handle situations of multiple scattering, where PML cannot be used, we have also derived the first local NBC for time-dependent multiple scattering problems. By using these new NBC, we now wish to include multiple-scattering effects into time-reversed absorbing conditions (TRAC), a completely new approach to recreate the past of a scattered field, but also to rapidly locate an unknown buried scatterer: both steps are crucial prior to any subsequent accurate numerical inversion procedure.
