APPROCEM

The Approximation Problem in Computational ElectroMagnetics

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 Obiettivo del progetto (Objective)

'The proposal is focused on computational electromagnetics (CEM) with emphasis on the moment method (MoM) and its application to the numerical solution of integral equations (IEs) arising in electromagnetic (EM) scattering by either perfectly-conducting or homogeneous dielectric objects in both 2-D and 3-D geometries. Particularly it addresses four research objectives (ROs). RO1 concerns a priori and a posteriori error analyses for the Reduced Basis Method on account of its application to popular IEs in EM scattering theory (i.e. EFIE, MFIE, and CFIE). RO2 is aimed to shed light on the intimate (but quite involved) relationship between properties of the relevant integral operators underlying our preferred EM models and analogous properties of finite ranked operators resulting from their MoM discretization. This is expected to provide insights for performance and reliability improvements of common iterative techniques used to solve the corresponding MoM system (i.e. CG, BiCG, and GMRES). RO3 points to contribute to the development and further understanding of the approximation problem in the light of Enflo’s and subsequent work on the existence of Banach spaces lacking Grothendieck’s approximation property. This will be done in regard to the sole EFIE. The issue seems almost completely unexplored, yet, it may be useful to know if and when MoM intrinsically fails to represent certain classes of operators, commonly encountered in EM scattering, by finite ranked operators (e.g. depending on the smoothness of scatterers). This is the deepest theoretical theme addressed by the proposal and represents one of the strongest motivations that has led the fellow to apply at LJLL on the consideration of its researchers’ expertise on foundations of numerical methods. Finally RO4 regards numerical experimentation. It will aid the fellow in the development and testing of novel MoM-based schemes for EM scattering problems, but also in the refinement of existing analog techniques.'

Introduzione (Teaser)

The development of antenna and radio-frequency technology requires firm understanding of electromagnetic phenomena. To accurately model the physics of the involved radiation and wave phenomena, EU-funded scientists have introduced new mathematical methods and procedures for numerical simulations.

Descrizione progetto (Article)

Of the most popular numerical methods for simulating electromagnetic fields, the 'The approximation problem in computational electromagnetics' (APPROCEM) project focused on the method of moments. This method is significantly more efficient in terms of computational resources as it requires calculating only boundary values, rather than values throughout space. It works by constructing a 'mesh' over the modelled area.

The problem of electromagnetic waves scattered by a conducting obstacle coated by a layer of dielectric material is described using coupled electric field integral equations. The latter contain expressions for both interior and exterior fields' values. By applying the method of moments with suitable basis functions, the integral equations are transformed into an equation of matrices that has to be solved.

The research work of the APPROCEM scientists provided insights into the relationship between properties of the relevant integral operators. The next step was to use their findings to improve commonly used iterative methods to solve the equation obtained from the moments' method. Iterative solvers are algorithms used to sequentially build up the solution.

Going far beyond the existing state of the art, the APPROCEM scientists provided new algorithms based on numerical simulations and assimilation of experimental measurements. Moreover, aspects of theories developed independently from each other were brought under a unified mathematical framework. This was achieved by the introduction of breakthrough ideas to evaluate the performance of methods used to approximate the equation's solution.

One of the fundamental drives behind the APPROCEM project was the ability to simulate the behaviour of devices and systems before they are actually built. This allows engineers to engage in levels of customisation and optimisation that is costly or even impossible if done experimentally. Insights into electromagnetic phenomena through the power of computation could lead to more advanced technology for radar antennas.