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Publication# Low frequency electromagnetic wave propagation in 3D plasma configurations

Abstract

We investigate low-frequency electromagnetic wave propagation and absorption properties in 2D and 3D plasma configurations. For these purposes, we have developed a new full-wave 3D code LEMan that determines a global solution of the wave equation in bounded stellarator plasmas excited with an external antenna. No assumption on the wavelength compared to the plasma size is made, all the effects of the 3D geometry and finite plasma extent are included. The equation is formulated in terms of electromagnetic potentials in order to avoid numerical pollution effects. The code utilises linear and Hermite cubic finite element discretisation in the radial direction and Fourier series in the poloidal and toroidal variables. The full cold plasma model including finite electron inertia and, thus, mode conversion effects is implemented. The code uses Boozer magnetic coordinates and has an interface to the TERPSICHORE code. Special care is taken to treat the magnetic axis and to ensure the unicity of the numerical solution. The discretisation, interpolation and numerical derivation methods specifically adapted for our problem avoid the energy sink in the origin and provide a very good local and global energy conservation. A special algorithm has been developed to analytically expand the wave equation coefficients in the full 3D stellarator geometry. The code has been specifically optimised for vector computing platform, reaching close to maximum average performances on the NEC SX5 machine. The code has been applied in 1D, 2D, and 3D geometries. No unphysical solutions have been observed. LEMan successfully recovers all the fundamental properties of the Alfvén spectrum (gaps, eigenmodes). Benchmarks have been made against the 2D LION code and JET experimental measurements, showing a good agreement between the results.

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The Uniformization Theorem due to Koebe and Poincaré implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

Benjamin Fuchs, Ruzica Golubovic, Juan Ramon Mosig, Anja Skrivervik Favre

A design procedure for spherical lens antennas is described. A particle swarm optimization (PSO) algorithm is coupled to a mode matching technique based on spherical wave expansion to analyze the lens antennas. The proposed methodology is applied to three optimization problems using real-number and binary PSO. First, the maximization of the directivity of Luneburg lens antennas is addressed. Then, amplitude shaped radiation patterns are synthesized by optimizing, both amplitude and position of each element of an array that illuminates a lens. Finally, a dual-beam reconfigurable lens antenna is optimized. By only switching properly the elements of an array, the lens antenna radiates either a directive or a sectoral beam. Numerical comparisons with a full wave commercial software successfully validate the proposed design procedure. (C) 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1655-1659, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.25278

This thesis concerns optimal packing problems of tubes, or thick curves, where thickness is defined as follows. Three points on a closed space curve define a circle. Taking the infimum over all radii of pairwise-distinct point triples defines the thickness Δ. A closed curve with positive thickness has a self-avoiding neighbourhood that consists of a disjoint union of normal disks with radius Δ, which is a tube. The thesis has three main parts. In the first, we study the problem of finding the longest closed tube with prescribed thickness on the unit two-sphere, and show that solutions exist. Furthermore, we give explicit solutions for an infinite sequence of prescribed thicknesses Θn = sin(π/2n). Using essentially basic geometric arguments, we show that these are the only solutions for prescribed thickness Θn, and count their multiplicity using algebraic arguments involving Euler's totient function. In the second part we consider tubes on the three-sphere S3. We show that thickness defined by global radius of curvature coincides with the notion of thickness based on normal injectivity radius in S3. Then three natural, but distinct, optimisation problems for knotted, thick curves in S3 are identified, namely, to fix the length of the curve and maximise thickness, to fix a minimum thickness and minimise length, or simply to maximise thickness with length left free. We demonstrate that optimisers, or ideal shapes, within a given knot type exist for each of these three problems. Finally, we propose a simple analytic form of a strong candidate for a thickness maximising trefoil in S3 and describe its interesting properties. The third and final part discusses numerical computations and their implications for ideal knot shapes in both R3 and S3. We model a knot in R3 as a finite sequence of coefficients in a Fourier representation of the centreline. We show how certain presumed symmetries pose restrictions on the Fourier coefficients, and thus significantly reduce the number of degrees of freedom. As a consequence our numerical technique of simulated annealing can be made much faster. We then present our numeric results. First, computations approach an approximation of an ideal trefoil in S3 close to the analytic candidate mentioned above, but, supporting its ideality, are still less thick. Second, for the ideal trefoil in R3, numerics suggest the existence of a certain closed cycle of contact chords, that allows us to decompose the trefoil knot into two base curves, which once determined, and taken together with the symmetry, constitute the ideal trefoil.