The Vlasov equation describes the evolution of the distribution function of particles in phase space (x,v), where the particles interact with long-range forces, but where shortrange "collisional" forces are neglected. A space plasma consists of low-mass electrically charged particles, and therefore the most important long-range forces acting in the plasma are the Lorentz forces created by electromagnetic fields. What makes the numerical solution of the Vlasov equation a challenging task is that the fully three-dimensional problem leads to a partial differential equation in the six-dimensional phase space, plus time, making it hard even to store a discretised solution in a computer’s memory. Solutions to the Vlasov equation have also a tendency of becoming oscillatory in velocity space, due to free streaming terms (ballistic particles), in which steep gradients are created and problems of calculating the v (velocity) derivative of the function accurately increase with time. In the present thesis, the numerical treatment is limited to one- and two-dimensional systems, leading to solutions in two- and four-dimensional phase space, respectively, plus time. The numerical method developed is based on the technique of Fourier transforming the Vlasov equation in velocity space and then solving the resulting equation, in which the small-scale information in velocity space is removed through outgoing wave boundary conditions in the Fourier transformed velocity space. The Maxwell equations are rewritten in a form which conserves the divergences of the electric and magnetic fields, by means of the Lorentz potentials. The resulting equations are solved numerically by high order methods, reducing the need for numerical over-sampling of the problem. The algorithm has been implemented in Fortran 90, and the code for solving the one-dimensional Vlasov equation has been parallelised by the method of domain decomposition, and has been implemented using the Message Passing Interface (MPI) method. The code has been used to investigate linear and non-linear interaction between electromagnetic fields, plasma waves, and particles.

The paper deals with a combination of the Fourier-finite-element method with the Nitsche-finite-element method (as a mortar method). The approach is applied to the Dirichlet problem of the Poisson equation in three-dimensional axisymmetric domains $\widehat\Omega$ with non-axisymmetric data. The approximating Fourier method yields a splitting of the 3D-problem into 2D-problems. For solving the 2D-problems on the meridian plane $\Omega_a$, the Nitsche-finite-element method with non-matching meshes is applied. Some important properties of the approximation scheme are derived and the rate of convergence in some $H^1$-like norm is proved to be of the type ${\mathcal O}(h+N^{-1})$ ($h$: mesh size on $\Omega_a$, $N$: length of the Fourier sum) in case of a regular solution of the boundary value problem. Finally, some numerical results are presented.

This paper is the last part of a three-fold article aimed at some efficient numerical methods for solving the Poisson problem in three-dimensional prismatic and axisymmetric domains. In the first and second parts, the Fourier singular complement method (FSCM) was introduced and analysed for prismatic and axisymmetric domains with reentrant edges, as well as for the axisymmetric domains with sharp conical vertices. In this paper we shall mainly conduct numerical experiments to check and compare the accuracies and efficiencies of FSCM and some other related numerical methods for solving the Poisson problem in the aforementioned domains. In the case of prismatic domains with a reentrant edge, we shall compare the convergence rates of three numerical methods: 3D finite element method using prismatic elements, FSCM, and the 3D finite element method combined with the FSCM. For axisymmetric domains with a non-convex edge or a sharp conical vertex we investigate the convergence rates of the Fourier finite element method (FFEM) and the FSCM, where the FFEM will be implemented on both quasi-uniform meshes and locally graded meshes. The complexities of the considered algorithms are also analysed.

The paper deals with a combination of the Fourier method with the Nitsche-finite-element method (as a mortar method). The approach is applied to the Dirichlet problem of the Poisson equation in threedimensional axisymmetric domains with reentrant edges generating singularities. The approximating Fourier method yields a splitting of the 3D problem into 2D problems on the meridian plane of the given domain. For solving the 2D problems bearing corner singularities, the Nitsche finite-element method with non-matching meshes and mesh grading near reentrant corners is applied. Using the explicit representation of singular functions, the rate of convergence of the Fourier-Nitsche-mortaring is estimated in some $H^1$-like norm as well as in the $L_2$-norm. Finally, some numerical results are presented.

In this paper, we study the partial Fourier method for treating the Lamé equations in three-­dimensional axisymmetric domains subjected to nonaxisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement u, the body force f \in (L_2)^3 and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three­dimensional boundary value problem to an infinite sequence of two­dimensional boundary value problems, whose solutions u_n (n = 0,1,2,...) are the Fourier coefficients of u. This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients u_n are proved, which are important for further numerical treatment, e.g. by the finite-element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two­dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients u_n and of the error of the partial Fourier approximation are given.

This paper is the last part of a three-fold article aimed at some efficient numerical methods for solving the Poisson problem in three-dimensional prismatic and axisymmetric domains. In the first and second parts, the Fourier singular complement method (FSCM) was introduced and analysed for prismatic and axisymmetric domains with reentrant edges, as well as for the axisymmetric domains with sharp conical vertices. In this paper we shall mainly conduct numerical experiments to check and compare the accuracies and efficiencies of FSCM and some other related numerical methods for solving the Poisson problem in the aforementioned domains. In the case of prismatic domains with a reentrant edge, we shall compare the convergence rates of three numerical methods: 3D finite element method using prismatic elements, FSCM, and the 3D finite element method combined with the FSCM. For axisymmetric domains with a non-convex edge or a sharp conical vertex we investigate the convergence rates of the Fourier finite element method (FFEM) and the FSCM, where the FFEM will be implemented on both quasi-uniform meshes and locally graded meshes. The complexities of the considered algorithms are also analysed.

The paper deals with a combination of the Fourier method with the Nitsche-finite-element method (as a mortar method). The approach is applied to the Dirichlet problem of the Poisson equation in threedimensional axisymmetric domains with reentrant edges generating singularities. The approximating Fourier method yields a splitting of the 3D problem into 2D problems on the meridian plane of the given domain. For solving the 2D problems bearing corner singularities, the Nitsche finite-element method with non-matching meshes and mesh grading near reentrant corners is applied. Using the explicit representation of singular functions, the rate of convergence of the Fourier-Nitsche-mortaring is estimated in some $H^1$-like norm as well as in the $L_2$-norm. Finally, some numerical results are presented.

Radars are used for range estimation of distant objects. They operate on the principle of sending electromagnetic pulses that are reflected off a target. This leads to the propagation of electromagnetic waves over large distances. As the waves propagate, they are affected by several aspects that decrease the performance of the radar system. In this master thesis, we derive a mathematical model that describes electromagnetic propagation in the troposphere. The model developed is based on a parabolic equation and uses the split-step Fourier method for its numerical solution. Using the model, we estimate the influence of a varying, complex, refractive index of the atmosphere, different lossy materials in the ground, terrain, and oceans. The terrain is described using a piecewise linear shift map method. The modelling of the ocean is done using a novel model which is a combination of terrain for large swells and Miller surface roughness for smaller waves, both based on a Pierson-Moskowitz sea spectrum. The model is validated and found to agree very well, with results found in the literature.

FundaÃÃo de Amparo à Pesquisa do Estado do Cearà / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho analisou-se a integridade mecÃnica do material compÃsito constituÃdo por uma matriz polimÃrica, resina epoxi DER 331, e por fibra de vidro tipo âEâ como agente de reforÃo. Esta anÃlise foi realizada utilizando-se o ensaio de emissÃo acÃstica, com o objetivo de correlacionar as formas de onda dos sinais com os mecanismos de falhas associados aos esforÃos de traÃÃo e flexÃo aplicados a espÃcimes fabricados com tal material. Na anÃlise dos sinais como funÃÃo do tempo foram utilizados os mÃtodos de contagem de caixas, que fornece a dimensÃo fractal, e o do intervalo re-escalado de Hurst, sendo tambÃm utilizado o mÃtodo espectral de Fourier para a anÃlise no domÃnio da freqÃÃncia. Mostrou-se que os expoentes calculados pelos mÃtodos espectral de Fourier e re-escalado de Hurst estÃo correlacionados com a dimensÃo fractal obtida pelo mÃtodo de contagem de caixa, e satisfazem as relaÃÃes previstas pelas leis de escala. Os resultados mostraram tambÃm a existÃncia de duas regiÃes de escala distintas, sendo uma caracterizada pela persistÃncia do sinal e outra por um comportamento aleatÃrio caracterÃstica do ruÃdo presente. As dimensÃes fractais obtidas apresentaram-se independentes da taxa de aquisiÃÃo da forma da onda emitida indicando a propriedade de auto-similaridade dos sinais estudados, o que confirma a sua caracterÃstica fractal. Finalmente, à conjecturado uma correlaÃÃo entre o coeficiente de Hurst/dimensÃo fractal e as falhas mecÃnicas observadas. / In this work it is analysed the mechanical integrity of the composite material constituted of a polymeric matrix, epoxy resin DER 331, reinforced by glass fiber type E. This analysis has been done by using the acoustic emission testing with the aim to correlating the wave-form of the pulse with the flaw mechanisms associated to the tensile and bending loads applied to the samples. The analysis of the pulses as a function of time has been made by using box counting method, which provides the fractal dimension, and the rescaled Hurst analysis. The analysis in the frequency domain has been made by using the spectral Fourier method It has been shown that the exponents obtained from the spectral Fourier method and the rescaled Hurst analysis are correlated to the box counting fractal dimension, and satisfy the known relations obtained from the scaling laws. The results have also shown the existence of two scaling regions, characterized by the persistence of the pulse and by a random behaviour, respectively. The fractal dimensions have also been shown to be independent of time acquisition of the emitted pulse, and this indication of self-similarity confirms its fractal characteristics. Finally, it is conjectured a correlation between the Hurst coefficient/fractal dimension and the mechanical flaws observed.

In this paper, we study the partial Fourier method for treating the Lamé equations in three-­dimensional axisymmetric domains subjected to nonaxisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement u, the body force f \in (L_2)^3 and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three­dimensional boundary value problem to an infinite sequence of two­dimensional boundary value problems, whose solutions u_n (n = 0,1,2,...) are the Fourier coefficients of u. This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients u_n are proved, which are important for further numerical treatment, e.g. by the finite-element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two­dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients u_n and of the error of the partial Fourier approximation are given.