An analysis of finite-difference and finite-volume formulations of conservation laws

Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations--potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomeclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out

Machine learning high multiplicity matrix elements for electron-positron and hadron-hadron colliders

The LHC is a large-scale particle collider experiment collecting vast quantities of experimental data to study the fundamental particles, and forces, of nature. Theoretical predictions made with the SM can be compared with observables measured at experiments. These predictions rely on the use of Monte Carlo event generators to simulate events which demand the evaluation of a matrix element. For high multiplicity processes this can take up a significant portion of the time spent simulating an event. In this thesis, we explore the usage of machine learning to accelerate the evaluation of matrix elements by introducing a factorisation-aware neural network model. Matrix elements are plagued with singular structures in regions of phase-space where particles become soft or collinear, however, the behaviour of the matrix element in these limits is well-understood. By exploiting the factorisation property of matrix elements in these limits, the model can learn how to best represent the approximation of the matrix elements as a linear combination of singular functions. We examine the application of the model to e−e+ annihilation matrix elements at tree-level and one-loop level, as well as to leading order pp collisions where the acceleration of event generation is critical for current experiments

Uncovering spatio-temporal patterns in semiconductor superlattices by efficient data processing tools

Time periodic patterns in a semiconductor superlattice, relevant to microwave generation, are obtained upon numerical integration of a known set of drift-diffusion equations. The associated spatiotemporal transport mechanisms are uncovered by applying (to the computed data) two recent data processing tools, known as the higher order dynamic mode decomposition and the spatiotemporal Koopman decomposition. Outcomes include a clear identification of the asymptotic self-sustained oscillations of the current density (isolated from the transient dynamics) and an accurate description of the electric field traveling pulse in terms of its dispersion diagram. In addition, a preliminary version of a data-driven reduced order model is constructed, which allows for extremely fast online simulations of the system response over a range of different configurations.The authors are indebted to two anonymous referees for some useful comments and suggestions on an earlier version of the paper. This work has been supported by the Fondo Europeo de Desarrollo Regional Ministerio de Ciencia, Innovación y Universidades–Agencia Estatal de Investigación, under Grants No. TRA2016-75075-R, No. MTM2017-84446-C2-2-R, and No. PID2020-112796RB-C22, and by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23) and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation)

Challenges in Computational Electromagnetics:Analysis and Optimization of Planar Multilayered Structures

To meet strict requirements of the information society technologies, antennas and circuit elements are becoming increasingly complex. Frequently, their electromagnetic (EM) properties cannot be anymore expressed in closed-form analytical expressions mainly because of the multitude of irregular geometries found in actual devices. Therefore, accurate and efficient (in terms of computational time and memory) electromagnetic models coupled with the robust optimization techniques, are needed in order to be able to predict and optimize the behavior of the innovative antennas in complex environments. The contribution of this thesis consists in the development and improvement of accurate electromagnetic modeling and optimization algorithms for an ubiquitous class of antennas, the planar printed antennas. The approach most commonly used to model and analyze this type of structures is the Integral Equation (IE) technique numerically solved using the Method of Moments (MoM). From the computational point of view, the main challenge is to develop techniques for efficient numerical evaluation of spatial-domain Green's functions, which are commonly expressed in terms of the well-known Sommerfeld integrals (SIs), i.e., semi-infinite range integrals with Bessel function kernels. Generally, the analytical solution of the SIs is not available, and their numerical evaluation is notoriously difficult and time-consuming because the integrands are both oscillatory and slowly decaying, and might possess singularities on and/or near the integration path. Due to the key role that SIs play in many EM problems, the development of fast and accurate techniques for their evaluation is of paramount relevance. This problem is studied in detail and several efficient methods are developed. Finally, the applicability of one of these methods, namely the Weighted Averages (WA) technique, is extended to the challenging case appearing in many practical EM problems: the evaluation of semi-infinite integrals involving products of Bessel functions. However, the development of effective analysis codes is only one aspect. At least equally important is the availability of reliable optimization techniques for an adequate design of antennas. For that purpose, the Particle Swarm Optimization (PSO) algorithm is introduced in the context of our analysis codes. Moreover, the innovative hybrid version of the PSO algorithm, called the Tournament Selection PSO, has been proposed with the aim of even further improving convergence performances of the classical PSO algorithm. Detailed theoretical description of this socially inspired evolutionary algorithm is given in the thesis. Finally, the characteristics of both algorithms are compared throughout several EM optimization problems

Fermionic SK-models with Hubbard interaction: Magnetism and electronic structure

Models with range-free frustrated Ising spin- and Hubbard interaction are treated exactly by means of the discrete time slicing method. Critical and tricritical points, correlations, and the fermion propagator, are derived as a function of temperature T, chemical potential \mu, Hubbard coupling U, and spin glass energy J. The phase diagram is obtained. Replica symmetry breaking (RSB)-effects are evaluated up to four-step order (4RSB). The use of exact relations together with the 4RSB-solutions allow to model exact solutions by interpolation. For T=0, our numerical results provide strong evidence that the exact density of states in the spin glass pseudogap regime obeys \rho(E)=const |E-E_F| for energies close to the Fermi level. Rapid convergence of \rho'(E_F) under increasing order of RSB is observed. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of localized disordered fermionic systems in 2D. Beyond half filling we obtain a quadratic dependence of the fermion filling factor on the chemical potential. We find a half filling transition between a phase for U>\mu, where the Fermi level lies inside the Hubbard gap, into a phase where \mu(>U) is located at the center of the upper spin glass pseudogap (SG-gap). For \mu>U the Hubbard gap combines with the lower one of two SG-gaps (phase I), while for \mu<U it joins the sole SG-gap of the half-filling regime (phase II). We predict scaling behaviour at the continuous half filling transition. Implications of the half-filling transition between the deeper insulating phase II and phase I for delocalization due to hopping processes in itinerant model extensions are discussed and metal-insulator transition scenarios described.Comment: 29 pages, 26 Figures, 4 jpeg- and 3 gif-Fig-files include