Coverage Optimization Using Lattice Flower Constellations

Recently developed satellite systems require a group of satellites acting in concert with one another to meet mission objectives. Specifying a constellation by defining all the orbit elements for each satellite is complex, inconvenient, and computationally impossible for constellations with many satellites. There are many degrees of freedom in the parameters for constellations such as number of orbital planes, number of satellites in the orbital plane, orbital inclination and altitude. Therefore, an efficient way to design to a constellation is to adopt some orbital elements with common value and some other derived by algorithms and various algorithms have been proposed. Among them, the Lattice Flower Constellations theory is more suitable to optimization of constellation design because it is a minimum parameterization theory and because this design methodology contains most of the existing methodologies as subsets. The main contributions of this dissertation are 1) the development of an algorithm which provides uniform points on a sphere for fast evaluation of coverage fitness functions, 2) the presentation of a set of three non-classical constellation missions using Lattice Flower Constellations, and 3) the investigation of a new class of orbits, called “J2-propelled,” and associated constellations which are particularly suitable in the three-dimensional lattice theory of flower constellations. For global coverage missions, fitness functions for constellation design are computed using globally distributed points. Most of the grid data sets are provided with a fixed step in latitude and longitude. Therefore, conventionally computed points are distributed with a fixed step in latitude and longitude. Since these are certainly not uniform distributions of points on the Earth, mainly due to the increase of point density at high latitude regions, converting these data into an “equivalent” distribution of points (with different weights) is needed. This will allow the data sets to be dramatically decreased to small amount data sets with appropriate values and, consequently, computational burden is then reduced using \equivalent" uniformly distributed points. For elliptical constellations, the Lattice Flower Constellations require nine design parameters of which six are integers. For circular constellations, there are five required design parameters of which three are integers. A general optimization technique implies finding the optimal values of these parameters. This dissertation introduces a general process to perform constellation optimization for any specific optimality definition, that is, for any specific space mission. To demonstrate the feasibility and the effectiveness of the proposed approach this optimization tool is applied to three distinct types of space missions: a) global radio occultation, b) interferometric imaging, and c) constrained communication missions. The results obtained validate the proposed methodology. A linear theory to design orbits and constellations where the Earth oblateness perturbation, the J2 perturbation, generates dynamics that are periodic in an inertial or in a rotating frame is presented. In J2-propelled orbits, the linear (secular) J2 effect is used instead of being fought to allow the satellites accessing specific 3-dimensional volumes around the Earth. The main motivation is to design space missions (satellites and constellations) able to measure physical quantities (e.g., magnetic or electric fields) in large space volumes by limiting the control costs to compensate the other gravitational and non-gravitational orbital perturbations only

Variational Methods in Shape Space

This dissertation deals with the application of variational methods in spaces of geometric shapes. In particular, the treated topics include shape averaging, principal component analysis in shape space, computation of geodesic paths in shape space, as well as shape optimisation. Chapter 1 provides a brief overview over the employed models of shape space. Geometric shapes are identified with two- or three-dimensional, deformable objects. Deformations will be described via physical models; in particular, the objects will be interpreted as consisting of either a hyperelastic solid or a viscous liquid material. Furthermore, the description of shapes via phase fields or level sets is briefly introduced. Chapter 2 reviews different and related approaches to shape space modelling. References to related topics in image segmentation and registration are also provided. Finally, the relevant shape optimisation literature is introduced. Chapter 3 recapitulates the employed concepts from continuum mechanics and phase field modelling and states basic theoretical results needed for the later analysis. Chapter 4 addresses the computation of shape averages, based on a hyperelastic notion of shape dissimilarity: The dissimilarity between two shapes is measured as the minimum deformation energy required to deform the first into the second shape. A corresponding phase-field model is introduced, analysed, and finally implemented numerically via finite elements. A principal component analysis of shapes, which is consistent with the previously introduced average, is considered in Chapter 5. Elastic boundary stresses on the average shape are used as representatives of the input shapes in a linear vector space. On these linear representatives, a standard principal component analysis can be performed, where the employed covariance metric should be properly chosen to depend on the input shapes. Chapter 6 interprets shapes as belonging to objects made of a viscous liquid and correspondingly defines geodesic paths between shapes. The energy of a path is given as the total physical dissipation during the deformation of an object along the path. A rigid body motion invariant time discretisation is achieved by approximating the dissipation along a path segment by the deformation energy of a small solid deformation. The numerical implementation is based on level sets. Chapter 7 is concerned with the optimisation of the geometry and topology of solid structures that are subject to a mechanical load. Given the load configuration, the structure rigidity, its volume, and its surface area shall be optimally balanced. A phase field model is devised and analysed for this purpose. In this context, the use of nonlinear elasticity allows to detect buckling phenomena which would be ignored in linearised elasticity

Technology for large space systems: A bibliography with indexes (supplement 16)

This bibliography lists 673 reports, articles and other documents introduced into the NASA scientific and technical information system between July 1, 1986 and December 31, 1986. Its purpose is to provide helpful information to the researcher, manager, and designer in technology development and mission design according to system interactive analysis and design, structural and thermal analysis and design, structural concepts and control systems, electronics, advanced materials, assembly concepts, propulsion, and solar power satellite systems

Towards a new spacetime paradigm : Gauge symmetries and post-Riemann geometries in gravitation

In this thesis the geometrical methods and symmetry principles in gravitation are explored motivating a new perspective into the spacetime paradigm. The effects of post-Riemann spacetime geometries with torsion are studied in applications to fundamental fermionic and bosonic fields, cosmology, astrophysics and gravitational waves. The physical implications and related phenomenological considerations of this study are addressed, and the fundamental ideas related to spacetime physics, motivated by geometrical methods and symmetry principles, are also discussed in the context of the possible routes towards a new spacetime paradigm in gravitation and unified field theories. We explore the analogies between the gauge approach to gravity and the pre-metric approach to electrodynamics, within the exterior calculus of forms. These analogies are developed, reinforcing the hypothesis of the primacy of the conformal structure over the metric structure. Since the conformal symmetries seem to be broken symmetries in nature, the Poincar´e gauge theories of gravity (PGTG) are taken into consideration. These presuppose a Riemann-Cartan (RC) spacetime geometry with curvature and torsion and motivate the search for torsion effects in physical systems. We study both minimal and non-minimal couplings of fermionic spinors to the background torsion and find changes in the energy levels (in the flat spacetime limit), including parity breaking effects. The Einstein-Cartan-Dirac-Maxwell theory is explored including its cosmological applications. The presence of minimal couplings to torsion induces non-linearities and non-minimal couplings in the matter fields dynamics and the resulting cosmological model is non-singular, including early and future bounces, early acceleration and torsion induced dark-energy due to fermionic vacuum condensates. Some potential astrophysical applications due to the torsion interaction with fermionic and bosonic fields are also considered as well as the effects of curvature in electromagnetic fields, including the extensions with inhomogeneous and anisotropic constitutive electromagnetic relations that respect the local isometries. In this context, the Parametrized Post-Newtonian (PPN) formalism is also implemented, making a bridge with the testing of different gravity theories. The effects of torsion are also analysed in gravitational wave (GW) physics, following the perturbations of a RC spacetime and in the field equations of a specific quadratic PGTG. The gravitational wave effects into electromagnetic fields are also studied with potential applications for non-standard detectors, which in principle could be extended for theories beyond GR, searches of extra polarizations and extra degrees of freedom

History of Mathematics: Models and Visualization in the Mathematical and Physical Sciences

This workshop brought together historians of mathematics and science as well as mathematicians to explore important historical developments connected with models and visual elements in the mathematical and physical sciences. It addressed the larger question of what has been meant by a model, a notion that has seldom been subjected to careful historical study. Most of the talks dealt with case studies from the period 1800 to 1950 that covered a number of analytical, geometrical, mechanical, astronomical, and physical phenomena. The workshop also considered the role of visual thinking as a component of mathematical creativity and understanding

Construction and dynamics of knotted fields in soft matter systems

Knotted fields are physical fields containing knotted, linked, or otherwise topologically interesting structure. They occur in a wide variety of physical systems — fluids, superfluids, electromagnetism, optics and high energy physics to name a few. Far from being passive structures, the occurrence of knotting in a physical field often modifies its overall properties, rendering their study interesting from both a theoretical and practical point of view. In this thesis, we focus on knotted fields in ‘soft matter’ systems, systems which may be loosely characterised as those in which geometry plays a fundamental role, and which undergo substantial deformations in response to external forces, changes in temperature etc. Such systems are often experimentally accessible, making them natural testbeds for exploring knotted fields in all their guises. After providing an introduction to knotted fields with a focus on soft matter in the first chapter, in the second we introduce a method of explicitly constructing such fields for any knotted curve based on Maxwell’s solid angle construction. We discuss its theory, emphasising a fundamental homotopy formula as unifying methods for computing the solid angle, as well as describing a naturally induced curve framing, which we show is related to the writhe of the curve before using it to characterise the local structure in the neighbourhood of the knot. We then discuss its practical implementation, giving examples of its use and providing C code. In subsequent chapters we use this methodology to initialise simulations in our study of knotted fields in two soft matter systems: excitable media and twist-bend nematics. In excitable media we provide a systematic survey of knot dynamics up to crossing number eight, finding generically unsteady behaviour driven by a wave-slapping mechanism. Nevertheless, we also find novel complex knotted structures and characterise their geometry and steady state motion, as well as greatly expanding upon previous evidence to demonstrate the ability of the dynamics to untangle geometries without reconnection. In twist-bend nematics we describe their fundamental geometry, that of bend. The zeros of bend are a set of lines with rich geometric and topological structure. We characterise their local structure, describe how they are canonically oriented and discuss a notion of their self-linking. We then describe their topological significance, showing that these zeros compute Skyrmion and Hopfion numbers, with accompanying simulations in twist-bend nematics

Are the waves detected by LIGO the waves according to Einstein, Pirani, Bondi, Trautmann, Kopeikin or what are they?

From the geometric formulation of gravity, according to the Einstein-Grosmann-Hilbert equations, of November 1915, as the geodesic movement in the semirimennian manifold of positive curvature, spacetime, where due to absence of symmetries, the conservation of energy-impulse is not possible taking together the material processes and that of the gravitational geometric field, however, given those symmetries in the flat Minkowski spacetime, using the De Sitter model, Einstein linearizing gravitation, of course, really in the absence of gravity, in 1916, purged of some mathematical errors in 1918, he introduced "gravitational waves" as disturbances in the curvature of space, and in the absence of knowing physically what spacetime is and philosophically in dispute, that previously in 1936 and definitively in 1937, Einstein showed they did not exist. It was through the works arising from the dynamics of academic discourse, from the perspective not of Einstein but of Weyl, that Bondi, Pirani, Robinson and Trautman, in the 1950s, after Einstein's death, "gravitational waves" were reintroduced and led to experimental search. In 2002, from Sergei Kopeikin's VLBI experiment, its supposed speed was established, without obtaining unanimous recognition from the community of scientists but rather dividing them. And it was in February 2016 that the aLIGO-aVirgo collaboration announced that they had detected them for the first time. In this work, the history that led to this supposed discovery is presented and it is stated that the waves detected are really from the quantum vacuum in which everything that exists is immersed, the author's thesis exposed immediately in response to that 2016 announcement

Are the waves detected by LIGO the waves according to Einstein, Pirani, Bondi, Trautmann, Kopeikin or what are they?

From the geometric formulation of gravity, according to the Einstein-Grosmann-Hilbert equations, of November 1915, as the geodesic movement in the semirimennian manifold of positive curvature, spacetime, where due to absence of symmetries, the conservation of energy-impulse is not possible taking together the material processes and that of the gravitational geometric field, however, given those symmetries in the flat Minkowski spacetime, using the De Sitter model, Einstein linearizing gravitation, of course, really in the absence of gravity, in 1916, purged of some mathematical errors in 1918, he introduced "gravitational waves" as disturbances in the curvature of space, and in the absence of knowing physically what spacetime is and philosophically in dispute, that previously in 1936 and definitively in 1937, Einstein showed they did not exist. It was through the works arising from the dynamics of academic discourse, from the perspective not of Einstein but of Weyl, that Bondi, Pirani, Robinson and Trautman, in the 1950s, after Einstein's death, "gravitational waves" were reintroduced and led to experimental search. In 2002, from Sergei Kopeikin's VLBI experiment, its supposed speed was established, without obtaining unanimous recognition from the community of scientists but rather dividing them. And it was in February 2016 that the aLIGO-aVirgo collaboration announced that they had detected them for the first time. In this work, the history that led to this supposed discovery is presented and it is stated that the waves detected are really from the quantum vacuum in which everything that exists is immersed, the author's thesis exposed immediately in response to that 2016 announcement