Spatiospectral concentration of vector fields on a sphere

We construct spherical vector bases that are bandlimited and spatially concentrated, or, alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function bases into arbitrarily shaped regions of interest on the sphere, and within certain bandlimits in the vector spherical-harmonic domain. As with the concentration problem for scalar functions on the sphere, which has been treated in detail elsewhere, a Slepian vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate problems for the radial and tangential components. For regions with advanced symmetry such as polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, lending itself to efficient diagonalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical-harmonic degree or bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over the entire sphere and over the geographic target region. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.Comment: Submitted to Applied and Computational Harmonic Analysi

Transcending the Rayleigh Hypothesis with multipolar sources distributed across the topological skeleton of a scatterer

There is an ever-growing need to study the optical response of complex photonic systems involving multi-scattering phenomena with strong near-field interactions. Since fully numerical methods often imply high computational costs, semi-analytical methods are preferred. However, most semi-analytical methods are commonly plagued by what is known as the problem of the Rayleigh Hypothesis: they typically use analytical representations of the scattered fields that are invalid in the near-field region of the scatterer. In this work, we present an alternative representation scheme for the scattered fields based on a distribution of multipolar sources across the topological skeleton of the scatterer. We demonstrate how such a representation overcomes the problem of the Rayleigh Hypothesis for scatterers of arbitrary geometry. In that regard, our work enriches the available toolkit of semi-analytical methods in light-scattering by pushing decisively against one of the fundamental limitations of the existing methods

Spherical Harmonics on constitutive equations for biological cells

Tese (doutorado)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Civil e Ambiental, 2019.Desenvolvem-se e avaliam-se neste trabalho modelos constitutivos não-lineares incluindo o estudo de grandes deformações com o objetivo de modelar células biológicas representadas por elementos de cascas finas. É utilizada como ponto de partida a formulação clássica de elementos de cascas finas, considerando as hipóteses de Kirchhoff que apresentam como mais importante característica a redução dimensional. Esta é atingida derivando tensões 2D como médias das tensões 3D pela integração direta sob a espessura da casca. Para a definição da deformação do continuo é utilizada uma descrição Lagrangiana. As células biológicas não podem ser modeladas de forma correta utilizando modelos constitutivos lineares. Especificamente no estudo dos glóbulos vermelhos devem ser considerados: o comportamento elástico não linear e o aporte da viscosidade da parede da célula. Consequentemente, neste trabalho, modelos hiperelasticos são implementados junto ao modelo de Kelvin-Voigth para obter um modelo viscoelástico. Na implementação computacional Funções de Esféricos Harmônicos são utilizadas para sintetizar as principais variáveis, esforços e deslocamentos. Isto se deve a que a geometria dos glóbulos vermelhos pode ser descrita de forma simples utilizando coordenadas esféricas. Resultando numa implementação de baixo custo computacional que consegue lidar com altas não linearidades. Este trabalho apresenta uma formulação de um método indireto pois consiste no cálculo de coeficientes da expansão de Esféricos Harmônicos, sendo que estes coeficientes não têm sentido físico. É importante mencionar que o projeto se encontra num estágio inicial e não foi encontrado na literatura uma aplicação utilizando teoria de cascas, Harmônicos Esféricos junto com modelos constitutivos lidando com grandes deformações. Finalmente o método é validado e estudado suas possíveis aplicações.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) e Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).In this work, constitutive models are developed and evaluated with the aim of modeling biological cells represented by thin shell elements in a second-order analysis. The classical formulation of thin shell elements is used while considering dimensional reduction, which is the main feature of the Kirchhoff hypotheses. This reduction is achieved by deriving two-dimensional stresses as averages of the true three-dimensional stresses by means of direct integration through the shell thickness. A Lagrangian description is used to define the deformation of the continuum. Biological cells cannot be correctly modeled using linear constitutive relations. Specifically, in the study of red blood cells, one should consider both their nonlinear elastic behavior and the contribution of the cell wall viscosity. Consequently, hyperelastic constitutive equations are implemented using the Kelvin-Voigt approach to obtain a viscoelastic model. In the computational implementation, spherical harmonic functions are used to synthesize the main variables, resultant forces and displacements since the geometry of red blood cells can be simply described using spherical coordinates. As a result, a low-cost computational implementation for highly nonlinear analyses is obtained. This work presents a formulation of an indirect method since consists on the calculation of the expansion coefficients of a Spherical Harmonic Analysis, these coefficients have no physical meaning. It is important to mention that this work is part of a project that is at an early stage. In the literature no application was found using shell theory, Spherical Harmonics with constitutive models dealing with large deformations. Finally, the method is validated and its possible applications are discussed

A time-dependent spectral point spread function for the OSIRIS optical spectrograph

The primary goal of the recently formed Absorption Cross Sections of Ozone (ACSO) Commission is to establish an international standard for the ozone cross section used in the retrieval of atmospheric ozone number density profiles. The Canadian instrument OSIRIS onboard the Swedish spacecraft Odin has produced high quality ozone profiles since 2002, and as such the OSIRIS research team has been asked to contribute to the ACSO Commission by evaluating the impact of implementing different ozone cross sections into SASKTRAN, the radiative transfer model used in the retrieval of OSIRIS ozone profiles. Preliminary analysis revealed that the current state of the OSIRIS spectral point spread function, an array of values describing the dispersion of light within OSIRIS, would make such an evaluation difficult. Specifically, the current spectral point spread function is time-independent and therefore unable to account for any changes in the optics introduced by changes in the operational environment of the instrument. Such a situation introduces systematic errors when modelling the atmosphere as seen by OSIRIS, errors that impact the quality of the ozone number density profiles retrieved from OSIRIS measurements and make it difficult to accurately evaluate the impact of using different ozone cross sections within the SASKTRAN model. To eliminate these errors a method is developed to calculate, for the 310-350 nm wavelength range, a unique spectral point spread function for every scan in the OSIRIS mission history, the end result of which is a time-dependent spectral point spread function. The development of a modelling equation is then presented, which allows for any noise present in the time-dependent spectral point spread function to be reduced and relates the spectral point spread function to measured satellite parameters. Implementing this modelled time-dependent spectral point spread function into OSIRIS ozone retrieval algorithms is shown to improve all OSIRIS ozone profiles by 1-2% for tangent altitudes of 35-48 km. Analysis is also presented that reveals a previously unaccounted for temperature-dependent altitude shift in OSIRIS measurements. In conjunction with the use of the time-dependent spectral point spread function, accounting for this altitude shift is shown to result in an almost complete elimination of the temperature-induced systematic errors seen in OSIRIS ozone profiles. Such improvements lead to improved ozone number density profiles for all times of the OSIRIS mission and make it possible to evaluate the use of different ozone cross sections as requested by the ACSO Commission