Descritores moleculares são essenciais em muitas aplicações de física e química computacional, como na análise de similaridade entre ligantes baseada em sua estrutura. Harmônicos esféricos têm sido utilizados como descritores da superfície molecular por serem uma forma compacta de descrição geométrica e por possuírem um descritor invariante por rotação. Assim, este trabalho propõe um método de análise de similaridade estrutural entre ligantes no qual se modela a superfície de uma molécula através de uma expansão em harmônicos esféricos realizada pelo programa LIRA. Os coeficientes encontrados são utilizados para percorrer o banco de dados DUD-E, com descritores previamente calculados, utilizando Distância Euclidiana e diversos valores de corte para selecionar compostos mais semelhantes. O potencial do método é avaliado usando o Ultrafast Shape Recognition (USR) como método padrão, pelo fato de ser uma excelente e rápida métrica para análise da similaridade de ligantes. Foram selecionadas 50 moléculas de diferentes tamanhos e composição de forma a representar todos os grupos moleculares presentes na DUD-E. Em seguida, cada molécula foi submetida à busca de similares variando-se valores de corte para o LIRA em que o conjunto de moléculas selecionadas foi comparado com as selecionadas pelo USR através de um processo de classificação binária e criação e interpretação de curvas ROC. Além do benchmarking, foi realizada a análise das componentes principais para determinar quais descritores são os mais importantes e carregam as melhores informações utilizadas na descrição da superfície da molécula. A partir das componentes principais, foi realizado um estudo do uso de funções peso, associando mais importância aos descritores adequados, e a redução da dimensionalidade do banco de dados, seleção de um novo conjunto de autovetores que formam as bases do espaço vetorial e uma nova descrição das moléculas para o novo espaço, no qual cada variação foi avaliada através de um novo benchmarking. O LIRA se mostrou tão rápido quanto o USR e apresentou grande potencial de seleção de moléculas similares, para a maioria das moléculas testadas, pois as curvas ROC apresentaram pontos acima da linha do aleatório. Tanto a redução da dimensionalidade quanto o uso de funções de ponderação agregaram valor à métrica deixando-a mais veloz, no caso da redução da quantidade de descritores, e seletiva, em ambos os casos. Dessa forma, o método proposto se mostrou eficiente em mensurar a similaridade entre ligantes de forma seletiva e rápida utilizando somente informações a respeito da superfície molecular. / Molecular descriptors are essential for many applications in computational chemistry and physics, such as ligand-based similarity searching. Spherical harmonics have previously been suggested as comprehensive descriptors of molecular structure due to their properties, orthonormality and rotationally invariant. Here we proposed a ligand similarity analysis method where molecule\'s surface is modeled by an expansion in Spherical Harmonics, called LIRA, whose coefficient are used to perform a search in DUD-E database, with all descriptors previously calculated, measured by Euclidian Distance and different cutoff\'s values to select similar compounds. Method\'s potential is evaluated against Ultrafast Shape Recognition (USR), due to it is an excellent a fast metric to ligand similarity analysis, in a benchmarking. Fifty molecules are selected varying chemical composition and size to represent all molecular groups of DUD-E. After that, which one was submitted in a search with different values of cutoff for LIRA and the subset selected was compared with the ones selected by USR through binary classification and ROC curves analysis. Beyond benchmarking, it was performed a principal component analysis to identify which are the most valuable coefficient for shape description. Using principal components two other studies are made, weight functions are applied to descriptors, providing more value for those carry more information, and dimensionality reduction, where a subset of eigenvectors are select to form the new basis of the vector space and the new molecule\'s description was made in the new space, which variation was tested in a new benchmarking. Lira showed to be as fast as USR and a big potential to select similar molecules, for the majority of the molecules tested, because ROC curves had points over the random line. Dimensionality reduction and weight functions improved LIRA results raising velocity, due to the use of less descriptors to model molecule\'s surface, and the selection power, for both cases. In summary, the proposed method showed to be an efficient and fast tool for measure similarity between ligands based in molecular shape.

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Descritores moleculares são essenciais em muitas aplicações de física e química computacional, como na análise de similaridade entre ligantes baseada em sua estrutura. Harmônicos esféricos têm sido utilizados como descritores da superfície molecular por serem uma forma compacta de descrição geométrica e por possuírem um descritor invariante por rotação. Assim, este trabalho propõe um método de análise de similaridade estrutural entre ligantes no qual se modela a superfície de uma molécula através de uma expansão em harmônicos esféricos realizada pelo programa LIRA. Os coeficientes encontrados são utilizados para percorrer o banco de dados DUD-E, com descritores previamente calculados, utilizando Distância Euclidiana e diversos valores de corte para selecionar compostos mais semelhantes. O potencial do método é avaliado usando o Ultrafast Shape Recognition (USR) como método padrão, pelo fato de ser uma excelente e rápida métrica para análise da similaridade de ligantes. Foram selecionadas 50 moléculas de diferentes tamanhos e composição de forma a representar todos os grupos moleculares presentes na DUD-E. Em seguida, cada molécula foi submetida à busca de similares variando-se valores de corte para o LIRA em que o conjunto de moléculas selecionadas foi comparado com as selecionadas pelo USR através de um processo de classificação binária e criação e interpretação de curvas ROC. Além do benchmarking, foi realizada a análise das componentes principais para determinar quais descritores são os mais importantes e carregam as melhores informações utilizadas na descrição da superfície da molécula. A partir das componentes principais, foi realizado um estudo do uso de funções peso, associando mais importância aos descritores adequados, e a redução da dimensionalidade do banco de dados, seleção de um novo conjunto de autovetores que formam as bases do espaço vetorial e uma nova descrição das moléculas para o novo espaço, no qual cada variação foi avaliada através de um novo benchmarking. O LIRA se mostrou tão rápido quanto o USR e apresentou grande potencial de seleção de moléculas similares, para a maioria das moléculas testadas, pois as curvas ROC apresentaram pontos acima da linha do aleatório. Tanto a redução da dimensionalidade quanto o uso de funções de ponderação agregaram valor à métrica deixando-a mais veloz, no caso da redução da quantidade de descritores, e seletiva, em ambos os casos. Dessa forma, o método proposto se mostrou eficiente em mensurar a similaridade entre ligantes de forma seletiva e rápida utilizando somente informações a respeito da superfície molecular. / Molecular descriptors are essential for many applications in computational chemistry and physics, such as ligand-based similarity searching. Spherical harmonics have previously been suggested as comprehensive descriptors of molecular structure due to their properties, orthonormality and rotationally invariant. Here we proposed a ligand similarity analysis method where molecule\'s surface is modeled by an expansion in Spherical Harmonics, called LIRA, whose coefficient are used to perform a search in DUD-E database, with all descriptors previously calculated, measured by Euclidian Distance and different cutoff\'s values to select similar compounds. Method\'s potential is evaluated against Ultrafast Shape Recognition (USR), due to it is an excellent a fast metric to ligand similarity analysis, in a benchmarking. Fifty molecules are selected varying chemical composition and size to represent all molecular groups of DUD-E. After that, which one was submitted in a search with different values of cutoff for LIRA and the subset selected was compared with the ones selected by USR through binary classification and ROC curves analysis. Beyond benchmarking, it was performed a principal component analysis to identify which are the most valuable coefficient for shape description. Using principal components two other studies are made, weight functions are applied to descriptors, providing more value for those carry more information, and dimensionality reduction, where a subset of eigenvectors are select to form the new basis of the vector space and the new molecule\'s description was made in the new space, which variation was tested in a new benchmarking. Lira showed to be as fast as USR and a big potential to select similar molecules, for the majority of the molecules tested, because ROC curves had points over the random line. Dimensionality reduction and weight functions improved LIRA results raising velocity, due to the use of less descriptors to model molecule\'s surface, and the selection power, for both cases. In summary, the proposed method showed to be an efficient and fast tool for measure similarity between ligands based in molecular shape.

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Neste trabalho estudaremos subconjuntos especiais da esfera unitária Sm-1 de Rm, m 2, comumente chamados na literatura de designs esféricos. Os objetivos principais são analisar várias equivalências para o conceito, suas conexões com ambos, rotações sobre Sm?1 e mergulhos em esferas de dimensão superior, e resultados sobre a cardinalidade dos designs esféricos / In this work we will study specials subsets of the unitary sphere Sm-1 of Rm, m 2, usually called in literature spherical designs. The main objectives are to analyze many equivalences for the concept, its connections with both, rotations on Sm?1 and embedded in spheres of higher dimension, and results on the cardinality of spherical designs

Neste trabalho estudaremos subconjuntos especiais da esfera unitária Sm-1 de Rm, m 2, comumente chamados na literatura de designs esféricos. Os objetivos principais são analisar várias equivalências para o conceito, suas conexões com ambos, rotações sobre Sm?1 e mergulhos em esferas de dimensão superior, e resultados sobre a cardinalidade dos designs esféricos / In this work we will study specials subsets of the unitary sphere Sm-1 of Rm, m 2, usually called in literature spherical designs. The main objectives are to analyze many equivalences for the concept, its connections with both, rotations on Sm?1 and embedded in spheres of higher dimension, and results on the cardinality of spherical designs

This research is concerned with the self-adaptive numerical solution of the neutral particle radiation transport problem. Radiation transport is an extremely challenging computational problem since the governing equation is seven-dimensional (3 in space, 2 in direction, 1 in energy, and 1 in time) with a high degree of coupling between these variables. If not careful, this relatively large number of independent variables when discretized can potentially lead to sets of linear equations of intractable size. Though parallel computing has allowed the solution of very large problems, available computational resources will always be finite due to the fact that ever more sophisticated multiphysics models are being demanded by industry. There is thus the pressing requirement to optimize the discretizations so as to minimize the effort and maximize the accuracy. One way to achieve this goal is through adaptive phase-space refinement. Unfortunately, the quality of discretization (and its solution) is, in general, not known a priori; accurate error estimates can only be attained via the a posteriori error analysis. In particular, in the context of the finite element method, the a posteriori error analysis provides a rigorous error bound. The main difficulty in applying a well-established a posteriori error analysis and subsequent adaptive refinement in the context of radiation transport is the strong coupling between spatial and angular variables. This research attempts to address this issue within the context of the second-order, even-parity form of the transport equation discretized with the finite-element spherical harmonics method. The objective of this thesis is to develop a posteriori error analysis in a coupled space-angle framework and an efficient adaptive algorithm. Moreover, the mesh refinement strategy which is tuned for minimizing the error in the target engineering output has been developed by employing the dual argument of the problem. This numerical framework has been implemented in the general-purpose neutral particle code EVENT for assessment.

Diffusion MRI (dMRI) measurements allow us to infer the microstructural properties of white matter and to reconstruct fiber pathways in-vivo. High angular diffusion imaging (HARDI) allows for the creation of more and more complex local models connecting the microstructure to the measured signal. One of the challenges is the derivation of meaningful metrics describing the underlying structure from the local models. The aim hereby is to increase the specificity of the widely used metric fractional anisotropy (FA) by using the additional information contained within the HARDI data. A local model which is connected directly to the underlying microstructure through the model of a single fiber population is spherical deconvolution. It produces a fiber orientation density function (fODF), which can often be interpreted as superposition of multiple peaks, each associated to one relatively coherent fiber population (bundle). Parameterizing these peaks one is able to disentangle and characterize these bundles. In this work, the fODF peaks are approximated by Bingham distributions, capturing first and second order statistics of the fiber orientations, from which metrics for the parametric quantification of fiber bundles are derived. Meaningful relationships between these measures and the underlying microstructural properties are proposed. The focus lies on metrics derived directly from properties of the Bingham distribution, such as peak length, peak direction, peak spread, integral over the peak, as well as a metric derived from the comparison of the largest peaks, which probes the complexity of the underlying microstructure. These metrics are compared to the conventionally used fractional anisotropy (FA) and it is shown how they may help to increase the specificity of the characterization of microstructural properties. Visualization of the micro-structural arrangement is another application of dMRI. This is done by using tractography to propagate the fiber layout, extracted from the local model, in each voxel. In practice most tractography algorithms use little of the additional information gained from HARDI based local models aside from the reconstructed fiber bundle directions. In this work an approach to tractography based on the Bingham parameterization of the fODF is introduced. For each of the fiber populations present in a voxel the diffusion signal and tensor are computed. Then tensor deflection tractography is performed. This allows incorporating the complete bundle information, performing local interpolation as well as using multiple directions per voxel for generating tracts. Another aspect of this work is the investigation of the spherical harmonic representation which is used most commonly for the fODF by means of the parameters derived from the Bingham distribution fit. Here a strong connection between the approximation errors in the spherical representation of the Dirac delta function and the distribution of crossing angles recovered from the fODF was discovered. The final aspect of this work is the application of the metrics derived from the Bingham fit to a number of fetal datasets for quantifying the brain’s development. This is done by introducing the Gini-coefficient as a metric describing the brain’s age.

Nuclear reactor design requires the calculation of integral core parameters and power and radiation profiles. These physical parameters are obtained by the solution of the linear neutron transport equation over the geometry of the reactor. In order to represent the fine structure of the nuclear core a very small geometrical mesh size should be used, but the computational capacity available these days is still not enough to solve these transport problems in the time range (hours-days) that would make the method useful as a design tool. This problem is traditionally solved by the solution of simple, smaller problems in specific parts of the core and then use a procedure known as homogenization to create average material properties and solve the full problem with a wider mesh size. The iterative multi-level solution procedure is inspired in this multi-stage approach, solving the problem at fuel-pin (cell) level, fuel assembly and nodal levels. The nested geometrical structure of the finite element representation of a reactor can be used to create a set of restriction/prolongation operators to connect the solution in the different levels. The procedure is to iterate between the levels, solving for the error in the coarse level using as source the restricted residual of the solution in the finer level. This way, the complete problem is only solved in the coarsest level and in the other levels only a pair of restriction/interpolation operations and a relaxation is required. In this work, a multigrid solver is developed for the in-moment equation of the spherical harmonics, finite element formulation of the second order transport equation. This solver is implemented as a subroutine in the code EVENT. Numerical tests are provided as a standalone diffusion solver and as part of a block Jacobi transport solver.

<p>Fans are often used in equipment such as home appliances andelectronic equipment where the margin of profit is small butcustomers demands on a low noise level are high. Therefore,methods for predicting the noise emitted by an applicationincluding one or several fans are desirable in order toimprove, accelerate and reduce the cost of low-noise design.The Noise Shaping Technology (NST) has been developed withinthe EC-project NABUCCO in order to fulfil the aboverequirements on a prediction method. According to NST, thenoise source (not necessary a fan) is described by one orseveral noise descriptors, CSSs, and the correspondingtransmission paths through the structure described by one orseveral transfer functions, ACFs. In this thesis, theapplicability of NST is evaluated on a cabinet for electronicequipment where small axial cooling fans constitute the primarysources of the airborne sound.</p><p>As an axial fan is a complex source of sound,simplifications are necessary when modelling its acousticproperties. Therefore, the sound radiation of an axial fan infree space was examined by expanding the generated soundpressure field into spherical harmonics. The conclusion on asource model for the cabinet example, where the fans are moreor less In-duct mounted, is a modified single axial dipole. Themodel is expected to be valid in the entire frequency range ofinterest except in the mid-frequency range where the modaldensity is low. In order to improve the source model in thisfrequency range, a future model based on a rotating dipole isproposed.</p><p>The sound power of a small axial fan is measured in an ISO10302 test-rig. In order to take account of flow conditions,acoustically transparent ducts have been developed. These shallbe attached to the test-rig when measuring the sound power ofthe fan. A simple but practical method of how to correct thesound power for the baffling effect of the test-rig has alsobeen developed. Finally, the sound power can be converted intodipole force, which is the airborne CSS corresponding to thesingle axial dipole model.</p><p>The corresponding airborne transfer function (ACF), i.e.,from dipole force at the source point to sound pressure at thereceiver point, is measured reciprocally by taking use ofLyamshevs reciprocity relation.</p><p>From multiplication of the CSS and the ACF, the soundpressure can be predicted. The prediction shows quite goodagreement with the measured values.</p><p><b>Keywords:</b>axial fan, airborne sound, sourcecharacterisation, transmission path analysis, In-duct,spherical harmonics, rotating dipole, installation effects, ISO10302, flow conditions, baffling effect, acousticallytransparent ducts, Lyamshevs reciprocity relation, reciprocity,CSS, ACF, GSM, NST.</p>