Dissertationen zum Thema „Spherical Harmonic method“

In this thesis, the parallel performance of the finite element-spherical harmonics (FE-PN) method implemented in the general-purpose radiation transport code EVENT is studied both analytically and empirically. EVENT solves the coupled set of space-angle discretized FE-PN equations using a parallel block-Jacobi domain decomposition method. As part of the analytical study, the thesis presents complexity results for EVENT when solving for a 3D criticality benchmark radiation transport problem in parallel. The empirical analysis is concerned with the impact of the main algorithmic factors affecting performance. Firstly, EVENT supports two solution strategies, namely MOD (Moments Over Domains) and DOM (Domains Over Moments), to solve the transport equation in parallel. The two strategies differ in the way they solve the multi-level space-angle coupled systems of equations. The thesis presents empirical evidence of which of the two solution strategies is more efficient. Secondly, different preconditioners are used in the Preconditioned Conjugate Gradient (PCG) inside EVENT. Performance of EVENT is compared when using three preconditioners, namely diagonal, SSOR(Symmetric Successive Over-Relaxation) and ILU. The other two factors, angular and spatial resolutions of the problem affect both the performance and precision of EVENT. The thesis presents comparative results on EVENTs performance as these two resolutions are increased. From the empirical performance study of EVENT, a bottleneck is identified that limits the improvement in performance as number of processors used by EVENT is increased. In some experiments, it is observed that uneven assignment of computational load among processors causes a significant portion of the total time being spent in synchronization among processors. The thesis presents two indicators that identify when such inefficiency occur; and in such a case, a load rebalancing strategy is applied that computes a new partition of the problem so that each partition corresponds to equal amount of computational load.

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Dissertacao (Mestrado)
IPEN/D
Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP

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.

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Tese (Doutoramento)
IPEN/T
Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP

Fast algorithms for potential evaluation in N-body problems often tend to be extremely abstract and complex. This thesis presents a simple, hierarchical approach to solving the potential evaluation problem in O(n) time. The approach is developed in the field of electrostatics and can be extended to N-body problems in general. Herein, the potential vector is expressed as a product of the potential matrix and the charge vector. The potential matrix itself is a product of component matrices. The potential function satisfies the Laplace equation and is hence expressed as a linear combination of spherical harmonics, which form the general solutions of the Laplace equation. The orthogonality of the spherical harmonics is exploited to reduce execution time. The duality of the various lists in the algorithm is used to reduce storage and computational complexity. A smart tree-construction strategy leads to efficient parallelism at computation intensive stages of the algorithm. The computational complexity of the algorithm is better than that of the Fast Multipole Algorithm, which is one of the fastest contemporary algorithms to solve the potential evaluation problem. Experimental results show that accuracy of the algorithm is comparable to that of the Fast Multipole Algorithm. However, this approach uses some implementation principles from the Fast Multipole Algorithm. Parallel efficiency and scalability of the algorithms are studied by the experiments on IBM p690 multiprocessors.

This thesis addresses a number of problems in computer vision, image processing, and geometry processing, and presents novel solutions to these problems. The overarching theme of the techniques presented here is a multi-scale approach, leveraging mathematical tools to represent images and surfaces at different scales, and methods that can be adapted from one type of domain (eg., the plane) to another (eg., the sphere). The main problem addressed in this thesis is known as stereo reconstruction: reconstructing the geometry of a scene or object from two or more images of that scene. We develop novel algorithms to do this, which work for both planar and spherical images. By developing a novel way to formulate the notion of disparity for spherical images, we are able effectively adapt our algorithms from planar to spherical images. Our stereo reconstruction algorithm is based on a novel application of distance transforms to multi-scale matching. We use matching information aggregated over multiple scales, and enforce consistency between these scales using distance transforms. We then show how multiple spherical disparity maps can be efficiently and robustly fused using visibility and other geometric constraints. We then show how the reconstructed point clouds can be used to synthesize a realistic sequence of novel views, images from points of view not captured in the input images, in real-time. Along the way to this result, we address some related problems. For example, multi-scale features can be detected in spherical images by convolving those images with a filterbank, generating an overcomplete spherical wavelet representation of the image from which the multiscale features can be extracted. Convolution of spherical images is much more efficient in the spherical harmonic domain than in the spatial domain. Thus, we develop a GPU implementation for fast spherical harmonic transforms and frequency domain convolutions of spherical images. This tool can also be used to detect multi-scale features on geometric surfaces. When we have a point cloud of a surface of a particular class of object, whether generated by stereo reconstruction or by some other modality, we can use statistics and machine learning to more robustly estimate the surface. If we have at our disposal a database of surfaces of a particular type of object, such as the human face, we can compute statistics over this database to constrain the possible shape a new surface of this type can take. We show how a statistical spherical wavelet shape prior can be used to efficiently and robustly reconstruct a face shape from noisy point cloud data, including stereo data.

Thesis (M.S.)--Nuclear and Radiological Engineering, Georgia Institute of Technology, 2008.
Committee Chair: Stacey, Weston M.; Committee Co-Chair: de Oliveira, Cassiano R.E.; Committee Member: Hertel, Nolan; Committee Member: van Rooijen, Wilfred F.G.

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Dissertacao (Mestrado)
IPEN/D
Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP

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Tese (Doutoramento)
IPEN/T
Instituto de Pesquisas Energeticas e Nucleares - IPEN/CNEN-SP

[EN] The neutron diffusion equation is an approximation of the neutron transport equation that describes the neutron population in a nuclear reactor core. In particular, we will consider here VVER-type reactors which use the neutron diffusion equation discretized on hexagonal meshes. Most of the simulation codes of a nuclear power reactor use the multigroup neutron diffusion equation to describe the neutron distribution inside the reactor core.To study the stationary state of a reactor, the reactor criticality is forced in artificial way leading to a generalized differential eigenvalue problem, known as the Lambda modes equation, which is solved to obtain the dominant eigenvalues of the reactor and their corresponding eigenfunctions. To discretize this model a finite element method with h-p adaptivity is used. This method allows to use heterogeneous meshes, and allows different refinements such as the use of h-adaptive meshes, reducing the size of specific cells, and p-refinement, increasing the polynomial degree of the basic functions used in the expansions of the solution in the different cells. Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration of the neutron diffusion equation. To simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. The spatial discretization of this equation is done using a finite element method that permits h-p refinements for different geometries. Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties and many techniques exist for the treatment of the rod cusping problem. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying different benchmark problems. For reactor calculations, the accuracy of a diffusion theory solution is limited for for complex fuel assemblies or fine mesh calculations. To improve these results a method that incorporates higher-order approximations for the angular dependence, as the simplified spherical harmonics (SPN ) method must be employed. In this work an h-p Finite Element Method (FEM) is used to obtain the dominant Lambda mode associated with a configuration of a reactor core using the SPN approximation. The performance of the SPN (N= 1, 3, 5) approximations has been tested for different reactor benchmarks.
[ES] La ecuación de la difusión neutrónica es una aproximación de la ecuación del transporte de neutrones que describe la población de neutrones en el núcleo de un reactor nuclear. En particular, consideraremos reactores de tipo VVER y para simular su comportamiento se utilizará la ecuación de la difusión neutrónica para cuya discretización se hace uso de mallas hexagonales. La mayoría de los códigos de simulación de reactores nucleares utilizan aproximación multigrupo de energía de la ecuación de la difusión neutrónica para describir la distribución de neutrones en el interior del núcleo del reactor. Para estudiar el estado estacionario del reactor, es posible forzar la criticidad del reactor de forma artificial modificando las secciones eficaces de forma que se obtiene un problema de valores propios diferencial, conocido como el problema de los Modos Lambda, que se resuelve para obtener los valores propios dominantes del reactor y sus correspondientes funciones propias. Para discretizar este modelo se ha hecho uso de un método de elementos finitos con adaptabilidad h-p. Este método permite el uso de mallas heterogéneas, y de diferentes refinamientos como el uso mallas h-adaptativas, reduciendo el tamaño de los distintos nodos, y el p-refinado, aumentando el grado del polinomio de las funciones básicas utilizado en los desarrollos de la solución en los diferentes nodos. Se ha desarrollado un código basado en un método de elementos finitos de alto orden para resolver el problema de los Modos Lambda en un reactor con geometría hexagonal y se han obtenido los Modos dominantes para distintos problemas de referencia. Una vez que se ha obtenido la solución para la distribución de neutrones en estado estacionario, ésta se utiliza como condición inicial para la integración de la ecuación de difusión neutrónica dependiente del tiempo. Para simular el comportamiento de un reactor nuclear para un determinado transitorio, es necesario ser capaz de integrar la ecuación de la difusión neutrónica dependiente del tiempo en el interior del núcleo del reactor. La discretización espacial de esta ecuación se hace usando un método de elementos finitos de alto orden que permite refinados de tipo h-p para distintas geometrías. Los transitorios que implican el movimiento de los bancos de las barras de control tienen el problema conocido como el efecto 'rod-cusping'. Estudios anteriores, por lo general, han abordado este problema utilizando una malla fija y definiendo propiedades promedio para los materiales correspondientes a las celdas donde se tiene la barra de control parcialmente insertada. En el presente trabajo se propone el uso de un esquema de malla móvil, de forma que en mallado espacial va cambiando con el movimiento de la barra de control, evitando la necesidad de utilizar secciones eficaces equivalentes para las celdas parcialmente insertadas. El funcionamiento de este esquema de malla móvil propuesto se estudia resolviendo distintos problemas tipo. La precisión obtenida mediante de la teoría de la difusión en los cálculos de reactores es limitada cuando se tienen elementos de combustible complejos o se pretenden realizar cálculos en malla fina. Para mejorar estos resultados, es necesario disponer de un método que incorpore aproximaciones de orden superior de la ecuación del transporte de neutrones. Una posibilidad es hacer uso de las ecuaciones PN simplificadas (SPN ). En este trabajo se utiliza un método de elementos finitos h-p para obtener los modos dominantes asociados con una configuración dada del núcleo de un reactor nuclear con geometría hexagonal usando la aproximación SPN . El funcionamiento de las aproximaciones SPN (N = 1, 3, 5) se ha estudiado para distintos problemas de referencia.
[CAT] L'equació de la difusió neutrònica és una aproximació de l'equació del transport de neutrons que descriu la població de neutrons en el nucli de un reactor nuclear. En particular, considerarem reactors de tipus VVER i per a simular el seu comportament s'utilitzarà l'equació de la difusió neutrónica que es discretitza fent ús de malles hexagonals. La majoria dels codis de simulació de reactors nuclears utilitzen l'aproximació multigrup d'energia de l'equació de la difusió neutrónica per a descriure la distribució de neutrons a l'interior del nucli del reactor. Per a estudiar l'estat estacionari del reactor, és possible forçar la seua criticitat de forma artificial modificant les seccions eficaces de manera que s'obté un problema de valors propis diferencial, conegut com el problema dels Modes Lambda, que es resol per a obtenir els valors propis dominants del reactor i les seues corresponents funcions pròpies. Per a discretitzar aquest model s'ha fet ús d'un mètode d'elements finits amb adaptabilitat h-p. Aquest mètode permet l'ús de malles heterogènies, i de diferents refinaments com l'ús malles h-adaptatives, reduint la grandària dels diferents nodes, i el p-refinat, augmentant el grau del polinomi de les funcions bàsiques utilitzat en els desenvolupaments de la solució en els diferents nodes. S'ha desenvolupat un codi basat en un mètode d'elements finits d'alt ordre per a resoldre el problema dels Modes Lambda en un reactor amb geometria hexagonal i s'han obtingut els Modes dominants per a diferents problemes de referència. Una vegada que s'ha obtingut la solució per a la distribució de neutrons en estat estacionari, aquesta s'utilitza com a condició inicial per a la integració de l'equació de difusió neutrònica depenent del temps. Per a simular el comportament d'un reactor nuclear per a un determinat transitori, és necessari ser capaç d'integrar l'equació de la difusió neutrónica depenent del temps a l'interior del nucli del reactor. La discretitzación espacial d'aquesta equació es fa usant un mètode d'elements finits d'alt ordre que permet refinats de tipus h-p per a diferents geometries. Els transitoris que impliquen el moviment dels bancs de les barres de control tenen el problema conegut com l'efecte 'rod-cusping'. Estudis anteriors, en general, han abordat aquest problema utilitzant una malla fixa i definint propietats equivalents per als materials corresponents a les cel·les on es té la barra de control parcialment inserida. En el present treball es proposa l'ús d'un esquema de malla mòbil, de manera que en mallat espacial va canviant amb el moviment de la barra de control, evitant la necessitat d'utilitzar seccions eficaces equivalents per a les cel·les parcialment inserides. El funcionament de aquest esquema de malla mòbil s'estudia resolent diferents problemes tipus. La precisió obtinguda mitjançant de la teoria de la difusió en els càlculs de reactors és limitada quan es tenen elements de combustible complexos o es pretenen realitzar càlculs en malla fina. Per a millorar aquests resultats, és necessari disposar d'un mètode que incorpore aproximacions d'ordre superior de l'equació del transport de neutrons. Una possibilitat és fer ús de les equacions PN simplificades (SPN ). En aquest treball s'utilitza un mètode d'elements finits h- p per a obtenir els modes dominants associats amb una configuració donada del nucli de un reactor amb geometria hexagonal usant l'aproximació SPN . El funcionament de les aproximacions SPN (N = 1, 3, 5) s'ha estudiat per a diferents problemes de referència.
Fayez Moustafa Moawad, R. (2016). Approximation of The Neutron Diffusion Equation on Hexagonal Geometries Using a h-p finite element method [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/65353
TESIS

Les performances des antennes Ultra Large Bande (ULB) les rendent appropriées pour de nombreuses applications. En radar à pénétration de surface (SPR), application visée de cette thèse, une telle bande passante offre un excellent compromis entre capacité de pénétration et résolution spatiale en imagerie micro-ondes. De plus, il a été démontré que la prise en compte du champ rayonné par l'antenne en présence de la surface améliore considérablement la qualité des images obtenues. Cette thèse aborde la problématique de la quantité de données représentant les antennes ULB. En effet, les descripteurs classiques d'antenne ne suffisent pas à caractériser l’évolution en fréquence de leurs performances. Le développement en harmoniques ou vecteurs sphériques est utilisé pour modéliser le diagramme de rayonnement d’antennes tout en réduisant le volume de données. D'autre part, les méthodes d'expansion en singularités modélisent la réponse en fréquence (ou impulsionnelle) de l'antenne par un ensemble de pôles de résonance. Le but de ce travail de thèse est d'établir un modèle compact qui représente avec précision le rayonnement d'antenne, et permette la connaissance du champ à différentes distances. A cette fin, plusieurs combinaisons des méthodes de caractérisation ont été étudiées. L'approche proposée est validée par la modélisation du diagramme de rayonnement simulé et mesuré d'une antenne Vivaldi (ETSA). Le modèle établi fournit le champ rayonné à différentes distances de l'antenne avec une erreur inférieure à 3% avec un taux de compression de 99%. La dernière partie de cette thèse présente une application de l'approche proposée au rayonnement d’antennes en présence d’interfaces
UWB antennas bandwidth makes them highly suitable for a number of applications. In surface penetrating radar (SPR) applications, which is the focus of our research, such a bandwidth range allows good signal penetration ability and fine space resolution for microwave imaging. In addition, it has been shown that the knowledge of the radiated field by the antenna enhances drastically the quality of the resulting images. The work reported in this thesis deals with the problematic of the huge amount of data representing UWB antennas. Indeed, due to the frequency dependence, the classical antenna parameters are not sufficient to characterize this type of antenna. The scalar or vector spherical wave expansion is widely used to expand the radiation pattern of a radiating antenna and permit a high compression data rate. On the other hand, the Singularity Expansion Methods are used in frequency/time domain to model the antenna response by a set of resonant poles. The purpose of this thesis is to establish a compact model representing accurately the antenna radiation characteristics, which also allows to find the field at various distances. To this end, several ways of combining the aforementioned methods have been investigated. The proposed approach is validated by modeling the simulated and measured radiation pattern of an Exponential Tapered Slot Antenna (ETSA) in free space. Furthermore, we verify that the established compact model provide radiated field at different distances from the antenna with a compression of the initial pattern up to 99% and an error below 3%. The last part of this thesis, present an application of the proposed methodology to SPR context

L’objet de ce travail est de proposer un modèle adapté pour la simulation du remplissage de moules qui réponde au meilleur compromis entre temps de calcul et précision des résultats. La difficulté est double. Il faut prendre en compte le phénomène de remplissage qui est un problème complexe à frontières libres et les spécificités liées au Verre : viscosité fortement thermodépendante et température de fusion élevée qui nécessite de prendre en compte le rayonnement. Le Chapitre I est consacrée à la partie écoulement du Verre liquide. La bibliothèque numérique Aquilon/Thétis, adaptée pour traiter ce type de problèmes et les couplages thermique air/verre/parois, a été utilisée (Méthode V.O.F pour le suivi de l’interface, méthodes de type Lagrangien augmenté/Projection vectorielle pour le couplage Vitesse-Pression). Pour l’aspect radiatif, différentes approches sont proposées : conductivité radiative équivalente (Chapitre II), méthode explicite directe pour la validation (Chapitre III) et méthode d’harmoniques sphériques ou méthode PN (Chapitre IV). Dans le Chapitre V, la méthode PN retenue est validée dans des cas simples et est appliquée ensuite à des cas avec couplage convectif en géométries complexes et obstacles semi-transparents (1D, 2D et 3D, 2D axi-symétrique et milieu non gris). Une version P1 modifiée est présentée. Les résultats sont assez proches de ceux donnés par la méthode P3 avec des temps de calcul modestes. L’intérêt de ce modèle est qu’il est facilement intégrable dans des codes numériques existants : une seule équation différentielle du second ordre stationnaire à résoudre en 3D
The aim of this study is to propose an adapted model for the simulation of mould filling that must be a compromise solution between computational time and results accuracy. The double difficulty is to take into account the filling phenomenon that is a complex problem due to the presence of free boundaries and to the Glass specificities: viscosity that is highly thermal dependant and high melting temperature that requires taking into account radiation effects. Chapter I is devoted to the melting Glass flow. The numerical libraries Aquilon/Thétis, adapted for solving such type of problems and the thermal coupling between Air/Glass/Walls, has been used. (V.O.F method for front tracking, Augmented Lagrangian/Vector Projection methods for solving Pressure/Velocity coupling). For radiative aspect, different approaches are proposed: equivalent radiative conductivity (Chapter II), direct explicit method for validation (Chapter III) and spherical harmonics method or PN method (Chapter IV). In the Chapter V, the selected PN method is validated through simple cases and is then applied in other cases with convective coupling in complex geometries including semi-transparent inclusions (1D, 2D and 3D, 2D axi-symmetric and non grey medium). A P1 modified version is presented. The results are close to those given by P3 method but with reduced computational time. The main interest of this model is that it can be easily implemented in existing numerical codes: a single stationary second order partial differential equation to solve in 3D

The main purpose of this thesis is to address some foundational questions regarding time-dependent spherical random fields and, then, to investigate some new estimation procedures for the class of spherical functional autoregressions.

碩士
國立成功大學
航空太空工程學系
81
1. 乙醇作為替代燃料的應用已有很久的歷史，而臺灣本身亞熱帶型的 氣候極適合種植高醣類的農作物，而利於乙醇的提煉。但乙醇與水存在一 共沸點的因素，因此在提煉高純度乙醇時，常需要更多的經費與時間，因 而造成高純度乙醇的價格昂貴。因此本文的主要目的是測試低純度乙醇在 小幅度修改下的商業用汽油引擎上的適用性。在經過適當修改的裕隆Ａ１ ２汽油引擎上，本實驗分別以汽油及不同純度的乙醇為燃料。來測試在不 同轉速及負荷下，其運轉的狀況、耗油量 、進排氣溫度 、汽缸內壓力及 廢氣污染…等變化。此外，並提高引擎壓縮比，再作一系列的觀察與測試 。進一步評估純酒精用於汽油引擎的優缺得失與可行性。結果顯示，現有 的汽油引擎化油器並無法提供足夠的乙醇以維持適當的油/氣比，利用調 整化油器之阻風門可獲得額外的注油，使得測試過程中λ值 (excess air ratio) 控制略大於1。此外因乙醇之高氣化熱，須添加額外的熱源幫 助乙醇汽化。並因乙醇之低熱含值及高汽化熱的性質，造成引擎操作溫度 較低，而在高轉速(3000 rpm)及高負荷(3.0 kgf-m)下無法使引擎順利運 轉，且乙醇的惰速限制高出汽油約 500 rpm。在熱效率的比較上，乙醇略 低於汽油1∼2﹪。在廢氣污染方面，汽油與乙醇在HC及CO的污染上並無太 大的差異。而在NO上，乙醇比起汽油約可降低70﹪∼85﹪的污染量。 2. 本文主要是應用多重格點法的數值計算方法，去處理一個二維矩型 封閉區間中，結合熱傳導及熱幅射的熱傳問題。而在熱幅射方面，我們所 採用的是球面諧合近似法中的 P-1 及 P-3 近似。因求解的問題中其邊界 條件是屬於 mixed type boundary，而在利用多重格點法求解時，則需加 上隱性邊界處理法。結果顯示，在 P-1 近似中，多重格點法比單層格點 可得到快約25倍的計算效率。而在 P-3 近似中，經過適當改良 V 循環後 ，多重格點法比單層格點則約可提高3倍的計算效率。