Rozprawy doktorskie na temat „P-finite element method”

[Truncated abstract] The scaled boundary finite element method is a novel computational method developed by Wolf and Song which reduces partial differential equations to a set of ordinary linear differential equations. The method, which is semi-analytical, is suitable for solving linear elliptic, parabolic and hyperbolic partial differential equations. The method has proved to be very efficient in solving various types of problems, including problems of potential flow and diffusion. The method out performs the finite element method when solving unbounded domain problems and problems involving stress singularities and discontinuities. The scaled boundary finite element method involves solution of a quadratic eigenproblem, the computational expense of which increases rapidly as the number of degrees of freedom increases. Consequently, to a greater extent than the finite element method, it is desirable to obtain solutions at a specified level of accuracy while using the minimum number of degrees of freedom necessary. In previous work, no systematic study had been performed so far into the use of elements of higher order, and no consideration made of p adaptivity. . . The primal problem is solved normally using the basic scaled boundary finite element method. The dual problem is solved by the new technique using the fundamental solution. A guaranteed upper error bound based on the Cauchy-Schwarz inequality is derived. A iv goal-oriented p-hierarchical adaptive procedure is proposed and implemented efficiently in the scaled boundary finite element method.

The vector finite element method has gained great attention since overcoming the deficiencies incurred by the scalar basis functions for the vector Helmholtz equation. Most implementations of vector FEM have been non-adaptive, where a mesh of the domain is generated entirely in advance and used with a constant degree polynomial basis to assign the degrees of freedom. To reduce the dependency on the users' expertise in analyzing problems with complicated boundary structures and material characteristics, and to speed up the FEM tool, the demand for adaptive FEM grows high. For efficient adaptive FEM, error estimators play an important role in assigning additional degrees of freedom. In this proposal study, hierarchical vector basis functions and four error estimators for p-refinement are investigated for electromagnetic applications.

[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

Abstract P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary re- sults for 1D case, condition number estimates and some inequalities for 2D reference element.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the p-version finite element matrixes. Also DD preconditioning is considered.

In this thesis, quadratic hexahedral edge elements have been applied to the three dimensional for open region electromagnetic scattering problems. For this purpose, a semi-automatic all-hexahedral mesh generation algorithm is developed and implemented. Material properties inside the elements and along the edges are also determined and prescribed during the mesh generation phase in order to be used in the solution phase. Based on the condition number quality metric, the generated mesh is optimized by means of the Particle Swarm Optimization (PSO) technique. A framework implementing hierarchical hexahedral edge elements is implemented to investigate the performance of linear and quadratic hexahedral edge elements. Perfectly Matched Layers (PMLs), which are implemented by using a complex coordinate transformation, have been used for mesh truncation in the software. Sparse storage and relevant efficient matrix ordering are used for the representation of the system of equations. Both direct and indirect sparse matrix solution methods are implemented and used. Performance of quadratic hexahedral edge elements is deeply investigated over the radar cross-sections of several curved or flat objects with or without patches. Instead of the de-facto standard of 0.1 wavelength linear element size, 0.3-0.4 wavelength quadratic element size was observed to be a new potential criterion for electromagnetic scattering and radiation problems.

This thesis provides a contribution towards a general procedure for solving robustly and efficiently limit and shakedown analyses of engineering structures within the static approach which has been chosen for its simplicity of implementation. Throughout the thesis, attempts at improving the robustness and efficiency of the computations are presented. Beginning with efforts to prevent volumetric locking, which is a severe shortcoming of traditional low order h-type displacement elements, the investigation proposes the use of the high order p-version of the finite element method. It is shown theoretically and confirmed numerically that this p-method is not only robust in preventing locking, but also provides very accurate results. However, the use of uniformly distributed high order p-elements may be computationally demanding when the size of the problem becomes large. This difficulty is tackled by two main approaches: use of a p-adaptive procedure at the elastic computation stage and use of approximate piecewise linear yield functions. The p-adaptive scheme produces a non-uniform p-distribution and helps to greatly reduce the number of degrees of freedom needed while still guaranteeing the required level of accuracy. The overall gain is that the sizes of the models are reduced significantly and hence also the computational effort. The adoption of piecewise linear yield surfaces helps to further increase the efficiency at the expense of possibly slightly less accurate, but still very acceptable, results. State-of-the-art linear programming solvers based on the very efficient interior point methodology are used. Significant gains in efficiency are achieved. A heuristic, semi-adaptive scheme to piecewise linearize the yield surfaces is then developed to further reduce the size of the underlying optimization problems. The results show additional gains in efficiency. Finally, major conclusions are summarized, and various aspects suitable for further research are highlighted.

Automation of finite element analysis based on a fully adaptive p-refinement procedure can reduce the magnitude of discretization error to the desired accuracy with minimum computational cost and computer resources. This study aims to 1) develop an efficient p-refinement procedure with a non-uniform p analysis capability for solving 2-D and 3-D elastostatic mechanics problems, and 2) introduce a stress error estimate. An element-by-element algorithm that decides the appropriate order for each element, where element orders can range from 1 to 8, is described. Global and element data structures that manage the complex data generated during the refinement process are introduced. These data structures are designed to match the concept of object-oriented programming where data and functions are managed and organized simultaneously. The stress error indicator introduced is found to be more reliable and to converge faster than the error indicator measured in an energy norm called the residual method. The use of the stress error indicator results in approximately 20% fewer degrees of freedom than the residual method. Agreement of the calculated stress error values and the stress error indicator values confirms the convergence of final stresses to the analyst. The error order of the stress error estimate is postulated to be one order higher than the error order of the error estimate using the residual method. The mapping of a curved boundary element in the working coordinate system from a square-shape element in the natural coordinate system results in a significant improvement in the accuracy of stress results. Numerical examples demonstrate that refinement using non-uniform p analysis is superior to uniform p analysis in the convergence rates of output stresses or related terms. Non-uniform p analysis uses approximately 50% to 80% less computational time than uniform p analysis in solving the selected stress concentration and stress intensity problems. More importantly, the non-uniform p refinement procedure scales the number of equations down by 1/2 to 3/4. Therefore, a small scale computer can be used to solve equation systems generated using high order p-elements. In the calculation of the stress intensity factor of a semi-elliptical surface crack in a finite-thickness plate, non-uniform p analysis used fewer degrees of freedom than a conventional h-type element analysis found in the literature.
Ph. D.

The analysis of the p- and hp-versions of the finite element methods has been studied in much detail for the Hilbert spaces W1,2 (omega). The following work extends the previous approximation theory to that of general Sobolev spaces W1,q(Q), q 1, oo . This extension is essential when considering the use of the p and hp methods to the non-linear a-Laplacian problem. Firstly, approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces W1,q(Q) are given. This analysis shows that the traditional view of avoiding the use of high order polynomial finite element methods is incorrect, and that the rate of convergence of the p version is always at least that of the h version (measured in terms of number of degrees of freedom). It is also shown that, if the solution has certain types of singularity, the rate of convergence of the p version is twice that of the h version. Numerical results are given, confirming the results given by the approximation theory. The p-version approximation theory is then used to obtain the hp approximation theory. The results obtained allow both non-uniform p refinements to be used, and the h refinements only have to be locally quasiuniform. It is then shown that even when the solution has singularities, exponential rates of convergence can be achieved when using the /ip-version, which would not be possible for the h- and p-versions.

Dissertation (PhD)--University of Stellenbosch, 2002.
ENGLISH ABSTRACT: This document presents a Galerkin FE formulation for the full-wave, frequency domain, electromagnetic analysis of three dimensional structures relevant to microwave engineering, together with the investigation of two techniques to enhance the formulation's computational efficiency. The first technique considered is the fast multi pole method (FMM) and the second technique is adaptive refinement of the discretization, based on a posteriori error estimation. Thus, the motivation for the work presented in this document is to increase the computational efficiency of the FE formulation considered. The FE formulation considered is widely used within the microwave engineering, finite element community. Tetrahedral, rectilinear, curl-conforming, mixed- and full order, hierarchical vector elements are used. The formulation is extended to incorporate a cavity backed aperture employing the appropriate half-space Green function within a BI boundary condition, which represents a specific member of a large class of hybrid FE-BI formulations. The formulation is also extended to model coaxial ports via a Neumann boundary condition, using a priori knowledge of the dominant modal fields. Results are presented in support of the formulation and its extensions, including novel results on the coupling between microstrip patch antennas on a perforated substrate. The FMM is investigated first, with the purpose of optimizing the non-local BI component of the cavity FE-BI formulation, in light of its coupling with the differential equation based, sparse FEM. The FMM results in a partly sparse factorization of the BI contribution to the system matrix. Error control schemes for the FMM are thoroughly reviewed and an additional, novel scheme is empirically devised. The second technique investigated, which is more directly related to the FEM and larger in scope, is the use of a posteriori error estimation, in order to optimize the FE discretization through adaptive refinement. A overview of available a posteriori error estimation techniques in the general FE literature is given as well as a survey of available techniques that are specifically tailored to Maxwell's equations. Two known approaches within the applied mathematics literature are adapted to the FE formulation at hand, resulting in two novel, residual based error estimation procedures for this FE formulation - one explicit in nature and the other implicit. The two error estimators are then used to drive a single p adaptive analysis cycle of the FE formulation, experimentally demonstrating their effectiveness. A quasi-static condition is introduced and successfully used to enhance the adaptive algorithm's effectiveness, independently of the error estimation procedure employed. The novel error estimation schemes and adaptive results represent the main research contributions of this study.
AFRIKAANSE OPSOMMING: Hierdie dokument beskryf 'n Galerkin eindige element (EE) formulering vir die volgolf, frekwensiegebied, elektromagnetiese analise van driedimensionele strukture relevant vir mikrogolfingenieurwese, saam met die ondersoek van twee tegnieke om die numeriese effektiwiteit van die formulering te verbeter. Die eerste tegniek wat ondersoek word, is die vinnige multipooi metode (VMM) en die tweede is die aanpasbare verfyning van die EE diskretisering, gebaseer op a posteriori foutberaming. Dus, die motivering vir hierdie werk is om die numeriese effektiwiteit van die genoemde EE formulering te verbeter. Die bogenoemde EE formulering word algemeen gebruik deur die mikrogolfingenieurswese, eindige element-gemeenskap. Tetrahedriese, reglynige, curl-ondersteunende, hierargiese vektorelemente van gemengde- en volledige ordes word gebruik. Die formulering word uitgebrei om holtes in 'n oneindige grondvlak te kan hanteer, deur gebruik te maak van die toepaslike Green funksie binne 'n grensintegraal (GI) grensvoorwaarde, wat 'n spesifieke lid is van 'n groot klas, hibriede, EE-GI formulerings. Die formulering word ook uitgebrei om koaksiale poorte to modelleer via 'n Neumann grensvoorwaarde, deur die gebruik van a priori kennis van die koaksiale, dominante modus-velde. Resultate word gelewer om die formulering, saam met die uitbreidings daarvan, te ondersteun, insluitende oorspronklike resultate in verband met die koppeling tussen mikrostrook plakantennes op 'n geperforeerde substraat. Die VMM word eerste ondersoek, met die doelom die nie-lokale, GI komponent van die EEGI formulering vir holtes te optimeer, weens die koppeling daarvan met die yl, differensiaalvergelyking- gebaseerde, eindige element-metode. Die VMM lei tot 'n gedeeltelik-yl faktorisering van die GI bydrae tot die algehele matriksvergelyking. Skemas om die VMM fout te beheer word deeglik ondersoek en 'n addisionele, oorspronklike skema word empiries ontwikkel. Die tweede tegniek wat ondersoek word, wat meer direk verband hou met die eindige elementmetode, en van groter omvang is, is die gebruik van a posteriori foutberaming om die EE diskretisasie te optimeer deur middel van aanpasbare verfyning. 'n Oorsig van beskikbare, a posteriori foutberamingstegnieke in die algemene EE literatuur word gegee, asook 'n opname van beskikbare tegnieke wat spesifiek gerig is op Maxwell se vergelykings. Twee bekende benaderings binne die toegepaste wiskunde-literatuur word aangepas by die bogenoemde EE formulering, wat lei tot twee oorspronklike residu-gebaseerde foutberamingstegnieke vir hierdie formulering - een van 'n eksplisiete aard en die ander implisiet. Die twee foutberamingstegnieke word gebruik om 'n enkel, p-aanpasbare analisesiklus aan te dryf, wat die effektiwiteit van die foutberamingstegnieke eksperimenteel demonstreer. 'n Kwasi-statiese vereiste word beskryf en suksesvol gebruik om die aanpasbare algoritme se effektiwiteit te verhoog, onafhanklik van die foutberamingstegniek wat gebruik word. Die oorspronklike foutberamingstegnieke en aanpasbare algoritme-resultate verteenwoordig die hoof navorsingsbydraes van hierdie studie.

Orientador: Marco Lúcio Bittencourt
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
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Resumo: Este trabalho tem como objetivo principal a implementação de uma arquitetura de software para o Método de Elementos Finitos de Alta Ordem (MEF-AO), baseando-se no paradigma de programação orientada a objeto (POO) e no uso de técnicas de otimização de código fonte. O software foi escrito em linguagem C++ e desenvolvido sobre um framework com ferramentas que auxiliaram no desenvolvimento. A modelagem do sistema foi realizada de forma a facilitar e promover o reuso e manutenção do código. Buscou-se, também, a flexibilidade e generalização do MEF-AO ao permitir a variação nos procedimentos da construção das equações e o uso de malhas p não-uniforme. Neste caso, cada elemento pode ser interpolado com uma ordem polinomial diferente, além de permitir o uso de um algoritmo local de solução. Tal característica pode diminuir o número de operações e de armazenamento, pois o número de funções de forma é aumentado apenas onde é necessário o uso de mais pontos para interpolação da malha de solução. No final, o software é avaliado aplicando o problema de projeção para malha de quadrados e hexaedros
Abstract: The main objective of this work is the implementation of a software architecture for the High-Order Finite Element Method (HO-FEM), based on the Object Oriented Paradigm (OOP) and on source-code optimization techniques. The software was written in C++ programming language and developed over a framework which provided tools that assisted the implementation. The system was modeled so to promote code reuse and maintainability. Furthermore, the system modeling also provided flexibility and generalization for the HO-FEM by allowing modifications on the procedures for equation assembling and the use of p-non-uniform meshes. In this case, each element can be interpolated with different polynomial order, and allows the application of an algorithm for local solution. Such features can reduce the number of operations for memory allocation, since the number of shape functions is increased only where a higher density of points is needed by the solution mesh. Finally, the software is assessed by applying the projection problem for meshes of squares and hexahedros
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Mestre em Engenharia Mecânica

A recent development and innovation in elevator technologies have sprawled interest in how these technologies would affect the forms and shapes of future high-rise buildings. The elevator that uses linear motors instead of ropes and can thus travel horizontally and on inclines is of particular interest. Once the vertical cores are no longer needed for the elevators, new and radical building forms and shapes are anticipated. It is expected that the buildings will have bridges and/or the buildings themselves will structurally perform more like bridges than buildings, therefore this study addresses the following topic - structural design and performance of tube mega frame in arch-shaped high-rise buildings. Evidently, for a structure of an arched shape, the conventional structural system used in high-rise buildings does not address the structural challenges. On the other hand, The Tubed Mega Frame system developed by Tyréns is designed to support a structural system for high-rise building without the central core, in which the purpose is to transfer all the loads to the ground via the perimeter of the building, making the structure more stable by maximizing the lever arm for the structure. The system has not yet been realized nor tested in realistic circumstances. This thesis aims at evaluating the efficiency of the Tubed Mega Frame system in arched shaped tall buildings. Multiple shapes and type of arches are evaluated to find the best possible selection. Structural behavior of different arch structures is studied using analytical tools and also finite element method in software SAP2000. The most efficient arch shape is sought to distribute the self-weight of the structure. The analysis shows that it is possible to accurately determine efficient arch shape based on a specific load distribution. Furthermore, continuing with the arch shape found in previous steps, a 3D finite element model is built and analyzed for linear static, geometric non-linearity (P-Delta) and linear dynamic cases in the ETABS software. For the given scope, the results of the analysis show that the Tubed Mega Frame structural system is potentially feasible and has relatively high lateral stiffness in the plane of the arch, while the out-of-plane lateral stiffness is comparatively smaller. For the service limit state, the maximum story drift ratio is within the limitation of 1/400 for in-plane deformations, while for out-of-plane the comfort criteria limit is exceeded.

O elemento hélio, encontrado principalmente em reservas de gás natural, entra em condensação à temperatura de 4,2K, e é a única substância conhecida que permanece no estado líquido até o zero absoluto. Na fase liquida, o hélio apresenta ainda, em K, outra mudança de fase, onde passa de líquido comum à superfluido, com viscosidade praticamente nula. Estas propriedades conferem ao hélio importantes aplicações. Uma hdas principais aplicações é como agente refrigerante em supercondutores, como por exemplo, no acelerador de partículas LHC, que está sendo construído na fronteira da França com a Suíça, em aparelhos de ressonância magnética, satélites artificiais, etc. Neste trabalho, são apresentados dois modelos matemáticos para a transferência de calor no hélio líquido. O primeiro modelo, considerando apenas movimentos macroscópicos, é derivado com base nas leis constitutivas de Fourier e de Gorter-Mellink. O segundo modelo, baseado nas técnicas de Fremond, inclui movimentos microscópicos e pode ser visto como uma regularização do primeiro modelo. Os dois modelos são governados por equações diferenciais fortemente não lineares resultantes da não linearidade da lei de Gorter-Mellink e da mudança de fase. Ambos os modelos podem ser considerados casos particulares do problema de Stefan de duas fases, sendo que em uma das fases o fluxo de calor é governado pela equação não-linear do problema conhecido como p-laplaciano, com p=4/3. São também apresentadas técnicas para resolver de forma eficiente o problema do p-laplaciano, tanto para valores grandes de p, p>>2, quanto para valores de p próximos à 1, que constituem importantes desafios numéricos. Para tanto são propostos dois métodos iterativos simples, um baseado no método de quase-Newton, com termo de relaxação e, outro através da decomposição de Helmholtz, gerando um sistema de equações cujas matrizes são constantes, o que diminui significativamente o custo computacional. Experimentos numéricos são realizados para testar a eficiência dos modelos numéricos propostos bem como dos algoritmos desenvolvidos para resolver os sistemas de equações algébricas não lineares resultantes das aproximações por elementos finitos. São apresentados resultados de estudos de convergência, mostrando taxas de convergência ótimas ou quase ótimas, comparáveis às das interpolantes. Para o problema com mudança de fase, devido à descontinuidade do gradiente da temperatura sobre a interface que separa as duas fases do hélio líquido, as taxas de convergência não são ótimas. Usando malhas adaptativas, consegue-se taxas ótimas também para o problema com mudança de fase. Usando dados experimentais, encontrados na literatura, para os parâmetros de condutividade térmica, densidade e calor específico, dependentes da temperatura, são também apresentados testes de validação do modelo e exemplos de possíveis aplicações. Nos testes de validação do modelo, compara-se a solução numérica do modelo matemático com resultados experimentais para a temperatura, encontrados na literatura.
The element helium, found mainly in natural gas reserves, condenses at temperature of 4.2K, and is the unique known substance that remains in liquid to absolute zero. In the liquid phase, the helium presents still another phase change in 2.19K, where passes of common liquid to superfluous liquid, with almost zero viscosity. These properties give the helium important applications. One of the major applications is as a coolant in superconductors, such as in the particle accelerator LHC, which is being built in the French border with Switzerland, in magnetic resonance devices, artificial satellites, etc.. In this paper, we present two mathematical models for heat transfer in liquid helium. The first model, considering only macroscopic movements, is derived based on constitutive laws of Fourier and Gorter-Mellink. The second model, based on techniques of Fremond, includes microscopic movements and can be seen as a regularization of the first model. Both models are governed by highly nonlinear differential equations resulting from the nonlinearity of the law of Gorter-Mellink and change of phase. Both models can be considered special cases of the Stefan problem in two phases, with phase one of the heat flux is governed by non-linear equation of the problem known as p-Laplacian, with p = 4/3. We also presented techniques to efficiently solve the problem of p-Laplacian, both for large values of p, p>> 2, and for values of p close to 1, which are major numerical challenges. Are proposed two simple iterative methods, one based on the method of quasi-Newton, with the relaxation term and the other by the Helmholtz decomposition, creating a system of equations whose matrices are constant, which reduces significantly the computational cost. Numerical experiments are conducted to test the efficiency of numerical models proposed and the algorithms developed for solving systems of nonlinear algebraic equations arising from approximations by finite elements. Are also presented results of studies of convergence, showing rates of optimal or near optimal convergence, comparable to that of interpolates. For the problem with phase change, due to the discontinuity of the gradient of temperature on the interface separating the two phases of liquid helium, the rate of convergence is not optimal. Using adaptive mesh, it is also great rates to the problem with change of phase. Using experimental data found in literature, for the parameters of thermal conductivity, density and specific heat, temperature dependent, are also presented for validation testing of the model and examples of possible applications. In tests for validating the model, compared to the numerical solution of the mathematical model with experimental results for the temperature found in literature.

In this paper, we consider domain decomposition preconditioners for a system of linear algebraic equations arising from the p-version of the fem. We propose several multi-level preconditioners for the Dirichlet problems in the sub-domains in two and three dimensions. It is proved that the condition number of the preconditioned system is bounded by a constant independent of the polynomial degree. The proof uses interpretations of the p-version element stiffness matrix and mass matrix on [-1,1] as h-version stiffness matrix and weighted mass matrix. The analysis requires wavelet methods.

For a class of two-dimensional boundary value problems including diffusion and elasticity problems it is proved that the constants in the corresponding strengthened Cauchy-Buniakowski-Schwarz (C.B.S.) inequality in the cases of h -hierarchical and p -hierarchical finite element discretizations with triangular meshes differ by the factor 0.75. For plane linear elasticity problems and triangulations with right isosceles tri- angles formulas are presented that show the dependence of the constant in the C.B.S. inequality on the Poisson's ratio. Furthermore, numerically determined bounds of the constant in the C.B.S. inequality are given for three-dimensional elasticity problems discretized by means of tetrahedral elements. Finally, the robustness of iterative solvers for elasticity problems is discussed briefly.

In this paper we consider the p-FEM for elliptic boundary value problems on tetrahedral meshes where the entries of the stiffness matrix are evaluated by numerical quadrature. Such a quadrature can be done by mapping the tetrahedron to a hexahedron via the Duffy transformation. We show that for tensor product Gauss-Lobatto-Jacobi quadrature formulas with q+1=p+1 points in each direction and shape functions that are adapted to the quadrature formula, one again has discrete stability for the fully discrete p-FEM. The present error analysis complements the work [Eibner/Melenk 2005] for the p-FEM on triangles/tetrahedra where it is shown that by adapting the shape functions to the quadrature formula, the stiffness matrix can be set up in optimal complexity.

A finite element analysis methodology employing p-adaptivity is proposed. Object-oriented design and methodologies are used to implement the finite element package. The p-refinement phase is facilitated through the addition and deletion of general serendipity element edge nodes. Zienkiewicz-Zhu (ZZ) error estimator is used to determine the localized error. A modified superconvergent patch recovery technique is implemented to recover highly accurate nodal gradients utilized in the error estimation phase. Another variation to the ZZ error estimator, suggested by Blacker (12), is also tested. The object-oriented design leads to easier maintainability and extensibility. The advantage of object-oriented design is the ability to try new solvers, new elements and new problem types with minimum programming effort and time. The source code for the thesis is written in Fortran 90, with a graphical user interface (GUI) written in Java. The user interface performs pre-processing and post-processing. The Java user interface can add networking capabilities to the program. This method has been successfully applied to some benchmark problems.

In the framework of the Jacobi-weighted Sobolev space, we design the a-posterior error estimators and error indicators associated with residuals and jumps of normal derivatives on internal edges with appropriate Jacobi weights for the p-version of the finite element method. With the help of quasi Jacobi projection operators, the upper bounds and the lower bounds of indicators and estimators are analyzed, which shows that such a-posteriori error estimation is quasi optimal. The indicators and estimators are computed for some model problems and programmed in C++. The numerical results show the reliability of our indicators and estimators.

碩士
國立中央大學
數學研究所
96
In this paper, we design two-dimensional p-version finite element using rectangular elements by C++. In design, we solve the following partial differential equations by using rectangular elements[sum^{n}_{l=1} { - igtriangledown cdot (A^{kl} igtriangledown u_{l}) + B^{kl} cdot igtriangledown u_{l} + C^{kl} u_{l} - igtriangledown cdot (D^{kl} u_{l}) } = f^{k} , , k = 1 cdots n]where n is the number of variables, k stands for the number of equations, so we have n equations.This paper will show the struct of program and hierarchical shape function.

碩士
國立中央大學
數學研究所
96
In this paper, we design two-dimensional p-version finite element using triangular elements by C++. In design, we solve the following partial differential equations by using triangular elements sum^{n}_{l=1} { - bigtriangledown cdot (A^{kl} bigtriangledown u_{l}) + B^{kl} cdot bigtriangledown u_{l}+ C^{kl} u_{l} - bigtriangledown cdot (D^{kl} u_{l}) } = f^{k} , k = 1 cdots n where n is the number of variables, k stands for the number of equations, so we have n equations. This paper will show the struct of program and hierarchical shape function.

In the framework of the Jacobi-weighted Besov and Sobolev spaces, we analyze the approximation to singular and smooth functions. We construct stable and compatible polynomial extensions from triangular and square faces to prisms, hexahedrons and pyramids, and introduce quasi Jacobi projection operators on individual elements, which is a combination of the Jacobi projection and the interpolation at vertices and on sides of elements. Applying these results we establish the convergence of the h-p version of the finite element method with quasi uniform meshes in three dimensions for elliptic problems with smooth solutions or singular solutions on polyhedral domains in three dimensions. The rate of convergence interms of h and p is proved to be the best.

Savings in computational resources have become a main subject of study in geomechanics due to the complexity of the problems analyzed. This thesis develops and evaluates the performance of a partial p-adaptive mesh optimization method for stress analysis of underground excavations. The method uses transition finite elements in order to connect a mesh of quadratic (triangular or quadrilateral) elements with a mesh of linear (triangular or quadrilateral) elements. The analysis is performed using SIMFEM, a computer program that solves plane strain problems and is able to handle this type of mixed meshes. After the formulation of 4 types of transition elements (5-node, 7-node quadrilateral, 4-node and 5-node triangle), which were incorporated into the code, 57 models (including the analysis of a pressurized cavity) were run, analyzed and for some of the models, the results obtained were compared to commercial software, as to ensure the correct behaviour of these elements in a finite element mesh. A final application was performed modeling a continuum with two circular excavations, surrounded by a linear elastic material in a biaxial stress field, obtaining favourable results. The global stiffness matrix size was reduced by 85.4% and by 74.1% for the models using triangles and quadrilaterals respectively, as a result, the calculation times were considerably reduced. The average percentage of error with respect to the models without optimization, measured at eight critical points, was 0.15% and 2.63% in the case of triangular and quadrilateral elements respectively.