Dissertations / Theses on the topic 'Multilevel Finite Element Method'
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1
Jung, M., and U. Rüde. "Implicit extrapolation methods for multilevel finite element computations." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800516.
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Extrapolation methods for the solution of partial differential equations are commonly based on the existence of error expansions for the approximate solution. Implicit extrapolation, in the contrast, is based on applying extrapolation indirectly, by using it on quantities like the residual. In the context of multigrid methods, a special technique of this type is known as \034 -extrapolation. For finite element systems this algorithm can be shown to be equivalent to higher order finite elements. The analysis is local and does not use global expansions, so that the implicit extrapolation technique may be used on unstructured meshes and in cases where the solution fails to be globally smooth. Furthermore, the natural multilevel structure can be used to construct efficient multigrid and multilevel preconditioning techniques. The effectivity of the method is demonstrated for heat conduction problems and problems from elasticity theory.
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Nepomnyaschikh, Sergey V. "Optimal Multilevel Extension Operators." Universitätsbibliothek Chemnitz, 2005. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200500971.
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In the present paper we suggest the norm-preserving
explicit operator for the extension of
finite-element functions from boundaries of domains
into the inside. The construction of this operator
is based on the multilevel decomposition of
functions on the boundaries and on the equivalent
norm for this decomposition. The cost of the action
of this operator is proportional to the number of
nodes.
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Green, Seth. "Multilevel, subdivision-based, thin shell finite elements : development and an application to red blood cell modeling /." Thesis, Connect to this title online; UW restricted, 2003. http://hdl.handle.net/1773/7110.
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Unwin, Helena Juliette Thomasin. "Uncertainty quantification of engineering systems using the multilevel Monte Carlo method." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/277877.
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This thesis examines the quantification of uncertainty in real-world engineering systems using the multilevel Monte Carlo method. It is often infeasible to use the traditional Monte Carlo method to investigate the impact of uncertainty because computationally it can be prohibitively expensive for complex systems. Therefore, the newer multilevel method is investigated and the cost of this method is analysed in the finite element framework. The Monte Carlo and multilevel Monte Carlo methods are compared for two prototypical examples: structural vibrations and buoyancy driven flows through porous media. In the first example, the impact of random mass density is quantified for structural vibration problems in several dimensions using the multilevel Monte Carlo method. Comparable eigenvalues and energy density approximations are found for the traditional Monte Carlo method and the multilevel Monte Carlo method, but for certain problems the expectation and variance of the quantities of interest can be computed over 100 times faster using the multilevel Monte Carlo method. It is also tractable to use the multilevel method for three dimensional structures, where the traditional Monte Carlo method is often prohibitively expensive. In the second example, the impact of uncertainty in buoyancy driven flows through porous media is quantified using the multilevel Monte Carlo method. Again, comparable results are obtained from the two methods for diffusion dominated flows and the multilevel method is orders of magnitude cheaper. The finite element models for this investigation are formulated carefully to ensure that spurious numerical artefacts are not added to the solution and are compared to an analytical model describing the long term sequestration of CO2 in the presence of a background flow. Additional cost reductions are achieved by solving the individual independent samples in parallel using the new podS library. This library schedules the Monte Carlo and multilevel Monte Carlo methods in parallel across different computer architectures for the two examples considered in this thesis. Nearly linear cost reductions are obtained as the number of processes is increased.
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Bängtsson, Erik. "Robust Preconditioners Based on the Finite Element Framework." Doctoral thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7828.
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Robust preconditioners on block-triangular and block-factorized form for three types of linear systems of two-by-two block form are studied in this thesis. The first type of linear systems, which are dense, arise from a boundary element type of discretization of crack propagation problems. Numerical experiment show that simple algebraic preconditioning strategies results in iterative schemes that are highly competitive with a direct solution method. The second type of algebraic systems, which are sparse, indefinite and nonsymmetric, arise from a finite element (FE) discretization of the partial differential equations (PDE) that describe (visco)elastic glacial isostatic adjustment (GIA). The Schur complement approximation in the block preconditioners is constructed by assembly of local, exactly computed Schur matrices. The quality of the approximation is verified in numerical experiments. When the block preconditioners for the indefinite problem are combined with an inner iterative scheme preconditioned by a (nearly) optimal multilevel preconditioner, the resulting preconditioner is (nearly) optimal and robust with respect to problem size, material parameters, number of space dimensions, and coefficient jumps. Two approaches to mathematically formulate the PDEs for GIA are compared. In the first approach the equations are formulated in their full complexity, whereas in the second their formulation is confined to the features and restrictions of the employed FE package. Different solution methods for the algebraic problem are used in the two approaches. Analysis and numerical experiments reveal that the first strategy is more accurate and efficient than the latter. The block structure in the third type of algebraic systems is due to a fine-coarse splitting of the unknowns. The inverse of the pivot block is approximated by a sparse matrix which is assembled from local, exactly inverted matrices. Numerical experiments and analysis of the approximation show that it is robust with respect to problem size and coefficient jumps.
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Karlsson, Christian. "A comparison of two multilevel Schur preconditioners for adaptive FEM." Thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-219939.
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There are several algorithms for solving the linear system of equations that arise from the finite element method with linear or near-linear computational complexity. One way is to find an approximation of the stiffness matrix that is such that it can be used in a preconditioned conjugate residual method, that is, a preconditioner to the stiffness matrix. We have studied two preconditioners for the conjugate residual method, both based on writing the stiffness matrix in block form, factorising it and then approximating the Schur complement block to get a preconditioner. We have studied the stationary reaction-diffusion-advection equation in two dimensions. The mesh is refined adaptively, giving a hierarchy of meshes. In the first method the Schur complement is approximated by the stiffness matrix at one coarser level of the mesh, in the second method it is approximated as the assembly of local Schur complements corresponding to macro triangles. For two levels the theoretical bound of the condition number is 1/(1-C²) for either method, where C is the Cauchy-Bunyakovsky-Schwarz constant. For multiple levels there is less theory. For the first method it is known that the condition number of the preconditioned stiffness matrix is O(l²), where l is the number of levels of the preconditioner, or, equivalently, the number mesh refinements. For the second method the asymptotic behaviour is not known theoretically. In neither case is the dependency of the condition number of C known. We have tested both methods on several problems and found the first method to always give a better condition number, except for very few levels. For all tested problems, using the first method it seems that the condition number is O(l), in fact it is typically not larger than Cl. For the second method the growth seems to be superlinear.
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Aghabarati, Ali. "Multilevel and algebraic multigrid methods for the higher order finite element analysis of time harmonic Maxwell's equations." Thesis, McGill University, 2014. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=121485.
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The Finite Element Method (FEM) applied to wave scattering and quasi-static vector field problems in the frequency domain leads to sparse, complex-symmetric, linear systems of equations. For large problems with complicated geometries, most of the computer time and memory used by FEM goes to solving the matrix equation. Krylov subspace methods are widely used iterative methods for solving large sparse systems. They depend heavily on preconditioning to accelerate convergence. However, application of conventional preconditioners to the "curl-curl" operator which arises in vector electromagnetics does not result in a satisfactory performance and specialized preconditioning techniques are required. This thesis presents effective Multilevel and Algebraic Multigrid (AMG) preconditioning techniques for p-adaptive FEM analysis. In p-adaption, finite elements of different polynomial orders are present in the mesh and the system matrix can be structured into blocks corresponding to the orders of the basis functions. The new preconditioners are based on a p-type multilevel Schwarz (pMUS) approximate inversion of the block structured system. A V-cycle multilevel correction starts by applying Gauss-Seidel to the highest block level, then the next level down, and so on. On the other side of the V, Gauss-Seidel iterations are applied in the reverse order. At the bottom of the cycle is the lowest order system, which is usually solved exactly with a direct solver. The proposed alternative is to use Auxiliary Space Preconditioning (ASP) at the lowest level and continue the V-cycle downwards, first into a set of auxiliary, node-based spaces, then through a series of progressively smaller matrices generated by an Algebraic Multigrid (AMG). The algebraic coarsening approach is especially useful for problems with fine geometric details, requiring a very large mesh in which the bulk of the elements remain at low order. In addition, for wave problems, a "shifted Laplace" technique is applied, in which part of the ASP/AMG algorithm uses a perturbed, complex frequency. A significant convergence acceleration is achieved. The performance of Krylov algorithms is further enhanced during p-adaption by incorporation of a deflation technique. This projects out from the preconditioned system the eigenvectors corresponding to the smallest eigenvalues. The construction of the deflation subspace is based on efficient estimation of the eigenvectors from information obtained when solving the first problem in a p-adaptive sequence. Extensive numerical experiments have been performed and results are presented for both wave and quasi-static problems. The test cases considered are complicated to solve and the numerical results show the robustness and efficiency of the new preconditioners. Deflated Krylov methods preconditioned with the current Multilevel/ASP/AMG approach are always considerably faster than the reference methods and speedups of up to 10 are achieved for some test problems.La méthode des éléments finis (FEM) appliquée à la dispersion des ondes et aux problèmes de champ de vecteurs quasi-statique dans le domaine fréquentiel mène à des systèmes d'équations linéaires rares, symétriques-complexes. Pour de grands problèmes ayant des géométries complexes, la plupart du temps et de la mémoire d'ordinateur utilisé par FEM va à la résolution de l'équation de la matrice. Les méthodes itératives de Krylov sont celles largement utilisées dans la résolution de grands systèmes creux. Elles dépendent fortement des préconditionnement qui accélèrent la convergence. Toutefois, l'application de préconditionnements conventionnels à l'opérateur "rot-rot" qui surgit en électromagnétisme vectoriel n'aboutit pas à des résultats satisfaisants et des techniques de préconditionnement spécialisés sont exigées.Cette thèse présente des techniques de préconditionnement efficaces multiniveau et multigrilles algébrique (AMG) pour l'analyse p-adaptative FEM. Dans la p-adaptation, des éléments finis de différents ordres polynomiaux sont présents dans le maillage et la matrice du système peut être structurée en blocs correspondant aux ordres des fonctions de base. Les nouveaux préconditionneurs sont basés sur un type d'inversion approximative à multiniveau p Schwarz (pMUS) du système structuré de bloc. Une correction à niveaux multiples en cycle V débute par l'application de Gauss-Seidel au niveau du bloc le plus élevé, suivi par le niveau inférieur, et ainsi de suite. De l'autre côté du V, des itérations de Gauss-Seidel sont appliquées en ordre inverse. Au bas du cycle se trouve le système d'ordre le plus bas, qui est habituellement résolu exactement avec un solveur direct. L'alternative proposée est d'utiliser l'espace auxiliaire de préconditionnement (ASP) au niveau le plus bas et de poursuivre le cycle en V vers le bas, d'abord en un ensemble d'auxiliaires, basé sur les espacements de nœuds, à travers une série de plus en plus petites de matrices générées par un multigrille algébrique (AMG). L'approche de grossissement algébrique est particulièrement utile aux problèmes ayant de fins détails géométriques, nécessitant une très grande maille dans laquelle la majeure partie des éléments restent à un niveau plus bas.En outre, pour des problèmes d'onde, la technique "décalé Laplace" est appliquée, dans laquelle une partie de l'algorithme ASP/AMG utilise une fréquence complexe perturbée. Une accélération de la convergence significative est atteinte. La performance des algorithmes de Krylov est davantage renforcée au cours du p-adaptation par l'incorporation d'une technique de déflation. Cette saillie fait dépasser hors du système préconditionné, les vecteurs propres correspondants aux plus petites valeurs propres. La construction du sous-espace de déflation est basée sur une estimation efficace des vecteurs propres à partir d'informations obtenues lors de la résolution du premier problème dans une séquence p-adaptatif. Des expériences numériques approfondies ont été effectuées et les résultats sont présentés à la fois aux problèmes d'onde et quasi-statiques. Les cas de test sont considérés comme compliqués à résoudre et les résultats numériques montrent la robustesse et l'efficacité des nouveaux préconditionnements. Les méthodes de Krylov de déflation préconditionnés par l'approche multiniveaux/ASP/AMG actuelle sont toujours considérablement plus rapides que les méthodes de référence et des accélérations allant jusqu'à 10 sont atteintes pour certains problèmes de test.
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Marquez, Damian Jose Ignacio. "Multilevel acceleration of neutron transport calculations." Thesis, Atlanta, Ga. : Georgia Institute of Technology, 2007. http://hdl.handle.net/1853/19731.
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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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Thess, M. "Parallel Multilevel Preconditioners for Problems of Thin Smooth Shells." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801416.
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In the last years multilevel preconditioners like BPX became more and more
popular for solving second-order elliptic finite element discretizations by iterative
methods. P. Oswald has adapted these methods for discretizations of the fourth
order biharmonic problem by rectangular conforming Bogner-Fox-Schmidt elements
and nonconforming Adini elements and has derived optimal estimates for the
condition numbers of the preconditioned linear systems. In this paper we generalize
the results from Oswald to the construction of BPX and Multilevel Diagonal
Scaling (MDS-BPX) preconditioners for the elasticity problem of thin smooth shells of
arbitrary forms where we use Koiter's equations of equilibrium for an homogeneous
and isotropic thin shell, clamped on a part of its boundary and loaded by a
resultant on its middle surface. We use the two discretizations mentioned above and the
preconditioned conjugate gradient method as iterative method. The parallelization
concept is based on a non-overlapping domain decomposition data structure. We
describe the implementations of the multilevel preconditioners. Finally, we show
numerical results for some classes of shells like plates, cylinders, and hyperboloids.
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Elfverson, Daniel. "Multiscale Methods and Uncertainty Quantification." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-262354.
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In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements. We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries. For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability.
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Jung, M., and J. F. Maitre. "Some Remarks on the Constant in the Strengthened C.B.S. Inequality: Application to $h$- and $p$-Hierarchical Finite Element Discretizations of Elasticity Problems." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801431.
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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.
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Globisch, G., and S. V. Nepomnyaschikh. "The hierarchical preconditioning having unstructured grids." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801398.
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In this paper we present two hierarchically preconditioned methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner by decomposing functions on it are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as the BPX-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods which can be of importance in the industrial engineering, where often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available.
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Jung, Michael, Aleksandr M. Matsokin, Sergey V. Nepomnyaschikh, and Yu A. Tkachov. "Multilevel preconditioning operators on locally modified grids." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601671.
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Systems of grid equations that approximate elliptic boundary value problems on locally modified grids are considered. The triangulation, which approximates the boundary with second order of accuracy, is generated from an initial uniform triangulation by shifting nodes near the boundary according to special rules. This "locally modified" grid possesses several significant features: this triangulation has a regular structure, the generation of the triangulation is rather fast, this construction allows to use multilevel preconditioning (BPX-like) methods. The proposed iterative methods for solving elliptic boundary value problems approximately are based on two approaches: The fictitious space method, i.e. the reduction of the original problem to a problem in an auxiliary (fictitious) space, and the multilevel decomposition method, i.e. the construction of preconditioners by decomposing functions on hierarchical grids. The convergence rate of the corresponding iterative process with the preconditioner obtained is independent of the mesh size. The construction of the grid and the preconditioning operator for the three dimensional problem can be done in the same way.
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Eibner, Tino, and Jens Markus Melenk. "Multilevel preconditioning for the boundary concentrated hp-FEM." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601662.
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The boundary concentrated finite element method
is a variant of the hp-version of the finite
element method that is particularly suited for
the numerical treatment of elliptic boundary
value problems with smooth coefficients and low
regularity boundary conditions. For this method
we present two multilevel preconditioners that
lead to preconditioned stiffness matrices with
condition numbers that are bounded uniformly in
the problem size N. The cost of applying the
preconditioners is O(N). Numerical examples
illustrate the efficiency of the algorithms.
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Kang, David Sung-Soo. "Hybrid stress finite element method." Thesis, Massachusetts Institute of Technology, 1986. http://hdl.handle.net/1721.1/14973.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1986.MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERO
Bibliography: leaves 257-264.
by David Sung-Soo Kang.
Ph.D.
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Sevilla, Cárdenas Rubén. "NURBS-Enhanced Finite Element Method (NEFEM)." Doctoral thesis, Universitat Politècnica de Catalunya, 2009. http://hdl.handle.net/10803/5857.
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Aquesta tesi proposa una millora del clàssic mètode dels elements finits (finite element method, FEM) per a un tractament eficient de dominis amb contorns corbs: el denominat NURBS-enhanced finite element method (NEFEM). Aquesta millora permet descriure de manera exacta la geometría mitjançant la seva representació del contorn CAD amb non-uniform rational B-splines (NURBS), mentre que la solució s'aproxima amb la interpolació polinòmica estàndard. Per tant, en la major part del domini, la interpolació i la integració numèrica són estàndard, retenint les propietats de convergència clàssiques del FEM i facilitant l'acoblament amb els elements interiors. Només es requereixen estratègies específiques per realitzar la interpolació i la integració numèrica en elements afectats per la descripció del contorn mitjançant NURBS.La implementació i aplicació de NEFEM a problemes que requereixen una descripció acurada del contorn són, també, objectius prioritaris d'aquesta tesi. Per exemple, la solució numèrica de les equacions de Maxwell és molt sensible a la descripció geomètrica. Es presenta l'aplicació de NEFEM a problemes d'scattering d'ones electromagnètiques amb una formulació de Galerkin discontinu. S'investiga l'habilitat de NEFEM per obtenir solucions precises amb malles grolleres i aproximacions d'alt ordre, i s'exploren les possibilitats de les anomenades malles NEFEM, amb elements que contenen singularitats dintre d'una cara o aresta d'un element. Utilitzant NEFEM, la mida de la malla no està controlada per la complexitat de la geometria. Això implica una dràstica diferència en la mida dels elements i, per tant, suposa un gran estalvi tant des del punt de vista de requeriments de memòria com de cost computacional. Per tant, NEFEM és una eina poderosa per la simulació de problemes tridimensionals a gran escala amb geometries complexes. D'altra banda, la simulació de problemes d'scattering d'ones electromagnètiques requereix mecanismes per aconseguir una absorció eficient de les ones scattered. En aquesta tesi es discuteixen, optimitzen i comparen dues tècniques en el context de mètodes de Galerkin discontinu amb aproximacions d'alt ordre.
La resolució numèrica de les equacions d'Euler de la dinàmica de gasos és també molt sensible a la representació geomètrica. Quan es considera una formulació de Galerkin discontinu i elements isoparamètrics lineals, una producció espúria d'entropia pot evitar la convergència cap a la solució correcta. Amb NEFEM, l'acurada imposició de la condició de contorn en contorns impenetrables proporciona resultats precisos inclús amb una aproximació lineal de la solució. A més, la representació exacta del contorn permet una imposició adequada de les condicions de contorn amb malles grolleres i graus d'interpolació alts. Una propietat atractiva de la implementació proposada és que moltes de les rutines usuals en un codi d'elements finits poden ser aprofitades, per exemple rutines per realitzar el càlcul de les matrius elementals, assemblatge, etc. Només és necessari implementar noves rutines per calcular les quadratures numèriques en elements corbs i emmagatzemar el valor de les funciones de forma en els punts d'integració. S'han proposat vàries tècniques d'elements finits corbs a la literatura. En aquesta tesi, es compara NEFEM amb altres tècniques populars d'elements finits corbs (isoparamètics, cartesians i p-FEM), des de tres punts de vista diferents: aspectes teòrics, implementació i eficiència numèrica. En els exemples numèrics, NEFEM és, com a mínim, un ordre de magnitud més precís comparat amb altres tècniques. A més, per una precisió desitjada NEFEM és també més eficient: necessita un 50% dels graus de llibertat que fan servir els elements isoparamètrics o p-FEM per aconseguir la mateixa precisió. Per tant, l'ús de NEFEM és altament recomanable en presència de contorns corbs i/o quan el contorn té detalls geomètrics complexes.
This thesis proposes an improvement of the classical finite element method (FEM) for an efficient treatment of curved boundaries: the NURBSenhanced FEM (NEFEM). It is able to exactly represent the geometry by means of the usual CAD boundary representation with non-uniform rational Bsplines (NURBS), while the solution is approximated with a standard piecewise polynomial interpolation. Therefore, in the vast majority of the domain, interpolation and numerical integration are standard, preserving the classical finite element (FE) convergence properties, and allowing a seamless coupling with standard FEs on the domain interior. Specifically designed polynomial interpolation and numerical integration are designed only for those elements affected by the NURBS boundary representation.
The implementation and application of NEFEM to problems demanding an accurate boundary representation are also primary goals of this thesis. For instance, the numerical solution of Maxwell's equations is highly sensitive to geometry description. The application of NEFEM to electromagnetic scattering problems using a discontinuous Galerkin formulation is presented. The ability of NEFEM to compute an accurate solution with coarse meshes and high-order approximations is investigated, and the possibilities of NEFEM meshes, with elements containing edge or corner singularities, are explored. With NEFEM, the mesh size is no longer subsidiary to geometry complexity, and depends only on the accuracy requirements on the solution, whereas standard FEs require mesh refinement to properly capture the geometry. This implies a drastic difference in mesh size that results in drastic memory savings, and also important savings in computational cost. Thus, NEFEM is a powerful tool for large-scale scattering simulations with complex geometries in three dimensions. Another key issue in the numerical solution of electromagnetic scattering problems is using a mechanism to perform the absorption of outgoing waves. Two perfectly matched layers are discussed, optimized and compared in a high-order discontinuous Galerkin framework.
The numerical solution of Euler equations of gas dynamics is also very sensitive to geometry description. Using a discontinuous Galerkin formulation and linear isoparametric elements, a spurious entropy production may prevent convergence to the correct solution. With NEFEM, the exact imposition of the solid wall boundary condition provides accurate results even with a linear approximation of the solution. Furthermore, the exact boundary representation allows using coarse meshes, but ensuring the proper implementation of the solid wall boundary condition. An attractive feature of the proposed implementation is that the usual routines of a standard FE code can be directly used, namely routines for the computation of elemental matrices and vectors, assembly, etc. It is only necessary to implement new routines for the computation of numerical quadratures in curved elements and to store the value of shape functions at integration points.
Several curved FE techniques have been proposed in the literature. In this thesis, NEFEM is compared with some popular curved FE techniques (namely isoparametric FEs, cartesian FEs and p-FEM), from three different perspectives: theoretical aspects, implementation and performance. In every example shown, NEFEM is at least one order of magnitude more accurate compared to other techniques. Moreover, for a desired accuracy NEFEM is also computationally more efficient. In some examples, NEFEM needs only 50% of the number of degrees of freedom required by isoparametric FEs or p-FEM. Thus, the use of NEFEM is strongly recommended in the presence of curved boundaries and/or when the boundary of the domain has complex geometric details.
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Valivarthi, Mohan Varma, and Hema Chandra Babu Muthyala. "A Finite Element Time Relaxation Method." Thesis, Högskolan i Halmstad, Sektionen för Informationsvetenskap, Data– och Elektroteknik (IDE), 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-17728.
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In our project we discuss a finite element time-relaxation method for high Reynolds number flows. The key idea consists of using local projections on polynomials defined on macro element of each pair of two elements sharing a face. We give the formulation for the scalar convection–diffusion equation and a numerical illustration.
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梁耀華 and Yew-wah Leung. "Finite element solution on microcomputers." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1989. http://hub.hku.hk/bib/B31209300.
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Leung, Yew-wah. "Finite element solution on microcomputers /." [Hong Kong] : University of Hong Kong, 1989. http://sunzi.lib.hku.hk/hkuto/record.jsp?B12754948.
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Charrier, Julia. "Analyse numérique d’équations aux dérivées aléatoires, applications à l’hydrogéologie." Thesis, Cachan, Ecole normale supérieure, 2011. http://www.theses.fr/2011DENS0030/document.
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Ce travail présente quelques résultats concernant des méthodes numériques déterministes et probabilistes pour des équations aux dérivées partielles à coefficients aléatoires, avec des applications à l'hydrogéologie. On s'intéresse tout d'abord à l'équation d'écoulement dans un milieu poreux en régime stationnaire avec un coefficient de perméabilité lognormal homogène, incluant le cas d'une fonction de covariance peu régulière. On établit des estimations aux sens fort et faible de l'erreur commise sur la solution en tronquant le développement de Karhunen-Loève du coefficient. Puis on établit des estimations d'erreurs éléments finis dont on déduit une extension de l'estimation d'erreur existante pour la méthode de collocation stochastique, ainsi qu'une estimation d'erreur pour une méthode de Monte-Carlo multi-niveaux. On s'intéresse enfin au couplage de l'équation d'écoulement considérée précédemment avec une équation d'advection-diffusion, dans le cas d'incertitudes importantes et d'une faible longueur de corrélation. On propose l'analyse numérique d'une méthode numérique pour calculer la vitesse moyenne à laquelle la zone contaminée par un polluant s'étend. Il s'agit d'une méthode de Monte-Carlo combinant une méthode d'élements finis pour l'équation d'écoulement et un schéma d'Euler pour l'équation différentielle stochastique associée à l'équation d'advection-diffusion, vue comme une équation de Fokker-PlanckThis work presents some results about probabilistic and deterministic numerical methods for partial differential equations with stochastic coefficients, with applications to hydrogeology. We first consider the steady flow equation in porous media with a homogeneous lognormal permeability coefficient, including the case of a low regularity covariance function. We establish error estimates, both in strong and weak senses, of the error in the solution resulting from the truncature of the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates, from which we deduce an extension of the existing error estimate for the stochastic collocation method along with an error estimate for a multilevel Monte-Carlo method. We finally consider the coupling of the previous flow equation with an advection-diffusion equation, in the case when the uncertainty is important and the correlation length is small. We propose the numerical analysis of a numerical method, which aims at computing the mean velocity of the expansion of a pollutant. The method consists in a Monte-Carlo method, combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation
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Müller, Björn. "Parallel Finite Element Method with FEAP 8.2." Darmstadt TU, Fachgebiet Numerische Berechnungsverfahren im Maschinenbau, 2009. http://tuprints.ulb.tu-darmstadt.de/1336/.
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Zhang, Lin. "Generalized finite element method for multiscale analysis." Diss., Texas A&M University, 2003. http://hdl.handle.net/1969.1/1141.
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This dissertation describes a new version of the Generalized Finite Element Method (GFEM), which is well suited for problems set in domains with a large number of internal features (e.g. voids, inclusions, etc.), which are practically impossible to solve using the standard FEM. The main idea is to employ the mesh-based handbook functions which are solutions of boundary value problems in domains extracted from vertex patches of the employed mesh and are pasted into the global approximation by the Partition of Unity Method (PUM). It is shown that the p-version of the Generalized FEM using mesh-based handbook functions is capable of achieving very high accuracy.
It is also analyzed that the effect of the main factors affecting the accuracy of the method namely: (a) The data and the buffer included in the handbook domains, and (b) The accuracy of the numerical construction of the handbook functions. The robustness of the method is illustrated by several model problems defined in domains with a large number of closely spaced voids and/or inclusions with various shapes, including the heat conduction problem defined on domains with porous media and/or a real composite material.
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Vu, Thu Hang. "Enhancing the scaled boundary finite element method." University of Western Australia. School of Civil and Resource Engineering, 2006. http://theses.library.uwa.edu.au/adt-WU2006.0068.
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[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.
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Davidsson, Johan. "Sobolev Spaces and the Finite Element Method." Thesis, Örebro universitet, Institutionen för naturvetenskap och teknik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:oru:diva-67470.
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In this essay we present the Sobolev spaces and some basic properties of them. The Sobolev spaces serve as a theoretical framework for studying solutions to partial differential equations. The finite element method is presented which is a numerical method for solving partial differential equations.
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Pack, Peter Michael Walter. "The finite element method in underwater acoustics." Thesis, University of Southampton, 1986. https://eprints.soton.ac.uk/52298/.
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A Finite Element Method (FEM) is developed to calculate rotationally symmetric acoustic propagation over short range intervals (0-5 km) in shallow oceans (0-200 m deep) at low frequencies (0-50 Hz). The method allows full two-way wave propagation in range dependent environments and includes coupling to a full elastic seabed. Numerical results from a computer program are presented for propagation upslope, downslope, over seamounts and across trenches in the seabed. The seabed is modelled as a pressure release surface, a fluid halfspace and an elastic, solid halfspace and the implications of each type of model are discussed. The halfspaces, being represented by a new set of infinite elements, are modelled without truncation. The results are presented primarily as plots of transmission loss against range for a fixed depth receiver. Subsidiary results show the effect of depth averaging the receiver location, and extract mode amplitude data to reveal the strength of mode coupling and backscatter in different environments.
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Liu, Yunshan. "P-adaptive hybrid/mixed finite element method /." The Ohio State University, 1998. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487950153602937.
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Lu, Chuan. "Generalized finite element method for electromagnetic analysis." Diss., Connect to online resource - MSU authorized users, 2008.
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Thesis (Ph. D.)--Michigan State University. Electrical and Computer Engineering, 2008.Title from PDF t.p. (viewed on Apr. 8, 2009) Includes bibliographical references (p. 148-153). Also issued in print.
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Gundu, Krishna Mohan. "hp-Finite Element Method for Photonics Applications." Diss., The University of Arizona, 2008. http://hdl.handle.net/10150/195940.
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A hp-finite element method is implemented to numerically study the modes of waveguides with two dimensional cross-section and to compute electromagnetic scattering from three dimensional objects. A method to control the chromatic dispersion properties of photonic crystal fibers using the selective hole filling technique is proposed. The method is based on a single hole-size fiber geometry, and uses an appropriate index-matching liquid to modify the effective size of the filled holes. The dependence of dispersion properties of the fiber on the design parameters such as the refractive index of the liquid, lattice constant and hole diameter are studied numerically. It is shown that very small dispersion values between 0±0.5ps/nm-km can be achieved over a bandwidth of 430-510nm in the communication wavelength region of 1300-1900nm. Three such designs are proposed with air hole diameters in the range 1.5-2.0μm. A novel multi-core fiber design strategy for obtaining a at in-phase supermode that optimizes utilization of the active medium inversion in the multiple cores is proposed. The spatially at supermode is achieved by engineering the fiber so that the total mutual coupling between neighboring active cores is equal. Different designs suitable for different fabrication processes such as stack-and-draw and drilling are proposed. An important improvement over previous methods is the design simplicity and better tolerance to perturbations. The optimal implementation of perfectly matched layer (PML) in terms of minimizing the computational overhead it introduces is studied. In one dimension it is shown that PML implementation with a single cell and a high order finite element produces minimal overhead. Estimates of optimal cell size and optimal finite element degree are given. Based on the single cell implementation of PML in three dimensions, field enhancement in metallic bowties is computed.
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Olivier, Albertus Hendrik. "Object-oriented finite element framework." Thesis, Stellenbosch : Stellenbosch University, 2002. http://hdl.handle.net/10019.1/52971.
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Thesis (MScEng)--Stellenbosch University, 2002.ENGLISH ABSTRACT: The role of the computer has changed from a calculation tool to a tool that supports human thinking. In this thesis fundamental aspects of the Finite Element method are mapped to an object model with a well defined structure which provides for local and distributed analysis work. To achieve this the following was investigated: • An object-oriented framework for the Finite Element analysis method • An associated graphical user interface that enables the user to create and modify Finite Element models in an effective way • Requirements for the sharing of analysis information in a communication network Proposed solutions are implemented in a pilot application which indicates their potential.
AFRIKAANSE OPSOMMING: Die rol van die rekenaar het verander vanaf 'n gereedskapstuk wat berekening doen na 'n gereedskapstuk wat menslike denke ondersteun. In hierdie tesis word die fundamentele aspekte van die Eindige Element metode oorgedra na 'n objek model met 'n goed gedefinieerde struktuur wat lokale en verspreide analisering werk ondersteun. Om dit te bereik is die volgende ondesoek: • 'n Objek orienteerde raamwerk vir die Eindige Element metode • 'n Geassosieerde grafiese raamwerk wat die gebruiker in staat stelom objekte te skep en te verander • Vereistes vir die deel van analise inligting in 'n kommunikasie netwerk Die voorgestelde oplossing is geimplimenteer in 'n loodsimplementering wat die voordele van die benadering uitlig.
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Xiao, Dong Wen. "Efficiency analysis on element decomposition method for stochastic finite element analysis." Thesis, University of Macau, 2000. http://umaclib3.umac.mo/record=b1636334.
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Irfanoglu, Bulent. "Boundary Element-finite Element Acoustic Analysis Of Coupled Domains." Phd thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/12605360/index.pdf.
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This thesis studies interactions between coupled acoustic domain(s) and enclosing rigid or elastic boundary. Boundary element-finite element (BE-FE) sound-structure interaction models are developed by coupling frequency domain BE acoustic and FE structural models using linear inviscid acoustic and elasticity theories.
Flexibility in analyses is provided by discontinuous triangular and quadrilateral elements in the BE method (BEM), and a rectangular plate and a triangular shell element in the FE method (FEM).
An analytical formulation is developed for an extended fundamental sound-structure interaction problem that involves locally reacting sound absorptive treatment on interior elastic boundary. This new formulation is built upon existing analytical solutions for a configuration known as the cavity-backed-plate problem. Results from developed analytical formulation are compared against those from independent BE-FE analyses.
Analytical and BE-FE analysis results for a selection of cavity-plate(s) interaction cases are given.
Single- and multi-domain BE analyses of cavity-Helmholtz resonator interaction are provided as an alternative to modal method of acoustoelasticity.
A discrete-form of the existing BE acoustic particle velocity formulation is presented and demonstrated on a basic case study. Both the existing and the discretized BE acoustic particle velocity formulations could be utilized in acoustic studies.
A selection of case studies involving fundamental configurations are studied both analytically and computationally (by BE or BE-FE methods). These studies could provide a basis for benchmark case development in the field of acoustics.
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Jung, Michael, and Todor D. Todorov. "On the Convergence Factor in Multilevel Methods for Solving 3D Elasticity Problems." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601510.
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The constant gamma in the strengthened Cauchy-Bunyakowskii-Schwarz inequality is a basic tool for constructing of two-level and multilevel preconditioning matrices. Therefore many authors consider estimates or computations of this quantity. In this paper the bilinear form arising from 3D linear elasticity problems is considered on a polyhedron. The cosine of the abstract angle between multilevel finite element subspaces is computed by a spectral analysis of a general eigenvalue problem. Octasection and bisection approaches are used for refining the triangulations. Tetrahedron, pentahedron and hexahedron meshes are considered. The dependence of the constant $\gamma$ on the Poisson ratio is presented graphically.
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Perry, William H. "Finite element analysis of polymer flows." Ohio : Ohio University, 1985. http://www.ohiolink.edu/etd/view.cgi?ohiou1184072781.
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Ayers, Christopher Lee. "Concurrent processing of finite element calculations." Thesis, Georgia Institute of Technology, 1989. http://hdl.handle.net/1853/13072.
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Heinrich, Bernd, and Beate Jung. "The Fourier-finite-element method with Nitsche-mortaring." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601493.
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The paper deals with a combination of the
Fourier-finite-element method with the
Nitsche-finite-element method (as a mortar method).
The approach is applied to the Dirichlet problem
of the Poisson equation in three-dimensional
axisymmetric domains $\widehat\Omega$ with
non-axisymmetric data. The approximating Fourier
method yields a splitting of the 3D-problem into
2D-problems. For solving the 2D-problems on the
meridian plane $\Omega_a$,
the Nitsche-finite-element method with
non-matching meshes is applied. Some important
properties of the approximation scheme are
derived and the rate of convergence in some
$H^1$-like norm is proved to be of the type
${\mathcal O}(h+N^{-1})$ ($h$: mesh size on
$\Omega_a$, $N$: length of the Fourier sum) in
case of a regular solution of the boundary value
problem. Finally, some numerical results are
presented.
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Starkloff, Hans-Jörg. "Stochastic finite element method with simple random elements." Universitätsbibliothek Chemnitz, 2008. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200800596.
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We propose a variant of the stochastic finite element method, where the random
elements occuring in the problem formulation are approximated by simple random
elements, i.e. random elements with only a finite number of possible values.
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Ahlbert, Gabriella. "Method Evaluation of Global-Local Finite Element Analysis." Thesis, Linköpings universitet, Hållfasthetslära, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-78103.
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When doing finite element analysis upon the structure of Saab’s aeroplanes a coarse global model of mainly shell elements is used to determine the load distribution for sizing the structure. At some parts of the aeroplane it is however desirable to implement a more detailed analysis. These areas are usually modelled with solid elements; the problem of connecting the fine local solid elements to the coarse global model will shell elements then arises. This master thesis is preformed to investigate possible Global-Local methods to use for the structural analysis on Gripen. First a literature study of current methods on the market is made, thereafter a few methods are implemented on a generic test structure and later on also tested on a real detail of Gripen VU. The methods tested in this thesis are Mesh refinement in HyperWorks, RBE3 in HyperWorks, Glue in MSC Patran/Nastran and DMIG in MSC Nastran. The software is however not evaluated in this thesis, and a further investigation is recommended to find the most fitting software for this purpose. All analysis are performed with linear assumptions. Mesh refinement is an integrated technique where the elements are gradually decreasing in size. Per definition, this technique cannot handle gaps, but it has almost identical results to the fine reference model. RBE3 is a type of rigid body elements with zero stiffness, and is used as an interface element. RBE3 is possible to use to connect both Shell-To-Shell and Shell-To-Solid, and can handle offsets and gaps in the boundary between the global and local model. Glue is a contact definition and is also available in other software under other names. The global respectively the local model is defined as contact bodies and a contact table is used to control the coupling. Glue works for both Shell-To-Shell and Shell-To-Solid couplings, but has problem dealing with offsets and gaps in the boundary between the global and local model. DMIG is a superelement technique where the global model is divided into smaller sub-models which are mathematically connected. DMIG is only possible to use when the nodes on the boundary on the local model have the same position as the nodes at the boundary of the global model. Thus, it is not possible to only use DMIG as a Global-Local method, but can advantageously be combined with other methods. The results indicate that the preferable method to use for Global-Local analysis is RBE3. To decrease the size of the files and demand of computational power, RBE3 can be combined with a superelement technique, for example DMIG. Finally, it is important to consider the size of the local model. There will inevitably be boundary effect when performing a Global-Local analysis of the suggested type, and it is therefore important to make the local model big enough so that the boundary effects have faded before reaching the area of interest.
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Ali, Hassan O. "Near-field computation using the finite element method." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ26122.pdf.
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Parvini, Mehdi. "Pavement deflection analysis using stochastic finite element method." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape11/PQDD_0014/NQ42757.pdf.
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Durrani, Jawad Nadeem. "Dynamics of pipelines with a finite element method." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/MQ65153.pdf.
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Parvini, Mehdi. "Pavement deflection analysis using stochastic finite element method /." *McMaster only, 1997.
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Savchuk, Tatyana. "The multiscale finite element method for elliptic problems." Ann Arbor, Mich. : ProQuest, 2007. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3245025.
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Thesis (Ph. D. in Applied Mathematics)--Southern Methodist University, 2007.Title from PDF title page (viewed Mar. 18, 2008). Source: Dissertation Abstracts International, Volume: 67-12, Section: B, page: 7120. Adviser: Zhangxin (John) Chen. Includes bibliographical references.
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Dewasurendra, Lohitha. "A finite element method for ring rolling processes." Ohio : Ohio University, 1998. http://www.ohiolink.edu/etd/view.cgi?ohiou1175098096.
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Druma, Adriana M. "Analysis of carbon foams by finite element method." Ohio : Ohio University, 2005. http://www.ohiolink.edu/etd/view.cgi?ohiou1177611967.
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Balland, Philippe. "The solenoidal finite element method and reservoir simulation." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.260727.
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Wong, Sze-chun, and 黃仕進. "Two level finite element method for structural analysis." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1988. http://hub.hku.hk/bib/B30425918.
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Sun, Huaiyang, and 孫懷洋. "Fractal finite element method for anisotropic crack problems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2003. http://hub.hku.hk/bib/B26720474.
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Chen, Yupeng. "FINITE ELEMENT METHOD MODELING OF ADVANCED ELECTRONIC DEVICES." Doctoral diss., University of Central Florida, 2006. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/3115.
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In this dissertation, we use finite element method together with other numerical techniques to study advanced electron devices. We study the radiation properties in electron waveguide structure with multi-step discontinuities and soft wall lateral confinement. Radiation mechanism and conditions are examined by numerical simulation of dispersion relations and transport properties. The study of geometry variations shows its significant impact on the radiation intensity and direction. In particular, the periodic corrugation structure exhibits strong directional radiation. This interesting feature may be useful to design a nano-scale transmitter, a communication device for future nano-scale system. Non-quasi-static effects in AC characteristics of carbon nanotube field-effect transistors are examined by solving a full time-dependent, open-boundary Schrödinger equation. The non-quasi-static characteristics, such as the finite channel charging time, and the dependence of small signal transconductance and gate capacitance on the frequency, are explored. The validity of the widely used quasi-static approximation is examined. The results show that the quasi-static approximation overestimates the transconductance and gate capacitance at high frequencies, but gives a more accurate value for the intrinsic cut-off frequency over a wide range of bias conditions. The influence of metal interconnect resistance on the performance of vertical and lateral power MOSFETs is studied. Vertical MOSFETs in a D2PAK and DirectFET package, and lateral MOSFETs in power IC and flip chip are investigated as the case studies. The impact of various layout patterns and material properties on RDS(on) will provide useful guidelines for practical vertical and lateral power MOSFETs design.Ph.D.
School of Electrical Engineering and Computer Science
Engineering and Computer Science
Electrical Engineering
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Pascoe, Steven Keith. "Contact stress analysis using the finite element method." Thesis, University of Liverpool, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240266.
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Herbert, Rolf China. "Modelling insect wings using the finite element method." Thesis, University of Exeter, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.370012.
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