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Dissertations / Theses on the topic 'Adaptive Multigrid'
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1
Brannick, James. "Adaptive algebraic multigrid coarsening strategies." Diss., Connect to online resource, 2005. http://wwwlib.umi.com/cr/colorado/fullcit?p3190384.
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2
Mayfield, Andrew James. "Adaptive mesh refinement." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.358687.
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3
Kahl, Karsten. "Adaptive Algebraic Multigrid Methods for Lattice QCD Computations." Wuppertal Universitätsbibliothek Wuppertal, 2010. http://d-nb.info/1000531767/34.
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Rottmann, Matthias [Verfasser]. "Adaptive Domain Decomposition Multigrid for Lattice QCD / Matthias Rottmann." Wuppertal : Universitätsbibliothek Wuppertal, 2016. http://d-nb.info/1093603240/34.
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Thorne, Jr Daniel Thomas. "Multigrid with Cache Optimizations on Adaptive Mesh Refinement Hierarchies." UKnowledge, 2003. http://uknowledge.uky.edu/gradschool_diss/325.
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This dissertation presents a multilevel algorithm to solve constant and variable coeffcient elliptic boundary value problems on adaptively refined structured meshes in 2D and 3D. Cacheaware algorithms for optimizing the operations to exploit the cache memory subsystem areshown. Keywords: Multigrid, Cache Aware, Adaptive Mesh Refinement, Partial Differential Equations, Numerical Solution.
6
Kahl, Karsten [Verfasser]. "Adaptive Algebraic Multigrid Methods for Lattice QCD Computations / Karsten Kahl." Wuppertal : Universitätsbibliothek Wuppertal, 2010. http://d-nb.info/1000531767/34.
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Vey, Simon. "Adaptive Finite Elements for Systems of PDEs: Software Concepts, Multi-level Techniques and Parallelization." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2008. http://nbn-resolving.de/urn:nbn:de:bsz:14-ds-1212489177096-59154.
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In the recent past, the field of scientific computing has become of more and more importance for scientific as well as for industrial research, playing a comparable role as experiment and theory do. This success of computational methods in scientific and engineering research is next to the enormous improvement of computer hardware to a large extend due to contributions from applied mathematicians, who have developed algorithms which make real life applications feasible. Examples are adaptive methods, high order discretization, fast linear and non-linear solvers and multi-level methods. The application of these methods in a large class of problems demands for suitable and robust tools for a flexible and efficient implementation. In order to play a crucial role in scientific and engineering research, besides efficiency in the numerical solution, also efficiency in problem setup and interpretation of simulation results is of utmost importance. As modeling and computing comes closer together, efficient computational methods need to be applied to new sets of equations. The problems to be addressed by simulation methods become more and more complicated, ranging over different scales, interacting on different dimensions and combining different physics. Such problems need to be implemented in a short period of time, solved on complicated domains and visualized with respect to the demand of the user. %Only a modular abstract simulation environment will fulfill these requirements and allow to setup, solve and visualize real-world problems appropriately. In this work, the concepts and the design of the C++ finite element toolbox AMDiS (adaptive multidimensional simulations) are described. It is shown, how abstract data structures and modern software concepts can help to design user-friendly finite element software, which provides large flexibility in problem definition while on the other hand efficiently solves these problems. Also systems of coupled problems can be solved in an intuitive way. In order to demonstrate its possibilities, AMDiS has been applied to several non-standard problems. The most time-consuming part in most simulations is the solution of linear systems of equations. Multi-level methods use discretization hierarchies to solve these systems in a very efficient way. In AMDiS, such multi-level techniques are implemented in the context of adaptive finite elements. Several numerical results are given which compare this multigrid solver with classical iterative methods. Besides the development of more efficient algorithms also the growing hardware capabilities lead to an improvement of simulation possibilities. Modern computing clusters contain more and more processors and also personal computers today are often equipped with multi-core processors. In this work, a new parallelization approach has been developed which allows the parallelization of sequential code in a very easy way and reduces the communication overhead compared to classical parallelization concepts.
8
Rosam, Jan. "A fully implicit, fully adaptive multigrid method for multiscale phase-field modelling." Thesis, University of Leeds, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.445357.
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9
Sanders, Geoffrey D. "Extensions to adaptive smooth aggregation (alphaSA) multigrid: Eigensolver initialization and nonsymmetric problems." Connect to online resource, 2008. 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:3337216.
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Vey, Simon. "Adaptive Finite Elements for Systems of PDEs: Software Concepts, Multi-level Techniques and Parallelization." Doctoral thesis, Technische Universität Dresden, 2007. https://tud.qucosa.de/id/qucosa%3A23684.
Full textAbstract:
In the recent past, the field of scientific computing has become of more and more importance for scientific as well as for industrial research, playing a comparable role as experiment and theory do. This success of computational methods in scientific and engineering research is next to the enormous improvement of computer hardware to a large extend due to contributions from applied mathematicians, who have developed algorithms which make real life applications feasible. Examples are adaptive methods, high order discretization, fast linear and non-linear solvers and multi-level methods. The application of these methods in a large class of problems demands for suitable and robust tools for a flexible and efficient implementation. In order to play a crucial role in scientific and engineering research, besides efficiency in the numerical solution, also efficiency in problem setup and interpretation of simulation results is of utmost importance. As modeling and computing comes closer together, efficient computational methods need to be applied to new sets of equations. The problems to be addressed by simulation methods become more and more complicated, ranging over different scales, interacting on different dimensions and combining different physics. Such problems need to be implemented in a short period of time, solved on complicated domains and visualized with respect to the demand of the user. %Only a modular abstract simulation environment will fulfill these requirements and allow to setup, solve and visualize real-world problems appropriately. In this work, the concepts and the design of the C++ finite element toolbox AMDiS (adaptive multidimensional simulations) are described. It is shown, how abstract data structures and modern software concepts can help to design user-friendly finite element software, which provides large flexibility in problem definition while on the other hand efficiently solves these problems. Also systems of coupled problems can be solved in an intuitive way. In order to demonstrate its possibilities, AMDiS has been applied to several non-standard problems. The most time-consuming part in most simulations is the solution of linear systems of equations. Multi-level methods use discretization hierarchies to solve these systems in a very efficient way. In AMDiS, such multi-level techniques are implemented in the context of adaptive finite elements. Several numerical results are given which compare this multigrid solver with classical iterative methods. Besides the development of more efficient algorithms also the growing hardware capabilities lead to an improvement of simulation possibilities. Modern computing clusters contain more and more processors and also personal computers today are often equipped with multi-core processors. In this work, a new parallelization approach has been developed which allows the parallelization of sequential code in a very easy way and reduces the communication overhead compared to classical parallelization concepts.
11
Muzaferija, Samir. "Adaptive finite volume method for flow prediction using unstructured meshes and multigrid approach." Thesis, Online version, 1994. http://ethos.bl.uk/OrderDetails.do?did=1&uin=uk.bl.ethos.311804.
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12
Lezeau, Patrick A. "An adaptive quasi-Newton coupled multigrid solver for the simulation of steady multiphase flows." Thesis, Cranfield University, 1997. http://hdl.handle.net/1826/4025.
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This thesis is concerned with the application of adaptive local quasi-Newton coupled
multigrid (ALQNMG) solvers to the numerical simulation of viscous incompressible
fluids, using the multi-fluid model.
The ALQNMG methodologyh as proven highly successful for single phase flows [1],
leading to solution algorithms which are: (i) robust, (ii) efficient and (iii) accurate.
Its extension to multiphase flows is very challenging because the governing equations
are mathematically complex and their solutions are subject to constraints.
The solver presented here has therefore required a considerable number of specific
algorithmic developments.
The outline of the thesis is as follows: firstly, the modelling and simulation of multiphase
flows are reviewed, together with the different numerical techniques implemented
in the solver. Finite volume discrete multiphase equations are then derived
on structured, staggered grids. Next, having specified the solution algorithm, we
consider the accuracy of the solver. Results from several test cases of varying complexity
are compared with those of a widely used commercial CFD package and good
agreementis obtained.
The question of performance is then addressed in detail, both in terms of robustness
and speed of convergence. Good accelerations are obtained using the multigrid
method but the convergence rates are often not grid-independent. The most likely
explanation is that the discrete operators are highly non-linear and therefore have
different characteristics on different grids. Furthermore, the solution algorithm is
shown to not handle certain multiphase diffusive terms very well. Convergence rates
are much faster than those achieved by single grid solvers and commercial codes
typically by one order of magnitude and often more, although the solver is not fully
optimal.
Finally, adaption is considered. Grids are generated automatically which facilitates
the use of the code and allows error control. It is confirmed that multigrid methods
offer a good framework for the implementation of adaption. Considerable gains in
speed and memory usage, by one further order of magnitude, are achieved.
13
Offermans, Nicolas. "Towards adaptive mesh refinement in Nek5000." Licentiate thesis, KTH, Mekanik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-217501.
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The development of adaptive mesh refinement capabilities in the field of computational fluid dynamics is an essential tool for enabling the simulation of larger and more complex physical problems. While such techniques have been known for a long time, most simulations do not make use of them because of the lack of a robust implementation. In this work, we present recent progresses that have been made to develop adaptive mesh refinement features in Nek5000, a code based on the spectral element method. These developments are driven by the algorithmic challenges posed by future exascale supercomputers. First, we perform the study of the strong scaling of Nek5000 on three petascale machines in order to assess the scalability of the code and identify the current bottlenecks. It is found that strong scaling limit ranges between 5, 000 and 220, 000 degrees of freedom per core depending on the machine and the case. The need for synchronized and low latency communication for efficient computational fluid dynamics simulation is also confirmed. Additionally, we present how Hypre, a library for linear algebra, is used to develop a new and efficient code for performing the setup step required prior to the use of an algebraic multigrid solver for preconditioning the pressure equation in Nek5000. Finally, the main objective of this work is to develop new methods for estimating the error on a numerical solution of the Navier–Stokes equations via the resolution of an adjoint problem. These new estimators are compared to existing ones, which are based on the decay of the spectral coefficients. Then, the estimators are combined with newly implemented capabilities in Nek5000 for automatic grid refinement and adaptive mesh adaptation is carried out. The applications considered so far are steady and two-dimensional, namely the lid-driven cavity at Re = 7, 500 and the flow past a cylinder at Re = 40. The use of adaptive mesh refinement techniques makes mesh generation easier and it is shown that a similar accuracy as with a static mesh can be reached with a significant reduction in the number of degrees of freedom.QC 20171114
14
Meneghello, Gianluca. "Stability and receptivity of the swept-wing attachment-line boundary layer : a multigrid numerical approach." Palaiseau, Ecole polytechnique, 2013. http://pastel.archives-ouvertes.fr/docs/00/79/55/43/PDF/tesi.pdf.
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Le but de cette étude est l'analyse de la stabilité et des propriétés de réceptivité de l'écoulement tridimensionnel au bord d'attaque d'une aile en flèche. Le projet est divisé en deux parties: (i) le calcul de l'écoulement de base stationnaire comme une solution de l'état d'équilibre de Navier-Stokes et (ii) l'étude du problème aux valeurs propres direct et adjoint obtenu en linéarisant les équations de Navier-Stokes autour de l'écoulement de base. Un code DNS a été développé sur la base d'un cadre multigrid. La solution des équations de Navier-Stokes non linéaires et stationnaires à différents nombres de Reynolds est obtenue à un coût de calcul de près de O(n), où n est le nombre de degrés de liberté du problème. L'étude de la stabilité et des propriétés de réceptivité est effectuée en résolvant numériquement le problème aux valeurs propres / vecteurs propres. Un algorithme de Krylov-Schur, couplé avec une transformation shift-invert, est utilisé pour extraire la partie la plus intéressante du spectre. Deux branches peuvent être identifiées et l'une d'elles est associée à des vecteurs propres montrant une connexion entre les modes caractéristique du bord d'attaque et les modes de type crossflow. Le wavemaker est localisé dans une région près du bord d'attaque. Les résultats numériques sont comparés qualitativement avec des observations expérimentales et des analyses de stabilité localeThe goal of this study is the analysis of the stability and receptivity properties of the three-dimensional flow past the leading edge of a swept wing. The project is divided into two parts: (i) the computation of the steady base flow as a solution of the steady-state Navier-Stokes equations and (ii) the study of the direct and adjoint eigenvalue problems obtained by linearizing the time-dependent Navier-Stokes equation around the base flow. In order to address the first part, a DNS code has been developed based on a multigrid framework. The solution of the non-linear steady-state Navier-Stokes equation at various Reynolds numbers is obtained by continuation at a computational cost of nearly O(n), where n is the number of degrees of freedom (dof) of the problem. The study of the stability and receptivity properties is performed by numerically solving the eigenvalue/eigenvector problem. A Krylov-Schur algorithm, coupled with a shift-invert spectral transformation, is used to extract part of the spectrum. Two branches may be identified and one of these is associated with eigenvectors displaying a connection between attachment line and cross-flow modes. The wave-maker region for these eigenvectors is shown to be located close to the attachment line by computing the corresponding solution to the adjoint eigenvalue problem. The numerical global results are compared qualitatively with existing experimental observations and local stability analysis
15
Villar, Millena Martins. "Análise numérica detalhada de escoamentos multifásicos bidimensionais." Universidade Federal de Uberlândia, 2007. https://repositorio.ufu.br/handle/123456789/14664.
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Conselho Nacional de Desenvolvimento Científico e TecnológicoThe mathematical modeling of multiphase flows involves the interaction between deformable and moving geometries with the fluid in which they are dispersed (immersed). This kind of interaction is present in many practical applications. A common approach to handle these problems is the so called Front-Tracking/Front-Capturing Hybrid Methods. This methodology consists in separating the problem into two domains: an eulerian domain, which is kept fixed and is used to discretize the fluid equations of both phases, and a lagrangian domain, which is used to solve the equations of motion of the interface. Since there is no geometric dependence between these two domains, the method can easily handle moving and deformable interfaces that are dispersed in the flow. Following this line of research, the present work aims to capture accurately details of such flows by resolving adequately the relevant physical scales in time and in space. This can be achieved by applying locally refined meshes which adapt dynamically to cover special flow regions, e.g. the vicinity of the fluid-fluid interfaces. To obtain the required resolution in time, a semi-implicit second order discretization to solve the Navier-Stokes equations is used. The turbulence modeling is introduced in the present work through Large Eddy Simulation. The eficiency and robustness of the methodology applied are verified via convergence analysis, as well as with simulations of one-phase and two-phase flows for several Reynolds numbers. The results of two-phase flows, with one bubble and with multiple bubbles, are presented. The results obtained for a single bubble case are compared with Clift's shape diagram (Clift et al., 1978).
A modelagem matemática de escoamentos multifásicos envolve a interação de geometrias móveis e deformáveis com o meio fluido que as envolve. Este tipo de interação faz parte de uma extensa lista de aplicações. Uma linha proposta para o tratamento num érico deste tipo de problema são os métodos híbridos Front-Tracking/Front-Capturing. Esta abordagem leva à separação do problema em dois domínios distintos (líquido/gás e líquido/líquido), um domínio fixo, euleriano, utilizado para discretizar as equações de ambas as fases, e outro móvel, lagrangiano, usado para as interfaces. Para o presente trabalho, na metologia utilizada, ambos os domínios são geometricamente independentes e não apresentam restrição quanto ao movimento e à deformação da fase dispersa. Seguindo esta linha, no presente trabalho propõe-se capturar detalhes deste de tipo escoamento, resolvendo adequadamente as escalas físicas do tempo e do espaço, utilizando malhas bloco estruturada refinadas localmente, as quais se adaptam dinamicamente para recobrir as regiões de interesse do escoamento (como, por exemplo, ao redor da interface fluido-fluido). Para se obter a resolução necessária no tempo, é usada uma discretização semi-implícita de segunda ordem para solucionar as equações de Navier-Stokes. A modelagem da turbulência é introduzida no presente trabalho via Simulação de Grandes Escalas. A eficiência e a robustez da metodologia implementada são verificadas via análise de convergência do método, bem como a simulação de escoamentos monofásicos e bifásicos para diferentes números Reynolds. São também apresentados resultados para escoamentos bifásicos com uma só bolha assim como para múltiplas bolhas. Os resultados de escoamentos mono-bolhas são comparados com o diagrama de forma de Clift et al. (1978).
Doutor em Engenharia Mecânica
16
Coco, Armando. "Finite-Difference Ghost-Cell Multigrid Methods for Elliptic problems with Mixed Boundary Conditions and Discontinuous Coefficients." Doctoral thesis, Università di Catania, 2012. http://hdl.handle.net/10761/1107.
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The work of this thesis is devoted to the development of an original and general numerical method for solving the elliptic equation in an arbitrary domain (described by a level-set function) with general boundary conditions (Dirichlet, Neumann, Robin, ...) using Cartesian grids. It can be then considered an immersed boundary method, and the scheme we use is based on a finite-difference ghost-cell technique. The entire problem is solved by an effective multigrid solver, whose components have been suitably constructed in order to be applied to the scheme.
The method is extended to the more challenging case of discontinuous coefficients, and the multigrid is suitable modified in order to attain the optimal convergence factor of the whole iteration procedure.
The development of the multigrid is an important feature of this thesis, since multigrid solvers for discontinuous coefficients maintaining the optimal convergence factor without depending on the jump in the coefficient and on the problem size is recently studied in literature.
The method is second order accurate in the solution and its gradient. A convergence proof for the first order scheme is provided, while second order is confirmed by several numerical tests.
17
Meneghello, Gianluca. "Réceptivité et sensibilité de la couche limite dans le bord d'attaque d'une aile en fleche : une approche multigrid." Phd thesis, Ecole Polytechnique X, 2013. http://pastel.archives-ouvertes.fr/pastel-00795543.
Full textAbstract:
Le but de cette étude est l'analyse de la stabilité et des propriétés de réceptivité de l'écoulement tridimensionnel au bord d'attaque d'une aile en flèche. Le projet est divisé en deux parties: (i) le calcul de l'écoulement de base stationnaire comme une solution de l'état d'équilibre de Navier-Stokes et (ii) l'étude du problème aux valeurs propres direct et adjoint obtenu en linéarisant les équations de Navier-Stokes autour de l'écoulement de base. Un code DNS a été développé sur la base d'un cadre multigrid. La solution des équations de Navier-Stokes non linéaires et stationnaires à différents nombres de Reynolds est obtenue à un coût de calcul de près de O(n), où n est le nombre de degrés de liberté du problème. L'étude de la stabilité et des propriétés de réceptivité est effectuée en résolvant numériquement le problème aux valeurs propres / vecteurs propres. Un algorithme de Krylov-Schur, couplé avec une transformation shift-invert, est utilisé pour extraire la partie la plus intéressante du spectre. Deux branches peuvent être identifiées et l'une d'elles est associée à des vecteurs propres montrant une connexion entre les modes caractéristique du bord d'attaque et les modes de type crossflow. Le wavemaker est localisé dans une région près du bord d'attaque. Les résultats numériques sont comparés qualitativement avec des observations expérimentales et des analyses de stabilité locale.
18
Löf, Henrik. "Iterative and Adaptive PDE Solvers for Shared Memory Architectures." Doctoral thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7136.
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Scientific computing is used frequently in an increasing number of disciplines to accelerate scientific discovery. Many such computing problems involve the numerical solution of partial differential equations (PDE). In this thesis we explore and develop methodology for high-performance implementations of PDE solvers for shared-memory multiprocessor architectures. We consider three realistic PDE settings: solution of the Maxwell equations in 3D using an unstructured grid and the method of conjugate gradients, solution of the Poisson equation in 3D using a geometric multigrid method, and solution of an advection equation in 2D using structured adaptive mesh refinement. We apply software optimization techniques to increase both parallel efficiency and the degree of data locality. In our evaluation we use several different shared-memory architectures ranging from symmetric multiprocessors and distributed shared-memory architectures to chip-multiprocessors. For distributed shared-memory systems we explore methods of data distribution to increase the amount of geographical locality. We evaluate automatic and transparent page migration based on runtime sampling, user-initiated page migration using a directive with an affinity-on-next-touch semantic, and algorithmic optimizations for page-placement policies. Our results show that page migration increases the amount of geographical locality and that the parallel overhead related to page migration can be amortized over the iterations needed to reach convergence. This is especially true for the affinity-on-next-touch methodology whereby page migration can be initiated at an early stage in the algorithms. We also develop and explore methodology for other forms of data locality and conclude that the effect on performance is significant and that this effect will increase for future shared-memory architectures. Our overall conclusion is that, if the involved locality issues are addressed, the shared-memory programming model provides an efficient and productive environment for solving many important PDE problems.
19
Nós, Rudimar Luiz. "\"Simulações de escoamentos tridimensionais bifásicos empregando métodos adaptativos e modelos de campo fase\"." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-08052007-143200/.
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Este é o primeiro trabalho que apresenta simulações tridimensionais completamente adaptativas de um modelo de campo de fase para um fluido incompressível com densidade de massa constante e viscosidade variável, conhecido como Modelo H. Solucionando numericamente as equações desse modelo em malhas refinadas localmente com a técnica AMR, simulamos computacionalmente escoamentos bifásicos tridimensionais. Os modelos de campo de fase oferecem uma aproximação física sistemática para investigar fenômenos que envolvem sistemas multifásicos complexos, tais como fluidos com camadas de mistura, a separação de fases sob forças de cisalhamento e a evolução de micro-estruturas durante processos de solidificação. Como as interfaces são substituídas por delgadas regiões de transição (interfaces difusivas), as simulações de campo de fase requerem muita resolução nessas regiões para capturar corretamente a física do problema em estudo. Porém essa não é uma tarefa fácil de ser executada numericamente. As equações que caracterizam o modelo de campo de fase contêm derivadas de ordem elevada e intrincados termos não lineares, o que exige uma estratégia numérica eficiente capaz de fornecer precisão tanto no tempo quanto no espaço, especialmente em três dimensões. Para obter a resolução exigida no tempo, usamos uma discretização semi-implícita de segunda ordem para solucionar as equações acopladas de Cahn-Hilliard e Navier-Stokes (Modelo H). Para resolver adequadamente as escalas físicas relevantes no espaço, utilizamos malhas refinadas localmente que se adaptam dinamicamente para recobrir as regiões de interesse do escoamento, como por exemplo, as vizinhanças das interfaces do fluido. Demonstramos a eficiência e a robustez de nossa metodologia com simulações que incluem a separação dos componentes de uma mistura bifásica, a deformação de gotas sob cisalhamento e as instabilidades de Kelvin-Helmholtz.This is the first work that introduces 3D fully adaptive simulations for a phase field model of an incompressible fluid with matched densities and variable viscosity, known as Model H. Solving numerically the equations of this model in meshes locally refined with AMR technique, we simulate computationally tridimensional two-phase flows. Phase field models offer a systematic physical approach to investigate complex multiphase systems phenomena such as fluid mixing layers, phase separation under shear and microstructure evolution during solidification processes. As interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations need great resolution in these regions to capture correctly the physics of the studied problem. However, this is not an easy task to do numerically. Phase field model equations have high order derivatives and intricate nonlinear terms, which require an efficient numerical strategy that can achieve accuracy both in time and in space, especially in three dimensions. To obtain the required resolution in time, we employ a semi-implicit second order discretization scheme to solve the coupled Cahn-Hilliard/Navier-Stokes equations (Model H). To resolve adequatly the relevant physical scales in space, we use locally refined meshes which adapt dynamically to cover special flow regions, e.g., the vicinity of the fluid interfaces. We demonstrate the efficiency and robustness of our methodology with simulations that include spinodal decomposition, the deformation of drops under shear and Kelvin-Helmholtz instabilities.
20
Le, Carlier de Veslud Christian. "Etude et réalisation de logiciels d'éléments finis sur micro-ordinateur : mailleurs automatiques résolution auto-adaptative de problèmes de mécanique utilisant la méthode multigrilles." Vandoeuvre-les-Nancy, INPL, 1991. http://docnum.univ-lorraine.fr/public/INPL_T_1991_LE_CARLIER_DE_VESLUD_C.pdf.
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Nous construisons un ensemble de logiciels permettant la modélisation par éléments finis de problèmes de mécanique et leur résolution par des méthodes multigrilles sur micro-ordinateur macintosh. Nous privilégions la simplicité, la sureté de l'emploi et l'automaticité des logiciels développés. Pour cela, 4 algorithmes de maillages automatiques bi et tridimensionnels (par bloc, et par une méthode de Delaunay en 2D, par bloc et par élévation de maillage plan en 3D) utilisant les plus récents développements ont été mis au point. La gestion graphique de ces logiciels utilise un grand nombre d'utilitaires (menus déroulants, saisie directe à l'écran, élimination des parties cachées, modification locale ou globale du maillage, etc. . . ). Ces stratégies sont appliquées aux ossatures planes linéaires de poutre avec ou sans cisaillement transverse et ainsi au cas de l'élasticité linéaire plane. Dans ce dernier cas, l'utilisation de la méthode multigrilles associée à des mailleurs auto-adaptatifs permet d'affiner l'étude là où c'est nécessaire, offrant ainsi un gain de précision et de rapidité
21
Cakmak, Mehtap. "Development Of A Multigrid Accelerated Euler Solver On Adaptively Refined Two- And Three-dimensional Cartesian Grids." Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/2/12610753/index.pdf.
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Cartesian grids offer a valuable option to simulate aerodynamic flows around complex geometries such as multi-element airfoils, aircrafts, and rockets. Therefore, an adaptively-refined Cartesian grid generator and Euler solver are developed. For the mesh generation part of the algorithm, dynamic data structures are used to determine connectivity information between cells and uniform mesh is created in the domain. Marching squares and cubes algorithms are used to form interfaces of cut and split cells. Geometry-based cell adaptation is applied in the mesh generation. After obtaining appropriate mesh around input geometry, the solution is obtained using either flux vector splitting method or Roe&rsquos approximate Riemann solver with cell-centered approach. Least squares reconstruction of flow variables within the cell is used to determine high gradient regions of flow. Solution based adaptation method is then applied to current mesh in order to refine these regions and also coarsened regions where unnecessary small cells exist. Multistage time stepping is used with local time steps to increase the convergence rate. Also FAS multigrid technique is used in order to increase the convergence rate. It is obvious that implementation of geometry and solution based adaptations are easier for Cartesian meshes than other types of meshes. Besides, presented numerical results show the accuracy and efficiency of the algorithm by especially using geometry and solution based adaptation. Finally, Euler solutions of Cartesian grids around airfoils, projectiles and wings are compared with the experimental and numerical data available in the literature and accuracy and efficiency of the solver are verified.
22
Jones, Alison Claire. "A projected multigrid method for the solution of non-linear finite element problems on adaptively refined grids." Thesis, University of Leeds, 2005. http://etheses.whiterose.ac.uk/1328/.
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This thesis describes the formulation and application of an adaptive multigrid method for the efficient solution of nonlinear elliptic and parabolic partial differential equations and systems. A continuous Galerkin finite-element method is combined with locally adaptive mesh refinement and an optimal multigrid solver to achieve this efficiency. The novel contribution of this work lies in the manner in which these two techniques are combined. In particular the multigrid solver provides a natural and simple method of handling grid points that are not fully connected, so called hanging nodes. This allows for a straightforward adaptive gridding scheme that does not need to take any special measures to repair these hanging nodes for a standard element-by-element implementation of the finite element assembly process. Specifically, on each element, only the usual finite element basis functions are required, even in the vicinity of hanging nodes. Furthermore the standard multigrid full approximation scheme (FAS) may be applied with only minor modifications to account for the presence of the hanging nodes. A wide cross-section of nonlinear elliptic and parabolic problems are used to demonstrate the performance of the proposed algorithm, which is shown to provide optimal accuracy at an optimal computational cost.
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Lin, Ming-Huei, and 林明輝. "On Adaptive Multigrid Method." Thesis, 1993. http://ndltd.ncl.edu.tw/handle/98070543798105564267.
Full textAbstract:
碩士國立交通大學
應用數學研究所
81
Adaptive multigrid methods are developed for the finite element ( FE ) and finite volume element ( FVE ) approximations of general elliptic boundary value problems.
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Stals, Linda. "Parallel multigrid on unstructured grids using adaptive finite element methods." Phd thesis, 1995. http://hdl.handle.net/1885/138505.
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Bai, Ji-Hong, and 白佶弘. "Adaptive multigrid-continuation methods for Bose-Einestein condensates in a periodic potential." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/72505662921024132017.
Full textAbstract:
碩士國立中興大學
應用數學系所
97
We study multigrid-continuation method for treating parameter-dependent problems. The proposed algorithm which can be flexibly implemented is a generalization of the two-grid discretization schemes [8]. That is, approximating points on a solution curve do not necessarily lie on the same fine grid. We apply the algorithm to compute energy levels and superfluid densities of Bose-Einstein condensates (BEC) in a periodic potential. For positive scattering length, if the chemical potential is large enough, and the domain is properly chosen, the results show that the number of peaks of the first few energy states of the 1D BEC and 2D BEC in a periodic potential depends on the wave number of the periodic potential. Moreover, for dark solitons the number of peaks of the ground state solutions is Π(2l/d_j), where the periodic potential is expressed in terms of the cosine functions. Finally, we study two-stage continuation algorithms for Bloch waves of 1D BEC in optical lattices. Our sample numerical result is reported.
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Son, Chiou Shean, and 邱顯森. "Application of Adaptive Multigrid Method to the Simulator of Sliders in Hard Disk Drives." Thesis, 2000. http://ndltd.ncl.edu.tw/handle/03711870134649964626.
Full textAbstract:
碩士國立臺灣大學
機械工程學研究所
88
An accurate and efficient numerical simulator is essential to the design and development of modern hard disk drives. The governing equation for the air bearing pressure between the slider and the disk is the generalized Reynolds equation. Due to the non-linearity of the Reynolds equation and the complexity of the slider surface, the air bearing pressure can only be solved numerically. The multigrid method has been proven to be a reliable scheme for solving the Reynolds equation. However, in order to describe the high-pressure gradient on the arbitrary shaped rails accurately, extremely fine mesh is required. This would significantly deteriorate the efficiency of the multigrid method especially when applied to the modern textured shaped sliders. In this thesis, we develop a numerical program based on the adaptive multigrid method to simulate the steady-state as well as the dynamical responses of sliders. The program automatically constructs high-level fine-mesh patches at and only at locations where the relative truncation errors exceed the required tolerance. This adaptive scheme can save computational resource and increase simulation efficiency dramatically. By using this program, we study the effects of the textured rails on the flying characteristics of an H-shaped negative pressure slider. It is found that the flying height increases with the number of landing pads.
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Lee, Wei-Jen, and 李偉任. "Multigrid and Adaptive Methods for Computing Singular Solutions of Laplace Equation on Corner Domains." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/77603040683327370289.
Full textAbstract:
碩士國立交通大學
應用數學系數學建模與科學計算碩士班
98
Elliptic boundary value problems on domain with corners have singular behavior near the corners. Such singular behavior affect the accuracy of the finite element method throughout the whole domain. For the Poisson equation with homogeneous Dirichlet boundary conditions defined on a polygonal domain with re-entrant corners, it is well known that the solution has the singular function u=w+ks representation ,where w is the regular part of the solution and s are known as singular functions that depend only on the corresponding re-entrant angles. Coefficients k known as the stress intensity factors in the context of mechanics can be expressed in terms of u by extraction formula, where s- are known as dual singular funciton. Accurate calculation of these quantities is of great importance in many practical engineering problems. Similar singular function representations hold for the solutions of interface,biharmonic,elasticity, and evolution problems in [1, 2].
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Halder, Soham. "Scalability Bottleneck Analysis of High Performance Applications." Thesis, 2020. https://etd.iisc.ac.in/handle/2005/4487.
Full textAbstract:
Obtaining high performance and scalability for high performance applications are challenging.
There are various bottlenecks including, higher rate of memory access, complex algorithm, high
rate of communication, big messages over communication, write-write con
ict etc. that ham-
per scalability. Consequently, there is a critical need for performance prediction and scalability
bottleneck analysis to enable scientists to understand impediments to performance on emerging
systems. Performance predictions for large problem sizes and processors using limited small
scale runs are useful for a variety of purposes including scalability projections, and help in min-
imizing the time taken for constructing training data for building performance models. Many
parallel applications often suffer from latent performance limitations that may prevent them
from scaling to larger machine sizes. Often, such scalability bugs manifest themselves only when
an attempt to scale the code is actually being made, where remediation can be substantially
difficult. Hence, performance predictions for such applications are challenging.
One method is to build analytical models based on the knowledge of the application charac-
teristics. This requires laborious constructions of the models, and the models are difficult to
build for large-scale parallel applications. Such models are also error-prone due to manual in-
tervention. Hence, the model generation in recent years is mainly based on empirical methods.
The model is generated using a base equation with some parameters and those parameters are
learned from some training runs on the cores which are already available. The trained model is
then used to predict the performance for higher cores. Thus, most of the existing methods rely
heavily on the basis the subtleties of extrapolations for achieving good prediction accuracies.
Techniques that can predict performance with reasonable accuracy may fail to capture the
scalability trends of the applications at higher scales. In general, if a scalability bottleneck is
hidden and can only manifest at larger core counts, it is hard for extrapolation techniques to
correctly capture the trend from only the performance of the training runs. Static code analysis
is needed to capture the application characteristics at different scales.
In this thesis, we develop strategies for both performance predictions and scalability projections
including for applications whose scalability trends can change at scale due to the presence of
scalability bottlenecks. For performance predictions, two strategies are explored for building
performance models using executions on smaller number of cores. In the rst strategy, the orig-
inal application is executed on smaller number of cores, and a performance model is developed
using these executions. In the second strategy, common benchmark kernels are executed on
smaller number of cores and a benchmark kernel that matches with the application in terms of
execution characteristics like instruction counts is chosen. The performance model of the chosen
benchmark kernel is then used to predict performance of the application for larger scales. For
application specifi c extrapolation the resuls show that the error percentage stays around 15%
but the prediction function fails to capture the trend shown by the applications. The correla-
tion coeffecient between actual and predicted behaviour of AMG (Adaptive Multigrid) is around
75%. For Benchmark based prediction, the error percentage is better that simple extrapolation
method and stays below 10% for SMG (Semicoarsening Multigrid) but this method also fails to
capture the trend shown by the application at high cores. The correlation coefficient between
actual and predicted behaviour of SMG is around 90%.
For scalability projections, the different phases of the application are instrumented to collect
various parameters including time stamps and number of invocations of the phase executions.
These phases could be function calls or loops. The phase and parameter values of a phase that
follows the scalability trend of the application for different number of cores are identi fied. For
example, the number of invocations of a particular function call may follow the scalability trend
of the overall application for different number of cores. Static analysis is performed to map the
identfii ed parameter values in terms of the input parameters, namely, problem size and number
of cores. A performance model is then built for the parameter value, e.g., number of invocations
of a function call, in terms of the input parameters. The performance model is then used to
project the scalability trend for larger number of cores. Since this performance model is built
for the parameter that follows the scalability trend of the application, the model can be reliably
used for scalability projections and identifying scalability bottlenecks. We demonstrated the
techniques with three scienti c applications, namely, SMG (Semi-Coarsening MultiGrid), AMG
(Adaptive MultiGrid) and GTC (Gyrokinetic Toroidal Code).
In application speci c metric based prediction, the results show prediction errors below 10%
for all the above applications considered and also show that that the the scalability trends of
the applications are captured well by the predictions. The correlation coefficient of the actual
and predicted trends at high core is at least 97% while for other methods also considered this
value is well less than 90% for the applications.
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Richter, Thomas [Verfasser]. "Parallel multigrid method for adaptive finite elements with application to 3D flow problems / vorgelegt von Thomas Richter." 2005. http://d-nb.info/97605843X/34.
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Isaac, Tobin Gregory. "Scalable, adaptive methods for forward and inverse problems in continental-scale ice sheet modeling." Thesis, 2015. http://hdl.handle.net/2152/31372.
Full textAbstract:
Projecting the ice sheets' contribution to sea-level rise is difficult because of the complexity of accurately modeling ice sheet dynamics for the full polar ice sheets, because of the uncertainty in key, unobservable parameters governing those dynamics, and because quantifying the uncertainty in projections is necessary when determining the confidence to place in them. This work presents the formulation and solution of the Bayesian inverse problem of inferring, from observations, a probability distribution for the basal sliding parameter field beneath the Antarctic ice sheet. The basal sliding parameter is used within a high-fidelity nonlinear Stokes model of ice sheet dynamics. This model maps the parameters "forward" onto a velocity field that is compared against observations. Due to the continental-scale of the model, both the parameter field and the state variables of the forward problem have a large number of degrees of freedom: we consider discretizations in which the parameter has more than 1 million degrees of freedom. The Bayesian inverse problem is thus to characterize an implicitly defined distribution in a high-dimensional space. This is a computationally demanding problem that requires scalable and efficient numerical methods be used throughout: in discretizing the forward model; in solving the resulting nonlinear equations; in solving the Bayesian inverse problem; and in propagating the uncertainty encoded in the posterior distribution of the inverse problem forward onto important quantities of interest. To address discretization, a hybrid parallel adaptive mesh refinement format is designed and implemented for ice sheets that is suited to the large width-to-height aspect ratios of the polar ice sheets. An efficient solver for the nonlinear Stokes equations is designed for high-order, stable, mixed finite-element discretizations on these adaptively refined meshes. A Gaussian approximation of the posterior distribution of parameters is defined, whose mean and covariance can be efficiently and scalably computed using adjoint-based methods from PDE-constrained optimization. Using a low-rank approximation of the covariance of this distribution, the covariance of the parameter is pushed forward onto quantities of interest.
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Pathak, Harshavardhana Sunil. "Adaptive Mesh Redistribution for Hyperbolic Conservation Laws." Thesis, 2013. http://etd.iisc.ac.in/handle/2005/3281.
Full textAbstract:
An adaptive mesh redistribution method for efficient and accurate simulation of multi dimensional hyperbolic conservation laws is developed. The algorithm consists of two coupled steps; evolution of the governing PDE followed by a redistribution of the computational nodes. The second step, i.e. mesh redistribution is carried out at each time step iteratively with the primary aim of adapting the grid to the computed solution in order to maximize accuracy while minimizing the computational overheads. The governing hyperbolic conservation laws, originally defined on the physical domain, are transformed on to a simplified computational domain where the position of the nodes remains independent of time. The transformed governing hyperbolic equations are recast in a strong conservative form and are solved directly on the computational domain without the need for interpolation that is typically associated with standard mesh redistribution algorithms. Several standard test cases involving numerical solution of scalar and system of hyperbolic conservation laws in one and two dimensions are presented in order to demonstrate the accuracy and computational efficiency of the proposed technique.
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Pathak, Harshavardhana Sunil. "Adaptive Mesh Redistribution for Hyperbolic Conservation Laws." Thesis, 2013. http://hdl.handle.net/2005/3281.
Full textAbstract:
An adaptive mesh redistribution method for efficient and accurate simulation of multi dimensional hyperbolic conservation laws is developed. The algorithm consists of two coupled steps; evolution of the governing PDE followed by a redistribution of the computational nodes. The second step, i.e. mesh redistribution is carried out at each time step iteratively with the primary aim of adapting the grid to the computed solution in order to maximize accuracy while minimizing the computational overheads. The governing hyperbolic conservation laws, originally defined on the physical domain, are transformed on to a simplified computational domain where the position of the nodes remains independent of time. The transformed governing hyperbolic equations are recast in a strong conservative form and are solved directly on the computational domain without the need for interpolation that is typically associated with standard mesh redistribution algorithms. Several standard test cases involving numerical solution of scalar and system of hyperbolic conservation laws in one and two dimensions are presented in order to demonstrate the accuracy and computational efficiency of the proposed technique.
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Nesme, Matthieu. "Milieu mécanique déformable multirésolution pour la simulation interactive." Phd thesis, 2008. http://tel.archives-ouvertes.fr/tel-00293531.
Full textAbstract:
Les modèles dynamiques sont incontournables en synthèse d'animations car ils permettent la simulation réaliste de phénomènes physiques et accordent une meilleure immersion dans un monde virtuel.Plusieurs approches performantes permettent l'animation d'objets déformables, mais les scènes sont souvent complexes à modéliser rendant leur utilisation difficile en pratique.
Dans cette thèse nous proposons une solution simplifiant l'animation physique interactive d'objets déformables. Nous suggérons de plonger et d'interpoler l'objet dans une grille déformable sur laquelle s'appliquent des lois mécaniques. Une méthode d'éléments finis rapides et robustes a été étendue afin de prendre en compte la répartition de la matière et plusieurs propriétés de matériaux à l'intérieur d'un élément, et ainsi offrir un comportement amélioré à des résolutions grossières. Afin de concentrer les calculs là où ils sont le plus nécessaires, une formulation multirésolution simple est proposée.
Puis nous analysons deux méthodes permettant d'améliorer la propagation des déformations pour des matériaux "mal conditionnés" : une formulation hiérarchique des élément finis, lourde à mettre en place mais permettant facilement la multirésolution, et une formulation multigrid, élégante et performante, mais plus difficile à décliner en multirésolution.
Enfin nous validons la précision de notre méthode en la soumettant à diverses expériences.
Il en résulte une méthode rapide, robuste, précise et facile d'utilisation aussi bien pour un infographiste, qui peut animer n'importe quel modèle sans connaissances préalables, que pour la modélisation individualisée d'un patient à partir d'images médicales segmentées.