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Wei, Fei. "Weighted least-squares finite element methods for PIV data assimilation." Thesis, Montana State University, 2011. http://etd.lib.montana.edu/etd/2011/wei/WeiF0811.pdf.
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The ability to diagnose irregular flow patterns clinically in the left ventricle (LV) is currently very challenging. One potential approach for non-invasively measuring blood flow dynamics in the LV is particle image velocimetry (PIV) using microbubbles. To obtain local flow velocity vectors and velocity maps, PIV software calculates displacements of microbubbles over a given time interval, which is typically determined by the actual frame rate. In addition to the PIV, ultrasound images of the left ventricle can be used to determine the wall position as a function of time, and the inflow and outflow fluid velocity during the cardiac cycle. Despite the abundance of data, ultrasound and PIV alone are insufficient for calculating the flow properties of interest to clinicians. Specifically, the pressure gradient and total energy loss are of primary importance, but their calculation requires a full three-dimensional velocity field. Echo-PIV only provides 2D velocity data along a single plane within the LV. Further, numerous technical hurdles prevent three-dimensional ultrasound from having a sufficiently high frame rate (currently approximately 10 frames per second) for 3D PIV analysis. Beyond microbubble imaging in the left ventricle, there are a number of other settings where 2D velocity data is available using PIV, but a full 3D velocity field is desired. This thesis develops a novel methodology to assimilate two-dimensional PIV data into a three-dimensional Computational Fluid Dynamics simulation with moving domains. To illustrate and validate our approach, we tested the approach on three different problems: a flap displaced by a fluid jut; an expanding hemisphere; and an expanding half ellipsoid representing the left ventricle of the heart. To account for the changing shape of the domain in each problem, the CFD mesh was deformed using a pseudo-solid domain mapping technique at each time step. The incorporation of experimental PIV data can help to identify when the imposed boundary conditions are incorrect. This approach can also help to capture effects that are not modeled directly like the impacts of heart valves on the flow of blood into the left ventricle.
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Bringmann, Philipp. "Adaptive least-squares finite element method with optimal convergence rates." Doctoral thesis, Humboldt-Universität zu Berlin, 2021. http://dx.doi.org/10.18452/22350.
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Die Least-Squares Finite-Elemente-Methoden (LSFEMn) basieren auf der Minimierung des Least-Squares-Funktionals, das aus quadrierten Normen der Residuen eines Systems von partiellen Differentialgleichungen erster Ordnung besteht. Dieses Funktional liefert einen a posteriori Fehlerschätzer und ermöglicht die adaptive Verfeinerung des zugrundeliegenden Netzes. Aus zwei Gründen versagen die gängigen Methoden zum Beweis optimaler Konvergenzraten, wie sie in Carstensen, Feischl, Page und Praetorius (Comp. Math. Appl., 67(6), 2014) zusammengefasst werden. Erstens scheinen fehlende Vorfaktoren proportional zur Netzweite den Beweis einer schrittweisen Reduktion der Least-Squares-Schätzerterme zu verhindern. Zweitens kontrolliert das Least-Squares-Funktional den Fehler der Fluss- beziehungsweise Spannungsvariablen in der H(div)-Norm, wodurch ein Datenapproximationsfehler der rechten Seite f auftritt. Diese Schwierigkeiten führten zu einem zweifachen Paradigmenwechsel in der Konvergenzanalyse adaptiver LSFEMn in Carstensen und Park (SIAM J. Numer. Anal., 53(1), 2015) für das 2D-Poisson-Modellproblem mit Diskretisierung niedrigster Ordnung und homogenen Dirichlet-Randdaten. Ein neuartiger expliziter residuenbasierter Fehlerschätzer ermöglicht den Beweis der Reduktionseigenschaft. Durch separiertes Markieren im adaptiven Algorithmus wird zudem der Datenapproximationsfehler reduziert.
Die vorliegende Arbeit verallgemeinert diese Techniken auf die drei linearen Modellprobleme das Poisson-Problem, die Stokes-Gleichungen und das lineare Elastizitätsproblem. Die Axiome der Adaptivität mit separiertem Markieren nach Carstensen und Rabus (SIAM J. Numer. Anal., 55(6), 2017) werden in drei Raumdimensionen nachgewiesen. Die Analysis umfasst Diskretisierungen mit beliebigem Polynomgrad sowie inhomogene Dirichlet- und Neumann-Randbedingungen. Abschließend bestätigen numerische Experimente mit dem h-adaptiven Algorithmus die theoretisch bewiesenen optimalen Konvergenzraten.The least-squares finite element methods (LSFEMs) base on the minimisation of the least-squares functional consisting of the squared norms of the residuals of first-order systems of partial differential equations. This functional provides a reliable and efficient built-in a posteriori error estimator and allows for adaptive mesh-refinement. The established convergence analysis with rates for adaptive algorithms, as summarised in the axiomatic framework by Carstensen, Feischl, Page, and Praetorius (Comp. Math. Appl., 67(6), 2014), fails for two reasons. First, the least-squares estimator lacks prefactors in terms of the mesh-size, what seemingly prevents a reduction under mesh-refinement. Second, the first-order divergence LSFEMs measure the flux or stress errors in the H(div) norm and, thus, involve a data resolution error of the right-hand side f. These difficulties led to a twofold paradigm shift in the convergence analysis with rates for adaptive LSFEMs in Carstensen and Park (SIAM J. Numer. Anal., 53(1), 2015) for the lowest-order discretisation of the 2D Poisson model problem with homogeneous Dirichlet boundary conditions. Accordingly, some novel explicit residual-based a posteriori error estimator accomplishes the reduction property. Furthermore, a separate marking strategy in the adaptive algorithm ensures the sufficient data resolution. This thesis presents the generalisation of these techniques to three linear model problems, namely, the Poisson problem, the Stokes equations, and the linear elasticity problem. It verifies the axioms of adaptivity with separate marking by Carstensen and Rabus (SIAM J. Numer. Anal., 55(6), 2017) in three spatial dimensions. The analysis covers discretisations with arbitrary polynomial degree and inhomogeneous Dirichlet and Neumann boundary conditions. Numerical experiments confirm the theoretically proven optimal convergence rates of the h-adaptive algorithm.
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Storn, Johannes. "Topics in Least-Squares and Discontinuous Petrov-Galerkin Finite Element Analysis." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/20141.
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Aufgrund der fundamentalen Bedeutung partieller Differentialgleichungen zur Beschreibung von Phänomenen in angewandten Wissenschaften ist deren Analyse ein Kerngebiet der Mathematik. Durch Computer lassen sich die Lösungen für eine Vielzahl dieser Gleichungen näherungsweise bestimmen. Die dabei verwendeten numerischen Verfahren sollen auf möglichst exakte Approximationen führen und deren Genauigkeit verifizieren. Die Least-Squares Finite-Elemente-Methode (LSFEM) und die unstetige Petrov-Galerkin (DPG) Methode sind solche Verfahren. Sie werden in dieser Dissertation untersucht.
Der erste Teil der Arbeit untersucht die Genauigkeit der mittels LSFEM berechneten Näherungen. Dazu werden Eigenschaften der zugrundeliegenden Differentialgleichungen mit den Eigenschaften der LSFEM kombiniert. Dies zeigt, dass die Abweichung der berechneten Näherung von der exakten Lösung einem berechenbaren Residuum asymptotisch entspricht. Ferner wird ein Verfahren zu Berechnung einer garantierten oberen Fehlerschranke eingeführt. Während etablierte Fehlerschätzer den Fehler signifikant überschätzt, zeigen numerische Experimente eine äußerst geringe Überschätzung des Fehlers mittels der neuen Fehlerschranke.
Die Analyse der Fehlerschranken für das Stokes-Problem offenbart ein Beziehung der LSFEM und der LBB Konstanten. Diese Konstante ist entscheidend für die Existenz und Stabilität von Lösungen in der Strömungslehre. Der zweite Teil der Arbeit nutzt diese Beziehung und entwickelt ein auf der LSFEM basierendes Verfahren zur numerischen Berechnung der LBB Konstanten.
Der dritte Teil der Arbeit untersucht die DPG Methode. Dabei werden existierende Anwendungen der DPG Methode zusammengefasst und analysiert. Diese Analyse zeigt, dass sich die DPG Methode als eine leicht gestörte LSFEM interpretieren lässt. Diese Interpretation erlaubt die Anwendung der Resultate aus dem ersten Teil der Arbeit und ermöglicht dadurch eine genauere Untersuchung existierender und die Entwicklung neuer DPG Methoden.The analysis of partial differential equations is a core area in mathematics due to the fundamental role of partial differential equations in the description of phenomena in applied sciences. Computers can approximate the solutions to these equations for many problems. They use numerical schemes which should provide good approximations and verify the accuracy. The least-squares finite element method (LSFEM) and the discontinuous Petrov-Galerkin (DPG) method satisfy these requirements. This thesis investigates these two schemes. The first part of this thesis explores the accuracy of solutions to the LSFEM. It combines properties of the underlying partial differential equation with properties of the LSFEM and so proves the asymptotic equality of the error and a computable residual. Moreover, this thesis introduces an novel scheme for the computation of guaranteed upper error bounds. While the established error estimator leads to a significant overestimation of the error, numerical experiments indicate a tiny overestimation with the novel bound. The investigation of error bounds for the Stokes problem visualizes a relation of the LSFEM and the Ladyzhenskaya-Babuška-Brezzi (LBB) constant. This constant is a key in the existence and stability of solution to problems in fluid dynamics. The second part of this thesis utilizes this relation to design a competitive numerical scheme for the computation of the LBB constant. The third part of this thesis investigates the DPG method. It analyses an abstract framework which compiles existing applications of the DPG method. The analysis relates the DPG method with a slightly perturbed LSFEM. Hence, the results from the first part of this thesis extend to the DPG method. This enables a precise investigation of existing and the design of novel DPG schemes.
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Akargun, Yigit Hayri. "Least-squares Finite Element Solution Of Euler Equations With Adaptive Mesh Refinement." Master's thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614138/index.pdf.
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Least-squares finite element method (LSFEM) is employed to simulate 2-D and axisymmetric flows governed by the compressible Euler equations. Least-squares formulation brings many advantages over classical Galerkin finite element methods. For non-self-adjoint systems, LSFEM result in symmetric positive-definite matrices which can be solved efficiently by iterative methods. Additionally, with a unified formulation it can work in all flight regimes from subsonic to supersonic. Another advantage is that, the method does not require artificial viscosity since it is naturally diffusive which also appears as a difficulty for sharply resolving high gradients in the flow field such as shock waves. This problem is dealt by employing adaptive mesh refinement (AMR) on triangular meshes. LSFEM with AMR technique is numerically tested with various flow problems and good agreement with the available data in literature is seen.
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Goktolga, Mustafa Ugur. "Simulation Of Conjugate Heat Transfer Problems Using Least Squares Finite Element Method." Master's thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614787/index.pdf.
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In this thesis study, a least-squares finite element method (LSFEM) based conjugate heat transfer solver was developed. In the mentioned solver, fluid flow and heat transfer computations were performed separately. This means that the calculated velocity values in the flow calculation part were exported to the heat transfer part to be used in the convective part of the energy equation. Incompressible Navier-Stokes equations were used in the flow simulations. In conjugate heat transfer computations, it is required to calculate the heat transfer in both flow field and solid region. In this study, conjugate behavior was accomplished in a fully coupled manner, i.e., energy equation for fluid and solid regions was solved simultaneously and no boundary conditions were defined on the fluid-solid interface. To assure that the developed solver works properly, lid driven cavity flow, backward facing step flow and thermally driven cavity flow problems were simulated in three dimensions and the findings compared well with the available data from the literature. Couette flow and thermally driven cavity flow with conjugate heat transfer in two dimensions were modeled to further validate the solver. Finally, a microchannel conjugate heat transfer problem was simulated. In the flow solution part of the microchannel problem, conservation of mass was not achieved. This problem was expected since the LSFEM has problems related to mass conservation especially in high aspect ratio channels. In order to overcome the mentioned problem, weight of continuity equation was increased by multiplying it with a constant. Weighting worked for the microchannel problem and the mass conservation issue was resolved. Obtained results for microchannel heat transfer problem were in good agreement in general with the previous experimental and numerical works.
In the first computations with the solverquadrilateral and triangular elements for two dimensional problems, hexagonal and tetrahedron elements for three dimensional problems were tried. However, since only the quadrilateral and hexagonal elements gave satisfactory results, they were used in all the above mentioned simulations.
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Johnsen, Eivind. "Application method of the least squares finite element method to fracture mechanics." Thesis, Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/16435.
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Danisch, Garvin. "Gemischte Finite-element-least-squares-Methoden für die Flachwassergleichungen mit kleiner Viskosität." [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=983833052.
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Dolan, P. S. "Viscous incompressible flow solutions via divergence free least squares finite element optimisation." Thesis, University of Hertfordshire, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.377903.
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Prabhakar, Vivek. "Least squares based finite element formulations and their applications in fluid mechanics." [College Station, Tex. : Texas A&M University, 2006. http://hdl.handle.net/1969.1/ETD-TAMU-1152.
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Bochev, Pavel B. "Least squares finite element methods for the Stokes and Navier-Stokes equations." Diss., This resource online, 1994. http://scholar.lib.vt.edu/theses/available/etd-06062008-165910/.
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Bringmann, Philipp [Verfasser]. "Adaptive least-squares finite element method with optimal convergence rates / Philipp Bringmann." Berlin : Humboldt-Universität zu Berlin, 2021. http://d-nb.info/1226153186/34.
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Zhu, Lei. "A discontinuous least-squares spatial discretization for the sn equations." [College Station, Tex. : Texas A&M University, 2008. http://hdl.handle.net/1969.1/ETD-TAMU-3026.
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Münzenmaier, Steffen [Verfasser]. "Least-squares finite element methods for coupled generalized Newtonian Stokes-Darcy flow / Steffen Münzenmaier." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2012. http://d-nb.info/1029514526/34.
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Furlan, Felipe Adolvando Correia. "Métodos locais de integração explícito e implícito aplicados ao método de elementos finitos de alta ordem." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/263464.
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Orientador: Marco Lucio BittencourtDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
Made available in DSpace on 2018-08-18T15:16:56Z (GMT). No. of bitstreams: 1 Furlan_FelipeAdolvandoCorreia_M.pdf: 1842661 bytes, checksum: 69ed6fc529cf4f757f3c8a2f42e20518 (MD5) Previous issue date: 2011
Resumo: O presente trabalho apresenta algoritmos locais de integração explícitos e implícitos aplicados ao método de elementos finitos de alta ordem, baseados na decomposição por autovetores das matrizes de massa e rigidez. O procedimento de solução é realizado para cada elemento da malha e os resultados são suavizados no contorno dos elementos usando a aproximação por mínimos quadrados. Consideraram-se os métodos de diferença central e Newmark para o desenvolvimento dos procedimentos de solução elemento por elemento. No algoritmo local explícito, observou-se que as soluções convergem para as soluções globais obtidas com a matriz de massa consistente. O algoritmo local implícito necessitou de subiterações para alcançar convergência. Exemplos bi e tridimensionais de elasticidade linear e não linear são apresentados. Os resultados mostraram precisão apropriada para problemas com solução analítica. Exemplos maiores também foram apresentados com resultados satisfatórios
Abstract: This work presents explicit and implicit local integration algorithms applied to the high-order finite element method, based on the eigenvalue decomposition of the elemental mass and stiffness matrices. The solution procedure is performed for each element of the mesh and the results are smoothed on the boundary of the elements using the least square approximation. The central difference and Newmark methods were considered for developing the element by element solution procedures. For the local explicit algorithm, it was observed that the solutions converge for the global solutions obtained with the consistent mass matrix. The local implicit algorithm required subiterations to achieve convergence. Two-dimensional and three-dimensional examples of linear and non-linear elasticity are presented. Results showed appropriate accuracy for problems with analytical solution. Larger examples are also presented with satisfactory results
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Mestre em Engenharia Mecânica
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Nisters, Carina [Verfasser], and Jörg [Akademischer Betreuer] Schröder. "Least-squares finite element methods with applications in fluid and solid mechanics / Carina Nisters ; Betreuer: Jörg Schröder." Duisburg, 2018. http://d-nb.info/1172634114/34.
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Lee, Hyesuk Kwon. "Optimization Based Domain Decomposition Methods for Linear and Nonlinear Problems." Diss., Virginia Tech, 1997. http://hdl.handle.net/10919/30696.
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Optimization based domain decomposition methods for the solution of partial differential equations are considered. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations.
First, we consider a linear constraint. The existence of optimal solutions for the optimization problem is shown as is its convergence to the exact solution of the given problem. We then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined. Finite element approximations to solutions of the optimality system are defined and analyzed as is an eminently parallelizable gradient method for solving the optimality system. The linear constraint minimization problem is also recast as a linear least squares problem and is solved by a conjugate gradient method.
The domain decomposition method can be extended to nonlinear problems such as the Navier-Stokes equations. This results from the fact that the objective functional for the minimization problem involves the jump in dependent variables across the interfaces between subdomains. Thus, the method does not require that the partial differential equations themselves be derivable through an extremal problem.
An optimality system is derived by applying a Lagrange multiplier rule to a constrained optimization problem. Error estimates for finite element approximations are presented as is a gradient method to solve the optimality system. We also use a Gauss-Newton method to solve the minimization problem with the nonlinear constraint.Ph. D.
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Pratt, Brittan Sheldon. "An assessment of least squares finite element models with applications to problems in heat transfer and solid mechanics." Texas A&M University, 2008. http://hdl.handle.net/1969.1/85941.
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Research is performed to assess the viability of applying the least squares model
to one-dimensional heat transfer and Euler-Bernoulli Beam Theory problems. Least
squares models were developed for both the full and mixed forms of the governing
one-dimensional heat transfer equation along weak form Galerkin models. Both least
squares and weak form Galerkin models were developed for the first order and second
order versions of the Euler-Bernoulli beams.
Several numerical examples were presented for the heat transfer and Euler-
Bernoulli beam theory. The examples for heat transfer included: a differential equation having the same form as the governing equation, heat transfer in a fin, heat
transfer in a bar and axisymmetric heat transfer in a long cylinder. These problems
were solved using both least squares models, and the full form weak form Galerkin
model. With all four examples the weak form Galerkin model and the full form least
squares model produced accurate results for the primary variables. To obtain accurate results with the mixed form least squares model it is necessary to use at least
a quadratic polynominal. The least squares models with the appropriate approximation functions yielde more accurate results for the secondary variables than the weak
form Galerkin.
The examples presented for the beam problem include: a cantilever beam with
linearly varying distributed load along the beam and a point load at the end, a simply
supported beam with a point load in the middle, and a beam fixed on both ends with a distributed load varying cubically. The first two examples were solved using the
least squares model based on the second order equation and a weak form Galerkin
model based on the full form of the equation. The third problem was solved with
the least squares model based on the second order equation. Both the least squares
model and the Galerkin model calculated accurate results for the primary variables,
while the least squares model was more accurate on the secondary variables.
In general, the least-squares finite element models yield more acurate results for
gradients of the solution than the traditional weak form Galkerkin finite element models. Extension of the present assessment to multi-dimensional problems and nonlinear
provelms is awaiting attention.
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Pontaza, Juan Pablo. "Least-squares variational principles and the finite element method: theory, formulations, and models for solid and fluid mechanics." Diss., Texas A&M University, 2003. http://hdl.handle.net/1969.1/288.
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We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.
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Hellwig, Friederike. "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/20034.
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Die vorliegende Arbeit "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods" beweist optimale Konvergenzraten für vier diskontinuierliche Petrov-Galerkin (dPG) Finite-Elemente-Methoden für das Poisson-Modell-Problem für genügend feine Anfangstriangulierung. Sie zeigt dazu die Äquivalenz dieser vier Methoden zu zwei anderen Klassen von Methoden, den reduzierten gemischten Methoden und den verallgemeinerten Least-Squares-Methoden. Die erste Klasse benutzt ein gemischtes System aus konformen Courant- und nichtkonformen Crouzeix-Raviart-Finite-Elemente-Funktionen. Die zweite Klasse verallgemeinert die Standard-Least-Squares-Methoden durch eine Mittelpunktsquadratur und Gewichtsfunktionen.
Diese Arbeit verallgemeinert ein Resultat aus [Carstensen, Bringmann, Hellwig, Wriggers 2018], indem die vier dPG-Methoden simultan als Spezialfälle dieser zwei Klassen charakterisiert werden. Sie entwickelt alternative Fehlerschätzer für beide Methoden und beweist deren Zuverlässigkeit und Effizienz.
Ein Hauptresultat der Arbeit ist der Beweis optimaler Konvergenzraten der adaptiven Methoden durch Beweis der Axiome aus [Carstensen, Feischl, Page, Praetorius 2014]. Daraus folgen dann insbesondere die optimalen Konvergenzraten der vier dPG-Methoden.
Numerische Experimente bestätigen diese optimalen Konvergenzraten für beide Klassen von Methoden. Außerdem ergänzen sie die Theorie durch ausführliche Vergleiche beider Methoden untereinander und mit den äquivalenten dPG-Methoden.The thesis "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods" proves optimal convergence rates for four lowest-order discontinuous Petrov-Galerkin methods for the Poisson model problem for a sufficiently small initial mesh-size in two different ways by equivalences to two other non-standard classes of finite element methods, the reduced mixed and the weighted Least-Squares method. The first is a mixed system of equations with first-order conforming Courant and nonconforming Crouzeix-Raviart functions. The second is a generalized Least-Squares formulation with a midpoint quadrature rule and weight functions. The thesis generalizes a result on the primal discontinuous Petrov-Galerkin method from [Carstensen, Bringmann, Hellwig, Wriggers 2018] and characterizes all four discontinuous Petrov-Galerkin methods simultaneously as particular instances of these methods. It establishes alternative reliable and efficient error estimators for both methods. A main accomplishment of this thesis is the proof of optimal convergence rates of the adaptive schemes in the axiomatic framework [Carstensen, Feischl, Page, Praetorius 2014]. The optimal convergence rates of the four discontinuous Petrov-Galerkin methods then follow as special cases from this rate-optimality. Numerical experiments verify the optimal convergence rates of both types of methods for different choices of parameters. Moreover, they complement the theory by a thorough comparison of both methods among each other and with their equivalent discontinuous Petrov-Galerkin schemes.
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Storn, Johannes [Verfasser], Carsten [Gutachter] Carstensen, Gerhard [Gutachter] Starke, and Dietmar [Gutachter] Gallistl. "Topics in Least-Squares and Discontinuous Petrov-Galerkin Finite Element Analysis / Johannes Storn ; Gutachter: Carsten Carstensen, Gerhard Starke, Dietmar Gallistl." Berlin : Humboldt-Universität zu Berlin, 2019. http://d-nb.info/1192302907/34.
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Camp, Brian David. "A Class of Immersed Finite Element Spaces and Their Application to Forward and Inverse Interface Problems." Diss., Virginia Tech, 2003. http://hdl.handle.net/10919/29923.
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A class of immersed finite element (IFE) spaces is developed for solving elliptic boundary value problems that have interfaces. IFE spaces are finite element approximation spaces which are based upon meshes that can be independent of interfaces in the domain. Three different quadratic IFE spaces and their related biquadratic IFE spaces are introduced here for the purposes of solving both forward and inverse elliptic interface problems in 1D and 2D. These different spaces are constructed by (i) using a hierarchical approach, (ii) imposing extra continuity requirements or (iii) using a local refinement technique. The interpolation properties of each space are tested against appropriate testing functions in 1D and 2D. The IFE spaces are also used to approximate the solution of a forward elliptic interface problem using the Galerkin finite element method and the mixed least squares finite element method. Finally, one appropriate space is selected to solve an inverse interface problem using either an output least squares approach or the least squares with mixed equation error method.Ph. D.
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Fave, Sebastian Philipp. "Investigative Application of the Intrinsic Extended Finite Element Method for the Computational Characterization of Composite Materials." Thesis, Virginia Tech, 2014. http://hdl.handle.net/10919/50483.
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Computational micromechanics analysis of carbon nanotube-epoxy nanocomposites, containing aligned nanotubes, is performed using the mesh independent intrinsic extended finite element method (IXFEM). The IXFEM employs a localized intrinsic enrichment strategy to treat arbitrary discontinuities defined through the level-set method separate from the problem domain discretization, i.e. the finite element (FE) mesh. A global domain decomposition identifies local subdomains for building distinct partition of unities that appropriately suit the approximation. Specialized inherently enriched shape functions, constructed using the moving least square method, enhance the approximation space in the vicinity of discontinuity interfaces, maintaining accuracy of the solution, while standard FE shape functions are used elsewhere. Comparison of the IXFEM in solving validation problems with strong and weak discontinuities against a standard finite element method (FEM) and analytic solutions validates the enriched intrinsic bases, and shows anticipated trends in the error convergence rates. Applying the IXFEM to model composite materials, through a representative volume element (RVE), the filler agents are defined as individual weak bimaterial interfaces. Though a series of RVE studies, calculating the effective elastic material properties of carbon nanotube-epoxy nanocomposite systems, the benefits in substituting the conventional mesh dependent FEM with the mesh independent IXFEM when completing micromechanics analysis, investigating effects of high filler count or an evolving microstructure, are demonstrated.Master of Science
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Krueger, Justin Michael. "Parameter Estimation Methods for Ordinary Differential Equation Models with Applications to Microbiology." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/78674.
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The compositions of in-host microbial communities (microbiota) play a significant role in host health, and a better understanding of the microbiota's role in a host's transition from health to disease or vice versa could lead to novel medical treatments. One of the first steps toward this understanding is modeling interaction dynamics of the microbiota, which can be exceedingly challenging given the complexity of the dynamics and difficulties in collecting sufficient data. Methods such as principal differential analysis, dynamic flux estimation, and others have been developed to overcome these challenges for ordinary differential equation models. Despite their advantages, these methods are still vastly underutilized in mathematical biology, and one potential reason for this is their sophisticated implementation. While this work focuses on applying principal differential analysis to microbiota data, we also provide comprehensive details regarding the derivation and numerics of this method. For further validation of the method, we demonstrate the feasibility of principal differential analysis using simulation studies and then apply the method to intestinal and vaginal microbiota data. In working with these data, we capture experimentally confirmed dynamics while also revealing potential new insights into those dynamics. We also explore how we find the forward solution of the model differential equation in the context of principal differential analysis, which amounts to a least-squares finite element method. We provide alternative ideas for how to use the least-squares finite element method to find the forward solution and share the insights we gain from highlighting this piece of the larger parameter estimation problem.Ph. D.
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Kouri, Jeffrey Victor. "Improved finite element analysis of thick laminated composite plates by the predictor corrector technique and approximation of C[superscript]1 continuity with a new least squares element." Diss., Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/20762.
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Müller, Benjamin Verfasser], Gerhard [Akademischer Betreuer] Starke, Jörg [Akademischer Betreuer] [Schröder, and Christian [Akademischer Betreuer] Meyer. "Mixed least squares finite element methods based on inverse stress-strain relations in hyperelasticity / Benjamin Müller. Gutachter: Jörg Schröder ; Christian Meyer. Betreuer: Gerhard Starke." Duisburg, 2015. http://d-nb.info/1071543490/34.
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Müller, Benjamin [Verfasser], Gerhard Akademischer Betreuer] Starke, Jörg [Akademischer Betreuer] [Schröder, and Christian [Akademischer Betreuer] Meyer. "Mixed least squares finite element methods based on inverse stress-strain relations in hyperelasticity / Benjamin Müller. Gutachter: Jörg Schröder ; Christian Meyer. Betreuer: Gerhard Starke." Duisburg, 2015. http://nbn-resolving.de/urn:nbn:de:hbz:464-20150522-081101-4.
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Ferreira, Sabrina dos Santos 1984. "Estudo do método dos elementos finitos de mínimos quadrados - LSFEM para resolução da equação de convecção - difusão bidimensional." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/265744.
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Orientador: Luiz Felipe Mendes de MouraDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
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Resumo: O objetivo deste trabalho foi o estudo da distribuição de temperatura em um domínio retangular, para tal foi resolvida a Equação de Convecção - Difusão Bidimensional via Método dos Elementos Finitos de Mínimos Quadrados - Least Squares Finite Element Method (LSFEM). Para discretização espacial foi utilizado elementos bidimensionais quadráticos, nesse caso os elementos quadriláteros com oito nós foram escolhidos. A discretização temporal nos casos transientes foi aproximada via Método de Crank - Nicolson. Para a o obtenção da matriz do elemento e o vetor do lado direito a quadratura de Gauss - Legendre foi empregada. A solução do sistema algébrico resultante foi obtida a partir do Método dos Gradientes Conjugados, um dos métodos iterativos mais eficientes na resolução de sistemas lineares quando a matriz é simétrica, esparsa e definida positiva, sendo essas características resultantes da formulação via LSFEM. É apresentada a formulação matemática do problema, a metodologia empregada na solução. Para a obtenção dos resultados um código em linguagem C foi implementado, por fim são apresentados os resultados obtidos, as conclusões e sugestões para trabalhos futuros
Abstract: The objective of this work was the study of temperature distribution in a rectangular domain, for that was solved Equation Convection - Diffusion Two-Dimensional via Least Squares Finite Element Method - LSFEM. For spatial discretization was used two-dimensional quadratic elements, in this case the quadrilateral elements with eight nodes were chosen. The time discretization in transient cases was approached via Crank - Nicolson Method. To obtain the element matrix and vector right side of the Gauss - Legendre quadrature was employed. The solution of resulting algebraic system was obtained using Conjugate Gradient Method, one of the most efficient iterative methods for solving linear sistenas when the matrix is ??symmetric, and positive definite sparse, and these resulting characteristics of the formulation via LSFEM. The mathematical formulation of the problem, the methodology used in the solution is shown. To obtain the results a code in C language was implemented, finally is presented results, conclusions and suggestions for future work
Mestrado
Termica e Fluidos
Mestra em Engenharia Mecânica
28
Johnson, Mikael. "Acoustic Emission in Composite Laminates - Numerical Simulations and Experimental Characterization." Doctoral thesis, KTH, Solid Mechanics, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3452.
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Reis, Luiz Antonio 1975. "Acoplamento MEC-MEF para análise de pórtico linear sobre base elástica." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/258076.
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Orientador: Leandro Palermo JuniorDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Civil, Arquitetura e Urbanismo
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Resumo: O presente trabalho está divido em quatro partes. Na primeira parte, utilizando o método dos elementos de contorno (MEC), se fez a análise de problemas bidimensionais com aproximação linear. Foi considerada a possibilidade de se aplicar a técnica de sub-regiões para se levar em conta a diversidade de materiais, bem como a suavização do contorno por mínimos quadrados para evitar a possíveis perturbações. Foi considerado a possibilidade de colocação de uma linha de carga no domínio. Na segunda parte, utilizando o método dos elementos finitos (MEF), se fez a análise linear de pórticos planos. Para este estudo foram utilizadas barras com dois nós e esses com três graus de liberdade. Na terceira parte, a análise elástica linear de meios contínuos (Estado Plano de Tensão Generalizado) enrijecidos com elementos lineares (barras) é estudada fazendo-se um acoplamento entre elementos modelados com o MEC e com o MEF. As fibras são modeladas pelo MEF com elementos lineares de três graus de liberdade por nó e quatro nós por barra. Os elementos planos são modelados pelo MEC com elementos isoparamétricos lineares no perímetro. É permitido o uso de sub-regiões com objetivo de generalizar o tratamento do meio elástico. Na quarta parte, utilizando o acoplamento MEF/MEF, se fez a análise linear de pórticos planos sobre base elástica. O acoplamento se dá entre as barras do pórtico e as barras introduzidas como enriquecedor no problema elástico bidimensional. Tendo em conta estes aspectos da formulação desenvolvida, alguns exemplos são apresentados para avaliação de seu desempenho nos problemas de engenharia
Abstract: This paper is divided into four parts . In the first part , using the boundary element method (BEM) , we did the analysis of two-dimensional problems with linear approximation . We considered the possibility of applying the technique of sub - regions to take into account different materials, as well as smoothing the contour by least squares to avoid possible disturbances . We considered the possibility of placing a load line in the field. In the second part, the linear analysis for plane frames was carried out with the finite element method (FEM). Bars with two nodes and three degrees of freedom were used in this study . In the third part, the linear elastic analysis of continuous media (Generalized Plane Stress problems) stiffened with one-dimensional elements (bars) is studied through between elements of the BEM and the FEM. The fibers are modeled by FEM with three degrees of freedom linear elements and using four-nodes. The plane domain is modeled with the BEM and using isoparametric elements. The use of sub - regions in order to generalize the treatment of the elastic medium is allowed. In the fourth part , using the FEM / FEM coupling , a linear analysis of plane frames on elastic foundation is carried out. Some examples are presented to evaluate the formulation behavior engineering problems
Mestrado
Estruturas
Mestre em Engenharia Civil
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Romão, Estaner Claro 1979. "Estudo numérico da aplicação do método dos elementos finitos de Galerkin e dos mínimos quadrados na solução da equação da convecção-difusão-reação tridimensional." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/263390.
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Orientador: Luiz Felipe Mendes de MouraTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
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Resumo: Este trabalho trata da aplicação do Método dos Elementos Finitos nas variantes Galerkin e Mínimos Quadrados com equações auxiliares para a solução numérica da equação diferencial parcial que modela a convecção-difusão-reação definida sobre um domínio tridimensional em regime permanente. Na discretização espacial foram utilizados elementos hexaedrais com oito (elemento linear) e vinte e sete (elemento quadrático) nós, no qual foram adotadas funções de interpolação de Lagrange nas coordenadas locais. Transformando toda a formulação do problema das coordenadas globais para as coordenadas locais, o Método da Quadratura de Gauss-Legendre foi utilizado para integração numérica dos coeficientes das matrizes dos elementos. Adicionalmente, à formulação pelos dois métodos, um código computacional foi implementado para simular o fenômeno proposto. Dispondo de soluções analíticas, várias análises de erro numérico foram realizadas a partir das normas L2 (erro médio no domínio) e L? (maior erro cometido no domínio), validando assim os resultados numéricos. Um caso real é proposto e analisado
Abstract: This paper the application of the Finite Element Method in variants Galerkin and Least Squares with auxiliary equations for the numerical solution of partial differential equation that models the convection-diffusion-reaction defined over a three-dimensional domain in steady state. In the spatial discretization were used hexahedrons elements with eight (linear element) and twenty-seven (quadratic element) nodes, which were adopted Lagrange interpolation functions in local coordinates. Transforming the problem of global coordinates to local coordinates, the method of Gauss-Legendre quadrature was used for numerical integration of the coefficients of the matrices of the elements. Additionally, the formulation by the two methods, a computer code was implemented to simulate the phenomenon proposed. Offering analytical solutions, several numerical error analysis were performed from L2 norms (average error in the domain) and L? (higher error in the domain), thus validating the numerical results. A real case is proposed and analyzed
Doutorado
Termica e Fluidos
Doutor em Engenharia Mecânica
31
Pereira, Vanessa Davanço [UNESP]. "Simulação numérica de escoamentos de fluidos pelo método de elementos finitos de mínimos quadrados." Universidade Estadual Paulista (UNESP), 2005. http://hdl.handle.net/11449/90836.
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Neste trabalho foram feitas simulações de escoamentos incompressiveis por um método de elementos finitos de mínimos quadrados (LSFEM – Least Squares Finite Element Method), usando as formulações velocidade-pressão-vorticidade e velocidade-pressão-tensão, denominadas na literatura de formulações u − p −ω e u = p −τ respectivamente. Estas formulações são preferidas por resultarem em sistemas de equações diferenciais de primeira ordem, o que é mais conveniente para implementação pelo LSFEM. O objetivo principal deste trabalho é a simulação computacional de escoamentos laminares, transicionais e turbulentos através da aplicação da metodologia de simulação de grandes escalas (LES – Large Eddy Simulation) com o modelo de viscosidade turbulenta de Smagorinky para modelar as tensões submalha. Alguns problemas padrões foram resolvidos para validar um código computacional desenvolvido e os resultados são apresentados e comparados com resultados disponíveis na literatura.
In this work simulations of incompressible fluid flows have been done by a Least Squares Finite Element Method (LSFEM) using the velocity-pressure-vorticity and velocity-pressurestress formulations, named, in the literature, u − p −ω and u = p −τ formulations respectively. These formulations are preferred because the resulting equations are partial differential equations of first order, which is more convenient for implementation by LSFEM. The main purpose of this work are the numerical computations of laminar, transitional and turbulent fluid flows through the application of large eddy simulation (LES) methodology using the LSFEM. The Navier- Stokes equations in u − p −ω and u = p −τ formulations are filtered and the eddy viscosity model of Smagorinsky is used for modeling the sub-grid-scale stresses. Some benchmark problems are solved for validate a developed numerical code and the preliminary results are presented and compared with available results from the literature.
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Szypowski, Ryan. "Least-squares finite elements and constrained evolution systems." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2008. http://wwwlib.umi.com/cr/ucsd/fullcit?p3320112.
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Thesis (Ph. D.)--University of California, San Diego, 2008.Title from first page of PDF file (viewed September 12, 2008). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 73-74).
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Pereira, Vanessa Davanço. "Simulação numérica de escoamentos de fluidos pelo método de elementos finitos de mínimos quadrados /." Ilha Solteira : [s.n.], 2005. http://hdl.handle.net/11449/90836.
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Orientador: João Batista Campos SilvaBanca: João Batista Aparecido
Banca: Luiz Felipe Mendes de Moura
Resumo: Neste trabalho foram feitas simulações de escoamentos incompressiveis por um método de elementos finitos de mínimos quadrados (LSFEM - Least Squares Finite Element Method), usando as formulações velocidade-pressão-vorticidade e velocidade-pressão-tensão, denominadas na literatura de formulações u − p −ω e u = p −τ respectivamente. Estas formulações são preferidas por resultarem em sistemas de equações diferenciais de primeira ordem, o que é mais conveniente para implementação pelo LSFEM. O objetivo principal deste trabalho é a simulação computacional de escoamentos laminares, transicionais e turbulentos através da aplicação da metodologia de simulação de grandes escalas (LES - Large Eddy Simulation) com o modelo de viscosidade turbulenta de Smagorinky para modelar as tensões submalha. Alguns problemas padrões foram resolvidos para validar um código computacional desenvolvido e os resultados são apresentados e comparados com resultados disponíveis na literatura.
Abstract: In this work simulations of incompressible fluid flows have been done by a Least Squares Finite Element Method (LSFEM) using the velocity-pressure-vorticity and velocity-pressurestress formulations, named, in the literature, u − p −ω and u = p −τ formulations respectively. These formulations are preferred because the resulting equations are partial differential equations of first order, which is more convenient for implementation by LSFEM. The main purpose of this work are the numerical computations of laminar, transitional and turbulent fluid flows through the application of large eddy simulation (LES) methodology using the LSFEM. The Navier- Stokes equations in u − p −ω and u = p −τ formulations are filtered and the eddy viscosity model of Smagorinsky is used for modeling the sub-grid-scale stresses. Some benchmark problems are solved for validate a developed numerical code and the preliminary results are presented and compared with available results from the literature.
Mestre
34
Steeger, Karl Verfasser], and Jörg [Akademischer Betreuer] [Schröder. "Least-squares mixed finite elements for geometrically nonlinear solid mechanics / Karl Steeger ; Betreuer: Jörg Schröder." Duisburg, 2017. http://d-nb.info/1136864016/34.
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Steeger, Karl [Verfasser], and Jörg [Akademischer Betreuer] Schröder. "Least-squares mixed finite elements for geometrically nonlinear solid mechanics / Karl Steeger ; Betreuer: Jörg Schröder." Duisburg, 2017. http://d-nb.info/1136864016/34.
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Pechmann, Patrick R. "Penalized Least Squares Methoden mit stückweise polynomialen Funktionen zur Lösung von partiellen Differentialgleichungen." kostenfrei, 2008. http://www.opus-bayern.de/uni-wuerzburg/volltexte/2008/2813/.
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Machado, Fernando Machado. "Aproximação de Galerkin mínimos-quadrados de escoamentos axissimétricos de fluido Herschel-Bulkley através de expansões abruptas." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2007. http://hdl.handle.net/10183/11964.
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O estudo de escoamentos de fluidos não-Newtonianos através de expansões desperta um grande interesse em pesquisadores nas diversas áreas da engenharia, devido a sua ampla aplicação em indústrias e no meio acadêmico. O objetivo principal desta Dissertação é simular problemas de escoamentos envolvendo fluidos viscoplásticos através de expansões axissimétricas abruptas. O modelo mecânico empregado é baseado nas equações de conservação de massa e de momentum para escoamentos isocóricos acoplados com a equação constitutiva de um Fluido Newtoniano Generalizada (GNL), com a função de viscosidade de Herschel-Bulkley regularizada pela equação de Papanastasiou. O modelo mecânico é aproximado por um modelo estabilizado de elementos finitos, denominado método Galerkin Mínimos-Quadrados, ou Galerkin Least-squares (GLS). Esse método (GLS) é usado a fim superar as dificuldades numéricas do modelo de Galerkin clássico: a condição de Babuška-Brezzi e a instabilidade inerente em regiões advectivas do escoamento. O método é construído adicionando termos de malha-dependentes a fim aumentar a estabilidade da formulação de Galerkin clássica sem danificar sua consistência. A formulação GLS é aplicada para estudar a influência do índice power-law, da tensão limite de cisalhamento e razão de aspecto na dinâmica do escoamento de fluidos de Herschel-Bulkley através de expansões axissimétricas abruptas de razão de aspecto 1:2 e 1:4. Os problemas que envolvem números de Reynolds desprezíveis, para uma escala do número de Herschel-Bulkley entre 0 e 100 e índice de comportamento entre 0,2 e 1,0 são apresentados. Os resultados são fisicamente detalhados e estão de acordo com a literatura.The study of non-Newtonian fluid flows in expansions is of great interest for researchers in the several branches of engineering, due to their wide application both in industry and academy. The objective of this Dissertation is to simulate flow problems involving a viscoplastic fluid through an axisymmetric abrupt expansion. The mechanical model employed is based on the mass and momentum conservative equations for isochoric flows coupled with the Generalized Newtonian Liquid (GNL) constitutive equation, with the Papanastasiou-regularized Herschel-Bulkley viscosity function. The mechanical model is approximated by a stabilized finite element scheme, namely the Galerkin Least-squares method. This method (GLS) is used in order to overcome the numerical difficulties of the classical Galerkin method: the Babuška- Brezzi condition and the inherent instability in advective flow regions. The method is built adding mesh-dependent terms in order to increase the stability of the classical Galerkin formulation without damaging its consistency. The GLS formulation is applied to study the influence of power-law index, yield stress and aspect reason in the flow dynamics of Herschel- Bulkley fluids through an axisymmetric abrupt expansions of aspect reason 1:2 and 1:4. Problems involving negligible Reynolds numbers, for a Herschel-Bulkley number range between 0 and 100 and e power-law index range between 0.2 and 1.0 are presented. The results are physically comprehensive and are in accordance with the literature.
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Gdhami, Asma. "Méthodes isogéométriques pour les équations aux dérivées partielles hyperboliques." Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4210/document.
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L’Analyse isogéométrique (AIG) est une méthode innovante de résolution numérique des équations différentielles, proposée à l’origine par Thomas Hughes, Austin Cottrell et Yuri Bazilevs en 2005. Cette technique de discrétisation est une généralisation de l’analyse par éléments finis classiques (AEF), conçue pour intégrer la conception assistée par ordinateur (CAO), afin de combler l’écart entre la description géométrique et l’analyse des problèmes d’ingénierie. Ceci est réalisé en utilisant des B-splines ou des B-splines rationnelles non uniformes (NURBS), pour la description des géométries ainsi que pour la représentation de champs de solutions inconnus.L’objet de cette thèse est d’étudier la méthode isogéométrique dans le contexte des problèmes hyperboliques en utilisant les fonctions B-splines comme fonctions de base. Nous proposons également une méthode combinant l’AIG avec la méthode de Galerkin discontinue (GD) pour résoudre les problèmes hyperboliques. Plus précisément, la méthodologie de GD est adoptée à travers les interfaces de patches, tandis que l’AIG traditionnelle est utilisée dans chaque patch. Notre méthode tire parti de la méthode de l’AIG et la méthode de GD.Les résultats numériques sont présentés jusqu’à l’ordre polynomial p= 4 à la fois pour une méthode deGalerkin continue et discontinue. Ces résultats numériques sont comparés pour un ensemble de problèmes de complexité croissante en 1D et 2DIsogeometric Analysis (IGA) is a modern strategy for numerical solution of partial differential equations, originally proposed by Thomas Hughes, Austin Cottrell and Yuri Bazilevs in 2005. This discretization technique is a generalization of classical finite element analysis (FEA), designed to integrate Computer Aided Design (CAD) and FEA, to close the gap between the geometrical description and the analysis of engineering problems. This is achieved by using B-splines or non-uniform rational B-splines (NURBS), for the description of geometries as well as for the representation of unknown solution fields.The purpose of this thesis is to study isogeometric methods in the context of hyperbolic problems usingB-splines as basis functions. We also propose a method that combines IGA with the discontinuous Galerkin(DG)method for solving hyperbolic problems. More precisely, DG methodology is adopted across the patchinterfaces, while the traditional IGA is employed within each patch. The proposed method takes advantageof both IGA and the DG method.Numerical results are presented up to polynomial order p= 4 both for a continuous and discontinuousGalerkin method. These numerical results are compared for a range of problems of increasing complexity,in 1D and 2D
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Bertrand, Fleurianne [Verfasser]. "Approximated flux boundary conditions for Raviart-Thomas finite elements on domains with curved boundaries and applications to first-order system least squares / Fleurianne Bertrand." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2014. http://d-nb.info/1063982103/34.
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Pereira, Vanessa Davanço. "Métodos de elementos finitos estabilizados em problemas de convecção-difusão." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/263418.
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Orientadores: Luiz Felipe Mendes de Moura, João Batista Campos SilvaTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
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Resumo: Neste trabalho foram desenvolvidos e aplicados códigos computacionais baseados em métodos de elementos finitos estabilizados, principalmente, a versão de mínimos quadrados (LSFEM - Least Squares Finite Element Method) que leva a sistemas algébricos sempre simétricos e positivos definidos, independentemente dos sistemas de equações diferenciais parciais dos casos considerados. Um método conhecido como CBS (Characteristic Based Split) também tem sido aplicado o qual possibilita a aplicação do método de Galerkin (GFEM - Galerkin Finite Element Method) para solução de problemas de escoamento de fluidos, sem oscilações na solução. Resultados têm sido obtidos para problemas de convecção-difusão bi e tridimensional discretizando os domínios por malhas estruturadas de elementos quadráticos, e para solução de escoamentos incompressíveis bidimensionais usando malhas não estruturadas de elementos finitos triangulares lineares
Abstract: In this study we developed and implemented computer codes based on stabilized finite element methods, especially the version of least squares (LSFEM - Least Squares Finite Element Method) that leads to algebraic systems always symmetric and positive defined, independently of the systems of partial differential equations of the cases considered. A method known as CBS (Characteristic Based Split) has also been implemented which allows the application of the Galerkin method (GFEM - Galerkin Finite Element Method) to solve problems of fluid flow without oscilations in the solution. Results have been obtained for two-and three-dimensional problems of convection-diffusion type discretizing the domains by structured meshes of quadratic elements and for solution of two-dimensional incompressible flows using unstructured meshes of linear triangles
Doutorado
Termica e Fluidos
Doutor em Engenharia Mecânica
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Nguyen, Minh Chien. "Modélisation et simulation multiphysique du bain de fusion en soudage à l'arc TIG." Thesis, Aix-Marseille, 2015. http://www.theses.fr/2015AIXM4749/document.
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Au cours de ce travail, un modèle physique et numérique 3D du procédé de soudage à l’arc TIG (Tungsten Inert Gas) a été développé dans l’objectif de prédire, en fonction des paramètres opératoires, les grandeurs utiles au concepteur d’assemblages soudés.Le modèle développé, à l’aide du code de calcul aux éléments finis Cast3M, traite les phénomènes physiques agissant dans la pièce et, plus particulièrement, dans le bain de soudage, l’arc étant traité comme une source. Pour ce faire, les équations non-linéaires de la thermohydraulique couplées à celles de l’électromagnétisme sont résolues en régime stationnaire avec un modèle prenant en compte la surface libre déformable du bain de soudage.Une première étape du développement a porté sur la modélisation des phénomènes électromagnétiques par deux méthodes numériques différentes, à comparer les résultats numériques obtenus avec ceux de la littérature. Ensuite, afin de valider le pouvoir prédictif du modèle, des simulations de différentes configurations de soudage d’intérêt ont été étudiées, en variant la composition chimique du matériau, la vitesse de défilement, la pression d’arc imposée et, plus particulièrement, la position de soudage. Des comparaisons avec des expériences et des modèles numériques de la littérature confirment les bonnes tendances obtenues. Enfin, une approche de la modélisation de l’apport de matière a été abordée et des résultats de cette approche ont été montrés. Notre modèle complet constitue donc une base solide pour le développement de modèles de simulation numérique du soudage (SNS) 3D totalement couplés avec l’arc dans le futur et sera intégré dans le logiciel métier WPROCESSIn this work, we develop a 3D physical and numerical model of the GTA (gas tungsten arc) welding process in order to predict, for given welding parameters, useful quantities for the designer of welded assembly.The model is developed in the Cast3M finite element software and takes into account the main physical phenomena acting in the workpiece and particularly in the weld pool, subject to source terms modeling the arc part of the welding process. A steady solution of this model is thought for and involves the coupling of the nonlinear thermohydaulics and electromagnetic equations together with the displacement of the deformable free surface of the weld pool.A first step in the development consisted in modeling the electromagnetic phenomena with two different numerical methods, in comparing the numerical results obtained with those of the literature. Then, in order to assess the predictive capability of the model, simulations of various welding configurations are performed : variation in the chemical composition of the material, of the welding speed, of the prescribed arc pressure and of the welding positions, which is a focus of this work, are studied. A good agreement is obtained between the results of our model and other experimental and numerical results of the literature. Eventually, a model accounting for metal filling is proposed and its results are discussed. Thus, our complete model can be seen as a solid foundation towards future totally-coupled 3D welding models including the arc and it will be included in the WPROCESS software dedicated to the numerical simulation of welding
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Oqielat, Moa'ath Nasser. "Modelling water droplet movement on a leaf surface." Queensland University of Technology, 2009. http://eprints.qut.edu.au/30232/.
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The central aim for the research undertaken in this PhD thesis is the development of a model for simulating water droplet movement on a leaf surface and to compare the model behavior with experimental observations. A series of five papers has been presented to explain systematically the way in which this droplet modelling work has been realised. Knowing the path of the droplet on the leaf surface is important for understanding how a droplet of water, pesticide, or nutrient will be absorbed through the leaf surface. An important aspect of the research is the generation of a leaf surface representation that acts as the foundation of the droplet model. Initially a laser scanner is used to capture the surface characteristics for two types of leaves in the form of a large scattered data set. After the identification of the leaf surface boundary, a set of internal points is chosen over which a triangulation of the surface is constructed. We present a novel hybrid approach for leaf surface fitting on this triangulation that combines Clough-Tocher (CT) and radial basis function (RBF) methods to achieve a surface with a continuously turning normal. The accuracy of the hybrid technique is assessed using numerical experimentation. The hybrid CT-RBF method is shown to give good representations of Frangipani and Anthurium leaves. Such leaf models facilitate an understanding of plant development and permit the modelling of the interaction of plants with their environment. The motion of a droplet traversing this virtual leaf surface is affected by various forces including gravity, friction and resistance between the surface and the droplet. The innovation of our model is the use of thin-film theory in the context of droplet movement to determine the thickness of the droplet as it moves on the surface. Experimental verification shows that the droplet model captures reality quite well and produces realistic droplet motion on the leaf surface. Most importantly, we observed that the simulated droplet motion follows the contours of the surface and spreads as a thin film. In the future, the model may be applied to determine the path of a droplet of pesticide along a leaf surface before it falls from or comes to a standstill on the surface. It will also be used to study the paths of many droplets of water or pesticide moving and colliding on the surface.
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Müller, Hannes. "Ein Konzept zur numerischen Berechnung inkompressibler Strömungen auf Grundlage einer diskontinuierlichen Galerkin-Methode in Verbindung mit nichtüberlappender Gebietszerlegung." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 1999. http://nbn-resolving.de/urn:nbn:de:swb:14-992350020281-96843.
Full textAbstract:
A new combination of techniques for the numerical computation of incompressible flow is presented. The temporal discretization bases on the discontinuous Galerkin-formulation. Both constant (DG(0)) and linear approximation (DG(1)) in time is discussed. In case of DG(1) an iterative method reduces the problem to a sequence of problems each with the dimension of the DG(0) approach. For the semi-discrete problems a Galerkin/least-squares method is applied. Furthermore a non-overlapping domain decomposition method can be used for a parallelized computation. The main advantage of this approach is the low amount of information which must be exchanged between the subdomains. Due to the slight bandwidth a workstation-cluster is a suitable platform. Otherwise this method is efficient only for a small number of subdomains. The interface condition is of the Robin/Robin-type and for the Navier-Stokes equation a formulation introducing a further pressure interface condition is used. Additionally a suggestion for the implementation of the standard k-epsilon turbulence model with special wall function is done in this context. All the features mentioned above are implemented in a code called ParallelNS. Using this code the verification of this approach was done on a large number of examples ranging from simple advection-diffusion problems to turbulent convection in a closed cavity.
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Taghaddosi, Farzad. "An adaptive least-squares finite element method for the compressible Euler equations." Thesis, 1996. http://spectrum.library.concordia.ca/4799/1/MM18448.pdf.
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Vallala, Venkat. "Alternative Least-Squares Finite Element Models of Navier-Stokes Equations for Power-Law Fluids." 2009. http://hdl.handle.net/1969.1/ETD-TAMU-2009-05-575.
Full textAbstract:
The Navier-Stokes equations can be expressed in terms of the primary variables
(e.g., velocities and pressure), secondary variables (velocity gradients, vorticity, stream
function, stresses, etc.), or a combination of the two. The Least-Squares formulations of
the original partial differential equations (PDE's) in terms of primary variables require
C1 continuity of the finite element spaces across inter-element boundaries. This higherorder
continuity requirement for PDE's in primary variables is a setback to Least-Squares
formulation when compared to the weak form Galerkin formulation. To overcome this
requirement, the PDE or PDE's are first transformed into an equivalent lower order
system by introducing additional independent variables, sometimes termed auxiliary
variables, and then formulating the Least-Squares model based on the equivalent lower
order system. These additional variables can be selected to represent physically
meaningful variables, e.g., fluxes, stresses or rotations, and can be directly approximated
in the model. Using these auxiliary variables, different alternative Least-Squares finite
element models are developed and investigated.
In this research, the vorticity and stress based alternative Least-Squares finite
element formulations of Navier-Stokes equations are developed and are verified with the benchmark problems. The Least-Squares formulations are developed for both the
Newtonian and non-Newtonian fluids (based on the Power-Law model) and the effects of
linearization before and after minimization are investigated using the benchmark
problems. For the non-Newtonian fluids both the shear thinning and shear thickening
fluids have been studied by varying the Power-Law index from 0.25 to 1.5. Also, the
traditional weak form based penalty method is formulated for the non-Newtonian case
and the results are compared with the Least-Squares formulation.
The results matched with the benchmark problems for Newtonian and non-
Newtonian fluids, irrespective of the formulation. There was no effect of linerization in
the case of Newtonian fluids. However for non-Newtonian fluids, there was some
tangible effect of linearization on the accuracy of the solution. The effect was more
pronounced for lower power-law indices compared to higher power-law indices. And
there seemed to have some kind of locking that caused the matrices to be ill-conditioned
especially for lower values of power-law indices.
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Lai, Qing-Xian, and 賴慶賢. "Simulation of Two-Dimension Shallow Water Equation By The Least-Squares Finite Element Method." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/52353771629982055017.
Full textAbstract:
碩士中原大學
土木工程研究所
90
ABSTRACT This study is focused on the numerical simulation of discontinuous free surface problems such as the supercritical shock waves flowing on various geometries and the flow field of a broken dam. The Least-square finite element method (LSFEM) is adopted for the simulation of two-dimensional nonlinear shallow water equation. The simulation of supercritical shock waves in the following five different geometries: one-side oblique contraction channel, one-side oblique expansion channel, a oblique contraction-expansion channel, curved contraction expansion and curved contraction channel are evaluated by the theoretical solution, and the simulated results of previous published data. The dam break problems are also simulated with the reservoir water depth ratio of 10/5 in one dimension dam break and 10/9, 2, and 1 in partial dam break. The simulated results are shown to be in good agreement with analytical solution and numerical results of other methods. This paper demonstrates that the LSFEM can effectively simulate the supercritical shock waves and dam break flow with discontinuous free surface. Key words：least-square finite element method、shallow water equation 、supercritical flow、partial dam break
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Danisch, Garvin [Verfasser]. "Gemischte Finite-element-least-squares-Methoden für die Flachwassergleichungen mit kleiner Viskosität / von Garvin Danisch." 2007. http://d-nb.info/983833052/34.
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Kumar, Rajeev. "A least-squares/Galerkin split finite element method for incompressible and compressible Navier-Stokes equations." 2008. http://hdl.handle.net/10106/1121.
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Wang, Yun-Tsz, and 王韻詞. "On Two Iterative Least-Squares Finite Element Schemes for Solving the Incompressible Navier-Stokes Equations." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/z3jr42.
Full textAbstract:
碩士國立中央大學
數學研究所
96
This thesis is devoted to a numerical study of two iterative least-squares finite element schemes on uniform meshes for solving the stationary incompressible Navier-Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, the Navier-Stokes problem can be recast as a first-order quasilinear velocity-vorticity-pressure system. Two Picard-type iterative least-squares finite element schemes are proposed for approximating the solution to the nonlinear first-order problem. In each iteration, we apply the usual L2 least-squares scheme or a weighted L2 least-squares scheme to solve the corresponding Oseen problem. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that, for the same test problem with smooth exact solution, the L2 least-squares solutions are more accurate than the weighted L2 least-squares solutions for low Reynolds number flows, while for flows with relatively higher Reynolds numbers the weighted L2 least-squares approximations seem to be better than the L2 least-squares approximations. Finally, numerical results for driven cavity flows are also given to demonstrate the effectiveness of the iterative least-squares finite element approach.
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Kao, Shih-Chao, and 高仕超. "Some Residual-Free Bubble Enrichment Least-Squares Finite Element Method for the Convection-Diffusion Equation." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/z55z24.
Full textAbstract:
碩士國立中央大學
數學研究所
96
In this thesis, we formulate the least-squares finite element method using piececewise linears to solve the convection-diffusion equation which is convection-dominated and we find that the solution is diffusive and the classical mesh refinement for the least-squares finite element method is not an economical method. Then we use the residual-free bubble method to enrich the least-squares finite element method. This is a new application of residual-free bubble method and we solve some test problems. The numerical results show that the residual-feee bubble method for the least-squares finite element method has a good effect of enrichment。