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1
Warren, Micah. "Special Lagrangian equations /." Thesis, Connect to this title online; UW restricted, 2008. http://hdl.handle.net/1773/5749.
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Haskins, Mark. "Constructing special Lagrangian cones /." Digital version accessible at:, 2000. http://wwwlib.umi.com/cr/utexas/main.
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Ware, Antony Frank. "A spectral Lagrange-Galerkin method for convection-dominated diffusion equations." Thesis, University of Oxford, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.302896.
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CAMALET, EUGENE. "Methodes de couplage euler-lagrange pour les equations d'euler-poisson." Paris 6, 1995. http://www.theses.fr/1995PA066276.
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Nous etudions dans une premiere partie la modelisation d'un plasma froid par les equations d'euler-poisson sans pression. La description lagrangienne des equations de convection permet de prendre en compte les phenomenes de deferlement (vitesses multivoques) apparaissant dans ce type de plasma. Les instabilites dues a la methode particule/maille sont resorbees par l'introduction d'une pression numerique. La seconde partie est consacree a la simulation de dispositifs semiconducteurs de type mesfet et diode par un modele hydrodynamique isotherme. Les collisions sont modelisees par un terme de relaxation en temps. On utilise la methode numerique developpee dans la premiere partie. Enfin on etudie un modele sans pression ou la vitesse derive d'un potentiel couple a l'equation de poisson. Dans le cadre gravitationnel on montre que les solutions sont caracterisees par un principe de minimisation de l'energie. Si la densite est bornee on montre que les vitesses gagnent en regularite dans le cadre electrostatique et gravitationnel
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Stoffel, Joshua David. "Lagrange-Chebyshev Based Single Step Methods for Solving Differential Equations." University of Akron / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=akron1335299082.
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Priestley, A. "Lagrange and characteristic Galerkin methods for evolutionary problems." Thesis, University of Oxford, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.376942.
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Baturin, Nickolay G. "Dynamics and effects of the tropical instability waves /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1997. http://wwwlib.umi.com/cr/ucsd/fullcit?p9737308.
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Danish, Md. "Lagrangian tracking, analysis and modeling of velocity-gradient dynamics in compressible turbulence." Thesis, IIT Delhi, 2017. http://localhost:8080/xmlui/handle/12345678/7060.
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Theron, Willem Frederick Daniel. "Analysis of the rolling motion of loaded hoops /." Link to the online version, 2008. http://hdl.handle.net/10019.1/1206.
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Baker, M. D. "A spectral Lagrange-Galerkin method for periodic/non-periodic convection-dominated diffusion problems." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240539.
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Schneider, David. "Nonholonomic Euler-Poincaré equations and stability in Chaplygin's sphere /." Thesis, Connect to this title online; UW restricted, 2000. http://hdl.handle.net/1773/5783.
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Barlow, Andrew. "An adaptive multi-material Arbitrary Lagrangian Eulerian algorithm for computational shock hydrodynamics." Thesis, Swansea University, 2002. https://cronfa.swan.ac.uk/Record/cronfa43081.
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Bessette, Gregory Carl. "Modeling of impact problems using an H-adaptive, explicit Lagrangian finite element method in three dimensions /." Full text (PDF) from UMI/Dissertation Abstracts International, 2000. http://wwwlib.umi.com/cr/utexas/fullcit?p3004213.
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Dean, Bruce H. "Variational results and solutions in gauge gravity and a bifurcation analysis of black hole orbital dynamics." Morgantown, W. Va. : [West Virginia University Libraries], 1999. http://etd.wvu.edu/templates/showETD.cfm?recnum=859.
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Thesis (Ph. D.)--West Virginia University, 1999.Title from document title page. Document formatted into pages; contains ix, 223 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 214-223) and index.
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Tröltzsch, Fredi. "On the Lagrange-Newton-SQP Method for the Optimal Control of Semilinear Parabolic Equations." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800995.
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A class of Lagrange-Newton-SQP methods is investigated for optimal control problems
governed by semilinear parabolic initial- boundary value problems. Distributed and boundary
controls are given, restricted by pointwise upper and lower bounds. The convergence of the method
is discussed in appropriate Banach spaces. Based on a weak second order sufficient optimality condition
for the reference solution, local quadratic convergence is proved. The proof is based on the
theory of Newton methods for generalized equations in Banach spaces.
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Bermejo, Rodolfo. "Analysis of a Galerkin-Characteristic algorithm for the potential vorticity-stream function equations." Thesis, University of British Columbia, 1990. http://hdl.handle.net/2429/30561.
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In this thesis we develop and analyze a
Galerkin-Characteristic method to integrate the potential
vorticity equations of a baroclinic ocean. The method proposed
is a two stage inductive algorithm. In the first stage the
material derivative of the potential vorticity is approximated
by combining Galerkin-Characteristic and Particle methods.
This yield a computationally efficient algorithm for this
stage. Such an algorithm consists of updating the dependent
variable at the grid points by cubic spline interpolation at
the foot of the characteristic curves of the advective
component of the equations. The algorithm is unconditionally
stable and conservative for Δt = O(h). The error analysis with respect to L² -norm shows that the algorithm converges with
order O(h); however, in the maximum norm it is proved that for
sufficiently smooth functions the foot of the characteristic
curves are superconvergent points of order O(h⁴ /Δt).
The second stage of the algorithm is a projection of
the Lagrangian representation of the flow onto the Cartesian
space-time Eularian representation coordinated with
Crank-Nicholson Finite Elements. The error analysis for this
stage with respect to L²-norm shows that the approximation
component of the global error is O(h²) for the free-slip boundary condition, and O(h) for the no-slip boundary condition. These estimates represent an improvement with respect to other estimates for the vorticity previously
reported in the literature. The evolutionary component of the
global error is equal to K(Δt² + h), where K is a constant that depends on the derivatives of the advective quantity along the Characteristic. Since the potential vorticity is a quasi-conservative quantitiy, one can conclude that K is in general small. Numerical experiments illustrate our theoretical results for both stages of the method.Science, Faculty of
Mathematics, Department of
Graduate
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Frink, Neal T. "Three-dimensional upward scheme for solving the Euler equations on unstructured tetrahedral grids." Diss., Virginia Tech, 1991. http://hdl.handle.net/10919/39423.
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A new upwind scheme is developed for solving the three-dimensional Euler equations on unstructured tetrahedral meshes. The method yields solution accuracy and efficiency comparable to that currently available from similar structured-grid codes. The key to achieving this result is a novel cell reconstruction process which is based on an analytical formulation for computing solution gradients within tetrahedral cells. Prior methodology requires the application of cumbersome numerical procedures to evaluate surface integrals around the cell volume. The result is that higher-order differences can now be constructed more efficiently to attain computational times per cell comparable to those of structured codes.
The underlying philosophy employed in constructing the basic flow solver is to draw on proven structured-grid technology whenever possible in order to reduce risk. Thus, spatial discretization is accomplished by a cell-centered finite-volume formulation using flux-difference splitting. Solutions are advanced in time by a 3- stage Runge-Kutta time-stepping scheme with convergence accelerated to steady state by local time stepping and implicit residual smoothing. The flow solver operates at a speed of 34 microseconds per cell per cycle on a CRAY-2S supercomputer and requires 64 words of memory per cell.
Transonic solutions are presented for a broad class of configurations to demonstrate the accuracy, speed, and robustness of the new scheme. Solutions are shown for the ONERA M6 wing, the Boeing 747-200 configuration, a low-wing transport configuration, a high-speed civil transport configuration, and the space shuttle ascent configuration. Computed surface pressure-coefficient distributions on the ONERA M6 wing are compared with structured-grid results as well as experimental data to quantify the accuracy. A further assessment of grid sensitivity and the effect of convergence acceleration parameters is also included for this configuration.
The more complex configurations serve to demonstrate the robustness and efficiency of the new method and its potential for performing routine aerodynamic analysis of full aircraft configurations. For example, the basic transonic flow features are well captured on the space shuttle ascent configuration with only 7 megawords of memory and 142 minutes of CRAY-YMP run time.Ph. D.
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COUSSEMENT, GREGORY. "Equations d'euler-lagrange d'une methode d'optimisation et d'adaptation de maillages et unification avec d'autres methodes." Paris 11, 1996. http://www.theses.fr/1996PA112063.
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Cette these a pour objet de contribuer a l'evolution des techniques de construction de maillages par l'etude et la mise en uvre des equations d'euler-lagrange tridimensionnelles d'une methode variationnelle d'optimisation et d'adaptation de maillages structures par sous-domaines ainsi que par l'unification de la methode avec d'autres methodes populaires variationnelles et elliptiques. L'etude generalisee des fondements de la methode d'optimisation et d'adaptation consideree met en evidence les hypotheses utilisees initialement par jacquotte pour mesurer la deformation directe du maillage vis a vis d'un maillage de reference. L'analyse de la formulation euler-lagrange realisee pour les points situes a l'interieur et situes sur les bords des sous-domaines permet un traitement general de tous les points y compris de ceux soumis a des conditions de raccord multidomaine et/ou des conditions aux limites de nuds libres, fixes ou glissant sur des courbes ou des surfaces. Au travers de la double formulation variationnelle et euler-lagrange, l'unification avec d'autres methodes populaires montre que toutes ces methodes peuvent s'interpreter comme des procedures de construction de maillages qui minimisent une mesure de la deformation appreciee par le biais de proprietes controlant les changements de longueur, d'aire, de volume et/ou d'orthogonalite. Sur base du logiciel optim3d developpe a l'onera pour l'optimisation et l'adaptation de maillages multidomaines structures, la forme euler-lagrange du principe a ete mis en uvre avec succes. Pour permettre la construction de maillages, le systeme complexe d'equations differentielles couplees est linearise et est discretise par des differences finies. Le systeme algebrique resultant est resolu avec un procede hybride qui conjugue une technique de relaxation quasi-optimale de type gauss-seidel symetrique pour les points interieurs aux sous-domaines et une technique de relaxation de type jacobi pour les points de bords soumis aux conditions aux limites et aux raccords multi-domaines. La construction de maillages tridimensionnels pour des cas didactiques et des cas industriels complexes demontre la generalite des developpements realises ainsi que la robustesse de l'algorithme qui s'accommode de mauvaise initialisation, meme aleatoire, du maillage
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Hu, Guanghui. "Numerical simulations of the steady Euler equations on unstructured grids." HKBU Institutional Repository, 2009. http://repository.hkbu.edu.hk/etd_ra/1106.
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McCabe, Ryan Matthew. "Small-scale coastal dynamics and mixing from a Lagrangian perspective /." Thesis, Connect to this title online; UW restricted, 2008. http://hdl.handle.net/1773/10963.
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Bagderina, Yulia, and Nikolai Tarkhanov. "Differential invariants of a class of Lagrangian systems with two degrees of freedom." Universität Potsdam, 2013. http://opus.kobv.de/ubp/volltexte/2013/6312/.
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We consider systems of Euler-Lagrange equations with two degrees of freedom and
with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lie's infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples.
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Layton, Richard A. "Analytical system dynamics /." Thesis, Connect to this title online; UW restricted, 1995. http://hdl.handle.net/1773/7131.
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Lu, Ming 1968. "A Lagrangian formulation of the Euler equations for subsonic flows /." Thesis, McGill University, 2007. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=103268.
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This thesis presents a Lagrangian formulation of the Euler equations for subsonic flows. A special coordinate transformation is used to define the Lagrangian coordinates, namely the stream function and the Lagrangian distance, in function of the Cartesian coordinates. This Lagrangian formulation introduces two new geometry state variables, and a Lagrangian behavior parameter defining a pseudo-Lagrangian time used during the iteration procedure to obtain the solution for subsonic flows.The eigenstructure and characteristics analysis for the new system of equations is based on a linear Jacobian matrix-mapping procedure, which starts from the well-known eigenstructure and characteristics in the Eulerian plane and uses the coordinate transformation to find their counterparts in the Lagrangian plane. This analysis studies the basic properties of the Euler equations in the Lagrangian formulation, such as hyperbolicity, homogeneity and rotational invariance. The Riemann problem in the Lagrangian plane is also studied. Those elements are used to construct the numerical scheme for solving the Euler equations in the Lagrangian formulation.
The numerical scheme is constructed using first and second-order dimensional-splitting with hybrid flux operators, based on flux vector splitting and Godunov methods, which include a 2-D Riemann solver in the Lagrangian plane. The numerical method is validated by comparing the present solutions with the results obtained with an Eulerian formulation for several internal flows.
This numerical method based on a Lagrangian formulation has also been extended for the solution of unsteady subsonic flows by using a dual time approach. The method validation in this case has been done by comparison with the Eulerian formulation solutions for several internal subsonic flows with oscillating boundaries.
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Bispo, Rafael Santana. "Simulação harmônica particionada usando um método baseado em multiplicadores de Lagrange." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/265157.
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Orientador: Renato PavanelloDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica
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Resumo: Atualmente, existe uma grande tendência no incremento da produção de energia elétrica através de fontes renováveis. Em especial, a geração de energia elétrica produzida através de parques eólicos tem sido bastante adotada. O projeto desses equipamentos envolve a modelagem dinâmica acoplada solo-fluido-estrutura que pode ser estudada usando-se a formulação particionada, onde o problema da interação entre os meios é tratado de maneira iterativa. Nesse tipo de técnica, é possível que modelos fisicamente heterogêneos, chamados de partições, possam utilizar diferentes técnicas de discretização, como por exemplo o domínio do fluido ser baseado em uma formulação de Elementos de Contorno e o domínio estrutura baseado em uma formulação em Elementos Finitos. Neste trabalho, é realizado um estudo dinâmico de turbinas eólicas, utilizando tratamento particionado e Multiplicadores de Lagrange afim de se obter as frequências características e as curvas de resposta em frequência do sistema em análise. A discretização do problema é realizada através do Método dos Elementos Finitos (FEM) utilizando elemento de pórtico e quadrilateral de Wilson. Desta forma, a resolução de problemas de interação, utilizando a formulação particionada, é estudada com a finalidade de avaliar a convergência e a viabilidade da técnica em problemas harmônicos estruturais
Abstract: Currently, there is a great tendency in increasing the production of electricity through renewable sources. In this context, the generation of electric energy produced by wind farms has been widely adopted. The design of these devices involves the dynamic modeling of coupling fluid-structure-soil that can be studied using the partitioned formulation, where the problem of interaction between the parties is iterative manner. In this type of technique, it is possible that physically heterogeneous models, called partitions, can use different discretization techniques, such as the domain of fluid is based on a formulation of boundary element or based on a finite element formulation. In this paper a harmonic simulation of wind turbines, using partitioned treatment and Lagrange multipliers is studied in order to obtain the characteristic frequencies and frequency response function of the system under analysis. The discretization of the problem is performed using the Finite Element Method (FEM), as well as beam elements and the quadrilateral Wilson element. Thus, the resolution of elastodynamics problems, using the partitioned formulation is studied with the aim of assessing the feasibility and convergence of this technique, applied to dynamic harmonic analysis
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Mestre em Engenharia Mecânica
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Oliveira, Anabela Pacheco de Pacheco de Oliveira Anabela De Oliveira Anabela Pacheco. "A comparison of Eulerian-Lagrangian methods for the solution of the transport equation /." Full text open access at:, 1994. http://content.ohsu.edu/u?/etd,208.
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DeVries, Mark R. "Vibration of a cantilever beam that slides axially in a rigid frictionless hole." Thesis, Monterey, California : Naval Postgraduate School, 1990. http://handle.dtic.mil/100.2/ADA241352.
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Thesis (M.S. in Mechanical Engineering)--Naval Postgraduate School, September 1990.Thesis Advisor(s): Salinas, David. "September 1990." Description based on title screen as viewed on December 17, 2009. DTIC Identifier(s): Vibration, sliding friction, guns barrels, cantilever beams, Euler equations, finite element analysis, transformations (mathematics), theses, frictionless holes, holes (openings), recoil. Author(s) subject terms: Cantilever beam, finite element method, axial motion, vibration, transient behavior. Includes bibliographical references (p. 110). Also available in print.
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Tang, Jiqing. "A Lagrangian heuristic for winner determination problem in combinatorial auctions /." View abstract or full-text, 2004. http://library.ust.hk/cgi/db/thesis.pl?IEEM%202004%20TANG.
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Hillmer, Kiekebusch Ingrid Andrea. "Scales of interactions between physical processes, primary producers and nutrients in aquatic ecosystems." University of Western Australia. Centre for Water Research, 2007. http://theses.library.uwa.edu.au/adt-WU2007.0177.
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[Truncated abstract] Estimating internal biogeochemical fluxes is essential to the understanding of the dynamic of aquatic ecosystems. Different ecological approaches have been used to gain insight into the internal cycling, but success has been limited. A critical point is the identification of the characteristic scales of patterns and the underlying processes affecting the behaviour of biological and chemical species. Failing to capture these scales leads to misinterpretation of field and numerical data. In this study, key aspects in the design of ecological surveys are identified to ensure that the internal biogeochemical processes are well represented. In the first part of this thesis, a 1D reaction-diffusion-advection equation is used to investigate the formation of patterns and relevant time and spatial scales. This is used to define an approach for the determination of a critical domain size that allows differentiation of the role of local and internal cycling from advective fluxes across the open boundaries in a shallow coastal ecosystem. By using a 3D numerical model, in conjunction with an extensive field data set, it is shown that domain sizes must be larger than this critical value in order to capture the patterns generated within the system. For smaller domains, transport processes control the evolution of the system across the boundaries misleading the interpretation of the internal ecological dynamics. The study of the influence of boundary fluxes on ecological patchiness was motivated by the need to define the size of the domain necessary for the assessment of the impact of a sewage outflow on a coastal regime. The quantification of biogeochemical processes has proven to be difficult to achieve especially under conditions of high spatial and temporal hydrodynamic and biogeochemical variability. In the second part of this thesis, a Lagrangian experimental design is employed to estimate biogeochemical rate coefficients in situ. A set of four drogues and a cross-transect sampling design is used to capture the patchy distribution of phytoplankton and nutrient species, and high transport and mixing rates. ... Total chlorophyll from both models shows similar behavior when the variability in the 3D model, expressed as Chlamax/Chlamin, is low. When Chlamax/Chlamin is high, the difference between the biomass predicted by the two models reaches 30% due to the generation of localised patches. Comparison of the 1D and 3D results highlights the need of using models that are able to resolve the spatial complexity to some extent, as the use of averaged properties may produce misleading results. This is especially important in the presence of patches with differential physiological and biogeochemical characteristics, and nonlinear processes, in which case biomass average is not necessarily linearly related to the averaged environment.
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Oliveira, Anabela Pacheco de. "Eulerian-Lagrangian analysis of transport and residence times in estuaries and coasts /." Full text open access at:, 1997. http://content.ohsu.edu/u?/etd,209.
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Thierauf, Rainer Georg. "A Lagrangian for a system of two dyons." PDXScholar, 1988. https://pdxscholar.library.pdx.edu/open_access_etds/3840.
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Maxwell's equations for the electromagnetic field are symmetrized by introducing magnetic charges into the formalism of electrodynamics. The symmetrized equations are solved for the fields and potentials of point particles. Those potentials, some of which are found to be singular along a line, are used to formulate the Lagrangian for a system of two dyons (particles with both electric and magnetic charge). The equations of motion are derived from the Lagrangian. It is shown that the dimensionality constants k and k * , which we r e introduced to define the units of the electromagnetic fields, have to be equal in order to avoid center of mass acceleration in the two dyon system.
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Maury, Bertrand. "Resolution numerique des equations de navier-stokes 2d et 3d avec surface libre par une methode euler-lagrange arbitraire." Paris 6, 1995. http://www.theses.fr/1995PA066773.
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Dans le cadre general de l'approche euler-lagrange arbitraire, une methode complete de resolution des equations de navier-stokes avec surface libre par elements finis est presentee. La prise en compte du terme d'inertie et la convection de la surface libre sont bases sur la methode des caracteristiques. Des proprietes de consistance et de conservation de la masse et de l'energie pour le schema propose sont demontrees. Diverses applications 2d et 3d confirment la validite et la stabilite de la methode
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Brown, Jaclyn Nicole School of Mathematics UNSW. "The kinematics and dynamics of cross-hemispheric flow in the Central and Eastern Equatorial Pacific." Awarded by:University of New South Wales. School of Mathematics, 2005. http://handle.unsw.edu.au/1959.4/20831.
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This thesis concerns two topics: the kinematics of Pacific cross-equatorial flow ??? the location, timing and magnitude of the flow; and their dynamics???what are the driving forces controlling the flow? Despite extensive observations in the central and eastern Pacific, observations of these flows remain contradictory. We use output from an Ocean General Circulation Model (OGCM) viewed from a Lagrangian framework on density layers. This addresses the problem of high variability due to features such as Tropical Instability Waves. The annual mean flow is found to be southward nearly everywhere, east of 140??W. Flow becomes stronger in the second half of the year due to a bolus transport of very light surface water, introduced by Tropical Instability Waves. A Tropical Cell pattern occurs along the equator that does not require diapycnal downwelling. From 160??E to 160??W the annual mean flow is northward, occurring mostly in the mixed layer, appearing to originate partly from the Equatorial Undercurrent surfacing in the east. The northward flow is strongest in March and becomes southward in September. The wind stress and nonlinear terms are shown to be the key driving features, with a prescribed biharmonic Smagorinsky horizontal friction scheme having negligible impact. From 160??E to 160??W, the flow is partly accounted for by an Ekman forcing, with the curl of the nonlinear term providing a crucial additional torque, more than doubling the magnitude in some instances. From 160??W to 120??W the wind stress curl provides a weak southward flow of about 1 Sv, which increases by the nonlinear addition to around 5 Sv. The curl of the steady component of the nonlinear term, derived from annual mean currents, is similar in structure to the total nonlinear term, but higher in magnitude. The structure of the variable term, which was mostly of opposite sign to the steady term, suggests damping occurs in place of friction. While our study is limited to an examination of the model's characteristics, our results provide important clues to the observed flow patterns not resolved by present-day measurements. This study also highlights the importance of time-space variability and both horizontal and vertical density structure in controlling the flow and its feedback on the system.
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Southern, James Alastair. "Mathematical and computational modelling of ultrasound elasticity imaging." Thesis, University of Oxford, 2006. http://ora.ox.ac.uk/objects/uuid:242fddf0-ef9c-4a90-88f5-c7b41f4bda5a.
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In this thesis a parameter recovery method for use in ultrasound elasticity imaging is developed. Elasticity imaging is a method for using a series of ultrasound images (and the displacement field between them) to estimate the spatial variation of the stiffness of the tissue being imaged. Currently iterative methods are used to do this: a model of tissue mechanics is assumed and a large number of simulations using varying parameters are compared to the actual displacement field. The aim of this work is to develop a solution method that works back from the known displacement field to determine the tissue properties, reducing the number of simulations that must be performed to one. The parameter recovery method is based on the formulation and direct solution of the 2-d linear elasticity inverse problem using finite element methods. The inverse problem is analyzed mathematically and the existence and uniqueness of solutions is described for varying numbers of displacement fields and appropriate boundary conditions. It is shown to be hyperbolic (and so difficult to solve numerically) and then reformulated as a minimization problem with hyperbolic Euler-Lagrange equations. A finite element solution of the minimization problem is developed and implemented. The results of the finite element implementation are shown to work well in recovering the parameters used in numerical simulations of the linear elasticity forward problem so long as these are continuous. The method is shown to be robust in dealing with small errors in displacement estimation and larger errors in the boundary values of the parameters. The method is also tested on displacement fields calculated from series of real ultrasound images. The validity of modelling the ultrasound elasticity imaging process as a 2-d problem is discussed. The assumption of plane strain is shown not to be valid and methods for extending the parameter recovery method to 3 dimensions once 3-d ultrasound becomes more widely used are described (but not implemented).
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Maroofi, Hamed. "Applications of the Monge - Kantorovich theory." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29197.
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Mendonça, Luziane Ferreira de. "Aceleração quase-Newton para problemas de minimização com restrições." [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306668.
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Orientadores: Vera Lucia da Rocha Lopes, Jose Mario MartinezTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Sistemas de Otimalidade (ou Sistemas KKT) são sistemas formados pelas condições primais-duais estacionárias para a solução de problemas de otimização. Sob hipóteses adequadas (condições de qualificação), os minimizadores locais de um problema de minimização satisfarão as equações e inequações KKT; entretanto, infelizmente, muitos outros pontos estacionários (incluindo maximizadores) também são soluções desse sistema não linear. Por essa razão, os métodos destinados à resolução de problemas de programação não-linear fazem uso constante da estrutura de minimização, e o uso simples de métodos destinados à resolução de sistemas não-lineares podem gerar soluções espúrias. Todavia, caso o método destinado à resolução do sistema KKT tenha um ponto inicial situado na região de atração para um minimizador, esse método pode vir a ser muito eficiente. Neste trabalho, os métodos quase-Newton para a resolução de sistemas não-lineares são usados como aceleradores de algoritmos de programação não-linear (Lagrangiano Aumentado) com restrições de igualdade, desigualdade e caixa. Utilizamos como acelerador o método simétrico inverso de correção de posto um (ISR1), o qual realiza reínicios periódicos e faz uso das estruturas esparsas das matrizes para armazenamento. São demonstrados resultados de convergência e são realizados vários experimentos numéricos que comprovam a eficiência desta estratégia para problemas de minimização com restrições de igualdade, e indicam outros caminhos para problemas de minimização com restrições gerais (igualdade, desigualdade e caixa)
Abstract: Optimality (or KKT) systems arise as primal-dual stationarity conditions for constrained optimization problems. Under suitable constraint qualifications, local minimizers satisfy KKT equations but, unfortunately, many other stationary points (including, perhaps, maximizers) may solve these nonlinear systems too. For this reason, nonlinear-programming solvers make strong use of the minimization structure and the naive use of nonlinear-system solvers in optimization may lead to spurious solutions. Nevertheless, in the basin of attraction of a minimizer, nonlinear-system solvers may be quite efficient. In this work quasi-Newton methods for solving nonlinear systems are used as accelerators of nonlinear-programming (augmented Lagrangian) algorithms. A periodically-restarted memoryless symmetric rank-one (SRI) correction method is introduced for that purpose. Convergence results are given. For problems with only equality constraints, numerical experiments that confirm that the acceleration is effective are presented. A bunch of problems with equalities, inequalities and box constraints is tested and several comments and suggestions for further work are presented
Doutorado
Doutor em Matemática Aplicada
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Altintan, Derya. "An Extension To The Variational Iteration Method For Systems And Higher-order Differential Equations." Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613864/index.pdf.
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It is obvious that differential equations can be used to model real-life problems. Although it is possible to obtain analytical solutions of some of them, it is in general
difficult to find closed form solutions of differential equations. Finding thus approximate solutions has been the subject of many researchers from different areas.
In this thesis, we propose a new approach to Variational Iteration Method (VIM) to obtain the solutions of systems of first-order differential equations. The main
contribution of the thesis to VIM is that proposed approach uses restricted variations only for the nonlinear terms and builds up a matrix-valued Lagrange multiplier that leads to the extension of the method (EVIM).
Close relation between the matrix-valued Lagrange multipliers and fundamental solutions of the differential equations highlights the relation between the extended version of the variational iteration method and the classical variation of parameters formula.
It has been proved that the exact solution of the initial value problems for (nonhomogenous) linear differential equations can be obtained by such a generalisation using
only a single variational step.
Since higher-order equations can be reduced to first-order systems, the proposed approach is capable of solving such equations tooindeed, without such a reduction, variational iteration method is also extended to higher-order scalar equations. Further, the close connection with the associated first-order systems is presented. Such extension of the method to higher-order equations is then applied to solve boundary value problems: linear and nonlinear ones. Although the corresponding Lagrange multiplier resembles the Green&rsquo
s function, without the need of the latter, the extended approach to the variational iteration method is systematically applied to solve boundary value problems, surely in the nonlinear case as well. In order to show the applicability of the method, we have applied the EVIM to various real-life problems: the classical Sturm-Liouville eigenvalue problems, Brusselator reaction-diffusion, and chemical master equations. Results show that the method is simple, but powerful and effective.
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Fourestey, Gilles. "Simulation numérique et contrôle optimal d'intéractions fluide incompressible / structure par une méthode de Lagrange-Galerkin d'ordre 2 : Applications aux ouvrages d'art." Marne-la-vallée, ENPC, 2002. http://www.theses.fr/2002ENPC0224.
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Dhaouadi, Firas. "An augmented lagrangian approach for Euler-Korteweg type equations." Thesis, Toulouse 3, 2020. http://www.theses.fr/2020TOU30139.
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On présente un modèle hyperbolique quasi-linéaire de premier ordre approximant les équations d'Euler-Korteweg (E-K), qui décrivent des écoulements de fluides compressibles dont l'énergie dépend du gradient de la densité. Le système E-K peut être vu comme les équations d'Euler-Lagrange d'un Lagrangien soumis à la conservation de la masse. Vu la présence du gradient de la densité dans le Lagrangien, des dérivées d'ordre élevé de la densité apparaissent dans les équations du mouvement. L'approche présentée ici permet d'obtenir un système d'équations hyperboliques qui approxime le système E-K. L'idée est d'introduire un nouveau paramètre d'ordre qui approxime la densité via une méthode de pénalisation classique. Le gradient de cette nouvelle variable remplace alors le gradient de la densité dans le Lagrangien, ce qui permet de construire le Lagrangien augmenté. Les équations d'Euler-Lagrange associées à celui-ci, sont des équations hyperboliques avec des termes sources raides et des vitesses de caractéristiques rapides. Ce système est analysé puis résolu numériquement en utilisant des schémas de type IMEX. En particulier, cette approche a été appliquée à l'équation de Schrödinger non-linéaire défocalisante (qui peut être réduite au système E-K via la transformée de Madelung), pour laquelle des comparaisons avec des solutions exactes et asymptotiques ont été faites, notamment pour des solitons gris et des ondes de choc dispersives. La même approche a été également appliquée aux équations de filmes minces avec capillarité, pour lesquelles une comparaison avec des résultats numériques de référence et des résultats expérimentaux a été faite. Il a été démontré que le modèle augmenté peut aussi bien s'appliquer pour des modèles dont le terme de capillarité est non-linéaire. Dans ce même cadre, une étude de gouttes stationnaires sur un substrat solide horizontal a été établie afin de classifier les profils possibles de gouttes selon leur énergie. Ceci a permis également de faire des comparaisons du modèle augmenté sur des solutions stationnaires. Enfin, une partie indépendante de ce travail est consacrée à l'étude des équations équivalentes associées aux schémas numériques, où l'on démontre que les conditions de stabilité qui dérivent d'une troncature de l'équation équivalente, n'a du sens que si la série correspondante dans l'espace de Fourier est convergente, sur les longueurs d'onde admissibles dans la pratiqueAn approximate first order quasilinear hyperbolic model for Euler-Korteweg (E-K) equations, describing compressible fluid flows whose energy depend on the gradient of density, is derived. E-K system can be seen as the Euler-Lagrange equations to a Lagrangian submitted to the mass conservation constraint. Due to the presence of the density gradient in the Lagrangian, one recovers high-order derivatives of density in the motion equations. The approach presented here permits us to obtain a system of hyperbolic equations that approximate E-K system. The idea is to introduce a new order parameter which approximates the density via a carefully chosen penalty method. The gradient of this new independent variable will then replace the original gradient of density in the Lagrangian, resulting in the so-called augmented Lagrangian. The Euler-Lagrange equations of the augmented Lagrangian result in a first order hyperbolic system with stiff source terms and fast characteristic speeds. Such a system is then analyzed and solved numerically by using IMEX schemes. In particular, this approach was applied to the defocusing nonlinear Schrödinger equation (which can be reduced to the E-K equations via the Madelung transform), for which a comparison with exact and asymptotic solutions, namely gray solitons and dispersive shock waves was performed. Then, the same approach was extended to thin film flows with capillarity, for which comparison of the numerical results with both reference numerical solutions and experimental results was performed. It was shown that the augmented model is also extendable to models with full nonlinear surface tension. In the same setting, a study of stationary droplets on a horizontal solid substrate was conducted in an attempt to classify droplet profiles depending on their energy forms. This also allowed to compare the augmented Lagrangian approach in the case of stationary solutions, and which showed excellent agreement with the reference solutions. Lastly, an independent part of this work is devoted to the study of modified equations associated to numerical schemes for stability purposes. It is shown that for a linear scheme, stability conditions which are obtained from a truncation of the associated modified equation, are only relevant if the corresponding series in Fourier space is convergent for the admissible wavenumbers
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Awi, Romeo Olivier. "Minimization problems involving polyconvex integrands." Diss., Georgia Institute of Technology, 2015. http://hdl.handle.net/1853/53901.
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This thesis is mainly concerned with problems in the areas of the Calculus of Variations and
Partial Differential Equations (PDEs).
The properties of the functional to minimize with respect to the given topology
play an important role in the existence of minimizers of integral problems. We will introduce
the important concepts of quasiconvexity and polyconvexity. Inspired by finite element methods from Numerical Analysis, we introduce a perturbed problem which has some surprising uniqueness properties.
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Ennaji, Hamza. "Variational methods for Hamilton-Jacobi equations and applications." Thesis, Limoges, 2021. http://www.theses.fr/2021LIMO0013.
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L’objectif de cette thèse est de proposer des méthodes variationnelles pour l’analyse mathématiques et numérique d’une classe d’équations d’HJ. Le caractère métrique de ces équations permet de caractériser l’ensemble des sous-solutions, à savoir, elles sont 1-Lipschitz par rapport à la distance Finslerienne associée au Hamiltonien. De manière équivalente, cela revient à dire que le gradient de ces fonctions appartient à une certaine boule Finslerienne. La solution recherchée est la sous-solution maximale, qui peut être décrite par une formule du type Hopf-Lax, qui résout un problème de maximisation avec contrainte sur le gradient. Nous dérivons un problème dual associé faisant intervenir la variation totale Finslerienne de mesures vectorielles avec contrainte divergente. Nous exploitons la structure de point-selle pour proposer une résolution numérique avec la méthode du Lagrangien augmenté. Cette caractérisation de l’équation d’HJ montre aussi le lien avec des problèmes de transport optimal vers/depuis le bord. Ce lien avec le transport optimal de masse nous amène à généraliser l’approche d’Evans-Gangbo. En effet, nous montrons que la sous-solution maximale de l’équation d’HJ s’obtient en faisant tendre p→∞ dans une classe de p-Laplaciens de type Finsler avec des obstacles sur le bord. Cela nous permet aussi de construire le flux optimal pour le problème de Beckmann associé. Parmi les applications que l’on regarde, le problème du Shape from Shading qui consiste à reconstruire la surface d’un objet en 3D à partir d’une image en nuances de gris de cet objetIn this thesis we propose some variational methods for the mathematical and numerical analysis of a class of HJ equations. Thanks to the metric character of these equations, the set of subsolution corresponds to the set of 1-Lipschitz functions with respect to the Finsler metric associated to the Hamiltonian. Equivalently, it corresponds to the set of functions whose gradient belongs to a Finsler ball. The solution we are looking for is the maximal one, which can be described via a Hopf-Lax formula, solves a maximization problem under gradient constraint. We derive the associated dual problem which involves the Finsler total variation of vector measures under a divergence constraint. We take advantage of this saddle-point structure to use the augmented Lagrangian method for the numerical approximation of HJ equation. This characterization of the HJ equation allows making the link with some optimal transport problems. This link with optimal transport leads us to generalize the Evans-Gangbo approach. In fact, we show that the maximal viscosity subsolution of the HJ equation can be recovered by taking p→ ∞ in a class of Finslerp-Laplace problems with boundary obstacles. In addition, this allows us to construct the optimal flow for the associated Beckmann problem. As an application, we use our variational approach for the Shape from Shading problem
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Danielson, Jon David. "Mobile boom cranes and advanced input shaping control." Thesis, Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24651.
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Tkachenko, Sergey. "Analytical and numerical study of a dispersive shallow water model." Thesis, Aix-Marseille, 2020. http://www.theses.fr/2020AIXM0114.
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Le présent manuscrit est dédié à la modélisation analytique et numérique des systèmes dispersifs non-linéaires, notamment les équations de Serre-Green-Naghdi (SGN) et le modèle du liquide à bulles de Iordanskii-Kogarko-Wijngaarden (IKW). Ces modèles partagent une structure commune. Tout d'abord, les pressions des deux modèles dépendent des dérivées matérielles de leurs variables macroscopiques. Ensuite, les équations de SGN et IKW sont les équations d'Euler-Lagrange qui viennent du principe d'Hamilton de l'action stationnaire, chacun avec son lagrangien naturel. Enfin, les équations sont invariantes par rapport à la transformation de Galilée, ce qui est nécessaire pour les modèles mathématiques physiquement pertinents. Dans une première partie nous dérivons les équations de modulation de Whitham pour le modèle de SGN et nous montrons qu'elles sont strictement hyperboliques quelque soit l'amplitude d'onde, i.e. que les paquets d'ondes périodiques sont modulationnellement stables. Les tests numériques pour le système complet SGN sont présentés, ils confirment les résultats théoriques. Dans une seconde partie, nous dérivons et étudions un modèle multidimensionnel qui se rapproche des deux modèles SGN et IKW, en utilisant l'approche du lagrangien étendu. Le nouveau modèle est hyperbolique et décrit précisément les phénomènes dispersifs, et de plus il possède les propriétés des équations d'Euler, ce qui permet d'imposer les conditions initiales discontinues et d'étudier les ondes de choc dispersifs. Les tests numériques pour les chocs dispersifs 1-D et 2-D sont présentés. Les résultats sont en bon accord avec ceux obtenus par des modèles numériques du système SGN exacteThe present thesis is devoted to analytical and numerical modeling of nonlinear dispersive systems, notably Serre-Green-Naghdi equations (SGN equations) and Iordanskii-Kogarko-Wijngaarden bubbly fluids model (IKW model). These models share the common structure. First, pressures of both models depend on the material derivatives of their macroscopic variables. Second, SGN and IKW equations are the Euler-Lagrange equations coming from Hamilton's principle of stationary action, each one with its natural Lagrangian. Finally, the equations are Galilean invariant which is necessary for physically relevant mathematical models. In the first part we derive the Whitham modulation equations for the SGN model and prove that these equations are strictly hyperbolic for any wave amplitude, i. e. that the periodic wave solutions of the SGN equations are modulationally stable. Numerical simulations of the full SGN equations are also shown, and these results confirm the theoretical stability results found by using the Whitham modulation theory. In the second part, we derive and study a multi-dimensional model which approximates both the SGN and IKW systems using the extended Lagrangian approach. The new model is hyperbolic and accurately describes dispersive phenomena while possessing the properties of classic Euler equations, which allows to impose discontinuous initial data and study dispersive shock waves. We take the approximate SGN system as an example for numerical simulations. Numerical tests for 1-D and 2-D dispersive shock waves are presented. The results are in a good agreement with those obtained via existing numerical models of the exact SGN system
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Li, Chunying Anna. "Theoretical modeling and experimental studies of particle-laden plumes from wastewater discharges." Click to view the E-thesis via HKUTO, 2006. http://sunzi.lib.hku.hk/hkuto/record/B37318640.
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Choi, Gyunghyun. "Nondifferentiable optimization algorithms with application to solving Lagrangian dual problems." Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/38657.
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Kallel, Achraf. "Une modélisation du contact par l'approche mortier : application à la mise en forme." Thesis, Compiègne, 2014. http://www.theses.fr/2014COMP2173/document.
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Cette thèse est située dans le cadre du projet FUI OASIS ayant comme objectif la modélisation d'un processus d'emboutissage optimisé. Le travail consiste essentiellement au développement des algorithmes de contact plus appropriés à ce type de mise en forme. Dans la littérature et pour plusieurs codes de calcul industriels, l'approche NTS (nœud à segment) demeure la plus utilisée pour la résolution d'un problème de contact. Dans certaine configuration, cette méthode présente des insuffisances et un manque de précision. On la remplaçant par l'approche mortier, on arrive à résoudre une gamme assez large de problèmes de contact. La méthode mortier, utilisée au initialement pour un calcul avec décomposition de domaine, a été le centre d'intérêt de plusieurs travaux de recherche pour la modélisation du contact. Dans ce travail, on va regrouper plusieurs méthodes de gestion du contact en les combinant avec l'approche mortier. L'algorithme de résolution, les éléments d'implémentation ainsi quelques exemples de validation présentant une critique des avantages et les limites de chaque techniques sont détaillés dans ce travail afin d'obtenir un support technique pour tous travail ultérieurs avec la méthode mortier. Le principal avantage de la méthode mortier se manifeste dans l'application des conditions de contact sous forme d'intégrale dans l'interface. Bien que cette méthode permette de réduire la différence des contraintes dans l'interface de contact d'un élément à un autre pour obtenir une meilleure continuité de la pression de contact, elle demeure insuffisante dans certaines applications en particulier pour les problèmes en grande déformation. Le lissage des surfaces de contact, qu'on peut appliquer par différentes techniques, présente une solution classique à ce genre de problème en mécanique de contact. L'originalité de ce travail, c'est la combinaison de l'utilisation des courbes B-Spline cubiques pour la description presque exacte de la surface de contact d'un côté avec une formulation avec l'approche mortier pour l'application des conditions de contact d'un autre côté. Cette combinaison forme un duo gagnant permettant de résoudre un problème de contact en grandes déformation. Les termes permettant l'implémentation des différentes techniques de lissage pour la résolution d'un problème de contact sont détaillés. Une attention particulière est accordée au lissage avec les B-Spline Cubiques.Tous les algorithmes détaillés dans ce travail sont implémentés dans un code maison FiEStA. C'est un code de calcul par éléments finis libre en langage C++. Certains développements concernant la loi de comportement hyper-élastique et l'intégralité du module du contact sont développés dans ce travail de thèseThis thesis is situated in the FUI OASIS project which the objective is the modeling of an optimized stamping process. The work mainly involves the development of the most appropriate contact algorithms such formatting. In the literature and several industrial computing codes, the NTS approach (node to segment) remains the most used for the resolution of a contact problem. In certain configuration, this method has shortcomings and a lack of precision. We replacing it with mortar approach, we manage to solve a broad range of contact problems. The mortar method, used for the initial for calculation using domain decomposition, was the focus of several research projects for the modeling of the contact. In this work, we will consolidate multiple contact formulation methods in combination with mortar approach. The resolution algorithm, the elements of implementation and some examples of validation with a review of the advantages and limitations of each technique are detailed in this work in order to get technical support for subsequent work with the mortar method. The main advantage of the mortar method is in the application of the contact conditions in integral form in the interface. Although this method reduces the difference of the stresses in the contact interface of a component to another to obtain a better continuity of the contact pressure, it is still insufficient in some applications, particularly for large deformation problems. The smoothing of contact surfaces, which can be applied by various techniques, presents a classic solution to this problem in mechanical contact. The originality of this work is the combination of using cubic B-Spline curves for the almost exact description of the contact surface on one side with the use of the mortar approach to the application of the contact conditions on the other hand. This combination forms a winning combination for solving a contact problem in large deformation. The terms allowing the implementation of the different smoothing techniques for solving a problem of contact are detailed. Particular attention is paid to smoothing with Cu bic B-Spline. All algorithms detailed in this work are implemented in a house code 'Fiesta'. This is a free finite elements computer code in C ++. Some developments in the law of hyper-elastic behavior and completeness of the contact module are developed in this thesis
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Theron, Willem F. D. "Analysis of the rolling motion of loaded hoops." Thesis, Stellenbosch : University of Stellenbosch, 2008. http://hdl.handle.net/10019.1/1206.
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Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2008.This dissertation contains a detailed report on the results of a research project on the behaviour of a dynamical system consisting of a hoop to which a heavy particle is fixed at the rim. This loaded hoop rolls on a rough surface while remaining in the vertical plane. The motion of the hoop consists of various, possibly alternating, phases consisting of rolling without slipping, spinning or skidding motion and in some cases ends by hopping off the surface. A general mathematical model is developed, consisting of a system of second order ordinary differential equations, one for each of the three degrees of freedom. Analytic solutions are obtained in some cases; otherwise numerical solutions are used. Three specific applications of the general model are dealt with. In the first application the problem of massless hoops is investigated. The main emphasis is on the somewhat controversial question of what happens after the normal reaction becomes zero in a position where the particle is still moving downwards. A new result shows that the hoop can continue to move horizontally in a motion defined as skimming. The second application deals with rigid hoops and a large number of detailed results are presented. Classification schemes for the different types of behaviour are introduced and summarised in the form of phase diagrams. Some emphasis is placed on the rather amazing number of different patterns of motion that can be obtained by varying the parameters. In the third application two elastic models are analysed, with the primary purpose of explaining one aspect of the reported behaviour of experimental hoops, namely hopping while the particle is moving downwards. A chapter on experimental models rounds off the project.
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Fourestey, Gilles. "Simulation numérique et contrôle optimal d'interactions fluide incompressible/structure par une méthode de Lagrange-Galerkin d'ordre 2. Applications aux ouvrages d'art." Phd thesis, Ecole des Ponts ParisTech, 2002. http://tel.archives-ouvertes.fr/tel-00005675.
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Le but de cette thèse est la construction et l'implémentation d'une méthode de Lagrange-Galerkin d'ordre élevé dans un code de simulation d'interactions fluide/structure. Cette méthode repose sur une formulation par éléments finis mixtes et une méthode des caractéristiques d'ordre 2 en maillage fixe ou mobile. La stabilité de ce schéma a été étudiée dans des cas simples. Des analyses aéroélastiques de structures généralement rencontrées dans le Génie Civil ont été effectuées à travers des tests numériques sur des coupes de ponts en mouvements forcés et libres. Les résultats obtenus ont été comparés à ceux obtenus avec la méthode de Lagrange-Galerkin d'ordre 1 ainsi qu'à des études réalisées en soufflerie expérimentale. Enfin, l'utilisation des méthodes de Lagrange-Galerkin dans des problèmes de contrôle optimal a été étudiée. Un schéma discret linéarisé basé sur la méthode des caractéristiques a été construit et quelques tests simples pour des problèmes de contrôle et d'identification en maillages fixes et mobiles ont été effectués.
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Craig, Rob James. "Modelling of a thermodynamically driven heat engine with application intended for water pumping." Thesis, Stellenbosch : Stellenbosch University, 2014. http://hdl.handle.net/10019.1/95923.
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Beersing-Vasquez, Kiran. "Suturing in Surgical Simulations." Thesis, KTH, Numerisk analys, NA, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-260254.
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The goal of this project is to develop virtual surgical simulation software in order to simulate the suturing and knot tying processes associated with surgical thread. State equations are formulated using Lagrangian mechanics, which is useful for the conservation of energy. Solver methods are developed with theory based in Differential Algebraic Equations (DAEs) which concern governing Ordinary Differential Equations (ODEs) that are constraint with Algebraic Equations (AE). An implicit integration scheme and Newton's method is used to solve the system in each step. Furthermore, a collision response process based on the Linear Complementarity Problem (LCP) is implemented to handle collisions and measure their forces. Models have been developed to represent the different types of objects. A spline model is used to represent the suture and mass-spring model for the tissue. They were both selected for their efficiency and base on real physical properties. The spline model was also chosen as it is continuous and can be evaluated at any point along the length. Other objects are also defined such as rigid bodies. The Lagrangian multiplier method is used to define the constraints in the model. This allows for the construction of complex models. An important constraint is the suturing constraint, which is created when a sufficient force is applied by the suture tip on to the tissue. This constraint allows only a sliding point along the suture to pass through a specific point on the tissue. This results in a virtual suturing model which can be built on for use in surgical simulations. Further investigations would be interesting to increase performance, accuracy and scope of the simulator.Det här projektet syftar till att utveckla mjukvara för virtuell simulering av kirurgi som involverar knytande av suturtråd. Lagranges ekvationer används för att härleda energibevarande tillståndsekvationer. Lösningsmetoderna grundar sig i teori från området Differential-Algebraiska Ekvationer (DAEer), som avser att kontrollera Ordinära Differentialekvationer (ODEer) med algebraiska bivillkor. Ett implicit integrationsschema och Newtons metod används för att lösa systemet i varje steg. Utöver det så implementeras en kollisionsrespons-process baserad på det linjära komplementaritetsproblemet (LCP) för att hantera kollisioner och mäta deras krafter. Modeller har utvecklats för att representera olika typer av objekt. En spline-modell används för att representera suturtråden och ett mass-fjäder system för vävnaden. Valet baserades på deras höga prestanda samt starka anknytning till objektens fysiska egenskaper. Spline-modellen valdes också då dess kontinuitet innebär att den går att evaluera för en godtycklig punkt inom dess domän. Andra objekt, såsom stela kroppar, finns också definierade. Lagrangemultiplikator används för att definiera bivillkor i modellen. Detta tillåter konstruktionen av komplexa modeller. Ett viktigt bivillkor är sutur-bivillkoret som uppstår när tillräcklig kraft från spetsen på den kirurgiska nålen appliceras på vävnaden. Detta bivillkor tillåter att endast en glidande punkt längsmed suturen passerar genom en specifik punkt på vävnaden. Detta resulterar i en virtuell modell för stygn som kan byggas vidare på för användning i kirurgiska simulationer. Det vore intressant med ytterligare undersökningar för att förbättra prestandan, precisionen och simulatorns omfattning.
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Hariz, Belgacem Khader. "Higher-order Embedding Formalism, Noether’s Theorem on Time Scales and Eringen’s Nonlocal Elastica." Electronic Thesis or Diss., Pau, 2022. https://theses.hal.science/tel-03981833.
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En mathématiques, le calcul des variations est un ensemble de méthodes permettant la détermination de solutions à des problèmes d'optimisation des quantités traduites en termes de fonctionnelle. De nombreuses applications existent, notamment dans la recherche de courbes ou de surfaces minimales. Les systèmes dynamiques considérés sont de natures diverses (équations différentielles, intégrales ou stochastiques) et modélisent des problèmes d'origines multiples : aérospatiale, automobile, biologie, économie, médecine, etc. Le théorème de Noether présente un fort intérêt puisqu'il propose une loi de conservation explicite (traduisant souvent une quantité physique comme l'énergie totale ou le moment angulaire en mécanique classique) qui permet de réduire ou d'intégrer l'équation différentielle associée par quadrature. L'objectif de ma thèse contient de nombreux thèmes, dans le premier but nous allons : *) donner le théorème de Noether discret dans le cadre ”time scale” (Le formalisme lagrangien et hamiltonien). Le passage de la nature discrète à la nature continue de la structure la morphologie est d'un intérêt primordial en physique pour comprendre comment la microstructure peut influencer les propriétés macroscopiques du matériau à plus grande échelle. Ce passage peut être modélisé par un système discret appelé 'Hencky's chain' et l'équation du mouvement est donnée par des équations aux différences non linéaires et cette équation ne possède pas de Lagrangien. Le deuxième but nous allons : *) donner les structures lagrangienne, hamiltonienne via le facteur intégrant et trouver la solution analytique de l'équation non locale au sens d'Eringen (nonlocalité différentielle d'Eringen, 1983). Le troisième but nous allons : *) étudier l'existence des formulations variationnelles via le principe de Brezis Ekeland-Nayroles (Gery de Saxce) - application sur la formulation 4D développée par E. Rouhaud pour l'étude des déformations des matériaux *) développer des schémas numériques qui respectent certaines particularités. En particulier, un schéma permettent de mettre en œuvre la théorie 4D développée par E. Rouhaud. *) applications numériques et théoriques sur le problème des déformations des matériauxThe aim of this thesis is to deal with the connection between continuous and discrete versions of a given object. This connection can be studied in two different directions: one going from a continuous setting to a discrete analogue, and in a symmetric way, from a discrete setting to a continuous one. The first procedure is typically used in numerical analysis in order to construct numerical integrators and the second one is typical of continuous modeling for the study of micro-structured materials.In this manuscript, we focus our attention on three distinct problems. In the first part, we propose a general framework precising different ways to derive a discrete version of a differential equation called discrete embedding formalism.More precisely, we exhibit three main discrete associate: the differential, integral or variational structure in both classical and high-order approximations.The second part focuses on the preservation of symmetries for discrete versions of Lagrangian and Hamiltonian systems, i.e., the discrete analogue of Noether's theorem.Finally, the third part applies these results in mechanics, i.e., the problem studied by N. Challamel, Kocsis and Wang called Eringen's nonlocal elastica equation which can beobtained by the continualization method. Precisely, we construct a discrete version of Eringen's nonlocal elastica then we study the difference with Challamel's proposal