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Flandoli, Franco, and Dejun Luo. "Euler-Lagrangian approach to 3D stochastic Euler equations." Journal of Geometric Mechanics 11, no. 2 (2019): 153–65. http://dx.doi.org/10.3934/jgm.2019008.
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Corona, Dario, and Fabio Giannoni. "A New Approach for Euler-Lagrange Orbits on Compact Manifolds with Boundary." Symmetry 12, no. 11 (November 20, 2020): 1917. http://dx.doi.org/10.3390/sym12111917.
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Consider a compact manifold with boundary, homeomorphic to the N-dimensional disk, and a Tonelli Lagrangian function defined on the tangent bundle. In this paper, we study the multiplicity problem for Euler-Lagrange orbits that satisfy the conormal boundary conditions and that lay on the boundary only in their extreme points. In particular, for suitable values of the energy function and under mild hypotheses, if the Tonelli Lagrangian is reversible then the minimal number of Euler-Lagrange orbits with prescribed energy that satisfies the conormal boundary conditions is N. If L is not reversible, then this number is two.
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Gavrilyuk, Sergey, and Keh-Ming Shyue. "Hyperbolic approximation of the BBM equation." Nonlinearity 35, no. 3 (February 18, 2022): 1447–67. http://dx.doi.org/10.1088/1361-6544/ac4c49.
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Abstract It is well known that the Benjamin–Bona–Mahony (BBM) equation can be seen as the Euler–Lagrange equation for a Lagrangian expressed in terms of the solution potential. We approximate the Lagrangian by a two-parameter family of Lagrangians depending on three potentials. The corresponding Euler–Lagrange equations can be then written as a hyperbolic system of conservations laws. The hyperbolic BBM system has two genuinely nonlinear eigenfields and one linear degenerate eigenfield. Moreover, it can be written in terms of Riemann invariants. Such an approach conserves the variational structure of the BBM equation and does not introduce the dissipation into the governing equations as it usually happens for the classical relaxation methods. The state-of-the-art numerical methods for hyperbolic conservation laws such as the Godunov-type methods are used for solving the ‘hyperbolized’ dispersive equations. We find good agreement between the corresponding solutions for the BBM equation and for its hyperbolic counterpart.
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Yang, Yue Neng, Jie Wu, and Wei Zheng. "Modeling Approach of a Near-Space Airship Using Newton-Euler and Lagrangian Formulation." Applied Mechanics and Materials 232 (November 2012): 553–60. http://dx.doi.org/10.4028/www.scientific.net/amm.232.553.
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This paper presents dynamics modeling approach of the near-space airship. First, reference frames and motion variables of the airship are defined, and dynamics model of an airship is derived from the Newton-Euler formulation. Second, the Lagrangian modeling approach in terms of quasi-coordinates is presented, and dynamics model is derived form the Lagrangian formulation, considering the airship and its ambient air as a “rigid body-fluid” system from an energy point of view. The added inertial, gravity, aerostatics, and dissipative forces are incorporated into the dynamics model, and are expressed as parameterized matrices in a common framework. Third, the validity of the proposed dynamics model is verified by comparing simulations with the dynamics model derived from Newton-Euler formulation.
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Tenebe, I. T., A. S. Ogbiye, D. O. Omole, and P. C. Emenike. "Parametric evaluation of the Euler-Lagrangian approach for tracer studies." DESALINATION AND WATER TREATMENT 109 (2018): 344–49. http://dx.doi.org/10.5004/dwt.2018.22213.
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Ciampa, Gennaro, Gianluca Crippa, and Stefano Spirito. "Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit." Archive for Rational Mechanics and Analysis 240, no. 1 (March 1, 2021): 295–326. http://dx.doi.org/10.1007/s00205-021-01612-z.
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AbstractIn this paper we prove the uniform-in-time $$L^p$$ L p convergence in the inviscid limit of a family $$\omega ^\nu $$ ω ν of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution $$\omega $$ ω of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $$\omega ^{\nu }$$ ω ν to $$\omega $$ ω in $$L^p$$ L p . Finally, we show that solutions of the Euler equations with $$L^p$$ L p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.
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Horsin, Thierry, and Otared Kavian. "Lagrangian controllability of inviscid incompressible fluids: a constructive approach." ESAIM: Control, Optimisation and Calculus of Variations 23, no. 3 (May 12, 2017): 1179–200. http://dx.doi.org/10.1051/cocv/2016043.
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We present here a constructive method of Lagrangian approximate controllability for the Euler equation. We emphasize on different options that could be used for numerical recipes: either, in the case of a bi-dimensionnal fluid, the use of formal computations in the framework of explicit Runge approximations of holomorphic functions by rational functions, or an approach based on the study of the range of an operator by showing a density result. For this last insight in view of numerical simulations in progress, we analyze through a simplified problem the observed instabilities.
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ШайдуровВ.В., ШайдуровВ В., and ЧередниченкоО М. ЧередниченкоО.М. "Semi-Lagrangian approximations of the convection operator in symmetric form." Вычислительные технологии, no. 3 (June 21, 2023): 101–16. http://dx.doi.org/10.25743/ict.2023.28.3.007.
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Рассмотрены два полулагранжевых численных метода для одномерного (по пространству) уравнения переноса с оператором в симметричной форме: эйлероволагранжев и лагранжево-эйлеров. Оба метода свободны от ограничения Куранта на соотношение шагов по времени и пространству. Причем во втором методе достигнут второй порядок аппроксимации для гладких решений и продемонстрировано отсутствие численной вязкости для разрывных решений. Purpose. The purpose of the study is the development and comparison of two numerical semi-Lagrangian methods with fulfillment of the conservation law at a discrete level. The approach is applied for the transport equation in the symmetric form, reflecting the law of conservation for the square of the transferred substance. The article presents the Euler – Lagrangian method, built on a rectangular difference grid that uses local values of characteristics to calculate the coefficients of difference equations. Lagrangian – Euler method is built on a spatial non-uniform grid obtained by crossing the characteristic trajectories of the equation with lines in time. Methodology. The integro-interpolation method is applied to derive approximations for the differential operator which allowed obtaining simple formulas connecting values of the grid function at the neighboring layers in time. Numerical calculations of characteristic trajectories are held by the Euler method or the Runge – Kutta method of the second order, depending on the required accuracy. Findings. Numerical methods with the mentioned properties are developed and numerically confirmed, convergence and discrete conservation laws for them are mathematically proved. The first order convergence for both time and space is proved for the Euler – Lagrange method. The second order convergence also in time and space is proved for the Lagrange – Euler method. Originality/value. The Euler – Lagrange and Lagrange – Euler methods for the numerical solution of the convection equation are developed. These methods induce differential conservation law at discrete level. The first and the second order of convergence correspondingly are mathematically proved for them. The Lagrange – Euler method has showed two improved aspects: firstly, it has greater order of convergence than the Euler – Lagrange one and secondly, it allows solving problems with the discontinuous solutions without smoothing them.
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Smojver, I., and D. Ivancevic. "Bird impact at aircraft structure – Damage analysis using Coupled Euler Lagrangian Approach." IOP Conference Series: Materials Science and Engineering 10 (June 1, 2010): 012050. http://dx.doi.org/10.1088/1757-899x/10/1/012050.
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Pedro, Josè C., and P. Sibanda. "An Algorithm for the Strong-Coupling of the Fluid-Structure Interaction Using a Staggered Approach." ISRN Applied Mathematics 2012 (June 20, 2012): 1–14. http://dx.doi.org/10.5402/2012/391974.
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We present a staggered approach for the solution of the piston fluid-structure problem in a time-dependent domain. The one-dimensional fluid flow is modelled using the nonlinear Euler equations. We investigate the time marching fluid-structure interaction and integrate the fluid and structure equations alternately using separate solvers. The Euler equations are written in moving mesh coordinates using the arbitrary Lagrangian-Eulerian (ALE) approach and discretised in space using the finite element method while the structure is integrated in time using an implicit finite difference Newmark-Wilson scheme. The influence of the time lag is studied by comparing two different structural predictors.
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Chen, Xi, Natalia Andronova, Bram Van Leer, Joyce E. Penner, John P. Boyd, Christiane Jablonowski, and Shian-Jiann Lin. "A Control-Volume Model of the Compressible Euler Equations with a Vertical Lagrangian Coordinate." Monthly Weather Review 141, no. 7 (July 1, 2013): 2526–44. http://dx.doi.org/10.1175/mwr-d-12-00129.1.
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Abstract Accurate and stable numerical discretization of the equations for the nonhydrostatic atmosphere is required, for example, to resolve interactions between clouds and aerosols in the atmosphere. Here the authors present a modification of the hydrostatic control-volume approach for solving the nonhydrostatic Euler equations with a Lagrangian vertical coordinate. A scheme with low numerical diffusion is achieved by introducing a low Mach number approximate Riemann solver (LMARS) for atmospheric flows. LMARS is a flexible way to ensure stability for finite-volume numerical schemes in both Eulerian and vertical Lagrangian configurations. This new approach is validated on test cases using a 2D (x–z) configuration.
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EL-NABULSI, AHMAD RAMI. "FRACTIONAL QUANTUM EULER–CAUCHY EQUATION IN THE SCHRÖDINGER PICTURE, COMPLEXIFIED HARMONIC OSCILLATORS AND EMERGENCE OF COMPLEXIFIED LAGRANGIAN AND HAMILTONIAN DYNAMICS." Modern Physics Letters B 23, no. 28 (November 10, 2009): 3369–86. http://dx.doi.org/10.1142/s0217984909021387.
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Fractional quantum Euler–Cauchy equation in the Schrödinger picture is derived from the fractional action-like variational approach recently introduced by the author. Many interesting consequences are revealed and explored, in particular the emergence of complexified harmonic oscillators, complexified Lagrangian and Hamiltonian and complexified fractional action integral.
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Tort, Marine, and Thomas Dubos. "Usual Approximations to the Equations of Atmospheric Motion: A Variational Perspective." Journal of the Atmospheric Sciences 71, no. 7 (June 20, 2014): 2452–66. http://dx.doi.org/10.1175/jas-d-13-0339.1.
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Abstract The usual geophysical approximations are reframed within a variational framework. Starting from the Lagrangian of the fully compressible Euler equations expressed in a general curvilinear coordinates system, Hamilton’s principle of least action yields Euler–Lagrange equations of motion. Instead of directly making approximations in these equations, the approach followed is that of Hamilton’s principle asymptotics; that is, all approximations are performed in the Lagrangian. Using a coordinate system where the geopotential is the third coordinate, diverse approximations are considered. The assumptions and approximations covered are 1) particular shapes of the geopotential; 2) shallowness of the atmosphere, which allows for the approximation of the relative and planetary kinetic energy; 3) small vertical velocities, implying quasi-hydrostatic systems; and 4) pseudoincompressibility, enforced by introducing a Lagangian multiplier. This variational approach greatly facilitates the derivation of the equations and systematically ensures their dynamical consistency. Indeed, the symmetry properties of the approximated Lagrangian imply the conservation of energy, potential vorticity, and momentum. Justification of the equations then relies, as usual, on a proper order-of-magnitude analysis. As an illustrative example, the asymptotic consistency of recently introduced shallow-atmosphere equations with a complete Coriolis force is discussed, suggesting additional corrections to the pressure gradient and gravity.
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Hu, Jianyu, Xiaoli Chen, and Jinqiao Duan. "An Onsager–Machlup approach to the most probable transition pathway for a genetic regulatory network." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 4 (April 2022): 041103. http://dx.doi.org/10.1063/5.0088397.
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We investigate a quantitative network of gene expression dynamics describing the competence development in Bacillus subtilis. First, we introduce an Onsager–Machlup approach to quantify the most probable transition pathway for both excitable and bistable dynamics. Then, we apply a machine learning method to calculate the most probable transition pathway via the Euler–Lagrangian equation. Finally, we analyze how the noise intensity affects the transition phenomena.
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Tuite, Michael, and Siddhartha Sen. "A String Motivated Approach to the Relativistic Point Particle." Modern Physics Letters A 18, no. 33n35 (November 20, 2003): 2483–90. http://dx.doi.org/10.1142/s0217732303012726.
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Using concepts developed in string theory, Cohen, Moore, Nelson and Polchinski calculated the propagator for a relativistic point particle. Following these authors we extend the technique to include the case of closed world lines. The partition function found corresponds to the Feynmann and Schwinger proper time formalism. We also explicitly verify that the partition function is equivalent to the usual path length action partition function. As an example of a sum over closed world lines, we compute the Euler-Heisenberg effective Lagrangian in a novel way.
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LEONTYEV, V. V. "ANALYSIS OF THE STRESS-STRAIN STATE OF A RIVET JOINT USING THE ABAQUS CAE SYSTEM." Fundamental and Applied Problems of Engineering and Technology 6 (2020): 51–57. http://dx.doi.org/10.33979/2073-7408-2020-344-6-51-57.
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The method for analyzing of stress-strain state characteristics of unloaded riveted joints performed with OST 1 11781-74 rivets has been developed using Coupled Euler-Lagrange finite element approach implemented in the CAD / CAE system Abaqus. A comparative analysis of the stress-strain state characteristics of the examined riveted joint’s finite element models using the Lagrangian and the Coupled Lagrangian-Eulerian finite element approaches has been conducted. A three- dimensional finite element model based on the CLE method has been proposed for further study of fatigue strength and durability of the loaded riveted joints.
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Adeniyi, A. A., H. P. Morvan, and K. A. Simmons. "A coupled Euler-Lagrange CFD modelling of droplets-to-film." Aeronautical Journal 121, no. 1246 (October 13, 2017): 1897–918. http://dx.doi.org/10.1017/aer.2017.107.
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ABSTRACTIn this paper, a droplet to film interaction model technique is presented. In the proposed approach, the liquid and gas continua are modelled using an enhanced Volume-of-Fluid (VoF) technique while the droplets are tracked using a Lagrangian framework and are coupled to the Eulerian phases using source terms. The eventual target application is an aeroengine bearing chamber in which oil is found as droplets, shed from the bearings, splashing on impact, separated from wall surfaces at obstacles or simply re-entrained, and as a continuum oil film coating the bearing chamber outer walls which it also cools. In finite volume Computational Fluid Dynamics (CFD) techniques, a prohibitively large number of cells would be required to describe the details of the droplet impact phenomenon. Based on published correlations, the splashing droplets are created and tracked as Lagrangian particles. The flowing film and the gas continua are handled with an enhanced VoF technique.
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Tokmurzin, D., and D. Adair. "Development of Euler-Lagrangian Simulation of a Circulating Fluidized Bed Reactor for Coal Gasification." Eurasian Chemico-Technological Journal, no. 1 (February 20, 2019): 45. http://dx.doi.org/10.18321/ectj789.
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A Computational Particle Fluid Dynamics (CPFD) model based on the Multiphase Particle in Cell (MP-PIC) approach is used for Shubarkol coal gasification simulation in an atmospheric circulating fluidized bed reactor. The simulation is developed on a basis of experimental data available from a biomass gasification process. The cross-section diameter of the reactor riser is 200 mm and the height is 6500 mm. The Euler-Lagrangian simulation is validated using experimental data available in the literature and also compared with an Euler-Euler simulation. The gasification reactions kinetics model is improved, and homogenous and heterogeneous chemistry are described by reduced-chemistry, with the reaction rates solved numerically using volume-averaged chemistry. The simulations reveal gas composition, temperature, and pressure interdependencies along the height of the reactor. The product gas composition compares well with the experiment and the temperature profile demonstrate good consistency with the experiment. The developed model is used for a case study of Shubarkol coal gasification in the circulating fluidized bed reactor.
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Cao, Kien Van, and Anh Pham Huy Ho. "Pendubot trajectory planning and control using virtual holonomic constraint approach." Science and Technology Development Journal 18, no. 3 (August 30, 2015): 76–85. http://dx.doi.org/10.32508/stdj.v18i3.887.
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In this paper, the virtual holonomic constraint approach is initiatively applied for the trajectory planning and control design of a typical double link underactuated mechanical system, called the Pendubot. The goal is to create synchronous oscillations of both links. After modeling the system using Euler-Lagrangian equations of motion, the parameters of the model are identified with optimization techniques. Using this model, the trajectory planning is done via Virtual Holonomic Constraint approach on the basis of re-parameterization of the motion according to geometrical relations among the generalized coordinates of the system.
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Soon, Genevieve, Hui Zhang, Adrian Wing-Keung Law, and Chun Yang. "Modelling of Melting in Packed Media due to Forced Air Convection with Higher Temperature using Euler-Euler-Lagrangian approach." International Journal of Heat and Mass Transfer 194 (September 2022): 123055. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2022.123055.
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GRABOWSKA, KATARZYNA, PAWEŁ URBAŃSKI, and JANUSZ GRABOWSKI. "GEOMETRICAL MECHANICS ON ALGEBROIDS." International Journal of Geometric Methods in Modern Physics 03, no. 03 (May 2006): 559–75. http://dx.doi.org/10.1142/s0219887806001259.
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A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler–Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.
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WANG, Y. J., N. ZHAO, C. W. WANG, and D. H. WANG. "A SECOND-ORDER ADAPTIVE ARBITRARY LAGRANGIAN–EULERIAN METHOD FOR THE COMPRESSIBLE EULER EQUATIONS." Modern Physics Letters B 23, no. 04 (February 10, 2009): 583–601. http://dx.doi.org/10.1142/s0217984909017923.
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Most of finite volume schemes in the Arbitrary Lagrangian–Eulerian (ALE) method are constructed on the staggered mesh, where the momentum is defined at the nodes and the other variables (density, pressure and specific internal energy) are cell-centered. However, this kind of schemes must use a cell-centered remapping algorithm twice which is very inefficient. Furthermore, there is inconsistent treatment of the kinetic and internal energies.1 Recently, a new class of cell-centered Lagrangian scheme for two-dimensional compressible flow problems has been proposed in Ref. 2. The main new feature of the algorithm is the introduction of four pressures on each edge, two for each node on each side of the edge. This scheme is only first-order accurate. In this paper, a second-order cell-centered conservative ENO Lagrangian scheme is constructed by using an ENO-type approach to extend the spatial second-order accuracy. Time discretization is based on a second-order Runge–Kutta scheme. Combining a conservative interpolation (remapping) method3,4 with the second-order Lagrangian scheme, a kind of cell-centered second-order ALE methods can be obtained. Some numerical experiments are made with this method. All results show that our method is effective and have second-order accuracy. At last, in order to further increase the resolution of shock regions, we use an adaptive mesh generation based on the variational principle5 as a rezoned strategy for developing a class of adaptive ALE methods. Numerical experiments are also presented to valid the performance of the proposed method.
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Hemami, Hooshang. "Towards a compact and computer-adapted formulation of the dynamics and stability of multi rigid body systems." Journal of Automatic Control 12, no. 1 (2002): 64–70. http://dx.doi.org/10.2298/jac0201064h.
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The dynamics of rigid bodies coupled by homonymic and non-homonymic constraints are formulated by the Newton - Euler method - employing a compact notation. The compact notation involves the use of two three by three matrices A and ? and the totality of constraint vector C. The Lagrangian and Newton - Euler methods are related for a one - link rigid body in order to introduce the methodology of the paper in full detail. Stability and control of the resulting nonlinear systems are investigated by the use of Lyapunov methods. Digital computer simulations of typical movements are carried out in order to demonstrate feasibility of the formulation and the approach.
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Andrianopoli, Laura, and Lucrezia Ravera. "On the Geometric Approach to the Boundary Problem in Supergravity." Universe 7, no. 12 (November 28, 2021): 463. http://dx.doi.org/10.3390/universe7120463.
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We review the geometric superspace approach to the boundary problem in supergravity, retracing the geometric construction of four-dimensional supergravity Lagrangians in the presence of a non-trivial boundary of spacetime. We first focus on pure N=1 and N=2 theories with negative cosmological constant. Here, the supersymmetry invariance of the action requires the addition of topological (boundary) contributions which generalize at the supersymmetric level the Euler-Gauss-Bonnet term. Moreover, one finds that the boundary values of the super field-strengths are dynamically fixed to constant values, corresponding to the vanishing of the OSp(N|4)-covariant supercurvatures at the boundary. We then consider the case of vanishing cosmological constant where, in the presence of a non-trivial boundary, the inclusion of boundary terms involving additional fields, which behave as auxiliary fields for the bulk theory, allows to restore supersymmetry. In all the cases listed above, the full, supersymmetric Lagrangian can be recast in a MacDowell-Mansouri(-like) form. We then report on the application of the results to specific problems regarding cases where the boundary is located asymptotically, relevant for a holographic analysis.
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Yaguchi, Takaharu. "Lagrangian approach to deriving energy-preserving numerical schemes for the Euler–Lagrange partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis 47, no. 5 (August 1, 2013): 1493–513. http://dx.doi.org/10.1051/m2an/2013080.
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Pakdemirli, M., and A. H. Nayfeh. "Nonlinear Vibrations of a Beam-Spring-Mass System." Journal of Vibration and Acoustics 116, no. 4 (October 1, 1994): 433–39. http://dx.doi.org/10.1115/1.2930446.
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The nonlinear response of a simply supported beam with an attached spring-mass system to a primary resonance is investigated, taking into account the effects of beam midplane stretching and damping. The spring-mass system has also a cubic nonlinearity. The response is found by using two different perturbation approaches. In the first approach, the method of multiple scales is applied directly to the nonlinear partial differential equations and boundary conditions. In the second approach, the Lagrangian is averaged over the fast time scale, and then the equations governing the modulation of the amplitude and phase are obtained as the Euler-Lagrange equations of the averaged Lagrangian. It is shown that the frequency-response and force-response curves depend on the midplane stretching and the parameters of the spring-mass system. The relative importance of these effects depends on the parameters and location of the spring-mass system.
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Kukavica, I., and W. S. Ożański. "Local-in-time existence of a free-surface 3D Euler flow with H 2+δ initial vorticity in a neighborhood of the free boundary." Nonlinearity 36, no. 1 (December 14, 2022): 636–52. http://dx.doi.org/10.1088/1361-6544/aca5e3.
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Abstract We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that u 0 ∈ H 2.5 + δ is such that c u r l u 0 ∈ H 2 + δ in an arbitrarily small neighborhood of the free boundary, and we use the Lagrangian approach to derive an a priori estimate that can be used to prove local-in-time existence and uniqueness of solutions under the Rayleigh–Taylor stability condition.
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QU, KUN, CHANG SHU, and YONG TIAN CHEW. "SIMULATION OF SHOCK-WAVE PROPAGATION WITH FINITE VOLUME LATTICE BOLTZMANN METHOD." International Journal of Modern Physics C 18, no. 04 (April 2007): 447–54. http://dx.doi.org/10.1142/s012918310701067x.
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A new approach was recently proposed to construct equilibrium distribution functions [Formula: see text] of the lattice Boltzmann method for simulation of compressible flows. In this approach, the Maxwellian function is replaced by a simple function which satisfies all needed relations to recover compressible Euler equations. With Lagrangian interpolation polynomials, the simple function is discretized onto a fixed velocity pattern to construct [Formula: see text]. In this paper, the finite volume method is combined with the new lattice Boltzmann models to simulate 1D and 2D shock-wave propagation. The numerical results agree well with available data in the literatures.
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Besse, Nicolas, and Uriel Frisch. "A Constructive Approach to Regularity of Lagrangian Trajectories for Incompressible Euler Flow in a Bounded Domain." Communications in Mathematical Physics 351, no. 2 (January 6, 2017): 689–707. http://dx.doi.org/10.1007/s00220-016-2816-3.
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Cheng, Juan, Chi-Wang Shu, and Qinghong Zeng. "A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations." Communications in Computational Physics 12, no. 5 (November 2012): 1307–28. http://dx.doi.org/10.4208/cicp.150311.090112a.
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AbstractLagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However such schemes are typically not conservative for the total energy. Recently, significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving, accuracy and non-oscillatory performance.
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Leprevost, Antonin, Vincent Faucher, and Maria Adela Puscas. "A Computationally Efficient Dynamic Grid Motion Approach for Arbitrary Lagrange–Euler Simulations." Fluids 8, no. 5 (May 16, 2023): 156. http://dx.doi.org/10.3390/fluids8050156.
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The present article addresses the topic of grid motion computation in Arbitrary Lagrange–Euler (ALE) simulations, where a fluid mesh must be updated to follow the displacements of Lagrangian boundaries. A widespread practice is to deduce the motion for the internal mesh nodes from a parabolic equation, such as the harmonic equation, introducing an extra computational cost to the fluid solver. An alternative strategy is proposed to minimize that cost by changing from the parabolic equation to a hyperbolic equation, implementing an additional time derivative term allowing an explicit solution of the grid motion problem. A fictitious dynamic problem is thus obtained for the grid, with dedicated material parameters to be carefully chosen to enhance the computational efficiency and preserve the mesh quality and the accuracy of the physical problem solution. After reminding the basics of the ALE expression of the Navier–Stokes equations and describing the proposed hyperbolic equation for the grid motion problem, the paper provides the necessary characterization of the influence of the fictitious grid parameters and the analysis of the robustness of the new approach compared to the harmonic reference equation on a significant 2D test case. A 3D test case is finally extensively studied in terms of computational performance to highlight and discuss the benefits of the hyperbolic equation for ALE grid motion.
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Lee, H. P. "Vibrations of a Beam Moving Over Supports with Clearance." Shock and Vibration 1, no. 6 (1994): 549–57. http://dx.doi.org/10.1155/1994/405058.
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The transverse vibration of a beam moving over two supports with clearance is analyzed using Euler beam theory. The equations of motion are formulated based on a Lagrangian approach and the assumed mode method. The supports with clearance are modeled as frictionless supports with piecewise-linear stiffness. A feature of the present formulation is that its complexity does not increase with increased number of supports. Results of numerical simulations are presented for various prescribed motions of the beam. The effect of support clearance on the stability of the beam is investigated.
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Koshelev, K. B., and S. V. Strijhak. "Modelling of pulverized coal injection combustion process in the validation test rig using the Euler-Lagrangian approach." Journal of Physics: Conference Series 1382 (November 2019): 012018. http://dx.doi.org/10.1088/1742-6596/1382/1/012018.
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Musilová, Jana, and Stanislav Hronek. "The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories." Communications in Mathematics 24, no. 2 (December 1, 2016): 173–93. http://dx.doi.org/10.1515/cm-2016-0012.
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Abstract As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals.It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-Lagrange equations of an appropriately formulated variational principle. There are several well established geometrical theories providing a general description of variational problems of different kinds. One of the most universal and comprehensive is the calculus of variations on fibred manifolds and their jet prolongations. Among others, it includes a complete general solution of the so-called strong inverse variational problem allowing one not only to decide whether a concrete equation of motion can be obtained from a variational principle, but also to construct a corresponding variational functional. Moreover, conservation laws can be derived from symmetries of the Lagrangian defining this functional, or directly from symmetries of the equations.In this paper we apply the variational theory on jet bundles to tackle some fundamental problems of physics, namely the questions on existence of a Lagrangian and the problem of conservation laws. The aim is to demonstrate that the methods are universal, and easily applicable to distinct physical disciplines: from classical mechanics, through special relativity, waves, classical electrodynamics, to quantum mechanics.
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Nikonov, Valeriy. "A Semi-Lagrangian Godunov-Type Method without Numerical Viscosity for Shocks." Fluids 7, no. 1 (December 30, 2021): 16. http://dx.doi.org/10.3390/fluids7010016.
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One of the most important and complex effects in compressible fluid flow simulation is a shock-capturing mechanism. Numerous high-resolution Euler-type methods have been proposed to resolve smooth flow scales accurately and to capture the discontinuities simultaneously. One of the disadvantages of these methods is a numerical viscosity for shocks. In the shock, the flow parameters change abruptly at a distance equal to the mean free path of a gas molecule, which is much smaller than the cell size of the computational grid. Due to the numerical viscosity, the aforementioned Euler-type methods stretch the parameter change in the shock over few grid cells. We introduce a semi-Lagrangian Godunov-type method without numerical viscosity for shocks. Another well-known approach is a method of characteristics that has no numerical viscosity and uses the Riemann invariants or solvers for water hammer phenomenon modeling, but in its formulation the convective terms are typically neglected. We use a similar approach to solve the one-dimensional adiabatic gas dynamics equations, but we split the equations into parts describing convection and acoustic processes separately, with corresponding different time steps. When we are looking for the solution to the one-dimensional problem of the scalar hyperbolic conservation law by the proposed method, we additionally use the iterative Godunov exact solver, because the Riemann invariants are non-conserved for moderate and strong shocks in an ideal gas. The proposed method belongs to a group of particle-in-cell (PIC) methods; to the best of the author’s knowledge, there are no similar PIC numerical schemes using the Riemann invariants or the iterative Godunov exact solver. This article describes the application of the aforementioned method for the inviscid Burgers’ equation, adiabatic gas dynamics equations, and the one-dimensional scalar hyperbolic conservation law. The numerical analysis results for several test cases (e.g., the standard shock-tube problem of Sod, the Riemann problem of Lax, the double expansion wave problem, the Shu–Osher shock-tube problem) are compared with the exact solution and Harten’s data. In the shock for the proposed method, the flow properties change instantaneously (with an accuracy dependent on the grid cell size). The iterative Godunov exact solver determines the accuracy of the proposed method for flow discontinuities. In calculations, we use the iteration termination condition less than 10−5 to find the pressure difference between the current and previous iterations.
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SOWARD, A. M., and P. H. ROBERTS. "The hybrid Euler–Lagrange procedure using an extension of Moffatt's method." Journal of Fluid Mechanics 661 (August 2, 2010): 45–72. http://dx.doi.org/10.1017/s0022112010002867.
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The hybrid Euler–Lagrange (HEL) description of fluid mechanics, pioneered largely by Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646), has had to face the fact, in common with all Lagrangian descriptions of fluid motion, that the variables used do not describe conditions at the coordinate x, upon which they depend, but conditions elsewhere at some displaced position xL(x, t) = x + ξ(x, t), generally dependent on time t. To address this issue, we employ ‘Lie dragging’ techniques of general tensor calculus to extend a method introduced by Moffatt (J. Fluid Mech., vol. 166, 1986, pp. 359–378) in the fluid dynamic context, whereby the point x is dragged to xL(x, t) by a ‘fictitious steady flow’ η*(x, t) in a unit of ‘fictitious time’. Whereas ξ(x, t) is a Lagrangian concept intimately linked to the location xL(x, t), the ‘dragging velocity’ η*(x, t) has an essentially Eulerian character, because it describes the fictitious velocity at x itself. For the case of constant-density fluids, we show, using solenoidal η*(x, t) instead of solenoidal ξ(x, t), how the HEL theory can be cast into Eulerian form. A useful aspect of this Eulerian development is that the mean flow itself remains solenoidal, a feature that traditional HEL theories lack. Our method realizes the objective sought by Holm (Physica D, vol. 170, 2002, pp. 253–286) in his derivation of the Navier–Stokes–α equation, which is the basis of one of the methods currently employed to represent the sub-grid scales in large-eddy simulations. His derivation, based on expansion to second order in ξ, contained an error which, when corrected, implied a violation of Kelvin's theorem on the constancy of circulation in inviscid incompressible fluid. We show that this is rectified when the expansion is in η* rather than ξ, Kelvin's theorem then being satisfied to all orders for which the expansion converges. We discuss the implications of our approach using η* for the Navier–Stokes–α theory.
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Wong, May, William C. Skamarock, Peter H. Lauritzen, Joseph B. Klemp, and Roland B. Stull. "A Compressible Nonhydrostatic Cell-Integrated Semi-Lagrangian Semi-Implicit Solver (CSLAM-NH) with Consistent and Conservative Transport." Monthly Weather Review 142, no. 4 (March 27, 2014): 1669–87. http://dx.doi.org/10.1175/mwr-d-13-00210.1.
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Abstract A cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic solver for the fully compressible moist Euler equations in two-dimensional Cartesian (x–z) geometry is presented. The semi-implicit CISL solver uses the inherently conservative semi-Lagrangian multitracer transport scheme (CSLAM) and a new flux-form semi-implicit formulation of the continuity equation that ensures numerically consistent transport. The flux-form semi-implicit formulation is based on a recent successful approach in a shallow-water equations (SWE) solver (CSLAM-SW). With the new approach, the CISL semi-implicit nonhydrostatic solver (CSLAM-NH) is able to ensure conservative and consistent transport by avoiding the need for a time-independent mean reference state. Like its SWE counterpart, the nonhydrostatic solver presented here is designed to be similar to typical semi-Lagrangian semi-implicit schemes, such that only a single linear Helmholtz equation solution and a single call to CSLAM are required per time step. To demonstrate its stability and accuracy, the solver is applied to a set of three idealized test cases: a density current (dry), a gravity wave (dry), and a squall line (moist). A fourth test case shows that shape preservation of passive tracers is ensured by coupling the semi-implicit CISL formulation with existing shape-preserving filters. Results show that CSLAM-NH solutions compare well with other existing solvers for the three test cases, and that it is shape preserving.
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Yıldız, Hüseyin Alpaslan, and Leyla Gören-Sümer. "Stabilization of a class of underactuated Euler Lagrange system using an approximate model." Transactions of the Institute of Measurement and Control 44, no. 8 (December 7, 2021): 1569–78. http://dx.doi.org/10.1177/01423312211058556.
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The energy shaping method, Controlled Lagrangian, is a well-known approach to stabilize the underactuated Euler Lagrange (EL) systems. In this approach, to construct a control rule, some nonlinear and nonhomogeneous partial differential equations (PDEs), which are called matching conditions, must be solved. In this paper, a method is proposed to obtain an approximate solution of these matching conditions for a class of underactuated EL systems. To develop this method, the potential energy matching condition is transformed to a set of linear PDEs using an approximation of inertia matrices. Hence, the assignable potential energy function and the controlled inertia matrix both are constructed as a common solution of these PDEs. Subsequently, the gyroscopic and dissipative forces are determined as the solution for kinetic energy matching condition. Conclusively, the control rule is constructed by adding energy shaping rule and additional dissipation injection to provide asymptotic stability. The stability analysis of the closed-loop system which used the control rule derived with the proposed method is also provided. To demonstrate the success of the proposed method, the stability problem of the inverted pendulum on a cart is considered.
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Cheng, Lei, Guo Jie Huang, Jian Wei Wang, Wei Xiao, and Shui Sheng Xie. "Numerical Simulation of Extrusion Process to Produce Complex Aluminum Profiles Using the ALE Approach." Advanced Materials Research 1004-1005 (August 2014): 1260–64. http://dx.doi.org/10.4028/www.scientific.net/amr.1004-1005.1260.
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Although still having certain limitations, the numerical simulation technology has been increasingly applied to aid in optimizing the aluminum extrusion process and die design. In the present research, numerical simulations of the profiles extrusion process were performed, using the Finite Volume Method (FVM) and Finite Element Method (FEM) to make use of the individual merits of the Euler approach and Lagrange approach, respectively. The application of the simulation technology to produce large, complex profiles has, however, been quite limited. In order to solve the limited, numerical simulation of aluminum profiles with large and complicated cross-section in extrusion process was achieved using Arbitrary Lagrangian-Eulerian (ALE) approach, and non-uniform velocities at the die exit, leading to extrudate distortions, were predicted. Extrusion experiments proved that the die with the optimized design could circumvent the distortion problem. The numerical simulation technology can indeed be effectively used to reduce the number of die trials and offer the potential to realize zero die trial.
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Parker, Michael C., and Chris Jeynes. "A Relativistic Entropic Hamiltonian–Lagrangian Approach to the Entropy Production of Spiral Galaxies in Hyperbolic Spacetime." Universe 7, no. 9 (August 31, 2021): 325. http://dx.doi.org/10.3390/universe7090325.
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Double-spiral galaxies are common in the Universe. It is known that the logarithmic double spiral is a Maximum Entropy geometry in hyperbolic (flat) spacetime that well represents an idealised spiral galaxy, with its central supermassive black hole (SMBH) entropy accounting for key galactic structural features including the stability and the double-armed geometry. Over time the central black hole must accrete mass, with the overall galactic entropy increasing: the galaxy is not at equilibrium. From the associated entropic Euler–Lagrange Equation (enabling the application of Noether’s theorem) we develop analytic expressions for the galactic entropy production of an idealised spiral galaxy showing that it is a conserved quantity, and we also derive an appropriate expression for its relativistic entropic Hamiltonian. We generalise Onsager’s celebrated expression for entropy production and demonstrate that galactic entropy production (entropy production corresponds to the intrinsic dissipation characteristics) is composed of two parts, one many orders of magnitude larger than the other: the smaller is comparable to the Hawking radiation of the central SMBH, while the other is comparable to the high entropy processes occurring within the accretion disks of real SMBHs. We conclude that galaxies cannot be isolated, since even idealised spiral galaxies intrinsically have a non-zero entropy production.
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Kopiev, V. F., and S. A. Chernyshev. "LAGRANGIAN FORMALISM IN PROBLEMS OF SMALL OSCILLATIONS OF VORTEX FLOWS AND ITS CONNECTION WITH THE VARIATIONAL PRINCIPLE FOR IDEAL INCOMPRESSIBLE HYDRODYNAMICS OF VORTEX LINES." XXII workshop of the Council of nonlinear dynamics of the Russian Academy of Sciences 47, no. 1 (April 30, 2019): 74–77. http://dx.doi.org/10.29006/1564-2291.jor-2019.47(1).21.
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The paper discusses the description of vortex flows of an ideal incompressible fluid based on the formalism of Lagrangian mechanics. Using the displacement field of liquid particles as a generalized coordinate, we write out the Lagrangian describing the dynamics of small perturbations (Kopiev, Chernyshev, 2018). The corresponding Lagrange equations are the equation for the displacement field (Drazim, Reid, 1981): This equation is equivalent to the Helmholtz equation for vorticity perturbations. The displacement field is defined as the difference in the positions of liquid particles on trajectories in disturbed and undisturbed flows. Although this definition is given in terms of Lagrangian variables associated with liquid particles, the displacement field itself is an Euler variable, expressed through velocity and vorticity perturbations. An example of using Lagrangian to solve the problem of conservation of the quadrupole moment of a vortex flow is considered. Using the Noether theorem, conditions on a stationary flow are obtained, under which the quadrupole moment of small perturbations of this flow is an integral of motion (Kopiev, Chernyshev, 2018). It is shown that these conditions are satisfied for the jet flows uniform along the longitudinal coordinate. The result obtained is important in aeroacoustics due to the fact that the quadrupole moment of the vortex flow represents the main term of the decomposition of a compact acoustic source in Machnumber (Lighthill, 1952; Crow, 1970; Kopiev, Chernyshev, 1995). The generalization of these results to the nonlinear case is considered. The Lagrangian is obtained for an arbitrary nonlinear displacement field: nowhere Gis Green’s function of the Laplace equation. The corresponding Lagrange equations coincide with the differential equations describing the nonlinear dynamics of the displacement field (Drazin, Reid, 1981). Expansion of the Lagrangian in small perturbations to quadratic terms gives the Lagrangian of the linear system. The question of the relationship of the proposed approach to the description of the dynamics of an incompressible fluid and known approaches based on the formalism of Lagrangian mechanics with the coordinates of liquid particles as generalized coordinates (Chapman, 1978; Goncharov, Pavlov, 2008; Kuznetsov, Ruban, 1998) is considered. It is shown that the transformation of the Lagrangian obtained in (Kuznetsov, Ruban, 1998) to the Lagrangian can be carried out by transforming Lagrangian variables (coordinates of liquid particles) to Eulerian variables (displacement field). This study was supported by the Russian Science Foundation, project No. 17-11-01271.
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Zisser, Eddie, Avishai Sintov, Amir Shapiro, and Raziel Riemer. "Position Control of a Pneumatic Actuator Under Varying External Force." Mechanics and Mechanical Engineering 22, no. 4 (September 2, 2020): 1157–74. http://dx.doi.org/10.2478/mme-2018-0091.
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AbstractIn this paper a high accuracy position control strategy for a pneumatic actuation system subjected to a varying external force is proposed. A novel approach for the mathematical modeling of the pneumatic actuator, based on energy methods, is presented. The Lagrangian is derived from combining the kinetic and potential energies, leading to formulation of the Euler-Lagrange equation of motion. The nonlinear backstepping method is applied to derive the control law, and the derivative of the potential energy is used as the controlled parameter. Experimental results show that tracking a sine wave of 0.1m magnitude produces a maximum error of ±0.008m while the actuator is subjected to a time varying external force with a magnitude ranging from 570N to 1150N.
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Kusaka, Takashi, and Takayuki Tanaka. "Partial Lagrangian for Efficient Extension and Reconstruction of Multi-DoF Systems and Efficient Analysis Using Automatic Differentiation." Robotics 11, no. 6 (December 9, 2022): 149. http://dx.doi.org/10.3390/robotics11060149.
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In the fields of control engineering and robotics, either the Lagrange or Newton–Euler method is generally used to analyze and design systems using equations of motion. Although the Lagrange method can obtain analytical solutions, it is difficult to handle in multi-degree-of-freedom systems because the computational complexity increases explosively as the number of degrees of freedom increases. Conversely, the Newton–Euler method requires less computation even for multi-degree-of-freedom systems, but it cannot obtain an analytical solution. Therefore, we propose a partial Lagrange method that can handle the Lagrange equation efficiently even for multi-degree-of-freedom systems by using a divide-and-conquer approach. The proposed method can easily handle system extensions and system reconstructions, such as changes to intermediate links, for multi-degree-of-freedom serial link manipulators. In addition, the proposed method facilitates the derivation of the equations of motion-by-hand calculations, and when combined with an analysis algorithm using automatic differentiation, it can easily realize motion analysis and control the simulation of multi-degree-of-freedom models. Using multiple pendulums as examples, we confirm the effectiveness of system expansion and system reconstruction with the partial Lagrangians. The derivation of their equations of motion and the results of motion analysis by simulation and motion control experiments are presented. The system extensions and reconstructions proposed herein can be used simultaneously with conventional analytical methods, allowing manual derivations of equations of motion and numerical computer simulations to be performed more efficiently.
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Xie, Fangfang, Dingyi Pan, Yao Zheng, and Jianfeng Zou. "Smoothed profile method and its applications in VIV." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 7 (July 3, 2017): 1623–35. http://dx.doi.org/10.1108/hff-12-2016-0503.
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Purpose The purpose of this paper is to propose a partitioned approach by coupling the smoothed profile method (SPM) and the Euler tension beam model in simulating a vortex-induced vibration of both rigid and flexible cylinders at various reduced velocities. Design/methodology/approach For the fluid part, SPM in the framework of the spectral element method is adopted to simulate the flow. The advantage of SPM lies in modelling multiple complex shapes as it uses a fixed computational mesh without conformation to the geometry of the particles. For the structure part, an elastic-mounted rigid cylinder is considered in two-dimensional (2D) simulations, while a flexible cylinder with a Euler tension beam model is used in three-dimensional simulations. Findings Firstly, in the flow past a freely vibrating cylinder, the maximum vibration responses of the cylinder are about 0.73D and 0.1D in the y and x directions, respectively, which occur at the point Ur = 5.75 and are much higher than Ur = 5 in 2D simulations. It is found that the numerical results from the SPM solver are very consistent with those from the NEKTAR-Arbitrary Lagrangian Eulerian method (NEKTAR-ALE) solver or the NEKTAR-Fourier solver. Furthermore, the flow past the tandem cylinders is also investigated, where the upstream cylinder is static while the downstream one is free to vibrate. Specifically, the beating behaviour is captured from the vibration response of the freely vibrating cylinder under the reduced velocity of Ur = 6 with a gap distance of L = 3.5D. Originality/value The originality of the paper lies in coupling the SEM with the Euler beam model in simulating the vortex induced vibration (VIV) of flexible cylinders.
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Tar, J. K., I. J. Rudas, and J. F. Bitó. "Group theoretical approach in using canonical transformations and symplectic geometry in the control of approximately modelled mechanical systems interacting with an unmodelled environment." Robotica 15, no. 2 (March 1997): 163–79. http://dx.doi.org/10.1017/s0263574797000192.
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In spite of its simpler structure than that of the Euler-Lagrange equations-based model, the Hamiltonian formulation of Classical Mechanics (CM) gained only limited application in the Computed Torque Control (CTC) of the rather conventional robots. A possible reason for this situation may be, that while the independent variables of the Lagrangian model are directly measurable by common industrial sensors and encoders, the Hamiltonian canonical coordinates are not measurable and can also not be computed in the lack of detailed information on the dynamics of the system under control. As a consequence, transparent and lucid mathematical methods bound to the Hamiltonian model utilizing the special properties of such concepts as Canonical Transformations, Symplectic Geometry, Symplectic Group, Symplectizing Algorithm, etc. remain out of the reach of Dynamic Control approaches based on the Lagrangian model. In this paper the preliminary results of certain recent investigations aiming at the introduction of these methods in dynamic control are summarized and illustrated by simulation results. The proposed application of the Hamiltonian model makes it possible to achieve a rigorous deductive analytical treatment up to a well defined point exactly valid for a quite wide range of many different mechanical systems. From this point on it reveals such an ample assortment of possible non-deductive, intuitive developments and approaches even within the investigations aiming at a particular paradigm that publication of these very preliminary and early results seems to have definite reason, too.
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Spandan, Vamsi, Rodolfo Ostilla-Mónico, Roberto Verzicco, and Detlef Lohse. "Drag reduction in numerical two-phase Taylor–Couette turbulence using an Euler–Lagrange approach." Journal of Fluid Mechanics 798 (June 6, 2016): 411–35. http://dx.doi.org/10.1017/jfm.2016.316.
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Two-phase turbulent Taylor–Couette (TC) flow is simulated using an Euler–Lagrange approach to study the effects of a secondary phase dispersed into a turbulent carrier phase (here bubbles dispersed into water). The dynamics of the carrier phase is computed using direct numerical simulations (DNS) in an Eulerian framework, while the bubbles are tracked in a Lagrangian manner by modelling the effective drag, lift, added mass and buoyancy force acting on them. Two-way coupling is implemented between the dispersed phase and the carrier phase which allows for momentum exchange among both phases and to study the effect of the dispersed phase on the carrier phase dynamics. The radius ratio of the TC setup is fixed to ${\it\eta}=0.833$, and a maximum inner cylinder Reynolds number of $Re_{i}=8000$ is reached. We vary the Froude number ($Fr$), which is the ratio of the centripetal to the gravitational acceleration of the dispersed phase and study its effect on the net torque required to drive the TC system. For the two-phase TC system, we observe drag reduction, i.e. the torque required to drive the inner cylinder is lower compared with that of the single-phase system. The net drag reduction decreases with increasing Reynolds number $Re_{i}$, which is consistent with previous experimental findings (Murai et al., J. Phys.: Conf. Ser., vol. 14, 2005, pp. 143–156; Phys. Fluids, vol. 20(3), 2008, 034101). The drag reduction is strongly related to the Froude number: for fixed Reynolds number we observe higher drag reduction when $Fr<1$ than for with $Fr>1$. This buoyancy effect is more prominent in low $Re_{i}$ systems and decreases with increasing Reynolds number $Re_{i}$. We trace the drag reduction back to the weakening of the angular momentum carrying Taylor rolls by the rising bubbles. We also investigate how the motion of the dispersed phase depends on $Re_{i}$ and $Fr$, by studying the individual trajectories and mean dispersion of bubbles in the radial and axial directions. Indeed, the less buoyant bubbles (large $Fr$) tend to get trapped by the Taylor rolls, while the more buoyant bubbles (small $Fr$) rise through and weaken them.
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Mauland, R., and Ø. Elgarøy. "Quantized vortices in superfluid dark matter." Journal of Cosmology and Astroparticle Physics 2022, no. 01 (January 1, 2022): 044. http://dx.doi.org/10.1088/1475-7516/2022/01/044.
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Abstract In 2015 Berezhiani & Khoury proposed a Superfluid Dark Matter (SFDM) model where dark matter condenses and forms a superfluid on galactic scales. In the superfluid state phonons interact with baryons, resulting in a behavior similar to that of Modified Newtonian Dynamics (MOND). If one assumes that the DM condensate rotates along with the galaxy, a grid of vortices should form throughout the superfluid component if the rotation is fast enough. We aim to investigate the size and impact of the vortices on surrounding baryons, and to further investigate the parameter space of the model. We also look for a possible vortex solution of the Lagrangian presented for the SFDM theory. We first take a simple approach and investigate vortex properties in a constant density DM halo, applying knowledge from condensed matter physics. We then use the zero-temperature condensate density profile as a template to vary the DM particle mass and the energy scale, Λ, of the SFDM model. Further, we attempt to find a vortex solution of the theory by extracting the Euler-Lagrange equation with respect to the modulus of the condensate wavefunction from the full relativistic SFDM Lagrangian. For the constant density approach we find that the vortices are on millimeter scale, and separated by distances ∼0.002 AU. The parameter space of the model is found to be substantial and a reduction in the DM particle mass leads to larger vortices with a higher energy. However, none of the parameter combinations explored here give both realistic values of Λ and vortices energetic enough to have an observational impact on the galaxy as a whole. The vortex equation extracted from the Lagrangian of the model is unstable, and no solution exhibiting the standard properties of a vortex solution is found.
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Holz, Simon, Samuel Braun, Geoffroy Chaussonnet, Rainer Koch, and Hans-Jörg Bauer. "Close Nozzle Spray Characteristics of a Prefilming Airblast Atomizer." Energies 12, no. 14 (July 23, 2019): 2835. http://dx.doi.org/10.3390/en12142835.
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The formation of pollutant emissions in jet engines is closely related to the fuel distribution inside the combustor. Hence, the characteristics of the spray formed during primary breakup are of major importance for an accurate prediction of the pollutant emissions. Currently, an Euler–Lagrangian approach for droplet transport in combination with combustion and pollutant formation models is used to predict the pollutant emissions. The missing element for predicting these emissions more accurately is well defined starting conditions for the liquid fuel droplets as they emerge from the fuel nozzle. Recently, it was demonstrated that the primary breakup can be predicted from first principles by the Lagrangian, mesh-free, Smoothed Particle Hydrodynamics (SPH) method. In the present work, 2D Direct Numerical Simulations (DNS) of a planar prefilming airblast atomizer using the SPH method are presented, which capture most of the breakup phenomena known from experiments. Strong links between the ligament breakup and the resulting spray in terms of droplet size, trajectory and velocity are demonstrated. The SPH predictions at elevated pressure conditions resemble quite well the effects observed in experiments. Significant interdependencies between droplet diameter, position and velocity are observed. This encourages to employ such multidimensional interdependence relations as a base for the development of primary atomization models.
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Farooq, M. "Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations." Symmetry 11, no. 2 (February 1, 2019): 162. http://dx.doi.org/10.3390/sym11020162.
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Coupled systems of Lane–Emden equations are of considerable interest as they model several physical phenomena, for instance population evolution, pattern formation, and chemical reactions. Assuming a complex variational structure, we classify the generalized system of Lane–Emden type equations in relation to Noether-like operators and associated first integrals. Various forms of functions appearing in the considered system are taken, and it is observed that the Noether-like operators form an Abelian algebra for the corresponding Euler–Lagrange-type systems. Interestingly, we find that in many cases, the Noether-like operators satisfy the classical Noether symmetry condition and become the Noether symmetries. Moreover, we observe that the classical Noetherian integrals and the first integrals we determine using the complex Lagrangian approach turn out to be the same for the underlying system of Lane–Emden equations.
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ARROYO, JOSU, ÓSCAR J. GARAY, and JOSE MENCÍA. "QUADRATIC CURVATURE ENERGIES IN THE 2-SPHERE." Bulletin of the Australian Mathematical Society 81, no. 3 (March 2, 2010): 496–506. http://dx.doi.org/10.1017/s0004972709001142.
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AbstractThe classical variational analysis of curvature energy functionals, acting on spaces of curves of a Riemannian manifold, is extremely complicated, and the procedure usually can not be completely developed under such a degree of generality. Sometimes this difficulty may be overcome by focusing on specific actions in real space forms. In this note, we restrict ourselves to quadratic Lagrangian energies acting on the space of closed curves of the 2-sphere. We solve the Euler–Lagrange equation and show that there exists a two-parameter family of closed critical curves. We also discuss the stability of the circular critical points. Since, even for this class of energies, the complete variational analysis is quite involved, we use instead a numerical approach to provide a useful method of visualization of relevant aspects concerning uniqueness, stability and explicit representation of the closed critical curves.