Добірка наукової літератури з теми "Explicit symplectic integrator"

In recent works by Wu and Wang a class of explicit symplectic integrators in curved spacetimes was presented. Different splitting forms or appropriate choices of time-transformed Hamiltonians are determined based on specific Hamiltonian problems. As its application, we constructed a suitable explicit symplectic integrator for surveying the dynamics of test particles in a magnetized Reissner–Nordström spacetime. In addition to computational efficiency, the scheme exhibits good stability and high precision for long-term integration. From the global phase-space structure of Poincaré sections, the extent of chaos can be strengthened when energy E, magnetic parameter B, or the charge q become larger. On the contrary, the occurrence of chaoticity is weakened with an increase of electric parameter Q and angular momentum L. The conclusion can also be supported by fast Lyapunov indicators.

The dynamic equation of a mass point in the circular restricted three-body problem is governed by Coriolis and centrifugal force, in addition to a co-rotating potential relative to the frame. In this paper, we provide an explicit, symmetric integrator for this problem. Such an integrator is more efficient than the symplectic Euler method and the Gauss Runge–Kutta method as regards this problem. In addition, we proved the integrator is symplectic by the discrete Hamilton’s principle. Several groups of numerical experiments demonstrated the precision and high efficiency of the integrator in the examples of the quadratic potential and the bounded orbits in the circular restricted three-body problem.

A brane-world metric with an external magnetic field is a modified theory of gravity. It is suitable for the description of compact sources on the brane such as stars and black holes. We design a class of explicit symplectic integrators for this spacetime and use one of the integrators to investigate how variations of the parameters affect the motion of test particles. When the magnetic field does not vanish, the integrability of the system is destroyed. Thus, the onset of chaos can be allowed under some circumstances. Chaos easily occurs when the electromagnetic parameter becomes large enough. Dark matter acts as a gravitational force, so that chaotic motion can become more obvious as dark matter increases. The gravity of the black hole is weakened with an increasing positive cosmological parameter; therefore, the extent of chaos can be also strengthened. The proposed symplectic integrator is applied to a ray-tracing method and the study of such chaotic dynamics will be a possible reference for future studies of brane-world black hole shadows with chaotic patterns of self-similar fractal structures based on the Event Horizon Telescope data for M87* and Sagittarius A*.

Abstract Following the latest work of Wu et al., we construct time-transformed explicit symplectic schemes for a Hamiltonian system on the description of charged particles moving around a deformed Schwarzschild black hole with an external magnetic field. Numerical tests show that such schemes have good performance in stabilizing energy errors without secular drift. Meantime, tangent vectors are solved from the variational equations of the system with the aid of an explicit symplectic integrator. The obtained tangent vectors are used to calculate several chaos indicators, including Lyapunov characteristic exponents, fast Lyapunov indicators, a smaller alignment index, and a generalized alignment index. It is found that the smaller alignment index and generalized alignment index are the fastest indicators for distinguishing between regular and chaotic cases. The smaller alignment index is applied to explore the effects of the parameters on the dynamical transition from order to chaos. When the positive deformation factor and angular momentum decrease, or when the energy, positive magnetic parameter, and the magnitude of the negative deformation parameter increase, chaos easily occurs for the appropriate choices of initial conditions and the other parameters.

Abstract Accurate N-body simulations of multiple systems such as binaries and triples are essential for understanding the formation and evolution of interacting binaries and binary mergers, including gravitational wave sources, blue stragglers, and X-ray binaries. The logarithmic time-transformed explicit symplectic integrator (LogH), also known as algorithmic regularization, is a state-of-the-art method for this purpose. However, we show that this method is accurate for isolated Kepler orbits because of its ability to trace Keplerian trajectories, but much less accurate for hierarchical triple systems. The method can lead to an unphysical secular evolution of inner eccentricity in Kozal–Lidov triples, despite a small energy error. We demonstrate that hybrid methods, which apply LogH to the inner binary and alternative methods to the outer bodies, are significantly more effective, though not symplectic. Additionally, we introduce a more efficient hybrid method, BlogH, which eliminates the need for time synchronization and is time symmetric. The method is implemented in the few-body code SDAR. We explore suitable criteria for switching between the LogH and BlogH methods for general triple systems. These hybrid methods have the potential to enhance the integration performance of hierarchical triples.

ABSTRACT An accurate and efficient method dealing with the few-body dynamics is important for simulating collisional N-body systems like star clusters and to follow the formation and evolution of compact binaries. We describe such a method which combines the time-transformed explicit symplectic integrator and the slow-down method. The former conserves the Hamiltonian and the angular momentum for a long-term evolution, while the latter significantly reduces the computational cost for a weakly perturbed binary. In this work, the Hamilton equations of this algorithm are analysed in detail. We mathematically and numerically show that it can correctly reproduce the secular evolution like the orbit averaged method and also well conserve the angular momentum. For a weakly perturbed binary, the method is possible to provide a few orders of magnitude faster performance than the classical algorithm. A publicly available code written in the c++ language, sdar, is available on github. It can be used either as a standalone tool or a library to be plugged in other N-body codes. The high precision of the floating point to 62 digits is also supported.

AbstractWe consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge–Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost.

I describe an approach to data assimilation making use of an explicit map that defines a coordinate system on the slow manifold in the semi-geostrophic scaling in Lagrangian coordinates, and apply the approach to a simple toy system that has previously been proposed as a low-dimensional model for the semi-geostrophic scaling. The method can be extended to Lagrangian particle methods such as Hamiltonian particle–mesh and smooth-particle hydrodynamics applied to the rotating shallow-water equations, and many of the properties will remain for more general Eulerian methods. Making use of Hamiltonian normal-form theory, it has previously been shown that, if initial conditions for the system are chosen as image points of the map, then the fast components of the system have exponentially small magnitude for exponentially long times as ϵ →0, and this property is preserved if one uses a symplectic integrator for the numerical time stepping. The map may then be used to parametrize initial conditions near the slow manifold, allowing data assimilation to be performed without introducing any fast degrees of motion (more generally, the precise amount of fast motion can be selected).

We mainly focus on the effects of small changes of parameters on the dynamics of charged particles around Kerr black holes surrounded by an external magnetic field, which can be considered as a tidal environment. The radial motions of charged particles on the equatorial plane are studied via an effective potential. It is found that the particle energies at the local maxima values of the effective potentials increase with an increase in the black hole spin and the particle angular momenta, but decrease with an increase of one of the inductive charge parameter and magnetic field parameter. The radii of stable circular orbits on the equatorial plane also increase, whereas those of the innermost stable circular orbits decrease. On the other hand, the effects of small variations of the parameters on the orbital regular and chaotic dynamics of charged particles on the non-equatorial plane are traced by means of a time-transformed explicit symplectic integrator, Poincaré sections and fast Lyapunov indicators. It is shown that the dynamics sensitivity depends on small variations in the inductive charge parameter, magnetic field parameter, energy, and angular momentum. Chaos occurs easily as each of the inductive charge parameter, magnetic field parameter, and energy increases but is weakened as the angular momentum increases. When the dragging effects of the spacetime increase, the chaotic properties are not always weakened under some circumstances.

Les pneumatiques sont complexes à simuler car les matériaux y sont hétérogènes, incompressibles et non-linéaires. De plus la géométrie descend jusqu’à l’échelle millimétrique pour les sculptures de la bande de roulement, ce qui requiert un maillage fin. Le modèle éléments finis présente donc un grand nombre de degrés de liberté, reliés par des équations non-linéaires. En dynamique, la simulation est d’autant plus compliquée avec des chocs. Néanmoins elle est cruciale dans le processus de conception pneumatique, où elle apporte une meilleur compréhension de la physique ceci sans tests réels. Les schémas explicites rendent possible les simulations de chocs, car ils résolvent facilement les non-linéarités avec un coup calcul bas. Associés à une formulation de contact précise , ils forment des schémas robustes, précis et efficaces pour la dynamique non-linéaire avec impacts. Ce travail vise à choisir et un tel schéma, et l’améliorer pour la simulation de chocs sur pneumatiques.La première partie est un benchmark identifiant le schéma CD-Lagrange. L’intégration temporelle est réalisée par le schéma de la différence centrée, et le contact imposé par multiplicateurs de Lagrange sur la vitesse. Deux possibilités d’amélioration sont identifiées. La première est d’atteindre un impact conservatif, seul instant où le schéma n’est pas symplectique. La seconde amélioration est d’étendre la formulation au contact déformable-déformable.La deuxième partie vise à atteindre la conservation de l’énergie à l’impact en adaptant la méthode de la masse singulière au CD-Lagrange. Une première formulation 1D est construite. Elle démontre une amélioration majeure du bilan d’énergie. Deux formulations 3D sont ensuite explorées.La troisième partie introduit les méthodes mortier dans le CD-Lagrange. Elles permettent de traiter un contact déformable-déformable de manière robuste, même en présence de friction et de grands glissements. Une technique d’accélération est proposée pour résoudre le problème de contact, ceci sans perte de précision
Tyres are complex structures to simulate. The materials are heterogeneous and incompressible with non-linear responses. The geometry goes to the millimetre scales for tread patterns. For a finite elements simulation a precise mesh is then required. The model has then a large number of degrees of freedom and non-linear material laws. In dynamics, the simulation becomes even more challenging especially with impacts. Nevertheless it is crucial in the tire design process because it brings a deeper comprehension of the tire and avoids test on real structures. The explicit time-integration make feasible the impact simulations. They handle easily the non-linearities with a very low computational cost for a time-step. Merged with a precise contact formulation, they form robust, accurate and efficient schemes for addressing impact simulations. This work aims to choose and improve an explicit scheme for non-linear dynamics with impacts. The first part is a benchmark for selecting a scheme and enhance its possibilities of improvement. The selected one is the CD-Lagrange: an explicit scheme based on central difference method, a contact enforcement by Lagrange multipliers, and a contact condition on velocity. Two mains improvements are identified and explored. Firstly, the energy conservation at impact would make the scheme symplectic for deformable bodies. Secondly the formulation must be enlarged to deformable–deformable contact. The second part aims then to achieve the conservation of energy by adapting the singular mass matrix to the CD-Lagrange. The formulation is firstly built in 1D, and shows a major improvement for the energy balance. Then two possible extensions are explored for the 3D cases. The third part presents the CD-Lagrange scheme with a mortar formulation for deformable-deformable contact. It handles with stability and accuracy large sliding and friction. An acceleration technique is proposed for solving the contact problem, without any loss of accuracy

Le pneumatique est un produit complexe soumis à de nombreuses contraintes. Il doit répondre à un compromis entre coût, performance, sécurité et recyclabilité. Il est formé d'une multitude de couches composées de différents matériaux entraînant des comportements complexes à étudier. Ainsi, le choix de la simulation numérique s’impose, permettant notamment l’étude de nombreux scénarios. Elle permet d'étudier l'impact de chaque étape de fabrication, et notamment celle du démoulage, qui a inspiré cette thèse. Ce problème non-régulier est associé à du contact et de l'endommagement, modélisés à l’aide de modèle de zones cohésives, et à de la dynamique rapide, phénomènes rarement combinés ensemble en simulation. Le problème à résoudre étant en dynamique transitoire, le choix d’un intégrateur temporel explicite s’impose. L'idée ici est d'utiliser un schéma explicite symplectique possédant ainsi de bonnes propriétés énergétiques en vérifiant les équations de conservation discrètes. Basé sur des travaux antérieurs, le choix est porté sur le schéma explicite CD-Lagrange. Ainsi, l'étude se concentre sur l'interface de contact entre un solide déformable, et un solide rigide. Une méthode pour résoudre en dynamique des problèmes d’interface est présentée. Un cadre thermodynamique et explicite de résolution est alors proposé, avec un traitement local des non-linéarités et des non-régularités conduisant à un algorithme de résolution "matrix-free". Les formulations sont basées sur le cadre thermodynamique des matériaux standards généralisés et de la mécanique non régulière. Ensuite, l'accent est mis sur les lois d'évolution thermodynamique en étudiant la non-localité temporelle pour limiter la localisation de l’endommagement sur l’interface. Des modèles à effet retard sont alors introduits. L'aspect modulaire de la résolution proposée est montré, avec l’application de plusieurs lois d’interface et de comportement volumique. L'application à des problèmes en grandes transformations est également fournie. Enfin, la faisabilité de l'approche est mise en évidence par son intégration dans un code semi-industriel, MEF++
The tire is a complex product subjected to numerous constraints, and the designer must find a compromise between cost, performance, safety and recyclability. It is composed of a multitude of overlayed layers of different materials, resulting in complex behaviors. Thus, numerical simulation is an obvious choice by allowing the study of a wide range of scenarios. It enables to study the impact of each manufacturing step, and in particular the unmolding tire process, which inspired this thesis. This non-regular problem is associated to contact and damage, described by a cohesive zone model, with fast dynamics phenomena, rarely combined together in simulation. Since the problem is a transient dynamics one, the choice of an explicit time integrator is natural. The proposed idea here is the use of an explicit symplectic scheme providing by definition good energy properties and discrete equations conservation. Based on previous work, the explicit CD-Lagrange scheme is chosen. Thus, the study is focused on the contact interface between a deformable solid and a rigid one. A method for solving interface problems in dynamics is presented. A thermodynamic and explicit resolution framework is then proposed, with local treatment of non-linearities and non-regularities leading to a matrix-free resolution algorithm. Formulations are based on the thermodynamic framework of generalized standard materials and non-regular mechanics. Next, the focus is set on the thermodynamic evolution laws by studying temporal non-locality in order to limit the damage localization on the interface. Delayed-effect models are then introduced. The modular aspect of the proposed resolution is shown, with application of several interface laws and bulk behaviors. Application to large transformation contact problems is also provided. Finally, the feasibility of the approach is demonstrated by its integration into a semi-industrial code, MEF++

Störmer-Verlet integration scheme has many attractive properties when dealing with ODE systems in linear spaces: it is explicit, 2nd order, linear/angular momentum preserving and it is symplectic for Hamiltonian systems. In this paper we investigate its application for numerical simulation of the multibody system dynamics (MBS) by formulating Störmer-Verlet algorithm for the rotational rigid body motion in Lie-group setting. Starting from the investigations on the single free rigid body rotational dynamics, the paper introduces modified RATTLE integration scheme with the direct SO(3) rotational update. Furthermore, non-canonical Lie-group Störmer-Verlet integration scheme is presented through the different derivation stages. Several presented numerical examples show excellent conservation properties of the proposed geometric algorithm.