Symplectic Integration for Multivariate Dynamic Spline-Based Model of Deformable Linear Objects

Deformable Linear Objects (DLOs) such as ropes, cables, and surgical sutures have a wide variety of uses in automotive engineering, surgery, and electromechanical industries. Therefore, modeling of DLOs as well as a computationally efficient way to predict the DLO behavior are of great importance, in particular to enable robotic manipulation of DLOs. The main motivation of this work is to enable efficient pre- diction of the DLO behavior during robotic manipulation. In this paper, the DLO is modeled by a multivariate dynamic spline, while a symplectic integration method is used to solve the model iteratively by interpolating the DLO shape during the manipulation process. Comparisons between the symplectic, Runge-Kutta and Zhai integrators are reported. The presented results show the capabilities of the symplectic integrator to overcome other integration methods in predicting the DLO behavior. Moreover, the results obtained with different sets of model parameters integrated by means of the symplectic method are reported to show how they influence the DLO behavior estimation.

Applying explicit symplectic integrator to study chaos of charged particles around magnetized Kerr black hole

In a recent work of Wu, Wang, Sun and Liu, a second-order explicit symplectic integrator was proposed for the integrable Kerr spacetime geometry. It is still suited for simulating the nonintegrable dynamics of charged particles moving around the Kerr black hole embedded in an external magnetic field. Its successful construction is due to the contribution of a time transformation. The algorithm exhibits a good long-term numerical performance in stable Hamiltonian errors and computational efficiency. As its application, the dynamics of order and chaos of charged particles is surveyed. In some circumstances, an increase of the dragging effects of the spacetime seems to weaken the extent of chaos from the global phase-space structure on Poincaré sections. However, an increase of the magnetic parameter strengthens the chaotic properties. On the other hand, fast Lyapunov indicators show that there is no universal rule for the dependence of the transition between different dynamical regimes on the black hole spin. The dragging effects of the spacetime do not always weaken the extent of chaos from a local point of view.

Close encounters with the Death Star: Interactions between collapsed bodies and the Solar System

Aims. We aim to investigate the consequences of a fast massive stellar remnant – a black hole (BH) or a neutron star (NS) – encountering a planetary system. Methods. We modelled a close encounter between the actual Solar System (SS) and a 2 M⊙ NS and a 10 M⊙ BH, using a few-body symplectic integrator. We used a range of impact parameters, orbital phases at the start of the simulation derived from the current SS orbital parameters, encounter velocities, and incidence angles relative to the plane of the SS. Results. We give the distribution of possible outcomes, such as when the SS remains bound, when it suffers a partial or complete disruption, and in which cases the intruder is able to capture one or more planets, yielding planetary systems around a BH or a NS. We also show examples of the long-term stability of the captured planetary systems.

Step-size effect in the time-transformed leapfrog integrator on elliptic and hyperbolic orbits

ABSTRACT A drift-kick-drift (DKD) type leapfrog symplectic integrator applied for a time-transformed separable Hamiltonian (or time-transformed symplectic integrator; TSI) has been known to conserve the Kepler orbit exactly. We find that for an elliptic orbit, such feature appears for an arbitrary step size. But it is not the case for a hyperbolic orbit: When the half step size is larger than the conjugate momenta of the mean anomaly, a phase transition happens and the new position jumps to the non-physical counterpart of the hyperbolic trajectory. Once it happens, the energy conservation is broken. Instead, the kinetic energy minus the potential energy becomes a new conserved quantity. We provide a mathematical explanation for such phenomenon. Our result provides a deeper understanding of the TSI method, and a useful constraint of the step size when the TSI method is used to solve the hyperbolic encounters. This is particular important when an (Bulirsch–Stoer) extrapolation integrator is used together, which requires the convergence of integration errors.

Multi-Symplectic Method for the Logarithmic-KdV Equation

The multi-symplectic integrator, as a numerical integration approach with symmetry, is known to have the characteristic of preserving the qualitative features and geometric properties of certain systems. Using the multi-symplectic integrator, the numerical simulation of the Gaussian solitary wave propagation of the logarithmic Korteweg–de Vries (logarithmic-KdV) equation was investigated. The multi-symplectic formulation of the logarithmic-KdV equation was explored by introducing some intermediate variables. A fully implicit version of the centered box scheme was used to discretize the multi-symplectic equations. In addition, numerical experiments were carried out to show the conservative properties of the proposed scheme.

Optimal Third-Order Symplectic Integration Modeling of Seismic Acoustic Wave Propagation

ABSTRACT Seismic wavefield modeling based on the wave equation is widely used in understanding and predicting the dynamic and kinematic characteristics of seismic wave propagation through media. This article presents an optimal numerical solution for the seismic acoustic wave equation in a Hamiltonian system based on the third-order symplectic integrator method. The least absolute truncation error analysis method is used to determine the optimal coefficients. The analysis of the third-order symplectic integrator shows that the proposed scheme exhibits high stability and minimal truncation error. To illustrate the accuracy of the algorithm, we compare the numerical solutions generated by the proposed method with the theoretical analysis solution for 2D and 3D seismic wave propagation tests. The results show that the proposed method reduced the phase error to the eighth-order magnitude accuracy relative to the exact solution. These simulations also demonstrated that the proposed third-order symplectic method can minimize numerical dispersion and preserve the waveforms during the simulation. In addition, comparing different central frequencies of the source and grid spaces (90, 60, and 20 m) for simulation of seismic wave propagation in 2D and 3D models using symplectic and nearly analytic discretization methods, we deduce that the suitable grid spaces are roughly equivalent to between one-fourth and one-fifth of the wavelength, which can provide a good compromise between accuracy and computational cost.

A slow-down time-transformed symplectic integrator for solving the few-body problem

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.