scholarly journals Hybrid symplectic integrators for planetary dynamics

2019 ◽  
Vol 485 (4) ◽  
pp. 5490-5497 ◽  
Author(s):  
Hanno Rein ◽  
David M Hernandez ◽  
Daniel Tamayo ◽  
Garett Brown ◽  
Emily Eckels ◽  
...  
Keyword(s):  
High Order ◽  
Symplectic Integrators ◽  
Heaviside Function ◽  
Integration Scheme ◽  
Switching Function ◽  
Wide Range ◽  
Switching Functions ◽  
Planetary Dynamics ◽  
Original Motivation ◽  
Speed And Accuracy

Abstract Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high-order integration scheme for the duration of the encounter while otherwise using a standard second-order Wisdom–Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high-order integrator. We show that the original motivation for choosing a polynomial switching function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching functions including an infinitely differentiable function and a Heaviside function. We find that using a Heaviside function leads to a significantly simpler scheme compared to MERCURY , while maintaining the same accuracy in short-term simulations.

scholarly journals High-order symplectic integrators for planetary dynamics and their implementation in rebound

2019 ◽  
Vol 489 (4) ◽  
pp. 4632-4640 ◽  
Author(s):  
Hanno Rein ◽  
Daniel Tamayo ◽  
Garett Brown
Keyword(s):  
Large Time ◽  
Good Accuracy ◽  
Long Term Evolution ◽  
High Order ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Available N ◽  
Term Evolution ◽  
Planetary Dynamics

ABSTRACT Direct N-body simulations and symplectic integrators are effective tools to study the long-term evolution of planetary systems. The Wisdom–Holman (WH) integrator in particular has been used extensively in planetary dynamics as it allows for large time-steps at good accuracy. One can extend the WH method to achieve even higher accuracy using several different approaches. In this paper, we survey integrators developed by Wisdom et al., Laskar & Robutel, and Blanes et al. Since some of these methods are harder to implement and not as readily available to astronomers compared to the standard WH method, they are not used as often. This is somewhat unfortunate given that in typical simulations it is possible to improve the accuracy by up to six orders of magnitude (!) compared to the standard WH method without the need for any additional force evaluations. To change this, we implement a variety of high-order symplectic methods in the freely available N-body integrator rebound. In this paper, we catalogue these methods, discuss their differences, describe their error scalings, and benchmark their speed using our implementations.

scholarly journals High-order and tunable balanced bandpass filters using mixed technology resonators

2021 ◽  
pp. 1-9
Author(s):  
Dakotah Simpson ◽  
Dimitra Psychogiou
Keyword(s):  
Frequency Tuning ◽  
Bandpass Filters ◽  
High Order ◽  
Integration Scheme ◽  
Differential Mode ◽  
Common Mode ◽  
Transmission Zeros ◽  
Wide Range ◽  
The Common ◽  
Multiple Levels

Abstract This paper reports on high-order balanced bandpass filters (BPFs) that are continuously tunable in terms of frequency and bandwidth and can be intrinsically switched-off. They use a hybrid integration scheme based on two different types of capacitively loaded resonators—ceramic coaxial and microstrip—that reduce the filter size, enhance its out-of-band selectivity and common-mode suppression, and allow for multiple levels of transfer function tuning. High selectivity is obtained in the differential mode due to the high number of poles and transmission zeros present. The common mode is highly suppressed through the introduction of additional transmission zeros and resistively loaded resonators. Furthermore, the use of ceramic coaxial resonators results in supplementary transmission zeros that are used to lower the out-of-band transmission in the differential mode. Multiple levels of tuning are obtained by reconfiguring only the frequency of the BPF's resonators. For experimental validation, a tunable mixed-technology microstrip prototype was manufactured and measured at S-band. It exhibited frequency tuning between 2.22 and 2.94 GHz, bandwidth tuning between 104 and 268 MHz, and an intrinsically switched-off mode with isolation >50 dB in the differential mode. For all states, the common mode was suppressed by at least 35 dB at the center frequency and within a wide range.

Exact Subthreshold Integration with Continuous Spike Times in Discrete-Time Neural Network Simulations

2007 ◽  
Vol 19 (1) ◽  
pp. 47-79 ◽  
Author(s):  
Abigail Morrison ◽  
Sirko Straube ◽  
Hans Ekkehard Plesser ◽  
Markus Diesmann
Keyword(s):  
Grid Point ◽  
Propagation Delay ◽  
Integration Scheme ◽  
The Novel ◽  
Neuron Models ◽  
Exact Integration ◽  
Wide Range ◽  
Network Simulations ◽  
Adequate Representation ◽  
Grid Points

Very large networks of spiking neurons can be simulated efficiently in parallel under the constraint that spike times are bound to an equidistant time grid. Within this scheme, the subthreshold dynamics of a wide class of integrate-and-fire-type neuron models can be integrated exactly from one grid point to the next. However, the loss in accuracy caused by restricting spike times to the grid can have undesirable consequences, which has led to interest in interpolating spike times between the grid points to retrieve an adequate representation of network dynamics. We demonstrate that the exact integration scheme can be combined naturally with off-grid spike events found by interpolation. We show that by exploiting the existence of a minimal synaptic propagation delay, the need for a central event queue is removed, so that the precision of event-driven simulation on the level of single neurons is combined with the efficiency of time-driven global scheduling. Further, for neuron models with linear subthreshold dynamics, even local event queuing can be avoided, resulting in much greater efficiency on the single-neuron level. These ideas are exemplified by two implementations of a widely used neuron model. We present a measure for the efficiency of network simulations in terms of their integration error and show that for a wide range of input spike rates, the novel techniques we present are both more accurate and faster than standard techniques.

scholarly journals An experimental decomposition of nonlinear forces on a surface-piercing column: Stokes-type expansions of the force harmonics

2018 ◽  
Vol 848 ◽  
pp. 42-77 ◽  
Author(s):  
L. F. Chen ◽  
J. Zang ◽  
P. H. Taylor ◽  
L. Sun ◽  
G. C. J. Morgan ◽  
...  
Keyword(s):  
High Order ◽  
Quality Data ◽  
Wave Group ◽  
Wave Loading ◽  
Harmonic Force ◽  
Wave Basin ◽  
Wide Range ◽  
Time Histories ◽  
Focused Wave ◽  
Sloping Bed

Wave loading on marine structures is the major external force to be considered in the design of such structures. The accurate prediction of the nonlinear high-order components of the wave loading has been an unresolved challenging problem. In this paper, the nonlinear harmonic components of hydrodynamic forces on a bottom-mounted vertical cylinder are investigated experimentally. A large number of experiments were conducted in the Danish Hydraulic Institute shallow water wave basin on the cylinder, both on a flat bed and a sloping bed, as part of a European collaborative research project. High-quality data sets for focused wave groups have been collected for a wide range of wave conditions. The high-order harmonic force components are separated by applying the ‘phase-inversion’ method to the measured force time histories for a crest focused wave group and the same wave group inverted. This separation method is found to work well even for locally violent nearly-breaking waves formed from bidirectional wave pairs. It is also found that the $n$th-harmonic force scales with the $n$th power of the envelope of both the linear undisturbed free-surface elevation and the linear force component in both time variation and amplitude. This allows estimation of the higher-order harmonic shapes and time histories from knowledge of the linear component alone. The experiments also show that the harmonic structure of the wave loading on the cylinder is virtually unaltered by the introduction of a sloping bed, depending only on the local wave properties at the cylinder. Furthermore, our new experimental results reveal that for certain wave cases the linear loading is actually less than 40 % of the total wave loading and the high-order harmonics contribute more than 60 % of the loading. The significance of this striking new result is that it reveals the importance of high-order nonlinear wave loading on offshore structures and means that such loading should be considered in their design.

scholarly journals Generating Bessel beams with broad depth-of-field by using phase-only acoustic holograms

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Sergio Jiménez-Gambín ◽  
Noé Jiménez ◽  
José M. Benlloch ◽  
Francisco Camarena
Keyword(s):  
Field Distribution ◽  
Focal Length ◽  
Bessel Beam ◽  
Transverse Field ◽  
High Order ◽  
Depth Of Field ◽  
Uniform Field ◽  
Axial Distribution ◽  
Bessel Beams ◽  
Wide Range

AbstractWe report zero-th and high-order acoustic Bessel beams with broad depth-of-field generated using acoustic holograms. While the transverse field distribution of Bessel beams generated using traditional passive methods is correctly described by a Bessel function, these methods present a common drawback: the axial distribution of the field is not constant, as required for ideal Bessel beams. In this work, we experimentally, numerically and theoretically report acoustic truncated Bessel beams of flat-intensity along their axis in the ultrasound regime using phase-only holograms. In particular, the beams present a uniform field distribution showing an elongated focal length of about 40 wavelengths, while the transverse width of the beam remains smaller than 0.7 wavelengths. The proposed acoustic holograms were compared with 3D-printed fraxicons, a blazed version of axicons. The performance of both phase-only holograms and fraxicons is studied and we found that both lenses produce Bessel beams in a wide range of frequencies. In addition, high-order Bessel beam were generated. We report first order Bessel beams that show a clear phase dislocation along their axis and a vortex with single topological charge. The proposed method may have potential applications in ultrasonic imaging, biomedical ultrasound and particle manipulation applications using passive lenses.

scholarly journals Regional wave propagation using the discontinuous Galerkin method

Solid Earth ◽  
2013 ◽  
Vol 4 (1) ◽  
pp. 43-57 ◽  
Author(s):  
S. Wenk ◽  
C. Pelties ◽  
H. Igel ◽  
M. Käser
Keyword(s):  
Wave Propagation ◽  
Discontinuous Galerkin ◽  
Regional Scale ◽  
Central Italy ◽  
High Order ◽  
Integration Scheme ◽  
Computational Domain ◽  
Dislocation Source ◽  
High Order Derivative ◽  
Dg Method

Abstract. We present an application of the discontinuous Galerkin (DG) method to regional wave propagation. The method makes use of unstructured tetrahedral meshes, combined with a time integration scheme solving the arbitrary high-order derivative (ADER) Riemann problem. This ADER-DG method is high-order accurate in space and time, beneficial for reliable simulations of high-frequency wavefields over long propagation distances. Due to the ease with which tetrahedral grids can be adapted to complex geometries, undulating topography of the Earth's surface and interior interfaces can be readily implemented in the computational domain. The ADER-DG method is benchmarked for the accurate radiation of elastic waves excited by an explosive and a shear dislocation source. We compare real data measurements with synthetics of the 2009 L'Aquila event (central Italy). We take advantage of the geometrical flexibility of the approach to generate a European model composed of the 3-D EPcrust model, combined with the depth-dependent ak135 velocity model in the upper mantle. The results confirm the applicability of the ADER-DG method for regional scale earthquake simulations, which provides an alternative to existing methodologies.

Comparison of CFD Predictions-of Multi-Mode Vortex-Induced Vibrations of a Tension Riser With Laboratory Measurements

2006 ◽  
Author(s):  
P. W. Bearman ◽  
J. R. Chaplin ◽  
E. Fontaine ◽  
J. M. R. Graham ◽  
K. Herfjord ◽  
...  
Keyword(s):  
Finite Volume ◽  
Cross Flow ◽  
Correction Method ◽  
Two Dimensions ◽  
Integration Scheme ◽  
Triangular Grid ◽  
Laboratory Measurements ◽  
Discrete Vortex ◽  
Vortex Induced Vibrations ◽  
Wide Range

This paper compares previously unpublished laboratory measurements of the vortex-induced vibrations of a vertical tension riser with predictions of five CFD-based riser codes. The experiments were carried out with a 13m long, 28mm diameter model riser of which the upper 7m was in air, and the remainder in water with a uniform flow generated by carriage motion at speeds up to 1m/s. In-line and cross-flow responses were computed from measurements of the curvature of the riser at 32 stations over its length. Key results are compared with blind CFD predictions in which the flow is computed in two dimensions on each of a number of horizontal planes. The CFD codes used in this exercise represent a wide range of approaches, represented by (1) a vorticity-stream function method using a finite volume technique for integrating the vorticity transport equation, (2) finite elements with linear interpolation functions on a triangular grid, (3) a pressure-correction method in a finite-volume integration scheme on hybrid unstructured grids, (4) a discrete vortex formulation incorporating the growing core size or core spread method to model diffusion of vorticity, (5) a velocity-vorticity method using a hybrid Eulerian–Lagrangian vortex-in-cell method with LES, diffusion computed on a grid, and convection of vorticity by discrete vortices. In individual test cases, maximum computed cross-flow responses were between 23% and 160% of measured values, and there was a much wider range in ratios of predicted to measured curvatures. Overall, average ratios of predicted to measured cross-flow responses were between 59% and 106% for the 5 different codes, and those for maximum in-line deflections between 71% and 109%. Predicted curvatures in either direction were on average between 30% and 170% of the measured values.

Timoshenko Beam Model for Buckling of Nanowires with High-Order Surface Stresses Effects

2012 ◽  
Vol 528 ◽  
pp. 281-284 ◽  
Author(s):  
Min Sen Chiu ◽  
Tung Yang Chen
Keyword(s):  
Timoshenko Beam ◽  
Surface Stress ◽  
Surface Effect ◽  
High Order ◽  
Mathematical Framework ◽  
Beam Model ◽  
Axial Buckling ◽  
Surface Stresses ◽  
Wide Range ◽  
Order Surface

High-order surface effects can have a significant effect in the mechanical behavior of micro- and nano-sized materials and structures. In the literature the mathematical framework of surface/interface stresses are generally described by generalized Young-Laplace equations based on membrane theory. A refined model of surface stress, counting into surface stresses as well as surface moments, collectively referred to as high-order surface stress, was recently derived by the authors. This framework allows us to simulate the interface between two neighboring media which may have varying in-plane stress through the thickness of the thin membrane. To illustrate surface stress effects, we consider the critical force of axial buckling of nanowires by accounting various degrees of surface stresses. Using the refined Timoshenko beam theory, we incorporate the high-order surface effect in the simulation of axial buckling of nanowires. The results are compared with the solutions based on conventional surface stress model as well as existing experimental data. This study might be helpful to characterize the mechanical properties of nanowires in a wide range of applications.

Sign in / Sign up

Export Citation Format

Share Document