Calculation Of Gravitational Forces Research Articles

Numerical modeling of collisions of spheroidal galaxies: Mass loss efficiency by baryon components

The dynamics of the collision of two spheroidal galaxies, each of which consists of stellar and gas components inside a live massive dark halo, is considered in detail based on numerical simulations. Dark matter and stars are collisionless components, so their dynamics is modeled by the system of N body particles, and the gas component is described by the equations of gas dynamics. The calculation of gravitational forces is based on the direct summation method of the gravitational contribution from each particle, which provides the highest possible accuracy. We vary the impact parameter, the initial collision velocity, the masses of the three components for each model of galaxies in order to estimate the relative fractions of the masses that are lost by both galaxies as a result of close interactions, as well as the masses of gas and stars that are exchanged between the two galaxies during the scattering process. Estimating this exchanged mass of gas requires a Lagrangian approach to simulate gas dynamics, and the Smoothed Particle Hydrodynamics method tracks the trajectory of each smoothed particle. There are two regions of parameters that separate the merging of two objects into one (large merging) and the case of scattering, when both gravitating systems move away from each other after interaction. Collisionless components (dark matter + stars) in the interaction of approximately the same galaxies lose their maximum mass at parameters close to the boundary between merging and scattering. If the initial masses of colliding galaxies differ greatly then the lost fraction of mass is large for a low-mass object and small for a massive galaxy. The formation of global shock waves in colliding systems is a key factor determining the efficiency of sweeping gas out of the gravitational potential well. The efficiency of the stellar component transition from one object to another is negligible. Gas exchange can exceed 10 percent of the mass for a certain range of values of the impact parameter and the initial collision velocity.

Major merging of galaxies in multicomponent numerical models: mass loss and exchange

The process of collision of two multicomponent galaxies is considered in detail based on numerical simulations of the dynamics of gravitating gas, stars and dark mass. To solve the equations of motion of the gas component, we use the Smoothed Particle Hydrodynamics method. Modeling of collisionless components is based on the N-body model. The computations of gravitational forces are carried out using both the approximate hierarchical TreeCode algorithm and the direct method of summing the gravitational contribution from all particles, which provides an accurate solution. This approach allows testing various models and evaluating the resulting errors associated with the calculation of gravitational forces and a finite number of particles in each of the components. Both methods for calculating gravity are software implemented as parallel codes for Nvidia Tesla GPUs. The estimates of the lost mass and the efficiency of matter exchange between galaxies are discussed depending on the model parameters.

Optimal Softening for Gravitational Force Calculations in N-body Dynamics

Abstract The choice of the optimal value of the softening length (ϵ i ) of each particle dealing with N-body simulations has a profound impact on error values in the gravitational force calculation. A slight deviation from its exact optimal value may result in a large error in the calculation. In this paper we augment the accuracy of the existing gravitational force calculation methods by providing a new technique to calculate the individual optimal values of ϵ i for various configurations of the Plummer density model. We have proposed an expression that relates each particle by considering the average characteristic length (λ i ) and density (ρ i ), unlike previous studies that considered ϵ i as an exclusive function of ρ i . We have performed gravitational force calculations for each and every particle from the Plummer density model using compact as well as noncompact gravitational force methods based on smoothed particle hydrodynamics. We have tested our new equation for the entire range of numerical simulations performed during the study. It is found that the errors in our force calculations are not only lower than those estimated from previous studies but also remain flat for various considerations of nearest neighboring particles (N neigh). The adjusted expression of ϵ λ,i in our study has less dependence on N neigh in the Plummer sphere. Finally, based on the results obtained using the method proposed in this study, we find that it remarkably improves both the accuracy as well as the stability of the gravitational force calculation.

Сравнительный анализ методов вычисления гравитационной силы частиц при численном моделировании задачи N-тел

One of the main problems of modern codes for solving the N-body problem is their great similarity. The immediate consequence of this is that the usual method of checking the simulation results by comparing independent approaches is hardly possible. The requirement for accurate calculation of gravitational force does not imply the need to use a unique approach. An approximate calculation of gravitational forces between N interacting particles can be used. For example, an approximate method with high accuracy is the fast multipole method (FMM) that requires only  operations instead of  to calculate the forces all N particles. FMM groups particles into spatially bounded cells and uses intercellular interactions to approximate forces at any position in the receiver cell using the Taylor extension obtained from the multipole extension of the original cell. Using the error estimate of this process it is possible to minimize the calculations while obtaining the relative errors N≳105. The article provides a brief analysis of N-body modeling methods and shows that the FMM approximates the calculation of mutual forces between particles and the problem under consideration is formulated much easier than, for example, in the case of softened gravity.

A vector field approach to calculating gravitational forces

Calculation of gravitational forces is essential for many fundamental measurements, such as determining the gravitational constant or investigating violations of the inverse square law. These calculations, even with modern computational power, are slow and tedious. Improved calculation efficiency allows an experimentalist to easily check the effect of possible systematic biases and to ease the process of instrument design. Many gravitational measurements are expanded in terms of multipole moments for efficient calculations, however for many experimental geometries these do not converge, leaving awkward sextuple integrals. In this work we introduce a modified approach to the calculation which reduces the force between a point mass and any arbitrary object to a sum of single integrals. The force between any two objects can then be calculated as a quadruple rather than a sextuple integral.

Multipole calculation of gravitational forces

Multipole calculation of gravitational forces

START: smoothed particle hydrodynamics with tree-based accelerated radiative transfer

We present a novel radiation hydrodynamics code, START, which is a smoothed particle hydrodynamics (SPH) scheme coupled with accelerated radiative transfer. The basic idea for the acceleration of radiative transfer is parallel to the tree algorithm that is hitherto used to speed up the gravitational force calculation in an N-body system. It is demonstrated that the radiative transfer calculations can be dramatically accelerated, where the computational time is scaled as Np log Ns for Np SPH particles and Ns radiation sources. Such acceleration allows us to readily include not only numerous sources but also scattering photons, even if the total number of radiation sources is comparable to that of SPH particles. Here, a test simulation is presented for a multiple source problem, where the results with START are compared to those with a radiation SPH code without tree-based acceleration. We find that the results agree well with each other if we set the tolerance parameter as < 1.0, and then it demonstrates that START can solve radiative transfer faster without reducing the accuracy. One of important applications with START is to solve the transfer of diffuse ionizing photons, where each SPH particle is regarded as an emitter. To illustrate the competence of START, we simulate the shadowing effect by dense clumps around an ionizing source. As a result, it is found that the erosion of shadows by diffuse recombination photons can be solved. Such an effect is of great significance to reveal the cosmic reionization process.

VINE—A NUMERICAL CODE FOR SIMULATING ASTROPHYSICAL SYSTEMS USING PARTICLES. I. DESCRIPTION OF THE PHYSICS AND THE NUMERICAL METHODS

We present a numerical code for simulating the evolution of astrophysical systems using particles to represent the underlying fluid flow. The code is written in Fortran 95 and is designed to be versatile, flexible and extensible, with modular options that can be selected either at the time the code is compiled or at run time through a text input file. We include a number of general purpose modules describing a variety of physical processes commonly required in the astrophysical community and we expect that the effort required to integrate additional or alternate modules into the code will small. In its simplest form the code can evolve the dynamical trajectories of a set of particles in two or three dimensions using a module which implements either a Leapfrog or Runge-Kutta-Fehlberg integrator, selected by the user at compile time. The user may choose to allow the integrator to evolve the system using individual timesteps for each particle or with a single, global time step for all. Particles may interact gravitationally as N-body particles, and all or any subset may also interact hydrodynamically, using the Smoothed Particle Hydrodynamic (SPH) method by selecting the SPH module. A third particle species can be included with a module to model massive point particles which may accrete nearby SPH or N-body particles. Such particles may be used to model, e.g., stars in a molecular cloud. Free boundary conditions are implemented by default, and a module may be selected to include periodic boundary conditions. We use a binary ‘Press’ tree to organize particles for rapid access in gravity and SPH calculations. Modules implementing an interface with special purpose ‘GRAPE’ hardware may also be selected to accelerate the gravity calculations. If available, forces obtained from the GRAPE coprocessors may be transparently substituted for those obtained from the tree, or both tree and GRAPE may be used as a combination GRAPE/tree code. The code may be run without modification on single processors or in parallel using OpenMP compiler directives on large scale, shared memory parallel machines. We present simulations of several test problems, including a merger simulation of two elliptical galaxies with 800000 particles. In comparison to the Gadget-2 code of Springel (2005), the gravitational force calculation, which is the most costly part of any simulation including self-gravity, is ∼ 4.6 − 4.9 times faster with VINE when tested on different snapshots of the elliptical galaxy merger simulation when run on an Itanium 2 processor in an SGI Altix. A full simulation of the same setup with 8 processors is a factor of 2.91 faster with VINE. The code is available to the public under the terms of the Gnu General Public License. Subject headings: methods: numerical — methods: N-body simulations — galaxies: interactions

Current Status of GRAPE Project

AbstractI'll summarize the current status of GRAPE project. GRAPE-6, completed in 2002, has been used by a number of people, for a wide variety of problems such as planet formation, star cluster dynamics, galactic nuclei, and cosmology. In 2004, we started the development of the next-generation machine, GRAPE-DR. GRAPE-DR has a architecture radically different from that of previous GRAPEs. It does not have hardwired pipeline for gravitational force calculation but a large number of small and simple programmable processors. This change made it possible to apply GRAPE-DR to a wide range of problems to which GRAPE was not efficient, and at the same time it helps us to explore new algorithms for N-body simulations. The GRAPE-DR chip was completed in 2006, and second prototype board was completed in May 2007. We hope to have full production-level board commercially available by the end of year 2007. A single board will offer the theoretical peak speed of 2 Tflops, about 20 times as that of a single PCI card version of GRAPE-6.

GRAPE: A parallel computer dedicated to astrophysical many-body problems

GRAPE (GRAvitational PipelinE) is a parallel computer dedicated to solve classical gravitational many-body problems in astrophysics. Its prototype was first designed in 1988, and the machine GRAPE-4 attained its performance at 1 teraflops equivalent in 1995. Now, a sub-petaflops machine is expected to be complete before the beginning of the 21st century by a group that is led by one of the members of the former GRAPE project. In the gravitational N-body problems, calculation of gravitational forces between pairs of bodies attracting each other is the heaviest part of computation. It requires calculations of the order of N 2 for one time-step, which is compared with the other calculations such as the time marching only of the order of N calculations. Therefore, in the GRAPE system, the former is performed on a dedicated computer and the latter on a general-purpose computer as a host machine. Only the former is parallelized; it is easy because we can compute forces between different pairs of bodies independently of each other when the positions of the bodies are given. Since details of GRAPE are already published elsewhere, we discuss in the present paper how such a concept of GRAPE is related to the nature of the problem. A heterogeneous system, in which problem-specific, ultra-speed machines are connected with general-purpose machines, will be one of the solutions for highly computation intensive problems in natural science. Cooperations between computer scientists and computational scientists are urgently needed for its realization.

The PCI Interface for GRAPE Systems: PCI-HIB

We developed a PCI interface for GRAPE systems. GRAPE (GRAvity piPE) is a special-purpose computer for gravitational N-body simulations. A GRAPE system consists of GRAPE processor boards and a host computer. GRAPE processors perform the calculation of gravitational forces between particles. The host computer performs the remaining calculations. The newest of the GRAPE machines, GRAPE-4, has achieved a peak performance of 1.08 Tflop. The GRAPE-4 system uses TURBOChannel for the interface to the host, which limits the selection of the host computer. The TURBOChannel bus is not supported by any of recent workstations. We have developed a new host interface board which adopts the PCI bus instead of the TURBOChannel. PCI is an I/O bus standard developed by Intel. It has a fairly high peak transfer speed, and is available on a wide range of computers, from PCs to supercomputers. Thus, the new interface allows us to connect GRAPE-4 to a wide variety of host computers. In test runs with a Barnes-Hut treecode, we found that the performance of the new system with the PCI interface is 40% better than that of the original system.