Fast vortex method calculation using a special-purpose computer

Abstract

A mathematical formulation for the 3D vortex method has been developed for calculation using a special-purpose computer MDGRAPE-2 that was originally designed for molecular dynamics simulations. We made an assessment of this hardware for a few representative problems and compared the results with and without it. It is found that the generation of appropriate function tables, which are used to call libraries, embedded in MDGRAPE-2 is of primary importance in order to retain accuracy. The error arising from the approximation is evaluated by calculating a pair of vortex rings impinging to themselves. Consequently, acceleration about 50 times greater is achieved by MDGRAPE-2 while the error in the statistical quantities such as kinetic energy and enstrophy remain negligible.

Introduction

N-Body simulations were devised in the 1950s and have been widely used since the 1970s when digital computers became powerful enough and affordable. Today it is considered to be an orthodox method for studying particle systems. The classical N-body problem simulates the evolution of a system of N bodies, where the force exerted on each body arises due to its interaction with all other bodies in the system. N-Body algorithms have numerous applications in areas such as astrophysics, molecular dynamics, plasma physics and computational fluid dynamics using the vortex method. For each of these computational problems the calculation takes on a slightly different form but each share common features. The simulation proceeds over time steps, with each step computing the net force on every body and thereby updating its position and other attributes. The cost of a direct summation algorithm for force calculation is O(N²), therefore calculation time grows rapidly as the number of bodies N increases.

There are two ways to reduce the force calculation cost of an N-body simulation. One is to use fast algorithms such as the tree code developed by Barnes and Hut [1] or the fast multipole method (FMM) by Greengard and Rokhlin [2]. The tree code is an O(Nlog N) algorithm based on a hierarchical octree representation of space in three dimensions. It computes interactions between distant particles and reduces the number of operations by means of a first-order approximation. Many existing implementations of tree code algorithms only use up to quadropole moments and calculation costs rise quickly when high accuracy is required. In the FMM, the long-range forces are approximated by multipole expansion truncated at a certain degree, while the contributions from particles within nearby regions are calculated directly in a usual manner without approximation. Including higher order terms in multipole approximations and/or increasing the size of a nearby region can improve the computational accuracy. However, either effort substantially increases the computation time. In particular, the computation of a high-order term is very expensive.

The other way is to execute the N-body simulation with special-purpose hardware such as MDGRAPE-2 developed by Susukita et al. [3]. MDGRAPE-2, one of the GRAPE (Gravity Pipe, developed by Sugimoto et al. [5]) series machines, is a special-purpose computer designed for force calculations between point-charge or point-mass particles. Its performance is much higher than ordinary computers. It can speed up force calculations about 10–100 times when compared to general-purpose (defined as ’host’ hereafter) computers of the same cost.

The vortex method solves time-dependent incompressible flow problems by discretizing the vorticity into vortex elements and following these elements in time. This results in a volume mesh-free algorithm and saves significant time in preprocessing when compared to the conventional Navier–Stokes approach where grids need to be generated. The vortex methods have been developed and applied for the analysis of complicated, unsteady and vortical flows related to a wide range of problems found in industry, as it consists of a simple algorithm based on the physics of flow. For details see [6], [7].

The main difficulty with vortex methods as originally formulated is that the cost of the evaluation of the velocity field induced by N vortices is O(N²). This is expensive, particularly in three dimensions where a large number of elements are computed simultaneously. In the calculation of vortex methods, the largest computational load occurs in the routine that calculates the Biot–Savart law and the stretching term in the vorticity equation. In regions of high strain the spatial resolution becomes worse because the distance of each element becomes large. It is required to split vortex elements to keep the spatial resolution in the direction of the vorticity vector of the element. The result is a N-body interaction calculation for millions of particles having calculation cost of O(N²) with growing N. Nevertheless, these calculations have the same mathematical architecture as a multibody problem, thus permitting the use of special-purpose computers in multibody problems, e.g., MDGRAPE-2.

Our long-term objective is to solve high Reynolds number turbulent flows for engineering problems via a reasonable computational effort. The use of the 3D vortex method is attractive because of its simple formulation and flexibility in moving and/or deforming boundary problems. The long computation time due to the above-mentioned O(N²) problem may be reduced when we apply a special-purpose computer.

The purpose of the present paper is, therefore, to address a few issues that hindrance the application of MDGRAPE-2 to the vortex method. First, because of the simple architecture, it is required to generate an optimum function table when the embedded libraries are called from the main routine. Second, the cross-product calculation which is not considered in the original command set must be handled in a proper manner, which is treated in some previous works, e.g. [8], [9], [10], [11]. These points are discussed one after another in the subsequent sections after an introduction to the basic mathematical formulae.