Introduction

As electronic components have grown smaller in size and power and have increased in complexity, their enhanced sensitivity to the space radiation environment and its effects has become a major source of concern for the spacecraft engineer. The three primary considerations in the design of spacecraft are the description of the sources of space radiation, the determination of how that radiation propagates through material, and, thirdly, how radiation affects specific circuit components. As the natural and man-made space radiation environments were introduced in Chapter 3, the objective of this chapter is to address the latter two aspects of the radiation problem. In particular, because the “ambient” environment is typically only relevant to the outer surface of a space vehicle, it is necessary to treat the propagation of the external environment through the complex surrounding structures to the point inside the spacecraft where knowledge of the internal radiation environment is required. Although it is not possible to treat in detail all aspects of the problem of the radiation environment within a spacecraft, by dividing the problem into three parts – external environment, propagation, and internal environment – a basis for understanding the process of protecting a spacecraft from radiation will be established that can be applied to a wide range of radiation problems.

Radiation Interactions with Matter

From the standpoint of radiation interactions with matter, three particle families need to be considered:

1. photons (primarily EUV, X rays, and gamma rays),

2. charged particles (protons, electrons, and heavy ions),

3. neutrons.

Particle Impacts on Spacecraft

In this chapter, the effects of space particulates – hypervelocity impacts and scattering – are considered. Hypervelocity impacts, the primary effect of meteoroids and space debris, can be roughly divided into effects on single surfaces (namely, cratering or penetration of single surfaces), spall formation, and double-wall (or Whipple) shield penetration. A major consideration for each of these is the target. For example, if the target is an optical surface, then the damage induced on single surfaces is important. If the target is a tank, then penetration and/or failure of the tank is important and the characteristics of its contents become crucial. For electronic components inside a box, the size and distribution of the spall or spall/impactor products coming off the wall of the box are important. Examples of these factors are discussed below with emphasis on the practical considerations that must be taken into account in arriving at an effective and economical (in mass) protection system. A particularly important issue for meteoroid shielding design has arisen in recent years because of the need to have interplanetary spacecraft carrying nuclear RTGs use the Earth for gravitational assists. To provide adequate safety margins for these missions, which often require hypervelocity flybys of the Earth at distances of 300 km, mission planners must target the vehicle so that it will be extremely unlikely for a meteoroid impact on the spacecraft to lead to an Earth-impact trajectory.

In this chapter we will present examples of so-called open boundary problems. By this we mean that the simulation contains all particles relevant to the problem, and the size of the simulation region is adjusted accordingly at each timestep. This type of problem is by far the easiest one to which one can apply hierarchical data structures. The additional difficulties posed by periodic boundaries will be considered in Chapters 5 and 6.

Gravitational Problems in Astrophysics

Hierarchical tree codes were first developed in the context of astrophysics. This is not surprising because there is a big discrepancy between the number of bodies one would like to study – for example, O (1011) for a galaxy – and the number one can afford to model with a standard N-body code – at present O(105). PIC codes, which employ a grid structure to represent the fields in space, usually cannot handle these problems for two reasons: the complex structure of the investigated object and the large density contrasts such as those found in galaxies. N-body codes are able to avoid the first of these difficulties, because they are gridless and can therefore cope with arbitrarily complicated structures. The second difficulty remains, however, because of the N2 scaling of computation time. This can have two consequences. If the number of simulation particles is too small, the spatial resolution of the simulation might not be good enough to reveal the real dynamic behaviour of the system.

In this chapter we will consider a variety of fields where the tree algorithm in combination with periodic boundaries can be applied and where the speedup in comparison to standard MD and MC codes enables previously inaccessible problems to be investigated. This set of applications is in no way exhaustive, but is intended to indicate the types of problems where the algorithm might best be put to use.

Practically every N-body MD or MC code could incorporate the tree algorithm. However, for systems with short-range potentials, like the Lennard–Jones potentials for the description of solids, this does not bring much of an advantage. Because the potential falls off very rapidly (see Fig. 6.1), a sharp cutoff can be used to exclude interactions of more distant particles whose contribution is negligible. The tree algorithm is mainly suited to systems with long-range forces such as Coulomb, where the summed effect of distant particles is important.

An ideal application is dense, fully-ionized plasmas. Here the particles are so closely packed that an analytical treatment is difficult, and many MD and MC calculations have been carried out to investigate their properties. Limitations in the number of simulation particles make some problems difficult to address due to a combination of poor statistics and small system size. The tree algorithm could be successfully applied here, because the particles interact purely through Coulomb forces.

In previous chapters we occasionally referred to an alternative type of tree code, namely the Fast Multipole Method (FMM). This technique, an elegant refinement of the basic Barnes–Hut algorithm, appears to be best suited to ‘static’ problems, where the particle distribution is more or less uniform. Although it has not been as widely used as the Barnes–Hut (BH) method for dynamic problems – because of either its increased mathematical complexity or the additional computational overhead – it may well become the basis of ‘multimillion’ N-body problems in the near future. We therefore include an introduction to FMM here, based primarily on works by Greengard (1987, 1988, 1990) and Schmidt and Lee (1991). At the same time, we will try to maintain a consistency of notation with the Barnes–Hut algorithm (hereafter referred to as the ‘tree method’ or ‘tree algorithm’), as described in Chapter 2.

Outline of the Fast Multipole Algorithm

The Fast Multipole Method makes use of the fact that a multipole expansion to infinite order contains the total information of a particle distribution. As in the BH algorithm, the interaction between near neighbours is calculated by direct particle–particle force summation, and more distant particles are treated separately. However, the distinction between these two contributions is obtained in a different way. In FMM the distant region is treated as a single ‘far-field’ contribution, which is calculated by a high-order multipole expansion.

The FMM was first formulated by Greengard and Rokhlin (1987).

The difficulty in writing a ‘how-to’ book on numerical methods is to find a form which is accessible to people from various scientific backgrounds. When we started this project, hierarchical N-body techniques were deemed to be ‘too new’ for a book. On the other hand, a few minutes browsing in the References will reveal that the scientific output arising from the original papers of Barnes and Hut (1986) and Greengard and Rohklin (1987) is impressive but largely confined to two or three specialist fields. To us, this suggests that it is about time these techniques became better known in other fields where N-body problems thrive, not least in our own field of computational plasma physics. This book is therefore an attempt to gather everything hierarchical under one roof, and then to indicate how and where tree methods might be used in the reader's own research field. Inevitably, this has resulted in something of a pot-pourri of techniques and applications, but we hope there is enough here to satisfy the beginners and connoisseurs alike.

Tree Construction

We have seen in the preceding chapter that in grid-based codes the particles interact via some averaged density distribution. This enables one to calculate the influence of a number of particles represented by a cell on its neighbouring cells. Problems occur if the density contrast in the simulation becomes very large or the geometry of the problem is very complex.

So why does one bother with a grid at all and not just calculate the interparticle forces? The answer is simply that the computational effort involved quite dramatically limits the number of particles that can be simulated. Particularly with 1/r-type potentials, calculating each particle–particle interaction requires an unnecessary amount of work because the individual contributions of distant particles is small. On the other hand, gridless codes cannot distinguish between near-neighbours and more distant particles; each particle is given the same weighting.

Ideally, the calculation would be performed without a grid in the usual sense, but with some division of the physical space that maintains a relationship between each particle and its neighbours. The force could then be calculated by direct integration while combining increasingly large groups of particles at larger distances. Barnes and Hut (1986) observed that this works in the same way that humans interact with neighbouring individuals, more distant villages, and larger states and countries. A resident of Lower-Wobbleton, Kent, England, is unlikely to undertake a trip to Oberfriedrichsheim, Bavaria, Germany, for a beer and to catch up on the local gossip.