Voro++
voro++.hh
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1 // Voro++, a 3D cell-based Voronoi library
2 //
3 // Author : Chris H. Rycroft (LBL / UC Berkeley)
4 // Email : chr@alum.mit.edu
5 // Date : August 30th 2011
6 
7 /** \file voro++.hh
8  * \brief A file that loads all of the Voro++ header files. */
9 
10 /** \mainpage Voro++ class reference manual
11  * \section intro Introduction
12  * Voro++ is a software library for carrying out three-dimensional computations
13  * of the Voronoi tessellation. A distinguishing feature of the Voro++ library
14  * is that it carries out cell-based calculations, computing the Voronoi cell
15  * for each particle individually, rather than computing the Voronoi
16  * tessellation as a global network of vertices and edges. It is particularly
17  * well-suited for applications that rely on cell-based statistics, where
18  * features of Voronoi cells (eg. volume, centroid, number of faces) can be
19  * used to analyze a system of particles.
20  *
21  * Voro++ is written in C++ and can be built as a static library that can be
22  * linked to. This manual provides a reference for every function in the class
23  * structure. For a general overview of the program, see the Voro++ website at
24  * http://math.lbl.gov/voro++/ and in particular the example programs at
25  * http://math.lbl.gov/voro++/examples/ that demonstrate many of the library's
26  * features.
27  *
28  * \section class C++ class structure
29  * The code is structured around several C++ classes. The voronoicell_base
30  * class contains all of the routines for constructing a single Voronoi cell.
31  * It represents the cell as a collection of vertices that are connected by
32  * edges, and there are routines for initializing, making, and outputting the
33  * cell. The voronoicell_base class form the base of the voronoicell and
34  * voronoicell_neighbor classes, which add specialized routines depending on
35  * whether neighboring particle ID information for each face must be tracked or
36  * not. Collectively, these classes are referred to as "voronoicell classes"
37  * within the documentation.
38  *
39  * There is a hierarchy of classes that represent three-dimensional particle
40  * systems. All of these are derived from the voro_base class, which contains
41  * constants that divide a three-dimensional system into a rectangular grid of
42  * equally-sized rectangular blocks; this grid is used for computational
43  * efficiency during the Voronoi calculations.
44  *
45  * The container_base, container, and container_poly are then derived from the
46  * voro_base class to represent a particle system in a specific
47  * three-dimensional rectangular box using both periodic and non-periodic
48  * boundary conditions. In addition, the container_periodic_base,
49  * container_periodic, and container_periodic_poly classes represent
50  * a particle system in a three-dimensional non-orthogonal periodic domain,
51  * defined by three periodicity vectors that represent a parallelepiped.
52  * Collectively, these classes are referred to as "container classes" within
53  * the documentation.
54  *
55  * The voro_compute template encapsulates all of the routines for computing
56  * Voronoi cells. Each container class has a voro_compute template within
57  * it, that accesses the container's particle system, and computes the Voronoi
58  * cells.
59  *
60  * There are several wall classes that can be used to apply certain boundary
61  * conditions using additional plane cuts during the Voronoi cell compution.
62  * The code also contains a number of small loop classes, c_loop_all,
63  * c_loop_subset, c_loop_all_periodic, and c_loop_order that can be used to
64  * iterate over a certain subset of particles in a container. The latter class
65  * makes use of a special particle_order class that stores a specific order of
66  * particles within the container. The library also contains the classes
67  * pre_container_base, pre_container, and pre_container_poly, that can be used
68  * as temporary storage when importing data of unknown size.
69  *
70  * \section voronoicell The voronoicell classes
71  * The voronoicell class represents a single Voronoi cell as a convex
72  * polyhedron, with a set of vertices that are connected by edges. The class
73  * contains a variety of functions that can be used to compute and output the
74  * Voronoi cell corresponding to a particular particle. The command init()
75  * can be used to initialize a cell as a large rectangular box. The Voronoi cell
76  * can then be computed by repeatedly cutting it with planes that correspond to
77  * the perpendicular bisectors between that particle and its neighbors.
78  *
79  * This is achieved by using the plane() routine, which will recompute the
80  * cell's vertices and edges after cutting it with a single plane. This is the
81  * key routine in voronoicell class. It begins by exploiting the convexity
82  * of the underlying cell, tracing between edges to work out if the cell
83  * intersects the cutting plane. If it does not intersect, then the routine
84  * immediately exits. Otherwise, it finds an edge or vertex that intersects
85  * the plane, and from there, traces out a new face on the cell, recomputing
86  * the edge and vertex structure accordingly.
87  *
88  * Once the cell is computed, there are many routines for computing features of
89  * the the Voronoi cell, such as its volume, surface area, or centroid. There
90  * are also many routines for outputting features of the Voronoi cell, or
91  * writing its shape in formats that can be read by Gnuplot or POV-Ray.
92  *
93  * \subsection internal Internal data representation
94  * The voronoicell class has a public member p representing the
95  * number of vertices. The polyhedral structure of the cell is stored
96  * in the following arrays:
97  *
98  * - pts: a one-dimensional array of floating point numbers, that represent the
99  * position vectors x_0, x_1, ..., x_{p-1} of the polyhedron vertices.
100  * - nu: the order of each vertex n_0, n_1, ..., n_{p-1}, corresponding to
101  * the number of other vertices to which each is connected.
102  * - ed: a two-dimensional table of edges and relations. For the ith vertex,
103  * ed[i] has 2n_i+1 elements. The first n_i elements are the edges e(j,i),
104  * where e(j,i) is the jth neighbor of vertex i. The edges are ordered
105  * according to a right-hand rule with respect to an outward-pointing normal.
106  * The next n_i elements are the relations l(j,i) which satisfy the property
107  * e(l(j,i),e(j,i)) = i. The final element of the ed[i] list is a back
108  * pointer used in memory allocation.
109  *
110  * In a very large number of cases, the values of n_i will be 3. This is because
111  * the only way that a higher-order vertex can be created in the plane()
112  * routine is if the cutting plane perfectly intersects an existing vertex. For
113  * random particle arrangements with position vectors specified to double
114  * precision this should happen very rarely. A preliminary version of this code
115  * was quite successful with only making use of vertices of order 3. However,
116  * when calculating millions of cells, it was found that this approach is not
117  * robust, since a single floating point error can invalidate the computation.
118  * This can also be a problem for cases featuring crystalline arrangements of
119  * particles where the corresponding Voronoi cells may have high-order vertices
120  * by construction.
121  *
122  * Because of this, Voro++ takes the approach that it if an existing vertex is
123  * within a small numerical tolerance of the cutting plane, it is treated as
124  * being exactly on the plane, and the polyhedral topology is recomputed
125  * accordingly. However, while this improves robustness, it also adds the
126  * complexity that n_i may no longer always be 3. This causes memory management
127  * to be significantly more complicated, as different vertices require a
128  * different number of elements in the ed[][] array. To accommodate this, the
129  * voronoicell class allocated edge memory in a different array called mep[][],
130  * in such a way that all vertices of order k are held in mep[k]. If vertex
131  * i has order k, then ed[i] points to memory within mep[k]. The array ed[][]
132  * is never directly initialized as a two-dimensional array itself, but points
133  * at allocations within mep[][]. To the user, it appears as though each row of
134  * ed[][] has a different number of elements. When vertices are added or
135  * deleted, care must be taken to reorder and reassign elements in these
136  * arrays.
137  *
138  * During the plane() routine, the code traces around the vertices of the cell,
139  * and adds new vertices along edges which intersect the cutting plane to
140  * create a new face. The values of l(j,i) are used in this computation, as
141  * when the code is traversing from one vertex on the cell to another, this
142  * information allows the code to immediately work out which edge of a vertex
143  * points back to the one it came from. As new vertices are created, the l(j,i)
144  * are also updated to ensure consistency. To ensure robustness, the plane
145  * cutting algorithm should work with any possible combination of vertices
146  * which are inside, outside, or exactly on the cutting plane.
147  *
148  * Vertices exactly on the cutting plane create some additional computational
149  * difficulties. If there are two marginal vertices connected by an existing
150  * edge, then it would be possible for duplicate edges to be created between
151  * those two vertices, if the plane routine traces along both sides of this
152  * edge while constructing the new face. The code recognizes these cases and
153  * prevents the double edge from being formed. Another possibility is the
154  * formation of vertices of order two or one. At the end of the plane cutting
155  * routine, the code checks to see if any of these are present, removing the
156  * order one vertices by just deleting them, and removing the order two
157  * vertices by connecting the two neighbors of each vertex together. It is
158  * possible that the removal of a single low-order vertex could result in the
159  * creation of additional low-order vertices, so the process is applied
160  * recursively until no more are left.
161  *
162  * \section container The container classes
163  * There are four container classes available for general usage: container,
164  * container_poly, container_periodic, and container_periodic_poly. Each of
165  * these represent a system of particles in a specific three-dimensional
166  * geometry. They contain routines for importing particles from a text file,
167  * and adding particles individually. They also contain a large number of
168  * analyzing and outputting the particle system. Internally, the routines that
169  * compute Voronoi cells do so by making use of the voro_compute template.
170  * Each container class contains routines that tell the voro_compute template
171  * about the specific geometry of this container.
172  *
173  * \section voro_compute The voro_compute template
174  * The voro_compute template encapsulates the routines for carrying out the
175  * Voronoi cell computations. It contains data structures suchs as a mask and a
176  * queue that are used in the computations. The voro_compute template is
177  * associated with a specific container class, and during the computation, it
178  * calls routines in the container class to access the particle positions that
179  * are stored there.
180  *
181  * The key routine in this class is compute_cell(), which makes use of a
182  * voronoicell class to construct a Voronoi cell for a specific particle in the
183  * container. The basic approach that this function takes is to repeatedly cut
184  * the Voronoi cell by planes corresponding neighboring particles, and stop
185  * when it recognizes that all the remaining particles in the container are too
186  * far away to possibly influence cell's shape. The code makes use of two
187  * possible methods for working out when a cell computation is complete:
188  *
189  * - Radius test: if the maximum distance of a Voronoi cell
190  * vertex from the cell center is R, then no particles more than a distance
191  * 2R away can possibly influence the cell. This a very fast computation to
192  * do, but it has no directionality: if the cell extends a long way in one
193  * direction then particles a long distance in other directions will still
194  * need to be tested.
195  * - Region test: it is possible to test whether a specific region can
196  * possibly influence the cell by applying a series of plane tests at the
197  * point on the region which is closest to the Voronoi cell center. This is a
198  * slower computation to do, but it has directionality.
199  *
200  * Another useful observation is that the regions that need to be tested are
201  * simply connected, meaning that if a particular region does not need to be
202  * tested, then neighboring regions which are further away do not need to be
203  * tested.
204  *
205  * For maximum efficiency, it was found that a hybrid approach making use of
206  * both of the above tests worked well in practice. Radius tests work well for
207  * the first few blocks, but switching to region tests after then prevent the
208  * code from becoming extremely slow, due to testing over very large spherical
209  * shells of particles. The compute_cell() routine therefore takes the
210  * following approach:
211  *
212  * - Initialize the voronoicell class to fill the entire computational domain.
213  * - Cut the cell by any wall objects that have been added to the container.
214  * - Apply plane cuts to the cell corresponding to the other particles which
215  * are within the current particle's region.
216  * - Test over a pre-computed worklist of neighboring regions, that have been
217  * ordered according to the minimum distance away from the particle's
218  * position. Apply radius tests after every few regions to see if the
219  * calculation can terminate.
220  * - If the code reaches the end of the worklist, add all the neighboring
221  * regions to a new list.
222  * - Carry out a region test on the first item of the list. If the region needs
223  * to be tested, apply the plane() routine for all of its particles, and then
224  * add any neighboring regions to the end of the list that need to be tested.
225  * Continue until the list has no elements left.
226  *
227  * The compute_cell() routine forms the basis of many other routines, such as
228  * store_cell_volumes() and draw_cells_gnuplot() that can be used to calculate
229  * and draw the cells in a container.
230  *
231  * \section walls Wall computation
232  * Wall computations are handled by making use of a pure virtual wall class.
233  * Specific wall types are derived from this class, and require the
234  * specification of two routines: point_inside() that tests to see if a point
235  * is inside a wall or not, and cut_cell() that cuts a cell according to the
236  * wall's position. The walls can be added to the container using the
237  * add_wall() command, and these are called each time a compute_cell() command
238  * is carried out. At present, wall types for planes, spheres, cylinders, and
239  * cones are provided, although custom walls can be added by creating new
240  * classes derived from the pure virtual class. Currently all wall types
241  * approximate the wall surface with a single plane, which produces some small
242  * errors, but generally gives good results for dense particle packings in
243  * direct contact with a wall surface. It would be possible to create more
244  * accurate walls by making cut_cell() routines that approximate the curved
245  * surface with multiple plane cuts.
246  *
247  * The wall objects can used for periodic calculations, although to obtain
248  * valid results, the walls should also be periodic as well. For example, in a
249  * domain that is periodic in the x direction, a cylinder aligned along the x
250  * axis could be added. At present, the interior of all wall objects are convex
251  * domains, and consequently any superposition of them will be a convex domain
252  * also. Carrying out computations in non-convex domains poses some problems,
253  * since this could theoretically lead to non-convex Voronoi cells, which the
254  * internal data representation of the voronoicell class does not support. For
255  * non-convex cases where the wall surfaces feature just a small amount of
256  * negative curvature (eg. a torus) approximating the curved surface with a
257  * single plane cut may give an acceptable level of accuracy. For non-convex
258  * cases that feature internal angles, the best strategy may be to decompose
259  * the domain into several convex subdomains, carry out a calculation in each,
260  * and then add the results together. The voronoicell class cannot be easily
261  * modified to handle non-convex cells as this would fundamentally alter the
262  * algorithms that it uses, and cases could arise where a single plane cut
263  * could create several new faces as opposed to just one.
264  *
265  * \section loops Loop classes
266  * The container classes have a number of simple routines for calculating
267  * Voronoi cells for all particles within them. However, in some situations it
268  * is desirable to iterate over a specific subset of particles. This can be
269  * achieved with the c_loop classes that are all derived from the c_loop_base
270  * class. Each class can iterate over a specific subset of particles in a
271  * container. There are three loop classes for use with the container and
272  * container_poly classes:
273  *
274  * - c_loop_all will loop over all of the particles in a container.
275  * - c_loop_subset will loop over a subset of particles in a container that lie
276  * within some geometrical region. It can loop over particles in a
277  * rectangular box, particles in a sphere, or particles that lie within
278  * specific internal computational blocks.
279  * - c_loop_order will loop over a specific list of particles that were
280  * previously stored in a particle_order class.
281  *
282  * Several of the key routines within the container classes (such as
283  * draw_cells_gnuplot and print_custom) have versions where they can be passed
284  * a loop class to use. Loop classes can also be used directly and there are
285  * some examples on the library website that demonstrate this. It is also
286  * possible to write custom loop classes.
287  *
288  * In addition to the loop classes mentioned above, there is also a
289  * c_loop_all_periodic class, that is specifically for use with the
290  * container_periodic and container_periodic_poly classes. Since the data
291  * structures of these containers differ considerably, it requires a different
292  * loop class that is not interoperable with the others.
293  *
294  * \section pre_container The pre_container classes
295  * Voro++ makes use of internal computational grid of blocks that are used to
296  * configure the code for maximum efficiency. As discussed on the library
297  * website, the best performance is achieved for around 5 particles per block,
298  * with anything in the range from 3 to 12 giving good performance. Usually
299  * the size of the grid can be chosen by ensuring that the number of blocks is
300  * equal to the number of particles divided by 5.
301  *
302  * However, this can be difficult to choose in cases when the number of
303  * particles is not known a priori, and in thes cases the pre_container classes
304  * can be used. They can import an arbitrary number of particle positions from
305  * a file, dynamically allocating memory in chunks as necessary. Once particles
306  * are imported, they can guess an optimal block arrangement to use for the
307  * container class, and then transfer the particles to the container. By
308  * default, this procedure is used by the command-line utility to enable it to
309  * work well with arbitrary sizes of input data.
310  *
311  * The pre_container class can be used when no particle radius information is
312  * available, and the pre_container_poly class can be used when radius
313  * information is available. At present, the pre_container classes can only be
314  * used with the container and container_poly classes. They do not support
315  * the container_periodic and container_periodic_poly classes. */
316 
317 #ifndef VOROPP_HH
318 #define VOROPP_HH
319 
320 #include "config.hh"
321 #include "common.hh"
322 #include "cell.hh"
323 #include "v_base.hh"
324 #include "rad_option.hh"
325 #include "container.hh"
326 #include "unitcell.hh"
327 #include "container_prd.hh"
328 #include "pre_container.hh"
329 #include "v_compute.hh"
330 #include "c_loops.hh"
331 #include "wall.hh"
332 
333 #endif