<p>Voro++ is a software library for carrying out three-dimensional computations of the Voronoi tessellation. A distinguishing feature of the Voro++ library is that it carries out cell-based calculations, computing the Voronoi cell for each particle individually, rather than computing the Voronoi tessellation as a global network of vertices and edges. It is particularly well-suited for applications that rely on cell-based statistics, where features of Voronoi cells (eg. volume, centroid, number of faces) can be used to analyze a system of particles.</p>
<p>Voro++ is written in C++ and can be built as a static library that can be linked to. This manual provides a reference for every function in the class structure. For a general overview of the program, see the Voro++ website at <a href="http://math.lbl.gov/voro++/">http://math.lbl.gov/voro++/</a> and in particular the example programs at <a href="http://math.lbl.gov/voro++/examples/">http://math.lbl.gov/voro++/examples/</a> that demonstrate many of the library's features.</p>
<h1><a class="anchor" id="class"></a>
C++ class structure</h1>
<p>The code is structured around several C++ classes. The voronoicell_base class contains all of the routines for constructing a single Voronoi cell. It represents the cell as a collection of vertices that are connected by edges, and there are routines for initializing, making, and outputting the cell. The voronoicell_base class form the base of the voronoicell and voronoicell_neighbor classes, which add specialized routines depending on whether neighboring particle ID information for each face must be tracked or not. Collectively, these classes are referred to as "voronoicell classes" within the documentation.</p>
<p>There is a hierarchy of classes that represent three-dimensional particle systems. All of these are derived from the voro_base class, which contains constants that divide a three-dimensional system into a rectangular grid of equally-sized rectangular blocks; this grid is used for computational efficiency during the Voronoi calculations.</p>
<p>The container_base, container, and container_poly are then derived from the voro_base class to represent a particle system in a specific three-dimensional rectangular box using both periodic and non-periodic boundary conditions. In addition, the container_periodic_base, container_periodic, and container_periodic_poly classes represent a particle system in a three-dimensional non-orthogonal periodic domain, defined by three periodicity vectors that represent a parallelepiped. Collectively, these classes are referred to as "container classes" within the documentation.</p>
<p>The voro_compute template encapsulates all of the routines for computing Voronoi cells. Each container class has a voro_compute template within it, that accesses the container's particle system, and computes the Voronoi cells.</p>
<p>There are several wall classes that can be used to apply certain boundary conditions using additional plane cuts during the Voronoi cell compution. The code also contains a number of small loop classes, c_loop_all, c_loop_subset, c_loop_all_periodic, and c_loop_order that can be used to iterate over a certain subset of particles in a container. The latter class makes use of a special particle_order class that stores a specific order of particles within the container. The library also contains the classes pre_container_base, pre_container, and pre_container_poly, that can be used as temporary storage when importing data of unknown size.</p>
<h1><a class="anchor" id="voronoicell"></a>
The voronoicell classes</h1>
<p>The voronoicell class represents a single Voronoi cell as a convex polyhedron, with a set of vertices that are connected by edges. The class contains a variety of functions that can be used to compute and output the Voronoi cell corresponding to a particular particle. The command init() can be used to initialize a cell as a large rectangular box. The Voronoi cell can then be computed by repeatedly cutting it with planes that correspond to the perpendicular bisectors between that particle and its neighbors.</p>
<p>This is achieved by using the plane() routine, which will recompute the cell's vertices and edges after cutting it with a single plane. This is the key routine in voronoicell class. It begins by exploiting the convexity of the underlying cell, tracing between edges to work out if the cell intersects the cutting plane. If it does not intersect, then the routine immediately exits. Otherwise, it finds an edge or vertex that intersects the plane, and from there, traces out a new face on the cell, recomputing the edge and vertex structure accordingly.</p>
<p>Once the cell is computed, there are many routines for computing features of the the Voronoi cell, such as its volume, surface area, or centroid. There are also many routines for outputting features of the Voronoi cell, or writing its shape in formats that can be read by Gnuplot or POV-Ray.</p>
<h2><a class="anchor" id="internal"></a>
Internal data representation</h2>
<p>The voronoicell class has a public member p representing the number of vertices. The polyhedral structure of the cell is stored in the following arrays:</p>
<ul>
<li>pts: a one-dimensional array of floating point numbers, that represent the position vectors x_0, x_1, ..., x_{p-1} of the polyhedron vertices.</li>
<li>nu: the order of each vertex n_0, n_1, ..., n_{p-1}, corresponding to the number of other vertices to which each is connected.</li>
<li>ed: a two-dimensional table of edges and relations. For the ith vertex, ed[i] has 2n_i+1 elements. The first n_i elements are the edges e(j,i), where e(j,i) is the jth neighbor of vertex i. The edges are ordered according to a right-hand rule with respect to an outward-pointing normal. The next n_i elements are the relations l(j,i) which satisfy the property e(l(j,i),e(j,i)) = i. The final element of the ed[i] list is a back pointer used in memory allocation.</li>
</ul>
<p>In a very large number of cases, the values of n_i will be 3. This is because the only way that a higher-order vertex can be created in the plane() routine is if the cutting plane perfectly intersects an existing vertex. For random particle arrangements with position vectors specified to double precision this should happen very rarely. A preliminary version of this code was quite successful with only making use of vertices of order 3. However, when calculating millions of cells, it was found that this approach is not robust, since a single floating point error can invalidate the computation. This can also be a problem for cases featuring crystalline arrangements of particles where the corresponding Voronoi cells may have high-order vertices by construction.</p>
<p>Because of this, Voro++ takes the approach that it if an existing vertex is within a small numerical tolerance of the cutting plane, it is treated as being exactly on the plane, and the polyhedral topology is recomputed accordingly. However, while this improves robustness, it also adds the complexity that n_i may no longer always be 3. This causes memory management to be significantly more complicated, as different vertices require a different number of elements in the ed[][] array. To accommodate this, the voronoicell class allocated edge memory in a different array called mep[][], in such a way that all vertices of order k are held in mep[k]. If vertex i has order k, then ed[i] points to memory within mep[k]. The array ed[][] is never directly initialized as a two-dimensional array itself, but points at allocations within mep[][]. To the user, it appears as though each row of ed[][] has a different number of elements. When vertices are added or deleted, care must be taken to reorder and reassign elements in these arrays.</p>
<p>During the plane() routine, the code traces around the vertices of the cell, and adds new vertices along edges which intersect the cutting plane to create a new face. The values of l(j,i) are used in this computation, as when the code is traversing from one vertex on the cell to another, this information allows the code to immediately work out which edge of a vertex points back to the one it came from. As new vertices are created, the l(j,i) are also updated to ensure consistency. To ensure robustness, the plane cutting algorithm should work with any possible combination of vertices which are inside, outside, or exactly on the cutting plane.</p>
<p>Vertices exactly on the cutting plane create some additional computational difficulties. If there are two marginal vertices connected by an existing edge, then it would be possible for duplicate edges to be created between those two vertices, if the plane routine traces along both sides of this edge while constructing the new face. The code recognizes these cases and prevents the double edge from being formed. Another possibility is the formation of vertices of order two or one. At the end of the plane cutting routine, the code checks to see if any of these are present, removing the order one vertices by just deleting them, and removing the order two vertices by connecting the two neighbors of each vertex together. It is possible that the removal of a single low-order vertex could result in the creation of additional low-order vertices, so the process is applied recursively until no more are left.</p>
<h1><a class="anchor" id="container"></a>
The container classes</h1>
<p>There are four container classes available for general usage: container, container_poly, container_periodic, and container_periodic_poly. Each of these represent a system of particles in a specific three-dimensional geometry. They contain routines for importing particles from a text file, and adding particles individually. They also contain a large number of analyzing and outputting the particle system. Internally, the routines that compute Voronoi cells do so by making use of the voro_compute template. Each container class contains routines that tell the voro_compute template about the specific geometry of this container.</p>
<h1><a class="anchor" id="voro_compute"></a>
The voro_compute template</h1>
<p>The voro_compute template encapsulates the routines for carrying out the Voronoi cell computations. It contains data structures suchs as a mask and a queue that are used in the computations. The voro_compute template is associated with a specific container class, and during the computation, it calls routines in the container class to access the particle positions that are stored there.</p>
<p>The key routine in this class is compute_cell(), which makes use of a voronoicell class to construct a Voronoi cell for a specific particle in the container. The basic approach that this function takes is to repeatedly cut the Voronoi cell by planes corresponding neighboring particles, and stop when it recognizes that all the remaining particles in the container are too far away to possibly influence cell's shape. The code makes use of two possible methods for working out when a cell computation is complete:</p>
<ul>
<li>Radius test: if the maximum distance of a Voronoi cell vertex from the cell center is R, then no particles more than a distance 2R away can possibly influence the cell. This a very fast computation to do, but it has no directionality: if the cell extends a long way in one direction then particles a long distance in other directions will still need to be tested.</li>
<li>Region test: it is possible to test whether a specific region can possibly influence the cell by applying a series of plane tests at the point on the region which is closest to the Voronoi cell center. This is a slower computation to do, but it has directionality.</li>
</ul>
<p>Another useful observation is that the regions that need to be tested are simply connected, meaning that if a particular region does not need to be tested, then neighboring regions which are further away do not need to be tested.</p>
<p>For maximum efficiency, it was found that a hybrid approach making use of both of the above tests worked well in practice. Radius tests work well for the first few blocks, but switching to region tests after then prevent the code from becoming extremely slow, due to testing over very large spherical shells of particles. The compute_cell() routine therefore takes the following approach:</p>
<ul>
<li>Initialize the voronoicell class to fill the entire computational domain.</li>
<li>Cut the cell by any wall objects that have been added to the container.</li>
<li>Apply plane cuts to the cell corresponding to the other particles which are within the current particle's region.</li>
<li>Test over a pre-computed worklist of neighboring regions, that have been ordered according to the minimum distance away from the particle's position. Apply radius tests after every few regions to see if the calculation can terminate.</li>
<li>If the code reaches the end of the worklist, add all the neighboring regions to a new list.</li>
<li>Carry out a region test on the first item of the list. If the region needs to be tested, apply the plane() routine for all of its particles, and then add any neighboring regions to the end of the list that need to be tested. Continue until the list has no elements left.</li>
</ul>
<p>The compute_cell() routine forms the basis of many other routines, such as store_cell_volumes() and draw_cells_gnuplot() that can be used to calculate and draw the cells in a container.</p>
<h1><a class="anchor" id="walls"></a>
Wall computation</h1>
<p>Wall computations are handled by making use of a pure virtual wall class. Specific wall types are derived from this class, and require the specification of two routines: point_inside() that tests to see if a point is inside a wall or not, and cut_cell() that cuts a cell according to the wall's position. The walls can be added to the container using the add_wall() command, and these are called each time a compute_cell() command is carried out. At present, wall types for planes, spheres, cylinders, and cones are provided, although custom walls can be added by creating new classes derived from the pure virtual class. Currently all wall types approximate the wall surface with a single plane, which produces some small errors, but generally gives good results for dense particle packings in direct contact with a wall surface. It would be possible to create more accurate walls by making cut_cell() routines that approximate the curved surface with multiple plane cuts.</p>
<p>The wall objects can used for periodic calculations, although to obtain valid results, the walls should also be periodic as well. For example, in a domain that is periodic in the x direction, a cylinder aligned along the x axis could be added. At present, the interior of all wall objects are convex domains, and consequently any superposition of them will be a convex domain also. Carrying out computations in non-convex domains poses some problems, since this could theoretically lead to non-convex Voronoi cells, which the internal data representation of the voronoicell class does not support. For non-convex cases where the wall surfaces feature just a small amount of negative curvature (eg. a torus) approximating the curved surface with a single plane cut may give an acceptable level of accuracy. For non-convex cases that feature internal angles, the best strategy may be to decompose the domain into several convex subdomains, carry out a calculation in each, and then add the results together. The voronoicell class cannot be easily modified to handle non-convex cells as this would fundamentally alter the algorithms that it uses, and cases could arise where a single plane cut could create several new faces as opposed to just one.</p>
<h1><a class="anchor" id="loops"></a>
Loop classes</h1>
<p>The container classes have a number of simple routines for calculating Voronoi cells for all particles within them. However, in some situations it is desirable to iterate over a specific subset of particles. This can be achieved with the c_loop classes that are all derived from the c_loop_base class. Each class can iterate over a specific subset of particles in a container. There are three loop classes for use with the container and container_poly classes:</p>
<ul>
<li>c_loop_all will loop over all of the particles in a container.</li>
<li>c_loop_subset will loop over a subset of particles in a container that lie within some geometrical region. It can loop over particles in a rectangular box, particles in a sphere, or particles that lie within specific internal computational blocks.</li>
<li>c_loop_order will loop over a specific list of particles that were previously stored in a particle_order class.</li>
</ul>
<p>Several of the key routines within the container classes (such as draw_cells_gnuplot and print_custom) have versions where they can be passed a loop class to use. Loop classes can also be used directly and there are some examples on the library website that demonstrate this. It is also possible to write custom loop classes.</p>
<p>In addition to the loop classes mentioned above, there is also a c_loop_all_periodic class, that is specifically for use with the container_periodic and container_periodic_poly classes. Since the data structures of these containers differ considerably, it requires a different loop class that is not interoperable with the others.</p>
<h1><a class="anchor" id="pre_container"></a>
The pre_container classes</h1>
<p>Voro++ makes use of internal computational grid of blocks that are used to configure the code for maximum efficiency. As discussed on the library website, the best performance is achieved for around 5 particles per block, with anything in the range from 3 to 12 giving good performance. Usually the size of the grid can be chosen by ensuring that the number of blocks is equal to the number of particles divided by 5.</p>
<p>However, this can be difficult to choose in cases when the number of particles is not known a priori, and in thes cases the pre_container classes can be used. They can import an arbitrary number of particle positions from a file, dynamically allocating memory in chunks as necessary. Once particles are imported, they can guess an optimal block arrangement to use for the container class, and then transfer the particles to the container. By default, this procedure is used by the command-line utility to enable it to work well with arbitrary sizes of input data.</p>
<p>The pre_container class can be used when no particle radius information is available, and the pre_container_poly class can be used when radius information is available. At present, the pre_container classes can only be used with the container and container_poly classes. They do not support the container_periodic and container_periodic_poly classes. </p>