1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
#![cfg_attr(feature = "dev", feature(plugin))]
#![cfg_attr(feature = "dev", plugin(clippy))]
#![warn(missing_docs)]
#![feature(vec_remove_item)]
extern crate nalgebra;
extern crate rand;
#[cfg(feature = "serde-1")]
extern crate serde;
pub mod shapes;
pub mod util;
#[cfg(feature = "serde-1")]
mod serialization;
use nalgebra::Point3;
use nalgebra::core::{Matrix, Matrix3};
use rand::Rng;
use rand::distributions::IndependentSample;
use std::iter::repeat;
use shapes::Sphere;
pub trait Container {
fn contains(&self, sphere: &Sphere) -> bool;
fn volume(&self) -> f32;
}
#[derive(Debug)]
pub struct PackedVolume<C> {
pub spheres: Vec<Sphere>,
pub container: C,
}
impl<C: Container> PackedVolume<C> {
pub fn new<R: IndependentSample<f64>>(
container: C,
mut size_distribution: &mut R,
) -> PackedVolume<C> {
PackedVolume::<C> {
spheres: pack_spheres::<C, R>(&container, &mut size_distribution),
container: container,
}
}
pub fn from_vec(spheres: Vec<Sphere>, container: C) -> PackedVolume<C> {
PackedVolume::<C> {
spheres: spheres,
container: container,
}
}
pub fn volume_fraction(&self) -> f32 {
let vol_spheres: f32 = self.spheres.iter().map(|sphere| sphere.volume()).sum();
vol_spheres / self.container.volume()
}
pub fn void_ratio(&self) -> f32 {
let vol_spheres: f32 = self.spheres.iter().map(|sphere| sphere.volume()).sum();
let vol_total = self.container.volume();
(vol_total - vol_spheres) / vol_spheres
}
pub fn coordination_number(&self) -> f32 {
let num_particles = self.spheres.len() as f32;
let mut coordinations = 0;
for idx in 0..self.spheres.len() {
coordinations += self.sphere_contacts_count(idx);
}
coordinations as f32 / num_particles
}
pub fn fabric_tensor(&self) -> Matrix3<f32> {
let phi = |i: usize, j: usize| {
let mut sum_all = 0.;
for idx in 0..self.spheres.len() {
let center = self.spheres[idx].center.coords;
let p_c = self.sphere_contacts(idx);
let m_p = p_c.len() as f32;
let mut sum_vec = 0.;
for c in p_c.iter() {
let vec_n_pc = Matrix::cross(¢er, &c.center.coords);
let n_pc = vec_n_pc / nalgebra::norm(&vec_n_pc);
sum_vec += n_pc[i] * n_pc[j];
}
sum_all += sum_vec / m_p;
}
1. / self.spheres.len() as f32 * sum_all
};
Matrix3::from_fn(|r, c| phi(r, c))
}
fn sphere_contacts(&self, sphere_idx: usize) -> Vec<Sphere> {
let center = self.spheres[sphere_idx].center;
let radius = self.spheres[sphere_idx].radius;
self.spheres
.iter()
.cloned()
.filter(|sphere| {
(nalgebra::distance(¢er, &sphere.center) - (radius + sphere.radius)).abs() <
0.001
})
.collect()
}
fn sphere_contacts_count(&self, sphere_idx: usize) -> usize {
let center = self.spheres[sphere_idx].center;
let radius = self.spheres[sphere_idx].radius;
self.spheres
.iter()
.filter(|sphere| {
(nalgebra::distance(¢er, &sphere.center) - (radius + sphere.radius)).abs() <
0.001
})
.count()
}
}
pub fn pack_spheres<C: Container, R: IndependentSample<f64>>(
container: &C,
size_distribution: &mut R,
) -> Vec<Sphere> {
let mut rng = rand::thread_rng();
let init_radii: [f32; 3] = [
size_distribution.ind_sample(&mut rng) as f32,
size_distribution.ind_sample(&mut rng) as f32,
size_distribution.ind_sample(&mut rng) as f32,
];
let mut spheres = init_spheres(&init_radii, container);
let mut front = spheres.clone();
let mut new_radius = size_distribution.ind_sample(&mut rng) as f32;
'outer: while !front.is_empty() {
let curr_sphere = rng.choose(&front).unwrap().clone();
let set_v = spheres
.iter()
.cloned()
.filter(|s_dash| {
s_dash != &curr_sphere &&
nalgebra::distance(&curr_sphere.center, &s_dash.center) <=
curr_sphere.radius + s_dash.radius + 2. * new_radius
})
.collect::<Vec<_>>();
for (s_i, s_j) in pairs(&set_v) {
let mut set_f = identify_f(&curr_sphere, s_i, s_j, container, &set_v, new_radius);
if !set_f.is_empty() {
let s_new = rng.choose(&set_f).unwrap();
front.push(s_new.clone());
spheres.push(s_new.clone());
new_radius = size_distribution.ind_sample(&mut rng) as f32;
continue 'outer;
}
}
front.remove_item(&curr_sphere);
}
spheres
}
fn init_spheres<C: Container>(radii: &[f32; 3], container: &C) -> Vec<Sphere> {
let mut init = Vec::new();
let radius_a = radii[0];
let radius_b = radii[1];
let radius_c = radii[2];
let distance_c = radius_a + radius_b;
let distance_b = radius_a + radius_c;
let distance_a = radius_c + radius_b;
let x = (distance_b.powi(2) + distance_c.powi(2) - distance_a.powi(2)) / (2. * distance_c);
let y = (distance_b.powi(2) - x.powi(2)).sqrt();
let perimeter = distance_a + distance_b + distance_c;
let incenter_x = (distance_b * distance_c + distance_c * x) / perimeter;
let incenter_y = (distance_c * y) / perimeter;
init.push(Sphere::new(
Point3::new(-incenter_x, -incenter_y, 0.),
radius_a,
));
init.push(Sphere::new(
Point3::new(distance_c - incenter_x, -incenter_y, 0.),
radius_b,
));
init.push(Sphere::new(
Point3::new(x - incenter_x, y - incenter_y, 0.),
radius_c,
));
assert!(init.iter().all(|sphere| container.contains(&sphere)));
init
}
fn identify_f<C: Container>(
s_1: &Sphere,
s_2: &Sphere,
s_3: &Sphere,
container: &C,
set_v: &Vec<Sphere>,
radius: f32,
) -> Vec<Sphere> {
let distance_14 = s_1.radius + radius;
let distance_24 = s_2.radius + radius;
let distance_34 = s_3.radius + radius;
let vector_u = s_1.center - s_2.center;
let unitvector_u = vector_u / nalgebra::norm(&vector_u);
let vector_v = s_1.center - s_3.center;
let unitvector_v = vector_v / nalgebra::norm(&vector_v);
let cross_uv = Matrix::cross(&vector_u, &vector_v);
let unitvector_t = cross_uv / nalgebra::norm(&cross_uv);
let vector_w = -2. * s_1.center.coords;
let distance_a = (distance_24.powi(2) - distance_14.powi(2) + s_1.center.x.powi(2) +
s_1.center.y.powi(2) + s_1.center.z.powi(2) -
s_2.center.x.powi(2) -
s_2.center.y.powi(2) - s_2.center.z.powi(2)) /
(2. * nalgebra::norm(&vector_u));
let distance_b = (distance_34.powi(2) - distance_14.powi(2) + s_1.center.x.powi(2) +
s_1.center.y.powi(2) + s_1.center.z.powi(2) -
s_3.center.x.powi(2) -
s_3.center.y.powi(2) - s_3.center.z.powi(2)) /
(2. * nalgebra::norm(&vector_v));
let distance_c = distance_14.powi(2) - s_1.center.x.powi(2) - s_1.center.y.powi(2) -
s_1.center.z.powi(2);
let dot_uv = nalgebra::dot(&unitvector_u, &unitvector_v);
let dot_wt = nalgebra::dot(&vector_w, &unitvector_t);
let dot_uw = nalgebra::dot(&unitvector_u, &vector_w);
let dot_vw = nalgebra::dot(&unitvector_v, &vector_w);
let alpha = (distance_a - distance_b * dot_uv) / (1. - dot_uv.powi(2));
let beta = (distance_b - distance_a * dot_uv) / (1. - dot_uv.powi(2));
let value_d = alpha.powi(2) + beta.powi(2) + 2. * alpha * beta * dot_uv + alpha * dot_uw +
beta * dot_vw - distance_c;
let dot_wt_2 = dot_wt.powi(2);
let value_4d = 4. * value_d;
let mut f = Vec::new();
if dot_wt_2 > value_4d {
let gamma_pos = 0.5 * (-dot_wt + (dot_wt.powi(2) - 4. * value_d).sqrt());
let gamma_neg = 0.5 * (-dot_wt - (dot_wt.powi(2) - 4. * value_d).sqrt());
let s_4_positive = Sphere::new(
Point3::from_coordinates(
alpha * unitvector_u + beta * unitvector_v + gamma_pos * unitvector_t,
),
radius,
);
let s_4_negative = Sphere::new(
Point3::from_coordinates(
alpha * unitvector_u + beta * unitvector_v + gamma_neg * unitvector_t,
),
radius,
);
if container.contains(&s_4_positive) && !set_v.iter().any(|v| v.overlaps(&s_4_positive)) {
f.push(s_4_positive);
}
if container.contains(&s_4_negative) && !set_v.iter().any(|v| v.overlaps(&s_4_negative)) {
f.push(s_4_negative);
}
}
f
}
fn pairs(set: &[Sphere]) -> Vec<(&Sphere, &Sphere)> {
let n = set.len();
if n == 2 {
let mut minimal = Vec::new();
minimal.push((&set[0], &set[1]));
minimal
} else {
let mut vec_pairs = Vec::new();
if n > 2 {
for k in 0..n - 1 {
let subset = &set[k + 1..n];
vec_pairs.append(&mut subset
.iter()
.zip(repeat(&set[k]).take(subset.len()))
.collect::<Vec<_>>());
}
}
vec_pairs
}
}