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1 {-# LANGUAGE BangPatterns #-}
2 -- | The Grid module just contains the Grid type and two constructors
3 -- for it. We hide the main Grid constructor because we don't want
4 -- to allow instantiation of a grid with h <= 0.
5 module Grid (
6 cube_at,
7 grid_tests,
8 make_grid,
9 slow_tests,
10 zoom
11 )
12 where
13
14 import qualified Data.Array.Repa as R
15 import Test.HUnit (Assertion, assertEqual)
16 import Test.Framework (Test, testGroup)
17 import Test.Framework.Providers.HUnit (testCase)
18 import Test.Framework.Providers.QuickCheck2 (testProperty)
19 import Test.QuickCheck ((==>),
20 Arbitrary(..),
21 Gen,
22 Positive(..),
23 Property,
24 choose)
25 import Assertions (assertAlmostEqual, assertTrue)
26 import Comparisons ((~=))
27 import Cube (Cube(Cube),
28 find_containing_tetrahedron,
29 tetrahedra,
30 tetrahedron)
31 import Examples (trilinear, trilinear9x9x9, zeros, naturals_1d)
32 import FunctionValues (make_values, value_at)
33 import Point (Point(..))
34 import ScaleFactor (ScaleFactor)
35 import Tetrahedron (Tetrahedron, c, polynomial, v0, v1, v2, v3)
36 import ThreeDimensional (ThreeDimensional(..))
37 import Values (Values3D, dims, empty3d, zoom_shape)
38
39
40 -- | Our problem is defined on a Grid. The grid size is given by the
41 -- positive number h. The function values are the values of the
42 -- function at the grid points, which are distance h from one
43 -- another in each direction (x,y,z).
44 data Grid = Grid { h :: Double, -- MUST BE GREATER THAN ZERO!
45 function_values :: Values3D }
46 deriving (Show)
47
48
49 instance Arbitrary Grid where
50 arbitrary = do
51 (Positive h') <- arbitrary :: Gen (Positive Double)
52 fvs <- arbitrary :: Gen Values3D
53 return (make_grid h' fvs)
54
55
56 -- | The constructor that we want people to use. If we're passed a
57 -- non-positive grid size, we throw an error.
58 make_grid :: Double -> Values3D -> Grid
59 make_grid grid_size values
60 | grid_size <= 0 = error "grid size must be positive"
61 | otherwise = Grid grid_size values
62
63
64
65 -- | Takes a grid and a position as an argument and returns the cube
66 -- centered on that position. If there is no cube there (i.e. the
67 -- position is outside of the grid), it will throw an error.
68 cube_at :: Grid -> Int -> Int -> Int -> Cube
69 cube_at !g !i !j !k
70 | i < 0 = error "i < 0 in cube_at"
71 | i >= xsize = error "i >= xsize in cube_at"
72 | j < 0 = error "j < 0 in cube_at"
73 | j >= ysize = error "j >= ysize in cube_at"
74 | k < 0 = error "k < 0 in cube_at"
75 | k >= zsize = error "k >= zsize in cube_at"
76 | otherwise = Cube delta i j k fvs' tet_vol
77 where
78 fvs = function_values g
79 (xsize, ysize, zsize) = dims fvs
80 fvs' = make_values fvs i j k
81 delta = h g
82 tet_vol = (1/24)*(delta^(3::Int))
83
84 -- The first cube along any axis covers (-h/2, h/2). The second
85 -- covers (h/2, 3h/2). The third, (3h/2, 5h/2), and so on.
86 --
87 -- We translate the (x,y,z) coordinates forward by 'h/2' so that the
88 -- first covers (0, h), the second covers (h, 2h), etc. This makes
89 -- it easy to figure out which cube contains the given point.
90 calculate_containing_cube_coordinate :: Grid -> Double -> Int
91 calculate_containing_cube_coordinate g coord
92 -- Don't use a cube on the boundary if we can help it. This
93 -- returns cube #1 if we would have returned cube #0 and cube #1
94 -- exists.
95 | coord < offset = 0
96 | coord == offset && (xsize > 1 && ysize > 1 && zsize > 1) = 1
97 | otherwise = (ceiling ( (coord + offset) / cube_width )) - 1
98 where
99 (xsize, ysize, zsize) = dims (function_values g)
100 cube_width = (h g)
101 offset = cube_width / 2
102
103
104 -- | Takes a 'Grid', and returns a 'Cube' containing the given 'Point'.
105 -- Since our grid is rectangular, we can figure this out without having
106 -- to check every cube.
107 find_containing_cube :: Grid -> Point -> Cube
108 find_containing_cube g (Point x y z) =
109 cube_at g i j k
110 where
111 i = calculate_containing_cube_coordinate g x
112 j = calculate_containing_cube_coordinate g y
113 k = calculate_containing_cube_coordinate g z
114
115
116 zoom_lookup :: Values3D -> ScaleFactor -> a -> (R.DIM3 -> Double)
117 zoom_lookup v3d scale_factor _ =
118 zoom_result v3d scale_factor
119
120
121 zoom_result :: Values3D -> ScaleFactor -> R.DIM3 -> Double
122 zoom_result v3d (sfx, sfy, sfz) (R.Z R.:. m R.:. n R.:. o) =
123 f p
124 where
125 g = make_grid 1 v3d
126 offset = (h g)/2
127 m' = (fromIntegral m) / (fromIntegral sfx) - offset
128 n' = (fromIntegral n) / (fromIntegral sfy) - offset
129 o' = (fromIntegral o) / (fromIntegral sfz) - offset
130 p = Point m' n' o'
131 cube = find_containing_cube g p
132 t = find_containing_tetrahedron cube p
133 f = polynomial t
134
135
136 zoom :: Values3D -> ScaleFactor -> Values3D
137 zoom v3d scale_factor
138 | xsize == 0 || ysize == 0 || zsize == 0 = empty3d
139 | otherwise =
140 R.compute $ R.unsafeTraverse v3d transExtent f
141 where
142 (xsize, ysize, zsize) = dims v3d
143 transExtent = zoom_shape scale_factor
144 f = zoom_lookup v3d scale_factor
145
146
147 -- | Check all coefficients of tetrahedron0 belonging to the cube
148 -- centered on (1,1,1) with a grid constructed from the trilinear
149 -- values. See example one in the paper.
150 --
151 -- We also verify that the four vertices on face0 of the cube are
152 -- in the correct location.
153 --
154 trilinear_c0_t0_tests :: Test.Framework.Test
155 trilinear_c0_t0_tests =
156 testGroup "trilinear c0 t0"
157 [testGroup "coefficients"
158 [testCase "c0030 is correct" test_trilinear_c0030,
159 testCase "c0003 is correct" test_trilinear_c0003,
160 testCase "c0021 is correct" test_trilinear_c0021,
161 testCase "c0012 is correct" test_trilinear_c0012,
162 testCase "c0120 is correct" test_trilinear_c0120,
163 testCase "c0102 is correct" test_trilinear_c0102,
164 testCase "c0111 is correct" test_trilinear_c0111,
165 testCase "c0210 is correct" test_trilinear_c0210,
166 testCase "c0201 is correct" test_trilinear_c0201,
167 testCase "c0300 is correct" test_trilinear_c0300,
168 testCase "c1020 is correct" test_trilinear_c1020,
169 testCase "c1002 is correct" test_trilinear_c1002,
170 testCase "c1011 is correct" test_trilinear_c1011,
171 testCase "c1110 is correct" test_trilinear_c1110,
172 testCase "c1101 is correct" test_trilinear_c1101,
173 testCase "c1200 is correct" test_trilinear_c1200,
174 testCase "c2010 is correct" test_trilinear_c2010,
175 testCase "c2001 is correct" test_trilinear_c2001,
176 testCase "c2100 is correct" test_trilinear_c2100,
177 testCase "c3000 is correct" test_trilinear_c3000],
178
179 testGroup "face0 vertices"
180 [testCase "v0 is correct" test_trilinear_f0_t0_v0,
181 testCase "v1 is correct" test_trilinear_f0_t0_v1,
182 testCase "v2 is correct" test_trilinear_f0_t0_v2,
183 testCase "v3 is correct" test_trilinear_f0_t0_v3]
184 ]
185 where
186 g = make_grid 1 trilinear
187 cube = cube_at g 1 1 1
188 t = tetrahedron cube 0
189
190 test_trilinear_c0030 :: Assertion
191 test_trilinear_c0030 =
192 assertAlmostEqual "c0030 is correct" (c t 0 0 3 0) (17/8)
193
194 test_trilinear_c0003 :: Assertion
195 test_trilinear_c0003 =
196 assertAlmostEqual "c0003 is correct" (c t 0 0 0 3) (27/8)
197
198 test_trilinear_c0021 :: Assertion
199 test_trilinear_c0021 =
200 assertAlmostEqual "c0021 is correct" (c t 0 0 2 1) (61/24)
201
202 test_trilinear_c0012 :: Assertion
203 test_trilinear_c0012 =
204 assertAlmostEqual "c0012 is correct" (c t 0 0 1 2) (71/24)
205
206 test_trilinear_c0120 :: Assertion
207 test_trilinear_c0120 =
208 assertAlmostEqual "c0120 is correct" (c t 0 1 2 0) (55/24)
209
210 test_trilinear_c0102 :: Assertion
211 test_trilinear_c0102 =
212 assertAlmostEqual "c0102 is correct" (c t 0 1 0 2) (73/24)
213
214 test_trilinear_c0111 :: Assertion
215 test_trilinear_c0111 =
216 assertAlmostEqual "c0111 is correct" (c t 0 1 1 1) (8/3)
217
218 test_trilinear_c0210 :: Assertion
219 test_trilinear_c0210 =
220 assertAlmostEqual "c0210 is correct" (c t 0 2 1 0) (29/12)
221
222 test_trilinear_c0201 :: Assertion
223 test_trilinear_c0201 =
224 assertAlmostEqual "c0201 is correct" (c t 0 2 0 1) (11/4)
225
226 test_trilinear_c0300 :: Assertion
227 test_trilinear_c0300 =
228 assertAlmostEqual "c0300 is correct" (c t 0 3 0 0) (5/2)
229
230 test_trilinear_c1020 :: Assertion
231 test_trilinear_c1020 =
232 assertAlmostEqual "c1020 is correct" (c t 1 0 2 0) (8/3)
233
234 test_trilinear_c1002 :: Assertion
235 test_trilinear_c1002 =
236 assertAlmostEqual "c1002 is correct" (c t 1 0 0 2) (23/6)
237
238 test_trilinear_c1011 :: Assertion
239 test_trilinear_c1011 =
240 assertAlmostEqual "c1011 is correct" (c t 1 0 1 1) (13/4)
241
242 test_trilinear_c1110 :: Assertion
243 test_trilinear_c1110 =
244 assertAlmostEqual "c1110 is correct" (c t 1 1 1 0) (23/8)
245
246 test_trilinear_c1101 :: Assertion
247 test_trilinear_c1101 =
248 assertAlmostEqual "c1101 is correct" (c t 1 1 0 1) (27/8)
249
250 test_trilinear_c1200 :: Assertion
251 test_trilinear_c1200 =
252 assertAlmostEqual "c1200 is correct" (c t 1 2 0 0) 3
253
254 test_trilinear_c2010 :: Assertion
255 test_trilinear_c2010 =
256 assertAlmostEqual "c2010 is correct" (c t 2 0 1 0) (10/3)
257
258 test_trilinear_c2001 :: Assertion
259 test_trilinear_c2001 =
260 assertAlmostEqual "c2001 is correct" (c t 2 0 0 1) 4
261
262 test_trilinear_c2100 :: Assertion
263 test_trilinear_c2100 =
264 assertAlmostEqual "c2100 is correct" (c t 2 1 0 0) (7/2)
265
266 test_trilinear_c3000 :: Assertion
267 test_trilinear_c3000 =
268 assertAlmostEqual "c3000 is correct" (c t 3 0 0 0) 4
269
270 test_trilinear_f0_t0_v0 :: Assertion
271 test_trilinear_f0_t0_v0 =
272 assertEqual "v0 is correct" (v0 t) (Point 1 1 1)
273
274 test_trilinear_f0_t0_v1 :: Assertion
275 test_trilinear_f0_t0_v1 =
276 assertEqual "v1 is correct" (v1 t) (Point 0.5 1 1)
277
278 test_trilinear_f0_t0_v2 :: Assertion
279 test_trilinear_f0_t0_v2 =
280 assertEqual "v2 is correct" (v2 t) (Point 0.5 0.5 1.5)
281
282 test_trilinear_f0_t0_v3 :: Assertion
283 test_trilinear_f0_t0_v3 =
284 assertEqual "v3 is correct" (v3 t) (Point 0.5 1.5 1.5)
285
286
287 test_trilinear_reproduced :: Assertion
288 test_trilinear_reproduced =
289 assertTrue "trilinears are reproduced correctly" $
290 and [p (Point i' j' k') ~= value_at trilinear i j k
291 | i <- [0..2],
292 j <- [0..2],
293 k <- [0..2],
294 c0 <- cs,
295 t <- tetrahedra c0,
296 let p = polynomial t,
297 let i' = fromIntegral i,
298 let j' = fromIntegral j,
299 let k' = fromIntegral k]
300 where
301 g = make_grid 1 trilinear
302 cs = [ cube_at g ci cj ck | ci <- [0..2], cj <- [0..2], ck <- [0..2] ]
303
304
305 test_zeros_reproduced :: Assertion
306 test_zeros_reproduced =
307 assertTrue "the zero function is reproduced correctly" $
308 and [p (Point i' j' k') ~= value_at zeros i j k
309 | i <- [0..2],
310 j <- [0..2],
311 k <- [0..2],
312 let i' = fromIntegral i,
313 let j' = fromIntegral j,
314 let k' = fromIntegral k,
315 c0 <- cs,
316 t0 <- tetrahedra c0,
317 let p = polynomial t0 ]
318 where
319 g = make_grid 1 zeros
320 cs = [ cube_at g ci cj ck | ci <- [0..2], cj <- [0..2], ck <- [0..2] ]
321
322
323 -- | Make sure we can reproduce a 9x9x9 trilinear from the 3x3x3 one.
324 test_trilinear9x9x9_reproduced :: Assertion
325 test_trilinear9x9x9_reproduced =
326 assertTrue "trilinear 9x9x9 is reproduced correctly" $
327 and [p (Point i' j' k') ~= value_at trilinear9x9x9 i j k
328 | i <- [0..8],
329 j <- [0..8],
330 k <- [0..8],
331 t <- tetrahedra c0,
332 let p = polynomial t,
333 let i' = (fromIntegral i) * 0.5,
334 let j' = (fromIntegral j) * 0.5,
335 let k' = (fromIntegral k) * 0.5]
336 where
337 g = make_grid 1 trilinear
338 c0 = cube_at g 1 1 1
339
340
341 -- | The point 'p' in this test lies on the boundary of tetrahedra 12 and 15.
342 -- However, the 'contains_point' test fails due to some numerical innacuracy.
343 -- This bug should have been fixed by setting a positive tolerance level.
344 --
345 -- Example from before the fix:
346 --
347 -- b1 (tetrahedron c 20) (0, 17.5, 0.5)
348 -- -0.0
349 --
350 test_tetrahedra_collision_sensitivity :: Assertion
351 test_tetrahedra_collision_sensitivity =
352 assertTrue "tetrahedron collision tests isn't too sensitive" $
353 contains_point t20 p
354 where
355 g = make_grid 1 naturals_1d
356 cube = cube_at g 0 18 0
357 p = Point 0 17.5 0.5
358 t20 = tetrahedron cube 20
359
360
361 prop_cube_indices_never_go_out_of_bounds :: Grid -> Gen Bool
362 prop_cube_indices_never_go_out_of_bounds g =
363 do
364 let delta = Grid.h g
365 let coordmin = negate (delta/2)
366
367 let (xsize, ysize, zsize) = dims $ function_values g
368 let xmax = delta*(fromIntegral xsize) - (delta/2)
369 let ymax = delta*(fromIntegral ysize) - (delta/2)
370 let zmax = delta*(fromIntegral zsize) - (delta/2)
371
372 x <- choose (coordmin, xmax)
373 y <- choose (coordmin, ymax)
374 z <- choose (coordmin, zmax)
375
376 let idx_x = calculate_containing_cube_coordinate g x
377 let idx_y = calculate_containing_cube_coordinate g y
378 let idx_z = calculate_containing_cube_coordinate g z
379
380 return $
381 idx_x >= 0 &&
382 idx_x <= xsize - 1 &&
383 idx_y >= 0 &&
384 idx_y <= ysize - 1 &&
385 idx_z >= 0 &&
386 idx_z <= zsize - 1
387
388
389 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). Note that the
390 -- third and fourth indices of c-t10 have been switched. This is
391 -- because we store the triangles oriented such that their volume is
392 -- positive. If T and T-tilde share \<v1,v2,v3\> and v0,v0-tilde point
393 -- in opposite directions, one of them has to have negative volume!
394 prop_c0120_identity :: Grid -> Property
395 prop_c0120_identity g =
396 and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
397 c t0 0 1 2 0 ~= (c t0 1 0 2 0 + c t10 1 0 0 2) / 2
398 where
399 fvs = function_values g
400 (xsize, ysize, zsize) = dims fvs
401 cube0 = cube_at g 1 1 1
402 cube1 = cube_at g 0 1 1
403 t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
404 t10 = tetrahedron cube1 10
405
406
407 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
408 -- 'prop_c0120_identity'.
409 prop_c0111_identity :: Grid -> Property
410 prop_c0111_identity g =
411 and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
412 c t0 0 1 1 1 ~= (c t0 1 0 1 1 + c t10 1 0 1 1) / 2
413 where
414 fvs = function_values g
415 (xsize, ysize, zsize) = dims fvs
416 cube0 = cube_at g 1 1 1
417 cube1 = cube_at g 0 1 1
418 t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
419 t10 = tetrahedron cube1 10
420
421
422 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
423 -- 'prop_c0120_identity'.
424 prop_c0201_identity :: Grid -> Property
425 prop_c0201_identity g =
426 and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
427 c t0 0 2 0 1 ~= (c t0 1 1 0 1 + c t10 1 1 1 0) / 2
428 where
429 fvs = function_values g
430 (xsize, ysize, zsize) = dims fvs
431 cube0 = cube_at g 1 1 1
432 cube1 = cube_at g 0 1 1
433 t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
434 t10 = tetrahedron cube1 10
435
436
437 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
438 -- 'prop_c0120_identity'.
439 prop_c0102_identity :: Grid -> Property
440 prop_c0102_identity g =
441 and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
442 c t0 0 1 0 2 ~= (c t0 1 0 0 2 + c t10 1 0 2 0) / 2
443 where
444 fvs = function_values g
445 (xsize, ysize, zsize) = dims fvs
446 cube0 = cube_at g 1 1 1
447 cube1 = cube_at g 0 1 1
448 t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
449 t10 = tetrahedron cube1 10
450
451
452 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
453 -- 'prop_c0120_identity'.
454 prop_c0210_identity :: Grid -> Property
455 prop_c0210_identity g =
456 and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
457 c t0 0 2 1 0 ~= (c t0 1 1 1 0 + c t10 1 1 0 1) / 2
458 where
459 fvs = function_values g
460 (xsize, ysize, zsize) = dims fvs
461 cube0 = cube_at g 1 1 1
462 cube1 = cube_at g 0 1 1
463 t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
464 t10 = tetrahedron cube1 10
465
466
467 -- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
468 -- 'prop_c0120_identity'.
469 prop_c0300_identity :: Grid -> Property
470 prop_c0300_identity g =
471 and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
472 c t0 0 3 0 0 ~= (c t0 1 2 0 0 + c t10 1 2 0 0) / 2
473 where
474 fvs = function_values g
475 (xsize, ysize, zsize) = dims fvs
476 cube0 = cube_at g 1 1 1
477 cube1 = cube_at g 0 1 1
478 t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
479 t10 = tetrahedron cube1 10
480
481
482 -- | All of the properties from Section (2.9), p. 80. These require a
483 -- grid since they refer to two adjacent cubes.
484 p80_29_properties :: Test.Framework.Test
485 p80_29_properties =
486 testGroup "p. 80, Section (2.9) Properties" [
487 testProperty "c0120 identity" prop_c0120_identity,
488 testProperty "c0111 identity" prop_c0111_identity,
489 testProperty "c0201 identity" prop_c0201_identity,
490 testProperty "c0102 identity" prop_c0102_identity,
491 testProperty "c0210 identity" prop_c0210_identity,
492 testProperty "c0300 identity" prop_c0300_identity ]
493
494
495 grid_tests :: Test.Framework.Test
496 grid_tests =
497 testGroup "Grid Tests" [
498 trilinear_c0_t0_tests,
499 p80_29_properties,
500 testCase "tetrahedra collision test isn't too sensitive"
501 test_tetrahedra_collision_sensitivity,
502 testProperty "cube indices within bounds"
503 prop_cube_indices_never_go_out_of_bounds ]
504
505
506 -- Do the slow tests last so we can stop paying attention.
507 slow_tests :: Test.Framework.Test
508 slow_tests =
509 testGroup "Slow Tests" [
510 testCase "trilinear reproduced" test_trilinear_reproduced,
511 testCase "trilinear9x9x9 reproduced" test_trilinear9x9x9_reproduced,
512 testCase "zeros reproduced" test_zeros_reproduced ]