1 -- | The Grid module just contains the Grid type and two constructors
2 -- for it. We hide the main Grid constructor because we don't want
3 -- to allow instantiation of a grid with h <= 0.
13 import Data.Array (Array, array, (!))
14 import qualified Data.Array.Repa as R
16 import Test.Framework (Test, testGroup)
17 import Test.Framework.Providers.HUnit (testCase)
18 import Test.Framework.Providers.QuickCheck2 (testProperty)
19 import Test.QuickCheck (Arbitrary(..), Gen, Positive(..), choose)
23 import Cube (Cube(Cube),
24 find_containing_tetrahedron,
30 import PolynomialArray (PolynomialArray)
32 import Tetrahedron (Tetrahedron, c, number, polynomial, v0, v1, v2, v3)
33 import ThreeDimensional
34 import Values (Values3D, dims, empty3d, zoom_shape)
37 type CubeGrid = Array (Int,Int,Int) Cube
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 cube_grid :: CubeGrid }
50 instance Arbitrary Grid where
52 (Positive h') <- arbitrary :: Gen (Positive Double)
53 fvs <- arbitrary :: Gen Values3D
54 return (make_grid h' fvs)
57 -- | The constructor that we want people to use. If we're passed a
58 -- non-positive grid size, we throw an error.
59 make_grid :: Double -> Values3D -> Grid
60 make_grid grid_size values
61 | grid_size <= 0 = error "grid size must be positive"
62 | otherwise = Grid grid_size values (cubes grid_size values)
65 -- | Returns a three-dimensional array of cubes centered on the grid
66 -- points (h*i, h*j, h*k) with the appropriate 'FunctionValues'.
67 cubes :: Double -> Values3D -> CubeGrid
69 = array (lbounds, ubounds)
74 let tet_vol = (1/24)*(delta^(3::Int)),
76 Cube delta i j k (make_values fvs i j k) tet_vol]
82 ubounds = (xmax, ymax, zmax)
83 (xsize, ysize, zsize) = dims fvs
86 -- | Takes a grid and a position as an argument and returns the cube
87 -- centered on that position. If there is no cube there (i.e. the
88 -- position is outside of the grid), it will throw an error.
89 cube_at :: Grid -> Int -> Int -> Int -> Cube
91 | i < 0 = error "i < 0 in cube_at"
92 | i >= xsize = error "i >= xsize in cube_at"
93 | j < 0 = error "j < 0 in cube_at"
94 | j >= ysize = error "j >= ysize in cube_at"
95 | k < 0 = error "k < 0 in cube_at"
96 | k >= zsize = error "k >= zsize in cube_at"
97 | otherwise = (cube_grid g) ! (i,j,k)
99 fvs = function_values g
100 (xsize, ysize, zsize) = dims fvs
102 -- The first cube along any axis covers (-h/2, h/2). The second
103 -- covers (h/2, 3h/2). The third, (3h/2, 5h/2), and so on.
105 -- We translate the (x,y,z) coordinates forward by 'h/2' so that the
106 -- first covers (0, h), the second covers (h, 2h), etc. This makes
107 -- it easy to figure out which cube contains the given point.
108 calculate_containing_cube_coordinate :: Grid -> Double -> Int
109 calculate_containing_cube_coordinate g coord
110 -- Don't use a cube on the boundary if we can help it. This
111 -- returns cube #1 if we would have returned cube #0 and cube #1
114 | coord == offset && (xsize > 1 && ysize > 1 && zsize > 1) = 1
115 | otherwise = (ceiling ( (coord + offset) / cube_width )) - 1
117 (xsize, ysize, zsize) = dims (function_values g)
119 offset = cube_width / 2
122 -- | Takes a 'Grid', and returns a 'Cube' containing the given 'Point'.
123 -- Since our grid is rectangular, we can figure this out without having
124 -- to check every cube.
125 find_containing_cube :: Grid -> Point -> Cube
126 find_containing_cube g p =
130 i = calculate_containing_cube_coordinate g x
131 j = calculate_containing_cube_coordinate g y
132 k = calculate_containing_cube_coordinate g z
135 {-# INLINE zoom_lookup #-}
136 zoom_lookup :: Grid -> PolynomialArray -> ScaleFactor -> a -> (R.DIM3 -> Double)
137 zoom_lookup g polynomials scale_factor _ =
138 zoom_result g polynomials scale_factor
141 {-# INLINE zoom_result #-}
142 zoom_result :: Grid -> PolynomialArray -> ScaleFactor -> R.DIM3 -> Double
143 zoom_result g polynomials (sfx, sfy, sfz) (R.Z R.:. m R.:. n R.:. o) =
144 (polynomials ! (i, j, k, (number t))) p
147 m' = (fromIntegral m) / (fromIntegral sfx) - offset
148 n' = (fromIntegral n) / (fromIntegral sfy) - offset
149 o' = (fromIntegral o) / (fromIntegral sfz) - offset
150 p = (m', n', o') :: Point
151 cube = find_containing_cube g p
152 -- Figure out i,j,k without importing those functions.
153 Cube _ i j k _ _ = cube
154 t = find_containing_tetrahedron cube p
157 zoom :: Grid -> PolynomialArray -> ScaleFactor -> Values3D
158 zoom g polynomials scale_factor
159 | xsize == 0 || ysize == 0 || zsize == 0 = empty3d
161 R.force $ R.traverse arr transExtent (zoom_lookup g polynomials scale_factor)
163 arr = function_values g
164 (xsize, ysize, zsize) = dims arr
165 transExtent = zoom_shape scale_factor
170 -- | Check all coefficients of tetrahedron0 belonging to the cube
171 -- centered on (1,1,1) with a grid constructed from the trilinear
172 -- values. See example one in the paper.
174 -- We also verify that the four vertices on face0 of the cube are
175 -- in the correct location.
177 trilinear_c0_t0_tests :: Test.Framework.Test
178 trilinear_c0_t0_tests =
179 testGroup "trilinear c0 t0"
180 [testGroup "coefficients"
181 [testCase "c0030 is correct" test_trilinear_c0030,
182 testCase "c0003 is correct" test_trilinear_c0003,
183 testCase "c0021 is correct" test_trilinear_c0021,
184 testCase "c0012 is correct" test_trilinear_c0012,
185 testCase "c0120 is correct" test_trilinear_c0120,
186 testCase "c0102 is correct" test_trilinear_c0102,
187 testCase "c0111 is correct" test_trilinear_c0111,
188 testCase "c0210 is correct" test_trilinear_c0210,
189 testCase "c0201 is correct" test_trilinear_c0201,
190 testCase "c0300 is correct" test_trilinear_c0300,
191 testCase "c1020 is correct" test_trilinear_c1020,
192 testCase "c1002 is correct" test_trilinear_c1002,
193 testCase "c1011 is correct" test_trilinear_c1011,
194 testCase "c1110 is correct" test_trilinear_c1110,
195 testCase "c1101 is correct" test_trilinear_c1101,
196 testCase "c1200 is correct" test_trilinear_c1200,
197 testCase "c2010 is correct" test_trilinear_c2010,
198 testCase "c2001 is correct" test_trilinear_c2001,
199 testCase "c2100 is correct" test_trilinear_c2100,
200 testCase "c3000 is correct" test_trilinear_c3000],
202 testGroup "face0 vertices"
203 [testCase "v0 is correct" test_trilinear_f0_t0_v0,
204 testCase "v1 is correct" test_trilinear_f0_t0_v1,
205 testCase "v2 is correct" test_trilinear_f0_t0_v2,
206 testCase "v3 is correct" test_trilinear_f0_t0_v3]
209 g = make_grid 1 trilinear
210 cube = cube_at g 1 1 1
211 t = tetrahedron cube 0
213 test_trilinear_c0030 :: Assertion
214 test_trilinear_c0030 =
215 assertAlmostEqual "c0030 is correct" (c t 0 0 3 0) (17/8)
217 test_trilinear_c0003 :: Assertion
218 test_trilinear_c0003 =
219 assertAlmostEqual "c0003 is correct" (c t 0 0 0 3) (27/8)
221 test_trilinear_c0021 :: Assertion
222 test_trilinear_c0021 =
223 assertAlmostEqual "c0021 is correct" (c t 0 0 2 1) (61/24)
225 test_trilinear_c0012 :: Assertion
226 test_trilinear_c0012 =
227 assertAlmostEqual "c0012 is correct" (c t 0 0 1 2) (71/24)
229 test_trilinear_c0120 :: Assertion
230 test_trilinear_c0120 =
231 assertAlmostEqual "c0120 is correct" (c t 0 1 2 0) (55/24)
233 test_trilinear_c0102 :: Assertion
234 test_trilinear_c0102 =
235 assertAlmostEqual "c0102 is correct" (c t 0 1 0 2) (73/24)
237 test_trilinear_c0111 :: Assertion
238 test_trilinear_c0111 =
239 assertAlmostEqual "c0111 is correct" (c t 0 1 1 1) (8/3)
241 test_trilinear_c0210 :: Assertion
242 test_trilinear_c0210 =
243 assertAlmostEqual "c0210 is correct" (c t 0 2 1 0) (29/12)
245 test_trilinear_c0201 :: Assertion
246 test_trilinear_c0201 =
247 assertAlmostEqual "c0201 is correct" (c t 0 2 0 1) (11/4)
249 test_trilinear_c0300 :: Assertion
250 test_trilinear_c0300 =
251 assertAlmostEqual "c0300 is correct" (c t 0 3 0 0) (5/2)
253 test_trilinear_c1020 :: Assertion
254 test_trilinear_c1020 =
255 assertAlmostEqual "c1020 is correct" (c t 1 0 2 0) (8/3)
257 test_trilinear_c1002 :: Assertion
258 test_trilinear_c1002 =
259 assertAlmostEqual "c1002 is correct" (c t 1 0 0 2) (23/6)
261 test_trilinear_c1011 :: Assertion
262 test_trilinear_c1011 =
263 assertAlmostEqual "c1011 is correct" (c t 1 0 1 1) (13/4)
265 test_trilinear_c1110 :: Assertion
266 test_trilinear_c1110 =
267 assertAlmostEqual "c1110 is correct" (c t 1 1 1 0) (23/8)
269 test_trilinear_c1101 :: Assertion
270 test_trilinear_c1101 =
271 assertAlmostEqual "c1101 is correct" (c t 1 1 0 1) (27/8)
273 test_trilinear_c1200 :: Assertion
274 test_trilinear_c1200 =
275 assertAlmostEqual "c1200 is correct" (c t 1 2 0 0) 3
277 test_trilinear_c2010 :: Assertion
278 test_trilinear_c2010 =
279 assertAlmostEqual "c2010 is correct" (c t 2 0 1 0) (10/3)
281 test_trilinear_c2001 :: Assertion
282 test_trilinear_c2001 =
283 assertAlmostEqual "c2001 is correct" (c t 2 0 0 1) 4
285 test_trilinear_c2100 :: Assertion
286 test_trilinear_c2100 =
287 assertAlmostEqual "c2100 is correct" (c t 2 1 0 0) (7/2)
289 test_trilinear_c3000 :: Assertion
290 test_trilinear_c3000 =
291 assertAlmostEqual "c3000 is correct" (c t 3 0 0 0) 4
293 test_trilinear_f0_t0_v0 :: Assertion
294 test_trilinear_f0_t0_v0 =
295 assertEqual "v0 is correct" (v0 t) (1, 1, 1)
297 test_trilinear_f0_t0_v1 :: Assertion
298 test_trilinear_f0_t0_v1 =
299 assertEqual "v1 is correct" (v1 t) (0.5, 1, 1)
301 test_trilinear_f0_t0_v2 :: Assertion
302 test_trilinear_f0_t0_v2 =
303 assertEqual "v2 is correct" (v2 t) (0.5, 0.5, 1.5)
305 test_trilinear_f0_t0_v3 :: Assertion
306 test_trilinear_f0_t0_v3 =
307 assertClose "v3 is correct" (v3 t) (0.5, 1.5, 1.5)
310 test_trilinear_reproduced :: Assertion
311 test_trilinear_reproduced =
312 assertTrue "trilinears are reproduced correctly" $
313 and [p (i', j', k') ~= value_at trilinear i j k
318 let p = polynomial t,
319 let i' = fromIntegral i,
320 let j' = fromIntegral j,
321 let k' = fromIntegral k]
323 g = make_grid 1 trilinear
327 test_zeros_reproduced :: Assertion
328 test_zeros_reproduced =
329 assertTrue "the zero function is reproduced correctly" $
330 and [p (i', j', k') ~= value_at zeros i j k
334 let i' = fromIntegral i,
335 let j' = fromIntegral j,
336 let k' = fromIntegral k]
338 g = make_grid 1 zeros
340 t0 = tetrahedron c0 0
344 -- | Make sure we can reproduce a 9x9x9 trilinear from the 3x3x3 one.
345 test_trilinear9x9x9_reproduced :: Assertion
346 test_trilinear9x9x9_reproduced =
347 assertTrue "trilinear 9x9x9 is reproduced correctly" $
348 and [p (i', j', k') ~= value_at trilinear9x9x9 i j k
353 let p = polynomial t,
354 let i' = (fromIntegral i) * 0.5,
355 let j' = (fromIntegral j) * 0.5,
356 let k' = (fromIntegral k) * 0.5]
358 g = make_grid 1 trilinear
362 -- | The point 'p' in this test lies on the boundary of tetrahedra 12 and 15.
363 -- However, the 'contains_point' test fails due to some numerical innacuracy.
364 -- This bug should have been fixed by setting a positive tolerance level.
366 -- Example from before the fix:
368 -- > b0 (tetrahedron c 15) p
369 -- -3.4694469519536365e-18
371 test_tetrahedra_collision_sensitivity :: Assertion
372 test_tetrahedra_collision_sensitivity =
373 assertTrue "tetrahedron collision tests isn't too sensitive" $
376 g = make_grid 1 naturals_1d
377 cube = cube_at g 0 17 1
378 p = (0, 16.75, 0.5) :: Point
379 t15 = tetrahedron cube 15
382 prop_cube_indices_never_go_out_of_bounds :: Grid -> Gen Bool
383 prop_cube_indices_never_go_out_of_bounds g =
386 let coordmin = negate (delta/2)
388 let (xsize, ysize, zsize) = dims $ function_values g
389 let xmax = delta*(fromIntegral xsize) - (delta/2)
390 let ymax = delta*(fromIntegral ysize) - (delta/2)
391 let zmax = delta*(fromIntegral zsize) - (delta/2)
393 x <- choose (coordmin, xmax)
394 y <- choose (coordmin, ymax)
395 z <- choose (coordmin, zmax)
397 let idx_x = calculate_containing_cube_coordinate g x
398 let idx_y = calculate_containing_cube_coordinate g y
399 let idx_z = calculate_containing_cube_coordinate g z
403 idx_x <= xsize - 1 &&
405 idx_y <= ysize - 1 &&
411 grid_tests :: Test.Framework.Test
413 testGroup "Grid Tests" [
414 trilinear_c0_t0_tests,
415 testCase "tetrahedra collision test isn't too sensitive"
416 test_tetrahedra_collision_sensitivity,
417 testCase "trilinear reproduced" test_trilinear_reproduced,
418 testCase "zeros reproduced" test_zeros_reproduced ]
421 -- Do the slow tests last so we can stop paying attention.
422 slow_tests :: Test.Framework.Test
424 testGroup "Slow Tests" [
425 testProperty "cube indices within bounds"
426 prop_cube_indices_never_go_out_of_bounds,
427 testCase "trilinear9x9x9 reproduced" test_trilinear9x9x9_reproduced ]