diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml index fb978d2..5437cb5 100644 --- a/.github/workflows/ci.yml +++ b/.github/workflows/ci.yml @@ -13,6 +13,7 @@ jobs: matrix: version: - '1.10' + - '1' os: - ubuntu-latest arch: @@ -31,7 +32,7 @@ jobs: - name: Plugins files: "stokes_blocks.jl poisson_amr.jl poisson_unfitted.jl" - name: Other - files: "validation.jl validation_DrWatson.jl interpolation_fe.jl poisson_dev_fe.jl geometry_dev.jl" + files: "validation.jl validation_DrWatson.jl interpolation_fe.jl poisson_dev_fe.jl geometry_dev.jl arrays_dev.jl reffes_dev.jl tensor_values_dev.jl" steps: - uses: actions/checkout@v6 - uses: julia-actions/setup-julia@v2 diff --git a/Project.toml b/Project.toml index 840726e..8bb16d8 100644 --- a/Project.toml +++ b/Project.toml @@ -1,7 +1,7 @@ name = "Tutorials" uuid = "ab87341f-2653-5384-8dc6-691ea82e91b3" +version = "0.20.0" authors = ["Santiago Badia ", "Francesc Verdugo "] -version = "0.19.9" [deps] BlockArrays = "8e7c35d0-a365-5155-bbbb-fb81a777f24e" @@ -36,17 +36,19 @@ SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b" Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" [compat] -BlockArrays = "1.6.3" -DataStructures = "0.18.22" -Gridap = "0.19.9" +BlockArrays = "1" +DataStructures = "0.19" +Gridap = "0.20" GridapDistributed = "0.4" +GridapEmbedded = "0.9" GridapGmsh = "0.7" GridapP4est = "0.3" GridapPETSc = "0.5" -GridapSolvers = "0.6" +GridapSolvers = "0.7" MPI = "0.20" +PartitionedArrays = "0.3" SpecialFunctions = "1" -julia = "1.6" +julia = "1.10" [extras] MPIPreferences = "3da0fdf6-3ccc-4f1b-acd9-58baa6c99267" diff --git a/deps/build.jl b/deps/build.jl index 829dcf0..1f7f8c5 100644 --- a/deps/build.jl +++ b/deps/build.jl @@ -30,8 +30,11 @@ files = [ "Poisson with HHO on polytopal meshes"=>"poisson_hho.jl", "Block assembly and solvers: Incompressible Stokes example"=>"stokes_blocks.jl", "Lagrange multipliers" => "lagrange_multipliers.jl", - "Low-level API - Poisson equation"=>"poisson_dev_fe.jl", + "Low-level API - Arrays"=>"arrays_dev.jl", "Low-level API - Geometry" => "geometry_dev.jl", + "Low-level API - ReferenceFEs" => "reffes_dev.jl", + "Low-level API - Poisson equation"=>"poisson_dev_fe.jl", + "Low-level API - TensorValues"=>"tensor_values_dev.jl", ] Sys.rm(notebooks_dir;recursive=true,force=true) diff --git a/docs/Project.toml b/docs/Project.toml index 626e449..c3b9bb4 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -4,4 +4,4 @@ Literate = "98b081ad-f1c9-55d3-8b20-4c87d4299306" Tutorials = "ab87341f-2653-5384-8dc6-691ea82e91b3" [compat] -Documenter = "0.24" +Documenter = "1" diff --git a/docs/make.jl b/docs/make.jl index 8bd6553..9abdd25 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -36,8 +36,8 @@ for (i,(title,filename)) in enumerate(Tutorials.files) tutorial_title = string("# # [Tutorial ", i, ": ", title, "](@id ", tutorial_id, ")") tutorial_file = string(tutorial_prefix,splitext(filename)[1]) notebook_filename = string(tutorial_file, ".ipynb") - binder_url = joinpath("@__BINDER_ROOT_URL__","notebooks", notebook_filename) - nbviwer_url = joinpath("@__NBVIEWER_ROOT_URL__","notebooks", notebook_filename) + binder_url = "https://mybinder.org/v2/gh/gridap/Tutorials/gh-pages?filepath=notebooks/" * notebook_filename + nbviwer_url = "https://nbviewer.jupyter.org/github/gridap/Tutorials/blob/gh-pages/notebooks/" * notebook_filename binder_badge = string("# [![](",binder_logo,")](",binder_url,")") nbviwer_badge = string("# [![](",nbviwer_logo,")](",nbviwer_url,")") diff --git a/models/elasticFlag.jl b/models/elasticFlag.jl index 8764714..2908f38 100644 --- a/models/elasticFlag.jl +++ b/models/elasticFlag.jl @@ -2,9 +2,9 @@ using Gridap using Gridap.Io using GridapGmsh -model = GmshDiscreteModel("elasticFlag.msh") +meshdir(args...) = joinpath(@__DIR__, args...) +model = GmshDiscreteModel(meshdir("elasticFlag.msh")) -writevtk(model,"elasticFlag") +writevtk(model,meshdir("elasticFlag")) -fn = "elasticFlag.json" -to_json_file(model,fn) +to_json_file(model,meshdir("elasticFlag.json")) diff --git a/models/elasticFlag.json b/models/elasticFlag.json index db2df9f..5f754cc 100644 --- a/models/elasticFlag.json +++ b/models/elasticFlag.json @@ -1 +1 @@ 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\ No newline at end of file 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\ No newline at end of file diff --git a/models/model.jl b/models/model.jl index d3bb53a..b468806 100644 --- a/models/model.jl +++ b/models/model.jl @@ -3,10 +3,9 @@ using Gridap using Gridap.Io using GridapGmsh -model = GmshDiscreteModel("model.msh") - -writevtk(model,"model") - -fn = "model.json" -to_json_file(model,fn) +meshdir(args...) = joinpath(@__DIR__, args...) +model = GmshDiscreteModel(meshdir("model.msh")) +writevtk(model,meshdir("model")) + +to_json_file(model,meshdir("model.json")) diff --git a/models/model.json b/models/model.json index ef0f778..28ceb81 100644 --- a/models/model.json +++ b/models/model.json @@ -1 +1 @@ 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2,4917,6617,7307,4042,4979,5546,7374,793,5793,7005,7400,5316,5694,6317,6724,5307,5603,6204,6676,2470,2471,2486,6111,3953,4210,4380,7309,5927,6070,6662,6725,1175,5616,6415,7057,2805,5878,6639,7056,915,1244,5946,6514,1036,1294,5965,6640,5808,6687,7277,7430,5375,5915,6930,7103,5416,6532,6712,7134,3701,4050,4848,5621,912,1429,2179,6157,3815,4965,6587,6970,2932,2933,5610,6602,5507,5805,6105,6788,898,1116,5643,6637,5299,5786,6354,7036,2109,5676,6925,7369,2852,2853,5776,6372,5265,5739,6668,6765,5296,6032,6328,6761,913,1191,1792,6485,5631,6028,6592,7064,2427,4988,5863,7091,5519,6193,7193,7306,5304,5512,6753,7126,945,1793,5818,6449,2399,3988,4579,5754,5629,6368,6574,6974,4142,4596,5735,6888,950,1902,6323,6915,4058,5800,6474,6947,2671,2672,2675,6393,5354,5574,7192,7403,776,1429,2078,6016,5302,6003,6446,6534,2121,3233,3237,7335,3807,4890,6747,7465,5352,5654,5999,6651,1777,2265,2268,7069,5492,6368,7148,7181,5339,6009,6543,6588,1149,5610,6338,7308,5488,6093,6648,6912,5384,5809,5995,7401,5394,6299,6445,7337,1080,1395,3630,6045,5240,5514,6597,6814,3484,3487,3499,5774,1139,5563,6248,7189,3955,4459,5115,6120,1464,6539,7087,7151,678,5740,6307,6911,5341,5842,5962,6922,913,1651,5881,6717,5322,5692,6408,6409,2424,5714,7245,7409,916,1230,1677,5874,717,2102,5815,6213,3952,4560,4567,7445,2567,2618,2620,7009,5333,5922,6400,7184,4090,4241,5076,6369,4426,4814,5984,6965,2368,2452,6313,6401,5232,5633,6459,6612,5255,5595,6074,6932,1059,1243,5815,6481,5862,6513,6656,7310,1708,6672,7312,7415,3721,4289,6161,6674,3958,4449,5588,7399,1684,2055,2542,6676,2731,2732,2735,7018,3713,4983,6075,6991,5384,6554,6961,7401,4172,4343,5938,7437,3782,4245,5078,6067,4511,4975,6554,6961,2436,2437,2438,6037,5457,6080,6659,6899,1321,1948,2484,6638,5245,5596,6375,6668,4019,5630,7063,7247,4258,4840,6465,6609,1112,1578,1659,5788,5239,5616,6520,6696,5225,5660,6283,6411,2788,2789,5883,6883,6064,6314,6759,6899,3006,3007,3008,7119,5405,5750,6359,6984,1322,1860,5880,7087,4797,5904,6557,7468,804,1282,2076,5823,5280,5856,5920,6385,3883,5863,6366,6773,5252,5977,6062,6329,1072,1430,1599,6000,3826,4319,4487,6056,831,1653,5443,7448,3694,5024,5690,7231,2953,5793,6262,6860,4065,4597,6590,6965,5569,6154,6274,6848,2704,2707,6279,6802,3719,4730,5832,6510,2661,2874,2887,6118,2660,2873,6536,7165,2707,2711,6208,6802,5547,6049,6334,7235,3318,6176,6191,7349,3775,4820,5669,7264,2572,4318,4942,6041,4372,5733,6901,7382,4298,6055,6433,7344,5312,5949,6313,6407,5382,5582,6189,7278,5572,6053,6577,7419,1418,2206,2747,6371,2585,4749,6630,7361,863,6036,6821,7372,5027,5507,6788,7214,684,3623,5146,5997,5304,5687,6416,7262,973,1357,1791,5885,5407,5932,6603,7447,5339,6009,6185,6543,5496,5578,6096,6906,5311,5880,6126,7089,5927,6475,6812,7077,5243,6058,7121,7459,875,1149,1609,5610,4163,5451,6923,7200,3808,4066,4374,5997,3980,4136,4828,6458,5330,5777,6127,6447,1873,5862,6513,6656,3005,5887,6635,6835,3921,4555,5921,6902,1534,2966,2969,6410,4494,6233,6990,7405,1014,1985,5584,7243,5569,6154,7400,7435,4193,5164,5557,7212,3972,5182,5969,7127,2572,2573,2575,6041,3048,3055,6097,6311,3449,6002,6629,6975,5261,5653,6394,6511,5336,5705,6845,7462,856,2033,6777,7228,864,1252,1425,5668,838,1611,5830,7392,5546,6021,6644,7299,5519,6275,7193,7307,2245,5613,7207,7270,3639,5905,6548,6787,5630,6004,6506,7063,2474,2477,2481,6389,5318,5750,6553,6984,5297,5850,6413,6852,1279,1702,6321,7373,983,1312,6098,7161,5306,6318,6326,7081,3233,5808,7287,7335,5302,5774,5966,6741,5530,6159,6971,7470,1814,2319,5868,7002,5382,5841,6189,6722,5406,5762,6488,7049,2425,4841,4853,7091,801,1213,1976,6799,3416,3417,6747,7135,5513,5680,6500,7226,3892,4101,6255,6907,5080,5883,6897,7004,5613,6401,6542,7455,859,1509,5749,6781,3707,4284,4528,6866,4305,5791,6162,6995,5492,5630,6506,7418,5351,5755,6230,7235,2835,5659,6938,7389,2279,2280,2283,5746,3131,3210,6686,7037,2008,5831,6829,7396,5443,6024,6191,7375,893,1101,1247,5525,5519,6193,6275,6850,2514,2517,2519,6451,5263,5835,6256,6752,5371,5998,6606,6694,3117,3124,5902,6369,1636,6141,6863,7318,1126,1629,2041,5746,3657,3663,3664,6424,1419,2223,5944,7330,847,1072,1895,6005,5384,5809,5995,6740,5817,6600,6957,6980,2452,2453,2454,5949,5234,5903,6454,6952,5276,5795,6040,6142,2630,5824,6579,7252,3064,3109,4469,6981,5666,6254,7052,7139,1372,2720,2723,6606,1129,1412,2259,5926,2345,2347,2350,6302,3825,4532,4985,6390,5246,5572,6092,7356,4047,4774,6171,7403,3623,3624,3631,5997,5585,6266,7171,7465,5296,6032,6761,7098,5218,6013,6795,7234,3902,6056,6284,6853,6038,6211,6561,6937,3624,3631,5997,6571,933,1419,2223,5944,5394,6445,7027,7337,5375,6633,6647,7326,2011,2024,5959,7380,3821,4733,5144,7193,3248,3249,6244,6641,5234,5903,6454,6867,4545,4996,6382,7136,3819,5007,5041,6433,3777,4074,4549,6133,1201,2051,5872,6179,918,1170,1789,6676,5348,6847,7074,7101,5900,6441,7102,7408,5322,5650,6259,6402,4306,5652,6017,6560,985,2056,2071,6117,3893,4667,4819,5810,2758,2761,2763,6301,5469,6437,6789,6867,5733,6609,7363,7365,1634,1825,2554,6663,3723,4308,5065,6257,5803,5926,6430,6704,1172,1962,2197,6208,5324,5806,6245,6327,5710,6163,65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\ No newline at end of file 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,5700,7063,5298,5782,6430,7320,2968,2969,2972,5749,667,5570,6427,6524,3229,3286,3295,5651,758,1794,1914,6141,3446,5204,5917,6809,3893,4667,5810,6811,823,2012,2048,6150,3985,5042,5835,6857,1978,5739,6310,7267,2312,2315,5868,6525,1321,2126,5950,6638,3907,4695,6500,7441,2655,2874,3042,6118,5349,6068,6602,6770,5618,5710,6163,6505,3876,4092,4670,6594,2427,2429,4988,5863,5261,5958,6518,7042,5273,6046,6305,6636,953,5785,6196,7016,5352,5999,6601,7354,5310,5678,6309,6435,3283,3285,6034,6880,3757,4520,4530,6076,3607,3609,3636,5635,5291,5768,6295,7085,724,1784,5884,7191,2408,2410,5906,7444,2608,2609,2610,5623,5281,5549,6284,6972,5351,5601,7082,7196,1512,6821,7209,7458,5295,5813,6038,7047,4017,4399,5192,7451,3364,3382,3403,5655,4430,4708,6002,7367,1115,1351,1594,6196,986,1686,1943,6431,5386,5566,6239,7167,5347,6149,7295,7322,3475,3510,3511,7089,5322,5528,6259,6566,5308,5900,6320,6452,1761,3218,6475,6958,5846,6419,6647,7326,5248,5732,6140,6503,2016,5650,6755,7034,5338,6018,6336,7169,5297,5733,6465,6791,1581,3244,3247,6244,1971,5812,6710,7048,5220,5758,6490,6585,1166,1620,3494,5966,1053,1534,2117,6410,893,1247,5525,6878,4856,5169,5778,7066,1536,1908,6042,6642,3190,3191,3192,5745,1659,2343,5788,6978,3966,4326,4884,6521,1450,1903,6186,7178,3752,4814,5984,7418,5324,5683,6042,6586,3241,3283,6034,6698,5298,5876,6013,7327,5264,5755,6175,7235,2013,2714,2717,5625,769,1686,6849,6864,5700,6004,6468,7063,2627,2628,6106,7391,5589,5590,6692,6764,857,2145,6645,7096,3632,5466,7204,7265,3058,3062,5721,6709,2630,2632,5824,6579,4034,4717,5873,7166,3696,4629,4802,6252,5408,5910,6499,7229,3291,3292,3294,6072,5343,5689,6483,6982,1296,1455,3392,6442,1280,2039,2337,7153,5256,6212,6391,6928,4013,4331,5114,6289,3699,4832,6284,7127,5322,5734,6409,7293,3435,3447,6002,6809,5307,5626,6490,6721,2698,2701,5960,6438,5282,5663,6346,6944,1113,6038,6211,6937,3181,3182,5796,7221,1100,2110,3068,6988,953,2097,6196,7016,5767,6266,6352,6615,5331,5667,6183,6683,4152,4821,5165,6174,662,663,3418,7135,5400,5655,6737,7209,3569,3587,4229,6874,5545,5698,6228,7050,647,6813,6956,7003,679,3562,3589,5682,5316,5694,6709,6724,5292,5541,6355,6976,5280,5620,6385,6881,5288,5693,6243,6622,1584,2073,3491,6157,2628,2636,6106,7391,1270,2023,3122,6620,835,1063,1368,5820,934,1828,5613,7270,1937,5642,6614,6628,3129,3145,3205,6222,5694,5721,6522,6724,2795,6392,7025,7284,5294,5613,6429,7455,5372,5726,5992,7238,5166,5936,6869,6943,5246,5897,6499,7356,3876,4421,4670,6183,5276,5842,6040,6539,665,3456,3457,5828,3984,4230,4967,6408,5264,5617,6427,6950,825,1193,1737,6095,3534,3540,3548,5982,3238,3239,3240,5612,4553,4877,5777,7210,1639,6224,6645,7101,5458,5811,6006,7226,4275,5500,6533,6705,979,1999,5532,6802,5286,5644,6801,6942,2424,2425,5714,7091,1086,3441,3462,6436,5312,5823,6313,6401,4262,4548,5074,6004,2959,5569,6603,6948,3720,4779,6156,6592,5301,5715,6146,7291,5461,6095,6211,6850,908,1198,1666,5632,715,1163,1511,5533,3692,4369,6113,7024,5477,5572,7085,7419,3846,4056,4485,6308,863,1276,2149,6036,2392,5986,6916,7157,2734,5602,6891,7389,5228,6078,6959,7383,5250,5748,6462,6648,3370,5976,7372,7458,770,1121,2002,6020,2499,6107,6638,6984,5304,5512,6181,7060,1049,1286,5644,6306,1297,2080,5487,6953,5715,5764,6711,6719,2855,5776,6301,7109,784,1873,1944,6098,1112,1659,1922,2346,4197,4490,5111,6257,5358,5917,6058,6679,3871,5829,5913,6466,5389,6099,7009,7395,4058,5099,5800,6947,3882,5133,6487,7033,3775,4820,5669,7286,4560,4847,5741,7159,664,4677,5982,7123,5717,6203,6583,7439,3046,6018,6823,7169,2526,2529,6827,7206,1260,2108,5593,6878,4982,5681,6474,6947,5500,5687,6497,7262,3047,3048,3054,6097,4200,4614,6287,7080,5317,5645,6763,7108,5238,5615,6359,6885,5823,6321,6720,7258,3992,4917,6617,7307,4042,4979,5546,7374,793,5793,7005,7400,5316,5694,6317,6724,5307,5603,6204,6676,2470,2471,2486,6111,3953,4210,4380,7309,5927,6070,6662,6725,1175,5616,6415,7057,2805,5878,6639,7056,915,1244,5946,6514,1036,1294,5965,6640,5808,6687,7277,7430,5375,5915,6930,7103,5416,6532,6712,7134,3701,4050,4848,5621,912,1429,2179,6157,3815,4965,6587,6970,2932,2933,5610,6602,5507,5805,6105,6788,898,1116,5643,6637,5299,5786,6354,7036,2109,5676,6925,7369,2852,2853,5776,6372,5265,5739,6668,6765,5296,6032,6328,6761,913,1191,1792,6485,5631,6028,6592,7064,2427,4988,5863,7091,5519,6193,7193,7306,5304,5512,6753,7126,945,1793,5818,6449,2399,3988,4579,5754,5629,6368,6574,6974,4142,4596,5735,6888,950,1902,6323,6915,4058,5800,6474,6947,2671,2672,2675,6393,5354,5574,7192,7403,776,1429,2078,6016,5302,6003,6446,6534,2121,3233,3237,7335,3807,4890,6747,7465,5352,5654,5999,6651,1777,2265,2268,7069,5492,6368,7148,7181,5339,6009,6543,6588,1149,5610,6338,7308,5488,6093,6648,6912,5384,5809,5995,7401,5394,6299,6445,7337,1080,1395,3630,6045,5240,5514,6597,6814,3484,3487,3499,5774,1139,5563,6248,7189,3955,4459,5115,6120,1464,6539,7087,7151,678,5740,6307,6911,5341,5842,5962,6922,913,1651,5881,6717,5322,5692,6408,6409,2424,5714,7245,7409,916,1230,1677,5874,717,2102,5815,6213,3952,4560,4567,7445,2567,2618,2620,7009,5333,5922,6400,7184,4090,4241,5076,6369,4426,4814,5984,6965,2368,2452,6313,6401,5232,5633,6459,6612,5255,5595,6074,6932,1059,1243,5815,6481,5862,6513,6656,7310,1708,6672,7312,7415,3721,4289,6161,6674,3958,4449,5588,7399,1684,2055,2542,6676,2731,2732,2735,7018,3713,4983,6075,6991,5384,6554,6961,7401,4172,4343,5938,7437,3782,4245,5078,6067,4511,4975,6554,6961,2436,2437,2438,6037,5457,6080,6659,6899,1321,1948,2484,6638,5245,5596,6375,6668,4019,5630,7063,7247,4258,4840,6465,6609,1112,1578,1659,5788,5239,5616,6520,6696,5225,5660,6283,6411,2788,2789,5883,6883,6064,6314,6759,6899,3006,3007,3008,7119,5405,5750,6359,6984,1322,1860,5880,7087,4797,5904,6557,7468,804,1282,2076,5823,5280,5856,5920,6385,3883,5863,6366,6773,5252,5977,6062,6329,1072,1430,1599,6000,3826,4319,4487,6056,831,1653,5443,7448,3694,5024,5690,7231,2953,5793,6262,6860,4065,4597,6590,6965,5569,6154,6274,6848,2704,2707,6279,6802,3719,4730,5832,6510,2661,2874,2887,6118,2660,2873,6536,7165,2707,2711,6208,6802,5547,6049,6334,7235,3318,6176,6191,7349,3775,4820,5669,7264,2572,4318,4942,6041,4372,5733,6901,7382,4298,6055,6433,7344,5312,5949,6313,6407,5382,5582,6189,7278,5572,6053,6577,7419,1418,2206,2747,6371,2585,4749,6630,7361,863,6036,6821,7372,5027,5507,6788,7214,684,3623,5146,5997,5304,5687,6416,7262,973,1357,1791,5885,5407,5932,6603,7447,5339,6009,6185,6543,5496,5578,6096,6906,5311,5880,6126,7089,5927,6475,6812,7077,5243,6058,7121,7459,875,1149,1609,5610,4163,5451,6923,7200,3808,4066,4374,5997,3980,4136,4828,6458,5330,5777,6127,6447,1873,5862,6513,6656,3005,5887,6635,6835,3921,4555,5921,6902,1534,2966,2969,6410,4494,6233,6990,7405,1014,1985,5584,7243,5569,6154,7400,7435,4193,5164,5557,7212,3972,5182,5969,7127,2572,2573,2575,6041,3048,3055,6097,6311,3449,6002,6629,6975,5261,5653,6394,6511,5336,5705,6845,7462,856,2033,6777,7228,864,1252,1425,5668,838,1611,5830,7392,5546,6021,6644,7299,5519,6275,7193,7307,2245,5613,7207,7270,3639,5905,6548,6787,5630,6004,6506,7063,2474,2477,2481,6389,5318,5750,6553,6984,5297,5850,6413,6852,1279,1702,6321,7373,983,1312,6098,7161,5306,6318,6326,7081,3233,5808,7287,7335,5302,5774,5966,6741,5530,6159,6971,7470,1814,2319,5868,7002,5382,5841,6189,6722,5406,5762,6488,7049,2425,4841,4853,7091,801,1213,1976,6799,3416,3417,6747,7135,5513,5680,6500,7226,3892,4101,6255,6907,5080,5883,6897,7004,5613,6401,6542,7455,859,1509,5749,6781,3707,4284,4528,6866,4305,5791,6162,6995,5492,5630,6506,7418,5351,5755,6230,7235,2835,5659,6938,7389,2279,2280,2283,5746,3131,3210,6686,7037,2008,5831,6829,7396,5443,6024,6191,7375,893,1101,1247,5525,5519,6193,6275,6850,2514,2517,2519,6451,5263,5835,6256,6752,5371,5998,6606,6694,3117,3124,5902,6369,1636,6141,6863,7318,1126,1629,2041,5746,3657,3663,3664,6424,1419,2223,5944,7330,847,1072,1895,6005,5384,5809,5995,6740,5817,6600,6957,6980,2452,2453,2454,5949,5234,5903,6454,6952,5276,5795,6040,6142,2630,5824,6579,7252,3064,3109,4469,6981,5666,6254,7052,7139,1372,2720,2723,6606,1129,1412,2259,5926,2345,2347,2350,6302,3825,4532,4985,6390,5246,5572,6092,7356,4047,4774,6171,7403,3623,3624,3631,5997,5585,6266,7171,7465,5296,6032,6761,7098,5218,6013,6795,7234,3902,6056,6284,6853,6038,6211,6561,6937,3624,3631,5997,6571,933,1419,2223,5944,5394,6445,7027,7337,5375,6633,6647,7326,2011,2024,5959,7380,3821,4733,5144,7193,3248,3249,6244,6641,5234,5903,6454,6867,4545,4996,6382,7136,3819,5007,5041,6433,3777,4074,4549,6133,1201,2051,5872,6179,918,1170,1789,6676,5348,6847,7074,7101,5900,6441,7102,7408,5322,5650,6259,6402,4306,5652,6017,6560,985,2056,2071,6117,3893,4667,4819,5810,2758,2761,2763,6301,5469,6437,6789,6867,5733,6609,7363,7365,1634,1825,2554,6663,3723,4308,5065,6257,5803,5926,6430,6704,1172,1962,2197,6208,5324,5806,6245,6327,5710,6163,6505,7156,3564,3565,3566,7148,5306,5617,6318,6950,3891,4382,4523,6051,3906,4383,5792,7090,894,5933,6212,7369,5354,5569,6410,7469,5407,5569,6848,6948,1022,2142,6223,6962,5371,5718,6001,7240,5446,7002,7087,7151,5500,6073,6533,6597,3467,3469,3480,5742,3756,4774,5920,7403,3219,3243,6475,7077,3451,3452,5738,6485,5340,5584,6936,7243,3856,4375,5930,6352,3835,4327,4892,6780,5299,5786,6354,6678,1446,5808,6687,7277,2713,2714,2717,5625,3698,4085,4552,7443,5236,5707,6591,7052,1626,7010,7031,7404,5349,5610,6158,6602,2644,5708,6723,7223,4071,4436,4986,5814,5294,5669,6103,6542,5427,5928,6236,7069,5551,5684,6566,7293,5732,5922,6400,6733,889,1663,5869,6944,5367,6475,6934,6958,3115,3116,3119,6620,5862,6098,6513,6996,1968,6684,7267,7410,988,1233,2016,5650,2796,4192,4663,7066,4135,4780,5894,6376,3238,3240,3241,6034,1235,2043,2492,6449,5724,6397,6680,7140,5423,5715,6300,6711,5232,5529,6675,6776,1600,6704,6884,6918,3818,4332,4349,5615,4822,6650,7033,7231,3711,4159,4506,6902,678,3556,5740,6911,5131,6856,6895,7105,3717,4061,4367,5660,1135,1543,6144,7368,1206,2207,6092,6815,1865,5577,6515,7343,5520,5945,6141,6863,3002,3004,4145,5887,5978,6666,7038,7454,5490,5959,6122,7380,2312,5868,6525,7422,663,5941,6657,7135,5458,6373,6766,7226,2372,2374,5741,7026,3230,3237,3238,7335,4074,4824,5024,5690,5317,5646,6763,7108,3352,5937,6219,7276,3924,5054,5570,7227,5967,641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3,2541,2544,5947,6851,2393,4787,5986,6824,5311,5647,6525,6584,3873,5150,5700,7063,5298,5782,6430,7320,2968,2969,2972,5749,667,5570,6427,6524,3229,3286,3295,5651,758,1794,1914,6141,3446,5204,5917,6809,3893,4667,5810,6811,823,2012,2048,6150,3985,5042,5835,6857,1978,5739,6310,7267,2312,2315,5868,6525,1321,2126,5950,6638,3907,4695,6500,7441,2655,2874,3042,6118,5349,6068,6602,6770,5618,5710,6163,6505,3876,4092,4670,6594,2427,2429,4988,5863,5261,5958,6518,7042,5273,6046,6305,6636,953,5785,6196,7016,5352,5999,6601,7354,5310,5678,6309,6435,3283,3285,6034,6880,3757,4520,4530,6076,3607,3609,3636,5635,5291,5768,6295,7085,724,1784,5884,7191,2408,2410,5906,7444,2608,2609,2610,5623,5281,5549,6284,6972,5351,5601,7082,7196,1512,6821,7209,7458,5295,5813,6038,7047,4017,4399,5192,7451,3364,3382,3403,5655,4430,4708,6002,7367,1115,1351,1594,6196,986,1686,1943,6431,5386,5566,6239,7167,5347,6149,7295,7322,3475,3510,3511,7089,5322,5528,6259,6566,5308,5900,6320,6452,1761,3218,6475,6958,5846,6419,6647,7326,5248,5732,6140,6503,2016,5650,6755,7034,5338,6018,6336,7169,5297,5733,6465,6791,1581,3244,3247,6244,1971,5812,6710,7048,5220,5758,6490,6585,1166,1620,3494,5966,1053,1534,2117,6410,893,1247,5525,6878,4856,5169,5778,7066,1536,1908,6042,6642,3190,3191,3192,5745,1659,2343,5788,6978,3966,4326,4884,6521,1450,1903,6186,7178,3752,4814,5984,7418,5324,5683,6042,6586,3241,3283,6034,6698,5298,5876,6013,7327,5264,5755,6175,7235,2013,2714,2717,5625,769,1686,6849,6864,5700,6004,6468,7063,2627,2628,6106,7391,5589,5590,6692,6764,857,2145,6645,7096,3632,5466,7204,7265,3058,3062,5721,6709,2630,2632,5824,6579,4034,4717,5873,7166,3696,4629,4802,6252,5408,5910,6499,7229,3291,3292,3294,6072,5343,5689,6483,6982,1296,1455,3392,6442,1280,2039,2337,7153,5256,6212,6391,6928,4013,4331,5114,6289,3699,4832,6284,7127,5322,5734,6409,7293,3435,3447,6002,6809,5307,5626,6490,6721,2698,2701,5960,6438,5282,5663,6346,6944,1113,6038,6211,6937,3181,3182,5796,7221,1100,2110,3068,6988,953,2097,6196,7016,5767,6266,6352,6615,5331,5667,6183,6683,4152,4821,5165,6174,662,663,3418,7135,5400,5655,6737,7209,3569,3587,4229,6874,5545,5698,6228,7050,647,6813,6956,7003,679,3562,3589,5682,5316,5694,6709,6724,5292,5541,6355,6976,5280,5620,6385,6881,5288,5693,6243,6622,1584,2073,3491,6157,2628,2636,6106,7391,1270,2023,3122,6620,835,1063,1368,5820,934,1828,5613,7270,1937,5642,6614,6628,3129,3145,3205,6222,5694,5721,6522,6724,2795,6392,7025,7284,5294,5613,6429,7455,5372,5726,5992,7238,5166,5936,6869,6943,5246,5897,6499,7356,3876,4421,4670,6183,5276,5842,6040,6539,665,3456,3457,5828,3984,4230,4967,6408,5264,5617,6427,6950,825,1193,1737,6095,3534,3540,3548,5982,3238,3239,3240,5612,4553,4877,5777,7210,1639,6224,6645,7101,5458,5811,6006,7226,4275,5500,6533,6705,979,1999,5532,6802,5286,5644,6801,6942,2424,2425,5714,7091,1086,3441,3462,6436,5312,5823,6313,6401,4262,4548,5074,6004,2959,5569,6603,6948,3720,4779,6156,6592,5301,5715,6146,7291,5461,6095,6211,6850,908,1198,1666,5632,715,1163,1511,5533,3692,4369,6113,7024,5477,5572,7085,7419,3846,4056,4485,6308,863,1276,2149,6036,2392,5986,6916,7157,2734,5602,6891,7389,5228,6078,6959,7383,5250,5748,6462,6648,3370,5976,7372,7458,770,1121,2002,6020,2499,6107,6638,6984,5304,5512,6181,7060,1049,1286,5644,6306,1297,2080,5487,6953,5715,5764,6711,6719,2855,5776,6301,7109,784,1873,1944,6098,1112,1659,1922,2346,4197,4490,5111,6257,5358,5917,6058,6679,3871,5829,5913,6466,5389,6099,7009,7395,4058,5099,5800,6947,3882,5133,6487,7033,3775,4820,5669,7286,4560,4847,5741,7159,664,4677,5982,7123,5717,6203,6583,7439,3046,6018,6823,7169,2526,2529,6827,7206,1260,2108,5593,6878,4982,5681,6474,6947,5500,5687,6497,7262,3047,3048,3054,6097,4200,4614,6287,7080,5317,5645,6763,7108,5238,5615,6359,6885,5823,6321,6720,7258,3992,4917,6617,7307,4042,4979,5546,7374,793,5793,7005,7400,5316,5694,6317,6724,5307,5603,6204,6676,2470,2471,2486,6111,3953,4210,4380,7309,5927,6070,6662,6725,1175,5616,6415,7057,2805,5878,6639,7056,915,1244,5946,6514,1036,1294,5965,6640,5808,6687,7277,7430,5375,5915,6930,7103,5416,6532,6712,7134,3701,4050,4848,5621,912,1429,2179,6157,3815,4965,6587,6970,2932,2933,5610,6602,5507,5805,6105,6788,898,1116,5643,6637,5299,5786,6354,7036,2109,5676,6925,7369,2852,2853,5776,6372,5265,5739,6668,6765,5296,6032,6328,6761,913,1191,1792,6485,5631,6028,6592,7064,2427,4988,5863,7091,5519,6193,7193,7306,5304,5512,6753,7126,945,1793,5818,6449,2399,3988,4579,5754,5629,6368,6574,6974,4142,4596,5735,6888,950,1902,6323,6915,4058,5800,6474,6947,2671,2672,2675,6393,5354,5574,7192,7403,776,1429,2078,6016,5302,6003,6446,6534,2121,3233,3237,7335,3807,4890,6747,7465,5352,5654,5999,6651,1777,2265,2268,7069,5492,6368,7148,7181,5339,6009,6543,6588,1149,5610,6338,7308,5488,6093,6648,6912,5384,5809,5995,7401,5394,6299,6445,7337,1080,1395,3630,6045,5240,5514,6597,6814,3484,3487,3499,5774,1139,5563,6248,7189,3955,4459,5115,6120,1464,6539,7087,7151,678,5740,6307,6911,5341,5842,5962,6922,913,1651,5881,6717,5322,5692,6408,6409,2424,5714,7245,7409,916,1230,1677,5874,717,2102,5815,6213,3952,4560,4567,7445,2567,2618,2620,7009,5333,5922,6400,7184,4090,4241,5076,6369,4426,4814,5984,6965,2368,2452,6313,6401,5232,5633,6459,6612,5255,5595,6074,6932,1059,1243,5815,6481,5862,6513,6656,7310,1708,6672,7312,7415,3721,4289,6161,6674,3958,4449,5588,7399,1684,2055,2542,6676,2731,2732,2735,7018,3713,4983,6075,6991,5384,6554,6961,7401,4172,4343,5938,7437,3782,4245,5078,6067,4511,4975,6554,6961,2436,2437,2438,6037,5457,6080,6659,6899,1321,1948,2484,6638,5245,5596,6375,6668,4019,5630,7063,7247,4258,4840,6465,6609,1112,1578,1659,5788,5239,5616,6520,6696,5225,5660,6283,6411,2788,2789,5883,6883,6064,6314,6759,6899,3006,3007,3008,7119,5405,5750,6359,6984,1322,1860,5880,7087,4797,5904,6557,7468,804,1282,2076,5823,5280,5856,5920,6385,3883,5863,6366,6773,5252,5977,6062,6329,1072,1430,1599,6000,3826,4319,4487,6056,831,1653,5443,7448,3694,5024,5690,7231,2953,5793,6262,6860,4065,4597,6590,6965,5569,6154,6274,6848,2704,2707,6279,6802,3719,4730,5832,6510,2661,2874,2887,6118,2660,2873,6536,7165,2707,2711,6208,6802,5547,6049,6334,7235,3318,6176,6191,7349,3775,4820,5669,7264,2572,4318,4942,6041,4372,5733,6901,7382,4298,6055,6433,7344,5312,5949,6313,6407,5382,5582,6189,7278,5572,6053,6577,7419,1418,2206,2747,6371,2585,4749,6630,7361,863,6036,6821,7372,5027,5507,6788,7214,684,3623,5146,5997,5304,5687,6416,7262,973,1357,1791,5885,5407,5932,6603,7447,5339,6009,6185,6543,5496,5578,6096,6906,5311,5880,6126,7089,5927,6475,6812,7077,5243,6058,7121,7459,875,1149,1609,5610,4163,5451,6923,7200,3808,4066,4374,5997,3980,4136,4828,6458,5330,5777,6127,6447,1873,5862,6513,6656,3005,5887,6635,6835,3921,4555,5921,6902,1534,2966,2969,6410,4494,6233,6990,7405,1014,1985,5584,7243,5569,6154,7400,7435,4193,5164,5557,7212,3972,5182,5969,7127,2572,2573,2575,6041,3048,3055,6097,6311,3449,6002,6629,6975,5261,5653,6394,6511,5336,5705,6845,7462,856,2033,6777,7228,864,1252,1425,5668,838,1611,5830,7392,5546,6021,6644,7299,5519,6275,7193,7307,2245,5613,7207,7270,3639,5905,6548,6787,5630,6004,6506,7063,2474,2477,2481,6389,5318,5750,6553,6984,5297,5850,6413,6852,1279,1702,6321,7373,983,1312,6098,7161,5306,6318,6326,7081,3233,5808,7287,7335,5302,5774,5966,6741,5530,6159,6971,7470,1814,2319,5868,7002,5382,5841,6189,6722,5406,5762,6488,7049,2425,4841,4853,7091,801,1213,1976,6799,3416,3417,6747,7135,5513,5680,6500,7226,3892,4101,6255,6907,5080,5883,6897,7004,5613,6401,6542,7455,859,1509,5749,6781,3707,4284,4528,6866,4305,5791,6162,6995,5492,5630,6506,7418,5351,5755,6230,7235,2835,5659,6938,7389,2279,2280,2283,5746,3131,3210,6686,7037,2008,5831,6829,7396,5443,6024,6191,7375,893,1101,1247,5525,5519,6193,6275,6850,2514,2517,2519,6451,5263,5835,6256,6752,5371,5998,6606,6694,3117,3124,5902,6369,1636,6141,6863,7318,1126,1629,2041,5746,3657,3663,3664,6424,1419,2223,5944,7330,847,1072,1895,6005,5384,5809,5995,6740,5817,6600,6957,6980,2452,2453,2454,5949,5234,5903,6454,6952,5276,5795,6040,6142,2630,5824,6579,7252,3064,3109,4469,6981,5666,6254,7052,7139,1372,2720,2723,6606,1129,1412,2259,5926,2345,2347,2350,6302,3825,4532,4985,6390,5246,5572,6092,7356,4047,4774,6171,7403,3623,3624,3631,5997,5585,6266,7171,7465,5296,6032,6761,7098,5218,6013,6795,7234,3902,6056,6284,6853,6038,6211,6561,6937,3624,3631,5997,6571,933,1419,2223,5944,5394,6445,7027,7337,5375,6633,6647,7326,2011,2024,5959,7380,3821,4733,5144,7193,3248,3249,6244,6641,5234,5903,6454,6867,4545,4996,6382,7136,3819,5007,5041,6433,3777,4074,4549,6133,1201,2051,5872,6179,918,1170,1789,6676,5348,6847,7074,7101,5900,6441,7102,7408,5322,5650,6259,6402,4306,5652,6017,6560,985,2056,2071,6117,3893,4667,4819,5810,2758,2761,2763,6301,5469,6437,6789,6867,5733,6609,7363,7365,1634,1825,2554,6663,3723,4308,5065,6257,5803,5926,6430,6704,1172,1962,2197,6208,5324,5806,6245,6327,5710,6163,6505,7156,3564,3565,3566,7148,5306,5617,6318,6950,3891,4382,4523,6051,3906,4383,5792,7090,894,5933,6212,7369,5354,5569,6410,7469,5407,5569,6848,6948,1022,2142,6223,6962,5371,5718,6001,7240,5446,7002,7087,7151,5500,6073,6533,6597,3467,3469,3480,5742,3756,4774,5920,7403,3219,3243,6475,7077,3451,3452,5738,6485,5340,5584,6936,7243,3856,4375,5930,6352,3835,4327,4892,6780,5299,5786,6354,6678,1446,5808,6687,7277,2713,2714,2717,5625,3698,4085,4552,7443,5236,5707,6591,7052,1626,7010,7031,7404,5349,5610,6158,6602,2644,5708,6723,7223,4071,4436,4986,5814,5294,5669,6103,6542,5427,5928,6236,7069,5551,5684,6566,7293,5732,5922,6400,6733,889,1663,5869,6944,5367,6475,6934,6958,3115,3116,3119,6620,5862,6098,6513,6996,1968,6684,7267,7410,988,1233,2016,5650,2796,4192,4663,7066,4135,4780,5894,6376,3238,3240,3241,6034,1235,2043,2492,6449,5724,6397,6680,7140,5423,5715,6300,6711,5232,5529,6675,6776,1600,6704,6884,6918,3818,4332,4349,5615,4822,6650,7033,7231,3711,4159,4506,6902,678,3556,5740,6911,5131,6856,6895,7105,3717,4061,4367,5660,1135,1543,6144,7368,1206,2207,6092,6815,1865,5577,6515,7343,5520,5945,6141,6863,3002,3004,4145,5887,5978,6666,7038,7454,5490,5959,6122,7380,2312,5868,6525,7422,663,5941,6657,7135,5458,6373,6766,7226,2372,2374,5741,7026,3230,3237,3238,7335,4074,4824,5024,56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\ No newline at end of file diff --git a/models/solid.jl b/models/solid.jl index 603f6ba..6641531 100644 --- a/models/solid.jl +++ b/models/solid.jl @@ -3,10 +3,9 @@ using Gridap using Gridap.Io using GridapGmsh -model = GmshDiscreteModel("solid.msh") +meshdir(args...) = joinpath(@__DIR__, args...) +model = GmshDiscreteModel(meshdir("solid.msh")) -writevtk(model,"solid") - -fn = "solid.json" -to_json_file(model,fn) +writevtk(model,meshdir("solid")) +to_json_file(model,meshdir("solid.json")) diff --git a/models/solid.json b/models/solid.json index 311e2ae..b27e8b7 100644 --- a/models/solid.json +++ b/models/solid.json @@ -1 +1 @@ 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,7962,8506,9109,8003,8506,9109,8207,8891,9498,4329,4330,4341,4329,4341,8702,4480,5775,8640,4480,5775,9005,1462,2336,2664,1462,2336,9488,2336,2664,9488,4857,5502,6191,4857,5502,8891,7950,8208,8840,7950,8208,9347,7950,8840,9347,8208,8840,9347,1108,1782,2037,1108,2037,9752,4122,4142,9168,4122,4142,9289,4142,9168,9289,3546,3551,8299,3546,3551,9727,3546,8299,9727,1546,1711,2866,1546,2866,8665,1711,2866,8665,1712,8198,9763,1712,9746,9763,8198,9746,9763,4611,5284,5584,4611,5284,9326,4611,5584,9326,2011,2599,3628,2011,2599,8453,2011,3628,8453,7849,8749,9577,2382,8211,9621,2382,9404,9621,8211,9404,9621,1420,1787,2509,1668,2201,3610,1668,2201,9293,2201,3610,9293,7802,8491,8848,8491,8788,8848,8357,8795,9395,4492,4994,5450,4492,4994,8660,1215,1717,2173,1215,1717,8659,2654,8654,9151,2654,9151,9630,4446,9081,9637,7891,9081,9637,1338,1818,9168,1818,9168,9289,740,3689,6117,3689,6117,8715,2035,8542,9408,7855,8542,9408,5312,8541,9409,7856,8541,9409,1435,2083,2453,2083,2453,9553,1900,2404,2714,1900,2404,9298,1900,2714,9298,1130,1932,2660,1130,1932,9321,7851,8552,9228,7851,8782,9228,8420,8852,9421,8420,8988,9421,8852,8988,9421,4841,5482,6127,4841,6127,8280,5482,6127,8280,7790,8270,8450,7790,8450,9071,8270,8450,9071,4882,6119,9613,6119,9131,9613,7816,8916,9759,7817,8917,9758,2005,2947,8242,2005,2947,9339,4640,5182,5262,4640,5182,9284,5182,5262,9284,1331,2222,9537,2222,8830,9537,4608,5499,9538,5499,8829,9538,8059,8630,8791,8059,8675,8791,7921,8902,8997,7921,8997,9227,8902,8997,9227,7814,8561,9309,8561,9309,9310,8374,9103,9398,8064,8830,9714,8477,8830,9714,8065,8829,9715,8476,8829,9715,4643,5196,6430,4643,5196,8564,1979,3048,9066,1979,3048,9416,4981,5803,6272,4981,5803,8607,1349,2676,8483,1349,2676,9201,2676,8483,9201,4433,5390,5550,4433,5550,9036,3847,3849,3881,3847,3881,9327,7966,8105,9683,7967,8104,9682,1838,9340,9452,7856,8174,8897,7856,8174,9462,7856,8897,9462,8174,8897,9462,7855,8175,8896,7855,8175,9463,7855,8896,9463,8175,8896,9463,7931,8669,9505,7844,8343,8971,8343,8681,8971,7846,8342,8972,8342,8682,8972,1532,2472,8367,1532,2472,9235,1115,2786,9042,1115,2786,9763,2786,9042,9763,1357,2297,2874,1357,2297,9772,1357,2874,9772,2297,2874,9772,4634,5574,6151,4634,5574,9773,4634,6151,9773,5574,6151,9773,7788,8300,8344,8300,8344,8570,7789,8301,8345,8301,8345,8571,4448,5462,9152,5462,8481,9152,7797,8554,9649,2422,8110,9766,2422,8844,9766,8110,8844,9766,5699,8112,9765,5699,8842,9765,8112,8842,9765,4549,5567,8256,4549,5567,9244,3446,8658,9598,3446,9031,9598,8658,9031,9598,7829,8195,8900,7829,8827,8900,5666,6147,8667,5666,6147,9222,8419,8853,9426,8419,8989,9426,8853,8989,9426,4542,4915,6108,4542,6108,8639,5265,8133,8783,5265,8133,8904,5265,8783,8904,8133,8783,8904,3095,8577,9786,2389,2870,8643,2389,2870,9223,1263,8712,9680,7927,8134,8887,7927,8134,9639,8083,8297,9144,8082,8296,9146,4923,5306,5989,4923,5989,8875,5306,5989,8875,1646,2029,2712,1646,2712,8876,2133,2992,8528,2133,2992,8963,4894,5401,6125,4894,6125,8330,5401,6125,8330,8236,8580,9391,8236,8580,9429,1719,2488,3740,1719,2488,9551,1366,2020,8586,1366,2020,9667,4904,6103,6235,4904,6103,8958,7769,8157,8725,7769,8157,8884,8157,8725,8884,7768,8158,8724,8158,8724,8885,7888,8178,9142,7888,9119,9142,8097,9108,9132,8097,9132,9694,4898,8212,9552,4898,9552,9749,5931,8655,9150,5931,9150,9589,4585,5460,5993,5460,5993,8436,1080,1806,2573,1080,2573,8241,3915,5155,5646,3915,5646,9284,5155,5646,9284,1295,1769,2148,1295,2148,9336,1769,2148,9336,5459,8517,8828,8109,8517,8828,5092,5459,6006,5092,5459,8517,5567,6060,8898,5567,6060,9244,2022,2153,2898,2153,2898,8389,5299,5430,6175,5430,6175,8390,3680,8993,9371,8178,8993,9371,1111,8209,9506,1111,8375,9506,8209,8375,9506,8025,8090,9725,8025,8770,9725,8090,8770,9725,1214,2700,2842,1214,2700,9119,1214,2842,9119,2700,2842,9119,3891,3905,9724,1691,3153,9248,3153,8123,9248,1213,2699,2841,1213,2699,9118,1213,2841,9118,2699,2841,9118,4434,5339,5716,1157,2062,2439,2147,3011,3674,2147,3011,9364,2147,3674,9364,7911,8535,8639,4669,5025,6005,4669,5025,9791,5025,6005,9791,1428,2423,2850,1428,2423,8316,1428,2850,8316,1116,2652,3135,1116,3135,8630,2652,3135,8630,7920,8928,9475,8820,8928,9475,4999,5691,6051,4999,6051,8628,1090,3170,8648,1090,3170,8856,3170,8648,8856,5093,5932,6383,5932,6383,8594,4849,5639,8177,4849,5639,8640,8261,8531,8867,8261,8531,9663,8261,8867,9663,8262,8530,8866,8262,8530,9664,8262,8866,9664,4576,5067,5228,7901,8263,9785,8263,8599,9785,4704,5257,5576,5257,5576,8456,1427,1980,2299,1980,2299,8455,7807,8217,8762,8217,8762,9115,5806,6211,8719,5806,6211,8821,3063,8633,9443,3063,8730,9443,7773,8620,9020,1396,1867,1937,1396,1937,8203,4842,5115,9472,4608,5132,5499,5132,5499,8476,4849,5639,9569,8133,8440,8749,8133,8749,9577,8440,8749,9577,3395,3396,3397,3395,3396,9592,3395,3397,9592,8633,9260,9699,4887,8414,9050,4887,9050,9094,8414,9050,9094,1505,2501,7870,2501,7870,9437,4395,5928,6414,5928,6414,8539,1414,2253,2383,2253,2383,8399,4324,4335,8559,4324,4335,9618,4324,8559,9618,4906,5810,5812,4906,5810,9072,5810,5812,9072,7889,9118,9138,4373,5151,6336,4373,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,3933,9243,4781,8451,9271,4781,8451,9700,4781,9271,9700,8451,9271,9700,8159,8596,8650,8596,8650,9243,1998,2810,8626,1998,2810,9464,7799,9120,9727,7799,9179,9727,9120,9179,9727,4917,5888,6016,5888,6016,8396,8099,8910,9660,8910,9660,9707,4529,4880,5089,4529,4880,8793,1252,1603,1812,1252,1603,8794,2527,3719,3722,2527,3719,9790,8412,8472,8857,8413,8473,8858,4098,4105,8486,4098,4105,9484,4813,5604,6198,4813,6198,8632,5604,6198,8632,1536,2327,2921,1536,2921,8631,2327,2921,8631,1442,2251,2963,1442,2251,9568,1442,2963,9568,4719,5528,6240,4719,5528,9567,4719,6240,9567,9392,9397,9719,1692,3134,4029,1692,3134,9755,3134,4029,9755,4863,5446,8800,4863,5446,9333,1586,2169,8798,1586,2169,9334,1670,2292,3025,2292,3025,9709,8311,8487,8947,4150,4151,8487,4150,4151,9492,4150,8487,9492,4374,5032,5306,4374,5032,8503,4398,5196,6203,4398,5196,8564,4398,6203,8564,1588,2408,3167,4865,5685,6444,1717,2613,8547,1717,2613,9104,1995,4150,9492,1995,8487,9492,1232,2080,3171,1232,2080,8826,4509,5357,6448,4509,5357,8825,4445,6269,8914,1761,3053,3502,5034,6338,9058,1625,2450,3513,1625,2450,9356,2450,3513,9356,3432,3502,3503,3432,3502,8367,2400,4212,4228,2400,4228,9629,4212,4228,9629,4688,5058,5775,5058,5775,9005,4564,8947,9126,8311,8947,9126,3929,3930,3931,3930,3931,8425,1759,2658,2929,1759,2658,9732,1759,2929,9732,5003,5333,6101,5003,6101,9018,5333,6101,9018,1726,2056,2824,1726,2824,9019,2056,2824,9019,5272,6001,9226,5272,6001,9484,5272,9226,9484,2205,2985,3521,2205,3521,9328,2985,3521,9328,8054,8422,9356,8422,9291,9356,1304,1746,2015,1746,2015,9169,4581,5023,5292,5023,5292,9170,5122,5757,8936,5122,5757,9681,5122,8936,9681,4114,4118,4126,4118,4126,8753,3090,3853,8577,3090,3853,9786,1125,2669,9748,2669,9087,9748,1766,3003,8644,1766,3003,9376,7926,9045,9223,1101,2338,8636,1101,2338,9209,4630,5405,6426,4630,5405,9350,5061,6270,8488,5061,6270,9310,4019,4033,8865,4019,4033,9367,4033,8865,9367,8632,9041,9741,4044,4046,9458,4046,8548,9458,3390,3392,3482,3390,3392,8489,1452,2192,2913,1452,2913,8801,4807,5622,5680,4807,5680,8686,1686,3099,8272,1686,3099,9487,1686,8272,9487,8001,8388,9522,1622,2390,3026,1622,3026,9038,1172,2255,9776,1528,1998,2810,1528,2810,8626,4552,5100,6188,4552,5100,8943,1168,2992,8963,2894,4087,4088,2894,4087,8643,4087,4088,8643,1995,2724,9225,1995,2724,9492,1995,9225,9492,9307,9393,9474,3760,3763,5449,3760,3763,8374,1840,8879,9410,1840,9410,9690,8879,9410,9690,5117,8880,9411,5117,9411,9691,8880,9411,9691,8069,9001,9781,1255,9041,9276,1255,9041,9794,9041,9276,9794,4485,6125,9419,1560,2413,8806,1560,2413,9224,5029,6075,6190,5029,6075,9754,1259,2920,8623,1259,2920,8633,742,3848,8574,742,3848,9185,4580,5308,5322,5308,5322,8340,1287,1994,8956,1994,8956,9383,5586,8240,9688,5586,8973,9688,8240,8973,9688,2309,8241,9689,2309,8974,9689,8241,8974,9689,7894,9573,9778,9132,9573,9778,3978,3979,8558,3978,3979,9030,1430,1932,2197,1430,1932,9321,1430,2197,9321,3762,3763,3765,3762,3763,8374,8480,8790,9255,4287,4290,8596,4287,4290,9381,8563,9182,9473,1525,2669,8760,1525,2669,9087,1124,2670,9753,2670,9105,9753,4518,4972,6338,4972,6338,9058,1429,1648,2490,1429,1648,8509,4706,4925,5767,4706,4925,8510,5272,8486,9226,8486,9226,9484,1415,2763,9022,2763,8567,9022,4692,6040,9021,6040,8568,9021,3462,8485,9193,8485,9191,9193,4679,5155,5646,4679,5646,9284,7921,8658,9598,4878,5978,8641,4878,5978,9231,1601,2701,8642,1601,2701,9230,1560,2185,2668,1560,2668,9224,2185,2668,9224,4572,9371,9742,2064,2710,3524,2064,3524,9720,4462,5440,5524,4462,5524,9649,8434,9276,9671,8435,9277,9670,3810,3813,3815,3810,3815,8730,3813,3815,8730,1995,8487,9225,1850,3098,8930,1850,3098,9157,1850,8930,9157,1708,1910,4036,1708,1910,8364,1708,4036,8364,1507,2721,3046,1507,2721,9774,4784,5998,6323,4784,5998,9775,8049,8228,9654,8049,9230,9654,8050,8229,9655,8050,9231,9655,4568,5280,8673,4568,5280,8911,8160,8571,9272,8161,8570,9273,2012,2600,3555,2012,2600,8341,2012,3555,8341,1131,1689,9091,1131,1689,9256,1689,9091,9256,4687,5067,8562,4687,5067,9673,3396,3398,3485,4680,5156,5647,4680,5156,9327,4680,5647,9327,4629,5492,6425,4629,5492,8386,4629,6425,8386,1526,2670,8759,1526,2670,9105,1185,2163,2247,1185,2247,9650,4629,6425,9520,1974,4281,9679,5948,8205,9166,5948,9166,9695,8205,9166,9695,2671,8206,9165,2671,9165,9696,8206,9165,9696,2157,2333,9729,2333,8331,9729,3792,3793,3800,3792,3800,9262,1583,2189,2855,1583,2855,9418,768,1774,4312,768,1774,8457,762,763,4225,762,763,8400,4864,8902,9198,4864,9031,9198,8902,9031,9198,4366,5374,6287,4366,5374,9614,4366,6287,9614,5374,6287,9614,1418,1920,2440,1418,1920,8656,1380,2326,8930,1380,2326,9444,2326,8930,9444,4559,5259,6399,4559,5259,9203,4559,6399,9203,1704,2232,2995,2232,2995,9611,4753,5322,8340,4753,5322,8755,4753,8340,8755,1353,2128,3149,1353,2128,9352,8296,9145,9169,8297,9143,9170,2384,4196,4197,4196,4197,8692,3919,3932,3933,3919,3933,8650,3795,3798,3806,3795,3798,9323,8388,8409,9117,8409,9117,9571,1121,1919,2926,1121,1919,8586,1121,2926,8586,3857,3900,5643,3857,3900,9583,3900,5643,9583,4640,5080,8494,4640,5080,8807,8205,8812,9166,8205,9166,9757,8206,8811,9165,8206,9165,9756,8228,8953,9654,8229,8954,9655,1435,2458,8415,1435,2458,8994,1270,1963,2788,1270,2788,9304,1597,2252,2665,1597,2665,9612,2252,2665,9612,8084,8400,8574,8400,8574,9185,5010,5615,8371,5010,5615,9167,4817,5462,8481,4817,5462,9152,2240,2430,3882,2430,3882,9331,1339,1731,2708,4478,5926,8531,5926,8531,9758,1201,2649,8530,2649,8530,9759,4815,5732,9206,5732,9206,9586,1538,2455,9207,2455,9207,9585,4556,4878,8641,4556,4878,8969,1279,1601,8642,1279,1601,8968,4738,5899,8767,4738,8767,8783,5899,8767,8783,4089,4103,4104,4089,4103,8785,5408,6449,9078,5408,6449,9121,3614,3616,3624,3614,3616,9676,5026,6291,6332,5026,6291,8495,6291,6332,8495,1749,3014,3055,1749,3014,8496,3014,3055,8496,4526,4925,5465,4925,5465,8612,1249,1648,2188,1648,2188,8611,4559,5002,5697,4559,5697,9203,3875,9315,9386,8182,9315,9386,1252,1812,2377,1812,2377,8535,1369,1954,2666,1369,2666,9616,1954,2666,9616,6118,6164,9133,6164,9133,9508,1604,2841,2886,1604,2886,9332,2841,2886,9332,4493,5773,5972,5773,5972,9183,4089,4103,5612,4103,5612,8785,1735,2840,3688,1735,2840,8253,3800,8327,9323,3800,9262,9323,8327,9262,9323,4724,5564,6336,4724,5564,8490,1183,2440,3079,1183,2440,9015,1359,1878,9386,1359,1878,9629,4563,5129,5640,1286,1852,2363,1901,3035,9680,1353,2215,3149,1353,2215,8380,1402,1878,2369,1402,2369,9324,7999,8985,9536,8985,9536,9781,1208,2123,3150,2123,3150,8762,4630,5491,6426,4630,5491,8381,7930,8668,9374,8365,9299,9735,7914,9402,9597,1681,2649,8530,4958,5926,8531,1404,1880,2371,1404,2371,9204,4466,5821,8440,4466,5821,9496,1373,2446,2945,2446,2945,8527,3778,3780,3781,3778,3780,8309,4608,5138,8581,4608,5138,9538,7861,8480,9255,8046,8209,9700,8046,9506,9700,8209,9506,9700,1313,1654,2929,1313,1654,8271,4397,5195,5938,698,4224,5119,698,5119,5603,3431,3433,6129,3431,3433,9740,3433,6129,9740,7899,8465,9611,8465,9560,9611,4671,5136,5210,4671,5210,9516,3828,3829,8280,3828,3829,9515,3828,8280,9515,2305,2746,9282,2305,2746,9539,5582,6023,9281,5582,6023,9540,1314,1655,2527,1314,2527,8575,4414,5820,5828,4414,5828,9495,4438,5180,6379,4438,6379,8317,5180,6379,8317,2587,3504,3505,3504,3505,8931,1119,1876,2410,1119,1876,8483,1876,2410,8483,8350,9055,9481,3066,3067,8751,3066,3067,9619,3067,8751,9619,6343,6344,8752,6343,6344,9620,6344,8752,9620,7964,8690,9535,7965,8689,9534,4847,5397,5704,5397,5704,8819,4466,5821,5940,5821,5940,8440,7958,8226,9171,3401,5658,6368,3401,5658,8638,3401,6368,8638,5658,6368,8638,8642,8780,8968,8641,8779,8969,1400,2356,9677,4718,5439,5501,4718,5501,8395,4992,5933,9013,4992,5933,9468,9089,9090,9569,2047,3827,9601,2047,8408,9601,3827,8408,9601,3474,3476,9547,3474,3476,9720,1156,1810,2417,1810,2417,9135,4627,5490,6424,4627,5490,9453,5490,6424,9453,8027,8363,8819,8027,8819,9012,1713,2098,2793,1713,2793,9011,2098,2793,9011,1763,2713,2953,1763,2953,9219,4602,5123,9148,4602,5123,9540,1325,1846,9147,1325,1846,9539,4680,5184,5264,4680,5184,9327,1880,3940,3943,1880,3943,9485,3940,3943,9485,5040,5990,6230,5040,6230,9228,5337,6003,8800,2060,2726,8798,1420,1830,2509,1420,1830,8404,9204,9284,9511,4961,5720,8701,4961,5720,9636,5402,5689,6204,5402,6204,9035,4858,5393,6368,4858,6368,8638,4231,8333,9181,4231,9181,9444,4886,5196,6167,4886,6167,8189,5196,6167,8189,5006,8263,9149,5006,9149,9767,8263,9149,9767,4355,5711,6443,4355,5711,9316,1137,2543,2551,1137,2551,9553,4122,4134,4138,4122,4134,8667,4724,5151,6336,4724,5151,8553,8799,9239,9713,4366,5374,8621,4366,8621,9614,1331,1861,8573,1331,1861,9537,4852,5793,5795,4852,5795,8781,7952,9190,9278,8065,8581,9456,8581,9456,9538,747,748,4131,747,748,9014,747,4131,9014,8031,8893,9567,8032,8892,9568,2319,3968,3969,2319,3969,8558,3968,3969,8558,766,4231,9181,766,8333,9181,4365,5337,6003,4365,6003,8800,1088,2060,2726,1088,2726,8798,3940,3943,3944,3943,3944,9485,8669,9360,9373,8668,9361,9374,8535,8639,9634,1844,2480,9424,1844,9424,9777,3891,9702,9708,8191,9702,9708,2023,2779,9158,2023,2779,9524,5300,6056,9159,5300,6056,9525,4250,4252,9516,4250,4252,9548,4250,9516,9548,4252,9516,9548,3451,3453,5519,3451,3453,8422,8236,9417,9429,9071,9417,9429,3973,5031,5597,5031,5597,8458,4748,5271,5866,4748,5271,9237,4748,5866,9237,5271,5866,9237,8533,8651,9355,4067,4068,4069,4067,4069,9297,4626,5401,5687,4626,5687,8864,1798,4064,4078,1798,4064,9657,4064,4078,9657,169,1885,2373,169,1885,3931,169,2373,3931,4852,5793,6264,4852,6264,9750,3840,3841,3843,3841,3843,8828,7790,8279,9519,1714,2099,2794,2099,2794,9037,4641,5625,6372,4714,5032,8503,4714,5032,8680,5032,8503,8680,4506,4980,5452,4980,5452,9435,3680,3684,8993,3680,3684,9371,2206,3810,3815,2206,3815,8730,1530,2345,2403,4659,5074,5609,4659,5074,9401,1297,1862,1921,1297,1921,9377,1862,1921,9377,4574,5139,5198,4574,5198,9378,5139,5198,9378,4076,4077,4090,4136,4147,4148,4136,4147,8756,5531,8890,9521,5531,9515,9521,8890,9515,9521,4560,4987,5891,4560,5891,8933,1283,1710,2614,1283,2614,8932,4115,4151,8487,4115,4151,9126,4409,4935,5590,4409,4935,9347,1308,2183,8749,1308,2183,9176,1308,8749,9176,2446,2945,9640,2446,8527,9640,8028,8645,9068,8029,8646,9069,8155,9014,9375,9014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989,9793,3373,3393,3394,3373,3393,9232,4402,4882,5350,4473,5857,6038,4473,5857,8901,1772,2211,8259,1772,2211,9404,4552,4849,5358,4552,5358,9090,3404,3488,3489,5049,5488,8260,5049,5488,9396,4491,5977,6119,5977,6119,9131,7838,8378,9111,7839,8377,9110,1422,1659,2500,1422,1659,9417,1238,2055,2892,2055,2892,8687,1196,2435,8903,1196,2435,9746,1840,2632,8879,1840,2632,9410,5117,5909,8880,5117,5909,9411,1388,2052,2060,2052,2060,9147,4665,5329,5337,5329,5337,9148,4490,5976,6118,5976,6118,9133,1196,2580,2761,1196,2580,8903,8132,8710,9488,4191,5169,6301,4191,6301,8845,5169,6301,8845,7997,8632,9741,4482,6035,6142,1205,2758,2865,1663,2240,3090,1663,2240,9786,2240,3090,9786,2018,2635,8928,2018,2635,9623,3754,5207,6170,3754,6170,9791,5207,6170,9791,4650,5548,5836,4425,5116,5812,5116,5812,8526,3825,9515,9521,4908,5166,5811,4908,5811,9085,4681,5263,8918,4681,5263,9393,4473,5712,8901,4473,5712,9744,2242,3804,3805,3804,3805,8478,4927,5379,6378,4927,5379,9259,4781,6182,9700,4394,6413,9184,4102,4103,5102,4102,4103,8784,4102,5102,8784,4384,6060,8585,4384,6060,8898,3430,3432,3502,8285,8902,9198,4115,4116,4135,4115,4135,9126,770,4270,5624,770,4270,8494,2388,2871,9053,2388,2871,9289,4212,4213,4228,4213,4228,9629,3377,5055,5358,5055,5358,9090,4210,4250,4252,4210,4250,9548,4210,4252,9548,4750,5369,5459,5369,5459,8828,1790,2300,3491,1790,2300,9532,2300,3491,9532,351,2248,4324,351,2248,9618,351,4324,9618,4443,5535,6401,4443,5535,9178,1398,1868,2368,1398,2368,8869,1486,2372,2441,1486,2372,8779,4763,5649,5718,4763,5649,8780,1271,2035,9408,4548,5312,9409,3595,3598,9062,3595,3598,9388,4367,6447,8649,4367,6447,9020,1633,1892,3024,1633,1892,8854,1633,3024,8854,2369,3850,3877,2369,3850,9324,3850,3877,9324,1355,1768,8993,1355,1768,9748,1501,2701,8744,1501,2701,9230,4778,5978,8743,4778,5978,9231,7801,8428,9622,4950,6427,9233,1423,2461,9299,1423,2461,9300,4629,5492,6049,4403,5723,6270,5723,6270,8488,2271,3708,8597,2271,3708,9345,2271,8597,9345,3708,8597,9345,1486,2441,2519,2441,2519,8515,4763,5718,5796,5718,5796,8514,1171,1834,9199,1171,1834,9311,7771,8537,9718,8537,8538,9718,5658,5863,8638,5658,5863,9261,4561,5105,5925,5105,5925,8627,4616,5008,5562,4616,5008,8418,3471,3473,6062,3471,3473,8576,3471,6062,8576,4406,5024,5816,4406,5024,9227,4406,5816,9227,5024,5816,9227,1818,2895,4117,1818,2895,9289,1413,1724,2221,1413,1724,8692,1724,2221,8692,1640,2540,4005,1640,2540,9579,2540,4005,9579,1145,2003,2523,1145,2003,9574,2003,2523,9574,1074,2095,2561,1074,2095,9480,4351,5372,5838,4351,5372,9479,1284,1828,2648,1828,2648,8629,1378,2272,9774,1378,8383,9774,4655,5549,9775,4655,8384,9775,1183,2266,3079,2266,3079,9061,1621,2136,3086,1621,2136,8446,5665,6148,9160,5665,6148,9297,7970,9394,9508,8584,9114,9700,8584,9114,9710,8584,9700,9710,9114,9700,9710,654,4120,5759,654,4913,5759,3394,3483,8561,3483,8561,9768,4581,5360,5730,4581,5360,9495,5227,6189,6371,5227,6371,9645,4648,5270,6405,4648,5270,9510,5270,6405,9510,7927,8622,9259,8622,8965,9259,4910,5169,6301,4910,5169,8845,4910,6301,8845,1399,1869,2367,1399,1869,9240,1752,2192,2913,1752,2192,8379,1307,2184,3157,2184,3157,8856,4957,5525,6007,4957,6007,8703,5525,6007,8703,7929,9052,9644,3970,3971,3972,4690,5001,5498,4690,5001,8698,5001,5498,8698,4663,6055,6184,4663,6055,9031,6055,6184,9031,1425,1801,2809,1425,2809,8559,1950,2912,3094,1950,3094,9669,1613,1886,2689,1886,2689,8675,1144,2024,2358,2024,2358,9608,1096,2643,8566,1096,2643,9029,4454,5601,6381,4454,6381,8619,1271,1958,2594,1271,2594,9257,4548,5235,5871,4548,5871,9258,4772,5504,5972,5504,5972,9183,567,3697,5507,567,3697,9092,567,5507,9092,1766,2470,3003,1766,2470,8644,3859,8869,9697,8869,9583,9697,4782,5134,5778,5134,5778,9760,1994,2682,4097,1994,2682,8956,2682,4097,8956,3851,3853,3882,3853,3882,9786,1143,2019,2354,1143,2354,8888,4420,5296,5631,4420,5631,8889,1534,2097,2696,2097,2696,8620,4811,5374,5973,5374,5973,8621,1724,2221,8944,1724,8692,8944,4503,5092,6383,4503,6383,8397,1432,1794,2567,1432,2567,8525,1726,1921,2718,1921,2718,9377,5003,5198,5995,5198,5995,9378,5015,5951,6022,5015,6022,8327,4746,5824,6229,4746,5824,9571,4746,6229,9571,1629,2526,2824,2526,2824,9019,4906,5803,6101,5803,6101,9018,1184,1811,2264,1184,1811,9554,1184,2264,9554,4461,5088,5541,4461,5088,9555,4461,5541,9555,4684,5369,5664,4684,5369,8828,637,4015,4016,637,4016,8822,4015,4016,8822,4657,5517,5707,4657,5707,9460,5517,5707,9460,1435,2083,2906,1435,2906,8994,2083,2906,8994,1353,2215,2773,4556,4878,5359,4878,5359,8641,1279,1601,2082,1601,2082,8642,4709,5071,5844,4709,5844,8521,4524,5065,6104,4630,5491,6050,1865,2838,9066,1865,2838,9529,4497,4900,5326,4497,5326,9406,4712,5360,6183,4712,6183,8827,5360,6183,8827,1365,1987,2349,1365,2349,8457,3575,3577,3580,3575,3580,9164,3419,3420,3421,1646,2076,8437,1646,2076,8876,4923,5353,8438,4923,5353,8875,5001,5498,8945,5001,8698,8945,1253,1657,2018,1253,2018,8928,1633,3080,8854,1633,3080,9048,1633,8854,9048,4372,5962,6176,5962,6176,8666,8080,8374,9103,8223,9190,9542,2914,3486,3487,2914,3486,9445,2914,3487,9445,3486,3487,9445,8704,9299,9478,8573,9208,9537,361,1830,3077,361,3077,4325,1457,2516,2816,1457,2516,8724,2516,2816,8724,4734,5793,6093,5793,6093,8725,4550,4856,5457,4558,5864,6090,4558,5864,8309,1946,2883,3116,1946,3116,9335,7997,8914,9741,3978,3987,8405,3978,3987,9030,3435,5223,5924,3435,5223,8871,3435,5924,8871,5223,5924,8871,4302,4930,5437,4302,4930,8789,4930,5437,8789,5368,5662,6140,5662,6140,9589,4708,5054,8501,4708,5054,9698,4708,8501,9698,2251,2754,8937,2251,2754,9568,5528,6031,8938,5528,6031,9567,1139,1646,2146,1646,2146,8814,4416,4923,5423,4923,5423,8813,1394,1859,1934,1394,1859,8425,5021,5782,9025,5021,5782,9026,5021,9025,9026,1744,2505,9023,1744,2505,9024,4396,5153,5687,4396,5687,9641,1620,1956,2958,1620,1956,9010,1386,2778,2907,2778,2907,9262,4568,5348,8673,4568,5348,8851,1256,1996,2253,1256,1996,9065,1256,2253,9065,4435,5363,6194,5363,6194,8977,1158,2086,2917,2086,2917,8976,4203,4205,4206,4205,4206,9438,4664,5135,5521,4664,5521,8419,5135,5521,8419,1387,1858,2244,1387,2244,8420,1858,2244,8420,5134,9595,9760,8365,9595,9760,364,1802,2289,364,2289,4303,4891,5967,6122,5967,6122,9531,8015,8357,8795,1354,1769,9062,1354,1769,9753,3393,5409,6275,3393,5409,9592,3393,6275,9592,5409,6275,9592,1250,1864,2853,1250,1864,9088,1127,2447,3048,1127,3048,9066,2332,4129,4131,4129,4131,8651,4837,5690,6110,5690,6110,8805,1560,2413,2833,2413,2833,8806,4910,6357,8845,4910,6357,9044,4910,8845,9044,1181,2477,2502,2477,2502,8690,4458,5754,5779,5754,5779,8689,4398,5939,6429,4398,5939,8564,4398,6429,8564,3393,5409,6295,3393,5409,9232,3393,6295,9232,5409,6295,9232,281,1636,4079,281,1636,4084,281,4079,4084,1385,1899,2226,1899,2226,9794,8307,8944,9438,4931,6206,8269,4931,6206,9668,5265,5821,6400,5265,6400,8904,1512,2050,2629,1512,2629,8255,4789,5327,5906,4789,5906,8254,4313,4314,4326,1482,2012,2600,1482,2012,8341,3611,3633,5309,3611,3633,9565,3611,5309,9565,3663,3664,3665,3663,3665,9155,1470,1913,2077,1470,2077,9217,4747,5190,5354,4747,5354,9215,4528,5009,5986,4528,5009,9319,3534,5310,9491,3534,9491,9727,1401,2062,2063,1401,2062,8859,4678,5339,5340,4678,5339,8860,4753,5308,5322,7967,8883,9682,4687,5067,5228,4687,5228,9673,7966,8882,9683,4608,5529,5942,5529,5942,8919,1074,2364,3042,2364,3042,8824,4351,5641,6319,5641,6319,8823,1251,1732,2709,1251,1732,9692,1568,1874,3092,1568,1874,9197,4533,4874,5942,4874,5942,8919,1702,2477,2502,4979,5754,5779,4843,6051,8628,4843,6051,9407,1717,2173,2613,2173,2613,9104,4708,5054,5737,4708,5737,8501,764,5051,5626,764,5051,8792,5051,5626,8792,4118,4127,9737,3569,3586,8905,3569,3586,9051,4954,5289,6226,4954,5289,9587,4954,6226,9587,5289,6226,9587,4033,5121,8865,4033,5121,9367,4274,4275,4277,4275,4277,9393,1308,1821,2183,1308,1821,8749,1468,2479,2942,1468,2479,9609,2479,2942,9609,4980,5932,8757,4980,5932,9435,4890,5152,5974,4890,5152,9184,1695,1757,3061,1695,1757,9014,1695,3061,9014,1794,2530,9330,1794,2530,9459,1134,2235,2380,1134,2235,8613,4411,5512,5657,4411,5512,8614,1273,1765,2867,1273,1765,9089,8736,9644,9781,4789,5327,6310,4789,6310,9268,1512,2050,3033,1512,3033,9269,1410,1794,2496,1794,2496,9330,1248,1900,3175,1900,3175,8818,1614,2690,2844,2690,2844,9562,4852,6274,8781,4852,6274,9750,1331,2222,2228,1331,2228,9537,4608,5499,5505,4608,5505,9538,5233,6396,8572,5233,6396,9789,4602,5123,5337,4602,5337,9148,1325,1846,2060,1325,2060,9147,4740,5722,6126,4740,6126,8983,1463,2445,2849,1463,2849,8984,1358,1948,9130,1358,1948,9610,5447,6341,9008,5447,6341,9016,2170,3064,9009,2170,3064,9017,1132,2028,8465,1132,2028,9611,4464,4911,5522,4464,4911,8634,1187,1634,2245,1187,1634,8635,4617,5840,8878,4617,5840,9518,4928,6064,6387,8101,9125,9527,1182,1684,2341,1182,1684,8685,3732,3733,3735,3732,3733,9183,4577,5333,9018,4577,5333,9561,1300,2056,9019,1300,2056,9560,1397,1894,1975,1397,1894,9702,5061,5723,6222,5061,5723,8488,2010,2700,2951,2010,2700,9784,1709,3985,3986,1709,3986,9447,2908,3077,4325,2908,3077,9397,4173,4183,4185,2075,2489,2866,2075,2866,8665,1274,1779,2102,1274,2102,9312,4418,5072,8669,1451,1682,1943,1682,1943,8589,4728,4959,5220,4959,5220,8590,1414,1996,2216,1414,2216,9631,5287,5976,6227,5287,5976,9762,4262,4274,4277,4262,4277,9307,1266,2503,2548,1266,2548,9597,4799,5461,6099,4799,5461,9020,5461,6099,9020,7790,8368,9429,4884,5158,9131,4884,5158,9186,8276,8542,9408,8277,8541,9409,4610,5479,9172,4610,9052,9172,3553,5288,5876,3553,5876,9653,5288,5876,9653,4373,5920,8565,4373,5920,8992,1403,1948,2583,1403,2583,9307,5352,5766,6143,5352,6143,8670,1299,1628,2150,1299,2150,9743,1359,1878,1947,1359,1947,9629,4306,4335,8789,4306,4335,9618,1632,2218,2375,1632,2218,8997,1348,2123,2409,1348,2409,9115,4564,5271,6002,5271,6002,8947,4588,5147,5311,4588,5311,9137,1311,1870,2034,1311,2034,9136,8107,8246,9410,8246,8879,9410,8106,8247,9411,8247,8880,9411,4600,4913,5441,4600,4913,8303,7970,8328,9288,7970,9288,9622,1511,2376,2956,2376,2956,8524,4788,5653,6233,5653,6233,8523,769,770,4270,4714,5635,8680,4714,5635,8907,4972,5034,6439,4972,6439,9058,5034,6439,9058,1470,2077,2829,4747,5354,6106,7970,8227,9508,2044,2350,8520,2044,2350,9599,2044,8520,9599,4970,5898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21,8453,8721,3889,3902,3902,8075,3902,9709,1526,2537,2537,7961,1526,8759,2537,8759,7961,8759,1998,3058,1998,8622,1998,9464,5085,5851,5085,9198,5085,9291,3148,7784,3148,8376,7784,8376,3148,9503,7784,9503,2509,9574,7798,8730,7798,9428,8730,9428,8730,9698,9428,9698,8721,9141,8692,8944,8944,9295,5093,5932,5093,7809,5932,7809,5093,8594,5932,8594,7809,8594,7853,8251,8251,9148,8251,9325,9148,9325,3436,8931,7909,8931,8931,9314,8061,9296,8061,9661,6279,8672,6279,9233,6279,9419,8672,9419,1133,3070,1133,7793,1133,8220,4410,6347,4410,7794,4410,8221,4515,6404,3936,3953,4264,4268,4264,4272,4268,4272,4264,8176,4268,8176,4272,8176,4239,4240,4240,8187,4240,8306,1444,9559,3035,9559,8117,9034,1752,8835,6447,8649,8632,8649,8649,9020,1177,9741,5565,8977,8151,8977,8977,9463,2288,8976,8152,8976,8976,9462,1177,2051,4471,6306,4471,8430,6306,8430,6306,8677,8430,8677,1194,3029,1194,8429,3029,8429,3029,8678,8429,8678,4718,5471,4718,8149,4718,8395,5068,5856,5856,8660,5856,9698,4550,8321,7991,8321,8321,8765,3721,3723,2925,7808,2925,8483,2925,8569,7808,8569,8483,8569,6317,8184,8184,8394,8184,8417,1210,2846,3391,6042,1154,2167,2167,8469,2167,8735,4124,9692,4125,9692,8017,9692,1352,1770,4374,8158,4783,9242,8392,9242,8592,9242,3868,9593,5650,9593,3385,8582,8011,8582,8563,8582,8620,9041,5118,7845,1841,7843,2725,9383,7805,9383,8312,9383,1346,2607,1346,7799,2607,7799,1346,8460,2607,8460,7799,8460,4669,8094,8094,9791,1303,2006,2006,2310,2006,9549,2138,9037,7887,9037,8227,9037,8698,8945,8945,9294,4814,9595,6241,9595,7854,9595,5940,8449,6099,8449,7849,8449,2663,8448,2822,8448,7850,8448,5962,9072,8034,9072,8526,9072,1194,9680,8363,9680,9365,9680,8316,8576,8388,8576,1625,9564,1713,2602,5715,8737,5715,9070,8737,9070,5715,9406,8737,9406,9070,9406,4533,9478,8362,9478,1321,2110,1989,2428,2428,7924,1989,8391,2428,8391,7924,8391,8483,8915,8483,8990,4624,5885,4624,7799,4624,8459,2931,9459,7896,9459,8839,9459,6158,8579,7816,8579,2881,8578,7817,8578,6425,7785,6425,8386,7785,8386,6425,9338,7785,9338,8386,9338,3685,5048,3685,8129,5048,8129,3685,8381,5048,8381,5391,7957,5391,8352,7957,8352,5391,9306,8352,9306,2114,7956,2114,8351,7956,8351,2114,9305,7956,9305,8351,9305,4620,5247,4620,5683,4620,8187,5037,5934,5314,5934,8418,9692,1405,2489,2489,8048,2489,8665,4780,8579,1503,8578,4629,5047,8029,9069,8859,9069,8029,9723,8859,9723,9069,9723,3698,9382,2013,7883,2013,8466,7883,8466,2013,8529,7883,8529,5140,8164,5140,9430,8164,9430,5140,9687,8164,9687,9430,9687,1863,8163,1863,9431,8163,9431,1863,9686,8163,9686,9431,9686,2478,8437,5755,8438,1373,7932,1373,8527,7932,8527,1373,8761,7932,8761,7906,8738,7906,9033,7906,9488,8738,9488,9033,9488,7904,8785,7904,9236,7904,9763,7989,8157,7989,8876,4753,7874,4753,8340,7874,8340,4753,9518,8340,9518,7765,8761,6265,8500,8500,8632,8500,9020,4364,5500,4364,6008,5500,6008,1087,2223,1087,2731,2223,2731,2571,4245,3367,3369,4758,8076,4758,8915,8076,8915,4758,9653,8076,9653,8915,9653,5885,8244,8244,8791,5885,9491,8244,9491,8791,9491,1335,2795,1335,2830,2795,2830,1335,8023,2795,8023,2830,8023,4612,6072,4612,6107,6072,6107,4612,8024,6072,8024,6107,8024,8925,9678,1500,3102,3102,8290,3102,8946,4838,5922,5922,5997,5922,7812,1561,2645,2645,2720,2645,7813,5435,6365,5435,8666,6365,8666,5435,8763,6365,8763,8666,8763,4803,8468,7963,8468,4803,9390,7963,9390,8468,9390,3659,9576,5316,9576,8508,9576,4682,5766,5766,8056,5766,8670,3370,5011,3370,8140,3370,8278,4836,9685,9025,9685,1559,9684,9023,9684,3378,3381,3378,7840,3381,7840,3378,9182,3381,9182,4698,9778,6022,9778,7894,9778,6002,8947,7806,8947,8311,8947,5361,8081,8081,8243,8081,8890,730,8011,730,8582,730,8807,8582,8807,5277,6304,5277,9162,6304,9162,5277,9723,6304,9723,9162,9723,8143,8269,8269,9344,8269,9790,1510,7842,4787,7841,2374,3865,3864,3865,3865,8137,1212,2949,2949,3149,2949,9617,4802,8467,7960,8467,4802,9384,7960,9384,8467,9384,3450,3452,3452,8422,3452,9356,2761,8245,8245,8903,8903,9006,8202,8215,8215,8602,8215,8793,8201,8214,8214,8603,8214,8794,5603,6386,5603,8921,5603,9181,2822,8604,7850,8604,8448,8604,6099,8605,7849,8605,8449,8605,7764,8212,8212,8498,8212,9552,4594,8078,4594,8324,8078,8324,4594,8390,8324,8390,1317,8077,1317,8325,8077,8325,1317,8389,8325,8389,6425,9520,7785,9520,4089,8785,5086,8785,3164,8558,8155,8558,8479,8558,3783,8447,8447,9189,1302,2963,2251,2963,2963,8032,4579,6240,5528,6240,6240,8031,8511,9029,3478,9547,1460,2685,1460,8407,2685,8407,1460,9075,2685,9075,8407,9075,1771,3599,1771,8130,3599,8130,1771,8380,3599,8380,8117,8721,8117,8970,8721,8970,8721,9034,8970,9034,2021,9313,4869,9126,5388,9126,8459,8460,6006,8517,7964,9666,8014,9666,8219,9666,7965,9665,8013,9665,8218,9665,4203,9438,1609,2926,1609,8216,1609,8733,1515,2204,1515,2990,2204,2990,1515,8207,2990,8207,5494,8186,4486,6123,4202,6237,2622,9091,7825,9091,8528,9091,8433,9520,1159,8292,8006,8292,8292,9348,1920,3079,3079,7880,1920,9061,3079,9061,7880,9061,2902,8853,8431,8853,4837,6110,6110,6179,1560,2833,2833,2902,3982,3994,3994,8405,3994,9248,8072,8403,8072,8864,8403,8874,8864,8874,3389,8177,3389,8489,3389,9089,8177,9089,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3,1794,2567,7802,8427,3837,3840,3406,3490,8276,9257,8277,9258,9318,9329,4747,5190,3678,8162,3678,8993,3678,9364,1230,2790,4507,6067,4631,8948,2166,2479,2479,8534,2479,9609,1608,2889,2889,8117,8105,9013,3671,3677,3677,5497,3677,9705,5809,8691,4711,5332,5332,8548,4711,9422,5332,9422,3154,9755,1409,2716,2051,2716,2716,9176,1631,9195,1496,1805,1496,2875,1805,2875,2875,8705,1849,2383,7856,8798,7855,8800,5215,5854,1739,4318,8052,9049,8052,9151,2183,8749,2183,9082,2183,9176,1599,2945,1599,3173,1599,9345,8274,9667,8586,9667,8586,9783,9667,9783,4173,5457,4222,7953,4222,9181,4222,9444,7953,9444,9181,9444,8902,9198,1730,2340,2340,2989,2340,8447,2046,9321,9321,9481,3566,5492,5492,8386,5492,9338,1336,1803,3152,7973,3152,8586,3152,9362,2011,3140,2011,3638,3140,3638,2011,9385,3140,9385,3638,9385,2551,7955,1303,1630,9397,9719,8028,9721,8029,9722,1317,2022,1317,2153,1470,1913,4616,6408,1910,2564,1910,8364,1910,9602,1094,9219,8699,9219,8645,9068,8646,9069,3847,3848,3847,8168,3847,9324,7898,9420,1513,8354,8354,9079,5313,9497,8601,9497,5442,9272,8571,9272,2165,9273,8570,9273,2587,2813,2587,8931,2587,9314,4258,4259,4258,5853,4258,8862,3699,3844,4270,5262,4270,5624,5262,5624,4270,8494,5262,8494,5624,8494,8346,8641,8641,8779,8641,9366,4067,4096,1242,3089,2812,3089,3089,8291,7849,8749,8440,8749,8440,9499,3748,3751,4220,5626,5264,5626,5626,8792,4371,9228,8320,9228,8700,9228,1890,2313,4696,9363,4771,6351,2945,8527,1088,1902,1088,2954,1902,2954,1902,9735,2954,9735,8347,8642,8642,8780,8642,9372,1313,1759,5262,8073,5262,9511,1539,9100,2672,9100,8540,8833,8864,9115,1174,2120,1870,9136,5147,9137,4557,5135,9117,9547,1947,4217,1947,9629,4217,9629,1947,9703,4217,9703,9629,9703,9101,9494,1368,1991,1368,8479,1368,8558,3448,3511,1831,2069,1831,9282,2069,9282,2069,9730,9282,9730,3851,9706,1412,1841,4689,5118,1280,1858,8051,9046,8051,9150,8488,9140,3612,6117,8903,9645,4380,4895,4895,8388,4895,9571,7913,8118,7912,8119,1654,2999,1654,3484,2999,3484,1654,8561,2999,8561,3484,8561,1618,2503,1618,8432,1618,9127,4336,4337,4337,4338,4337,8308,3773,9196,8439,9196,4266,4280,3718,6276,3718,8269,6276,8269,3718,9668,6276,9668,4907,5322,1993,8990,9302,9659,1085,2372,1688,2372,2372,8779,4362,5649,4965,5649,5649,8780,4611,5097,5097,5533,5097,9411,1334,1820,1820,2256,1820,9410,6069,6377,2792,3100,4600,5964,3630,3632,4608,5132,8651,9355,9192,9747,8401,8861,4817,9152,3406,3408,1478,1887,1478,2608,1887,2608,1478,8538,1598,2944,1598,3172,1598,9768,5071,5844,8368,9519,8744,9230,8743,9231,1262,2997,1355,8993,1132,2313,1132,8101,1132,9125,8617,9239,3979,3988,8606,8785,2716,8529,4718,6039,1978,3060,1978,8610,1978,9107,4497,4900,9460,9706,1613,1991,4489,5975,5975,6121,3392,3482,3392,3527,3392,8489,8160,8345,8161,8344,4632,9705,4365,5337,5337,8800,5337,9148,1088,2060,2060,8798,2060,9147,4674,5251,5172,5251,5251,8170,4430,5170,5170,9519,2895,4117,2895,4149,4117,4149,4117,9289,4149,9289,4676,5144,5144,5216,1908,2795,1908,9554,5185,6072,5185,9555,8408,9521,3022,9667,8016,9667,9667,9777,2038,2171,9202,9559,4705,5063,8258,9123,8257,9122,5690,8805,7992,8805,4291,4297,4293,4297,4297,9751,1370,1992,1370,3127,1370,9047,1171,2442,2442,9199,2442,9464,2266,2827,4458,6311,4458,8205,6311,8205,4458,8689,6311,8689,8205,8689,1181,3034,1181,8206,3034,8206,1181,8690,3034,8690,8206,8690,3915,9284,5155,9284,8176,9284,4962,8674,1164,2402,1164,8111,2402,8111,1164,8959,2402,8959,8111,8959,4441,5679,4441,8113,5679,8113,4441,8961,5679,8961,8113,8961,8415,8939,8416,8941,4284,5214,4061,9434,8396,9434,8661,8756,3027,8080,3971,3974,3974,4976,3974,8417,4829,5843,1552,2566,4515,5331,5331,5842,5331,9094,3051,3154,3051,8007,3829,3832,3829,5482,3829,8280,3938,3942,3938,8154,3938,8918,4516,5231,3601,3604,4601,4966,8964,9156,4274,4276,4276,9229,4276,9393,4238,4240,1723,2831,7980,9140,6224,8244,6224,9338,6224,9531,4485,6427,6279,6427,6427,8672,1465,8727,2104,8727,4742,8726,5381,8726,7907,8173,1338,9168,9324,9703,8153,9331,1337,1683,1683,2617,1683,9554,4614,4960,4960,5894,4960,9555,7955,9441,4542,5740,5740,8214,5740,9496,1250,1731,1731,8357,1731,9657,6385,9203,3616,8850,8901,9669,5301,8671,8540,8931,2011,8453,3638,8453,8178,9748,4680,5225,4680,9315,1164,8389,8111,8389,8389,8959,4441,8390,8113,8390,8390,8961,4927,9572,7927,9572,8644,9472,3984,3985,3598,8165,3598,9062,3598,9336,8232,9497,2264,9554,5541,9555,5154,5483,5321,7883,5321,9054,4339,4347,1568,2609,1568,3092,1568,8840,2046,2284,1654,8271,3484,8271,8560,8637,1744,2058,1744,9715,1835,9772,8113,9772,9153,9772,5112,9773,8111,9773,9154,9773,1265,2463,2463,8215,2463,9501,1492,2532,2532,3097,2532,9428,4606,9092,2776,8855,8599,8855,2776,9785,8599,9785,3610,9293,1237,2054,2054,2564,2054,9134,2435,9763,2017,2416,5294,5693,4699,5777,5777,9391,5777,9429,5675,8710,5675,8983,8710,8983,5675,9545,4948,5858,1671,2581,4599,5432,5432,8356,5432,9546,3398,3485,3485,3486,3485,8271,4205,8307,1567,9719,2257,2491,5196,6430,8290,8772,5284,5584,5284,8669,5284,9326,8523,8589,8524,8590,4594,5299,4594,5430,1375,2328,2328,9143,2328,9696,4652,5605,5605,9145,5605,9695,4547,5201,5201,645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203,9644,8552,9228,8782,9228,2011,2697,2011,2948,1314,1760,9197,9347,3991,5592,2568,3512,4808,5526,4808,6395,1531,2249,1531,3118,7888,8178,7965,8689,7964,8690,1554,8232,1554,9088,1554,9640,4077,4079,4077,4081,4792,5481,4792,6104,2555,9650,9443,9650,3465,8543,2088,2089,2089,8258,2089,9449,5365,5366,5366,8257,5366,9448,4866,5617,5617,5970,5617,8273,3786,3791,3789,3791,3791,8758,1966,2402,5243,5679,4743,5411,1466,2134,7931,8669,3581,6034,5047,6034,6034,8905,5559,6358,3954,3956,3954,3960,3954,9241,3601,3603,3601,9762,1171,9311,8141,8318,1125,1916,1916,8284,1916,8760,4119,8248,1612,3136,3136,8970,3136,9034,8436,9255,9091,9255,5331,9050,3654,3672,3654,9067,3654,9212,1400,2356,8562,9673,1095,1920,1770,2756,2756,3656,2756,8967,8707,9106,4633,9069,1667,2200,4623,6348,2435,9746,2127,8426,2127,9503,2127,9562,4050,4052,1824,2473,2767,4126,4114,4126,4126,8753,4624,6349,4895,5780,2967,9255,5832,9649,9454,9649,8705,8999,8788,8848,8788,9141,3466,3468,6118,9508,768,769,768,4312,768,8457,5549,8384,2272,8383,1441,2224,1441,9301,1441,9337,4539,5815,4995,5514,4890,6124,4890,6166,6124,6166,6124,8637,6166,8637,4746,5681,5681,9149,5681,9647,1263,3114,1263,8712,5016,5280,5280,8673,1858,2656,5135,5933,1363,1905,1905,1985,8112,8113,8110,8111,1124,1917,4862,6068,5911,6068,1585,2791,2634,2791,3841,5369,1450,2465,1450,8599,1450,9095,2465,9095,8641,9231,8642,9230,5294,6024,2017,2747,2538,8310,2538,8817,2538,9038,8310,9038,4654,5610,5434,5610,5610,8332,4369,5327,7782,8341,4739,8697,1132,8465,8465,9125,8231,9553,4875,6449,6449,9121,6449,9726,4163,4168,3081,9738,1195,2915,4453,5108,1176,1831,1176,2069,4961,8701,1523,2624,2624,9136,2624,9686,4800,5901,5901,9137,5901,9687,8626,9545,2574,3798,3798,3805,3798,9323,4472,6192,1567,2053,8351,9674,8352,9675,5864,9189,1943,8570,5220,8571,4462,5440,4462,8554,4462,9649,1905,3850,3850,3855,3850,9324,4731,4957,4584,8133,4584,8500,4584,8783,759,760,759,4167,759,8559,4541,5604,4541,6134,1264,2327,1264,2857,1474,8080,1474,9549,1474,9739,9549,9739,4968,6411,5301,5768,1565,2473,1622,2538,3888,9331,3888,9460,4433,5583,5583,5865,3095,9444,3095,9786,1366,1919,1919,3153,1718,2237,4666,5570,4666,9654,1389,2293,1389,9655,5093,5508,1923,2541,3702,3716,1318,1761,1761,3053,3533,3551,3533,5478,3533,9727,4388,6187,2176,8465,2176,9560,2176,9729,8465,9729,1396,1867,1396,8203,1396,9405,3795,3796,1431,9193,4553,5347,4553,6371,5347,6371,5347,9645,6371,9645,9324,9327,1157,2062,2062,2775,4434,5339,5339,6052,1898,2980,1898,3175,2980,3175,2474,3089,5098,5460,5460,6333,4263,4270,4263,5262,4263,9511,7943,8401,1420,1787,1420,2523,6183,9495,4070,4078,4070,4092,2447,2994,2447,9066,2447,9140,3452,3512,4598,9626,760,4335,4734,5795,4697,5016,4697,9267,4738,5627,5076,5627,5627,9054,1185,2163,1185,8555,1185,9650,3891,3905,3891,9708,3905,9708,7992,8356,4338,4340,1588,2165,1588,2859,1588,8570,4865,5442,4865,6136,4865,8571,8391,9626,4993,6451,6451,8217,6451,9419,5124,5926,5926,6023,5926,9758,1847,2649,2649,2746,2649,9759,1074,9480,4351,9479,1730,3791,4410,4960,1133,1683,1393,2047,1901,2621,2487,2653,2487,8412,2487,8857,5764,5930,5764,8413,5764,8858,3173,9344,9344,9640,757,4155,2016,2548,5175,6257,5175,6452,1487,2746,4764,6023,7901,8855,7901,9785,1492,2555,1108,2749,1108,8302,9336,9788,5782,6150,2505,2873,4936,5634,1694,2734,4971,6011,8507,9304,3430,3432,3432,9162,1463,2398,1463,8984,1463,9625,4974,5237,4974,9236,4974,9603,8365,9300,8365,9735,4776,6160,1697,1960,1697,9238,1697,9604,7967,8104,8420,9421,1085,2405,1688,2405,2405,9655,4362,5682,4965,5682,5682,9654,1138,9674,4415,9675,4046,4055,1524,1935,2255,8815,5532,8816,2073,2481,4972,5596,4972,6441,4972,9058,1595,9680,4057,4058,1991,2652,1709,2322,1709,3985,1709,9447,4684,5745,5064,5745,7966,8105,8419,9426,5555,5557,5555,8279,5555,8695,9082,9646,1474,9103,2054,9096,5197,8657,5197,8957,5197,9761,5130,9391,8134,9391,5360,5730,2330,2550,2550,8616,2550,9082,4823,6143,4988,6143,6143,8670,1714,3576,5372,6457,5919,6457,2095,3180,2642,3180,2522,4316,4307,4316,4316,9621,2206,3063,3063,8633,3063,8730,2072,2480,8718,9010,9010,9552,1472,2574,7861,8480,1659,2500,4499,5025,5025,5485,5025,9791,2192,2913,2080,2627,5357,5904,6103,6235,759,2809,1907,3939,3974,5597,4329,4341,4480,5775,1462,2336,4857,5502,7950,8208,8840,9347,1108,2037,4122,4142,4142,9168,4142,9289,3546,3551,3546,8299,3546,9727,1546,2866,1711,2866,2866,8665,1712,9763,4611,5284,4611,5584,2011,2599,2011,3628,2382,9621,1668,2201,2201,3610,2201,9293,4492,4994,1215,1717,2654,9151,1818,9168,3689,6117,8542,9408,8541,9409,2083,2453,1900,2714,1900,9298,1130,1932,4841,6127,5482,6127,8450,9071,6119,9613,2005,2947,4640,5182,5182,5262,2222,9537,5499,9538,8902,8997,8997,9227,8830,9714,8829,9715,4643,5196,1979,3048,4981,5803,1349,2676,2676,8483,2676,9201,4433,5550,3847,3881,7856,8174,7855,8175,1532,2472,1115,2786,2786,9042,2786,9763,1357,2874,2297,2874,2297,9772,2874,9772,4634,6151,5574,6151,5574,9773,6151,9773,5462,9152,2422,9766,8110,9766,5699,9765,8112,9765,4549,5567,3446,9598,5666,6147,4542,6108,5265,8133,5265,8783,5265,8904,8133,8904,2389,2870,7927,8134,4923,5989,1646,2712,2133,2992,4894,6125,5401,6125,6125,8330,1719,2488,1366,2020,4904,6103,7769,8157,8097,9132,4898,9552,5931,9150,5460,5993,1080,2573,3915,5646,5155,5646,5646,9284,1295,2148,1769,2148,2148,9336,5092,5459,5567,6060,2153,2898,5430,6175,1111,9506,8770,9725,1214,2700,1214,2842,2700,2842,2842,9119,3153,9248,1213,2699,1213,2841,2699,2841,2841,9118,2147,3011,2147,3674,4669,5025,5025,6005,1428,2423,1428,2850,1116,3135,2652,3135,3135,8630,4999,6051,1090,3170,5932,6383,4849,5639,8261,9663,8262,9664,8263,9785,5257,5576,1980,2299,5806,6211,3063,9443,1396,1937,5132,5499,8133,8749,3395,3396,3395,3397,3395,9592,4887,9050,2501,7870,5928,6414,2253,2383,4324,4335,4324,8559,4324,9618,4906,5810,5810,5812,4373,5151,1268,2883,2883,9335,1277,2232,1704,2232,2232,9611,1111,2910,5126,5751,7952,8223,2076,2478,5353,5755,3560,3561,3560,8380,3560,9617,3817,5054,5054,8501,5054,9698,5010,5115,1424,2323,4399,4887,1171,2174,1212,2128,4557,5320,4818,5537,9638,9743,4533,5942,1267,1795,1795,8668,4544,5072,5072,8669,4968,6430,3426,3499,3459,3461,1280,2043,5711,6443,763,4182,5220,6444,1943,3167,7900,8192,3693,3694,4174,4183,4928,6387,1115,1712,1620,2135,1822,2881,1822,8578,1822,9437,5099,6158,5099,8579,5099,9436,2316,3896,2316,9386,3896,9386,2316,9702,3896,9702,1556,2980,1719,1984,3053,3502,3501,3502,4309,4317,1130,2046,2289,4303,2289,4312,4303,4312,2289,8421,3848,3850,1868,2368,1868,3890,2368,3890,2368,8869,3890,8869,3444,3446,3622,3648,1361,2318,3531,3547,3547,8186,3547,9653,4737,5435,9023,9538,9024,9538,8064,9025,8064,9026,9025,9537,9026,9537,4905,5390,4579,5366,5366,8449,1302,2089,2089,8448,4898,6236,3767,3770,3768,3770,3770,9196,1465,1763,1355,9748,4742,5040,4075,4085,5914,6138,5914,9422,4831,6154,5314,6154,6154,9205,1353,2773,4897,5233,4897,8235,4897,9789,2949,8341,5071,9483,5690,6187,4895,5168,5457,6185,5320,8710,4503,6006,2110,2377,2186,2275,5463,5552,3476,3477,8212,8718,8212,9010,1691,3133,2043,8709,4630,6050,9129,9130,1439,1788,1522,1988,4769,5832,761,4181,761,5525,761,8671,4996,5765,4397,5938,4397,6428,5938,6428,5938,9288,4686,5328,5328,5680,5328,9190,1411,9433,3422,3423,3973,3983,3973,5031,1354,9753,1409,2345,2004,2591,2004,3544,1565,2646,2646,3030,2646,8770,1180,2186,1180,8728,1180,8934,4457,5463,4457,8729,4457,8935,1352,2772,2214,3148,2853,9088,1208,2848,2716,3107,8718,9375,4715,5904,1438,2627,1277,1719,5412,6362,5412,9656,6362,9656,5412,9789,6362,9789,9656,9789,7901,9446,9095,9446,9446,9647,4392,4989,4389,5607,4389,5827,4389,9211,8790,9255,1338,2709,2709,3130,2709,8870,4635,5224,5410,6265,7762,9375,9010,9375,1366,2661,2020,2661,2661,8586,1402,9324,4028,9047,8822,9047,8951,9673,3429,5242,2529,2934,8411,8678,8678,9354,8410,8677,8677,9353,4800,5785,5785,8592,5785,9525,1523,2508,2508,8593,2508,9524,9493,9530,2427,9321,4915,5821,1678,3155,2219,3155,1678,3572,3155,3572,1415,1766,1974,9679,9405,9679,3973,3984,1239,9502,4123,4133,4123,9504,4133,9504,8533,9504,5173,5253,3669,3686,3669,9784,1393,2631,2121,2631,4740,5229,5229,5675,5229,9545,8392,8908,4563,5538,4563,5640,1286,2261,1286,2363,2233,2530,1729,2404,5367,9398,7794,8082,7794,8728,7793,8083,7793,8729,4425,5261,1579,2329,1579,8313,1579,9719,4005,4011,4005,4012,4005,9579,8894,8978,8894,9227,8978,9227,1231,3121,2503,3121,3923,3925,6126,8739,2849,8741,2031,2238,5308,5515,4922,6195,5572,9528,9081,9528,3875,3876,3875,8182,3875,9315,3998,5887,4916,5887,5887,8398,4413,5632,3498,3499,4629,6049,1395,1933,1860,1933,1933,9745,1188,2112,1188,2871,4584,5461,5461,8500,4584,9020,5461,9020,3621,5868,3621,8832,5868,8832,3621,9161,5868,9161,5739,6210,7824,8500,753,754,753,4037,754,4037,753,8591,754,8591,1122,1610,1176,1643,5937,9528,1507,3029,4784,6306,4576,4905,2826,3086,3086,8446,3086,9607,1630,2572,2572,3758,2572,8360,7929,9172,1578,2402,1578,9766,2402,9766,8111,9766,4855,5679,4855,9765,5679,9765,8113,9765,4215,4229,4221,4229,4229,9575,3376,3384,3376,5540,3376,9182,5532,6332,2255,3055,3492,9743,5328,5993,5328,8754,4078,4091,1146,2323,2323,9296,2323,9298,1331,8573,8573,9537,2160,4161,2160,4162,4161,4162,2160,8532,4161,8532,3434,3504,3503,3504,3504,8931,3452,3454,1611,2892,8611,8973,8612,8974,4571,5045,4571,8165,4571,9584,8561,9768,1705,2227,2227,3489,2227,8863,1798,4078,1446,2306,4649,9665,1372,9666,3671,4956,3671,6432,4956,6432,5497,6432,1910,3181,1646,8437,4923,8438,4591,6207,3703,3704,1802,2289,3542,3544,3543,3544,2501,9776,9437,9776,4564,6002,4564,8947,4746,5991,5991,8224,5991,9647,4735,6189,8004,8133,1450,2655,1450,3106,4448,5719,5719,8772,5719,9152,1550,2299,2299,2819,4827,5576,5576,6096,4647,5928,766,4269,5290,9542,5971,9542,2867,9089,5103,9688,3115,3504,5356,6283,4694,5771,5095,9395,4625,5952,1144,2444,2093,2742,2093,8424,2093,9588,4185,5107,4185,9267,1562,2491,2444,2491,1916,3043,1916,9787,1610,9096,1368,3050,5492,6425,4819,9761,9312,9391,1324,2280,1400,2394,2394,3104,1940,2880,1940,8083,1940,9063,5217,6157,5217,8082,5217,9064,4848,5476,4848,8739,4848,9634,3537,3538,1280,2656,1280,8946,4644,5298,4644,5938,5298,5938,5938,8328,1241,1695,3745,3747,1277,1984,4423,5600,5600,9005,870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\ No newline at end of file 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378,1694,1378,1971,4655,4971,4655,5248,3747,3750,8402,8625,5125,5385,1848,2108,4452,5978,5978,8641,5978,8743,1175,2701,2701,8642,2701,8744,9286,9694,4604,9085,9085,9406,4638,5174,5174,5254,9045,9223,1952,2978,1952,8709,1952,9625,3464,9191,8485,9191,9191,9716,1398,1976,2382,8211,2382,9290,2382,9404,5549,5672,5549,9027,5549,9274,2272,2395,2272,9028,2272,9275,4841,5482,4841,6262,5482,6262,4841,9515,5482,9515,6262,9515,5149,5567,5010,8371,5010,9472,8872,9158,4575,9159,8873,9159,8484,8513,2345,2350,1434,1924,9232,9592,8703,9407,3403,5481,9263,9710,1269,2740,2740,9394,2740,9579,4784,5988,1507,2711,3680,3681,8720,9637,8148,9170,2847,2889,1086,1624,4363,4901,7824,8133,8133,8776,8133,9176,2306,2609,8280,9515,8280,9601,9515,9601,1581,2586,5258,9521,1194,2091,8366,9210,8226,8923,5671,6210,3678,9371,3684,9371,1694,9465,4971,9466,4934,5864,5864,6090,5864,8309,9424,9777,4866,9658,8273,9658,4330,4341,4697,9494,8066,9072,763,4220,1288,1593,4565,4870,3857,5643,5143,5643,3857,9583,5643,9583,4354,6345,4354,9025,4354,9714,1077,3068,1077,9023,1077,9715,8067,9075,1669,9764,2933,8583,4311,4319,4319,5623,1454,2382,1454,8211,1454,9290,1643,2461,1643,8704,1643,9299,1769,9753,9062,9753,9690,9792,9691,9793,1082,2139,4359,5416,1163,2537,1714,2099,7847,8581,1126,2837,2837,9107,2837,9254,1402,1947,1402,8182,1402,9703,6265,6269,4743,6209,1466,2932,4067,4068,4068,4074,4068,9297,1220,2438,1220,8556,4771,4990,4771,5880,4209,4210,4210,9548,4210,9728,748,4131,9373,9652,9502,9651,3848,3849,8736,9644,3641,8381,3641,9385,3641,9587,9385,9587,2413,2910,2910,9505,3666,8964,3666,9155,8964,9155,3666,9156,1685,2749,1685,8302,8703,9608,1166,2258,1166,3124,2258,3124,2258,8965,4505,5057,4505,6133,5057,6133,4505,9482,5057,9482,1403,1879,1879,1948,5057,5432,1105,8941,4382,8939,4109,5867,5190,5867,5867,9484,7987,8704,8704,9478,4843,6145,6145,8628,6145,8787,4101,8785,5195,6204,5534,5691,5481,6104,6104,8454,6104,9261,5900,9277,2217,3618,3618,3629,3618,8929,4000,4003,3648,3659,3368,3369,2069,8704,8704,9730,3431,9162,5005,9162,4514,5332,5754,8689,2477,8690,8865,9367,6427,9233,1677,3149,1677,8380,1677,9617,738,2840,738,3610,2840,3610,738,8253,2840,8253,8879,9410,8880,9411,1221,2674,1221,9348,1221,9662,8227,8948,5012,5421,3073,8749,4715,5850,1438,2573,8286,9216,5339,5340,2062,2063,8287,9218,3480,3527,9024,9456,7901,8224,7901,8599,7901,9095,4617,9518,9573,9694,4378,5093,1500,1952,5114,8265,5733,5759,1772,9404,1096,1874,1189,2663,1189,8480,1189,9449,2085,2616,7769,8266,8289,9750,8993,9371,7901,9149,9149,9647,9149,9767,4828,5362,5362,5893,5259,9710,9106,9151,1338,1818,1818,2388,1818,9289,2275,8336,2275,8934,2275,9207,5552,8335,5552,8935,5552,9206,4282,9610,8169,9610,9610,9679,1981,9556,3571,3574,3571,9051,3571,9213,4891,5967,5164,5967,5967,9531,7766,8430,8430,8455,8430,9274,7767,8429,8429,8456,4417,9108,4417,9132,9108,9694,3816,3817,3817,8501,3817,9698,4400,4888,4094,4099,4099,4104,4099,8606,2848,8742,4428,4938,4428,6260,4428,8726,1151,1661,1151,2983,1151,8727,1765,2230,8773,9530,4712,5360,5360,8827,5360,9495,8490,9432,1414,2253,1996,2253,3675,9364,9364,9705,3559,6418,4314,4326,6236,9375,8212,9375,6236,9552,9375,9552,5046,9423,5046,9742,1988,2967,2723,3031,6000,6308,9322,9390,8438,9048,8438,9174,2047,2161,2047,3824,2047,8408,1178,2423,5013,6311,1736,3034,2184,2988,2184,3170,2184,8856,7807,8217,8217,8864,8217,9115,9170,9306,9306,9689,9305,9688,2004,3141,2004,3556,2004,8676,6412,8595,4393,6348,4446,5572,4425,5116,6350,8790,3863,3894,9087,9742,732,3538,2839,3538,3538,8299,1158,9462,8174,9462,4435,9463,8175,9463,2340,3786,2091,2385,8136,8680,8437,9044,8437,9173,4414,8827,5662,5728,9303,9384,1274,1903,4841,8280,4552,6188,4552,9090,4552,9569,9079,9150,3609,9565,1397,1866,1866,2365,1866,9724,9103,9398,2034,9136,5311,9137,4550,5042,4069,9297,8643,9297,2444,2497,2883,2923,2883,8370,2883,8750,5368,5662,1403,9307,3862,3863,1614,2690,1887,2690,2690,9562,8131,9625,4608,5942,5132,5942,5942,8919,4256,9308,5036,9668,4145,8756,5772,6286,8257,9009,8258,9008,4039,9602,6187,8356,6187,9546,6187,9594,8391,8664,1253,1758,1253,2109,1253,8928,8808,9683,5002,5808,5808,8775,5808,9203,3156,3664,4590,5392,4857,5392,5758,9367,1414,2383,5039,6182,6182,8451,6182,9700,9199,9311,4551,5180,4720,9265,1443,9264,4928,5450,4928,8238,4928,9699,6152,8541,4631,9451,5249,5858,5249,8697,5249,9421,1094,2640,4180,4183,4180,9358,4180,9418,1376,1890,8224,9149,8599,9149,4371,5917,8057,9076,9076,9396,8065,8581,8913,8965,1595,3005,1595,7988,1595,9365,1489,2142,2142,8322,2142,9465,4766,5419,5419,8323,5419,9466,5378,6061,2050,2629,2050,8255,4793,6187,4829,5498,1552,2221,4058,4060,2200,3545,3545,8299,3545,9120,4323,4327,1861,2262,5392,6191,4842,6307,4842,9472,5093,6383,4851,5699,1574,2422,5299,6175,6175,8390,6175,8980,2022,2898,2898,8389,2898,8981,1251,8870,3991,4008,3991,4009,3991,9440,5138,5539,4113,4121,4113,4155,4113,9084,1629,2526,3434,3436,9570,9637,1790,1951,1790,8188,1790,9743,3874,3876,3874,9315,3874,9327,2157,8908,5434,8909,1189,9501,2875,8542,9493,9638,4491,5406,5406,6422,5406,8923,4568,6053,5742,6053,6053,94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175,8267,3175,8818,4615,5095,5095,5665,5095,9297,1446,9659,1878,3877,3877,8182,1878,9324,3877,9324,8182,9324,1171,9224,5663,8780,2386,8779,5105,5237,3592,8975,3592,9163,8975,9163,3592,9164,8625,9712,1439,2601,1439,9342,4684,5064,5064,8061,5064,8517,1902,8618,4653,5590,8709,8927,8927,9625,4498,5951,4498,9329,4498,9778,1111,2140,8209,9271,4878,8641,1601,8642,4630,5048,4630,8381,4630,9350,5359,6019,5359,9208,745,3873,2030,2354,5307,5631,4632,5406,1828,1960,4354,6150,1077,2873,4889,5163,5163,5966,5163,8595,1137,2906,1137,8231,2906,8231,1137,8994,2906,8994,5318,5494,3937,3944,3814,6064,3814,8238,6064,8238,6064,9698,4426,5428,4426,5636,1149,2151,1149,2359,7974,8379,8379,8834,1092,9079,4635,5172,4635,8484,4635,9308,1790,9530,8321,9624,4849,9569,8177,9569,1455,2653,1455,8412,1455,8893,4732,5930,4732,8413,4732,8892,9208,9366,4542,4915,5333,9561,2056,9560,1424,9296,9253,9296,9088,9731,4831,5314,4502,6331,6041,6331,6331,9198,8834,9192,6134,9000,8068,9000,2857,9001,8069,9001,4172,4175,4809,5038,4809,5802,5038,5802,5038,8439,4833,5600,5600,9647,5600,9736,1403,2370,1403,9474,8697,9682,7796,8261,8261,9758,8496,9758,7795,8262,8262,9759,8495,9759,4111,4123,4111,5608,4123,5608,4111,8533,4123,8533,5608,8533,4188,4198,4521,6206,4521,9078,4521,9497,3666,9784,8964,9784,1595,2197,3613,3632,3613,5527,3613,8217,1790,3492,1790,9532,4344,4346,1530,2394,2394,2933,2394,8583,5607,5827,5827,9003,5827,9211,4631,5407,2375,9348,8038,9348,4746,9149,4746,9647,8224,9647,1982,9580,8350,9481,1913,2590,2590,4157,2590,9492,3852,3853,3853,8577,3853,9786,1867,1937,6227,9762,4141,4143,4143,4148,4143,8661,5745,8061,5745,9101,5745,9494,4181,4957,4957,5525,2092,3526,1629,2533,7929,9274,7929,9644,2951,9784,1859,1934,8666,9459,4503,5508,4503,8397,4503,9402,2392,2463,1650,2904,1650,9312,1650,9639,2904,9639,3367,3368,4741,8551,1293,1771,8133,8500,4570,5048,4342,4346,1464,8552,1968,2986,5245,6263,1261,9109,7962,9109,2439,2733,2733,8366,2733,9489,9024,9123,9026,9122,4240,4242,4044,4055,4042,8822,1811,8989,1811,9412,1811,9554,5088,8988,5088,9413,5088,9555,1624,1908,4901,5185,8362,8704,1411,2498,4401,5218,5218,6029,5218,9451,4876,8803,5669,5740,4065,4073,4065,4922,4065,8652,2285,4087,4087,4091,4087,8643,1219,2376,1219,8345,1219,9153,4496,5653,4496,8344,4496,9154,4066,4076,4076,6044,4076,8732,6044,8732,3577,3580,3427,3429,3428,3429,3429,9162,5056,5379,4065,4086,4909,5652,4754,5163,5163,5884,1858,9467,8420,9467,5135,9468,8419,9468,4699,4936,4936,5777,4936,9429,1347,2426,1347,8684,1347,9293,2426,9293,4118,9084,4673,5252,4673,8306,4673,9583,4782,5134,1156,2113,1156,9487,1156,9659,7889,8179,5154,6340,1435,2083,2083,8994,2083,9553,4417,5734,5734,9108,5734,9132,1645,4121,4936,8368,7848,8573,4679,9284,8494,8582,1093,2105,1093,9151,1093,9630,1540,2810,734,3551,734,6116,3551,6116,734,8278,6116,8278,1292,1770,3972,3985,4679,5155,1226,1786,1250,1798,1731,1798,1798,9657,4516,5232,5232,6165,5232,8394,6443,8401,6443,9070,9070,9316,1325,2052,4602,5329,3907,3939,3909,3939,3939,9204,1688,2977,2372,2977,2977,9366,6095,6261,2818,2984,1137,9553,2906,9553,4482,6142,1205,2865,3452,3513,1535,2520,3851,3852,4760,5516,4760,8878,4760,9111,1483,2239,1483,8877,1483,9110,4625,5318,5374,5790,1734,2839,4544,5097,5097,5284,1267,1820,1820,2007,4950,5494,4950,8672,4950,9233,7815,9121,7815,9344,9121,9344,9344,9345,1266,1541,1266,2014,1266,9597,4965,6254,5649,6254,6254,9372,5639,8177,5639,8640,5639,9392,8177,9392,1695,2319,1695,3164,1695,9014,2076,2768,4900,5479,6350,9614,8072,8217,5230,6347,1953,3070,5639,9569,9392,9569,4638,5595,4638,9129,2521,9296,4402,5219,5219,6030,5219,9423,5286,5977,5286,6228,1260,9171,1358,9610,1974,9610,1385,1926,4352,6011,4352,8550,4352,9466,1075,2734,1075,8549,1075,9465,1538,2397,1538,8336,1538,9585,4815,5674,4815,8335,4815,9586,4812,5797,2009,2699,2009,2950,1987,2349,2349,4267,2349,8457,4627,5689,8283,9626,5353,6045,2690,2947,2690,8242,2608,3072,3072,8538,3072,8684,1350,2411,5041,5962,5041,8551,5041,9072,1104,2516,1104,2987,1104,8922,4177,5620,5016,5620,5620,9267,4369,9106,3457,3459,4836,5740,1559,2463,1332,1981,1872,8920,4628,5688,1629,2824,2824,3052,2824,9019,4906,6101,6101,6329,6101,9018,1764,2685,1764,8552,1764,9075,1728,9196,2494,9015,4515,5269,3915,3945,3944,3945,3945,8484,4436,5433,1115,9763,1083,2678,1083,9038,1083,9192,2365,3892,3892,3904,3892,9702,1183,2266,760,4167,5716,6010,6010,8374,6010,9398,1351,2412,8095,8420,8246,8420,8420,8988,8096,8419,8247,8419,8419,8989,4168,4185,4184,4185,4185,9701,8944,9438,1530,2403,3780,8309,3780,9189,3780,9335,8309,9335,9189,9335,2647,3776,2336,2664,9173,9295,9174,9294,3624,3625,3625,3626,3625,9587,5282,6224,4782,5778,5778,5902,5778,9760,4028,4039,8229,9655,8228,9654,7912,8348,7913,8349,8349,8663,1677,2949,4574,9378,5139,9378,1297,9377,1862,9377,1956,8479,1437,1752,3594,3598,4392,6063,3592,9762,8975,9762,4532,5503,5503,8435,5503,9536,4058,9434,4491,6119,1445,2088,2088,9449,4722,5365,5365,9448,2449,2904,1487,2483,4764,5760,1655,2527,2527,8575,2527,9790,1555,9732,1097,2431,4374,5708,4575,5550,4963,5550,7936,9551,9342,9551,2097,2513,1369,3139,3139,8184,3139,9615,4989,5712,1430,2427,4991,5879,4439,5814,5814,6020,5814,8230,1346,3071,5698,8729,8729,9620,2421,8728,8728,9619,5266,6009,6009,8338,4564,5388,4564,8311,4564,9126,9306,9337,5101,5292,2657,8232,2657,9344,2657,9752,9516,9548,1891,3121,3121,8432,3121,8644,2200,3538,3538,3545,4440,5813,5813,6021,5813,8233,2527,3722,4731,5819,5819,8249,4731,8703,5819,8703,8965,9259,8120,9309,2287,3059,2287,8511,2287,9029,4490,6118,2496,9532,7952,8450,8270,8450,8450,9542,3711,6294,5547,6294,6294,9121,1294,1713,8408,8890,4889,5267,5946,9451,7943,9070,8556,9070,8556,9172,9021,9678,1504,2856,1504,2896,2856,2896,1504,8748,3651,9067,1784,2994,1784,9140,1784,9309,1572,2362,2362,2885,2362,9661,4751,8079,4751,9563,4751,9723,9563,9723,8899,8994,4489,5405,1832,2353,1832,4059,1832,9787,1675,2004,1164,1966,1966,8389,4441,5243,5243,8390,8470,8817,1153,2357,2357,8236,2357,9071,1237,1992,3707,3711,3707,9121,5112,5892,1835,2615,2456,9711,2580,9711,8245,9711,1757,3061,4078,4092,4078,9319,4092,9319,8357,9319,4682,5612,4838,5500,1561,2223,1514,2203,2203,9643,4705,5508,5508,8926,1215,2579,3640,4954,3561,3564,1174,2819,2819,2937,4451,6096,6096,6214,4119,4132,4119,4155,4132,4155,4155,8303,4673,5143,1621,1957,1621,8446,1621,8610,8573,8641,8345,9272,8344,9273,1645,4113,4894,5232,1612,2889,2280,9278,2280,9599,9278,9599,1712,2154,1712,8198,1712,9746,5068,6183,1845,2481,1845,9434,8391,8927,4672,5211,4672,5848,4672,9112,1371,3040,3829,6262,3829,9515,4388,5417,1367,3111,2189,4180,1263,1983,1263,3035,1263,9680,8708,9079,4989,5431,1405,2335,9309,9310,5632,6297,1245,2657,3945,5155,4234,4242,4242,5852,5466,9569,3373,3394,3373,8561,3394,8561,3394,9232,5473,5826,1098,2869,2445,2869,2869,9625,4895,9117,1589,2989,5652,9778,8039,9778,1383,2030,4660,5307,8420,8852,8419,8853,1768,1941,1941,9087,1941,9748,4539,6274,5601,6381,4581,5292,4581,5828,4581,8297,1304,2015,1304,2551,1304,8296,4749,6041,3730,3732,3978,9030,8558,9030,8246,9467,8247,9468,5081,6384,6384,9190,6384,9542,8731,9425,4716,5765,4681,8918,4506,4980,1230,2473,2473,8982,2473,9452,7901,8263,8263,8599,1656,2808,4933,6085,1972,2581,1972,8808,1972,9426,4744,6280,6280,9113,6280,9725,1906,1986,1906,9474,1410,9330,1410,9530,1410,9532,8916,8984,8917,8983,8500,8776,1358,2317,1358,9130,3953,3959,3958,3959,3959,9450,3464,3465,3465,3466,3465,9716,5200,6454,9132,9573,5703,8717,9081,9570,3877,9386,1727,2387,2387,3527,2387,9707,5057,6173,1923,9602,4062,9602,4476,5542,1199,2265,2522,4307,4307,9618,4307,9621,3442,3508,8699,8957,1893,2117,1253,2813,3763,5449,5449,6161,6161,8374,3477,5560,751,4038,1614,3137,3137,8537,3137,8538,5081,5993,1167,2923,4815,5552,4815,9206,1538,2275,1538,9207,2135,2425,2135,3085,4898,6102,4898,8212,4898,9749,6102,9749,4242,4243,1219,1841,4496,5118,1338,8870,4520,6403,6403,8160,6403,9272,1243,3126,3126,8161,3126,9273,3046,8429,3046,9275,3046,9774,9275,9774,6323,8430,6323,9274,6323,9775,9274,9775,2650,9301,1763,2640,5733,9737,5857,9737,8248,9737,5040,5917,1769,1942,1942,9105,1942,9753,8085,8232,8085,9088,8232,9088,1298,2273,1810,2273,2273,9487,4567,5189,1290,1912,5404,8414,5404,9520,5404,9531,5402,5689,5020,6162,4699,5634,1507,3046,4784,6323,1248,3081,3081,3175,1119,3138,4737,5810,5810,5962,5810,9072,2533,2685,8766,8777,1517,2797,4794,6074,2457,8997,3408,3490,1132,9611,4945,5478,4751,9210,9210,9563,8276,9408,9408,9463,9408,9469,8277,9409,9409,9462,9409,9470,1638,2544,2544,3123,2544,9255,1729,2282,1729,2879,5006,5559,5006,6156,5006,8263,2274,9187,5551,9188,1239,1955,8955,9510,4672,5161,4672,9359,9309,9732,9229,9610,3579,6433,5107,5786,5107,9267,5107,9701,4426,5304,1149,2027,5410,6269,8003,9427,9508,9681,4779,5473,1502,2196,5067,5228,5067,9673,8755,9197,3983,5031,8883,9590,8882,9591,8469,8781,1438,2146,4715,5423,1710,2702,4987,5979,1092,2050,3938,9474,4610,5203,5203,5936,5203,9644,8552,9228,8782,9228,2011,2697,2011,2948,1314,1760,9197,9347,3991,5592,2568,3512,4808,5526,4808,6395,1531,2249,1531,3118,7888,8178,7965,8689,7964,8690,1554,8232,1554,9088,1554,9640,4077,4079,4077,4081,4792,5481,4792,6104,2555,9650,9443,9650,3465,8543,2088,2089,2089,8258,2089,9449,5365,5366,5366,8257,5366,9448,4866,5617,5617,5970,5617,8273,3786,3791,3789,3791,3791,8758,1966,2402,5243,5679,4743,5411,1466,2134,7931,8669,3581,6034,5047,6034,6034,8905,5559,6358,3954,3956,3954,3960,3954,9241,3601,3603,3601,9762,1171,9311,8141,8318,1125,1916,1916,8284,1916,8760,4119,8248,1612,3136,3136,8970,3136,9034,8436,9255,9091,9255,5331,9050,3654,3672,3654,9067,3654,9212,1400,2356,8562,9673,1095,1920,1770,2756,2756,3656,2756,8967,8707,9106,4633,9069,1667,2200,4623,6348,2435,9746,2127,8426,2127,9503,2127,9562,4050,4052,1824,2473,2767,4126,4114,4126,4126,8753,4624,6349,4895,5780,2967,9255,5832,9649,9454,9649,8705,8999,8788,8848,8788,9141,3466,3468,6118,9508,768,769,768,4312,768,8457,5549,8384,2272,8383,1441,2224,1441,9301,1441,9337,4539,5815,4995,5514,4890,6124,4890,6166,6124,6166,6124,8637,6166,8637,4746,5681,5681,9149,5681,9647,1263,3114,1263,8712,5016,5280,5280,8673,1858,2656,5135,5933,1363,1905,1905,1985,8112,8113,8110,8111,1124,1917,4862,6068,5911,6068,1585,2791,2634,2791,3841,5369,1450,2465,1450,8599,1450,9095,2465,9095,8641,9231,8642,9230,5294,6024,2017,2747,2538,8310,2538,8817,2538,9038,8310,9038,4654,5610,5434,5610,5610,8332,4369,5327,7782,8341,4739,8697,1132,8465,8465,9125,8231,9553,4875,6449,6449,9121,6449,9726,4163,4168,3081,9738,1195,2915,4453,5108,1176,1831,1176,2069,4961,8701,1523,2624,2624,9136,2624,9686,4800,5901,5901,9137,5901,9687,8626,9545,2574,3798,3798,3805,3798,9323,4472,6192,1567,2053,8351,9674,8352,9675,5864,9189,1943,8570,5220,8571,4462,5440,4462,8554,4462,9649,1905,3850,3850,3855,3850,9324,4731,4957,4584,8133,4584,8500,4584,8783,759,760,759,4167,759,8559,4541,5604,4541,6134,1264,2327,1264,2857,1474,8080,1474,9549,1474,9739,9549,9739,4968,6411,5301,5768,1565,2473,1622,2538,3888,9331,3888,9460,4433,5583,5583,5865,3095,9444,3095,9786,1366,1919,1919,3153,1718,2237,4666,5570,4666,9654,1389,2293,1389,9655,5093,5508,1923,2541,3702,3716,1318,1761,1761,3053,3533,3551,3533,5478,3533,9727,4388,6187,2176,8465,2176,9560,2176,9729,8465,9729,1396,1867,1396,8203,1396,9405,3795,3796,1431,9193,4553,5347,4553,6371,5347,6371,5347,9645,6371,9645,9324,9327,1157,2062,2062,2775,4434,5339,5339,6052,1898,2980,1898,3175,2980,3175,2474,3089,5098,5460,5460,6333,4263,4270,4263,5262,4263,9511,7943,8401,1420,1787,1420,2523,6183,9495,4070,4078,4070,4092,2447,2994,2447,9066,2447,9140,3452,3512,4598,9626,760,4335,4734,5795,4697,5016,4697,9267,4738,5627,5076,5627,5627,9054,1185,2163,1185,8555,1185,9650,3891,3905,3891,9708,3905,9708,7992,8356,4338,4340,1588,2165,1588,2859,1588,8570,4865,5442,4865,6136,4865,8571,8391,9626,4993,6451,6451,8217,6451,9419,5124,5926,5926,6023,5926,9758,1847,2649,2649,2746,2649,9759,1074,9480,4351,9479,1730,3791,4410,4960,1133,1683,1393,2047,1901,2621,2487,2653,2487,8412,2487,8857,5764,5930,5764,8413,5764,8858,3173,9344,9344,9640,757,4155,2016,2548,5175,6257,5175,6452,1487,2746,4764,6023,7901,8855,7901,9785,1492,2555,1108,2749,1108,8302,9336,9788,5782,6150,2505,2873,4936,5634,1694,2734,4971,6011,8507,9304,3430,3432,3432,9162,1463,2398,1463,8984,1463,9625,4974,5237,4974,9236,4974,9603,8365,9300,8365,9735,4776,6160,1697,1960,1697,9238,1697,9604,7967,8104,8420,9421,1085,2405,1688,2405,2405,9655,4362,5682,4965,5682,5682,9654,1138,9674,4415,9675,4046,4055,1524,1935,2255,8815,5532,8816,2073,2481,4972,5596,4972,6441,4972,9058,1595,9680,4057,4058,1991,2652,1709,2322,1709,3985,1709,9447,4684,5745,5064,5745,7966,8105,8419,9426,5555,5557,5555,8279,5555,8695,9082,9646,1474,9103,2054,9096,5197,8657,5197,8957,5197,9761,5130,9391,8134,9391,5360,5730,2330,2550,2550,8616,2550,9082,4823,6143,4988,6143,6143,8670,1714,3576,5372,6457,5919,6457,2095,3180,2642,3180,2522,4316,4307,4316,4316,9621,2206,3063,3063,8633,3063,8730,2072,2480,8718,9010,9010,9552,1472,2574,7861,8480,1659,2500,4499,5025,5025,5485,5025,9791,2192,2913,2080,2627,5357,5904,6103,6235,759,2809,1907,3939,3974,5597,4329,4341,4480,5775,1462,2336,4857,5502,7950,8208,8840,9347,1108,2037,4122,4142,4142,9168,4142,9289,3546,3551,3546,8299,3546,9727,1546,2866,1711,2866,2866,8665,1712,9763,4611,5284,4611,5584,2011,2599,2011,3628,2382,9621,1668,2201,2201,3610,2201,9293,4492,4994,1215,1717,2654,9151,1818,9168,3689,6117,8542,9408,8541,9409,2083,2453,1900,2714,1900,9298,1130,1932,4841,6127,5482,6127,8450,9071,6119,9613,2005,2947,4640,5182,5182,5262,2222,9537,5499,9538,8902,8997,8997,9227,8830,9714,8829,9715,4643,5196,1979,3048,4981,5803,1349,2676,2676,8483,2676,9201,4433,5550,3847,3881,7856,8174,7855,8175,1532,2472,1115,2786,2786,9042,2786,9763,1357,2874,2297,2874,2297,9772,2874,9772,4634,6151,5574,6151,5574,9773,6151,9773,5462,9152,2422,9766,8110,9766,5699,9765,8112,9765,4549,5567,3446,9598,5666,6147,4542,6108,5265,8133,5265,8783,5265,8904,8133,8904,2389,2870,7927,8134,4923,5989,1646,2712,2133,2992,4894,6125,5401,6125,6125,8330,1719,2488,1366,2020,4904,6103,7769,8157,8097,9132,4898,9552,5931,9150,5460,5993,1080,2573,3915,5646,5155,5646,5646,9284,1295,2148,1769,2148,2148,9336,5092,5459,5567,6060,2153,2898,5430,6175,1111,9506,8770,9725,1214,2700,1214,2842,2700,2842,2842,9119,3153,9248,1213,2699,1213,2841,2699,2841,2841,9118,2147,3011,2147,3674,4669,5025,5025,6005,1428,2423,1428,2850,1116,3135,2652,3135,3135,8630,4999,6051,1090,3170,5932,6383,4849,5639,8261,9663,8262,9664,8263,9785,5257,5576,1980,2299,5806,6211,3063,9443,1396,1937,5132,5499,8133,8749,3395,3396,3395,3397,3395,9592,4887,9050,2501,7870,5928,6414,2253,2383,4324,4335,4324,8559,4324,9618,4906,5810,5810,5812,4373,5151,1268,2883,2883,9335,1277,2232,1704,2232,2232,9611,1111,2910,5126,5751,7952,8223,2076,2478,5353,5755,3560,3561,3560,8380,3560,9617,3817,5054,5054,8501,5054,9698,5010,5115,1424,2323,4399,4887,1171,2174,1212,2128,4557,5320,4818,5537,9638,9743,4533,5942,1267,1795,1795,8668,4544,5072,5072,8669,4968,6430,3426,3499,3459,3461,1280,2043,5711,6443,763,4182,5220,6444,1943,3167,7900,8192,3693,3694,4174,4183,4928,6387,1115,1712,1620,2135,1822,2881,1822,8578,1822,9437,5099,6158,5099,8579,5099,9436,2316,3896,2316,9386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837,9258,6074,8706,9258,7837,8706,9258,4374,6440,8158,4374,6440,8885,6440,8158,8885,1481,1875,2675,1481,1875,8072,1481,2675,8072,5139,6437,8163,5139,6437,8809,2249,8424,8833,7777,8424,8833,1862,3160,8164,1862,3160,8810,3960,5174,9129,3960,5174,9241,5174,9129,9241,4107,4110,9266,4107,4110,9484,4107,9266,9484,3464,3466,9556,3464,3466,9716,3464,9556,9716,3466,9556,9716,2503,7914,9597,2503,8432,9597,7914,8432,9597,7857,8145,8990,7857,8145,9184,4595,5339,8080,4595,5339,8860,4595,8080,8860,4395,6397,8718,4395,6397,9552,6397,8718,9552,5212,5529,8581,5212,5529,8802,4707,5474,8600,2199,7910,8530,2199,7910,9684,2199,8530,9684,7910,8530,9684,5476,7911,8531,5476,7911,9685,5476,8531,9685,7911,8531,9685,3819,8081,8501,3819,8501,9428,8081,8501,9428,2780,8758,9286,7862,8758,9286,1739,4322,8404,1739,8404,9574,1175,2486,2736,1175,2486,9456,2486,2736,9456,4624,9491,9727,7799,9491,9727,1479,2040,2598,1479,2040,8138,1479,2598,8138,8085,8357,9507,8085,9507,9731,8357,9507,9731,4595,8080,9476,1201,7795,8530,1201,8530,9759,7795,8530,9759,4478,7796,8531,4478,8531,9758,7796,8531,9758,8055,8124,9199,8055,8317,9199,8124,8317,9199,4817,7907,8481,4817,7907,8697,4817,8481,8697,7907,8481,8697,1670,3025,9331,1670,3025,9709,3025,9331,9709,2900,7872,9135,2900,8393,9135,7872,8393,9135,1345,1934,9582,1345,8144,9582,2461,7869,9282,2461,8704,9282,7869,8704,9282,7875,8094,8360,7875,8094,8679,2013,7883,9346,2013,9346,9599,7883,9346,9599,7897,8525,8526,7897,8525,9459,7897,8526,9459,8525,8526,9459,2797,7836,8705,2797,7836,9257,2797,8705,9257,7836,8705,9257,8005,8298,8705,8005,8298,9318,8005,8705,9318,1997,2390,3026,1997,3026,9038,2390,3026,9038,7932,8527,8598,7932,8527,9726,7932,8598,9726,8527,8598,9726,7892,8668,8805,7892,8805,9410,8668,8805,9410,7893,8669,8806,7893,8806,9411,8669,8806,9411,757,4073,4086,757,4086,8180,4073,4086,8180,7799,8299,9120,1488,2297,2437,2297,2437,8962,4765,5574,5714,5574,5714,8960,5853,8170,8862,5853,8170,9308,8170,8862,9308,9251,9361,9374,9252,9360,9373,2799,8055,8622,2799,8055,8887,2799,8622,8887,8055,8622,8887,6456,8625,9172,6456,8625,9316,6456,9172,9316,8625,9172,9316,8081,8501,8555,5002,7930,9361,5002,9361,9594,7930,9361,9594,5179,5652,8039,5179,5652,9132,5652,8039,9132,1792,2703,8998,1792,2703,9334,1792,8998,9334,5069,5980,8999,5069,5980,9333,5069,8999,9333,7895,8454,8838,8454,8838,9261,3957,3964,9240,3964,8170,9240,1767,3383,8563,1767,3383,8582,1658,3062,9124,1658,3062,9125,1658,9124,9125,8033,8552,8782,8033,8552,9075,8033,8782,9075,5730,7829,9495,5730,8827,9495,7829,8827,9495,1656,2090,2775,1656,2090,8325,1656,2775,8325,4933,5367,6052,4933,5367,8324,4933,6052,8324,1109,1783,2036,1109,2036,8041,8152,8797,8798,8152,8798,8897,8797,8798,8897,8151,8799,8800,8151,8800,8896,8799,8800,8896,1909,7948,8461,1909,8461,9514,7948,8461,9514,5186,7949,8462,5186,8462,9513,7949,8462,9513,7811,8326,8382,7811,8326,9195,8326,8382,9195,3564,3565,3604,3564,3604,8380,732,733,3537,732,733,8299,732,3537,8299,733,3537,8299,1109,1783,3130,1109,3130,8906,7765,8446,9607,7765,8761,9607,2185,2902,8431,2185,8431,9224,4814,7854,9760,4814,8816,9760,7854,8816,9760,3145,8178,9119,3145,8178,9155,3145,9119,9155,8178,9119,9155,1519,2454,8001,1519,2454,9340,1519,8001,9340,2454,8001,9340,5607,7969,9211,5607,8557,9211,7969,8557,9211,5903,8988,9421,7967,8988,9421,4204,4995,6237,4204,4995,8307,4775,5070,5726,5070,5726,9642,1126,7968,9640,1126,8527,9640,7968,8527,9640,3751,5556,6094,3751,5556,8234,5556,6094,8234,1821,8749,9082,7860,8749,9082,4773,6152,8706,4773,6152,9633,4773,8706,9633,6152,8706,9633,1800,1803,8084,1800,1803,9185,4519,8820,9475,7920,8820,9475,4513,5195,6166,4513,6166,9425,8125,9110,9655,8125,9110,9721,8125,9655,9721,8126,9111,9654,8126,9111,9722,8126,9654,9722,2017,8493,9379,5294,8492,9380,2626,8989,9426,7966,8989,9426,746,5553,6367,746,6367,8239,5553,6367,8239,8098,8517,9494,8098,9296,9494,8517,9296,9494,4831,5934,9205,4831,5934,9310,3623,8242,8832,3623,8242,9339,3623,8832,9339,8242,8832,9339,1126,2993,9640,2993,7968,9640,4891,8059,9050,4891,9050,9531,8059,9050,9531,4573,6036,8463,1296,2759,8464,4375,8338,9634,7906,8338,9634,4426,5199,5570,4426,5570,8953,5199,5570,8953,1149,1922,2293,1149,2293,8954,1922,2293,8954,8022,8265,8991,4667,8506,8936,7962,8506,8936,2516,2518,8817,2516,2518,9200,2518,8817,9200,5231,6416,8026,5231,6416,8647,5231,8026,8647,4431,5113,5444,5113,5444,8470,1154,1836,2167,1836,2167,8469,3756,3758,8360,3756,8360,9791,1762,2905,3108,1762,2905,8046,1762,3108,8046,2905,3108,8046,4619,5214,5684,4619,5214,8862,5214,5684,8862,2005,2592,3637,2005,2592,8242,2005,3637,8242,1797,2854,8017,1797,2854,9529,1797,8017,9529,2854,8017,9529,1465,2104,8121,1465,2104,9225,2104,8121,9225,4742,5381,8122,4742,5381,9226,5381,8122,9226,7782,8734,9617,7782,9453,9617,1614,1887,8538,7955,8149,8982,7955,8395,8982,5981,8160,8590,5981,8590,8980,8160,8590,8980,2704,8161,8589,2704,8589,8981,8161,8589,8981,7783,8381,8502,7783,8502,8788,8381,8502,8788,5083,5850,6448,5083,5850,9678,5083,6448,9678,8032,8413,8937,8315,8413,8937,8031,8412,8938,8314,8412,8938,7821,8019,8587,7821,8019,8996,7821,8587,8996,8019,8587,8996,7820,8018,8588,7820,8018,8995,7820,8588,8995,8018,8588,8995,4976,6416,8282,4976,6416,8417,3672,3673,9364,3672,8146,9364,7926,8085,9507,7926,8357,9507,6307,8040,9472,6307,9080,9472,8040,9080,9472,8131,8697,8883,8131,8697,9421,8131,8883,9421,4168,7830,9267,4168,9267,9701,7830,9267,9701,4442,5302,5670,4442,5302,8624,5302,5670,8624,1165,2025,2393,1165,2025,8623,5131,5386,9349,5131,5386,9363,5131,9349,9363,2831,9255,9501,7861,9255,9501,2139,7842,9264,2139,8493,9264,5416,7841,9265,5416,8492,9265,1964,2789,2861,1964,2789,9394,1783,8870,9285,8041,8870,9285,6200,7865,9249,6200,8474,9249,7865,8474,9249,1748,1930,2893,1748,1930,9302,1930,2893,9302,5181,6189,8719,5181,7890,8719,3977,3994,8123,3977,3994,9783,4590,5618,9636,4590,5618,9668,4590,9636,9668,1400,7997,9677,1400,8326,9677,7997,8326,9677,4743,5484,8246,4743,5484,8988,4743,8246,8988,1466,2207,8247,1466,2207,8989,1466,8247,8989,3023,9292,9541,7984,9292,9541,1421,8658,8894,7921,8658,8894,2994,7980,9066,2994,7980,9309,7980,9066,9309,7955,8296,8982,7955,8296,9553,7955,8982,9553,5013,8689,8973,7977,8689,8973,1736,8690,8974,7979,8690,8974,1590,2596,3836,1590,2596,8716,1590,3836,8716,2596,3836,8716,5660,8114,9227,5660,8820,9227,8114,8820,9227,4599,5422,8086,4599,5422,8293,7839,8078,8324,7838,8077,8325,2448,8114,8894,2448,8114,9596,2448,8894,9596,8114,8894,9596,4734,5336,5741,4734,5336,7964,5336,5741,7964,1457,2059,2464,1457,2059,7965,2059,2464,7965,7954,8406,9080,2344,2880,9143,2344,2880,9144,2344,9143,9144,5621,6157,9145,5621,6157,9146,5621,9145,9146,1683,2795,2830,1683,2795,8023,1683,2830,8023,4960,6072,6107,4960,6072,8024,4960,6107,8024,4531,5202,6141,5202,6141,8096,1254,1925,2864,1925,2864,8095,1087,8126,9230,1087,8378,9230,8126,8378,9230,4364,8125,9231,4364,8377,9231,8125,8377,9231,8136,8922,9290,8136,8922,9351,8136,9290,9351,8922,9290,9351,4073,4086,8652,4073,8180,8652,1408,1925,2618,1408,1925,8024,4685,5202,5895,4685,5202,8023,1374,8831,9140,8140,8831,9140,2549,8369,9432,7876,8369,9432,3130,7925,9168,3130,8870,9168,7925,8870,9168,4419,5501,8090,4419,5501,9113,4419,8090,9113,4239,4240,4249,4240,4249,8187,4528,8015,8601,4528,8015,9401,4528,8601,9401,8015,8601,9401,4851,6234,7881,4851,6234,8842,1574,2957,7882,1574,2957,8844,5313,8085,8601,5313,8085,8795,5313,8601,8795,8085,8601,8795,4758,5974,9653,5665,7926,9297,5665,7926,9395,5665,9297,9395,7926,9297,9395,1780,2896,8748,1780,2896,9506,2896,8748,9506,3970,3984,8458,3970,3984,9447,4757,5316,5875,4757,5316,8156,4757,5875,8156,8020,8037,8920,8020,8581,8920,8037,8581,8920,3040,8026,9615,3040,9502,9615,8026,9502,9615,4184,9253,9418,7830,9253,9418,1746,8296,8728,1746,8296,9169,1746,8728,9169,5023,8297,8729,5023,8297,9170,5023,8729,9170,7866,9055,9116,7866,9055,9589,7866,9116,9589,9055,9116,9589,4844,6083,8167,4844,6083,9382,4844,8167,9382,3942,8154,8918,3942,8154,9129,3942,8918,9129,8154,8918,9129,2162,3171,9337,2162,7957,9337,1453,2210,2235,1453,2210,8707,4730,5487,5512,4730,5487,8708,1983,3035,9559,8023,8220,9063,8023,8752,9063,8220,8752,9063,8024,8221,9064,8024,8751,9064,8221,8751,9064,4955,5496,9163,4955,8147,9163,5042,6144,9092,5042,7991,9092,6144,7991,9092,4901,7856,8897,4901,7856,9409,7856,8897,9409,1624,7855,8896,1624,7855,9408,7855,8896,9408,4409,5305,5509,4409,5305,8895,4409,5509,8895,4668,8507,8865,7959,8507,8865,4669,5485,5486,4669,5486,9398,5485,5486,9398,5528,8031,9567,5528,8938,9567,8031,8938,9567,2251,8032,9568,2251,8937,9568,8032,8937,9568,8425,9112,9751,6085,6268,8324,6085,8324,8377,2808,2991,8325,2808,8325,8378,4437,7847,9123,4437,9123,9538,7847,9123,9538,1160,7848,9122,1160,9122,9537,7848,9122,9537,7896,8526,8666,7896,8526,8763,1413,2221,2478,2221,2478,9295,4690,5498,5755,5498,5755,9294,7829,8083,9206,8083,9206,9586,7828,8082,9207,8082,9207,9585,4667,5757,8506,4667,5757,8936,2644,8065,8920,2644,8065,9456,2644,8920,9456,8065,8920,9456,4601,5321,5557,4601,5321,8767,5321,5557,8767,4756,5976,8130,4756,5976,9762,4756,8130,9762,5292,8297,9170,5292,9170,9194,8297,9170,9194,2039,3645,3686,2039,3686,8129,3645,3686,8129,3457,5483,8512,3457,5483,8691,5483,8512,8691,1781,2260,8061,1781,2260,9600,2260,8061,9600,7822,8512,8691,8512,8691,9454,1628,2113,8088,1628,2113,9659,2113,8088,9659,4705,5482,5792,4705,5482,8280,3956,3960,8169,3956,3960,9241,3956,8169,9241,2059,2672,8218,2059,2672,9200,2672,8218,9200,5119,6372,6386,4896,5628,5963,4896,5628,8124,5628,5963,8124,7771,8235,9789,7771,8572,9789,8235,8572,9789,3735,3738,8196,3735,3738,8562,3735,8196,8562,2593,2970,9122,2593,7885,9122,5870,6247,9123,5870,7884,9123,1378,8008,8322,1378,8008,8383,1378,8322,8383,8008,8322,8383,4655,8009,8323,4655,8009,8384,4655,8323,8384,8009,8323,8384,1678,2219,9213,1678,8179,9213,7896,8181,8863,7896,8181,9330,7896,8863,9330,8181,8863,9330,5944,6413,9184,5944,6413,9425,6413,9184,9425,5039,8046,9700,5039,8451,9700,8046,8451,9700,4723,5556,6094,4723,5556,8234,4723,6094,8234,1225,2764,9318,1225,8053,9318,1706,2549,7876,1706,2549,8519,1706,7876,8519,4983,5826,7874,4983,5826,8518,4983,7874,8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05,9655,4362,4965,5682,4362,5682,9654,4965,5682,9654,1138,8351,9674,1138,9174,9674,8351,9174,9674,4415,8352,9675,4415,9173,9675,8352,9173,9675,4046,4050,4055,4046,4055,8548,1524,1935,3076,1524,1935,8515,1996,8399,9065,8399,9065,9631,2255,8251,8815,2255,8815,9606,8251,8815,9606,5532,8250,8816,5532,8816,9605,8250,8816,9605,2073,2481,8284,2073,2481,9292,4972,5596,6441,4972,5596,9058,4972,6441,9058,1595,2197,9680,1595,7988,9680,7939,8625,9712,1446,9302,9659,4020,4057,4058,4057,4058,8396,8288,8468,8759,7801,8328,8687,7801,8687,9035,1991,2652,8479,1991,2652,8630,1709,2322,3985,1709,2322,9447,1709,3985,9447,4684,5064,5745,4684,5745,8061,5064,5745,8061,7966,8105,8419,7966,8105,9426,7966,8419,9426,8105,8419,9426,1303,1630,2045,5555,5557,8279,5555,5557,8695,5555,8279,8695,7860,9082,9646,8639,9082,9646,7898,8272,8846,8272,8607,8846,4756,5317,9762,1474,9103,9549,1474,9103,9739,9103,9549,9739,1237,2054,9096,2054,9096,9134,8391,8664,9626,8391,8927,9626,8664,8927,9626,5197,8657,8957,5197,8657,9761,5197,8957,9761,8657,8957,9761,1480,2039,9784,1500,2428,2691,5130,8134,9391,5130,9391,9642,8134,9391,9642,4712,5360,5730,5360,5730,9495,2330,2550,8616,2330,2550,9082,2550,8616,9082,7938,8155,8558,7808,8186,8915,4823,4988,6143,4823,6143,8670,4988,6143,8670,1714,2603,3576,1714,3576,9336,5372,5919,6457,5372,6457,9272,5919,6457,9272,2095,2642,3180,2095,3180,9273,2642,3180,9273,2522,4307,4316,2522,4316,9621,4307,4316,9621,7951,9125,9527,2206,3063,8633,2206,3063,8730,3063,8633,8730,2072,2480,8288,2072,2480,9332,7770,8718,9010,7770,9010,9552,8718,9010,9552,1472,1808,2574,1472,2574,8478,7861,8202,8480,7861,8480,8790,1659,2500,8965,1659,2500,9417,4499,5025,5485,4499,5025,9791,5025,5485,9791,1459,2456,2482,2192,2913,8379,2192,2913,8801,1552,2080,2627,2080,2627,8924,4829,5357,5904,5357,5904,8925,6103,6235,8235,6103,6235,8958,759,2809,4167,759,2809,8559,8177,8313,9569,8177,9089,9569,7889,8179,9118,7889,8179,9213,7889,9118,9213,1907,3909,3939,1907,3939,9204,7809,8757,9435,8264,8757,9435,3971,3974,5597,3974,4976,5597,7962,8506,9109,8003,8506,9109,8207,8891,9498,4329,4330,4341,4329,4341,8702,4480,5775,8640,4480,5775,9005,1462,2336,2664,1462,2336,9488,2336,2664,9488,4857,5502,6191,4857,5502,8891,7950,8208,8840,7950,8208,9347,7950,8840,9347,8208,8840,9347,1108,1782,2037,1108,2037,9752,4122,4142,9168,4122,4142,9289,4142,9168,9289,3546,3551,8299,3546,3551,9727,3546,8299,9727,1546,1711,2866,1546,2866,8665,1711,2866,8665,1712,8198,9763,1712,9746,9763,8198,9746,9763,4611,5284,5584,4611,5284,9326,4611,5584,9326,2011,2599,3628,2011,2599,8453,2011,3628,8453,7849,8749,9577,2382,8211,9621,2382,9404,9621,8211,9404,9621,1420,1787,2509,1668,2201,3610,1668,2201,9293,2201,3610,9293,7802,8491,8848,8491,8788,8848,8357,8795,9395,4492,4994,5450,4492,4994,8660,1215,1717,2173,1215,1717,8659,2654,8654,9151,2654,9151,9630,4446,9081,9637,7891,9081,9637,1338,1818,9168,1818,9168,9289,740,3689,6117,3689,6117,8715,2035,8542,9408,7855,8542,9408,5312,8541,9409,7856,8541,9409,1435,2083,2453,2083,2453,9553,1900,2404,2714,1900,2404,9298,1900,2714,9298,1130,1932,2660,1130,1932,9321,7851,8552,9228,7851,8782,9228,8420,8852,9421,8420,8988,9421,8852,8988,9421,4841,5482,6127,4841,6127,8280,5482,6127,8280,7790,8270,8450,7790,8450,9071,8270,8450,9071,4882,6119,9613,6119,9131,9613,7816,8916,9759,7817,8917,9758,2005,2947,8242,2005,2947,9339,4640,5182,5262,4640,5182,9284,5182,5262,9284,1331,2222,9537,2222,8830,9537,4608,5499,9538,5499,8829,9538,8059,8630,8791,8059,8675,8791,7921,8902,8997,7921,8997,9227,8902,8997,9227,7814,8561,9309,8561,9309,9310,8374,9103,9398,8064,8830,9714,8477,8830,9714,8065,8829,9715,8476,8829,9715,4643,5196,6430,4643,5196,8564,1979,3048,9066,1979,3048,9416,4981,5803,6272,4981,5803,8607,1349,2676,8483,1349,2676,9201,2676,8483,9201,4433,5390,5550,4433,5550,9036,3847,3849,3881,3847,3881,9327,7966,8105,9683,7967,8104,9682,1838,9340,9452,7856,8174,8897,7856,8174,9462,7856,8897,9462,8174,8897,9462,7855,8175,8896,7855,8175,9463,7855,8896,9463,8175,8896,9463,7931,8669,9505,7844,8343,8971,8343,8681,8971,7846,8342,8972,8342,8682,8972,1532,2472,8367,1532,2472,9235,1115,2786,9042,1115,2786,9763,2786,9042,9763,1357,2297,2874,1357,2297,9772,1357,2874,9772,2297,2874,9772,4634,5574,6151,4634,5574,9773,4634,6151,9773,5574,6151,9773,7788,8300,8344,8300,8344,8570,7789,8301,8345,8301,8345,8571,4448,5462,9152,5462,8481,9152,7797,8554,9649,2422,8110,9766,2422,8844,9766,8110,8844,9766,5699,8112,9765,5699,8842,9765,8112,8842,9765,4549,5567,8256,4549,5567,9244,3446,8658,9598,3446,9031,9598,8658,9031,9598,7829,8195,8900,7829,8827,8900,5666,6147,8667,5666,6147,9222,8419,8853,9426,8419,8989,9426,8853,8989,9426,4542,4915,6108,4542,6108,8639,5265,8133,8783,5265,8133,8904,5265,8783,8904,8133,8783,8904,3095,8577,9786,2389,2870,8643,2389,2870,9223,1263,8712,9680,7927,8134,8887,7927,8134,9639,8083,8297,9144,8082,8296,9146,4923,5306,5989,4923,5989,8875,5306,5989,8875,1646,2029,2712,1646,2712,8876,2133,2992,8528,2133,2992,8963,4894,5401,6125,4894,6125,8330,5401,6125,8330,8236,8580,9391,8236,8580,9429,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669,9087,1124,2670,9753,2670,9105,9753,4518,4972,6338,4972,6338,9058,1429,1648,2490,1429,1648,8509,4706,4925,5767,4706,4925,8510,5272,8486,9226,8486,9226,9484,1415,2763,9022,2763,8567,9022,4692,6040,9021,6040,8568,9021,3462,8485,9193,8485,9191,9193,4679,5155,5646,4679,5646,9284,7921,8658,9598,4878,5978,8641,4878,5978,9231,1601,2701,8642,1601,2701,9230,1560,2185,2668,1560,2668,9224,2185,2668,9224,4572,9371,9742,2064,2710,3524,2064,3524,9720,4462,5440,5524,4462,5524,9649,8434,9276,9671,8435,9277,9670,3810,3813,3815,3810,3815,8730,3813,3815,8730,1995,8487,9225,1850,3098,8930,1850,3098,9157,1850,8930,9157,1708,1910,4036,1708,1910,8364,1708,4036,8364,1507,2721,3046,1507,2721,9774,4784,5998,6323,4784,5998,9775,8049,8228,9654,8049,9230,9654,8050,8229,9655,8050,9231,9655,4568,5280,8673,4568,5280,8911,8160,8571,9272,8161,8570,9273,2012,2600,3555,2012,2600,8341,2012,3555,8341,1131,1689,9091,1131,1689,9256,1689,9091,9256,4687,5067,8562,4687,5067,9673,3396,3398,3485,4680,5156,5647,4680,5156,9327,4680,5647,9327,4629,5492,6425,4629,5492,8386,4629,6425,8386,1526,2670,8759,1526,2670,9105,1185,2163,2247,1185,2247,9650,4629,6425,9520,1974,4281,9679,5948,8205,9166,5948,9166,9695,8205,9166,9695,2671,8206,9165,2671,9165,9696,8206,9165,9696,2157,2333,9729,2333,8331,9729,3792,3793,3800,3792,3800,9262,1583,2189,2855,1583,2855,9418,768,1774,4312,768,1774,8457,762,763,4225,762,763,8400,4864,8902,9198,4864,9031,9198,8902,9031,9198,4366,5374,6287,4366,5374,9614,4366,6287,9614,5374,6287,9614,1418,1920,2440,1418,1920,8656,1380,2326,8930,1380,2326,9444,2326,8930,9444,4559,5259,6399,4559,5259,9203,4559,6399,9203,1704,2232,2995,2232,2995,9611,4753,5322,8340,4753,5322,8755,4753,8340,8755,1353,2128,3149,1353,2128,9352,8296,9145,9169,8297,9143,9170,2384,4196,4197,4196,4197,8692,3919,3932,3933,3919,3933,8650,3795,3798,3806,3795,3798,9323,8388,8409,9117,8409,9117,9571,1121,1919,2926,1121,1919,8586,1121,2926,8586,3857,3900,5643,3857,3900,9583,3900,5643,9583,4640,5080,8494,4640,5080,8807,8205,8812,9166,8205,9166,9757,8206,8811,9165,8206,9165,9756,8228,8953,9654,8229,8954,9655,1435,2458,8415,1435,2458,8994,1270,1963,2788,1270,2788,9304,1597,2252,2665,1597,2665,9612,2252,2665,9612,8084,8400,8574,8400,8574,9185,5010,5615,8371,5010,5615,9167,4817,5462,8481,4817,5462,9152,2240,2430,3882,2430,3882,9331,1339,1731,2708,4478,5926,8531,5926,8531,9758,1201,2649,8530,2649,8530,9759,4815,5732,9206,5732,9206,9586,1538,2455,9207,2455,9207,9585,4556,4878,8641,4556,4878,8969,1279,1601,8642,1279,1601,8968,4738,5899,8767,4738,8767,8783,5899,8767,8783,4089,4103,4104,4089,4103,8785,5408,6449,9078,5408,6449,9121,3614,3616,3624,3614,3616,9676,5026,6291,6332,5026,6291,8495,6291,6332,8495,1749,3014,3055,1749,3014,8496,3014,3055,8496,4526,4925,5465,4925,5465,8612,1249,1648,2188,1648,2188,8611,4559,5002,5697,4559,5697,9203,3875,9315,9386,8182,9315,9386,1252,1812,2377,1812,2377,8535,1369,1954,2666,1369,2666,9616,1954,2666,9616,6118,6164,9133,6164,9133,9508,1604,2841,2886,1604,2886,9332,2841,2886,9332,4493,5773,5972,5773,5972,9183,4089,4103,5612,4103,5612,8785,1735,2840,3688,1735,2840,8253,3800,8327,9323,3800,9262,9323,8327,9262,9323,4724,5564,6336,4724,5564,8490,1183,2440,3079,1183,2440,9015,1359,1878,9386,1359,1878,9629,4563,5129,5640,1286,1852,2363,1901,3035,9680,1353,2215,3149,1353,2215,8380,1402,1878,2369,1402,2369,9324,7999,8985,9536,8985,9536,9781,1208,2123,3150,2123,3150,8762,4630,5491,6426,4630,5491,8381,7930,8668,9374,8365,9299,9735,7914,9402,9597,1681,2649,8530,4958,5926,8531,1404,1880,2371,1404,2371,9204,4466,5821,8440,4466,5821,9496,1373,2446,2945,2446,2945,8527,3778,3780,3781,3778,3780,8309,4608,5138,8581,4608,5138,9538,7861,8480,9255,8046,8209,9700,8046,9506,9700,8209,9506,9700,1313,1654,2929,1313,1654,8271,4397,5195,5938,698,4224,5119,698,5119,5603,3431,3433,6129,3431,3433,9740,3433,6129,9740,7899,8465,9611,8465,9560,9611,4671,5136,5210,4671,5210,9516,3828,3829,8280,3828,3829,9515,3828,8280,9515,2305,2746,9282,2305,2746,9539,5582,6023,9281,5582,6023,9540,1314,1655,2527,1314,2527,8575,4414,5820,5828,4414,5828,9495,4438,5180,6379,4438,6379,8317,5180,6379,8317,2587,3504,3505,3504,3505,8931,1119,1876,2410,1119,1876,8483,1876,2410,8483,8350,9055,9481,3066,3067,8751,3066,3067,9619,3067,8751,9619,6343,6344,8752,6343,6344,9620,6344,8752,9620,7964,8690,9535,7965,8689,9534,4847,5397,5704,5397,5704,8819,4466,5821,5940,5821,5940,8440,7958,8226,9171,3401,5658,6368,3401,5658,8638,3401,6368,8638,5658,6368,8638,8642,8780,8968,8641,8779,8969,1400,2356,9677,4718,5439,5501,4718,5501,8395,4992,5933,9013,4992,5933,9468,9089,9090,9569,2047,3827,9601,2047,8408,9601,3827,8408,9601,3474,3476,9547,3474,3476,9720,1156,1810,2417,1810,2417,9135,4627,5490,6424,4627,5490,9453,5490,6424,9453,8027,8363,8819,8027,8819,9012,1713,2098,2793,1713,2793,9011,2098,2793,9011,1763,2713,2953,1763,2953,9219,4602,5123,9148,4602,5123,9540,1325,1846,9147,1325,1846,9539,4680,5184,5264,4680,5184,9327,1880,3940,3943,1880,3943,9485,3940,3943,9485,5040,5990,6230,5040,6230,9228,5337,6003,8800,2060,2726,8798,1420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5,1399,1869,2367,1399,1869,9240,1752,2192,2913,1752,2192,8379,1307,2184,3157,2184,3157,8856,4957,5525,6007,4957,6007,8703,5525,6007,8703,7929,9052,9644,3970,3971,3972,4690,5001,5498,4690,5001,8698,5001,5498,8698,4663,6055,6184,4663,6055,9031,6055,6184,9031,1425,1801,2809,1425,2809,8559,1950,2912,3094,1950,3094,9669,1613,1886,2689,1886,2689,8675,1144,2024,2358,2024,2358,9608,1096,2643,8566,1096,2643,9029,4454,5601,6381,4454,6381,8619,1271,1958,2594,1271,2594,9257,4548,5235,5871,4548,5871,9258,4772,5504,5972,5504,5972,9183,567,3697,5507,567,3697,9092,567,5507,9092,1766,2470,3003,1766,2470,8644,3859,8869,9697,8869,9583,9697,4782,5134,5778,5134,5778,9760,1994,2682,4097,1994,2682,8956,2682,4097,8956,3851,3853,3882,3853,3882,9786,1143,2019,2354,1143,2354,8888,4420,5296,5631,4420,5631,8889,1534,2097,2696,2097,2696,8620,4811,5374,5973,5374,5973,8621,1724,2221,8944,1724,8692,8944,4503,5092,6383,4503,6383,8397,1432,1794,2567,1432,2567,8525,1726,1921,2718,1921,2718,9377,5003,5198,5995,5198,5995,9378,5015,5951,6022,5015,6022,8327,4746,5824,6229,4746,5824,9571,4746,6229,9571,1629,2526,2824,2526,2824,9019,4906,5803,6101,5803,6101,9018,1184,1811,2264,1184,1811,9554,1184,2264,9554,4461,5088,5541,4461,5088,9555,4461,5541,9555,4684,5369,5664,4684,5369,8828,637,4015,4016,637,4016,8822,4015,4016,8822,4657,5517,5707,4657,5707,9460,5517,5707,9460,1435,2083,2906,1435,2906,8994,2083,2906,8994,1353,2215,2773,4556,4878,5359,4878,5359,8641,1279,1601,2082,1601,2082,8642,4709,5071,5844,4709,5844,8521,4524,5065,6104,4630,5491,6050,1865,2838,9066,1865,2838,9529,4497,4900,5326,4497,5326,9406,4712,5360,6183,4712,6183,8827,5360,6183,8827,1365,1987,2349,1365,2349,8457,3575,3577,3580,3575,3580,9164,3419,3420,3421,1646,2076,8437,1646,2076,8876,4923,5353,8438,4923,5353,8875,5001,5498,8945,5001,8698,8945,1253,1657,2018,1253,2018,8928,1633,3080,8854,1633,3080,9048,1633,8854,9048,4372,5962,6176,5962,6176,8666,8080,8374,9103,8223,9190,9542,2914,3486,3487,2914,3486,9445,2914,3487,9445,3486,3487,9445,8704,9299,9478,8573,9208,9537,361,1830,3077,361,3077,4325,1457,2516,2816,1457,2516,8724,2516,2816,8724,4734,5793,6093,5793,6093,8725,4550,4856,5457,4558,5864,6090,4558,5864,8309,1946,2883,3116,1946,3116,9335,7997,8914,9741,3978,3987,8405,3978,3987,9030,3435,5223,5924,3435,5223,8871,3435,5924,8871,5223,5924,8871,4302,4930,5437,4302,4930,8789,4930,5437,8789,5368,5662,6140,5662,6140,9589,4708,5054,8501,4708,5054,9698,4708,8501,9698,2251,2754,8937,2251,2754,9568,5528,6031,8938,5528,6031,9567,1139,1646,2146,1646,2146,8814,4416,4923,5423,4923,5423,8813,1394,1859,1934,1394,1859,8425,5021,5782,9025,5021,5782,9026,5021,9025,9026,1744,2505,9023,1744,2505,9024,4396,5153,5687,4396,5687,9641,1620,1956,2958,1620,1956,9010,1386,2778,2907,2778,2907,9262,4568,5348,8673,4568,5348,8851,1256,1996,2253,1256,1996,9065,1256,2253,9065,4435,5363,6194,5363,6194,8977,1158,2086,2917,2086,2917,8976,4203,4205,4206,4205,4206,9438,4664,5135,5521,4664,5521,8419,5135,5521,8419,1387,1858,2244,1387,2244,8420,1858,2244,8420,5134,9595,9760,8365,9595,9760,364,1802,2289,364,2289,4303,4891,5967,6122,5967,6122,9531,8015,8357,8795,1354,1769,9062,1354,1769,9753,3393,5409,6275,3393,5409,9592,3393,6275,9592,5409,6275,9592,1250,1864,2853,1250,1864,9088,1127,2447,3048,1127,3048,9066,2332,4129,4131,4129,4131,8651,4837,5690,6110,5690,6110,8805,1560,2413,2833,2413,2833,8806,4910,6357,8845,4910,6357,9044,4910,8845,9044,1181,2477,2502,2477,2502,8690,4458,5754,5779,5754,5779,8689,4398,5939,6429,4398,5939,8564,4398,6429,8564,3393,5409,6295,3393,5409,9232,3393,6295,9232,5409,6295,9232,281,1636,4079,281,1636,4084,281,4079,4084,1385,1899,2226,1899,2226,9794,8307,8944,9438,4931,6206,8269,4931,6206,9668,5265,5821,6400,5265,6400,8904,1512,2050,2629,1512,2629,8255,4789,5327,5906,4789,5906,8254,4313,4314,4326,1482,2012,2600,1482,2012,8341,3611,3633,5309,3611,3633,9565,3611,5309,9565,3663,3664,3665,3663,3665,9155,1470,1913,2077,1470,2077,9217,4747,5190,5354,4747,5354,9215,4528,5009,5986,4528,5009,9319,3534,5310,9491,3534,9491,9727,1401,2062,2063,1401,2062,8859,4678,5339,5340,4678,5339,8860,4753,5308,5322,7967,8883,9682,4687,5067,5228,4687,5228,9673,7966,8882,9683,4608,5529,5942,5529,5942,8919,1074,2364,3042,2364,3042,8824,4351,5641,6319,5641,6319,8823,1251,1732,2709,1251,1732,9692,1568,1874,3092,1568,1874,9197,4533,4874,5942,4874,5942,8919,1702,2477,2502,4979,5754,5779,4843,6051,8628,4843,6051,9407,1717,2173,2613,2173,2613,9104,4708,5054,5737,4708,5737,8501,764,5051,5626,764,5051,8792,5051,5626,8792,4118,4127,9737,3569,3586,8905,3569,3586,9051,4954,5289,6226,4954,5289,9587,4954,6226,9587,5289,6226,9587,4033,5121,8865,4033,5121,9367,4274,4275,4277,4275,4277,9393,1308,1821,2183,1308,1821,8749,1468,2479,2942,1468,2479,9609,2479,2942,9609,4980,5932,8757,4980,5932,9435,4890,5152,5974,4890,5152,9184,1695,1757,3061,1695,1757,9014,1695,3061,9014,1794,2530,9330,1794,2530,9459,1134,2235,2380,1134,2235,8613,4411,5512,5657,4411,5512,8614,1273,1765,2867,1273,1765,9089,8736,9644,9781,4789,5327,6310,4789,6310,9268,1512,2050,3033,15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62,9513,7847,9024,9456,9538,4813,5604,8632,8649,1536,2327,8631,8648,4603,4923,5306,8813,1326,1646,2029,8814,4931,5408,6276,8269,1445,2088,2089,9449,4722,5365,5366,9448,1547,2735,3081,9738,7804,8189,8564,9134,7990,8136,8503,8680,4600,5285,8180,9528,4525,5177,5681,9647,4017,4023,4048,9651,8025,8316,9113,9340,5400,5686,6201,9632,1868,3871,3890,9697,7900,8818,8911,9298,762,1803,2347,8400,8322,8383,8713,9028,8323,8384,8714,9027,2001,2229,8475,9068,5278,5506,8474,9069,1345,1934,2360,9582,5709,5903,8988,9421,1621,2826,3086,8446,7941,9005,9509,9736,1250,1798,2853,9657,3754,5207,9461,9791,2044,2280,9278,9599,5029,5469,6190,9754,5465,5918,8612,9558,2188,2641,8611,9557,2734,2872,8549,8971,6011,6149,8550,8972,7986,8256,9065,9612,8016,8288,9332,9424,7995,8256,8688,9065,2432,2626,8989,9426,5297,5939,6429,8564,7815,9344,9345,9640,1239,1955,2888,8569,4765,8953,9111,9654,1488,8954,9110,9655,1523,2508,2624,9686,4800,5785,5901,9687,8081,8501,8555,9428,1635,2246,2835,8699,4912,5523,6112,8700,4670,5438,8543,9716,3942,3946,5157,9129,7944,8728,8746,9557,7945,8729,8747,9558,2144,2446,3049,8527,8159,8425,9359,9751,8142,8556,9052,9097,7988,8819,9365,9680,2016,2714,8192,9600,3575,3576,3577,9164,1177,2394,2403,8583,1278,2013,2694,9346,2373,3931,3967,8425,3414,3493,3494,9743,1527,2194,2762,9301,3477,3479,5560,8910,3593,4990,5880,9388,1266,2014,8716,9597,1855,2334,8477,8578,5132,5611,8476,8579,1422,3101,8236,9639,3566,3587,5492,8386,7883,8223,9054,9599,1611,2845,2892,8560,4524,5543,6365,8838,1096,2198,8566,9197,7819,8855,9767,9785,7901,8855,9767,9785,8017,8193,8651,9355,1167,2001,2229,8475,4444,5278,5506,8474,1701,2537,8227,9109,3891,3892,3904,9702,8004,8616,9082,9083,7887,8147,9163,9164,1362,3078,9239,9713,1360,1896,2292,9709,2041,3619,3631,8850,1538,2186,2275,9207,4815,5463,5552,9206,8086,8356,8952,9546,7869,8365,9299,9478,4864,6184,8902,9031,4414,5820,5923,9080,4429,5355,6105,8722,1152,2078,2828,8723,4791,5480,6283,9343,4790,5244,8353,9106,1695,2319,3981,9014,5164,5885,6349,8791,8139,8369,8490,9398,7933,8111,8959,9766,7934,8113,8961,9765,7769,8628,8787,9750,7892,8786,8805,9341,4628,6423,8508,9576,4482,6035,8353,8681,1205,2758,8354,8682,5049,5090,8787,9544,4024,4025,4062,8275,7901,8237,8268,9446,1421,2065,2811,8894,7794,8082,8746,9064,7793,8083,8747,9063,6111,6339,8504,9347,1422,2357,8236,9417,4320,5589,8943,9473,4357,5083,5850,9678,1369,1699,3139,8184,4859,5744,6128,8472,1582,2467,2851,8473,4595,5038,8439,9476,7765,8253,8761,9607,4564,5380,6002,8311,1287,2103,2725,8312,8089,8834,9192,9263,1307,2184,2988,8528,1194,2091,2520,9139,1341,2905,3108,9580,4522,5409,9310,9592,2122,2538,8310,9038,8039,8617,9239,9713,4549,5149,5921,8256,3814,3817,5054,9698,1137,2084,8231,9441,1654,2528,2999,3484,1489,2142,2471,8322,4766,5419,5748,8323,5134,5571,6220,8365,7863,8309,9189,9335,8131,8693,8883,9682,4603,5779,6440,8689,1326,2502,3163,8690,4379,8664,8927,9626,8064,8256,8573,9612,8132,8694,8882,9683,2555,2818,9443,9650,4621,5216,8596,9243,1667,2839,3038,8683,4422,5064,5800,8517,7884,8202,9008,9449,7885,8201,9009,9448,7943,8401,8861,9365,1254,1840,8879,9690,4531,5117,8880,9691,1668,2840,3610,8253,1457,2464,2816,8724,4734,5741,6093,8725,4448,5111,5719,8772,3412,8188,9638,9743,4600,4922,5285,8180,2600,3555,8341,9201,763,4225,4233,8400,7975,8470,8817,8836,7903,8197,9043,9238,7989,9076,9396,9407,8053,8966,9318,9329,4585,5098,5460,8790,4881,6118,6164,9508,4887,6168,8414,9050,4789,6035,6214,8411,1512,2758,2937,8410,3860,3862,3905,9724,5948,6311,8205,9695,2671,3034,8206,9696,8055,8622,8887,9572,7937,8185,8822,9047,4594,5367,8324,8490,8136,8503,8680,8907,7949,8415,8940,9585,7948,8416,8942,9586,4707,5323,5704,8819,1573,3069,3070,9165,4850,6346,6347,9166,8006,8618,9280,9348,1131,2967,9091,9255,1791,2458,8659,8994,8111,8441,8981,9154,8113,8442,8980,9153,1095,1920,2858,8656,1867,2302,3919,8650,1531,2645,2720,8424,4808,5922,5997,8423,4601,5555,5557,8279,2004,3544,3556,8676,2212,3642,3645,9385,3413,3415,4964,9717,1634,2245,2858,8656,4911,5522,6135,8657,5455,8580,9391,9566,7831,8334,8672,9233,3417,5511,6213,9717,3839,3840,3842,8716,7762,8533,9375,9442,7851,8359,8635,9073,7852,8358,8634,9074,4745,6219,8584,9652,4443,6401,8580,9178,4585,5328,5993,8754,7960,8162,9087,9742,3714,3717,3718,8269,1594,2070,9084,9669,754,4083,4085,8591,2095,2561,8342,9480,5372,5838,8343,9479,4875,5547,6221,8598,7927,8622,8887,9572,1200,2045,2962,8679,4626,5527,8217,9676,1231,1773,8644,9581,7968,8232,9088,9640,4391,5105,5237,8627,6036,6435,8463,8996,2759,3158,8464,8995,7859,8316,8388,9522,763,8400,8792,9771,5127,6375,6402,8921,4778,5307,5631,9231,1501,2030,2354,9230,4700,5738,9108,9325,7829,8416,8827,9586,7886,8967,9067,9212,4030,8355,9458,9755,7848,8573,9366,9537,4377,5176,8373,9368,3410,3491,3492,9532,4286,5211,5638,8596,7973,8274,9322,9667,2131,3172,8561,9768,1532,1761,2472,8367,4495,6329,8588,9072,8280,8644,9581,9601,2378,8201,9009,9499,5655,8202,9008,9500,1441,2650,9301,9337,3921,3922,3934,9243,3627,3638,3641,9587,4837,5690,8786,8805,1154,1725,2531,8735,1111,2140,2910,9505,1218,3052,8587,9075,1543,2487,2653,8857,4820,5764,5930,8858,1587,2907,8966,9262,7920,8978,9598,9658,3541,3542,3543,9338,4471,5368,5797,9116,1114,1828,1960,8629,8088,8208,8846,9487,3739,5067,5577,8562,4880,5089,8793,9211,3378,3385,8563,9182,1373,2144,2446,8527,754,4037,4049,9058,7870,8251,9325,9776,7909,8273,8928,9314,4626,6451,8217,9419,7941,8640,9392,9719,4673,5143,5643,9583,7835,8635,9160,9223,4892,5975,6121,8403,4682,5236,6361,8785,7989,8157,8787,9002,7974,8379,8835,9764,5232,6202,8394,9641,8068,8434,9000,9414,8069,8435,9001,9415,8053,8966,9329,9778,1171,2174,2668,9224,2394,2933,8583,9677,4781,6133,6173,8451,2544,8480,9255,9501,4195,4197,4198,8692,3434,3436,3504,8931,7970,8199,8328,9427,4381,5090,8725,8787,4050,4052,4055,8204,2447,3048,9066,9140,7876,8369,8553,8679,1568,2306,2609,8840,4946,5259,6399,9263,1487,1681,2649,9759,4764,4958,5926,9758,1112,2330,2550,9082,4731,5819,8249,9407,4727,5932,6383,8594,1838,2454,9340,9452,4492,6387,8660,9698,3458,3460,3516,8225,4904,5421,8778,8958,2020,2661,3152,8586,1083,2122,2678,9038,8115,8633,9260,9699,4558,5344,6298,8309,7847,8581,9372,9538,4950,5494,6427,8672,4858,5619,5863,8638,3851,3852,3853,9786,4493,5773,8847,9183,3424,3426,8366,9782,7851,8320,8782,9228,3577,8147,9336,9788,3793,3796,5303,8327,1332,2084,2646,9441,4641,5263,5625,8615,7890,8245,8652,9645,7936,8196,9342,9551,4902,5433,5743,8298,3542,3544,8676,9179,7916,8251,9148,9606,7915,8250,9147,9605,4257,4291,4293,9582,1359,1894,1975,9702,4962,5771,6026,8222,7810,8568,8636,9209,1094,2233,8319,9219,1128,1926,2395,9275,7841,8163,8809,9686,7842,8164,8810,9687,4528,5141,6153,8601,4263,4270,4280,9511,4687,5773,8562,8847,2156,2613,2820,9104,4078,4091,4092,8357,8101,8234,9124,9527,5971,8450,8695,9542,5576,6096,8456,9354,2299,2819,8455,9353,7816,8579,9282,9436,7817,8578,9281,9437,5326,5545,9085,9406,4310,4311,8404,9509,4258,4284,5214,8862,4371,5510,8320,9228,1816,2882,9209,9522,7855,8799,8800,9333,7856,8797,8798,9334,1245,2657,2930,9752,7888,8178,9155,9156,1779,1989,2732,9490,1528,1998,8626,9464,1795,8668,9374,9609,1117,2924,8970,9616,1610,2891,8426,9096,4622,5136,5210,9516,4118,4120,4127,8248,1371,1709,2322,9447,3451,5519,5851,8422,4174,4180,4183,9358,4004,4009,8199,9579,2999,3483,3484,8561,1633,2429,3037,8854,1716,2843,3002,8929,4836,5740,6037,8214,1559,2463,2760,8215,4275,4276,5860,9393,4444,5506,5796,8474,1167,2229,2519,8475,2115,2528,3485,8271,3999,4000,4003,8398,2651,8537,8849,9718,4596,6341,9008,9016,1319,3064,9009,9017,1631,2356,2462,8326,3828,3831,8280,9601,7891,8248,9084,9669,1306,1625,2141,9564,2284,2529,9055,9570,1146,1781,2323,9296,8070,8254,8411,8678,8071,8255,8410,8677,5266,5705,5968,8445,3975,3976,3977,8274,5293,8224,9117,9370,1389,2170,2262,9017,4666,5447,5539,9016,7803,8216,8414,9094,4906,5810,6329,9072,8045,8198,8627,9236,1742,2391,2962,8360,1662,2436,2722,9734,4939,5713,5999,9733,4847,5711,5729,9316,1323,2008,2660,8774,1645,2008,2918,8774,4327,8421,8943,9473,1629,2533,3052,9075,8120,8561,9309,9310,8064,9026,9208,9537,7819,8308,8945,9439,3638,3641,9385,9587,1103,1618,1891,8432,3875,3877,8182,9386,1598,2944,8831,9768,1588,1943,3167,8570,4865,5220,6444,8571,1405,1959,3084,8756,7816,8262,8294,9759,7817,8261,8295,9758,7762,8193,8533,9442,1286,1852,8277,9470,4563,5129,8276,9469,1238,2055,8687,9517,7974,8379,8834,9764,1388,2052,2152,8693,4665,5329,5429,8694,1217,2616,8995,9246,1569,1899,8382,9794,1253,1854,2109,8928,1226,2064,2729,9720,4714,8503,8680,8907,1606,1881,2700,9313,7870,8339,9108,9325,4494,5893,8996,9250,8044,8197,8629,9238,1355,8178,8993,9748,2787,3515,3516,8225,765,2348,4269,8333,3778,3781,5864,8309,1532,2067,3021,8931,8062,8616,8913,9455,1523,2673,8593,9040,4800,5950,8592,9039,3442,3507,3508,9623,7866,8354,9079,9269,1595,1901,9365,9680,4910,5706,6314,8845,4871,5347,6371,9645,7992,8786,8805,9341,1444,1983,3035,9559,1266,2503,3036,9597,3790,3792,3793,9247,1130,1904,8774,9570,3876,8182,9324,9327,2408,2859,8570,9040,5685,6136,8571,9039,1594,2070,3094,9669,4477,5322,6239,8755,4387,5150,6277,8713,1110,1873,3000,8714,4145,4147,4148,8756,5159,5976,9133,9288,4662,5203,6098,9644,8180,8303,9175,9528,3553,3554,3558,8341,7797,8194,8659,9454,7866,8796,9079,9150,7807,8970,9616,9641,1507,2520,3029,8678,4784,5797,6306,8677,4590,6025,8143,9636,7918,8385,8766,8777,5092,6043,6383,8594,4524,5543,6104,8454,3785,3790,6251,9247,8040,8644,9376,9472,1554,2037,2657,9088,762,1803,4182,8400,1909,2418,8942,9514,5186,5695,8940,9513,4904,5256,6325,8482,3662,3679,3680,9156,1665,2389,9223,9507,2037,2657,9088,9752,1282,1725,2420,9114,7901,8263,9767,9785,5756,6219,8584,9652,7869,8365,9299,9300,1847,2649,8916,9759,5124,5926,8917,9758,4582,5709,8852,9421,4632,5497,6047,9705,7969,8283,9178,9490,4229,4234,5614,9575,7869,8250,9300,9760,7933,8111,8959,8960,7934,8113,8961,8962,1419,2290,9596,9730,7807,8217,8864,9419,4856,5606,8321,9701,1396,8203,9405,9679,4422,5092,6043,8517,7881,8412,8472,9567,7882,8413,8473,9568,7968,8527,8778,9726,7815,9078,9344,9640,4195,8307,8692,9438,8109,8397,9402,9597,1330,2168,8815,9606,4607,5445,8816,9605,4558,6090,8309,8771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\ No newline at end of file diff --git a/src/arrays_dev.jl b/src/arrays_dev.jl new file mode 100644 index 0000000..6602b7c --- /dev/null +++ b/src/arrays_dev.jl @@ -0,0 +1,475 @@ +# In this tutorial, we will learn +# +# - How the `Map` interface works and how to define custom maps +# - How `Broadcasting` and `Operation` compose maps lazily +# - How `lazy_map` defers computation and what type it returns for different inputs +# - What `Fill` is and how `lazy_map` propagates it without creating a `LazyArray` +# - How `LazyArray` stores expression trees and how to traverse one with zero allocation +# - How `CompressedArray` stores data and how `lazy_map` preserves its structure +# - How `Table` encodes lists-of-lists and the key operations on it +# - How `Reindex` and `PosNegReindex` provide lazy fancy indexing + +using Gridap +using Gridap.Arrays +using FillArrays +using Test + +# +# ## The Map interface +# +# A `Map` in Gridap is any callable that participates in the caching and lazy +# evaluation machinery. The interface has four methods: +# +# | Method | Purpose | +# |--------|---------| +# | `return_cache(f, x...)` | allocate reusable workspace | +# | `evaluate!(cache, f, x...)` | compute, reusing the workspace | +# | `evaluate(f, x...)` | allocate cache + evaluate (one-shot) | +# | `return_type(f, x...)` | infer return type without computing | +# +# Plain Julia functions are Maps by default: `evaluate!(c, f, x...) = f(x...)` +# with `nothing` as the cache. The cache matters for operations that write into +# pre-allocated buffers — the same workspace is reused across calls, so no +# allocation occurs inside a hot loop. + +f = x -> 2x +cache = return_cache(f, 3) +@test isnothing(cache) +@test evaluate!(cache, f, 3) == 6 +@test return_type(f, 3) == Int + +# `Broadcasting(g)` wraps a scalar function `g` so that it broadcasts +# element-wise over arrays. Its cache is a pre-allocated output array. + +bm = Broadcasting(+) +a = rand(3, 2) +b = rand(3, 2) +cache = return_cache(bm, a, b) +c = evaluate!(cache, bm, a, b) +@test c ≈ a .+ b + +# Calling `evaluate!` a second time reuses `cache`'s buffer — no new allocation. +# This is how Gridap avoids allocating at every cell during assembly. + +c2 = evaluate!(cache, bm, a, b) +@test c2 === c # same array object, not a copy + +# `Operation(f)(g, h)` composes: it represents `x ↦ f(g(x), h(x))`. + +fa(x) = 2 .* x +fb(x) = sqrt.(x) +fab = Operation(fa)(fb) +x = fill(4.0, 3, 3) +@test evaluate(fab, x) ≈ fa(fb(x)) + +# ## Array interfaces +# +# ### lazy_map +# +# `lazy_map(f, a, b, ...)` is the cell-array analogue of `map`: it applies `f` +# element-wise, but **defers computation**. The concrete return type is not +# always `LazyArray` — Gridap inspects the input types and applies structural +# optimisations. +# We will see it in action later on in the tutorial. +# +# ### The Gridap Array API +# +# Beyond the standard Julia `AbstractArray` interface (`size`, `getindex`, …), +# Gridap adds three methods that enable the zero-allocation traversal used +# throughout assembly: +# +# | Method | Purpose | +# |--------|---------| +# | `array_cache(a)` | allocate a reusable workspace for element access | +# | `getindex!(cache, a, i)` | fetch element `i`, writing into the cache buffer | +# | `testitem(a)` | return a representative element (used for type inference) | +# +# The default implementations work for any `AbstractArray`, but custom types +# can override them to avoid per-access allocation entirely. +# +# To see this in action we implement `MyScaledArray`: an array whose element +# `i` is `scale * data[i]`, where `data[i]` is itself a vector. The cache +# pre-allocates one output buffer so that `getindex!` never allocates. + +struct MyScaledArray{T} <: AbstractVector{Vector{T}} + data :: Vector{Vector{T}} + scale :: T +end + +Base.size(a::MyScaledArray) = size(a.data) +Base.IndexStyle(::Type{<:MyScaledArray}) = IndexLinear() +Base.getindex(a::MyScaledArray, i::Integer) = a.scale .* a.data[i] # allocates + +# The cache is a single pre-allocated output vector. `getindex!` writes into +# it and returns it — the caller must not hold on to the reference across calls. +# We provide `CachedArray` as a convenient wrapper that handles resizing when needed. +# The cache is reusable across calls to `getindex!`, and can be resized with `setsize!`. + +Arrays.array_cache(a::MyScaledArray) = CachedArray(similar(a.data[1])) + +function Arrays.getindex!(cache, a::MyScaledArray, i::Integer) + di = a.data[i] + setsize!(cache, size(di)) # resize cache if needed + r = cache.array + r .= a.scale .* a.data[i] + return r +end + +# `testitem` provides a representative element for type inference when the +# array might be empty. Defaults to `first(a)`. + +Arrays.testitem(a::MyScaledArray) = a.scale .* a.data[1] + +# Construction and basic access: + +raw = [rand(i) for i in 1:8] +msa = MyScaledArray(raw, 3.0) +@test msa[2] ≈ 3.0 .* raw[2] + +# Zero-allocation traversal: the same buffer is returned every iteration. + +buf = array_cache(msa) +for i in 1:length(msa) + v = getindex!(buf, msa, i) # no allocation; v is buf + @test v ≈ 3.0 .* raw[i] +end + +# Because `MyScaledArray` implements the interface, `lazy_map` can use it as +# an argument and will call `array_cache`/`getindex!` on it internally: + +doubled = lazy_map(v -> 2 .* v, msa) +@test doubled[3] ≈ 6.0 .* raw[3] + +# +# ### Fill arrays and the Fill optimisation +# +# `Fill(v, n)` (from FillArrays.jl) is a length-`n` array whose every element +# is `v`. It stores `v` exactly once. +# +# When all inputs to `lazy_map` are `Fill`, Gridap evaluates the function +# **once** on the stored value and returns a new `Fill` — no `LazyArray` is +# created. + +c = lazy_map(fa, Fill(4.0, 10)) +@test isa(c, Fill) +@test c[1] == fa(4.0) +@test c[7] == fa(4.0) + +# The same optimisation applies when mixing `Fill` with maps: + +maps = Fill(Broadcasting(+), 10) +xs = Fill(rand(2, 3), 10) +ys = Fill(rand(2, 3), 10) +v = lazy_map(evaluate, maps, xs, ys) +@test isa(v, Fill) + +# This is why evaluating shape functions at quadrature points on a uniform mesh +# yields a `Fill`: the same reference element and the same quadrature rule are +# reused for every cell, so the result is the same matrix everywhere. + +# +# ### LazyArray +# +# When inputs are not uniform, `lazy_map` returns a `LazyArray`. +# A `LazyArray{G,T,N,F}` stores: +# - `maps`: an array of `Map`s (often a `Fill` of a single map) +# - `args`: a tuple of argument arrays +# +# Entries are computed on demand. Calling `a[i]` allocates a fresh result; +# for hot loops, use the **cache interface** instead. + +a = [rand(3) for _ in 1:6] +b = [rand(3) for _ in 1:6] +c = lazy_map(Broadcasting(+), a, b) +d = lazy_map(Broadcasting(*), a, c) + +# `d` is the expression tree `a .* (a .+ b)`, unevaluated. + +@test d isa LazyArray + +# `array_cache` builds the cache for a (possibly nested) `LazyArray`. +# `getindex!` evaluates entry `i` into the cache's buffer — zero allocation +# after the first call. + +cache = array_cache(d) +for i in 1:length(d) + di = getindex!(cache, d, i) # result lives in cache buffer; no allocation + @test di ≈ a[i] .* (a[i] .+ b[i]) +end + +# Calling `getindex!` twice for the same index returns the **same array object** +# without recomputing — the cache memoises the last result. + +cache = array_cache(d) +@test getindex!(cache, d, 1) === getindex!(cache, d, 1) + +# When `a` appears at multiple nodes of the expression tree (in both the sum +# and the product above), the cache ensures `a[i]` is read only once per index. +# This structural sharing is the main reason cell-local assembly in Gridap is +# efficient: shape functions evaluated at quadrature points are computed once +# and reused across all integration terms. +# +# Multi-dimensional `LazyArray`s support both linear and Cartesian indexing: + +s = (3, 4) +x = rand(s...) +y = rand(s...) +z = lazy_map(+, x, y) +@test size(z) == s +@test z[2, 3] == x[2, 3] + y[2, 3] + +# `print_op_tree(d)` is a useful debugging utility that prints the full +# expression tree of a `LazyArray`, showing how maps and arguments are composed. + +# +# ### CompressedArray +# +# `CompressedArray(values, ptrs)` stores a short `values` vector and an integer +# `ptrs` vector. Element `i` is `values[ptrs[i]]`. This arises naturally +# whenever many cells share the same data — for example, on a uniform mesh all +# cells of the same type share the same reference FE. + +values = [10, 20, 31] +ptrs = [1, 2, 3, 3, 2, 2] +a = CompressedArray(values, ptrs) +@test a[1] == 10 +@test a[4] == 31 # ptrs[4] == 3 → values[3] == 31 +@test a[5] == 20 # ptrs[5] == 2 → values[2] == 20 + +# When **all** inputs to `lazy_map` are `CompressedArray`s with **identical +# `ptrs`**, Gridap applies the operation only to `values` and returns a new +# `CompressedArray`. The output has exactly as many unique entries as the +# input, regardless of the array length. + +b = lazy_map(-, a) +@test isa(b, CompressedArray) +@test b.ptrs === a.ptrs # ptrs object is reused +@test b.values == -values + +c = lazy_map(*, a, b) +@test isa(c, CompressedArray) +@test c.values == values .* (-values) + +# Mixing a `CompressedArray` with a `Fill` also works: the `Fill` is +# conceptually "compressed" to align with the `CompressedArray`'s structure. + +f = Fill(4, length(ptrs)) +c = lazy_map(*, a, f) +@test isa(c, CompressedArray) +@test c.ptrs === a.ptrs + +# Two `CompressedArray`s with **different** `ptrs` cannot share structure — +# Gridap falls back to a `LazyArray`. + +values2 = [10, 20] +ptrs2 = [1, 2, 1, 1, 2, 2] +b2 = CompressedArray(values2, ptrs2) +c = lazy_map(*, a, b2) +@test !isa(c, CompressedArray) +@test isa(c, LazyArray) + +# +# ## Tables +# +# `Table(data, ptrs)` is a compressed list-of-lists. Row `i` spans +# `data[ptrs[i] : ptrs[i+1]-1]`. The `Table` type is used throughout Gridap +# for mesh connectivity: cell-to-node maps, cell-to-DOF maps, etc. + +vv = [[1, 2, 3], [2, 3], [5, 8], Int[], [1, 2, 4]] +t = Table(vv) +@test length(t) == 5 +@test t[1] == [1, 2, 3] +@test t[4] == Int[] + +# `view(t, i)` returns a view of row `i` into the flat `data` array (no copy). +# Slicing with a range or vector of indices returns a new `Table`. + +@test view(t, 1) == [1, 2, 3] +@test view(t, 1:3) isa Table + +# Three helpers give direct access to the underlying flat storage: + +@test datarange(t, 1) == 1:3 # index range into t.data +@test dataview(t, 1) == [1, 2, 3] # view into t.data +@test dataview(t, 1:3) == [1, 2, 3, 2, 3, 5, 8] # contiguous slice + +# `dataiterator` yields `(row, local_position, value)` triples — the workhorse +# for iterating mesh connectivity without per-row allocation. + +for (i, j, v) in dataiterator(t) + @test t[i][j] == v +end + +# `generate_data_and_ptrs` reconstructs the flat (data, ptrs) pair from a +# vector of vectors: + +data, ptrs = generate_data_and_ptrs(vv) +@test data == [1, 2, 3, 2, 3, 5, 8, 1, 2, 4] +@test ptrs == [1, 4, 6, 8, 8, 11] + +# ### Pointer arithmetic helpers +# +# `length_to_ptrs!` converts a lengths vector to an exclusive prefix-sum +# (the standard CSR pointer format). + +lens = [0, 2, 4, 2] +length_to_ptrs!(lens) +@test lens == [1, 3, 7, 9] + +# ### inverse_table +# +# `inverse_table(a_to_b)` inverts an index map stored as a `Table`: given a +# cell-to-nodes table, it returns a node-to-cells table. The output is itself +# a `Table` because multiple cells may share a node. + +cell_to_nodes = Table([[1, 2, 3], [1, 2, 3]]) # two cells sharing nodes 1–3 +node_to_cells = inverse_table(cell_to_nodes) +@test node_to_cells == [[1, 2], [1, 2], [1, 2]] + +# ### append_tables_globally and append_tables_locally +# +# `append_tables_globally` stacks two tables vertically (more rows): + +t1 = Table([[1, 2, 3], [2, 3], [5, 8], Int[], [1, 2, 4]]) +t2 = Table([[1, 3], [4, 2, 3], Int[], Int[], [1, 2, 4]]) +t3 = append_tables_globally(t1, t2) +@test t3 == vcat(t1, t2) + +# `append_tables_locally((offsets...), (tables...))` merges corresponding rows +# horizontally, adding an integer offset to each table's entries. The canonical +# use is building a combined DOF map from per-field contributions: each field's +# local-to-global map is offset by the cumulative DOF count of previous fields. + +t4 = append_tables_locally((0, 5), (t1, t2)) +@test t4[1] == [1, 2, 3, 6, 8] # t1[1] ++ (t2[1] .+ 5) +@test t4[2] == [2, 3, 9, 7, 8] # t1[2] ++ (t2[2] .+ 5) +@test t4[3] == [5, 8] # t2[3] is empty, no change +@test t4[5] == [1, 2, 4, 6, 7, 9] + +# ### flatten_partition +# +# Given a partition table `a_to_bs` where each `a` owns a set of `b` indices, +# `flatten_partition` inverts it to a flat `b_to_a` vector. + +a_to_bs = Table([[1, 2, 3], [7, 8], [4, 5, 6]]) +b_to_a = flatten_partition(a_to_bs) +@test b_to_a == [1, 1, 1, 3, 3, 3, 2, 2] + +# +# ## Reindexing arrays lazyly +# +# `Reindex(a)` is a `Map` that performs a single index lookup: `Reindex(a)(i) = a[i]`. +# `lazy_map(Reindex(a), indices)` gives lazy fancy indexing — the result type +# mirrors the structure of `a`. + +src = [[1, 2, 4, 5], [2, 4, 6, 7], [4, 3, 5, 1], [2, 3]] +idx = [3, 1, 2] +c = lazy_map(Reindex(src), idx) +@test c[1] == src[3] +@test c[2] == src[1] + +# `Reindex(Fill(...))` → `Fill` (only value is ever read): + +c = lazy_map(Reindex(Fill(30.0, 10)), idx) +@test isa(c, Fill) +@test c[1] == 30.0 + +# `Reindex(CompressedArray(...))` → `CompressedArray` (ptrs reindexed, +# `values` left intact): + +ca = CompressedArray([30, 40, 10, 20, 30], [1, 2, 3, 5, 3, 1, 4, 2]) +c = lazy_map(Reindex(ca), idx) +@test isa(c, CompressedArray) +@test c.values === ca.values # same values; only ptrs reindexed + +# `Reindex(LazyArray(...))` → `LazyArray` (maps and args each reindexed separately, +# preserving the expression tree structure): + +la = lazy_map(-, [1, 2, 3, 5, 3, 1, 4, 2]) +c = lazy_map(Reindex(la), idx) +@test isa(c, LazyArray) +@test c[1] == la[3] + +# ### Broadcasting(Reindex(...)) on a Table +# +# A ubiquitous Gridap pattern: apply `Broadcasting(Reindex(global_vals))` to a +# `Table` of local-to-global ids. For each row (cell), this gathers the global +# values indexed by that row's entries — the cell-local node coordinates, +# DOF values, etc. + +p = VectorValue(1.0, 0.0) +gid_to_val = [p, 2p, 3p, -p, p] +lid_to_gid = Table([[2, 3, 1], [3, 4, 5], [1, 2], Int[]]) +ca = lazy_map(Broadcasting(Reindex(gid_to_val)), lid_to_gid) + +@test ca[1] == [2p, 3p, p] +@test ca[2] == [3p, -p, p] +@test ca[3] == [ p, 2p] +@test ca[4] == VectorValue{2,Float64}[] + +# +# ## PosNegReindex for signed reindexing +# +# `PosNegReindex(values_pos, values_neg)` generalises `Reindex` to a **signed** +# index convention: a positive index `i` reads `values_pos[i]`, and a negative +# index `i` reads `values_neg[-i]`. +# +# This is the mechanism Gridap uses throughout the FE machinery to distinguish +# **free DOFs** (positive global ids) from **Dirichlet-constrained DOFs** +# (negative ids), without any conditional in the inner loop. + +values_pos = Float64[40, 30, 10] +values_neg = -Float64[40, 30] +indices = [1, 3, -1, 2, -2] + +c = lazy_map(PosNegReindex(values_pos, values_neg), indices) +@test c[1] == 40.0 # index 1 → values_pos[1] +@test c[3] == -40.0 # index -1 → values_neg[1] +@test c[5] == -30.0 # index -2 → values_neg[2] + +# In the finite element context, `get_cell_dof_ids` returns a `Table` of +# **signed** global DOF ids: positive for free DOFs, negative for Dirichlet. +# Applying `Broadcasting(PosNegReindex(free_dof_vals, dirichlet_vals))` to +# this table extracts the DOF values for every cell — free values from the +# current iterate, Dirichlet values from the boundary data. + +# ### PosNegPartition +# +# When the pattern of positive and negative entries is known ahead of time, +# `PosNegPartition` pre-computes the split for more efficient dispatch. +# `PosNegPartition(ipos_to_i, n)` says that among `n` total entries, the +# positions listed in `ipos_to_i` hold positive indices and the rest negative. + +part = PosNegPartition([1, 4, 2], 5) # entries 1,2,4 are positive; 3,5 are negative +c = lazy_map(PosNegReindex(values_pos, values_neg), part) +r = [values_pos[i] for i in [1, 3, -1, 2, -2] if i > 0] # reference positives +@test c[[1, 2, 4]] ≈ r # the three positive-index slots + +# ### lazy_map preserves PosNegReindex structure +# +# When operands are `PosNegReindex`-indexed with the same index pattern, +# `lazy_map` applies the operation **separately** on the positive and negative +# value arrays and returns a new `PosNegReindex`-based structure. The +# positive and negative branches are never interleaved. + +a_pos = Float64[40, 30, 10]; a_neg = -Float64[40, 30] +b_pos = Float64[43, 50, 60]; b_neg = -Float64[41, 30] +a = lazy_map(PosNegReindex(a_pos, a_neg), indices) +b = lazy_map(PosNegReindex(b_pos, b_neg), indices) + +c = lazy_map(+, a, b) +@test collect(c) ≈ [a[i] + b[i] for i in 1:length(indices)] + +# The decomposition lets Gridap process free-DOF and Dirichlet-DOF contributions +# in separate, homogeneous batches during matrix assembly — avoiding +# branch-per-entry overhead and enabling structure-preserving optimisations. + +# +# ## Further reading +# +# - Gridap.jl/src/Arrays/Maps.jl — Map abstract type, Broadcasting, Operation +# - Gridap.jl/src/Arrays/LazyArrays.jl — LazyArray, lazy_map, array_cache, getindex! +# - Gridap.jl/src/Arrays/CompressedArrays.jl — CompressedArray and lazy_map specialisations +# - Gridap.jl/src/Arrays/Tables.jl — Table, inverse_table, append_tables_*, flatten_partition +# - Gridap.jl/src/Arrays/Reindexing.jl — Reindex, PosNegReindex, PosNegPartition +# - Gridap.jl/src/Arrays/Arrays.jl — module entry point; full list of exports diff --git a/src/darcy.jl b/src/darcy.jl index 69421ab..b8c0bea 100644 --- a/src/darcy.jl +++ b/src/darcy.jl @@ -63,14 +63,18 @@ model = CartesianDiscreteModel(domain,partition) # Next, we build the FE spaces. We consider the first order RT space for the flux and the discontinuous pressure space as described above. This mixed FE pair satisfies the inf-sup condition and, thus, it is stable. order = 1 +h = 1/100 V = FESpace(model, ReferenceFE(raviart_thomas,Float64,order), - conformity=:HDiv, dirichlet_tags=[5,6]) + conformity=:HDiv, dirichlet_tags=[5,6], scale_dof=true, global_meshsize=h) Q = FESpace(model, ReferenceFE(lagrangian,Float64,order), - conformity=:L2) + conformity=:L2, scale_dof=true, global_meshsize=h) -# Note that the Dirichlet boundary for the flux are the bottom and top sides of the squared domain (identified with the boundary tags 5, and 6 respectively), whereas no Dirichlet data can be imposed on the pressure space. We select `conformity=:HDiv` for the flux (i.e., shape functions with $H^1(\mathrm{div};\Omega)$ regularity) and `conformity=:L2` for the pressure (i.e. discontinuous shape functions). +# We select `conformity=:HDiv` for the flux (i.e., shape functions with $H^1(\mathrm{div};\Omega)$ regularity) and `conformity=:L2` for the pressure (i.e. discontinuous shape functions). +# Note that the Dirichlet boundary for the flux are the bottom and top sides of the squared domain (identified with the boundary tags 5, and 6 respectively), whereas no Dirichlet data can be imposed on the pressure space. +# +# The `scale_dof=true` argument is given to cancel the scaling of the DoFs with the meshsize `h` that is introduced by the piola map, e.g. `h`$^{2-1}$ (2 is the dimension) for the div-conforming contra-variant Piola map. This is usefull for mixed elements of different conformity that use small or non-uniform meshsize. On a uniform Cartesian mesh like `model` used here, we also passed `global_meshsize=1/100`. This argument fixes the local meshsize estimate needed to re-scale the DoFs, avoiding the computation of the volume of each face of the mesh that own a DoF. But this should only be used on quasi-uniform and shape-regular meshes. # # From these objects, we construct the trial spaces. Note that we impose homogeneous boundary conditions for the flux. diff --git a/src/inc_navier_stokes.jl b/src/inc_navier_stokes.jl index 5ce7f9f..d1ccafe 100644 --- a/src/inc_navier_stokes.jl +++ b/src/inc_navier_stokes.jl @@ -105,6 +105,8 @@ P = TrialFESpace(Q) Y = MultiFieldFESpace([V, Q]) X = MultiFieldFESpace([U, P]) +# `scale_dof` is not used here, because currently $H^1$ and $L^2$ conforming elements both use the same Piola map in Gridap (the trivial one). + # ## Triangulation and integration quadrature # # From the discrete model we can define the triangulation and integration measure diff --git a/src/poisson_hho.jl b/src/poisson_hho.jl index e358179..1e5862a 100644 --- a/src/poisson_hho.jl +++ b/src/poisson_hho.jl @@ -191,7 +191,7 @@ R = reconstruction_operator(ptopo,order,Y,Ωp,Γp,dΩp,dΓp) # - The stabilization term `s` # - The rhs term `l` -hTinv = CellField(1 ./ collect(get_array(∫(1)dΩp)), Ωp) +hTinv = CellField(1 ./ collect(get_array(∫(1)dΩp) .^ (1/2)), Ωp) function a(u,v) Ru_Ω, Ru_Γ = R(u) diff --git a/src/poisson_unfitted.jl b/src/poisson_unfitted.jl index 371cec1..4a59067 100644 --- a/src/poisson_unfitted.jl +++ b/src/poisson_unfitted.jl @@ -129,7 +129,7 @@ writevtk(Ω,"output_path/phys_trian") # # In the third plot, we show the region represented by physical triangulation (shaded in gray) embedded in the active grid (black-contoured cells). # -# ![fig5](../assets/unfitted_poisson/fig_physical_trian_3.png" width="420") +# ![fig5](../assets/unfitted_poisson/fig_physical_trian_3.png) # # **In a nutshell,** to define the unfitted FE formulation of the problem we need the \"active\" and \"physical\" triangulations of the domain. The former triangulation is used to define the FE spaces, whereas the latter is used to integrate the weak form. We use a level-set function to derive both of them. # diff --git a/src/reffes_dev.jl b/src/reffes_dev.jl new file mode 100644 index 0000000..1ec2870 --- /dev/null +++ b/src/reffes_dev.jl @@ -0,0 +1,319 @@ +# In this tutorial, we will learn +# +# - How to define a `ReferenceFEName` type to tag a new element family +# - How to construct a polynomial prebasis using `MonomialBasis` +# - How to encode moment-based DOFs as weighted integrals over element faces +# - How to implement a moment integrand σ(φ,μ,ds) using Gridap's `Field` algebra +# - How to assemble a complete `ReferenceFE` via `MomentBasedReferenceFE` +# - How to plug the element into the standard `ReferenceFE` dispatch system +# - How to verify the implementation against the built-in CR element and via convergence + +using Gridap +using Gridap.ReferenceFEs +using Gridap.Polynomials +using Gridap.Fields + +# +# ## What constitutes a ReferenceFE +# +# A `ReferenceFE{D}` in Gridap encodes everything the assembly loop needs to +# know about a single element type on a reference polytope: +# +# | Field | Role | +# |-------|------| +# | `prebasis` | polynomial space $\mathcal{P}$, $\dim\mathcal{P} = n$ | +# | `dofs` | $n$ linear functionals $\sigma_1,\ldots,\sigma_n$ on $\mathcal{P}$ | +# | `shapefuns` | dual basis $\varphi_j$: $\sigma_i(\varphi_j)=\delta_{ij}$ | +# | `face_own_dofs` | which DOF indices belong to each face of the polytope | +# | `conformity` | how adjacent cells share DOFs across facets | +# +# The shape functions are the columns of $\mathbf{M}^{-1}$ where +# $M_{ij}=\sigma_i(p_j)$ is the Vandermonde matrix of the prebasis $\{p_j\}$. +# Gridap's `MomentBasedReferenceFE` constructor handles the inversion and +# assembles the full struct, provided you supply the prebasis and a description +# of the DOFs as moment integrals over faces. + +# +# ## The Crouzeix–Raviart element +# +# The Crouzeix–Raviart (CR) element [1] is the canonical lowest-order +# nonconforming scalar element on simplices. On a $D$-simplex $K$ it pairs +# $\mathcal{P}_1(K)$ (dimension $D+1$) with one DOF per facet $F_i$: +# +# ```math +# \sigma_{F_i}(\varphi) = \frac{1}{|F_i|}\int_{F_i}\varphi\;dF, +# \qquad i=1,\ldots,D+1. +# ``` +# +# A simplex has exactly $D+1$ facets, so the set is unisolvent. +# Global CR functions are only weakly continuous across inter-element boundaries +# — adjacent cells agree on facet averages, not on pointwise values. +# This places global functions in the broken space $H^1_h(\mathcal{T}_h)$ +# rather than $H^1(\Omega)$. + +# +# ## Step 1 — The ReferenceFEName type +# +# Every element family is identified by a singleton subtype of `ReferenceFEName`. +# The type carries no data; it is a pure dispatch tag. +# The constants you already know from high-level drivers — `lagrangian`, +# `raviart_thomas`, `nedelec` — are exactly this: singleton instances of their +# respective `ReferenceFEName` subtypes, used as the second argument to +# `ReferenceFE(polytope, name, T, order)`. + +struct MyCrouzeixRaviart <: ReferenceFEName end +const my_crouzeix_raviart = MyCrouzeixRaviart() + +# +# ## Step 2 — The polynomial prebasis +# +# `MonomialBasis(Val(D), T, order, filter)` builds the basis of scalar monomials +# in `D` variables of type `T` up to degree `order`, keeping only monomials +# $x^\alpha$ for which `filter(α,order)` is true. The standard +# complete-polynomial filter `Polynomials._p_filter` accepts $|\alpha|\le\text{order}$, +# spanning $\mathcal{P}_\text{order}$. Two other built-in filters are +# `Polynomials._q_filter` (tensor-product $\mathcal{Q}_\text{order}$: each +# component $\alpha_i\le\text{order}$, used for Lagrangian elements on quads/hexes) +# and `Polynomials._s_filter` (serendipity space: drops high-degree corner monomials +# to reduce the DOF count on quads/hexes while retaining full approximation order). + +D = 2 +T = Float64 +p = TRI + +prebasis = MonomialBasis(Val(D), T, 1, Polynomials._p_filter) + +# On a triangle $\mathcal{P}_1$ has dimension $D+1=3$, spanned by +# $\{1,\,x_1,\,x_2\}$: + +length(prebasis) + +# Evaluating at the centroid $(1/3,1/3)$ of the reference triangle confirms +# the three monomials: + +evaluate(prebasis, [Point(1/3, 1/3)]) + +# +# ## Step 3 — The moment test basis on facets +# +# Each DOF integral $\int_F \varphi\cdot\mu\,dF$ involves a test polynomial +# $\mu$ defined on the facet $F$. For CR, $\mu=1$ ($\mathcal{P}_0$), so we +# need one constant basis function per facet. Since a facet is a +# $(D-1)$-simplex, we build a `MonomialBasis` in $D-1$ variables of degree 0: + +fb = MonomialBasis(Val(D-1), T, 0, Polynomials._p_filter) +length(fb) + +# There is exactly one moment per facet, so the total DOF count is 3 — matching +# the three edges of `TRI` and the dimension of $\mathcal{P}_1$. + +# +# ## Step 4 — The moment integrand +# +# A moment descriptor is a triplet `(face_range, σ, μ)` where +# - `face_range` selects which faces of the polytope carry the moments +# (an index range into the global face list of `p`), +# - `σ(φ,μ,ds)` is the integrand (linear in both `φ` and `μ`), +# - `μ` is the test basis on the face. +# +# When `MomentBasedReferenceFE` builds the DOF basis it calls `σ` with: +# - `φ`: the prebasis restricted and pulled back to the current face, +# - `μ`: the test basis in the face reference domain, +# - `ds`: a `FaceMeasure` carrying `cpoly` (cell polytope), `face` (local face +# index), and the quadrature and face-to-cell coordinate map. +# +# `σ` must return a `Field` (or array of `Field`s) representing the integrand. +# No numerical values are computed here; the framework evaluates the result at +# quadrature points internally. +# +# The CR integrand $\sigma_F(\varphi) = |F|^{-1}\int_F\varphi\,dF$ is: + +function fmom(φ, μ, ds) + D = num_dims(ds.cpoly) + facet_meas = ReferenceFEs._get_dfaces_measure(ds.cpoly, D-1) # vector of facet measures |F| + inv_meas = Fields.ConstantField(1 / facet_meas[ds.face]) + φμ = Broadcasting(Operation(⋅))(φ, μ) # pointwise product of Field arrays + Broadcasting(Operation(*))(φμ, inv_meas) # scale by 1/|F| +end + +# `_get_dfaces_measure` is an internal Gridap helper that returns the +# $d$-volume of each $d$-face of a polytope (edge lengths, face areas, etc.). +# +# `Broadcasting(Operation(f))` lifts a binary scalar operation `f` to act +# element-wise on arrays of `Field`s, returning a new array of `Field`s. +# The computation is deferred: the resulting expression tree is only evaluated +# at quadrature points when `MomentBasedDofBasis` assembles the Vandermonde matrix. + +# +# ## Step 5 — Assembling the ReferenceFE +# +# `get_dimrange(p, d)` returns the range of face indices for all $d$-faces of +# `p`; passing `D-1` selects all edges (facets of a triangle). + +moments = Tuple[ + (get_dimrange(p, D-1), fmom, fb), +] + +my_cr = ReferenceFEs.MomentBasedReferenceFE(my_crouzeix_raviart, p, prebasis, moments, L2Conformity()) + +# Inspecting the face-DOF ownership confirms one DOF per edge (faces 4–6 in +# the TRI face numbering, after 3 vertices and the cell interior) and none on +# vertices or interior: + +get_face_own_dofs(my_cr) + +# The CR shape functions are the unique degree-1 polynomials whose average +# over each edge is a Kronecker delta: $\sigma_i(\varphi_j)=\delta_{ij}$. +# We verify this by evaluating the DOF functionals against the shape functions +# — the result should be the $3\times 3$ identity: + +dof_basis = get_dof_basis(my_cr) +shp = get_shapefuns(my_cr) +evaluate(dof_basis, shp) + +# It can also be instructive to compare the shape functions against the prebasis. +# The Vandermonde matrix $M_{ij} = \sigma_i(p_j)$ (DOFs applied to monomials) +# encodes how the shape functions are formed as linear combinations of monomials: + +evaluate(dof_basis, prebasis) + +# +# ## Comparing with the built-in implementation +# +# Gridap ships its own `CrouzeixRaviartRefFE`. Both DOF bases must produce the +# same Vandermonde matrix when applied to the common prebasis: + +cr_builtin = CrouzeixRaviartRefFE(Float64, TRI, 1) +dofs_builtin = get_dof_basis(cr_builtin) + +M_mine = evaluate(dof_basis, prebasis) +M_ref = evaluate(dofs_builtin, prebasis) +@assert M_mine ≈ M_ref + +# +# ## Step 6 — Registering the dispatch hook +# +# For the element to be usable through the standard `ReferenceFE(polytope, name, T, order)` +# API (and hence in `FESpace`, `TestFESpace`, etc.) we add one method. + +function ReferenceFE(p::Polytope, ::MyCrouzeixRaviart, ::Type{T}, order) where T + @assert is_simplex(p) && order == 1 "MyCrouzeixRaviart is only defined for simplices at order 1" + D = num_dims(p) + pre = MonomialBasis(Val(D), T, order, Polynomials._p_filter) + fb = MonomialBasis(Val(D-1), T, 0, Polynomials._p_filter) + function σ(φ, μ, ds) + d = num_dims(ds.cpoly) + fm = ReferenceFEs._get_dfaces_measure(ds.cpoly, d-1) + scale = Fields.ConstantField(1 / fm[ds.face]) + Broadcasting(Operation(*))(Broadcasting(Operation(⋅))(φ, μ), scale) + end + moms = Tuple[(get_dimrange(p, D-1), σ, fb)] + ReferenceFEs.MomentBasedReferenceFE(my_crouzeix_raviart, p, pre, moms, L2Conformity()) +end + +# ## Step 7 — Global Conformity +# +# Locally, everything is in place. However, an `FESpace` requires gluing local +# spaces together accross cell boundaries. Although we provide certain pre-set +# conformities, depending on how complicated your element is you might have +# to tinker a bit. +# +# In our case, although CR is discontinuous, we still want to enforce continuity of +# DoFs accross faces. Thus, we also need a `get_face_own_dofs` override for `L2Conformity`. +# Without it, the default `L2Conformity` dispatch places all DOFs on the cell interior, +# which would make adjacent cells independent and break the nonconforming +# inter-element coupling that CR relies on. + +function ReferenceFEs.get_face_own_dofs(reffe::GenericRefFE{MyCrouzeixRaviart}, ::L2Conformity) + get_face_own_dofs(reffe) +end + +# This one-liner redirects to the stored `face_own_dofs` (the moment-derived +# facet ownership), so that when the `FESpace` machinery builds the global DOF +# numbering, it recognises that the edge DOF is shared between the two adjacent +# cells — exactly the nonconforming H1 interpretation. +# +# Another thing you might have to do is provide a `Pullback` that maps reference +# shapefunctions to physical ones (beyond mapping quadrature points). +# This is important for HDiv, HCurl and other vector-valued elements with nontrivial +# Piola transforms, but for CR the standard pullback suffices. +# To learn more about pullbacks, see the following files: +# - `src/ReferenceFEs/Pullbacks.jl` — the abstract `Pullback` type and local behaviour +# - `src/FESpaces/Pullbacks.jl` — space-level pullback behaviour + +# +# ## Verification: Poisson convergence +# +# We solve +# ```math +# -\Delta u = f \quad\text{in } \Omega=[0,1]^2, \qquad u=0 \quad\text{on }\partial\Omega, +# ``` +# with manufactured solution $u(x)=\sin(\pi x_1)\sin(\pi x_2)$ and +# $f=2\pi^2 u$. For the nonconforming CR discretisation (broken $H^1$ bilinear +# form, no penalty), the expected convergence rates are $O(h^2)$ in $L^2$ and +# $O(h)$ in the broken $H^1$ seminorm [1]. + +u_ex(x) = sin(π*x[1]) * sin(π*x[2]) +f_ex(x) = 2π^2 * sin(π*x[1]) * sin(π*x[2]) + +function solve_cr_poisson(n) + model = simplexify(CartesianDiscreteModel((0,1,0,1),(n,n))) + reffe = ReferenceFE(TRI, my_crouzeix_raviart, Float64, 1) + V = TestFESpace(model, reffe; conformity=:L2, dirichlet_tags="boundary") + U = TrialFESpace(V, u_ex) + + Ω = Triangulation(model) + dΩ = Measure(Ω, 4) + + a(u,v) = ∫( ∇(u)⊙∇(v) )dΩ + l(v) = ∫( f_ex * v )dΩ + + op = AffineFEOperator(a, l, U, V) + uh = solve(op) + + e = u_ex - uh + el2 = sqrt(sum( ∫(e*e)dΩ )) + eh1 = sqrt(sum( ∫(∇(e)⊙∇(e))dΩ )) + return el2, eh1 +end + +# Run on a sequence of successively refined meshes and compute convergence rates: + +ns = [4, 8, 16] +errs = [solve_cr_poisson(n) for n in ns] + +for i in 2:length(ns) + r_l2 = log(errs[i-1][1]/errs[i][1]) / log(2) + r_h1 = log(errs[i-1][2]/errs[i][2]) / log(2) + println("n=$(ns[i]): L² rate ≈ $(round(r_l2,digits=2)), broken H¹ rate ≈ $(round(r_h1,digits=2))") +end + +# +# ## References +# +# [1] M. Crouzeix, P.-A. Raviart. *Conforming and nonconforming finite element +# methods for solving the stationary Stokes equations I.* +# RAIRO Anal. Numér., 7(R-3):33–75, 1973. +# doi:[10.1051/m2an/197307R300331](http://dx.doi.org/10.1051/m2an/197307R300331) + +# +# ## Further reading +# +# The following source files in `Gridap.jl` are the most relevant to browse +# alongside this tutorial: +# +# - `src/ReferenceFEs/MomentBasedReferenceFEs.jl` — `MomentBasedDofBasis`, +# `FaceMeasure`, and `MomentBasedReferenceFE`; the core machinery used here. +# - `src/ReferenceFEs/CrouzeixRaviartRefFEs.jl` — the built-in CR element +# whose logic this tutorial re-derives step by step. +# - `src/ReferenceFEs/RaviartThomasRefFEs.jl` — a more involved moment-based +# element: vector-valued prebasis, $H(\text{div})$ conformity, and normal-trace +# moments instead of plain averages. +# - `src/ReferenceFEs/NedelecRefFEs.jl` — $H(\text{curl})$ analogue of RT; +# tangential-trace moments, useful as a contrast to the normal-trace case. +# - `src/ReferenceFEs/LagrangianRefFEs.jl` — point-evaluation DOFs (the +# `LagrangianDofBasis` counterpart to `MomentBasedDofBasis`). +# - `src/ReferenceFEs/ReferenceFEInterfaces.jl` — abstract interface: +# `ReferenceFE`, `ReferenceFEName`, `Conformity`, and `GenericRefFE`. +# - `src/Polynomials/MonomialBases.jl` — `MonomialBasis` and the three +# polynomial filters (`_p_filter`, `_q_filter`, `_s_filter`). diff --git a/src/tensor_values_dev.jl b/src/tensor_values_dev.jl new file mode 100644 index 0000000..a49eec5 --- /dev/null +++ b/src/tensor_values_dev.jl @@ -0,0 +1,386 @@ +# In this tutorial, we will learn +# +# - What the type parameters of `MultiValue{S,T,N,L}` encode and why `L` may be +# strictly less than `prod(S)` +# - How to define a concrete `MultiValue` subtype with constrained storage +# - How to implement the `getindex` and array-conversion interface so that generic +# operations work without further code +# - Which arithmetic, contraction, and linear-algebra operations Gridap provides +# for free once the interface is satisfied +# - How to add type-preserving operations that exploit a type's algebraic structure +# - What the built-in `MultiValue` subtypes are and how their storage layouts differ + +using Gridap +using Gridap.TensorValues +using LinearAlgebra: I +using Base: @propagate_inbounds +using Test + +# +# ## The `MultiValue` type hierarchy +# +# `Gridap.TensorValues` revolves around the abstract type +# +# ```julia +# MultiValue{S, T, N, L} <: Number +# ``` +# +# Its four type parameters mirror those of `StaticArrays.SArray`: +# +# | Parameter | Meaning | Example | +# |-----------|---------|---------| +# | `S` | `Tuple` type encoding the full tensor shape | `Tuple{3,3}` for a 3×3 matrix | +# | `T` | scalar component type | `Float64` | +# | `N` | tensor order (length of `S`) | `2` | +# | `L` | number of **stored** components | ≤ `prod(S)` | +# +# The crucial design freedom is `L ≤ prod(S)`. A `TensorValue{3,3}` stores all 9 +# entries; a `SymTensorValue{3}` stores only the 6 upper-triangle entries and +# reconstructs the lower triangle via symmetry on `getindex`; a +# `SkewSymTensorValue{3}` stores 3 entries, recovers the diagonal (zero) and +# lower triangle (negation) at index time. +# +# Subtypes of `MultiValue` subtype `Number`, not `AbstractArray`. This makes +# them behave as scalars under Julia's broadcasting: an array of `VectorValue`s +# participates in the same broadcast machinery as an array of `Float64`s. +# +# ## Implementing a custom `CirculantValue` tensor type +# +# A **circulant matrix** is fully determined by its first row +# $c = (c_1, c_2, \ldots, c_D)$. Every subsequent row is a cyclic shift: +# +# ```math +# A_{ij} = c_{\;\mathrm{mod}_1(j - i + 1,\; D)} +# ``` +# +# For $D = 3$: +# +# ```math +# A = \begin{pmatrix} c_1 & c_2 & c_3 \\ c_3 & c_1 & c_2 \\ c_2 & c_3 & c_1 \end{pmatrix} +# ``` +# +# A $D \times D$ circulant is a `Tuple{D,D}`-shaped second-order tensor whose +# $D^2$ entries depend on only $D$ independent values, so `L = D`. Circulant +# matrices arise in periodic-boundary convolution operators and are the matrices +# simultaneously diagonalised by the discrete Fourier transform. + +struct CirculantValue{D,T} <: MultiValue{Tuple{D,D},T,2,D} + data::NTuple{D,T} + function CirculantValue{D,T}(data::NTuple{D,T}) where {D,T} + new{D,T}(data) + end +end + +# The `.data` field holds the first row — the $D$ independent components. +# Gridap's generic machinery **assumes this field exists** and treats it as the +# canonical independent-component vector, so `get_indep_components` and `Tuple` +# already work correctly without any override. +# +# ### Constructors +# +# We follow the same pattern as `VectorValue` and `SymTensorValue`: a primary +# `NTuple` form, a promoted `Tuple` form, and a vararg convenience form. + +CirculantValue(data::NTuple{D,T}) where {D,T} = CirculantValue{D,T}(data) +CirculantValue{D}(data::NTuple{D,T}) where {D,T} = CirculantValue{D,T}(data) +CirculantValue{D,T1}(data::NTuple{D,T2}) where {D,T1,T2} = CirculantValue{D,T1}(NTuple{D,T1}(data)) + +CirculantValue(data::Tuple) = CirculantValue(promote(data...)) +CirculantValue{D}(data::Tuple) where {D} = CirculantValue{D}(promote(data...)) +CirculantValue{D,T}(data::Tuple) where {D,T} = CirculantValue{D,T}(NTuple{D,T}(data)) + +CirculantValue(data::Number...) = CirculantValue(data) +CirculantValue{D}(data::Number...) where {D} = CirculantValue{D}(data) +CirculantValue{D,T}(data::Number...) where {D,T} = CirculantValue{D,T}(data) + +# ### Completing the mandatory interface +# +# Three methods form the non-negotiable minimum; without any one of them, +# core generic operations silently break. + +import Gridap.TensorValues: change_eltype, num_indep_components + +# **`change_eltype`** — returns the parameterised type with a different scalar +# component type. The generic fallback `change_eltype(::Type{<:Number>, T) = T` +# would return `Float64` instead of `CirculantValue{D,Float64}`, making scalar +# multiplication (`2.0 * a`) crash when it tries `Float64((2.0, 4.0, 6.0))`. + +change_eltype(::Type{<:CirculantValue{D}}, ::Type{T2}) where {D,T2} = CirculantValue{D,T2} + +# **`num_indep_components`** — the fallback `num_components(V) = prod(S) = D²` +# gives the wrong count, breaking `zero(a)` (which calls +# `V(tfill(zero(T), Val(num_indep_components(V))))` and tries to build a +# `CirculantValue` with D² zeros) and `conj`/`real`/`imag` (which slice +# `Tuple(a)[1:Li]` with `Li = D²` even though `.data` has only D entries). + +num_indep_components(::Type{<:CirculantValue{D}}) where {D} = D + +# **`get_array`** — many higher-level operations (contraction, determinant, +# eigenvalues, `outer`) delegate to `get_array`, which must return an `SMatrix`. +# We expand all $D^2$ entries from the $D$-element first row. The `ntuple` +# expression is type-stable and produces zero allocations: + +import Gridap.TensorValues: get_array + +function get_array(arg::CirculantValue{D,T}) where {D,T} + # Expand the D independent entries into D² column-major data, then delegate + # to TensorValue's own get_array (which returns the StaticArrays SMatrix). + data = ntuple(Val(D*D)) do idx + j = (idx - 1) ÷ D + 1 # 1-based column index + i = (idx - 1) % D + 1 # 1-based row index + @inbounds arg.data[mod1(j - i + 1, D)] + end + get_array(TensorValue{D,D,T}(data)) +end + +# ### `getindex` +# +# For scalar two-index access, the circulant formula maps any `(i,j)` to a +# single lookup in the $D$-element `.data` tuple: + +@propagate_inbounds function Base.getindex(a::CirculantValue{D}, i::Integer, j::Integer) where {D} + @boundscheck checkbounds(a, i, j) + @inbounds a.data[mod1(j - i + 1, D)] +end + +# Confirm the layout for a concrete 3×3 example: + +a = CirculantValue(1.0, 2.0, 3.0) + +@test a[1,1] == 1.0 && a[1,2] == 2.0 && a[1,3] == 3.0 +@test a[2,1] == 3.0 && a[2,2] == 1.0 && a[2,3] == 2.0 +@test a[3,1] == 2.0 && a[3,2] == 3.0 && a[3,3] == 1.0 + +@test Matrix(get_array(a)) == [1.0 2.0 3.0 + 3.0 1.0 2.0 + 2.0 3.0 1.0] + +# ## Operations provided for free +# +# Because `CirculantValue` satisfies the `MultiValue` interface, Gridap's generic +# implementations for arithmetic, contractions, and linear algebra apply +# immediately — no additional code required. +# +# ### Arithmetic on independent components +# +# Addition, subtraction, and scalar multiplication act on the $D$ stored entries +# and return a new `CirculantValue`. The cost is $O(D)$, not $O(D^2)$. + +b = CirculantValue(4.0, 5.0, 6.0) + +@test a + b === CirculantValue(5.0, 7.0, 9.0) +@test b - a === CirculantValue(3.0, 3.0, 3.0) +@test 2.0 * a === CirculantValue(2.0, 4.0, 6.0) +@test a / 2.0 === CirculantValue(0.5, 1.0, 1.5) + +# `zero` constructs the zero circulant (all-zero first row) via +# `num_indep_components`, and works without any override: + +@test iszero(zero(a)) +@test zero(CirculantValue{3,Float64}) === CirculantValue(0.0, 0.0, 0.0) + +# ### Frobenius inner product and norm +# +# `inner(a, b)` or `a ⊙ b` is a full contraction over all index pairs, +# accessing all $D^2$ entries through `getindex`: + +@test a ⊙ a ≈ sum(a[i,j]^2 for i in 1:3, j in 1:3) +@test norm(a) ≈ sqrt(a ⊙ a) + +# ### Outer product +# +# `a ⊗ b` yields a fourth-order `HighOrderTensorValue{Tuple{3,3,3,3}}`: + +c = a ⊗ b +@test c isa HighOrderTensorValue +@test c[1,2,1,3] ≈ a[1,2] * b[1,3] + +# ### Matrix–vector contraction +# +# Contracting `a ⋅ v` against a `VectorValue` performs the matrix–vector product +# and returns a `VectorValue`: + +v = VectorValue(1.0, 0.0, 0.0) +w = a ⋅ v +@test w isa VectorValue{3,Float64} +@test w ≈ VectorValue(a[1,1], a[2,1], a[3,1]) # multiplication by the first unit vector extracts the first column + +# ### Trace and determinant +# +# Both delegate to `getindex` or `get_array` internally. For a circulant, +# every diagonal entry equals $c_1$, so $\mathrm{tr}(A) = D \cdot c_1$: + +@test tr(a) ≈ 3 * a.data[1] +@test det(a) ≈ det(Matrix(get_array(a))) + +# ## Type-preserving extensions +# +# Generic contractions return the most general matching type. Contracting two +# `CirculantValue`s with `⋅` uses `contracted_product`, which returns a +# `TensorValue` — the circulant structure is lost: + +ab = a ⋅ b +@test ab isa TensorValue{3,3,Float64} + +# The matrix entries are correct, but all $D^2 = 9$ values are now stored +# instead of $D = 3$. We recover the compact form by exploiting the core +# algebraic property: **the product of two circulants is again circulant**, with +# first row equal to the cyclic convolution of the two input rows: +# +# ```math +# (A B)_{1,n} = \sum_{k=1}^{D} c^a_k \; c^b_{\mathrm{mod}_1(n-k+1,\,D)} +# ``` + +function circ_mul(a::CirculantValue{D,Ta}, b::CirculantValue{D,Tb}) where {D,Ta,Tb} + T = promote_type(Ta, Tb) + row = ntuple(Val(D)) do n + sum(a.data[k] * b.data[mod1(n - k + 1, D)] for k in 1:D) + end + CirculantValue{D,T}(row) +end + +@test circ_mul(a, b) isa CirculantValue{3,Float64} +@test get_array(circ_mul(a, b)) ≈ get_array(a) * get_array(b) + +# `circ_mul` stores only $D$ values and computes in $O(D^2)$ — the same +# arithmetic as the general product, but with $D$ rather than $D^2$ storage. +# +# ### Transpose +# +# For a circulant with first row $(c_1, c_2, \ldots, c_D)$, the transpose has +# first row $(c_1, c_D, c_{D-1}, \ldots, c_2)$ — position 1 is fixed and +# positions 2 through $D$ are reversed. Gridap's fallback for +# `MultiValue{Tuple{D,D}}` is `@notimplemented`, so we provide a specialisation: + +function Base.transpose(a::CirculantValue{D,T}) where {D,T} + data = ntuple(Val(D)) do k + k == 1 ? a.data[1] : a.data[D - k + 2] + end + CirculantValue{D,T}(data) +end + +at = transpose(a) +@test at isa CirculantValue{3,Float64} +@test get_array(at) ≈ transpose(Matrix(get_array(a))) + +# A circulant is symmetric iff its first row is a palindrome: + +s = CirculantValue(1.0, 2.0, 2.0) +@test transpose(s) == s + +# ### Mixed-precision arithmetic (optional extension) +# +# By default, `promote_rule` for two different-T `MultiValue`s returns `Union{}` +# — meaning Julia raises a method error rather than trying to promote. Adding +# `promote_rule` and a matching `convert` unlocks operations like +# `CirculantValue{3,Float64} + CirculantValue{3,Int64}`. +# +# The `convert` must be explicit: the generic `convert(::Type{T<:Number}, x::Number) = T(x)` +# would call `CirculantValue{3,Float64}(x::CirculantValue{3,Int64})`, which +# matches the vararg constructor with a 1-element argument and then fails +# constructing `NTuple{3,Float64}` from a 1-element tuple. + +Base.promote_rule(::Type{<:CirculantValue{D,Ta}}, ::Type{<:CirculantValue{D,Tb}}) where {D,Ta,Tb} = + CirculantValue{D, promote_type(Ta,Tb)} + +Base.convert(::Type{<:CirculantValue{D,T}}, arg::CirculantValue{D}) where {D,T} = + CirculantValue{D,T}(Tuple(arg)) +Base.convert(::Type{<:CirculantValue{D,T}}, arg::CirculantValue{D,T}) where {D,T} = arg + +@test a + CirculantValue(1, 0, 0) === CirculantValue(2.0, 2.0, 3.0) + +# ### Identity +# +# The $D \times D$ identity matrix is circulant with first row $(1, 0, \ldots, 0)$. +# Overriding `one` lets generic code that calls `one(a)` or `one(CirculantValue{D,T})` +# get the compact representation rather than an error: + +function Base.one(::Type{V}) where {D, T, V <: CirculantValue{D,T}} + CirculantValue{D,T}(ntuple(i -> i == 1 ? one(T) : zero(T), Val(D))) +end +Base.one(a::CirculantValue) = one(typeof(a)) + +id = one(CirculantValue{3,Float64}) +@test get_array(id) ≈ Matrix(1.0*I, 3, 3) +@test circ_mul(a, id) == a + +# ## Built-in `MultiValue` types +# +# Gridap ships nine concrete subtypes covering the tensors most common in +# continuum mechanics and electromagnetics. +# +# | Type | Shape `S` | `L` | Storage | +# |------|-----------|-----|---------| +# | `VectorValue{D,T}` | `Tuple{D}` | `D` | all D components | +# | `TensorValue{D1,D2,T}` | `Tuple{D1,D2}` | `D1·D2` | all, column-major | +# | `SymTensorValue{D,T}` | `Tuple{D,D}` | `D(D+1)/2` | upper triangle, row ≤ col | +# | `SymTracelessTensorValue{D,T}` | `Tuple{D,D}` | `D(D+1)/2 − 1` | upper triangle; last diagonal recovered as `−(sum of others)` | +# | `SkewSymTensorValue{D,T}` | `Tuple{D,D}` | `D(D−1)/2` | strict upper triangle; diagonal zero, lower triangle negated | +# | `ThirdOrderTensorValue{D1,D2,D3,T}` | `Tuple{D1,D2,D3}` | `D1·D2·D3` | all, column-major | +# | `HighOrderTensorValue{S,T,N}` | any `S` | `prod(S)` | all, `SArray` | +# | `SymFourthOrderTensorValue{D,T}` | `Tuple{D,D,D,D}` | `(D(D+1)/2)²` | both index pairs symmetric: `ijkl=jikl=ijlk` | +# +# The independent-component counts for $D = 3$: + +@test num_indep_components(VectorValue{3,Float64}) == 3 +@test num_indep_components(TensorValue{3,3,Float64}) == 9 +@test num_indep_components(SymTensorValue{3,Float64}) == 6 +@test num_indep_components(SymTracelessTensorValue{3,Float64}) == 5 +@test num_indep_components(SkewSymTensorValue{3,Float64}) == 3 +@test num_indep_components(ThirdOrderTensorValue{2,2,2,Float64}) == 8 +@test num_indep_components(SymFourthOrderTensorValue{2,Float64}) == 9 + +# **`VectorValue{D,T}`** — a plain $D$-vector. + +u = VectorValue(1.0, 2.0, 3.0) +@test u ⋅ u ≈ 14.0 + +# **`TensorValue{D1,D2,T}`** — a general $D_1 \times D_2$ matrix with no symmetry +# imposed. The constructor fills entries in column-major order. + +F = TensorValue(1.0, 0.0, 0.0, # column 1 + 0.0, 2.0, 0.0, # column 2 + 0.0, 0.0, 3.0) # column 3 +@test F[1,1] == 1.0 && F[2,2] == 2.0 && F[3,3] == 3.0 +@test det(F) ≈ 6.0 + +# **`SymTensorValue{D,T}`** — symmetric second-order tensor; stores only the +# upper triangle. Entry (i,j) with i > j is recovered as (j,i). + +σ = SymTensorValue(1.0, 0.5, 0.0, # row 1: σ₁₁, σ₁₂, σ₁₃ + 2.0, 0.0, # σ₂₂, σ₂₃ + 3.0) # σ₃₃ +@test σ[2,1] == σ[1,2] == 0.5 + +# **`SkewSymTensorValue{D,T}`** — anti-symmetric; diagonal always zero, lower +# triangle recovered as the negation of the upper triangle. + +Ω = SkewSymTensorValue(0.1, 0.2, 0.3) # upper-triangle entries for D=3 +@test Ω[2,1] == -Ω[1,2] +@test iszero(Ω[1,1]) + +# **`SymTracelessTensorValue{D,T}`** (alias `QTensorValue`) — symmetric and +# trace-free; used for liquid-crystal order-parameter tensors. The final +# diagonal entry is not stored; it is recovered as minus the sum of the others. + +Q = SymTracelessTensorValue(0.3, 0.0, -0.1, 0.0, 0.0) # 5 entries for D=3 +@test tr(Q) ≈ 0.0 + +# **`ThirdOrderTensorValue{D1,D2,D3,T}`** — third-order tensor, alias for +# `HighOrderTensorValue{Tuple{D1,D2,D3}}`, no symmetry assumed. + +G = ThirdOrderTensorValue{2,2,2,Float64}(ntuple(i -> Float64(i), 8)) +@test size(G) == (2, 2, 2) + +# **`SymFourthOrderTensorValue{D,T}`** — fourth-order tensor with the two minor +# symmetries $\mathbb{C}_{ijkl} = \mathbb{C}_{jikl} = \mathbb{C}_{ijlk}$ that +# characterise the elasticity tensor. + +𝕔 = SymFourthOrderTensorValue{2,Float64}(ntuple(i -> Float64(i), 9)) +@test 𝕔[1,2,1,1] == 𝕔[2,1,1,1] # minor symmetry ij↔ji + +# Contracting the elasticity tensor with a symmetric strain gives a stress: + +ε = SymTensorValue(0.1, 0.0, 0.2) # 2×2 symmetric strain +τ = 𝕔 ⋅² ε # double contraction: stress +@test τ isa SymTensorValue{2,Float64}