Engineering domain: Structure Design

## Task Description

In this task, you are required to determine the optimal material distribution within a 2D design domain to minimize structural compliance under a given set of physical constraints. The domain is a fixed-size grid of **64 × 64 elements**. Each design must satisfy the specified boundary conditions, load configurations, and a volume fraction constraint.

## Input

The input consists of a 5-channel tensor, where each channel is a **64 × 64 NumPy array (.npy format)** representing:

- **Channel 1**: volume fraction, prescribed uniformly or non-uniformly across the grid.  
- **Channel 2**: von Mises stress under the given loading and boundary conditions.  
- **Channel 3**: Strain energy distribution under the same conditions.  
- **Channel 4**: x-direction load magnitude at each node (0 if none). 
- **Channel 5**: y-direction load magnitude at each node (0 if none).  

The inputs are specified in five 64x64 numpy array:
- channel 1: A 64x64 matrix with each element equal 0.48
- channel 2:[[3.02378885e-06 4.47539433e-06 7.50072817e-06 1.20186311e-05
  1.80109097e-05 2.54543330e-05 3.43215513e-05 4.45822821e-05
  5.62047910e-05 6.91577125e-05 8.34122703e-05 9.89449821e-05
  1.15740968e-04 1.33798037e-04 1.53131780e-04 1.73782043e-04
  1.95821249e-04 2.19365323e-04 2.44588274e-04 2.71741967e-04
  3.01183312e-04 3.33411876e-04 3.69121785e-04 4.09269653e-04
  4.55158574e-04 5.08461270e-04 5.71189253e-04 6.43753731e-04
  7.24444883e-04 8.61603091e-04 7.41751328e-03 1.26088212e-02
  2.28558406e-02 1.90377994e-02 2.29648198e-03 5.74503268e-04
  1.70806693e-04 5.59472407e-05 2.40936179e-05 2.09785195e-05
  2.83383866e-05 3.92710245e-05 5.09601691e-05 6.22304338e-05
  7.26109646e-05 8.19471512e-05 9.02286714e-05 9.75098238e-05
  1.03871446e-04 1.09402429e-04 1.14190775e-04 1.18319405e-04
  1.21864365e-04 1.24894237e-04 1.27470146e-04 1.29646034e-04
  1.31469059e-04 1.32980001e-04 1.34213660e-04 1.35199214e-04
  1.35960520e-04 1.36516369e-04 1.36880691e-04 1.35381261e-04]
 [9.99441870e-06 1.12235038e-05 1.38634008e-05 1.78071154e-05
  2.30355207e-05 2.95237637e-05 3.72418210e-05 4.61551870e-05
  5.62256893e-05 6.74124195e-05 7.96727639e-05 9.29635022e-05
  1.07241926e-04 1.22466889e-04 1.38599632e-04 1.55604117e-04
  1.73446380e-04 1.92092012e-04 2.11500144e-04 2.31610822e-04
  2.52319841e-04 2.73429263e-04 2.94549893e-04 3.14907379e-04
  3.32953288e-04 3.45612379e-04 3.47097711e-04 3.29221765e-04
  3.12111234e-04 6.08510000e-04 3.59270247e-03 5.18087526e-03
  1.32168194e-02 1.43017050e-02 4.15618859e-03 1.41871086e-03
  5.49137011e-04 2.31669836e-04 1.07999778e-04 5.93808033e-05
  4.26299521e-05 4.00876239e-05 4.39482247e-05 5.05486821e-05
  5.81038451e-05 6.57295301e-05 7.29931730e-05 7.96957838e-05
  8.57614630e-05 9.11794279e-05 9.59727398e-05 1.00181083e-04
  1.03851151e-04 1.07031244e-04 1.09768251e-04 1.12105984e-04
  1.14084301e-04 1.15738676e-04 1.17100023e-04 1.18194668e-04
  1.19044384e-04 1.19666461e-04 1.20073785e-04 1.20337789e-04]
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  3.41460670e-05 3.93967071e-05 4.56253381e-05 5.27931715e-05
  6.08552585e-05 6.97604009e-05 7.94508362e-05 8.98615592e-05
  1.00919068e-04 1.12539220e-04 1.24623702e-04 1.37054362e-04
  1.49684198e-04 1.62323090e-04 1.74715204e-04 1.86503157e-04
  1.97171221e-04 2.05956267e-04 2.11713348e-04 2.12736162e-04
  2.06620301e-04 1.90685736e-04 1.65312082e-04 1.51134218e-04
  2.63334631e-04 9.20941119e-04 2.44746839e-03 3.68516374e-03
  5.18872672e-03 6.36497638e-03 4.67760144e-03 2.25661676e-03
  1.08103309e-03 5.35911749e-04 2.79099804e-04 1.54861824e-04
  9.41517471e-05 6.52404603e-05 5.27898861e-05 4.90300096e-05
  4.98626330e-05 5.30562941e-05 5.73679019e-05 6.20960998e-05
  6.68450510e-05 7.13951408e-05 7.56300484e-05 7.94945337e-05
  8.29694529e-05 8.60566908e-05 8.87699298e-05 9.11289252e-05
  9.31559195e-05 9.48733840e-05 9.63025915e-05 9.74627149e-05
  9.83702607e-05 9.90387162e-05 9.94783327e-05 9.97591586e-05]
 [4.16295394e-05 4.23891394e-05 4.40496624e-05 4.65237958e-05
  4.97912972e-05 5.38249610e-05 5.85903740e-05 6.40455261e-05
  7.01402094e-05 7.68151090e-05 8.40004479e-05 9.16139943e-05
  9.95581552e-05 1.07715765e-04 1.15944019e-04 1.24065760e-04
  1.31857062e-04 1.39029690e-04 1.45206841e-04 1.49891012e-04
  1.52425636e-04 1.51962080e-04 1.47475242e-04 1.37966593e-04
  1.23280995e-04 1.06798444e-04 1.03599745e-04 1.62552795e-04
  4.06523400e-04 9.86394132e-04 1.78374670e-03 2.44852323e-03
  3.09814326e-03 3.61013837e-03 3.30263275e-03 2.31434577e-03
  1.38602652e-03 8.04681374e-04 4.70304174e-04 2.82003698e-04
  1.75940806e-04 1.16167394e-04 8.28164575e-05 6.47941377e-05
  5.57792162e-05 5.20891459e-05 5.15337521e-05 5.27874077e-05
  5.50369816e-05 5.77795579e-05 6.07034385e-05 6.36165751e-05
  6.64026399e-05 6.89935579e-05 7.13520420e-05 7.34603211e-05
  7.53127692e-05 7.69110295e-05 7.82607593e-05 7.93694404e-05
  8.02449011e-05 8.08943190e-05 8.13235561e-05 8.15972663e-05]
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  1.01576278e-04 1.06392396e-04 1.10942558e-04 1.15019509e-04
  1.18367091e-04 1.20668876e-04 1.21537814e-04 1.20513575e-04
  1.17084793e-04 1.10778431e-04 1.01417249e-04 8.97817565e-05
  7.92062724e-05 7.91830708e-05 1.12506768e-04 2.24775284e-04
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  6.45712985e-05 6.51434252e-05 6.55242540e-05 6.57699295e-05]
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  4.73105995e-04 7.44987039e-04 1.02115664e-03 1.26404811e-03
  1.48750640e-03 1.66928411e-03 1.71064384e-03 1.55490925e-03
  1.26543758e-03 9.51567463e-04 6.83433932e-04 4.80834304e-04
  3.36772910e-04 2.37459280e-04 1.70093164e-04 1.24843620e-04
  9.47045888e-05 7.48415115e-05 6.19596809e-05 5.38197404e-05
  4.88961722e-05 4.61450322e-05 4.48481892e-05 4.45091888e-05
  4.47834349e-05 4.54310914e-05 4.62850648e-05 4.72290501e-05
  4.81823462e-05 4.90892683e-05 4.99117104e-05 5.06238947e-05
  5.12086554e-05 5.16548189e-05 5.19553835e-05 5.21558529e-05]
 [1.09995072e-04 1.10055300e-04 1.10260289e-04 1.10552438e-04
  1.10912671e-04 1.11314695e-04 1.11724215e-04 1.12097926e-04
  1.12382233e-04 1.12511723e-04 1.12407371e-04 1.11974584e-04
  1.11101250e-04 1.09656207e-04 1.07488953e-04 1.04432174e-04
  1.00310074e-04 9.49580202e-05 8.82635871e-05 8.02469800e-05
  7.12121341e-05 6.20199264e-05 5.45591786e-05 5.24983455e-05
  6.23136825e-05 9.42061151e-05 1.61499614e-04 2.75859632e-04
  4.37162452e-04 6.25517604e-04 8.11038178e-04 9.78734036e-04
  1.13128329e-03 1.25484628e-03 1.30460798e-03 1.24561986e-03
  1.09128298e-03 8.90542783e-04 6.91177899e-04 5.20106518e-04
  3.85148286e-04 2.83709608e-04 2.09598063e-04 1.56373036e-04
  1.18577749e-04 9.19755466e-05 7.34150254e-05 6.06020212e-05
  5.18834588e-05 4.60729358e-05 4.23194166e-05 4.00116502e-05
  3.87097191e-05 3.80963535e-05 3.79423532e-05 3.80819893e-05
  3.83954437e-05 3.87962229e-05 3.92221022e-05 3.96285939e-05
  3.99842387e-05 4.02672316e-05 4.04630454e-05 4.06054038e-05]
 [1.34271035e-04 1.34096754e-04 1.33814665e-04 1.33377704e-04
  1.32768609e-04 1.31963557e-04 1.30931467e-04 1.29633156e-04
  1.28020365e-04 1.26034740e-04 1.23606905e-04 1.20655871e-04
  1.17089256e-04 1.12805095e-04 1.07696630e-04 1.01662410e-04
  9.46256145e-05 8.65690750e-05 7.75963688e-05 6.80350069e-05
  5.86045481e-05 5.06772072e-05 4.66501910e-05 5.03992811e-05
  6.76357157e-05 1.05676268e-04 1.71761945e-04 2.69275748e-04
  3.93213421e-04 5.29590167e-04 6.62797101e-04 7.85258425e-04
  8.95578332e-04 9.84539893e-04 1.02972329e-03 1.01077905e-03
  9.27723408e-04 8.01463281e-04 6.59841821e-04 5.24784817e-04
  4.08053901e-04 3.13196190e-04 2.39068301e-04 1.82580011e-04
  1.40244777e-04 1.08883844e-04 8.58599853e-05 6.90907827e-05
  5.69766766e-05 4.83080573e-05 4.21780848e-05 3.79099583e-05
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  3.07315406e-05 3.05706739e-05 3.05514382e-05 3.06113953e-05
  3.07051458e-05 3.08003835e-05 3.08750760e-05 3.09509514e-05]
 [1.58147346e-04 1.57740032e-04 1.56974536e-04 1.55814875e-04
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  1.23330952e-04 1.16465743e-04 1.08758987e-04 1.00195450e-04
  9.08172600e-05 8.07685963e-05 7.03652855e-05 6.01993668e-05
  5.12882653e-05 4.52704134e-05 4.46234565e-05 5.28207475e-05
  7.42324562e-05 1.13451720e-04 1.73751313e-04 2.54883967e-04
  3.51542736e-04 4.54522988e-04 5.54964970e-04 6.48023026e-04
  7.31091113e-04 7.97809541e-04 8.35949162e-04 8.33674732e-04
  7.88208403e-04 7.08282621e-04 6.09261305e-04 5.06293024e-04
  4.10148520e-04 3.26488793e-04 2.57054527e-04 2.01247227e-04
  1.57373218e-04 1.23413697e-04 9.74259140e-05 7.77137701e-05
  6.28731828e-05 5.17778260e-05 4.35415014e-05 3.74753306e-05
  3.30478827e-05 2.98511722e-05 2.75729917e-05 2.59750066e-05
  2.48757001e-05 2.41372357e-05 2.36554154e-05 2.33520562e-05
  2.31692524e-05 2.30651129e-05 2.30106677e-05 2.30162206e-05]
 [1.81186691e-04 1.80550446e-04 1.79310760e-04 1.77443324e-04
  1.74938244e-04 1.71782062e-04 1.67957638e-04 1.63444162e-04
  1.58217453e-04 1.52250780e-04 1.45516556e-04 1.37989466e-04
  1.29651801e-04 1.20502230e-04 1.10569689e-04 9.99348328e-05
  8.87622853e-05 7.73477602e-05 6.61844024e-05 5.60513640e-05
  4.81223529e-05 4.40786948e-05 4.61849560e-05 5.72423592e-05
  8.02878702e-05 1.17904624e-04 1.71151884e-04 2.38486452e-04
  3.15468961e-04 3.95958950e-04 4.74376263e-04 5.47124691e-04
  6.11476478e-04 6.62921642e-04 6.94316854e-04 6.98639363e-04
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  3.97628522e-04 3.26534558e-04 2.64336409e-04 2.11918727e-04
  1.68909839e-04 1.34298576e-04 1.06841087e-04 8.52939849e-05
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  1.71747750e-05 1.68721239e-05 1.66886041e-05 1.66242740e-05]
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  3.76083179e-04 3.16951760e-04 2.62836772e-04 2.15304762e-04
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  3.22571340e-05 2.69665513e-05 2.28874457e-05 1.97602481e-05
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  1.25280795e-05 1.20390676e-05 1.17338811e-05 1.16039171e-05]
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  9.44707021e-05 1.22666640e-04 1.57672915e-04 1.97869515e-04
  2.41029601e-04 2.84708897e-04 3.26629593e-04 3.64842746e-04
  3.97599626e-04 4.23109720e-04 4.39492688e-04 4.45093658e-04
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  1.72313560e-04 1.42983678e-04 1.17660057e-04 9.61581405e-05
  7.81449159e-05 6.32184245e-05 5.09609969e-05 4.09719026e-05
  3.28852362e-05 2.63779303e-05 2.11715873e-05 1.70307307e-05
  1.37592108e-05 1.11958665e-05 9.21010896e-06 7.69780902e-06
  6.57768561e-06 5.78828675e-06 5.28559147e-06 5.04907683e-06]
 [2.60575971e-04 2.59126934e-04 2.56212727e-04 2.51859308e-04
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  0.02560814 0.02524332 0.02492158 0.0246391  0.02439216 0.02417719
  0.02399083 0.02382994 0.02369163 0.02357323 0.02347235 0.02338682
  0.02331472 0.02325434 0.02320418 0.02316297 0.0231296  0.02310315
  0.02308287 0.02306817 0.02305861 0.02305235]
 [0.00894815 0.00920584 0.00973248 0.01049445 0.01146889 0.01263607
  0.01398106 0.01549318 0.01716419 0.01898583 0.02094711 0.02303157
  0.02521468 0.02746192 0.02972796 0.03195749 0.03408806 0.03605463
  0.03779516 0.03925617 0.04039741 0.04119479 0.04164178 0.04174923
  0.04154374 0.04106468 0.04036016 0.03948227 0.03848267 0.03740889
  0.03630196 0.03519524 0.03411438 0.03307801 0.03209869 0.03118413
  0.03033826 0.02956216 0.0288549  0.0282141  0.02763647 0.02711813
  0.02665488 0.02624242 0.02587645 0.02555281 0.0252675  0.02501676
  0.02479707 0.0246052  0.02443815 0.02429323 0.02416798 0.02406021
  0.02396796 0.02388951 0.02382334 0.02376816 0.02372284 0.02368645
  0.02365824 0.0236376  0.02362409 0.02361562]
 [0.0071093  0.00740516 0.00800216 0.00885242 0.00992452 0.01119607
  0.01265454 0.01429497 0.01611674 0.01812011 0.02030244 0.02265433
  0.0251556  0.02777175 0.0304517  0.03312783 0.03571922 0.03813796
  0.04029763 0.04212206 0.04355259 0.04455288 0.04511122 0.04524065
  0.04497686 0.04437383 0.04349745 0.04241817 0.04120413 0.03991607
  0.03860427 0.03730778 0.03605495 0.03486498 0.03374969 0.03271527
  0.03176382 0.03089453 0.03010468 0.02939031 0.02874674 0.02816894
  0.02765177 0.02719014 0.02677914 0.02641412 0.02609069 0.0258048
  0.02555269 0.02533093 0.0251364  0.02496628 0.02481802 0.02468936
  0.02457828 0.02448301 0.02440199 0.02433389 0.02427755 0.02423202
  0.02419652 0.02417043 0.02415329 0.02414279]
 [0.0052161  0.00556405 0.00624933 0.00719579 0.00835976 0.0097184
  0.01126555 0.01300609 0.01495104 0.01711327 0.01950306 0.02212304
  0.02496233 0.02799048 0.03115246 0.03436661 0.03752761 0.04051512
  0.04320677 0.04549226 0.04728508 0.04853015 0.04920774 0.04933426
  0.04895976 0.04816111 0.04703155 0.04566866 0.04416368 0.04259436
  0.04102177 0.03949035 0.03803    0.03665897 0.03538679 0.03421678
  0.03314804 0.03217701 0.03129847 0.03050633 0.02979412 0.02915534
  0.02858363 0.02807294 0.02761759 0.0272123  0.02685219 0.02653281
  0.0262501  0.02600037 0.02578031 0.02558694 0.02541758 0.02526986
  0.02514168 0.02503119 0.02493678 0.02485706 0.02479085 0.02473713
  0.02469512 0.02466416 0.02464379 0.02463149]
 [0.00333089 0.00376653 0.00457538 0.00562347 0.00685906 0.00826865
  0.00985997 0.01165279 0.0136738  0.01595274 0.01851804 0.02139099
  0.02457767 0.02805912 0.03178129 0.03564825 0.03952299 0.0432386
  0.0466186  0.04950038 0.05175462 0.05329737 0.05409646 0.05417412
  0.0536041  0.05250068 0.05100028 0.04924095 0.0473457  0.04541318
  0.04351537 0.04170022 0.03999639 0.03841841 0.03697109 0.03565293
  0.03445859 0.03338057 0.03241038 0.03153919 0.0307583  0.0300594
  0.02943469 0.02887697 0.02837963 0.02793665 0.02754259 0.02719252
  0.02688201 0.02660709 0.0263642  0.02615017 0.02596218 0.02579771
  0.02565458 0.02553083 0.0254248  0.02533503 0.02526028 0.02519951
  0.02515189 0.02511676 0.02509361 0.02507978]
 [0.00163042 0.00224056 0.00320305 0.0043135  0.00555572 0.00694317
  0.0085036  0.01027399 0.0122984  0.01462625 0.01730925 0.02039547
  0.02391927 0.0278862  0.03225452 0.03691828 0.04170097 0.04636812
  0.05065957 0.05433047 0.05718454 0.05909278 0.06000356 0.05994956
  0.05904582 0.05747017 0.05542881 0.05312008 0.05070902 0.04831713
  0.04602424 0.04387668 0.04189669 0.04009068 0.03845533 0.03698181
  0.03565856 0.03447293 0.03341227 0.03246441 0.03161801 0.03086267
  0.03018895 0.02958834 0.02905321 0.02857674 0.02815283 0.02777606
  0.0274416  0.02714516 0.02688291 0.02665149 0.02644789 0.02626949
  0.02611396 0.02597928 0.02586369 0.02576567 0.02568394 0.02561743
  0.02556525 0.02552671 0.0255013  0.02548625]
 [0.00116384 0.00171071 0.00256573 0.00354011 0.00463315 0.0058686
  0.00728301 0.00892349 0.0108478  0.01312461 0.01583248 0.01905521
  0.02287096 0.02733158 0.0324311  0.0380691  0.04402509 0.04996605
  0.05549807 0.06024231 0.06389569 0.0662579  0.06724546 0.06691081
  0.06544467 0.06313768 0.06031085 0.05725041 0.05417204 0.05121571
  0.04845935 0.04593746 0.04365751 0.04161159 0.03978401 0.03815576
  0.03670718 0.03541926 0.03427439 0.0332566  0.03235159 0.03154667
  0.03083062 0.03019358 0.02962683 0.02912272 0.02867451 0.02827627
  0.02792275 0.02760934 0.02733197 0.02708707 0.02687147 0.02668241
  0.02651745 0.0263745  0.02625171 0.0261475  0.02606056 0.02598975
  0.02593417 0.0258931  0.02586602 0.02585008]
 [0.00213265 0.0023765  0.00284792 0.00346983 0.00423676 0.00516715
  0.00629552 0.00767139 0.00936097 0.01144966 0.01404432 0.01727277
  0.02127604 0.02618551 0.0320761  0.03889487 0.04638857 0.0540828
  0.06136013 0.06761279 0.07236377 0.07529786 0.07627587 0.07538755
  0.07297257 0.06953093 0.06557338 0.06150648 0.05759302 0.05397204
  0.05069798 0.0477767  0.0451893  0.04290617 0.04089462 0.03912271
  0.03756104 0.03618333 0.03496648 0.03389044 0.0329378  0.03209357
  0.03134474 0.0306801  0.03008991 0.02956573 0.02910019 0.02868689
  0.02832022 0.02799528 0.02770777 0.02745393 0.02723046 0.02703447
  0.02686345 0.0267152  0.02658784 0.02647973 0.02638951 0.02631601
  0.02625832 0.02621568 0.02618756 0.02617109]
 [0.00333928 0.00342028 0.00360987 0.00390342 0.00431654 0.00487533
  0.00561936 0.00660495 0.0079095  0.00963699 0.01192507 0.0149525
  0.0189424  0.02414936 0.03080688 0.03900796 0.04853102 0.05871438
  0.06854898 0.07700601 0.08331963 0.08699048 0.08776243 0.08580358
  0.08176963 0.07656744 0.07101487 0.06565176 0.06074902 0.05639411
  0.05258165 0.04926601 0.04638877 0.04389134 0.04172019 0.03982848
  0.03817615 0.03672925 0.03545917 0.0343418  0.03335682 0.03248702
  0.03171783 0.0310368  0.03043331 0.02989823 0.0294237  0.02900291
  0.02862996 0.02829972 0.02800772 0.02775006 0.02752332 0.02732456
  0.02715116 0.0270009  0.02687184 0.02676231 0.02667092 0.02659648
  0.02653805 0.02649488 0.02646641 0.02644981]
 [0.00422335 0.00425169 0.00432662 0.00444839 0.00463062 0.00489547
  0.00527795 0.00583213 0.00663986 0.00782195 0.00955103 0.0120661
  0.01568995 0.02084468 0.02803121 0.0376904  0.04985001 0.06368346
  0.07745898 0.08929377 0.09793801 0.10259001 0.10272073 0.09865313
  0.09180834 0.08391157 0.07620557 0.06929603 0.0633225  0.05823506
  0.05392121 0.05026067 0.04714466 0.04448095 0.04219347 0.04022018
  0.03851061 0.03702366 0.03572573 0.03458923 0.03359133 0.03271309
  0.03193863 0.03125462 0.03064977 0.03011446 0.02964048 0.02922078
  0.02884926 0.02852067 0.02823042 0.02797454 0.02774958 0.02755253
  0.02738076 0.02723201 0.02710432 0.02699603 0.02690571 0.02683219
  0.0267745  0.02673188 0.02670379 0.02668747]
 [0.00467043 0.00469073 0.00474272 0.00482081 0.00492637 0.00506378
  0.00524383 0.00548991 0.00584903 0.00641049 0.00733651 0.00890669
  0.01156469 0.01595333 0.02295652 0.03367208 0.04897477 0.06827346
  0.08855187 0.1058616  0.11817017 0.12431252 0.12270196 0.11429795
  0.10253862 0.09067911 0.0803962  0.07187864 0.06491261 0.05920976
  0.05451226 0.05061216 0.0473475  0.0445932  0.04225262 0.04025053
  0.03852797 0.03703825 0.0357441  0.03461544 0.03362782 0.03276117
  0.03199891 0.0313272  0.03073443 0.03021078 0.02974792 0.0293387
  0.02897701 0.02865757 0.02837578 0.0281277  0.02790987 0.02771929
  0.02755337 0.02740985 0.02728678 0.02718251 0.02709563 0.02702495
  0.02696953 0.02692862 0.02690165 0.02688605]
 [0.00461015 0.00465384 0.00475191 0.00489297 0.00507027 0.0052754
  0.00549868 0.00573044 0.00596452 0.00620807 0.00650508 0.00698469
  0.00796121 0.01017754 0.01513353 0.02510049 0.04287252 0.07008567
  0.10200122 0.12887463 0.14763549 0.15624296 0.15009441 0.13209201
  0.11174985 0.0952497  0.0825223  0.07272585 0.06508627 0.05903291
  0.05416177 0.05018675 0.04690268 0.04416002 0.0418479  0.03988282
  0.03820089 0.03675257 0.03549891 0.03440893 0.03345772 0.03262501
  0.03189417 0.03125144 0.0306853  0.0301861  0.02974561 0.02935687
  0.02901387 0.02871146 0.02844518 0.02821116 0.02800604 0.0278269
  0.0276712  0.02753675 0.02742164 0.02732426 0.02724323 0.0271774
  0.02712584 0.0270878  0.02706275 0.0270483 ]
 [0.00400406 0.00411526 0.00435016 0.00468185 0.00509379 0.00557057
  0.00609743 0.00665813 0.00723165 0.00778771 0.00828208 0.00865851
  0.00889196 0.00917207 0.01011043 0.01341459 0.02677122 0.06125853
  0.1162629  0.16030552 0.19158705 0.21046445 0.18647204 0.14466476
  0.11540243 0.09547258 0.08144926 0.07120386 0.06347012 0.05747435
  0.05272269 0.04888781 0.04574538 0.0431372  0.04094891 0.039096
  0.03751485 0.03615668 0.03498356 0.03396552 0.03307862 0.0323035
  0.03162432 0.031028   0.03050365 0.03004211 0.02963564 0.02927763
  0.02896242 0.02868512 0.02844151 0.02822794 0.0280412  0.02787852
  0.02773749 0.02761601 0.02751226 0.02742468 0.02735196 0.027293
  0.02724688 0.02721291 0.02719056 0.02717772]
 [0.00286901 0.00315649 0.00370335 0.00440595 0.00522548 0.00614658
  0.007166   0.00828695 0.00951551 0.01085696 0.01231003 0.01385519
  0.01542859 0.01686313 0.01776104 0.01793951 0.02206899 0.03889525
  0.12949873 0.19243208 0.27995462 0.30889352 0.19755023 0.14044966
  0.10925212 0.08965714 0.07641055 0.06692623 0.05985649 0.05441791
  0.05012888 0.04667772 0.04385471 0.04151393 0.03955093 0.03788917
  0.03647128 0.03525349 0.03420183 0.03328951 0.03249514 0.03180142
  0.03119419 0.03066173 0.03019428 0.02978361 0.02942272 0.02910564
  0.02882724 0.02858307 0.02836928 0.02818249 0.02801977 0.02787855
  0.0277566  0.02765195 0.02756292 0.02748804 0.02742607 0.02737598
  0.02733691 0.02730819 0.02728932 0.02727851]
 [0.00181297 0.00238365 0.00331567 0.00440021 0.0055991  0.00691361
  0.00836194 0.00997336 0.01178789 0.01385833 0.01625445 0.01906906
  0.02242724 0.02649207 0.03148512 0.03749572 0.04470774 0.05375671
  0.18208981 0.29559719 0.39820416 0.36825408 0.18676491 0.12959535
  0.10079519 0.08296934 0.0710817  0.06262618 0.05634626 0.05152063
  0.04771377 0.04464701 0.04213444 0.04004742 0.03829425 0.03680777
  0.03553773 0.03444575 0.03350201 0.03268296 0.03196975 0.03134707
  0.03080237 0.03032523 0.02990693 0.02954008 0.02921838 0.02893645
  0.02868961 0.0284738  0.02828548 0.02812157 0.02797935 0.02785642
  0.0277507  0.02766037 0.02758382 0.02751969 0.0274668  0.02742417
  0.02739099 0.02736661 0.02735058 0.02733755]]
- channel 4: A 64×64 matrix of zeros with a single non-zero value 0.5 located at position (63, 20)
- channel 5: A 64×64 matrix representing a y-direction load distribution, with a single point load of 0.8660254 located at position (63, 19)

###Task
Based on these five input channels, compute and return the following outputs:
- C_y_hat (float): The compliance (objective value) of the optimized material layout.
- VF_y_hat (float): The volume fraction of the optimized layout you generated, calculated as the proportion of material (1's) in y_hat.

#### Constraints
- The design must minimize compliance.
- All boundary conditions and external loads must be satisfied.
- The volume fraction constraint must not be violated.
- The final outputs should be provided according to the specified field descriptions.