LTPP Computed Parameter: Dynamic Modulus
APPENDIX D: ANN MODEL FACTORS
D.1 INTRODUCTION
The ANN model developed herein contains a mapping ANN architecture and is based on supervised learning. In the developed network, the learning method used is a feed forward back propagation, which is one of the best known types of ANN models. The sigmoidal function, which is shown in equation 101, was chosen as the transfer function. After an indepth investigation of network configurations, it was found that the three–layer network with equal nodes in the first two layers is the most appropriate configuration. All three ANN models share some basic functions, which are shown in equations 101–105. Equations specific to the MR ANN are equations 106–108, and equations specific to the GV ANN and VV ANN models are equations 109–111. For equations 101–111, it should be understood that when a single index is used, it indicates an array. When dual indices are used, they represent a matrix with the first letter indicating the values in the row and the second letter indicating the values in the column. Index i represents the number of input parameters, index k represents the number of nodes in the first hidden layer, and index j represents the number of nodes in the second hidden layer. For the MR ANN, l represents the number of output values. All inputs are scaled to have a value between −1 and 1. The normalization equation is shown in equation 112. Also, note that the MR model produces coefficients for the sigmoidal function and is provided in equation 113. The units of the prediction are in megapascals as compared to pounds per square inch, which has been used elsewhere in this report.
 |
(101) |
 |
(102) |
 |
(103) |
 |
(104) |
 |
(105) |
 |
(106) |
 |
(107) |
 |
(108) |
 |
(109) |
 |
(110) |
 |
(111) |
Where:
T |
= |
The placeholder variable. |
f(T) |
= |
The transfer function. |
 |
= |
The value of the nodes at the first hidden layer. |
 |
= |
The transferred value of the nodes at the first hidden layer. |
 |
= |
The value of the nodes at the second hidden layer. |
 |
= |
The transferred value of the nodes at the second hidden layer. |
 |
= |
The value of output node l (MR ANN). |
 |
= |
The value of the output node (GV ANN and VV ANN). |
 |
= |
The transferred value of the output node l, normalized output (MR ANN). |
 |
= |
The transferred value of the output node, normalized output (GV ANN and VV ANN). |
Pi |
= |
The input variables. |
 |
= |
The weight factors for the first hidden layer. |
 |
= |
The weight factors for the second hidden layer. |
 |
= |
The bias factors for the first layer. |
 |
= |
The bias factors for the second layer. |
 |
= |
The bias factors for the outer layer (MR ANN). |
B0 |
= |
The bias factor for the outer layer (GV ANN and VV ANN). |
(Sigl)max |
= |
Maximum value for sigmoidal coefficient l in the trained data (MR ANN). |
(Sigl)min |
= |
Minimum value for sigmoidal coefficient l in the trained data (MR ANN). |
log |E*|max |
= |
Maximum log|E*| of the trained data (GV ANN and VV ANN). |
log |E*|min |
= |
Minimum log|E*| of the trained data (GV ANN and VV ANN). |
m |
= |
The number of nodes in the first hidden layer (see table 21). |
n |
= |
The number of nodes in the second hidden layer (see table 21). |
 |
(112) |
Where:
Pi |
= |
Given input variable, i. |
 |
= |
Normalized value for given input variable, i. |
MINi |
= |
Minimum value of i in calibration dataset. |
MAXi |
= |
Maximum value of i in the calibration dataset. |
 |
(113) |
Where:
fR |
= |
Reduced frequency (hertz). |
Sig1 |
= |
First sigmoidal function coefficient. |
Sig2 |
= |
Second sigmoidal function coefficient. |
Sig3 |
= |
Third sigmoidal function coefficient. |
Sig4 |
= |
Fourth sigmoidal function coefficient. |
Three ANN models have been developed from this architecture: (1) MR ANN, (2) VV ANN, and (3) |G*| ANN. In the following sections, the value of the weight factors, bias factors, input parameters, and normalization parameters are given for each of these models.
D.2 MR ANN
 |
(114) |
Where:
 |
= |
Resilient modulus at 41 °F (5 °C) (megapascals). |
 |
= |
Resilient modulus at 77 °F (25 °C) (megapascals). |
 |
= |
Resilient modulus at 104 °F (40 °C) (megapascals). |
α1 |
= |
Shift factor coefficient 1 (0.0007). |
α2 |
= |
Shift factor coefficient 2 (-0.1646). |
α3 |
= |
Shift factor coefficient 3 (0.806). |
Table 50. Wik1 matrix elements for MR ANN (part 1).
Element k |
Element i |
1 |
2 |
3 |
1 |
-0.00200156589300513 |
0.00128692125170315 |
0.00284438217881075 |
2 |
0.28516256606761700 |
0.49118675435971600 |
-0.01820933290109810 |
3 |
-0.13012155717600300 |
0.07853842949381770 |
0.17680064334424700 |
4 |
-0.29231959611729100 |
0.17617282979184900 |
0.39775504273252700 |
5 |
-1.41445013310154000 |
6.44297995264973000 |
-1.77482643703439000 |
6 |
-0.21241167223734400 |
0.12821712282974400 |
0.28884600944520300 |
7 |
0.03400370099657090 |
-0.02101635307328680 |
-0.04672053052579720 |
8 |
0.09548185985586570 |
-0.05752189953709870 |
-0.13026426280596500 |
9 |
0.32507208708348800 |
-0.19505459789439700 |
-0.44164369167303300 |
10 |
0.28899439562768500 |
1.14183643175563000 |
1.48347522466600000 |
11 |
-0.98864001747551200 |
-2.86851244653187000 |
-3.23450809016547000 |
12 |
1.62092571912098000 |
-4.58381702398853000 |
2.21293392742914000 |
Table 51. Wik1 matrix elements for MR ANN (part 2).
Element k |
Element i |
4 |
5 |
6 |
1 |
-712.8628402976380000 |
9.83219882009042000 |
-1.94319401017490000 |
2 |
753.53741515309100000 |
7.67671366971338000 |
-0.84350062295273200 |
3 |
-1093.068754047720000 |
9.23892947309088000 |
0.14382032104724800 |
4 |
-1129.557247424060000 |
-1.50590848559491000 |
-0.66955349262730300 |
5 |
-1.83158558608035000 |
3.97456523446851000 |
7.12208818273337000 |
6 |
-839.5292060420890000 |
8.73889517406220000 |
0.67603083268390200 |
7 |
755.74883976821200000 |
-12.71270279414850000 |
-1.34117207352107000 |
8 |
847.61568238744000000 |
-12.59290188069720000 |
-0.47503813284310300 |
9 |
410.33781806362100000 |
-7.80175950866107000 |
-2.36148957427579000 |
10 |
-805.5026910683050000 |
-4.21610143211805000 |
4.07885029040178000 |
11 |
471.04240973832900000 |
13.45559381618210000 |
-3.61642908665572000 |
12 |
28.37591023338010000 |
12.28550446563640000 |
0.70805040715617700 |
Table 52. Bk1 vector elements for MR ANN (transposed for convenience).
Element k |
B1 |
1 |
6.17091399170251000 |
2 |
7.71811183199239000 |
3 |
5.04220257285617000 |
4 |
1.36211474573909000 |
5 |
1.27586815658043000 |
6 |
1.24112853958781000 |
7 |
-0.55920953360997600 |
8 |
-1.54793233282465000 |
9 |
-2.38955526082692000 |
10 |
-5.31749985483251000 |
11 |
6.04452696422904000 |
12 |
2.43088472177158000 |
Table 53. Wkj2 matrix elements for MR ANN (part 1).
Element
j |
Element k |
1 |
2 |
3 |
4 |
1 |
-1.27212265453448000 |
0.98941325757367200 |
-1.90058367559899000 |
-1.69308092227318000 |
2 |
2.24917988742036000 |
-1.47046066978322000 |
2.81430927085362000 |
2.35680695833274000 |
3 |
0.90938132410845800 |
-1.87010703846175000 |
1.72113608938929000 |
0.53365095186003000 |
4 |
0.71420875730581000 |
-2.02463155580309000 |
1.24092642037176000 |
-0.75971225627055200 |
5 |
1.02573805084948000 |
-5.05227027803185000 |
-0.50811201125450400 |
0.12398342947210100 |
6 |
2.87344198982080000 |
1.1825109019882500 |
1.83686226283920000 |
1.07343492432610000 |
7 |
-6.84673965801500000 |
8.75898275822440000 |
-6.07092164029969000 |
-5.61770119661654000 |
8 |
-0.27215636118304200 |
-0.71265020644285800 |
0.10337231975487200 |
-0.50732389147556800 |
9 |
-1.24280778319817000 |
-0.42688724228155600 |
-1.49197352443203000 |
-1.1916709449455000 |
10 |
-0.87186156614646800 |
-0.55253243107747700 |
0.01014336151285740 |
-1.38109064819317000 |
11 |
5.05262125180162000 |
-3.58899791830666000 |
5.32477118646701000 |
6.08771475391962000 |
12 |
1.94315627250074000 |
-11.72058718677100000 |
1.02780825311488000 |
-0.07727872946434780 |
Table 54. Wkj2 matrix elements for MR ANN (part 2).
Element j |
Element k |
5 |
6 |
7 |
8 |
1 |
1.72171850082868000 |
-1.22361303255407000 |
0.26380146379060500 |
1.59052924999161000 |
2 |
-0.69983955029987400 |
2.15750218152474000 |
-2.72092291591448000 |
-2.56417332772541000 |
3 |
5.68303374707809000 |
-0.22567435819557400 |
-0.38683461689596400 |
-1.33743701004047000 |
4 |
-5.29792907293055000 |
-0.79868120409271800 |
0.56639895589096100 |
-0.15145370730044200 |
5 |
-6.9600412768491900 |
-0.25943738604012600 |
-0.07507398274261910 |
-0.60082153969042900 |
6 |
-6.90406457489009000 |
0.66048638607472900 |
-0.74734595052120800 |
-1.77757048725853000 |
7 |
14.28996210073420000 |
-4.93956480500064000 |
5.47678679301124000 |
5.04474379279507000 |
8 |
5.33401793234324000 |
-0.67814558115517500 |
1.35435682469839000 |
0.67517616517050400 |
9 |
6.13017525412482000 |
-0.25895030883091000 |
0.76688050726348400 |
0.59567627875874100 |
10 |
-4.52656667316437000 |
-0.61198000964802400 |
0.66990865283891800 |
1.79648574970781000 |
11 |
61.00473788943800000 |
5.94808205391293000 |
-6.49782178526526000 |
-5.63585227166289000 |
12 |
-0.74313876146962200 |
-0.49502882731012100 |
-0.04343884290084600 |
0.43382323116577400 |
Table 55. Wkj2 matrix elements for MR ANN (part 3).
Element j |
Element k |
9 |
10 |
11 |
12 |
1 |
1.28982526693395000 |
-1.36139827942535000 |
0.81242090814931100 |
0.75421177520166000 |
2 |
-1.98856516178736000 |
2.88665638759158000 |
-2.22345527946482000 |
47.32908703710720000 |
3 |
-1.46019991113224000 |
0.83288304533622300 |
-0.18737329755176000 |
7.94189735755131000 |
4 |
-0.62488286109702500 |
0.99358597783727000 |
-1.80707863873754000 |
-1.51697406306899000 |
5 |
-0.53491093234660500 |
0.03718852912964090 |
0.50412006696695100 |
-2.27324267108375000 |
6 |
-1.75495744905123000 |
0.04848011387017500 |
0.65137190325020200 |
-19.45130367879080000 |
7 |
6.34621810767813000 |
-4.33945264323816000 |
8.48179242696090000 |
-55.79280738270100000 |
8 |
-0.02703184429355360 |
-1.38007437316169000 |
-2.03641139073515000 |
7.15173377918400000 |
9 |
0.18219945071653500 |
-0.43950447308134000 |
0.88864139595275200 |
17.00276947159220000 |
10 |
0.18331101506990700 |
-1.29082311881848000 |
0.17907218849290100 |
0.00442335303199802 |
11 |
-4.70890802094776000 |
5.58102070625218000 |
-4.42512396747533000 |
34.44356640107110000 |
12 |
-0.58348656053904200 |
11.76004227042600000 |
-18.62573141905100000 |
40.36522874277380000 |
Table 56. Bj2 vector elements for MR ANN (transposed for convenience).
Element j |
B2 |
1 |
0.69029100571690900 |
2 |
24.78083328475230000 |
3 |
-14.21641765871200000 |
4 |
0.59454387354854500 |
5 |
0.83441149973420900 |
6 |
2.00770642875969000 |
7 |
-6.80461833224959000 |
8 |
-3.04506035154618000 |
9 |
-2.81699338543501000 |
10 |
-0.01908957707522740 |
11 |
-23.82659200096020000 |
12 |
-2.84950882850698000 |
Table 57. Wjl3 vector elements for MR ANN (part 1).
Element l |
Element j |
1 |
2 |
3 |
4 |
1 |
1.55631817035049000 |
-6.71717964044816000 |
5.52866041884021000 |
-23.13533591138010000 |
2 |
-2.10815560445854000 |
5.53651802240772000 |
-4.20046449739353000 |
13.84767577640070000 |
3 |
10.42186407217690000 |
1.07658778061795000 |
-1.00952254919933000 |
1.35728376824998000 |
4 |
12.53270054290200000 |
-2.82666156182142000 |
0.36140292673336200 |
11.63883002692080000 |
Table 58. Wjl3 vector elements for MR ANN (part 2).
Element l |
Element j |
5 |
6 |
7 |
8 |
1 |
7.89233240566186000 |
3.41508482981355000 |
0.23067944090008000 |
-6.35835335516061000 |
2 |
-4.35253958227055000 |
-4.33105623723765000 |
-0.23450370767118300 |
4.90555584894330000 |
3 |
-0.53624851772844700 |
4.26302489876063000 |
0.18477243720173400 |
0.95437352582270800 |
4 |
-5.27378018192719000 |
12.58074218873790000 |
0.30883922900382400 |
-0.81566323458280500 |
Table 59. Wjl3 vector elements for MR ANN (part 3).
Element l |
Element j |
9 |
10 |
11 |
12 |
1 |
3.12448098517872000 |
7.76943031096589000 |
0.16915092735543200 |
17.81482352191000000 |
2 |
-4.10144987129725000 |
-8.41207948330685000 |
-0.13394241862122600 |
-14.80599826102690000 |
3 |
4.33224573662362000 |
-12.80943716195050000 |
0.05065733872668660 |
-2.21601216037733000 |
4 |
12.62825381229430000 |
2.74268434100584000 |
0.08908525498240080 |
8.18232817352028000 |
Table 60. Normalization parameters for MR ANN.
Parameter |
Maximum |
Minimum |
MR5 (MPa) |
34053.0 |
4800.3 |
MR25 (MPa) |
15411.0 |
1081 |
MR40 (MPa) |
6863.7 |
378.9 |
α1 (1/ °C2) |
0.002400 |
-0.000194 |
α2 (1/ °C) |
-0.098 |
-0.300 |
α3 |
1.430 |
0.490 |
Sig1 (MPa) |
2.660 |
-0.043 |
Sig2 |
4.700 |
1.500 |
Sig3 |
4.100 |
0.650 |
Sig4 |
0.850 |
0.260 |
 |
(115) |
D.3 VV ANN
 |
(116) |
Where:
f |
= |
Frequency, Hz. |
η |
= |
Viscosity, 109 P (108 Pas). |
Table 61. Wik1 matrix elements for VV ANN.
Element k |
Element i |
1 |
2 |
3 |
4 |
1 |
0.00275807935276415 |
9.65414347487492000 |
2.03545176382323000 |
9.60483606710040000 |
2 |
0.00699289293419363 |
23.57848634613580000 |
-14.68700301452580000 |
-7.85111160151172000 |
3 |
-29.85518987292190000 |
-0.01748605711868700 |
-0.05226790767927880 |
0.10630299509262800 |
4 |
0.00272712204182532 |
0.22756591526942600 |
-0.44128805965281300 |
0.51123020782505100 |
5 |
0.00885326169582901 |
2.81905088512526000 |
-16.74274657827420000 |
-11.51343150553350000 |
6 |
-0.00604496464330366 |
-3.83323067510033000 |
-3.76213295881730000 |
-4.29370215749809000 |
7 |
0.00001271343965432 |
178.92786633284100000 |
-0.00411385898011919 |
0.00960133688419267 |
8 |
0.01990111297026090 |
-1.16149781971796000 |
3.28168963629711000 |
7.53282726564379000 |
9 |
0.01005974523187870 |
-0.39351960943073500 |
-2.51237935759967000 |
-4.29116433585095000 |
10 |
2.20552340599479000 |
-0.04056518032735070 |
0.20605359978971700 |
0.15739161949478100 |
11 |
-0.02404084443474640 |
-6.58972054945535000 |
-0.71981980149770700 |
1.52555084062215000 |
12 |
-0.00480809655510197 |
-0.27885745601162500 |
0.20260632406742000 |
-0.77108867937747000 |
13 |
0.00078471111967578 |
28.44415144259920000 |
-0.87571840373706600 |
1.95186725224564000 |
14 |
0.00771761131363787 |
5.33676984624904000 |
-11.97622550219350000 |
7.71358914882133000 |
Table 62. Bk1 vector elements for VV ANN (transposed for convenience).
Element k |
B1 |
1 |
15.74835436269630000 |
2 |
12.95405722517280000 |
3 |
-32.89896917533040000 |
4 |
-1.10052248959346000 |
5 |
-10.78091133520380000 |
6 |
-6.05302142493261000 |
7 |
179.14709718593800000 |
8 |
0.65710599534106200 |
9 |
0.97891317883012100 |
10 |
4.72223785338294000 |
11 |
-6.55165918673815000 |
12 |
0.16731428319782600 |
13 |
27.93047751952220000 |
14 |
-1.37408939580495000 |
Table 63. Wkj2 matrix elements for VV ANN (part 1).
Element j |
Element k |
1 |
2 |
3 |
4 |
1 |
0.10511744512798600 |
0.10087889704665500 |
-6.08319321387257000 |
-6.59951922338895000 |
2 |
0.48485201906871100 |
-0.98061150567901500 |
5.28213726518926000 |
-0.99307488071221600 |
3 |
0.38666686767509600 |
0.14407696249404200 |
-0.64821237624430900 |
-0.68553604874521800 |
4 |
-0.29103025921087400 |
-0.12300943951262000 |
0.20660407620275500 |
0.84431006573009900 |
5 |
7.20145641626984000 |
151.58382852395500000 |
-35.04225274891410000 |
50.31324421340920000 |
6 |
57.80869276380200000 |
45.53819717827600000 |
34.31952056143690000 |
24.37102641478090000 |
7 |
0.04766296493976920 |
-0.00042062272941119 |
-1.64780015541921000 |
-0.14767680975504800 |
8 |
1.33157778260189000 |
-0.00864139577441407 |
-10.53694387134640000 |
-28.77316964591470000 |
9 |
0.95432357054777500 |
-1.02389021831262000 |
46.82499405300610000 |
61.27640626961720000 |
10 |
0.09476608746471290 |
-0.04013797559617900 |
15.08416115539120000 |
1.26248388850460000 |
11 |
0.65673641456535000 |
25.30163159128400000 |
6.86855242012598000 |
52.55865424871390000 |
12 |
-57.90977231272230000 |
57.64668837199080000 |
-36.95301005956680000 |
-51.19423424496710000 |
13 |
2.27010107107278000 |
-2.76393198240337000 |
14.26953262603660000 |
20.57219630346000000 |
14 |
-0.81442427169913900 |
-1.58075985235425000 |
20.32130385637230000 |
-21.37864285760230000 |
Table 64. Wkj2 matrix elements for VV ANN (part 2).
Element j |
Element k |
5 |
6 |
7 |
8 |
1 |
0.10958125840908900 |
0.17103742786951300 |
7.53738934561000000 |
0.32157150848996400 |
2 |
1.10538840213556000 |
-0.06005243494707140 |
-0.46816179656466200 |
-0.11362528740921000 |
3 |
-0.03668587237246070 |
-0.20195215673052800 |
0.50583875586658800 |
0.04117830356601680 |
4 |
0.04165310223654260 |
0.12412276818401500 |
-0.71560911337291800 |
-0.06643717805416860 |
5 |
-135.8256006242760000 |
4.39896372970834000 |
2.12048232621756000 |
1.61468697764029000 |
6 |
12.18509505034930000 |
4.51309256843370000 |
7.19546156774681000 |
11.77248078904480000 |
7 |
0.00470501377909039 |
-0.00810109614328989 |
-0.20468271977533900 |
-0.00831305598572143 |
8 |
0.14429797312686300 |
0.32443645525838900 |
-18.31175213933740000 |
-0.17988695146765000 |
9 |
0.48668670982841300 |
-1.06169701219564000 |
-0.47416257725070700 |
2.01336853261287000 |
10 |
0.02124957257916830 |
-0.01491154156235320 |
214.20333131224300000 |
0.05344273444888970 |
11 |
-16.55295761437330000 |
31.36367340855210000 |
0.14069168225602600 |
20.67506639536700000 |
12 |
105.32395565518000000 |
-68.59510275333000000 |
-0.57180295407110300 |
24.60816018017130000 |
13 |
3.20913352965092000 |
-1.46922615656989000 |
-0.45618908696155800 |
-0.74587534667358800 |
14 |
2.14653394183588000 |
2.86656902371875000 |
-1.38951291360800000 |
1.51498306935364000 |
Table 65. Wkj2 matrix elements for VV ANN (part 3).
Element j |
Element k |
9 |
10 |
11 |
12 |
1 |
0.67436953255572400 |
1.52993707048353000 |
-1.56378559364968000 |
-5.97316328323617000 |
2 |
0.83884786099458000 |
0.72056843262828500 |
0.33823343591634200 |
0.32995739963357400 |
3 |
0.13129904885190400 |
0.03393257336078320 |
-0.33495422276929500 |
1.21059369021165000 |
4 |
-0.08374413143914640 |
-0.14011290475212000 |
0.40330307484415800 |
-0.92068329846481200 |
5 |
-4.74183346589286000 |
-7.18458646900432000 |
-8.20175738069755000 |
21.46322182793860000 |
6 |
-2.22050947171933000 |
-3.19929780229875000 |
43.60123067158340000 |
-38.15447962643450000 |
7 |
0.04970822735474000 |
0.38186826752702800 |
0.02975701201504030 |
-0.07187684022207080 |
8 |
18.96102127708150000 |
0.34577779423701500 |
0.10112166202643200 |
-4.34637124684574000 |
9 |
136.10585968855900000 |
3.75489444791017000 |
5.73368672918881000 |
35.37279289616110000 |
10 |
-0.36747370936357000 |
1.90705035404089000 |
-1.42366525777415000 |
6.11736464999887000 |
11 |
-20.86114381473210000 |
4.50957253702840000 |
0.73758107479932900 |
-3.21435115847231000 |
12 |
-19.00772620196210000 |
44.26713771722670000 |
38.62379852233980000 |
-1.87378784809568000 |
13 |
0.17584310006536200 |
3.18528344468959000 |
0.62107582578786300 |
8.04752096583409000 |
14 |
2.97468553092343000 |
-4.21086927843771000 |
2.73490619946452000 |
-3.67293829703328000 |
Table 66. Wkj2 matrix elements for VV ANN (part 4).
Element j |
Element k |
13 |
14 |
1 |
-1.12245731810529000 |
0.67867713986347200 |
2 |
0.41929338131065000 |
-1.37665914977400000 |
3 |
0.58160889599262400 |
0.36777315706927400 |
4 |
-0.51666057998473900 |
-0.35135512626691600 |
5 |
3.27948738968729000 |
1.94926465685885000 |
6 |
-45.89506402313440000 |
70.49683570639200000 |
7 |
0.01612131389179630 |
-0.01152122049407600 |
8 |
3.77564707007114000 |
-4.91896963383303000 |
9 |
3.89920658003942000 |
-49.40155143998830000 |
10 |
2.24644797717875000 |
-0.11299018277251500 |
11 |
-9.63227261518900000 |
-119.1073499783990000 |
12 |
-41.84530118313840000 |
-35.82982088240670000 |
13 |
0.56754251531620400 |
-3.01209559093347000 |
14 |
2.43256581696669000 |
-1.47837832821775000 |
Table 67. Bj2 vector elements for VV ANN (transposed for convenience).
Element j |
B2 |
1 |
-12.23193846569030000 |
2 |
0.61294509854138200 |
3 |
-2.26121589380006000 |
4 |
1.99708243932415000 |
5 |
22.07400990717020000 |
6 |
-31.86954422722210000 |
7 |
-2.41399982513437000 |
8 |
-55.27590960198670000 |
9 |
-106.0101552356340000 |
10 |
-30.89403826224180000 |
11 |
-76.24782342311330000 |
12 |
69.28157710468700000 |
13 |
20.21945409941280000 |
14 |
6.12083493792229000 |
Table 68. Wj3 vector elements for VV ANN.
Element j |
W3 |
1 |
1.15373219918479000 |
2 |
-6.43559921490721000 |
3 |
-27.68360314597900000 |
4 |
-29.84574981371640000 |
5 |
-0.19407309170597600 |
6 |
0.11995708036519800 |
7 |
37.96893504107600000 |
8 |
34.00719477253400000 |
9 |
0.43014683810727000 |
10 |
58.47886817741680000 |
11 |
11.94908594315590000 |
12 |
-0.07969689689272150 |
13 |
1.28313108040340000 |
14 |
0.53296310828566400 |
Table 69. Normalization parameters for VV ANN.
Parameter |
Maximum |
Minimum |
Frequency (Hz) |
25 |
0.01 |
Viscosity (109 P) |
1.99 x 10-6 |
27.00 |
VMA (percent) |
34.64 |
9.51 |
VFA (percent) |
95.07 |
32.82 |
|E*| (psi) |
6.77 |
3.52 |
 |
(117) |
D.4 |G*|-BASED ANN (GV ANN)
 |
(118) |
Table 70. Wik1 matrix elements for GV ANN.
Element k |
Element i |
1 |
2 |
3 |
1 |
-0.00024607061578008 |
0.39573112252617300 |
0.03137522820868100 |
2 |
0.06369415599503260 |
1.20396485280366000 |
-0.41366999743116300 |
3 |
0.12560166488947000 |
12.23092512080170000 |
-15.40428508265840000 |
4 |
144.78479285235900000 |
0.03602175009356860 |
-0.01486430375847190 |
5 |
-0.88590000706402800 |
-6.64620868848810000 |
6.76784019433308000 |
6 |
-0.24480066614146600 |
-175.126570586870000 |
110.38396928012700000 |
7 |
0.35658487276453800 |
-1.13633711409192000 |
-2.69698652853612000 |
8 |
0.05160380299493010 |
0.96914775793747200 |
-0.45058047297787600 |
9 |
-0.11763211836169300 |
-2.07308576925671000 |
0.85862809150608600 |
10 |
-0.10260986412141000 |
-1.89647330970902000 |
0.63576557888985500 |
11 |
0.83832125440623900 |
5.01864835388400000 |
-7.07830843630593000 |
12 |
-22.27016740128570000 |
2.82680472067850000 |
-1.10402495920697000 |
Table 71. Bk1 vector elements for GV ANN (transposed for convenience).
Element k |
B1 |
1 |
-13.23020275429390000 |
2 |
-1.15912963101547000 |
3 |
4.49668100338684000 |
4 |
144.90818089626700000 |
5 |
0.92374996199841500 |
6 |
187.12386013330800000 |
7 |
-0.64852041864182200 |
8 |
0.79297555520979200 |
9 |
1.27505357580263000 |
10 |
1.56260322810567000 |
11 |
-1.80964836750601000 |
12 |
-24.23180989095970000 |
Table 72. Wkj2 matrix elements for GV ANN (part 1).
Element j |
Element k |
1 |
2 |
3 |
4 |
1 |
1.68662477353906000 |
0.81911931325403900 |
-0.03005180291517210 |
0.00569345005879186 |
2 |
-4.86360723468991000 |
105.69622449695400000 |
-2.47849339660309000 |
0.18455777935409300 |
3 |
91.15855920616590000 |
92.66954611901590000 |
-40.91260113059330000 |
-1.67503899696897000 |
4 |
106.61660556715000000 |
22.90152433894880000 |
229.56133863086000000 |
-0.21741320127254800 |
5 |
0.73120568649091500 |
-2.04683544732935000 |
0.02907899740128110 |
-0.00680956721358145 |
6 |
20.01334655907270000 |
-124.1136850495880000 |
-12.93483076443820000 |
0.16320420801843400 |
7 |
11.03261781901980000 |
-55.35942281688430000 |
-0.25840813426995300 |
2.36249059396197000 |
8 |
6.30522135450740000 |
-72.37646707852670000 |
-2.13173103347156000 |
23.41004001398310000 |
9 |
-34.10766755925970000 |
26.04890333595410000 |
0.13622620891216500 |
-534.4781944799400000 |
10 |
-18.68321667137760000 |
103.90356571837300000 |
-2.46159833726818000 |
0.18433171253622200 |
11 |
19.14302423015890000 |
3.59998964741180000 |
0.25399603119775500 |
-44.54673896061580000 |
12 |
-0.33713658604817700 |
12.87184727785630000 |
0.13134300289504600 |
6.54659549746008000 |
Table 73. Wkj2 matrix elements for GV ANN (part 2).
Element j |
Element k |
5 |
6 |
7 |
8 |
1 |
0.01859496903873790 |
7.79141326044631000 |
0.01459371336584290 |
-4.01406752777277000 |
2 |
30.71090633980400000 |
5.37671972945662000 |
3.18886351743859000 |
-2.85108340883822000 |
3 |
87.71359895456880000 |
-77.08967734352910000 |
22.84350940748950000 |
335.75204711704400000 |
4 |
68.74692689690920000 |
-45.87194063996940000 |
12.79313464511960000 |
-37.77625414085690000 |
5 |
-0.0094809359102436 |
-1.94014227142969000 |
-0.01256576806807960 |
4.16910051124176000 |
6 |
-37.35043967525970000 |
5.27365300784224000 |
1.31038988899583000 |
43.16167413535480000 |
7 |
91.10288269578740000 |
-29.48793850026970000 |
-0.04849305692638800 |
3.23868144074583000 |
8 |
3.53501265914675000 |
0.83322882269153700 |
1.62246929641914000 |
6.69105720723216000 |
9 |
-0.09814064912135180 |
-0.02099590067358970 |
-0.19796523923434900 |
17.65358100702660000 |
10 |
55.61599363732450000 |
-5.35164182127019000 |
3.15862176137102000 |
-2.82977620499617000 |
11 |
-0.30334853272738200 |
43.07269450273440000 |
-0.22059949976226000 |
1.05780537005262000 |
12 |
0.11377535318423100 |
0.11441693675638400 |
0.15289220845030100 |
-1.13970260089778000 |
Table 74. Wkj2 matrix elements for GV ANN (part 3).
Element j |
Element k |
9 |
10 |
11 |
12 |
1 |
3.47596025162077000 |
-3.82827454649034000 |
0.01169888530415500 |
0.12954405222771200 |
2 |
94.55869709663890000 |
-7.34809568152935000 |
10.41344790169180000 |
21.32866301200200000 |
3 |
205.23211172111800000 |
-34.98627953103440000 |
76.29312010940660000 |
-30.59094661558560000 |
4 |
64.60376447639590000 |
-21.68091976305070000 |
66.41011966659030000 |
-26.22110679142560000 |
5 |
0.08966925765449130 |
-0.90205636358616400 |
-0.00585495017589258 |
0.00011267457317091 |
6 |
12.38797404837280000 |
-96.26501641581090000 |
-37.17032464251540000 |
0.12957951502862800 |
7 |
-50.14112337006290000 |
-8.38137267965061000 |
40.93959435883270000 |
-0.32474898284932000 |
8 |
-55.82517744922910000 |
-57.75954730123160000 |
2.17269724264550000 |
-54.42459642341290000 |
9 |
-1.13861767294711000 |
6.61476050123734000 |
0.25450956423993600 |
-1.38236982388117000 |
10 |
93.41332136138760000 |
-9.48311646261710000 |
49.01504550800940000 |
20.79054924536240000 |
11 |
2.55740586147654000 |
2.61695086136455000 |
-0.09940520779407240 |
3.00291187179692000 |
12 |
4.67064477635346000 |
8.20560279728423000 |
0.10724854309492400 |
3.33502620583327000 |
Table 75. Bj2 vector elements for GV ANN (transposed for convenience).
Element j |
B2 |
1 |
-3.02263601315923000 |
2 |
5.97704828771739000 |
3 |
-91.56648216379910000 |
4 |
-106.5989770883490000 |
5 |
-0.36740268358330700 |
6 |
-20.04954785032860000 |
7 |
-10.42449435680280000 |
8 |
-5.32258083806398000 |
9 |
32.81032420706510000 |
10 |
17.56993360741350000 |
11 |
-20.65675932973890000 |
12 |
2.35956011260892000 |
Table 76. Wj3 vector elements for GV ANN.
Element j |
W3 |
1 |
-134.7651195030360000 |
2 |
92.96416099463190000 |
3 |
0.07847134028534860 |
4 |
0.58725325729413100 |
5 |
-146.9138139427070000 |
6 |
-12.15444670634670000 |
7 |
34.34304769221520000 |
8 |
-63.09541362586340000 |
9 |
-76.62381419958460000 |
10 |
-93.21865080014820000 |
11 |
-29.95120632059920000 |
12 |
0.71757242259101300 |
Table 77. Normalization parameters for GV ANN.
Parameter |
Maximum |
Minimum |
|G*| (psi) |
676,000 |
0.0293 |
VMA (percent) |
22.21 |
9.51 |
VFA (percent) |
95.07 |
32.82 |
|E*| (psi) |
6.81 |
3.52 |
 |
(119) |