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.
![Equation 101. Definition of sigmoidal transfer function used in ANN model. f parenthesis T end parenthesis equals 2 divided by the sum of 1 plus exponential of the product of superscript −2 times T, all minus 1.](images/equation101.gif) |
(101) |
![Equation 102. Definition of vector H for the value of the nodes at the first hidden layer. H superscript 1subscript k equals B subscript k, superscript 1 plus the summation from i equals 1 to m of the product of matrix W for W superscript 1subscript ik and vector P for the normalized value for given input variable i, P hat subscript 1.](images/equation102.gif) |
(102) |
![Equation 103. Definition of vector H hat for the transferred value of the nodes at the first hidden layer. H hat superscript 1 subscript k equals f times parenthesis H superscript 1 subscript k end parenthesis.](images/equation103.gif) |
(103) |
![Equation 104. Definition of vector H for the value of the nodes at the second hidden layer. H superscript 2 subscript j equals B superscript 2 subscript j plus the summation from k equals 1 to n of the product of H hat superscript 1 subscript k and matrix W for W superscript 2 subscript kj.](images/equation104.gif) |
(104) |
![Equation 105. Definition of vector H hat for the transferred value of the nodes at the second hidden layer. H hat superscript 2 subscript j equals f times parenthesis H superscript 2 subscript j end parenthesis.](images/equation105.gif) |
(105) |
![Equation 106. Definition of vector H for the value of output node. H superscript 3 subscript l equals B superscript 0 subscript l plus the summation from j equals 1 to n of the product of vector H hat for H hat superscript 2 subscript j and matrix W for W superscript 3 subscript jl.](images/equation106.gif) |
(106) |
![Equation 107. Definition of vector H hat for the transferred value of the output node. H hat superscript 3 subscript l equals f parenthesis H superscript 3 subscript l end parenthesis.](images/equation107.gif) |
(107) |
![Equation 108. Definition of trained output for MR ANN model. Sig subscript l equals the sum of parenthesis H hat superscript 3 subscript l plus 1 end parenthesis, parenthesis parenthesis Sig subscript l end parenthesis subscript max minus parenthesis Sig subscript l end parenthesis subscript min end parenthesis, divided by 2, all plus parenthesis Sig subscript l end parenthesis subscript min.](images/equation108.gif) |
(108) |
![Equation 109. Definition of vector H for the value of the output node. H superscript 3 equals B superscript 0 plus the summation from j equals 1 to n of the product of H hat superscript 2 subscript j and matrix W for W superscript 3 subscript j.](images/equation109.gif) |
(109) |
![Equation 110. Definition of vector H hat for the transferred value of the output node. H hat superscript 3 equals f times parenthesis H superscript 3 end parenthesis.](images/equation110.gif) |
(110) |
![Equation 111. Definition of trained output for VV and GV ANN models. The logarithm base 10 of vertical line E superscript star vertical line equals the sum of parenthesis H hat superscript 3 plus 1 end parenthesis times parenthesis the logarithmic base 10 of vertical line E superscript star vertical line subscript max minus the logarithmic base 10 of vertical line E superscript star vertical line subscript min end parenthesis divided by 2, all plus the logarithmic base 10 of vertical line E superscript star vertical line subscript min.](images/equation111.gif) |
(111) |
Where:
T |
= |
The placeholder variable. |
f(T) |
= |
The transfer function. |
![Value of the nodes at the first hidden layer](images/h1k.gif) |
= |
The value of the nodes at the first hidden layer. |
![Transferred value of the nodes at the first hidden layer](images/H1Kb.gif) |
= |
The transferred value of the nodes at the first hidden layer. |
![Value of the nodes at the second hidden layer](images/H2J.gif) |
= |
The value of the nodes at the second hidden layer. |
![Transferred value of the nodes at the second hidden layer](images/H2Jb.gif) |
= |
The transferred value of the nodes at the second hidden layer. |
![Value of output node l (MR ANN)](images/H3L.gif) |
= |
The value of output node l (MR ANN). |
![Value of the output node (GV ANN and VV ANN)](images/h3.gif) |
= |
The value of the output node (GV ANN and VV ANN). |
![Transferred value of the output node l, normalized output (MR ANN)](images/h3lb.gif) |
= |
The transferred value of the output node l, normalized output (MR ANN). |
![Transferred value of the output node, normalized output (GV ANN and VV ANN)](images/H3b.gif) |
= |
The transferred value of the output node, normalized output (GV ANN and VV ANN). |
Pi |
= |
The input variables. |
![Weight factors for the first hidden layer](images/w1ik.gif) |
= |
The weight factors for the first hidden layer. |
![Weight factors for the second hidden layer](images/w2kj.gif) |
= |
The weight factors for the second hidden layer. |
![Bias factors for the first layer](images/b1k.gif) |
= |
The bias factors for the first layer. |
![Bias factors for the second layer](images/b2j.gif) |
= |
The bias factors for the second layer. |
![Bias factors for the outer layer (MR ANN)](images/b0l.gif) |
= |
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). |
![Equation 112. Definition of normalized value for given input variable. P hat subscript i equals the product of 2 times parenthesis P subscript i minus MIN subscript i end parenthesis divided by parenthesis MAX subscript i minus MIN subscript i end parenthesis, all minus 1.](images/equation112.gif) |
(112) |
Where:
Pi |
= |
Given input variable, i. |
![Normalized value for given input variable, i](images/normalized_value.gif) |
= |
Normalized value for given input variable, i. |
MINi |
= |
Minimum value of i in calibration dataset. |
MAXi |
= |
Maximum value of i in the calibration dataset. |
![Equation 113. Definition of trained output for VV and GV ANN models. The logarithm base 10 of vertical line E superscript star vertical line equals Sig subscript 1 plus Sig subscript 2 divided by the sum of 1 plus 1 divided by the exponential of superscript Sig subscript 3 plus Sig subscript 4 times the logarithmic base 10 of parenthesis f subscript R end parenthesis.](images/equation113.gif) |
(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
![Equation 114. Input variable vector for resilient modulus based model in terms of mixture properties. P subscript i equals 1x6 vector where the vector element is equal to parenthesis M superscript 5subscript R, M superscript 25 subscript R, M superscript 40 subscript R, alpha subscript 1, alpha subscript 2, and alpha subscript 3 end parenthesis.](images/equation114.gif) |
(114) |
Where:
![Resilient modulus at 41° F (5 °C), MPa](images/m5r.gif) |
= |
Resilient modulus at 41 °F (5 °C) (megapascals). |
![Resilient modulus at 77° F (25 °C), MPa](images/m25r.gif) |
= |
Resilient modulus at 77 °F (25 °C) (megapascals). |
![Resilient modulus at 104° F (40 °C), MPa](images/m40r.gif) |
= |
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 |
![Equation 115. The bias factors for the outer layer vector for resilient modulus based model in terms of coefficient values. B superscript 0 subscript l equals 1x4 vector where the vector element is equal to parenthesis 12.88483251257, −10.23705702365, −1.957618903369, and 3.273305236416 end parenthesis.](images/equation115.gif) |
(115) |
D.3 VV ANN
![Equation 116. Input variable vector for viscosity based model in terms of binder and mixture properties. P subscript i equals 1x4 vector where the vector element is equal to parenthesis f, eta, VMA, and VFA end parenthesis.](images/equation116.gif) |
(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 |
![Equation 117. The bias factor for the outer layer for viscosity based model in terms of coefficient value. B superscript 0 equals −3.484481025467.](images/equation117.gif) |
(117) |
D.4 |G*|-BASED ANN (GV ANN)
![Equation 118. Input variable vector for binder shear modulus based model in terms of binder and mixture properties. i, P subscript i equals to 1x3 vector, where the vector element is equal to parenthesis vertical line G superscript star vertical line, VMA, and VFA end parenthesis.](images/equation118.gif) |
(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 |
![Equation 119. The bias factor for the outer layer for binder shear modulus based model in terms of coefficient value. B superscript 0 equals −8.469734576039.](images/equation119.gif) |
(119) |