digital image correlation analysis of displacement based...

Research ArticleDigital Image Correlation Analysis of Displacement Based onCorrected Three Surface Fitting Algorithm

Tong-bin Zhao 123 Wei Zhang123 and Wei-yao Guo 123

1State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Provinceand the Ministry of Science and Technology Shandong University of Science and Technology Qingdao 266590 China2College of Mining and Safety Engineering Shandong University of Science and Technology Qingdao 266590 China3National Demonstration Center for Experimental Mining Engineering EducationShandong University of Science and Technology Qingdao 266590 China

Correspondence should be addressed to Wei-yao Guo 363216782qqcom

Received 8 July 2019 Accepted 9 September 2019 Published 30 September 2019

Academic Editor Sarangapani Jagannathan

Copyright copy 2019 Tong-bin Zhao et alis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

SFA (Surface Fitting Algorithm) for continuous displacement is an important method for digital image correlation with antinoiseability and computational eciency advantages in practical applications In order to improve the algorithm accuracy and expandits application range this paper tries to improve the SFA and studies the modied cubic surface tting algorithm CTSFA(Corrected ree Surface Fitting Algorithm) which is suitable for solving the initial value of continuous displacement Bilinearinterpolation and adjacent interpolation are used to analyze the gray level at any integer-pixel position in the displacement matrixand the weight coecient is givene distance-weighted method is used to approximate the true initial displacement value of thecontinuum and the algorithm suitable for digital image processing is extended to the continuum displacement solutione cubicsurface expression of the CTSFA programmatic application is solved by the least squares method and the correlation coecient ofthe power basis function is calculated In the computer simulation of speckle test the comparison between CTSFA and SFA on thecalculation results of linear and nonlinear displacement elds shows that the calculated amount of CTSFA is basically the same asthat of SFA but the calculation accuracy is doubled e study of analysing the Brazilian splitting test using CTSFA and SFAreveals that CTSFA is better than SFA in observing the development of cracks

1 Introduction

As a simple and direct method of calculating displacementDigital Image Correlation (DIC) has the advantages ofnoncontact full-eld measurement simple experimentalequipment and low environmental requirements which canuse an objectrsquos own surfacersquos natural texture to obtainspeckle images of deformation [1ndash5] After the DIC methodwas rst proposed [6 7] numerous scholars conductedresearch on the accuracy and computational eciency ofdierent algorithms in-depth and the advantages and dis-advantages of dierent methods had been pointed out[8ndash12]

SFA is one of themost commonly usedmethods for DICIt performs surface tting on the correlation coecient

matrix obtaining the subpixel displacement of the point onthe surface by tting the correlation coecient In theorythe method has been researched from dierent levels indepth [13ndash15] and the displacement positioning accuracy ofthe iniexcluence of various iniexcluencing factors has been eval-uated through experiments [16ndash19] In practical applica-tions matching the center of the subarea window bycalculating the correlation coecient through the SFA islimited to the integer pixel precision without locating at theactual displacement point which resulted in the calculationof the correlation coecient is not accurate enough Inresponse to this problem the initial displacement of SFA hasbeen tried to solve by using linear interpolation and gradientmethod to obtained high precision continuum displacement[20ndash23]

HindawiComplexityVolume 2019 Article ID 4620858 9 pageshttpsdoiorg10115520194620858

-is paper introduces the concept of distance weightingin digital image processing to modify SFA of initial dis-placement determination by CTSFA CTSFA adopts the graylevel interpolation algorithm which combines the nearestneighbor interpolation with bilinear interpolation Bymoving the fitting center and calculating the weightedweights the points participating in the surface fitting cal-culation become closer to the actual initial point For thecubic surface fitting function of CTSFA the least squaremethod is used to solve the extreme coordinates of thecorrelation coefficient -e accuracy of the algorithm isverified by computer-generated simulated speckle imagesand cracked Brazilian disc tests before application

2 Algorithm Principle

21 Basic eory of SFA Based on the actual data the SFAobtains the analytical expression between the functionf(x y) and the variables x and y and it can pass or ap-proximate all data points which can be distributed in thefunction f(x y) on the spatial surface represented Takingthe two polynomials as an example the fitting function is[24]

C(x y) a0 + a1x + a2y + a3x2

+ a4xy + a5y2 (1)

-e function C(x y) should satisfy the equations at theextreme points of the quadric

zC(x y)

zx a1 + 2a3x + a4y 0

zC(x y)

zy a2 + 2a5y + a4x 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

-en according to equation (2) [24] the position of theextreme point of the fitting surface is obtained [24]

x 2a1a5 minus a2a4

a24 minus 4a3a5

y 2a2a3 minus a1a4

a24 minus 4a3a5

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)

After finding the location of the extreme point thedifference between the coordinates of the initial value pointand the initial value point is a continuous displacement-erefore obtaining high-precision initial displacement andcalculating the coefficients of polynomial fitting function arethe two important of CTSFA

22 Determination of Initial Value of CTSFA DisplacementAt first CTSFA calculates the interpolation results of bi-linear nearest neighbor points and the gray variance of fourneighbor pixels around the interpolation point which basedon the traditional linear interpolation method when de-termining the initial displacement value -en it constructsthe weighting coefficients through the gray variance Finallythe two interpolation results are weighted together to get the

final interpolation results CTSFA not only considers thedistance between interpolation points and neighboringpoints but also takes the gray distribution characteristics ofneighboring points into account which can improve thesubpixel accuracy of initial displacement effectively

-e specific steps for determining initial values of dis-placement in CTSFA are as follows

(1) Calculating bilinear interpolation Gd(x y) and ad-jacent interpolation GN(x y) of the interpolationpoints (x y)As shown in Figure 1 bilinear interpolation uses thegray-scale weighted interpolation of four neighbor-ing points as the gray-scale value of the point whichcan be decomposed into two one-dimensional linearinterpolations expressed by

f R1( 1113857 x2 minus x

x2 minus x1f Q11( 1113857 +

x minus x1

x2 minus x1f Q21( 1113857

f R2( 1113857 x2 minus x

x2 minus x1f Q12( 1113857 +

x minus x1

x2 minus x1f Q22( 1113857

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(4)

f(P) y2 minus y

y2 minus y1f R1( 1113857 +

y minus y1

y2 minus y1f R2( 1113857 (5)

Among the four adjacent pixels of the pixel whichwill be sought the nearest neighbor pixel gray isassigned Define i + u and j + v (i and j are positiveintegers u and v are decimal greaters whose valuemore than 0 as well as less than 1) are the nonintegerpixel coordinates the value of the pixel whose grayf(i + u j + v) is solved and shown in Figure 2If (i + u j + v) falls in area A that means ult 05 andvlt 05 the gray value of the pixel in the upper leftcorner is assigned to the pixel to be sought Similarlythe falling in the B area gives the upper right cornerand so on

(2) Calculating the standard deviationH of four adjacentpoints fi(x y) (i 1 2 3 4) around the in-terpolation point and H is expressed by

H

14

1113944

4

i1fi(x y) minus f(x y)1113960 1113961

2

11139741113972

(6)

-e f(x y) is expressed by

f(x y) 14

1113944

4

i1fi(x y) (7)

(3) Weighting fusion of Gd(x y) and Gn(x y) isexpressed by equation (8) [25 26]

2 Complexity

g(x y) Wd times Wd(x y) + Wπ times Wπ(x y) (8)

In the equation (8)Wd andWn are weighted coefficientsand calculated by nonlinear weighting which can be shownby

Wd 12

e((minus H)100)

+ 05

Wπ 1 minus Wd

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)

-e interpolation result g(x y) is the initial value of theactual displacement required

In Figure 3 linear interpolation in x direction is carriedout at four points a1 a2 a3 and a4 to get A1 and A2 andcarried out at two points A1 and A2 to get A in y directionand the gray value of A is the result of bilinear interpolationGd(x y) After the result of four-point interpolation wascalculatedGn(x y) is got though the four points of b1 b2 b3and b4 interpolate adjacent using the above method and theGn(x y) is the point B in Figure 3 -en theWd andWn arecalculated by using point A and point B Finally G(x y)

(point C in Figure 3) is obtained through equations (8)-(9)and the point D is the true initial displacement -e flowchart of CTSFA calculation is shown in Figure 4

23 Calculation of CTSFA Fitting In order to improve thefitting effect of polynomial function this paper adopts three

times surface fitting function -e correlation coefficient ofthe target matching point (x y) and the points in sur-rounding neighborhood which are commonly used by anycubic surface function can be expressed by power basisfunction by

f(x y) a0 + a1x + a2y + a3x2

+ a4xy + a5y2

+ a6x3

+ a7x2y + a8xy

2+ a9y

3

(10)

In order to obtain the maximum value of the correlationcoefficient and the corresponding coordinates it is necessaryto calculate the cubic surface fitting polynomial -e leastsquares method is an approximation theory that the surfaceis generally not obtained by a known point but by mini-mizing the sum of the squares of the difference between the

x

y

a1a2

a3a4

A1

A2

A

b1

b2 b3

b4

B

Bilinear interpolation weight

Adjacent interpolation

weight

CD

Figure 3 CTSFA calculation of the initial value of displacement

O x1

Q12

Q11

Q22

Q21

R2

R1

P

y1

y2

y

x2x

Figure 1 Bilinear interpolation

A B

C D

u lt 05v lt 05

u gt 05v lt 05

u lt 05v gt 05

u gt 05v gt 05

u

v

(i j) (i + 1 j)

(i + 1 j + 1)(i j + 1)

Figure 2 Neighbor interpolation

Read the reference image and contrast image

Select calculation area

Set calculating window

Calculate integer pixels bysurface fitting method

Calculate subpixels by CTSFA

Calculate displacement

(1) Bilinear interpolation(2) Neighbor interpolation(3) Interpolation by weight

Figure 4 Flow chart of CTSFA calculation

Complexity 3

value of the sampled surface and the actual value of the fittedsurface Its main idea is that the sum of squared deviations ofprediction data and actual values is minimal -us the cubicsurface equation of equation (11) is expressed as

z 11139440

i31113944

0

ij

aijminus ixiy

jminus i⎛⎝ ⎞⎠ (11)

-is paper gets equation (12) through the given set ofdata points (xk yk) (k 1 2 N) m times (mN)polynomials

z 1113944

0

jm

1113944

0

ij

aijminus ixiy

jminus i⎛⎝ ⎞⎠ (12)

After obtaining equation (12) the square deviation Q ofthe predicted data and the real value are calculated and Q isshown by

Q 1113944N

k1zk minus 1113944

0

jm

1113944

0

ij

aijminus ixiky

jminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

2

(13)

Equation (14) is obtained by transforming the con-struction problem of the fitting polynomial of equation (10)into the extremum problem of themultivariate function andcijminus i satisfies zQzcijminus i 0

1113944

N

n1zk minus 1113944

0

jm

1113944

0

ij

aijminus ixkyjminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦x

iky

jminus i

k

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 0 (14)

Equation (15) is simplified by equation (14)

1113944

N

k11113944

0

jm

1113944

0

ij

aijminus ixiky

jminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ 1113944

N

k1zkx

iky

jminus i

k (15)

Equation (16) is further sorted out through equation(15)

1113944

0

jm

1113944

0

ij

aijminus i 1113944

N

k1x

iky

jminus i

k⎡⎣ ⎤⎦

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113944

N

k1zkx

iky

jminus i

k (16)

-e power basis function of CTSFA is solved after cijminus i isdetermined and themaximum of the correlation coefficientsand the position coordinates at the maximum can be ob-tained -en the actual displacement can be obtained bytaking the difference between the maximum and the initialdisplacement

3 Experimental Verification

31 Analysis of Digital Speckle Computer Simulation Inorder to verify the performance of CTSFA the accuracy istested by using the computer simulated speckle experimentA digital speckle image is selected as the reference framewith the preset theoretical displacement field given CTSFAand SFA are used for analyzing the experimental resultsthrough the reconstructed speckle field which reconstructedas the calculation frame

-e displacement of the speckle in the image can beprecisely controlled by using the numerical method tosimulate the speckle image of the uniform and nonuniformtheoretical deformation field -e simulated speckle imagesize in Figure 5 is 128times128 pixels and the intensity of thespeckle is Gaussian -e speckle size is 3 pixels and thenumber of speckles is 300 -e size of calculating window inSFA and CTSFA is 41times 41

311 Uniform Deformation Field Test -e effectiveness ofCTSFA and SFA is tested by using the uniform deformationfield speckle image of Figure 5(b) -e displacement of thespeckle field in Figure 5(b) is shown by

u 02x

v 0(17)

Corresponding calculations are performed using CTSFAand SFA respectively -e image analysis area is x 1sim100pixels and y 1sim100 pixels Figure 6 shows the lateral dis-placement field obtained by CTSFA and SFA

It is found from Figure 6 that the displacements inCTSFA and SFA are stratified and the stratification valuesare consistent with the applied displacement function in-dicating that both CTSFA and SFA can calculate the uniformdisplacement field-e displacement boundary in the u-fieldcloud map of CTSFA is smoother than that of SFA and thedisplacement boundary of SFA is irregularly jagged in-dicating that CTSFA is more accurate than SFA on calcu-lation of uniform displacement field

In Figure 7(a) the lateral displacement calculated byCTSFA is closer to the theoretical value and the calculationresult of SFA has a certain deviation from the theoreticalvalue indicating that the calculation accuracy and stabilityof CTSFA are better than SFA In Figure 7(b) the relativeerror of CTSFA in homogeneous deformation field is about12 which is half of that of SFA

312 Nonuniform Deformation Field Test -e speckle im-age of nonuniform deformation fields is simulated numeri-cally by CTSFA and SFA which is shown in Figure 5(c) -edisplacement of the speckle field in Figure 5(c) is shown by

u 100

2π

radic middot eminus (xminus 50)2( )8( )

v 0

(18)

CTSFA and SFA are used respectively to calculatenonuniform deformation fields and the area analyzed isx 44sim56 pixel and y 1sim100 pixel Figure 8 shows thelateral displacement field calculated by CTSFA and SFAFigure 9(a) shows the results of theoretical CTSFA andSFA Figure 9(b) shows the error analysis of CTSFA andSFA

In the u-field nephogram the displacement boundaryof CTSFA is smoother than that of SFA which indicatesthat CTSFA is better than SFA in dealing with non-uniform displacement field Especially in the case of large

4 Complexity

displacement the value of CTSFA is closer to the theo-retical value while the value of SFA is lower the mainreason being that the initial value of displacement selectedby CTSFA is more accurate

-e data on the y 40 pixel interface are selected forerror analysis -e abscissa coordinates in Figure 9 representthe pixel coordinates of x and the ordinate coordinatesrepresent the value of u

(a)

u = 02x

(b)

minus(x minus 50)282π middot eu = 100

(c)

Figure 5 Images of simulation generates speckle (a) Initial reference image (b) Uniform deformation field (c) Inhomogeneous de-formation field

100

80

60

40

20

20 40 60 80 100

18

2

6

10

14

x (pixel)

y (pi

xel)

(a)

x (pixel)44 47 50 53 56

35

3

11

19

27

100

80

60

40

20

y (pi

xel)

(b)

x (pixel)20 40 60 80 100

18

2

6

10

14

100

80

60

40

20

y (pi

xel)

(c)

Figure 6 Uniform deformation field obtained different algorithms (a) Deformation theory (b) CTSFA (c) SFA

CTSFASFATheoretical

u (p

ixel

)

10 20 30 40 50 60 70 80 90 1000x (pixel)

02468

101214161820

(a)

Rela

tive e

rror

()

CTSFASFA

20 30 40 50 60 70 80 9010x (pixel)

12

14

16

18

20

22

24

26

(b)

Figure 7 Results of uniform displacement field (a) Comparison of displacement values (b) Comparison of relative error

Complexity 5

In Figure 9(a) at the position x 50 pixel where thegradient of deformation is large the calculated result of SFAdeviates from the theoretical value greatly and CTSFA iscloser to the theoretical value indicating that the spatialresolution and displacement resolution of CTSFA are higherthan that of SFA and CTSFA is more suitable for analyzingin the region of the large gradient of displacementFigure 9(b) shows that the relative error of CTSFA is stable atabout 3 and the relative error of SFA is 3 times that ofCTSFA

32 Test of Cracked Brazil Disk In order to evaluate theaccuracy and efficiency of CTSFA and SFA further the testof cracked Brazilian disk taken by industrial CCD cameraunder static loading is carried out using CTSFA and SFAand the evolution of lateral displacement field is analyzed-e experimental specimens choose sandstone specimenswith diameter of 50mm and thickness of 25mm A crack of10mm in length is prefabricated at the center of the spec-imen and four sets of experiments are carried out-e speed

of collecting photos by industrial CCD camera is 10 framesper second and the testing machine is RLJW-2000 -eloading rate of the test is 005mmmin -e picture of thestate of rock failure in the rock mechanical test is shown inFigure 10 and the curve of load-time obtained by the testingmachine is shown in Figure 11

Based on the characteristics of the loading curve 7typical moments in the whole loading process are selectedfor identification -e digital image correlation calculationbased on CTSFA and SFA is used to calculate the above timepoints and compare the results Among them the markingpoint a is the reference point analyzed by digital imagecorrelation the marking points b and c are located in thelinear elastic phase of the test the marking points d e and fare located in the plastic phase of the test and the markingpoint g is located at the peak point of the test

-e calculation area and time of CTSFA and SFA aresame and the results are shown in Table 1 In the linearelastic stage CTSFA observes the precursor of crack for-mation and SFA only observes the vertical compression ofthe specimen without showing the crack location In the

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 563

11

19

27

35

(a)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(b)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(c)

Figure 8 Inhomogeneous deformation field obtained different algorithms (a) Deformation theory (b) CTSFA (c) SFA

44 46 48 50 52 54 560

5

10

15

20

25

30

35

40

CTSFASFATheoretical

x (pixel)

u (p

ixel

)

(a)

46 48 50 52 54

2

4

6

8

CTSFASFA

x (pixel)

u (p

ixel

)

(b)

Figure 9 Results of nonuniform displacement field (a) Comparison of displacement values (b) Comparison of relative error

6 Complexity

(a) (b)

Figure 10 -e pictures in cracked Brazilian test (a) Before rock failure (b) After rock failure

a

b

c

d

ef

g

0 001 002 003 004Time (h)

Load

(kN

)

005 006 007

1

2

3

4

Figure 11 Load-time curve of cracked Brazilian test

Table 1 u field of CTSFA and SFA at different times

b(t 78 s) c(t 106 s) d(t 130 s)

CTSFA

0050

ndash0010

002000100

00300040

mm

0040

ndash0020

00100ndash0010

00200030

mm0060

ndash0030

00150ndash0015

00300045

mm

SFA

mm0070

0010

004000300020

00500060

mm0040

ndash0020

00100ndash0010

00200030

mm

0030

ndash0030

0ndash0010ndash0020

00100020

Complexity 7

plastic stage both CTSFA and SFA observe basically theextension of cracks but the stratification of the lateraldisplacement of CTSFA is more obvious At the peak loadpoint the crack propagation pattern observed by CTSFAconforms to the actual damage of the specimen -e crackobserved by SFA appears jagged in the upper and lower sidesof the precrack indicating that the observation effect ofCTSFA is better than SFA

4 Conclusion

-is paper comes up with CTSFA based on the idea ofdistance weight the validity of CTSFA is verified by digitalspeckle simulation and Brazilian disk tests and the mainconclusions are as follows

(1) With the characteristics of high computation effi-ciency and maneuverability it is easy for CTSFA toimplement programmed application Computersimulation of digital speckle shows that CTSFA hasgood adaptability and stability and it is suitable forboth uniform and inhomogeneous deformationfields -e relative error of CTSFA has a low fluc-tuation which is kept at 12 in the uniform de-formation field and at 30 in the nonuniformdeformation field Additionally the measurementaccuracy of CTSFA is 3 times as high as that of SFA

(2) For the region with large displacement gradient in thecontinuum deformation field the calculation result ofSFA is lower than that of the theoretical value and therelative error is larger -e relative error of CTSFAhowever is closer to the theoretical value -e spatialresolution of CTSFA is higher than that of SFA

(3) Using the programmed CTSFA and SFA to analyzethe same Brazilian disk test CTSFA can observe thecrack propagation at the early stage of continuum

failure and the observed crack morphology is ingood agreement with the actual situation In otherwords the performance of CTSFA in programmedapplication is better than that of SFA

Data Availability

No data were used to support this study

Conflicts of Interest

-e authors declare that they have no competing financialinterests

Acknowledgments

-e research described in this paper was financially sup-ported by the Shandong Provincial Natural Science Foun-dation (no ZR2019QEE026) National Natural ScienceFoundation of China (no 51674160) Tairsquoshan Scholar En-gineering Construction Fund of Shandong Province ofChina (no ts201511026) and Taishan Scholar Talent TeamSupport Plan for Advantaged amp Unique Discipline Areas

References

[1] B Pan H Xie Z Wang K Qian and Z Wang ldquoStudy onsubset size selection in digital image correlation for specklepatternsrdquo Optics Express vol 16 no 10 pp 7037ndash7048 2008

[2] B Pan ldquoRecent progress in digital image correlationrdquo Ex-perimental Mechanics vol 51 no 7 pp 1223ndash1235 2011

[3] B Pan and K Li ldquoA fast digital image correlation method fordeformation measurementrdquo Optics and Lasers in Engineeringvol 49 no 7 pp 841ndash847 2011

[4] P C Hung and A S Voloshin ldquoIn-plane strain measurementby digital image correlationrdquo Journal of the Brazilian Society

Table 1 Continuede(t 150 s) f(t 168 s) g(t 175 s)

CTSFA

mm

0040

ndash0080

ndash0020ndash0040ndash0060

00020

mm

0080

ndash0160


00040

mm0300

ndash0300

0ndash0100ndash0200

01000200

SFA

mm

0020

ndash0040


00010

mm0150

ndash0150

0ndash0050ndash0100

00500100

mm0400

ndash0200

01000ndash0100

02000300

8 Complexity

of Mechanical Sciences and Engineering vol 25 no 3pp 215ndash221 2003

[5] T Hua H Xie S Wang Z Hu P Chen and Q ZhangldquoEvaluation of the quality of a speckle pattern in the digitalimage correlation method by mean subset fluctuationrdquo Opticsamp Laser Technology vol 43 no 1 pp 9ndash13 2011

[6] Z Luo H Zhu J Chu L Shen and L Hu ldquoStrain mea-surement by ultrasonic speckle technique based on adaptivegenetic algorithmrdquo e Journal of Strain Analysis for Engi-neering Design vol 48 no 7 pp 446ndash456 2013

[7] G Chen Q B Feng K Q Ding and Z Gao ldquoSubpixeldisplacement measurement method based on the combina-tion of particle swarm optimization and gradient algorithmrdquoOptical Engineering vol 56 no 10 pp 3ndash5 2017

[8] W-Y Guo T-B Zhao Y-L Tan F-H Yu S-C Hu andF-Q Yang ldquoProgressive mitigation method of rock burstsunder complicated geological conditionsrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 96pp 11ndash22 2017

[9] W Y Guo Q H Gu Y L Tan and S C Hu ldquoCase studies ofrock bursts in tectonic areas with facies changerdquo Energiesvol 12 no 7 p 1330 2019

[10] W Y Guo Y L Tan F H Yu et al ldquoMechanical behavior ofrock-coal-rock specimens with different coal thicknessesrdquoGeomechanics and Engineering vol 15 no 4 pp 1017ndash10272018

[11] W-Y Guo F-H Yu Y-L Tan and T-B Zhao ldquoExperi-mental study on the failure mechanism of layer-crackstructurerdquo Energy Science amp Engineering pp 1ndash22 2019

[12] R Barati ldquoApplication of excel solver for parameter esti-mation of the nonlinear Muskingum modelsrdquo KSCE Journalof Civil Engineering vol 17 no 5 pp 1139ndash1148 2013

[13] G Crammond S W Boyd and J M Dulieu-Barton ldquoSpecklepattern quality assessment for digital image correlationrdquoOptics and Lasers in Engineering vol 51 no 12 pp 1368ndash1378 2013

[14] M S Kirugulige H V Tippur and T S Denney ldquoMea-surement of transient deformations using digital imagecorrelation method and high-speed photography applicationto dynamic fracturerdquo Applied Optics vol 46 no 22pp 5083ndash5096 2007

[15] M Jerabek Z Major and R W Lang ldquoStrain determinationof polymeric materials using digital image correlationrdquoPolymer Testing vol 29 no 3 pp 407ndash416 2010

[16] B Pan ldquoBias error reduction of digital image correlation usingGaussian pre-filteringrdquo Optics and Lasers in Engineeringvol 51 no 10 pp 1161ndash1167 2013

[17] B Pan L Yu and D Wu ldquoHigh-accuracy 2D digital imagecorrelation measurements with bilateral telecentric lenseserror analysis and experimental verificationrdquo ExperimentalMechanics vol 53 no 9 pp 1719ndash1733 2013

[18] B Shen and G H Paulino ldquoDirect extraction of cohesivefracture properties from digital image correlation a hybridinverse techniquerdquo Experimental Mechanics vol 51 no 2pp 143ndash163 2011

[19] Z Jiang Q Kemao H Miao J Yang and L Tang ldquoPath-independent digital image correlation with high accuracyspeed and robustnessrdquo Optics and Lasers in Engineeringvol 65 pp 93ndash102 2015

[20] Z Hu H Xie J Lu T Hua and J Zhu ldquoStudy of the per-formance of different subpixel image correlation methods in3D digital image correlationrdquo Applied Optics vol 49 no 21pp 4044ndash4051 2010

[21] B T Fricke ldquo-ree-dimensional deformation field mea-surement with digital speckle correlationrdquo Applied Opticsvol 42 no 34 pp 533ndash542 2003

[22] K Fang T Zhao Y Zhang Y Qiu and J Zhou ldquoRock conepenetration test under lateral confining pressurerdquo In-ternational Journal of Rock Mechanics and Mining Sciencesvol 119 pp 149ndash155 2019

[23] J Wang J Ning J Q Jiang and T Bu ldquoStructural charac-teristics of strata overlying of a fully mechanized longwall facea case studyrdquo Journal of the Southern African Institute ofMining and Metallurgy vol 118 no 11 pp 1195ndash1204 2018

[24] H Wang H Xie X Dai and J Zhu ldquoFabrication of a DICsensor for in-plane deformation measurementrdquoMeasurementScience and Technology vol 24 no 6 Article ID 065402 2013

[25] J Zhu G Yan G He and L Chen ldquoFabrication andoptimization of micro-scale speckle patterns for digitalimage correlationrdquo Measurement Science and Technologyvol 27 no 1 Article ID 015203 2016

[26] W J Qian J Li J G Zhu W F Hao and L Chen ldquoDis-tortion correction of a microscopy lens system for de-formation measurements based on speckle pattern andgratingrdquo Optics and Lasers in Engineering vol 124 Article ID105804 2019

Complexity 9

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


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-is paper introduces the concept of distance weightingin digital image processing to modify SFA of initial dis-placement determination by CTSFA CTSFA adopts the graylevel interpolation algorithm which combines the nearestneighbor interpolation with bilinear interpolation Bymoving the fitting center and calculating the weightedweights the points participating in the surface fitting cal-culation become closer to the actual initial point For thecubic surface fitting function of CTSFA the least squaremethod is used to solve the extreme coordinates of thecorrelation coefficient -e accuracy of the algorithm isverified by computer-generated simulated speckle imagesand cracked Brazilian disc tests before application

2 Algorithm Principle

21 Basic eory of SFA Based on the actual data the SFAobtains the analytical expression between the functionf(x y) and the variables x and y and it can pass or ap-proximate all data points which can be distributed in thefunction f(x y) on the spatial surface represented Takingthe two polynomials as an example the fitting function is[24]

C(x y) a0 + a1x + a2y + a3x2

+ a4xy + a5y2 (1)

-e function C(x y) should satisfy the equations at theextreme points of the quadric

zC(x y)

zx a1 + 2a3x + a4y 0

zC(x y)

zy a2 + 2a5y + a4x 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(2)

-en according to equation (2) [24] the position of theextreme point of the fitting surface is obtained [24]

x 2a1a5 minus a2a4

a24 minus 4a3a5

y 2a2a3 minus a1a4

a24 minus 4a3a5

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)

After finding the location of the extreme point thedifference between the coordinates of the initial value pointand the initial value point is a continuous displacement-erefore obtaining high-precision initial displacement andcalculating the coefficients of polynomial fitting function arethe two important of CTSFA

22 Determination of Initial Value of CTSFA DisplacementAt first CTSFA calculates the interpolation results of bi-linear nearest neighbor points and the gray variance of fourneighbor pixels around the interpolation point which basedon the traditional linear interpolation method when de-termining the initial displacement value -en it constructsthe weighting coefficients through the gray variance Finallythe two interpolation results are weighted together to get the

final interpolation results CTSFA not only considers thedistance between interpolation points and neighboringpoints but also takes the gray distribution characteristics ofneighboring points into account which can improve thesubpixel accuracy of initial displacement effectively

-e specific steps for determining initial values of dis-placement in CTSFA are as follows

(1) Calculating bilinear interpolation Gd(x y) and ad-jacent interpolation GN(x y) of the interpolationpoints (x y)As shown in Figure 1 bilinear interpolation uses thegray-scale weighted interpolation of four neighbor-ing points as the gray-scale value of the point whichcan be decomposed into two one-dimensional linearinterpolations expressed by

f R1( 1113857 x2 minus x

x2 minus x1f Q11( 1113857 +

x minus x1

x2 minus x1f Q21( 1113857

f R2( 1113857 x2 minus x

x2 minus x1f Q12( 1113857 +

x minus x1

x2 minus x1f Q22( 1113857

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(4)

f(P) y2 minus y

y2 minus y1f R1( 1113857 +

y minus y1

y2 minus y1f R2( 1113857 (5)

Among the four adjacent pixels of the pixel whichwill be sought the nearest neighbor pixel gray isassigned Define i + u and j + v (i and j are positiveintegers u and v are decimal greaters whose valuemore than 0 as well as less than 1) are the nonintegerpixel coordinates the value of the pixel whose grayf(i + u j + v) is solved and shown in Figure 2If (i + u j + v) falls in area A that means ult 05 andvlt 05 the gray value of the pixel in the upper leftcorner is assigned to the pixel to be sought Similarlythe falling in the B area gives the upper right cornerand so on

(2) Calculating the standard deviationH of four adjacentpoints fi(x y) (i 1 2 3 4) around the in-terpolation point and H is expressed by

H

14

1113944

4

i1fi(x y) minus f(x y)1113960 1113961

2

11139741113972

(6)

-e f(x y) is expressed by

f(x y) 14

1113944

4

i1fi(x y) (7)

(3) Weighting fusion of Gd(x y) and Gn(x y) isexpressed by equation (8) [25 26]

2 Complexity



Wd 12

e((minus H)100)

+ 05

Wπ 1 minus Wd

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)






f(x y) a0 + a1x + a2y + a3x2

+ a4xy + a5y2

+ a6x3

+ a7x2y + a8xy

2+ a9y

3

(10)


x

y

a1a2

a3a4

A1

A2

A

b1

b2 b3

b4

B



weight

CD


O x1

Q12

Q11

Q22

Q21

R2

R1

P

y1

y2

y

x2x


A B

C D

u lt 05v lt 05

u gt 05v lt 05

u lt 05v gt 05

u gt 05v gt 05

u

v

(i j) (i + 1 j)

(i + 1 j + 1)(i j + 1)










Complexity 3


z 11139440

i31113944

0

ij

aijminus ixiy

jminus i⎛⎝ ⎞⎠ (11)


z 1113944

0

jm

1113944

0

ij

aijminus ixiy

jminus i⎛⎝ ⎞⎠ (12)


Q 1113944N

k1zk minus 1113944

0

jm

1113944

0

ij

aijminus ixiky

jminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

2

(13)


1113944

N

n1zk minus 1113944

0

jm

1113944

0

ij


k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦x

iky

jminus i

k

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 0 (14)


1113944

N

k11113944

0

jm

1113944

0

ij

aijminus ixiky

jminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ 1113944

N

k1zkx

iky

jminus i

k (15)


1113944

0

jm

1113944

0

ij

aijminus i 1113944

N

k1x

iky

jminus i

k⎡⎣ ⎤⎦

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113944

N

k1zkx

iky

jminus i

k (16)






u 02x

v 0(17)





u 100

2π


v 0

(18)



4 Complexity



(a)

u = 02x

(b)


(c)


100

80

60

40

20

20 40 60 80 100

18

2

6

10

14

x (pixel)

y (pi

xel)

(a)

x (pixel)44 47 50 53 56

35

3

11

19

27

100

80

60

40

20

y (pi

xel)

(b)

x (pixel)20 40 60 80 100

18

2

6

10

14

100

80

60

40

20

y (pi

xel)

(c)


CTSFASFATheoretical

u (p

ixel

)

10 20 30 40 50 60 70 80 90 1000x (pixel)

02468

101214161820

(a)

Rela

tive e

rror

()

CTSFASFA

20 30 40 50 60 70 80 9010x (pixel)

12

14

16

18

20

22

24

26

(b)


Complexity 5






x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 563

11

19

27

35

(a)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(b)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(c)


44 46 48 50 52 54 560

5

10

15

20

25

30

35

40

CTSFASFATheoretical

x (pixel)

u (p

ixel

)

(a)

46 48 50 52 54

2

4

6

8

CTSFASFA

x (pixel)

u (p

ixel

)

(b)


6 Complexity

(a) (b)


a

b

c

d

ef

g

0 001 002 003 004Time (h)

Load

(kN

)

005 006 007

1

2

3

4



b(t 78 s) c(t 106 s) d(t 130 s)

CTSFA

0050

ndash0010

002000100

00300040

mm

0040

ndash0020

00100ndash0010

00200030

mm0060

ndash0030

00150ndash0015

00300045

mm

SFA

mm0070

0010

004000300020

00500060

mm0040

ndash0020

00100ndash0010

00200030

mm

0030

ndash0030

0ndash0010ndash0020

00100020

Complexity 7


4 Conclusion






Data Availability




Acknowledgments


References






CTSFA

mm

0040

ndash0080


00020

mm

0080

ndash0160


00040

mm0300

ndash0300

0ndash0100ndash0200

01000200

SFA

mm

0020

ndash0040


00010

mm0150

ndash0150

0ndash0050ndash0100

00500100

mm0400

ndash0200

01000ndash0100

02000300

8 Complexity
























Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018










Wd 12

e((minus H)100)

+ 05

Wπ 1 minus Wd

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)






f(x y) a0 + a1x + a2y + a3x2

+ a4xy + a5y2

+ a6x3

+ a7x2y + a8xy

2+ a9y

3

(10)


x

y

a1a2

a3a4

A1

A2

A

b1

b2 b3

b4

B



weight

CD


O x1

Q12

Q11

Q22

Q21

R2

R1

P

y1

y2

y

x2x


A B

C D

u lt 05v lt 05

u gt 05v lt 05

u lt 05v gt 05

u gt 05v gt 05

u

v

(i j) (i + 1 j)

(i + 1 j + 1)(i j + 1)










Complexity 3


z 11139440

i31113944

0

ij

aijminus ixiy

jminus i⎛⎝ ⎞⎠ (11)


z 1113944

0

jm

1113944

0

ij

aijminus ixiy

jminus i⎛⎝ ⎞⎠ (12)


Q 1113944N

k1zk minus 1113944

0

jm

1113944

0

ij

aijminus ixiky

jminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

2

(13)


1113944

N

n1zk minus 1113944

0

jm

1113944

0

ij


k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦x

iky

jminus i

k

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 0 (14)


1113944

N

k11113944

0

jm

1113944

0

ij

aijminus ixiky

jminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ 1113944

N

k1zkx

iky

jminus i

k (15)


1113944

0

jm

1113944

0

ij

aijminus i 1113944

N

k1x

iky

jminus i

k⎡⎣ ⎤⎦

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113944

N

k1zkx

iky

jminus i

k (16)






u 02x

v 0(17)





u 100

2π


v 0

(18)



4 Complexity



(a)

u = 02x

(b)


(c)


100

80

60

40

20

20 40 60 80 100

18

2

6

10

14

x (pixel)

y (pi

xel)

(a)

x (pixel)44 47 50 53 56

35

3

11

19

27

100

80

60

40

20

y (pi

xel)

(b)

x (pixel)20 40 60 80 100

18

2

6

10

14

100

80

60

40

20

y (pi

xel)

(c)


CTSFASFATheoretical

u (p

ixel

)

10 20 30 40 50 60 70 80 90 1000x (pixel)

02468

101214161820

(a)

Rela

tive e

rror

()

CTSFASFA

20 30 40 50 60 70 80 9010x (pixel)

12

14

16

18

20

22

24

26

(b)


Complexity 5






x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 563

11

19

27

35

(a)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(b)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(c)


44 46 48 50 52 54 560

5

10

15

20

25

30

35

40

CTSFASFATheoretical

x (pixel)

u (p

ixel

)

(a)

46 48 50 52 54

2

4

6

8

CTSFASFA

x (pixel)

u (p

ixel

)

(b)


6 Complexity

(a) (b)


a

b

c

d

ef

g

0 001 002 003 004Time (h)

Load

(kN

)

005 006 007

1

2

3

4



b(t 78 s) c(t 106 s) d(t 130 s)

CTSFA

0050

ndash0010

002000100

00300040

mm

0040

ndash0020

00100ndash0010

00200030

mm0060

ndash0030

00150ndash0015

00300045

mm

SFA

mm0070

0010

004000300020

00500060

mm0040

ndash0020

00100ndash0010

00200030

mm

0030

ndash0030

0ndash0010ndash0020

00100020

Complexity 7


4 Conclusion






Data Availability




Acknowledgments


References






CTSFA

mm

0040

ndash0080


00020

mm

0080

ndash0160


00040

mm0300

ndash0300

0ndash0100ndash0200

01000200

SFA

mm

0020

ndash0040


00010

mm0150

ndash0150

0ndash0050ndash0100

00500100

mm0400

ndash0200

01000ndash0100

02000300

8 Complexity
























Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018









z 11139440

i31113944

0

ij

aijminus ixiy

jminus i⎛⎝ ⎞⎠ (11)


z 1113944

0

jm

1113944

0

ij

aijminus ixiy

jminus i⎛⎝ ⎞⎠ (12)


Q 1113944N

k1zk minus 1113944

0

jm

1113944

0

ij

aijminus ixiky

jminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦

2

(13)


1113944

N

n1zk minus 1113944

0

jm

1113944

0

ij


k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦x

iky

jminus i

k

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 0 (14)


1113944

N

k11113944

0

jm

1113944

0

ij

aijminus ixiky

jminus i

k⎛⎝ ⎞⎠⎡⎢⎢⎣ ⎤⎥⎥⎦ 1113944

N

k1zkx

iky

jminus i

k (15)


1113944

0

jm

1113944

0

ij

aijminus i 1113944

N

k1x

iky

jminus i

k⎡⎣ ⎤⎦

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ 1113944

N

k1zkx

iky

jminus i

k (16)






u 02x

v 0(17)





u 100

2π


v 0

(18)



4 Complexity



(a)

u = 02x

(b)


(c)


100

80

60

40

20

20 40 60 80 100

18

2

6

10

14

x (pixel)

y (pi

xel)

(a)

x (pixel)44 47 50 53 56

35

3

11

19

27

100

80

60

40

20

y (pi

xel)

(b)

x (pixel)20 40 60 80 100

18

2

6

10

14

100

80

60

40

20

y (pi

xel)

(c)


CTSFASFATheoretical

u (p

ixel

)

10 20 30 40 50 60 70 80 90 1000x (pixel)

02468

101214161820

(a)

Rela

tive e

rror

()

CTSFASFA

20 30 40 50 60 70 80 9010x (pixel)

12

14

16

18

20

22

24

26

(b)


Complexity 5






x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 563

11

19

27

35

(a)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(b)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(c)


44 46 48 50 52 54 560

5

10

15

20

25

30

35

40

CTSFASFATheoretical

x (pixel)

u (p

ixel

)

(a)

46 48 50 52 54

2

4

6

8

CTSFASFA

x (pixel)

u (p

ixel

)

(b)


6 Complexity

(a) (b)


a

b

c

d

ef

g

0 001 002 003 004Time (h)

Load

(kN

)

005 006 007

1

2

3

4



b(t 78 s) c(t 106 s) d(t 130 s)

CTSFA

0050

ndash0010

002000100

00300040

mm

0040

ndash0020

00100ndash0010

00200030

mm0060

ndash0030

00150ndash0015

00300045

mm

SFA

mm0070

0010

004000300020

00500060

mm0040

ndash0020

00100ndash0010

00200030

mm

0030

ndash0030

0ndash0010ndash0020

00100020

Complexity 7


4 Conclusion






Data Availability




Acknowledgments


References






CTSFA

mm

0040

ndash0080


00020

mm

0080

ndash0160


00040

mm0300

ndash0300

0ndash0100ndash0200

01000200

SFA

mm

0020

ndash0040


00010

mm0150

ndash0150

0ndash0050ndash0100

00500100

mm0400

ndash0200

01000ndash0100

02000300

8 Complexity
























Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018










(a)

u = 02x

(b)


(c)


100

80

60

40

20

20 40 60 80 100

18

2

6

10

14

x (pixel)

y (pi

xel)

(a)

x (pixel)44 47 50 53 56

35

3

11

19

27

100

80

60

40

20

y (pi

xel)

(b)

x (pixel)20 40 60 80 100

18

2

6

10

14

100

80

60

40

20

y (pi

xel)

(c)


CTSFASFATheoretical

u (p

ixel

)

10 20 30 40 50 60 70 80 90 1000x (pixel)

02468

101214161820

(a)

Rela

tive e

rror

()

CTSFASFA

20 30 40 50 60 70 80 9010x (pixel)

12

14

16

18

20

22

24

26

(b)


Complexity 5






x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 563

11

19

27

35

(a)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(b)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(c)


44 46 48 50 52 54 560

5

10

15

20

25

30

35

40

CTSFASFATheoretical

x (pixel)

u (p

ixel

)

(a)

46 48 50 52 54

2

4

6

8

CTSFASFA

x (pixel)

u (p

ixel

)

(b)


6 Complexity

(a) (b)


a

b

c

d

ef

g

0 001 002 003 004Time (h)

Load

(kN

)

005 006 007

1

2

3

4



b(t 78 s) c(t 106 s) d(t 130 s)

CTSFA

0050

ndash0010

002000100

00300040

mm

0040

ndash0020

00100ndash0010

00200030

mm0060

ndash0030

00150ndash0015

00300045

mm

SFA

mm0070

0010

004000300020

00500060

mm0040

ndash0020

00100ndash0010

00200030

mm

0030

ndash0030

0ndash0010ndash0020

00100020

Complexity 7


4 Conclusion






Data Availability




Acknowledgments


References






CTSFA

mm

0040

ndash0080


00020

mm

0080

ndash0160


00040

mm0300

ndash0300

0ndash0100ndash0200

01000200

SFA

mm

0020

ndash0040


00010

mm0150

ndash0150

0ndash0050ndash0100

00500100

mm0400

ndash0200

01000ndash0100

02000300

8 Complexity
























Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018













x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 563

11

19

27

35

(a)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(b)

x (pixel)

y (pi

xel)

100

80

60

40

20

44 47 50 53 56

35

3

11

19

27

(c)


44 46 48 50 52 54 560

5

10

15

20

25

30

35

40

CTSFASFATheoretical

x (pixel)

u (p

ixel

)

(a)

46 48 50 52 54

2

4

6

8

CTSFASFA

x (pixel)

u (p

ixel

)

(b)


6 Complexity

(a) (b)


a

b

c

d

ef

g

0 001 002 003 004Time (h)

Load

(kN

)

005 006 007

1

2

3

4



b(t 78 s) c(t 106 s) d(t 130 s)

CTSFA

0050

ndash0010

002000100

00300040

mm

0040

ndash0020

00100ndash0010

00200030

mm0060

ndash0030

00150ndash0015

00300045

mm

SFA

mm0070

0010

004000300020

00500060

mm0040

ndash0020

00100ndash0010

00200030

mm

0030

ndash0030

0ndash0010ndash0020

00100020

Complexity 7


4 Conclusion






Data Availability




Acknowledgments


References






CTSFA

mm

0040

ndash0080


00020

mm

0080

ndash0160


00040

mm0300

ndash0300

0ndash0100ndash0200

01000200

SFA

mm

0020

ndash0040


00010

mm0150

ndash0150

0ndash0050ndash0100

00500100

mm0400

ndash0200

01000ndash0100

02000300

8 Complexity
























Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018








(a) (b)


a

b

c

d

ef

g

0 001 002 003 004Time (h)

Load

(kN

)

005 006 007

1

2

3

4



b(t 78 s) c(t 106 s) d(t 130 s)

CTSFA

0050

ndash0010

002000100

00300040

mm

0040

ndash0020

00100ndash0010

00200030

mm0060

ndash0030

00150ndash0015

00300045

mm

SFA

mm0070

0010

004000300020

00500060

mm0040

ndash0020

00100ndash0010

00200030

mm

0030

ndash0030

0ndash0010ndash0020

00100020

Complexity 7


4 Conclusion






Data Availability




Acknowledgments


References






CTSFA

mm

0040

ndash0080


00020

mm

0080

ndash0160


00040

mm0300

ndash0300

0ndash0100ndash0200

01000200

SFA

mm

0020

ndash0040


00010

mm0150

ndash0150

0ndash0050ndash0100

00500100

mm0400

ndash0200

01000ndash0100

02000300

8 Complexity
























Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018









4 Conclusion






Data Availability




Acknowledgments


References






CTSFA

mm

0040

ndash0080


00020

mm

0080

ndash0160


00040

mm0300

ndash0300

0ndash0100ndash0200

01000200

SFA

mm

0020

ndash0040


00010

mm0150

ndash0150

0ndash0050ndash0100

00500100

mm0400

ndash0200

01000ndash0100

02000300

8 Complexity
























Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018































Complexity 9








Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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