ResearchonHyperbolaFittingAlgorithmforTurbulenceLevel ...ResearchArticle ResearchonHyperbolaFittingAlgorithmforTurbulenceLevel MeasurementTestData YufengDu ,1 LongWu,2 XunnianWang,3

Research ArticleResearch on Hyperbola Fitting Algorithm for Turbulence LevelMeasurement Test Data

Yufeng Du 1 Long Wu2 Xunnian Wang3 Jun Lin1 and Neng Xiong1

1High Speed Aerodynamics Institute China Aerodynamics Research and Development Center Mianyang 621000 China2Science and Technology on Scramjet Laboratory China Aerodynamics Research and Development CenterMianyang 621000 China3State Key Laboratory of Aerodynamics China Aerodynamics Research and Development Center Mianyang 621000 China

Correspondence should be addressed to Yufeng Du 1415776643qqcom

Received 24 April 2020 Revised 19 June 2020 Accepted 1 July 2020 Published 23 October 2020

Academic Editor Luis J Yebra

Copyright copy 2020 Yufeng Du et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Hyperbola fitting of test data is an extremely important process in turbulence level measurement test in wind tunnelse solutionof the overdetermined equations (SOE) method is often used to solve hyperbola fitting parameters to obtain turbulence levelHowever due to unsteady flow characteristics the SOEmethod often results in overfitting phenomena whichmakes it impossibleto solve turbulence level accuratelyis paper proposes using the constrained least-squares (CLS) method to convert the problemof hyperbola fitting of test data into the inequality constrained optimization problem and then using the Lagrange programmingneural network (LPNN) method to solve turbulence level iteratively e stability of the LPNNmethod is analysed and three setsof typical turbulence level measurement test data are processed using the LPNN method e results verify the feasibility ofapplying the LPNN method to iteratively solve the turbulence level of wind tunnels

1 Introduction

Wind tunnel test is the most effective method for aerody-namic research Even if the computer-based numericalsimulation technology and the model flight test technologyare rapidly improving wind tunnel test is still an indis-pensable method for the research of complex aerodynamiccharacteristics during the research and development ofaircrafte precise design of advanced aircraft requires highaccuracy of wind tunnel test results However there existmany factors and phenomena which finally result in theinaccuracy of measurements in the wind tunnel such as theinfluence of flow qualities Reynolds and Mach numbersstandard model test results wind tunnel walls and sup-porting system interference in 2D and 3D tests [1ndash7]

As an important flow quality in wind tunnels turbulencelevel will affect the accuracy of wind tunnel test results suchas calculation of aircraft force and moment coefficients [1]measurement of test model attitude angle [8] and mea-surement of transition characteristics of the boundary layer

on test model surface [9] For aircraft design the errors inwind tunnel tests mean that there will be design errors inaerodynamic parameters such as lift and drag coefficientse design errors will lead to estimation errors in aircraftweight which severely restricts the economy and safety ofaircraft [10] erefore it is very important to evaluate theturbulence level in wind tunnels accurately andquantitatively

Hot-wire anemometry (HWA) is currently the mostwidely used method for turbulence level measurement dueto its advantages such as high-frequency response highsensitivity and cost effectiveness [11ndash14] In the com-pressible flow the hot-wire response function derived by thechanging overheat ratio method conforms to a hyperbolicrelationship erefore the problem of solving turbulencelevel can be converted into the problem of solving hyperbolafitting parameters of a set of two-dimensional scatteredpoints [15] However in the actual measurement and dataacquisition process the two-dimensional scattered pointsmay deviate from hyperbolic distribution due to unsteady

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 5620195 10 pageshttpsdoiorg10115520205620195

flow characteristics In some cases there are even straightlines and other conic distributions which make it difficult tosolve turbulence level precisely

Many researchers have studied hyperbola fitting andthe methods utilized differ due to different applicationscenarios Hough transform (HF) is a method of graphdetection and fitting based on pattern recognition e HFmethod converts scattered points in the image space tothose in the parameter space and converts the problem ofscattered points fitting to the problem of solving extremevalue in the parameter space thereby realizing the detec-tion and fitting of the hyperbola e HF method has theadvantages of strong robustness but its shortcomings oflow computational efficiency limit its real-time application[16ndash20] e least-squares (LS) method is the most com-monly used method for hyperbola fitting e key idea ofthe LS method is to establish a suberror function for eachscattered point and use the sum of the squares of allsuberror functions as the error function of hyperbola fit-tinge suberror function can be expressed by residuals ororthogonal distances between scattered points and fittedhyperbolic curves Hyperbolic curves are obtained bycalculating the extreme value of the error function e LSmethod has been widely used due to its ease of applicationand high computational efficiency [21ndash25] Both Lebiga VA[12 26 27] and Radespiel R [28ndash31] have done a lot ofresearch in the field of turbulence level measurement andobtaining the fitting solution in compressible flow eyconducted a comprehensive and thorough analysis offluctuation measurement results which provided datasupport for the uncertainty analysis of subsequent windtunnel tests However they have not conducted too muchresearch on the hyperbola fitting method Instead thesolution of the overdetermined equations (SOE) method isused directly to calculate the hyperbolic parameters tocomplete the hyperbola fitting e SOE method is basedon the LS method which means it has high computationalefficiency while the SOE method cannot obtain preciseturbulence level results due to overfitting if the scatteredpoints deviate from hyperbolic distribution

In order to solve the problem mentioned above thispaper proposes using the constrained least-squares (CLS)method to convert the problem of hyperbola fitting to theinequality constrained optimization problem and then usingthe Lagrange programming neural network (LPNN) methodto solve turbulence level iterativelye results show that theLPNN method is superior to the traditional SOE methodwhich verify the feasibility of the LPNN method for solvingturbulence level in wind tunnels

2 The SOE Method

According to the literature [26ndash32] the response function ofthe constant temperature hot-wire anemometer (CTA) incompressible flow is

ΔEE

FCTAΔmm

minus GCTAΔT0

T0 (1)

where E is the output voltage of CTA m and T0 are the gasmass flow rate and total temperature of the hot-wire probemeasuring point and FCTA and GCTA are the mass flow rateand total temperature sensitivity coefficients of CTA re-spectively FCTA andGCTA are only related to the overheatingratio namely the working temperature of the hot-wireprobe under a fixed flow condition Dividing equation (1) byGCTA and defining θ 1GCTAΔEE as the response functionand r FCTAGCTA as the independent variable we can get

θ Δmm

r minusΔT0

T0 (2)

Take the mean square value of equation (2)

θ21113969

1113888 1113889

2

Δmm

1113874 11138752r2

minus 2Δmm

1113874 1113875ΔT0

T01113888 1113889r +

ΔT0

T01113888 1113889

2

(3)

From equation (3) we can know that the responsefunction of CTA conforms to the hyperbolic relationship

with r as the independent variable and

θ21113969

as the dependentvariable e two-dimensional scattered points (RΘ) to befitted can be obtained by continuously changing the overheatratio n times and recording the output voltage of CTA at thesame time

R r1 r2 rn1113858 1113859T

Θ

θ211113969

θ221113969

θ2n1113969

1113876 1113877T

(4)

Construct the coefficient matrix A the variable vector Xto be sought and the nonhomogeneous term vector Π ofoverdetermined equations

A

r21 r1 1

r22 r2 1

⋮ ⋮ ⋮

r2n rn 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

X Δmm

1113874 11138752minus 2Δmm

1113874 1113875ΔT0

T01113888 1113889

ΔT0

T01113888 1113889

2⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

T

Π

θ211113969

1113874 11138752

θ221113969

1113874 11138752

θ2n1113969

1113874 11138752

1113890 1113891

T

(5)

erefore the flow fluctuations can be obtained bysolving the following overdetermined equations

A middot X Π (6)

Since A is not a square matrix it is impossible to solvethe overdetermined equations by solving the inverse matrixof A Instead the generalized inverse matrix of A can besolved by the singular value decomposition (SVD) methodto solve the overdetermined equations Any real matrix A(ntimes q) can be decomposed

2 Mathematical Problems in Engineering

A U middotWz 0

0 01113890 1113891 middot VT

(7)

where U(ntimesn) and V(qtimesq) are orthogonal matricesWz [diag(w1 w2 wz)] w1 w2 wz are the non-zero singular values of matrix A and w1 gew2 ge middot middot middot gewz gt 0z is the rank ofA When the rank ofA is full that is z q theSVD form of A is

A U middotWz

01113890 1113891 middot VT

(8)

According to equation (8) the generalized left inversematrix of A is

A+ V middot Wminus1

z 01113960 1113961 middot UT (9)

which satisfies A+ middot A I erefore the solution of theoverdetermined equations (6) using the SVD method is

X A+middot Π (10)

After obtaining the flow fluctuations the followingequations can be used to solve the turbulence level Tu of theflow

Δmm

Δ(ρu)

ρuΔρρ

+Δuu

ΔT0

T0 αΔTT

+ βΔuu

α 1

1 +(c minus 12)M2

β (c minus 1)M

2

1 +(c minus 12)M2

Δpp

Δρρ

+ΔTT

Δρρ

1c

Δpp

(11)

where ρ u T and p are the gas density velocity statictemperature and static pressure of the hot-wire probemeasuring point respectively and M is the Mach numberTurbulence level Tu can be solved by the following equations

Tu Δuu

1113874 11138752

⎛⎝ ⎞⎠

12

H2 HG G21113858 1113859 middot X( 111385712

(12)

H 1

1 minus M2

G M

2

1 minus M2 middot

1β

(13)

In the actual measurement process conventional windtunnels cannot guarantee the strict steadiness of flow due tothe long duration of the turbulence level measurement test

by using the changing overheat ratio method ereforethere will be cases where the two-dimensional scatteredpoints do not strictly obey hyperbolic distribution and eventhe distribution is similar to straight lines and other coniccurves e schematic diagram is shown in Figure 1

e points and the solid blue line in Figure 1 are thescattered points under ideal conditions and the hyperbolaobtained by the SOE method and the + points and thedotted red line are the scattered points with flow distur-bances and the curve obtained by the SOE method As aresult of overfitting the fitting curve is approximately astraight line which has deviated from the ideal hyperbola Inthis situation the SOE method will not accurately solve theturbulence level

3 Proposed Algorithm for SolvingTurbulence Level

In order to solve the problem mentioned above the CLSmethod is used to convert the problem of hyperbola fitting tothe inequality constrained optimization problem and thenthe LPNN method is used to obtain turbulence leveliteratively

31 1e CLS Method Compared with the traditional LSmethod the CLS method can limit the parameters to befitted within a certain range to prevent overfitting Supposethat the fitted hyperbolic equation of the two-dimensionalscattered points (RΘ) is

f2(r) c1r

2+ c2r + c3 (14)

where r is the independent variable f(r) is the dependentvariable and c1 c2 and c3 are the hyperbola fitting coeffi-cients According to equation (14) the dependent variablefitting values F(R) and residuals E are

F(R) f r1( 1113857 f r2( 1113857 f rn( 11138571113858 1113859T

E Θ minus F(R) e1 e2 en1113858 1113859T

ei

θ2i

1113969

minus f ri( 1113857 (i 1 2 n)

(15)

Select the error function S of the CLS method as the sumof squares of the residuals of each scatter

S 1113944n

i1e2i (16)

Comparing equations (3) and (14) it can be seen that thehyperbola fitting coefficients c1 c2 and c3 in the CLS methodare the flow fluctuations to be sought namely

C c1 c2 c31113858 1113859T

1113954X (17)

According to the physical meanings of the variables inequations (5) and (17) the constraints are established asfollows

(1) Since the mean square values of m and T0 fluctua-tions are always positive we can say c1 gt 0 and c3 gt 0

Mathematical Problems in Engineering 3

(2) e variances of m and T0 their covariance andcorrelation coefficient are defined respectively asfollows

σ2m (Δm)2

σ2T0 ΔT0( 1113857

2

σmT0 (Δm) ΔT0( 1113857

(18)

ρmT0

σmT0

σmσT0

minus1le ρmT0le 1 (19)

From equations (18) and (19) the correlation coefficientρmT0

can be rewritten as

ρmT0

(Δm) ΔT0( 1113857

(Δm)2

1113969

ΔT0( 11138572

1113969 minus12

minus2(Δmm) ΔT0T0( 1113857

(Δmm)2

1113969

ΔT0T0( 11138572

1113969

(20)

Constraints can be established based on the range of thecorrelation coefficient ρmT0

c22 minus 4c1c3 le 0 (21)

In summary the problem of hyperbola fitting of two-dimensional scattered points can be converted to the fol-lowing optimization problem

argminC

S(C)

st c1 gt 0amp c3 gt 0amp c22 minus 4c1c3 le 0

(22)

32 1e LPNN Method To solve the nonlinear inequalityconstrained optimization problem in equation (22) theLPNNmethod is applied [33]e inequality constraints canbe rewritten as follows

g1(C) minusc1 + τ le 0


g3(C) c22 minus 4c1c3 le 0

(23)

where τ is the correction value of coefficients c1 and c3 andτ 10minus 10 Its value is determined according to the flowquality values of the conventional wind tunnels eLagrange function L with inequality constraints is defined as

L(C ν λ) S(C) + 11139443

i1λi gi(C) + ]2i1113960 1113961 (24)

where λ [λ1 λ2 λ3]T are the constraint coefficients and

λi ge 0 for i 1 2 3] []1 ]2 ]3]T are the relaxation vari-

ables ere are three types of neurons in the LPNNmethodvariable neurons control C Lagrange neurons control λ andrelaxation neurons control ] According to the KKT con-dition the state equations of neurons can be obtained asfollows

dCdt

minuszL(C ν λ)

zC

dνdt

minuszL(C ν λ)

zν

dλdt

zL(C ν λ)

zλ

(25)

where t is characteristic time e Euler iteration equation is

Ck+1 Ck

minuszL Ck

νk λk

1113872 1113873

zCkmiddot Δt

νk+1 νk

minuszL Ck

νk λk

1113872 1113873

zνkmiddot Δt

λk+1 λk

+zL Ck

νk λk

1113872 1113873

zλkmiddot Δt

(26)

where k is the number of iterations and Δt is the iterationtime step e component form of equation (26) is

ck+1i c

ki minus

zS Ck1113872 1113873

zcki

+ 11139443

j1λk

j

zgj Ck1113872 1113873

zcki

⎡⎢⎢⎣ ⎤⎥⎥⎦ middot Δt i 1 2 3

]k+1l ]k

l minus 2λkl ]

kl middot Δt l 1 2 3

λk+1j λk

j + gj Ck1113872 1113873 + ]k

j1113872 11138732

1113876 1113877 middot Δt i 1 2 3

(27)

e neural network architecture diagram of the entireiteration process is shown in Figure 2

33 Stability Analysis of the LPNNMethod According to theliterature [33] when performing stability analysis it isnecessary to verify that the iterative equilibrium point

01 02 03 04 050r

5

6

7

8

9

10

11θ2

times10ndash3

Figure 1 Schematic diagram of scatters and fitted curves of idealand disturbed conditions


obtained by the LPNN method is the asymptotically stablepoint of the neural network Let the iterative equilibriumpoint be (Clowast ]lowast λlowast) and this point can be obtained by thefollowing equation

dCdt

dλdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minusnabla2CCL Clowast νlowast λlowast( 1113857 nablaCg Clowast νlowast( 1113857

minusnablaCg Clowast νlowast( 1113857 0

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ middot

C minus Clowast

λ minus λlowast⎡⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎦

minusG middotC minus Clowast


⎤⎥⎥⎥⎥⎥⎦

(28)

whereC [CT ]T]T and gi(C ]) gi(C) + ]2i for i 1 2 3Additionally we have

nablaCg Clowast νlowast( 1113857 nablaCg Clowast νlowast( 1113857

nablaνg Clowast νlowast( 1113857⎡⎣ ⎤⎦

minus1 0 minus4clowast3

0 0 2clowast2

0 minus1 minus4clowast1

2]lowast1 0 0

0 2]lowast2 0

0 0 2]lowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

nablaCg(Clowast ]lowast) can be easily verified to be a full columnrank To be more precise the gradients above at the equi-librium point are linearly independent which means that Clowast

is a regular point and (Clowast ]lowast λlowast) is a KuhnndashTucker point ofthe Lagrange function L [33] namely the following equationholds

nabla2CCL Clowast νlowast λlowast( 1113857gt 0 (30)

e specific form of nabla2CC

L(Clowast ]lowast λlowast) in matrix G inequation (28) is

nabla2CCL Clowast νlowast λlowast( 1113857

nabla2CCL Clowast νlowast λlowast( 1113857 0 0 0

0 2λlowast1 0 0

0 0 2λlowast2 0

0 0 0 2λlowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)

According to the results of equation (30) and λi ge 0 fori 1 2 3 we can say that nabla2CCL(Clowast ]lowast λlowast) is a strict positivedefinite matrix and the coefficient matrix G in equation (28)is a negative semidefinite matrix which means that theequilibrium point (Clowast ]lowast λlowast) is the asymptotically stablepoint of the network

4 Results and Analysis

41 Results of the Proposed Algorithm ree sets of typicalturbulence level measurement test results are selected to beprocessed e independent variable and dependent variabledata in equation (3) are shown in Table 1

e LPNN method is used to iteratively solve the tur-bulence level corresponding to the three sets of test dataabove in Table 1 Considering convergence speed the initialvalue of iteration is selected as 10minus4 10minus5 10minus6 and 10minus7respectively and the iteration time step is Δt 10minus 8 eiteration results of variable C are shown in Figure 3 eiteration results of variables ] and λ have no concern withsolving the turbulence level so they are not listed here

From Figure 3 we can easily find that variable C con-verges to a fixed value within approximate 4 times 106 steps andconsistent results can be obtained with different initialvalues e results indicate good and stable convergence ofthe LPNN method

g1 (C) ΣΔt

Δt

Δt

(v1k)2

v1k

λ1k

cik

()2 zndash1

zndash1

zndash1+

+

Σ+

+Σ

ndashndash

Σ+

+

Σ Σ+

+ndash

дg1 (C)дci

дS (C)дci

2

C

дgj (Ck)дci

Σ λjk

j=1

3

Figure 2 Architecture diagram of LPNN for inequality constrained optimization problem


Table 1 Data on turbulence level measurement test

M 0449 M 0516 M 0599

R

θ21113969

r

θ21113969

r

θ21113969

0146 0093 0147 0058 0146 00740177 0110 0179 0068 0178 00730214 0112 0217 0080 0215 00790257 0111 0261 0092 0259 01000413 0181 0426 0159 0421 01800742 0314 0788 0273 0773 0279

25

2

15

1

05

0

C1

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times10ndash4

times104

times106

18

16

14

12

1

08

06

04

02

00 05 1 15 2

(a)

C2

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8

1

05

0

ndash05

ndash1

ndash15

ndash2

(b)

C3

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

10

8

6

4

2

0

ndash2

(c)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(d)

Figure 3 Continued


times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420

ndash2ndash4ndash6ndash8 1 15 2

(e)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(f)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(g)

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(h)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(i)

Figure 3 Results of variables from the iteration process (a) C1 forM 0449 (b) C2 forM 0449 (c) C3 forM 0449 (d) C1 forM 0516(e) C2 for M 0516 (f ) C3 for M 0516 (g) C1 for M 0599 (h) C2 for M 0599 (i) C3 for M 0599


42 Comparison with the SOE Method e comparisonresults of the fitted hyperbola obtained by the LPNNmethodand the traditional SOE method are shown in Figure 4 eturbulence level and other flow qualities by the two methodsare shown in Table 2

For the case M 0449 Figure 4(a) shows that thescattered points of turbulence level measurement test resultsbasically conform to the hyperbolic distribution and thefitted hyperbolae obtained by the LPNN method and thetraditional SOE method basically coincide e results inTable 2 show that the turbulence level and other flowqualities are basically equal for M 0449 erefore in thecase when the scattered points conform to the hyperbolicdistribution both the LPNN method and the SOE methodcan be used to solve the turbulence level accurately

For the cases M 0516 and M 0599 Figures 4(b) and4(c) show that the scattered points deviate from the hy-perbolic distribution due to the unsteadiness of the flow andthere are even straight lines and other conic curves for thescattered points distribution e fitting curve obtained bythe traditional SOE method is an ellipse which is incon-sistent with the theoretical results from equation (3) Table 2shows the high goodness of fitting values and negative C3values for the SOE results of M 0516 and M 0599 C3represents the mean square value of T0 thus C3 should bepositive for all situations according to its physical meaningHowever the results of C3 obtained by the SOE method arenegative which indicate that overfitting has occurred forM 0516 and M 0599 e scattered points are overfittedinto ellipses by the SOEmethod which results in negative C3

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(a)

0

005

01

015

02

025

03

035

04

0 01 02 03 04 05 06 07 08 09r

θ2

PointLPNNSOE

(b)

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(c)

Figure 4 Comparison of hyperbola fitting from LPNN and SOE results (a) M 0449 (b) M 0516 (c) M 0599


values In this situation although equations (12) and (13) canstill be used to solve turbulence levels the results are not truevalues

e fitting curve for the cases M 0516 and M 0599obtained by the LPNN method is still hyperbola with theinfluence of the unsteadiness of the flow and the goodness offitting values are high which is consistent with the theo-retical results From the results we can easily find that theLPNN method is better than the SOE method and theresults verify the feasibility of applying the proposed algo-rithm to solve the turbulence level in wind tunnels

5 Conclusions

In this paper we propose a new hyperbola fitting algorithmfor turbulence level measurement test data based on theLagrange programming neural network method to solve theturbulence level in wind tunnels iteratively e LPNNmethod is stable and the equilibrium point is proved to beasymptotically stable e results show that better than theSOE method the LPNN method can be utilized to solveturbulence level and the results will not be affected by theunsteadiness of the flow e results of this paper will bemainly used to solve the turbulence level more precisely toensure the high accuracy of wind tunnel test results

Future research will continue to further develop thehyperbola fitting algorithm on the basis of the LPNNmethod to improve the accuracy of turbulence level resultsIn addition the LPNN method will also be improved to beapplied in data processing for wind tunnel test results

Data Availability

e data used to support the findings of this study are in-cluded within the tables of this article More turbulence levelmeasurement test data are available from the correspondingauthor upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] W D Harvey P C Stainback and F K Owen Evaluation ofFlow Quality in Two Large Nasa Wind Tunnels at TransonicSpeeds NASA-TP-1737 NASA Hampton Virginia 1980

[2] B Rasuo ldquoe influence of Reynolds and Mach numbers ontwo-dimensional wind-tunnel testing an experiencerdquo 1eAeronautical Journal vol 115 no 1166 pp 249ndash254 2011

[3] B Rasuo ldquoScaling between wind tunnels-results accuracy intwo-dimensional testingrdquo Transactions of the Japan Society forAeronautical and Space Sciences vol 55 no 2 pp 109ndash1152012

[4] D Damljanovic J Isakovic and B Rasuo ldquoT-38 wind-tunneldata quality assurance based on testing of a standard modelrdquoJournal of Aircraft vol 50 no 4 pp 1141ndash1149 2013

[5] D Damljanovic B Rasuo D Vukovic et al ldquoHypervelocityballistic reference models as experimental supersonic testcasesrdquo Aerospace Science and Technology vol 52 no 5pp 189ndash197 2016

[6] D Damljanovic D Vukovic G Ocokoljic et al ldquoA study ofwall-interference effects in wind-tunnel testing of a standardmodel at transonic speedsrdquo in Proceedings of the 30th Congressof the International Council of the Aeronautical SciencesDaejeon Korea September 2016

[7] G Ocokoljic B Rasuo and M Kozic ldquoSupporting systeminterference on aerodynamic characteristics of an aircraftmodel in a low-speed wind tunnelrdquo Aerospace Science andTechnology vol 64 pp 133ndash146 2017

[8] F K Owen T K McDevitt and D G MorganWind TunnelAngle of Attack Measurements Using an Optical Model Atti-tude System AIAA-2000-0414 AIAA Reno Nevada 2000

[9] F K Owen and A K Owen Effects of Freestream FlowQualityon Boundary Layer Transition in the National TransonicFacility AIAA-2013-1135 AIAA Grapevine TX USA 2013

[10] S L Treon F W Steinle and W R Hofstetter Data Cor-relation from Investigations of a High-Subsonic Speed Trans-port Aircraft Model in 1ree Major Transonic Wind TunnelsAIAA-69-794 AIAA Los Angeles CA USA 1969

[11] H L Dryden G B Schubauer and W C Mock Measure-ments of Intensity and Scale of Wind-Tunnel Turbulence andtheir Relation to the Critical Reynolds Number of SpheresNaca-Report-581 National Advisory Committee for Aero-nautics Washington DC USA 1937

[12] V Zinoviev and V Lebiga Application of Hot-Wire Tech-nology in a Blowdown Type Transonic Wind Tunnel AIAA-2001-0308 AIAA Reno Nevada 2001

[13] H Quix J Quest and C Brzek Hot-Wire Measurements inCryogenic Environment AIAA-2011-880 AIAA Orlando FLUSA 2011

[14] D Masutti E Spinosa O Chazot and M CarbonaroldquoDisturbance level characterization of a hypersonic blowdownfacilityrdquo AIAA Journal vol 50 no 12 pp 2720ndash2730 2012

[15] Y F Du J Lin and H S Ma ldquoMeasurement technique forturbulence level in compressible fluid by changing overheatratio of hot-wire anemometerrdquo Acta Aeronautica et Astro-nautica Sinica vol 38 no 11 p 121236 2017

Table 2 Comparison of flow quality from LPNN and SOE results

Method (Δmm)2 (C1) minus2(Δmm)(ΔT0T0) (C2) (ΔT0T0)2 (C3) Tu () Goodness of fitting

M 0449 LPNN 203 times 10minus 5 minus297 times 10minus 6 927 times 10minus 7 054 09981SOE 202 times 10minus 5 minus290 times 10minus 6 911 times 10minus 7 054 09981

M 0516 LPNN 126 times 10minus 5 minus451 times 10minus 7 154 times 10minus 7 044 09974SOE 940 times 10minus 6 241 times 10minus 6 minus293 times 10minus 7 046 09992

M 0599 LPNN 133 times 10minus 5 936 times 10minus 8 161 times 10minus 7 048 09864SOE 628 times 10minus 6 629 times 10minus 6 minus792 times 10minus 7 052 09936


[16] R O Duda and P E Hart ldquoUse of the Hough transformationto detect lines and curves in picturesrdquo Communications of theACM vol 15 no 1 pp 11ndash15 1972

[17] D H Ballard ldquoGeneralizing the Hough transform to detectarbitrary shapesrdquo Pattern Recognition vol 13 no 2pp 111ndash122 1981

[18] J Princen J Illingworth and J Kittler ldquoA formal definition ofthe Hough transform properties and relationshipsrdquo Journalof Mathematical Imaging and Vision vol 1 no 2 pp 153ndash1681992

[19] G Borgioli L Capineri P Falorni S Matucci andC G Windsor ldquoe detection of buried pipes from time-of-flight radar datardquo IEEE Transactions on Geoscience and Re-mote Sensing vol 46 no 8 pp 2254ndash2266 2008

[20] Q M Zhao W Li and H L Zhou ldquoHyperbola fitting anddimension inversion of cylindrical target based on modifiedHough modelrdquo Journal of PLA University of Science andTechnology (Natural Science Edition) vol 13 no 4 pp 365ndash370 2012

[21] P D Sampson ldquoFitting conic sections to ldquovery scatteredrdquo dataan iterative refinement of the bookstein algorithmrdquo ComputerGraphics and Image Processing vol 18 no 1 pp 97ndash108 1982

[22] R Safaee-Rad I Tchoukanov B Benhabib and K C SmithldquoAccurate parameter estimation of quadratic curves fromgrey-level imagesrdquo CVGIP Image Understanding vol 54no 2 pp 259ndash274 1991

[23] K Kanatani ldquoStatistical bias of conic fitting and renormali-zationrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 3 pp 320ndash326 1994

[24] S J Ahn W Rauh and H-J Warnecke ldquoLeast-squares or-thogonal distances fitting of circle sphere ellipse hyperbolaand parabolardquo Pattern Recognition vol 34 no 12pp 2283ndash2303 2001

[25] X Tu X Lei W Ma X Zuo and R Man ldquoHyperbolaminimum-zone evaluation and fitting based on geometry-optimised search algorithmrdquo Measurement vol 127pp 205ndash209 2018

[26] V A Lebiga V N Zinovrsquoev and A Y Pak ldquoUsing a hot-wireanemometer for measurement of characteristics of a randomacoustic field in compressible flowsrdquo Journal of AppliedMechanics and Technical Physics vol 43 no 3 pp 488ndash4922002

[27] V N Zinovev V A Lebiga and A Y Pak ldquoPreliminaryresults of flow fluctuation measurements in the cryogenictransonic wind tunnelrdquo Progress in Flight Physics vol 3pp 3ndash14 2012

[28] S R C Ali J Wu and R Radespiel High-frequency Mea-surements of Acoustic and Entropy Disturbances in a Hyper-sonic Wind Tunnel AIAA-2014-2644 AIAA Atlanta GAUSA 2014

[29] J Wu P Zamre and R Radespiel ldquoFlow quality experimentin a tandem nozzle wint tunnel at Mach 3rdquo Experiments inFluids vol 56 p 20 2015

[30] T Schilden W Schroder and S R C Ali ldquoAnalysis ofacoustic and entropy disturbances in a hypersonic windtunnelrdquo Physics of Fluids vol 28 p 56104 2016

[31] F Munoz J Wu and R Radespiel Freestream DisturbancesCharacterization in Ludwieg Tubes at Mach 6 AIAA SciTechForum San Diego CA USA 2019

[32] Y F Du J Lin and X N Wang ldquoMeasurement techniqueoptimization of turbulence level in compressible fluid bychanging overheat ratio of hot wire anemometerrdquo ActaAeronautica et Astronautica Sinica vol 40 no 12 Article ID123067 2019

[33] S Zhang and A G Constantinides ldquoLagrange programmingneural networksrdquo IEEE Transactions on Circuits and SystemsII Analog and Digital Signal Processing vol 39 no 7pp 441ndash452 1992


flow characteristics In some cases there are even straightlines and other conic distributions which make it difficult tosolve turbulence level precisely

Many researchers have studied hyperbola fitting andthe methods utilized differ due to different applicationscenarios Hough transform (HF) is a method of graphdetection and fitting based on pattern recognition e HFmethod converts scattered points in the image space tothose in the parameter space and converts the problem ofscattered points fitting to the problem of solving extremevalue in the parameter space thereby realizing the detec-tion and fitting of the hyperbola e HF method has theadvantages of strong robustness but its shortcomings oflow computational efficiency limit its real-time application[16ndash20] e least-squares (LS) method is the most com-monly used method for hyperbola fitting e key idea ofthe LS method is to establish a suberror function for eachscattered point and use the sum of the squares of allsuberror functions as the error function of hyperbola fit-tinge suberror function can be expressed by residuals ororthogonal distances between scattered points and fittedhyperbolic curves Hyperbolic curves are obtained bycalculating the extreme value of the error function e LSmethod has been widely used due to its ease of applicationand high computational efficiency [21ndash25] Both Lebiga VA[12 26 27] and Radespiel R [28ndash31] have done a lot ofresearch in the field of turbulence level measurement andobtaining the fitting solution in compressible flow eyconducted a comprehensive and thorough analysis offluctuation measurement results which provided datasupport for the uncertainty analysis of subsequent windtunnel tests However they have not conducted too muchresearch on the hyperbola fitting method Instead thesolution of the overdetermined equations (SOE) method isused directly to calculate the hyperbolic parameters tocomplete the hyperbola fitting e SOE method is basedon the LS method which means it has high computationalefficiency while the SOE method cannot obtain preciseturbulence level results due to overfitting if the scatteredpoints deviate from hyperbolic distribution

In order to solve the problem mentioned above thispaper proposes using the constrained least-squares (CLS)method to convert the problem of hyperbola fitting to theinequality constrained optimization problem and then usingthe Lagrange programming neural network (LPNN) methodto solve turbulence level iterativelye results show that theLPNN method is superior to the traditional SOE methodwhich verify the feasibility of the LPNN method for solvingturbulence level in wind tunnels

2 The SOE Method

According to the literature [26ndash32] the response function ofthe constant temperature hot-wire anemometer (CTA) incompressible flow is

ΔEE

FCTAΔmm

minus GCTAΔT0

T0 (1)

where E is the output voltage of CTA m and T0 are the gasmass flow rate and total temperature of the hot-wire probemeasuring point and FCTA and GCTA are the mass flow rateand total temperature sensitivity coefficients of CTA re-spectively FCTA andGCTA are only related to the overheatingratio namely the working temperature of the hot-wireprobe under a fixed flow condition Dividing equation (1) byGCTA and defining θ 1GCTAΔEE as the response functionand r FCTAGCTA as the independent variable we can get

θ Δmm

r minusΔT0

T0 (2)

Take the mean square value of equation (2)

θ21113969

1113888 1113889

2

Δmm

1113874 11138752r2

minus 2Δmm

1113874 1113875ΔT0

T01113888 1113889r +

ΔT0

T01113888 1113889

2

(3)

From equation (3) we can know that the responsefunction of CTA conforms to the hyperbolic relationship

with r as the independent variable and

θ21113969

as the dependentvariable e two-dimensional scattered points (RΘ) to befitted can be obtained by continuously changing the overheatratio n times and recording the output voltage of CTA at thesame time

R r1 r2 rn1113858 1113859T

Θ

θ211113969

θ221113969

θ2n1113969

1113876 1113877T

(4)

Construct the coefficient matrix A the variable vector Xto be sought and the nonhomogeneous term vector Π ofoverdetermined equations

A

r21 r1 1

r22 r2 1

⋮ ⋮ ⋮

r2n rn 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

X Δmm

1113874 11138752minus 2Δmm

1113874 1113875ΔT0

T01113888 1113889

ΔT0

T01113888 1113889

2⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

T

Π

θ211113969

1113874 11138752

θ221113969

1113874 11138752

θ2n1113969

1113874 11138752

1113890 1113891

T

(5)

erefore the flow fluctuations can be obtained bysolving the following overdetermined equations

A middot X Π (6)

Since A is not a square matrix it is impossible to solvethe overdetermined equations by solving the inverse matrixof A Instead the generalized inverse matrix of A can besolved by the singular value decomposition (SVD) methodto solve the overdetermined equations Any real matrix A(ntimes q) can be decomposed


A U middotWz 0

0 01113890 1113891 middot VT

(7)


A U middotWz

01113890 1113891 middot VT

(8)


A+ V middot Wminus1

z 01113960 1113961 middot UT (9)


X A+middot Π (10)


Δmm

Δ(ρu)

ρuΔρρ

+Δuu

ΔT0

T0 αΔTT

+ βΔuu

α 1

1 +(c minus 12)M2

β (c minus 1)M

2

1 +(c minus 12)M2

Δpp

Δρρ

+ΔTT

Δρρ

1c

Δpp

(11)


Tu Δuu

1113874 11138752

⎛⎝ ⎞⎠

12

H2 HG G21113858 1113859 middot X( 111385712

(12)

H 1

1 minus M2

G M

2

1 minus M2 middot

1β

(13)







f2(r) c1r

2+ c2r + c3 (14)


F(R) f r1( 1113857 f r2( 1113857 f rn( 11138571113858 1113859T

E Θ minus F(R) e1 e2 en1113858 1113859T

ei

θ2i

1113969

minus f ri( 1113857 (i 1 2 n)

(15)


S 1113944n

i1e2i (16)


C c1 c2 c31113858 1113859T

1113954X (17)





σ2m (Δm)2

σ2T0 ΔT0( 1113857

2

σmT0 (Δm) ΔT0( 1113857

(18)

ρmT0

σmT0

σmσT0



can be rewritten as

ρmT0

(Δm) ΔT0( 1113857

(Δm)2

1113969

ΔT0( 11138572

1113969 minus12


(Δmm)2

1113969

ΔT0T0( 11138572

1113969

(20)




argminC

S(C)


(22)





(23)


L(C ν λ) S(C) + 11139443

i1λi gi(C) + ]2i1113960 1113961 (24)




dCdt

minuszL(C ν λ)

zC

dνdt

minuszL(C ν λ)

zν

dλdt

zL(C ν λ)

zλ

(25)


Ck+1 Ck

minuszL Ck

νk λk

1113872 1113873

zCkmiddot Δt

νk+1 νk

minuszL Ck

νk λk

1113872 1113873

zνkmiddot Δt

λk+1 λk

+zL Ck

νk λk

1113872 1113873

zλkmiddot Δt

(26)


ck+1i c

ki minus

zS Ck1113872 1113873

zcki

+ 11139443

j1λk

j

zgj Ck1113872 1113873

zcki

⎡⎢⎢⎣ ⎤⎥⎥⎦ middot Δt i 1 2 3

]k+1l ]k

l minus 2λkl ]


λk+1j λk

j + gj Ck1113872 1113873 + ]k

j1113872 11138732

1113876 1113877 middot Δt i 1 2 3

(27)



01 02 03 04 050r

5

6

7

8

9

10

11θ2

times10ndash3




dCdt

dλdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ middot

C minus Clowast


⎤⎥⎥⎥⎥⎥⎦



⎤⎥⎥⎥⎥⎥⎦

(28)





0 0 2clowast2


2]lowast1 0 0

0 2]lowast2 0

0 0 2]lowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)








0 2λlowast1 0 0

0 0 2λlowast2 0

0 0 0 2λlowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)






g1 (C) ΣΔt

Δt

Δt

(v1k)2

v1k

λ1k

cik

()2 zndash1

zndash1

zndash1+

+

Σ+

+Σ

ndashndash

Σ+

+

Σ Σ+

+ndash

дg1 (C)дci

дS (C)дci

2

C

дgj (Ck)дci

Σ λjk

j=1

3




M 0449 M 0516 M 0599

R

θ21113969

r

θ21113969

r

θ21113969

0146 0093 0147 0058 0146 00740177 0110 0179 0068 0178 00730214 0112 0217 0080 0215 00790257 0111 0261 0092 0259 01000413 0181 0426 0159 0421 01800742 0314 0788 0273 0773 0279

25

2

15

1

05

0

C1

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times10ndash4

times104

times106

18

16

14

12

1

08

06

04

02

00 05 1 15 2

(a)

C2

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8

1

05

0

ndash05

ndash1

ndash15

ndash2

(b)

C3

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

10

8

6

4

2

0

ndash2

(c)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(d)

Figure 3 Continued


times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(e)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(f)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(g)

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(h)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(i)






04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(a)

0

005

01

015

02

025

03

035

04

0 01 02 03 04 05 06 07 08 09r

θ2

PointLPNNSOE

(b)

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(c)





5 Conclusions



Data Availability




References









































A U middotWz 0

0 01113890 1113891 middot VT

(7)


A U middotWz

01113890 1113891 middot VT

(8)


A+ V middot Wminus1

z 01113960 1113961 middot UT (9)


X A+middot Π (10)


Δmm

Δ(ρu)

ρuΔρρ

+Δuu

ΔT0

T0 αΔTT

+ βΔuu

α 1

1 +(c minus 12)M2

β (c minus 1)M

2

1 +(c minus 12)M2

Δpp

Δρρ

+ΔTT

Δρρ

1c

Δpp

(11)


Tu Δuu

1113874 11138752

⎛⎝ ⎞⎠

12

H2 HG G21113858 1113859 middot X( 111385712

(12)

H 1

1 minus M2

G M

2

1 minus M2 middot

1β

(13)







f2(r) c1r

2+ c2r + c3 (14)


F(R) f r1( 1113857 f r2( 1113857 f rn( 11138571113858 1113859T

E Θ minus F(R) e1 e2 en1113858 1113859T

ei

θ2i

1113969

minus f ri( 1113857 (i 1 2 n)

(15)


S 1113944n

i1e2i (16)


C c1 c2 c31113858 1113859T

1113954X (17)





σ2m (Δm)2

σ2T0 ΔT0( 1113857

2

σmT0 (Δm) ΔT0( 1113857

(18)

ρmT0

σmT0

σmσT0



can be rewritten as

ρmT0

(Δm) ΔT0( 1113857

(Δm)2

1113969

ΔT0( 11138572

1113969 minus12


(Δmm)2

1113969

ΔT0T0( 11138572

1113969

(20)




argminC

S(C)


(22)





(23)


L(C ν λ) S(C) + 11139443

i1λi gi(C) + ]2i1113960 1113961 (24)




dCdt

minuszL(C ν λ)

zC

dνdt

minuszL(C ν λ)

zν

dλdt

zL(C ν λ)

zλ

(25)


Ck+1 Ck

minuszL Ck

νk λk

1113872 1113873

zCkmiddot Δt

νk+1 νk

minuszL Ck

νk λk

1113872 1113873

zνkmiddot Δt

λk+1 λk

+zL Ck

νk λk

1113872 1113873

zλkmiddot Δt

(26)


ck+1i c

ki minus

zS Ck1113872 1113873

zcki

+ 11139443

j1λk

j

zgj Ck1113872 1113873

zcki

⎡⎢⎢⎣ ⎤⎥⎥⎦ middot Δt i 1 2 3

]k+1l ]k

l minus 2λkl ]


λk+1j λk

j + gj Ck1113872 1113873 + ]k

j1113872 11138732

1113876 1113877 middot Δt i 1 2 3

(27)



01 02 03 04 050r

5

6

7

8

9

10

11θ2

times10ndash3




dCdt

dλdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ middot

C minus Clowast


⎤⎥⎥⎥⎥⎥⎦



⎤⎥⎥⎥⎥⎥⎦

(28)





0 0 2clowast2


2]lowast1 0 0

0 2]lowast2 0

0 0 2]lowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)








0 2λlowast1 0 0

0 0 2λlowast2 0

0 0 0 2λlowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)






g1 (C) ΣΔt

Δt

Δt

(v1k)2

v1k

λ1k

cik

()2 zndash1

zndash1

zndash1+

+

Σ+

+Σ

ndashndash

Σ+

+

Σ Σ+

+ndash

дg1 (C)дci

дS (C)дci

2

C

дgj (Ck)дci

Σ λjk

j=1

3




M 0449 M 0516 M 0599

R

θ21113969

r

θ21113969

r

θ21113969

0146 0093 0147 0058 0146 00740177 0110 0179 0068 0178 00730214 0112 0217 0080 0215 00790257 0111 0261 0092 0259 01000413 0181 0426 0159 0421 01800742 0314 0788 0273 0773 0279

25

2

15

1

05

0

C1

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times10ndash4

times104

times106

18

16

14

12

1

08

06

04

02

00 05 1 15 2

(a)

C2

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8

1

05

0

ndash05

ndash1

ndash15

ndash2

(b)

C3

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

10

8

6

4

2

0

ndash2

(c)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(d)

Figure 3 Continued


times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(e)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(f)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(g)

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(h)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(i)






04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(a)

0

005

01

015

02

025

03

035

04

0 01 02 03 04 05 06 07 08 09r

θ2

PointLPNNSOE

(b)

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(c)





5 Conclusions



Data Availability




References










































σ2m (Δm)2

σ2T0 ΔT0( 1113857

2

σmT0 (Δm) ΔT0( 1113857

(18)

ρmT0

σmT0

σmσT0



can be rewritten as

ρmT0

(Δm) ΔT0( 1113857

(Δm)2

1113969

ΔT0( 11138572

1113969 minus12


(Δmm)2

1113969

ΔT0T0( 11138572

1113969

(20)




argminC

S(C)


(22)





(23)


L(C ν λ) S(C) + 11139443

i1λi gi(C) + ]2i1113960 1113961 (24)




dCdt

minuszL(C ν λ)

zC

dνdt

minuszL(C ν λ)

zν

dλdt

zL(C ν λ)

zλ

(25)


Ck+1 Ck

minuszL Ck

νk λk

1113872 1113873

zCkmiddot Δt

νk+1 νk

minuszL Ck

νk λk

1113872 1113873

zνkmiddot Δt

λk+1 λk

+zL Ck

νk λk

1113872 1113873

zλkmiddot Δt

(26)


ck+1i c

ki minus

zS Ck1113872 1113873

zcki

+ 11139443

j1λk

j

zgj Ck1113872 1113873

zcki

⎡⎢⎢⎣ ⎤⎥⎥⎦ middot Δt i 1 2 3

]k+1l ]k

l minus 2λkl ]


λk+1j λk

j + gj Ck1113872 1113873 + ]k

j1113872 11138732

1113876 1113877 middot Δt i 1 2 3

(27)



01 02 03 04 050r

5

6

7

8

9

10

11θ2

times10ndash3




dCdt

dλdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ middot

C minus Clowast


⎤⎥⎥⎥⎥⎥⎦



⎤⎥⎥⎥⎥⎥⎦

(28)





0 0 2clowast2


2]lowast1 0 0

0 2]lowast2 0

0 0 2]lowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)








0 2λlowast1 0 0

0 0 2λlowast2 0

0 0 0 2λlowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)






g1 (C) ΣΔt

Δt

Δt

(v1k)2

v1k

λ1k

cik

()2 zndash1

zndash1

zndash1+

+

Σ+

+Σ

ndashndash

Σ+

+

Σ Σ+

+ndash

дg1 (C)дci

дS (C)дci

2

C

дgj (Ck)дci

Σ λjk

j=1

3




M 0449 M 0516 M 0599

R

θ21113969

r

θ21113969

r

θ21113969

0146 0093 0147 0058 0146 00740177 0110 0179 0068 0178 00730214 0112 0217 0080 0215 00790257 0111 0261 0092 0259 01000413 0181 0426 0159 0421 01800742 0314 0788 0273 0773 0279

25

2

15

1

05

0

C1

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times10ndash4

times104

times106

18

16

14

12

1

08

06

04

02

00 05 1 15 2

(a)

C2

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8

1

05

0

ndash05

ndash1

ndash15

ndash2

(b)

C3

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

10

8

6

4

2

0

ndash2

(c)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(d)

Figure 3 Continued


times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(e)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(f)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(g)

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(h)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(i)






04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(a)

0

005

01

015

02

025

03

035

04

0 01 02 03 04 05 06 07 08 09r

θ2

PointLPNNSOE

(b)

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(c)





5 Conclusions



Data Availability




References










































dCdt

dλdt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦



⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦ middot

C minus Clowast


⎤⎥⎥⎥⎥⎥⎦



⎤⎥⎥⎥⎥⎥⎦

(28)





0 0 2clowast2


2]lowast1 0 0

0 2]lowast2 0

0 0 2]lowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)








0 2λlowast1 0 0

0 0 2λlowast2 0

0 0 0 2λlowast3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)






g1 (C) ΣΔt

Δt

Δt

(v1k)2

v1k

λ1k

cik

()2 zndash1

zndash1

zndash1+

+

Σ+

+Σ

ndashndash

Σ+

+

Σ Σ+

+ndash

дg1 (C)дci

дS (C)дci

2

C

дgj (Ck)дci

Σ λjk

j=1

3




M 0449 M 0516 M 0599

R

θ21113969

r

θ21113969

r

θ21113969

0146 0093 0147 0058 0146 00740177 0110 0179 0068 0178 00730214 0112 0217 0080 0215 00790257 0111 0261 0092 0259 01000413 0181 0426 0159 0421 01800742 0314 0788 0273 0773 0279

25

2

15

1

05

0

C1

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times10ndash4

times104

times106

18

16

14

12

1

08

06

04

02

00 05 1 15 2

(a)

C2

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8

1

05

0

ndash05

ndash1

ndash15

ndash2

(b)

C3

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

10

8

6

4

2

0

ndash2

(c)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(d)

Figure 3 Continued


times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(e)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(f)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(g)

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(h)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(i)






04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(a)

0

005

01

015

02

025

03

035

04

0 01 02 03 04 05 06 07 08 09r

θ2

PointLPNNSOE

(b)

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(c)





5 Conclusions



Data Availability




References










































M 0449 M 0516 M 0599

R

θ21113969

r

θ21113969

r

θ21113969

0146 0093 0147 0058 0146 00740177 0110 0179 0068 0178 00730214 0112 0217 0080 0215 00790257 0111 0261 0092 0259 01000413 0181 0426 0159 0421 01800742 0314 0788 0273 0773 0279

25

2

15

1

05

0

C1

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times10ndash4

times104

times106

18

16

14

12

1

08

06

04

02

00 05 1 15 2

(a)

C2

0 05 1 15 2 25 3 35 4tΔt

times10ndash4

times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

ndash4

ndash6

ndash8

1

05

0

ndash05

ndash1

ndash15

ndash2

(b)

C3

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash5

0 05 1 15 2

10

8

6

4

2

0

ndash2

10

8

6

4

2

0

ndash2

(c)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(d)

Figure 3 Continued


times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(e)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(f)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(g)

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(h)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(i)






04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(a)

0

005

01

015

02

025

03

035

04

0 01 02 03 04 05 06 07 08 09r

θ2

PointLPNNSOE

(b)

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(c)





5 Conclusions



Data Availability




References









































times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(e)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(f)

25

2

15

1

05

0

C1

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

times104

times10ndash4

0 05 1 15 2

18

16

14

12

1

08

06

04

02

0

(g)

times10ndash4

0 05 1 15 2 25 3 35 4tΔt times106

C2

1

05

0

ndash05

ndash1

ndash15

ndash2times104

times10ndash5

0 05

1086420


(h)

times10ndash5

0 05 1 15 2 25 3 35 4tΔt times106

C3

10

8

6

4

2

0

ndash2

times104

times10ndash5

0 05

10

8

6

4

2

0

ndash21 15 2

(i)






04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(a)

0

005

01

015

02

025

03

035

04

0 01 02 03 04 05 06 07 08 09r

θ2

PointLPNNSOE

(b)

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(c)





5 Conclusions



Data Availability




References












































04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(a)

0

005

01

015

02

025

03

035

04

0 01 02 03 04 05 06 07 08 09r

θ2

PointLPNNSOE

(b)

04

035

03

025

02

015

01

005

00 01 02 03 04 05 06 07 08 09

r

θ2

PointLPNNSOE

(c)





5 Conclusions



Data Availability




References











































5 Conclusions



Data Availability




References




























































Documents

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