Improved FEM model for defect-shape constructionfrom MFL signal by using genetic algorithm

K.C. Hari, M. Nabi and S.V. Kulkarni

Abstract: In-line inspection of ferromagnetic gas or oil pipe lines having pipe wall defects istypically accomplished using magnetic flux leakage (MFL) technique. An efficient modellingand computational scheme for forward model, during the process of solving inverse problems inmagnetostatic non-destructive evaluation using finite-element method is presented. The shape,size and place of defect are determined considering the nonlinearity of the pipe material usinggenetic algorithm as the optimisation technique. It is shown that the reduced model improvesthe FE computations significantly. The methodology for construction of defect shapes fromparticular MFL signals has been explained.

1 Introduction

Electromagnetic non-destructive evaluation (NDE) methodscan be classified into eddy-current- and magnetic fluxleakage (MFL)-based methods. The eddy-current-basedmethods are used to probe defects in conducting specimens.These rely on the change in the impedance seen by a coilthat has constant excitation. The defect region acts as sec-ondary to the coil. The MFL method is popular forfinding defects in ferromagnetic oil and gas pipelines.This relies on the change in the MFL in the vicinity of thedefect inside the pipe wall.

One of the principal challenges in the NDE is to deter-mine the shape, size and location of the defect in the speci-men, based on impedance or MFL signal. In the past,inverse NDE problems were solved using a mesh updateprocedure [1], neural networks [2] and gradient-basedoptimisation methods [3]. In [4], the eddy-current-basedfinite-element forward model with edge elements wasused. In [5, 6], a radial basis function neural network wasused as a forward model. The features of an integral formu-lation in terms of a two component electric vector potentialexpanded over the edge elements was used in [7] for theeddy-current-based model. In [8], the eddy-current testingsignals were assumed to be picture images and a templatematching method with the help of genetic algorithm (GA)was used to characterise cracks. In [4–8], the GA wasused as an optimisation technique to obtain a globaloptimum. In [4, 6], the GA is improved by using a simulatedannealing technique in the perturbation process.

Neural networks are advantageous in cases where rapidinversions are required. However, their main drawback isthat they require a large database for training. Anotherdrawback is that the performance depends on the dataused in training and testing. When the test signal is no

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-smt:20060069

Paper first received 26th May and in revised form 23rd December 2006

K.C. Hari and S.V. Kulkarni are with the Electrical Engineering Department,Indian Institute of Technology-Bombay, Powai, Mumbai 400076, India.

M. Nabi is with The Electrical Engineering Department, Indian Institute ofTechnology-Delhi, Hauz Khas, New Delhi, 110016, India

E-mail: svk@ee.iitb.ac.in

196

longer similar to the training data, performance degrades.In contrast, methods embedding the physical model intothe signal inversion process do not require a large database.The physical model and the optimisation procedure arecrucial for these methods. On the issue of convergence,gradient-based optimisation often fails to converge to theglobal optimum in the presence of multiple local optima.The GA-based approach, on the other hand, begins with alarge set of initial search points (population) using well-defined probabilistic tools to guide a search towardsregions in the search space that are more likely to containthe global optimum [9]. Consequently, it has shown betterrobustness and likelihood of convergence to the globaloptimum for a wide range of applications and is henceused in the present work as an optimisation approach.In this work, a reduced model derived from the

finite-element method (FEM) physical model of a ferromag-netic pipe with defect and nonlinearity is combined with GAoptimisation for efficient defect-shape reconstruction. Theregion with defect and nonlinearity is small comparedwith the whole domain. This allows development of thereduced-order model to increase computational efficiencyand reduce memory requirements. The proposed metho-dology of combining FE model reduction and GA optimi-sation is demonstrated for six different defects.

2 Problem statement and formulation

The MFL phenomenon can be explained by the change inpermeability distribution in the pipe in the presence ofdefects. Magnetic fields in a ferromagnetic material canbe generated either by currents or permanent magnets. Asthe velocity effects that arise from the movement of theinspection tool are generally neglected, the problemreduces to a magnetostatic problem.

2.1 Problem domain description

The MFL inspection tool along with the cross-section of thepipe is shown in Fig. 1. The tool consists of one backingiron (I), two permanent magnets (M) and two magnetbrushes (B). The cross-sectional region of the pipe isP. The area immediately around the defect constitutes the‘region of interest’ (ROI). The dashed line shows an array

IET Sci. Meas. Technol., 2007, 1, (4), pp. 196–200

of 30 Hall sensors placed below the ROI, which sense eitherthe Bx or By profile over their surface for any defect.The inspection tool and the pipe section along with the

surrounding air form the electromagnetic field region ofthe problem. The pipe section constitutes region V2,whereas all other regions together constitute V1. The inter-face between V1 and V2 is denoted by G12. The entiredomain V ¼ V1 < V2 is enclosed by a boundary Gd. Themagnetic vector is specified to be zero (Dirichlet boundarycondition). The forward computational problem consists ofusing the reduced FE model to efficiently obtain the fieldprofile for any defect in the pipe’s cross-section.For modelling different crack shapes, the ROI is divided

into an array of 710 rectangular cells as shown in Fig. 2.The reluctivity n of each cell can be made equal to that of airor iron, resulting in different geometries of the crack, asillustrated in the figure.

2.2 Formulation of the inverse problem

The inverse problem here consists of a defect-shape recon-struction that is achieved by minimising an objective func-tion, representing the difference between the FE modelpredicted signal and the measured signal. The variousissues related with the formulation are described below.

† Input signal to the problem: As both components of fluxdensity, Bx and By, carry the information of defect profile,any one can be chosen as the input signal to the optimisationprocedure of the inverse problem.† Defect geometry parameters: A particular defect in thewhole defect area is characterised by a set of 10 depths:d1, d2, . . . , d10, where di [ f0; 1; . . . ; 7g. Thus the valueof a particular depth is encoded as a 3 bit binary string,and the parameter set can be represented by a 30 bitbinary string, as shown in Fig. 2.† Objective function formulation: As in [4], the mini-misation of an error function between measured andmodel predicted signal can be recast as maximising of the

7 mm

20 mm

0 3 4 5 6 6 5 4 3 0

pipeair

000100 110 100 011101110101011000

Fig. 2 Defect characterisation

I

B

M M

B

283 mm

7 mm

20 mm

10 mm

20 mm

10 mm

P

40 mm 40 mm 10 mm 20 mm

Ω

Γ

Γ1

2

12

d

Hall sensors

ROI

Ω

20 mm (10 sections)

Fig. 1 Problem domain

IET Sci. Meas. Technol., Vol. 1, No. 4, July 2007

following fitness function

F ¼1

1þ CPm

i¼1 kBmeasuredi BFEM

i kð1Þ

where m is the number of points taken on the signal and C isa constant. The global maximum value of F is 1, corre-sponding to the case J ¼

Pmi¼1 kB

measuredi B

FEMi k ¼ 0.

Consequently, 0 , F 1. Finally, it may be noted that incase of the error J reaching a local minimum other thanits global minimum of zero, the relative ratio between thecorresponding local and global maximas of F is determinedby the constant C.† Optimisation procedure: TheGA [9] usually begins with arandomly generated set of potential solutions, called theinitial population. A fitness function is used as a measureof closeness of each member in the population to theglobal optimum solution. Subsequently, a new populationis generated by applying genetic operators such as reproduc-tion, crossover and mutation on the previous population. Theselection mechanism for reproduction often favours thehighly fit members in such a way that the members moreclose to the global optimum are assigned higher probabilitiesfor producing children. Crossover operations ensure that thenew population inherits highly fit features, while mutationmay add previously unexplored features into the new popu-lation. With this, the population drifts to a global or nearglobal solution after a few number of generations. In thisiterative process, the size of the system of equations (FEMmodel) to be solved is decreased using a reduced model,which will be presented in Section 3. Based on the issuesdiscussed for the formulation of the inverse problem, a flow-chart for the defect-shape construction is shown in Fig. 3.

In this problem, reproduction was implemented usingbiased Roulette-Wheel algorithmwith a two-point crossover.It has been generally observed that the value of C has asignificant effect in controlling premature convergence.The mutation probability is varied between 0.3 and 0.5.

2.3 Governing equation

As the effect of velocity due to the movement of inspectiontool is neglected, the problem effectively reduces to a magne-tostatic problem with permanent-magnet excitation. For suchproblems, the governing equation can be written as [10–13]

r1

mr ~A

¼ r ðnm0

~MÞ ð2Þ

where ~A is the magnetic vector potential, n ¼ (1/m) is reluc-tivity, m0 the permeability of air and ~M in (Amp/m) is theremanent coercive magnetisation intensity vector for the

Favg> = Fmax

Measuredsignal

StopYes

No

FEMreduced

Updateddefect profile

parameters

Initial defectprofile

parameters

Solution of

model

GA operatorsApply

Fig. 3 Flowchart

197

magnets. For the present 2D case, ~A has only the z component,whereas ~M can have bothMx andMy components. The wholedomain is divided into rectangular elements. The density ofFEs is high in and around the pipe region as compared withthe rest of the region, so as to have more accurate fieldvalues in the pipe (V2) region. Following the standardGalerkin method [10–14] and assembling elementalmatrices, the global FE model is given by

Kx ¼ f ð3Þ

where x is the vector of unknown nodal values of ~A.

3 Reduced-order forward model

Structure of the FE mode. In [15], an efficient algorithm forFE modelling of a composite domain consisting of the mag-netic regionV2 and the air regionV1 has been proposed. Forsuch problems, the structure of the global FE model (3) hasbeen shown to be

D1 D2 0

DT2 D3 þ F1ðnÞ F2ðnÞ

0 FT2 ðnÞ F3ðnÞ

24

35 xV1

xG12

xV2

24

35 ¼

f V1

f G12

f V2

24

35 ð4Þ

where the vector x of nodal values of ~A has been partitionedinto nodes inside V1, on interface G12 and inside V2,respectively. The coefficient matrix is also partitionedaccordingly. In particular, the entries dependent on thereluctivity n are shown explicitly as the submatrices F.

The reduced FE model. As illustrated in [15], the FEsolution for problems can be characterised as follows.Let x be the solution of (3) for a particular reluctivitydistribution overV2, denoted by n

. Then, for any other arbi-trary distribution n over V2, the solution x is given by [15]

x ¼ x þ

P1 0

P2 0

0 I

24

35z ð5Þ

where P1 and P2 are such that P2 is a full rank square matrixof the same size as nodes on G12, and D1P1þ D2P2 ¼ 0, sothat the whole matrix formed by them on the right-hand sideof (5) represents ker [D1 D2 0].

The vector z is a smaller vector of the same size as thenumber of nodes in fG12 <V2g and can be obtained bysolving the smaller system of equations

M 021 þM 0

23ðnÞ

~z ¼ G023ðn

Þ G0

23ðnÞ

x

ð6Þ

where

M021 ¼

PT1D

T1P1 þ PT

2DT3P2 0

0 0

" #ð7Þ

M023 ¼

PT2F1P2 PT

2F2

FT2P2 F3

" #ð8Þ

and

G023 ¼

0 PT2F1 P

T2F2

0 FT2 F3

ð9Þ

Clearly, the smaller system of (6), along with (5), can betaken as the reduced FE model for the problem. In particu-lar, as discussed in [15], the simple choice of

P1 ¼ D11 D2 and P2 ¼ I ð10Þ

reduces the above smaller model to that obtained from thesubstructural-frontal method.

198

3.1 Nonlinear model

If the reluctivity n over the pipe region V2 is a function ofthe magnetic vector potential ~A or x, then the problembecomes a nonlinear one. For such problems, nonlinear iter-ations have to be carried out by coupling the reduced modelwith the Newton–Raphson (N–R) method. This is achievedas follows.Using the simple choice of P matrices given in (10) for

the present problem, the field solution over the entiredomain as given in (5) can be written more elaborately as

xV1

xG12

xV2

24

35 ¼

xV1

xG12

xV2

24

35þ

D11 D2 0

I 0

0 I

24

35z ð11Þ

where, as usual, the suffixes denoted correspond to thevarious regions in the domain. If the regionV2 and its bound-ary G12 is considered together as ~V2, then the last two blocksof (11) can be simply written as

z ¼ x ~V2 x~V2

ð12Þ

Again, considering the right-hand side of (6), it can be easilyshown using (8) and (9) that

G23ðnÞx¼ M

023ðnÞx

~V2

ð13Þ

G23ðnÞx ¼ M 0

23ðnÞx~V2

ð14Þ

Finally, substituting (12–14) in (6), yields

M021 þM

023ðnÞ

x ~V2

¼ M021 þM

023ðn

Þ

x~V2

ð15Þ

The above equation represents a set of nonlinear equations(n is function of x ~V2

) to be solved using N–R method.

Now, linearising (15), the corresponding set of linearequations are given by

½J n Dx ~V2

h in¼ ½Rn ð16Þ

where

½J n ¼ ½M021 þ

@

@x ~V2

M023ðnÞx ~V2

h i" #n

ð17Þ

Here, n is the index of nonlinear iterations, J the Jacobian andR the residual. Elemental matrix of the second term in J is

n

k11 k12 k13 k14

k21 k22 k23 k24

k31 k32 k33 k34

k41 k42 k43 k44

26664

37775

þ@n

@B2

X4i¼1

k1ixi k1ixi k1ixi k1ixi

k2ixi k2ixi k2ixi k2ixi

k3ixi k3ixi k3ixi k3ixi

k4ixi k4ixi k4ixi k4ixi

26664

37775

@B2

@x1@B2

@x2@B2

@x3@B2

@x4

26666666666664

37777777777775

ð18Þ

IET Sci. Meas. Technol., Vol. 1, No. 4, July 2007

Computations can be further reduced by taking only the firstterm in (18) for kDx ~V2

k . e. The complete model is con-sidered for kDx ~V2

k , e. Nonlinearity of n is taken as

n ¼ C1eC2B

2

þ C3 ð19Þ

The values of the constants are: C1 ¼ 49.4, C2 ¼ 1.46 andC3 ¼ 520.6.

4 Results

Within the total pipe wall cross-section of 180 7 mm,ROI of 20 7 mm is modelled by a 2-D FE code usingrectangular elements to simulate the pipeline inspectionprocess. The number of nodes in the pipe region is 915and the total number of nodes is 7107. Using the proposedmethod, it is sufficient to solve 915 915 system oflinear equations instead of 7107 7107. The source ofmagnetisation is simulated by permanent magnets(My ¼ 800 000 A/m). Out of 270 total possible defects(surface and internal defects) in ROI, only 810 internalsurface defects are considered.A database of 1000 random defects is used to generate a

set of 10 initial defects (initial population). With this

0.1 0.11 0.120.06

0.062

0.064

0.066

Defect 1

Fmax

= 0.95

0.1 0.11 0.120.06

0.062

0.064

0.066

Defect 2

0.1 0.11 0.120.06

0.062

0.064

0.066

Defect 3

0.1 0.11 0.120.06

0.062

0.064

0.066

Defect 5

0.1 0.11 0.120.06

0.062

0.064

0.066

Defect 1

Fmax

= 0.99

0.1 0.11 0.120.06

0.062

0.064

0.066

Defect 2

0.1 0.11 0.120.06

0.062

0.064

0.066

Defect 4

0.1 0.11 0.120.06

0.062

0.064

0.066

Defect 6

Actual defectImproved method

Fmax

= 0.95

Fmax

= 0.95

Fmax

= 0.95

Fmax

= 0.99

Fmax

= 0.99

Fmax

= 0.99

Fig. 4 Reconstruction results for different defects

0 5 10 15 20 25 30−0.5

0

0.5

Bx profiles

By profiles

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

increase in defect depth

increase in defect depth

Fig. 5 Bx, By, (Wb/m2) over Hall sensor array for various defectdepths

IET Sci. Meas. Technol., Vol. 1, No. 4, July 2007

approach, a starting average fitness of defect sets is obtainedin between 0.8 and 0.9. The entire process of updating theM-matrices and obtaining the whole solution for nonlinearmodel using N–R method took 20 s (for MATLAB 6.5running on LINUX platform with Pentium IV). For a fullmodel, the respective time is 2 min. Results predictingdifferent defect profiles for maximum fitness values of0.95 and 0.99 are presented in Fig. 4. In this figure, ROIof Fig. 1 is shown in magnification. In Fig. 4, the valueson x-axis and y-axis represent corresponding x and y coordi-nates. It can be seen from the results for defects 1 and 2 thatwith an increase in the maximum fitness value, the error inthe defect reconstruction reduces. Figs. 5 and 6 show thevariation of the leakage magnetic flux density profile overthe array of 30 Hall sensors (numbered left to right) forvarious defect depths and widths. Finally, the accuracy ofdefect construction can be further improved by takingfiner mesh in ROI.

5 Conclusion

Inverse algorithms potentially provide a powerful means forthe detection and characterisation of defects in MFL dataanalysis. The challenge in an inverse algorithm is todevelop a computationally efficient forward model. In thispaper, a scheme for fast forward simulation of MFLsignals using a reduced nonlinear FE forward model is pro-posed. This model was applied to the reconstruction of theinternal surface defect shape with the use of GA. The algor-ithm has been tested for various defect constructions and theresults thus obtained are quite accurate with high simulationspeed. The reduction in CPU time for the simulation can beattributed to the reduction in the size of the FE system to besolved. The modelling of discontinuous bubbles can alsobe realised by different binary string representations.

It may be noted that the method allows remeshing of ROIfor better construction of the defect. However, an analysisof the exact effect of changing the mesh inside the ROIon the overall accuracy of the result is an important areaof future investigation. The developed efficient algorithmcan be extended to 3-D applications. Future work willalso concentrate on reducing computational costs usingnonlinear model order reduction methods such as orthog-onal decomposition and GA coupled to gradient-basedoptimisation techniques.

0 5 10 15 20 25 30−0.4

−0.2

0

0.2

0.4

Bx profiles

By profiles

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

increase in defect width

increase in defect width

Fig. 6 Bx, By, (Wb/m2) over Hall sensor array for various defectwidths

199

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