Model Based Inversion Using the Element-Free Galerkin Method€¦ · for discontinuity profile reconstruction using the element-free Galerkin (EFG) method is presented in this paper.

Submitted April 2008

Model Based Inversion Usingthe Element-FreeGalerkin Method

by Xin Liu,*Yiming Deng,+Zhiwei Zeng,+Lalita Udpat and Jeremy S. Knopp+

ABSTRACTA modelbasediterative inversion techniquein nondestructive testing

for discontinuity profile reconstruction using the element-freeGalerkin(EFG) methodispresentedin this paper.Theadvantageof theEFG methodover thetraditional finite elementmodelis that it relieson a setof nodesin-steadof a complexmeshto discretize the solution domain. Consequently,only a small number of nodesthat representthe discontinuity profile needto beupdatedin eachiteration. This in turn avoidstheuseofa densemeshor labor-intensive remeshingprocedureand results in increasedefficacyand accuracyof solution. Theformulation of theforward EFG modelandresults of discontinuity profiling using an approachbasedon statespacesearcharepresented.Keywords: nondestructive testing, eddy current, meshlessmethods,element-freeGalerkin method,inverseproblem,statespacerepresentation.

INTRODUCTIONA critical issue in nondestructive testing (NDT) is the inverse

problem, which involves estimating the profile of the detectedcrack based on information contained in the probe measurements.Inverse problems are generally ill-posed and complete analyticalsolutions are seldom tractable. Practical solutions ranging fromsimple calibration methods to pattern recognition and constrainedsearch techniques are typically used in these problems (Udpa andUdpa, 1997).This paper presents a model based approach to the so-lution of inverse problems in eddy current testing using tree searchtechniques.

The model based approach (Li, 2001;Yan et al., 1998)to inverseproblems employs a forward model that solves the underlying gov-erning equation and predicts the measured signal in an iterativeframework, as shown in Figure 1.The iteration starts with an initial

Figure1 - Schematicrepresentationof theiterativeapproachof theinverseproblem(Li,2001).

* Department of Electrical and Computer Engineering, Michigan StateUniversity; 2120 Engineering, East Lansing, MI 48823; e-mail<[email protected]>.

t Department of Electrical and Computer Engineering, Michigan StateUniversity; 2120 Engineering, East Lansing, MI 48823.

:j:US Air Force Research Laboratory (AFRL!MLLP), Wright-Patterson AirForce Base, OH 45433.

740 Materials Evaluation/July2008

estimation of the discontinuity parameters, and the forward prob-lem is solved to determine the corresponding signal. The cost func-tion defining the error between the measured and predicted signalsis minimized iteratively by updating the discontinuity profile.When the error is below a predefined threshold, the procedurestops and the updated discontinuity parameters represent the de-sired solution.

Generally, in these methods the accuracy of the solution de-pends on the accuracy of the forward model used. Three criticalcomponents of the iterative procedure are: the forward model;discontinuity updating scheme; and choice of cost function thatis minimized.

Previous work using finite element method (FEM) based itera-tive techniques for inverting eddy current data has been reportedby Liu et al. (2001).This paper uses a numerical model based on theelement-free Galerkin (EFG) method (Belytschko et aI., 1994; Be-lytschko et al., 1996;Dolbow and Belytschko, 1998)as the forwardmodel. The advantage of the EFG method is that it relies only on aset of nodes instead of a complex mesh to discretize the solution do-main. In the inversion procedure, only a small number of nodesthat represent the discontinuity profile need to be updated. Thisscheme of discretization avoids use of a dense mesh in modelingtight cracks and labor-intensive remeshing in the iterative inversionprocedure.

This paper is organized as follows:. The next section gives a brief description of the forward eddycurrent model using the EFG method. The forward model is thenused in inversion procedures for estimating two-dimensionaldepth profiles of discontinuities.. The following section describes the iterative procedure based ona state space representation of the discontinuity profile, which isoptimized using a constrained tree search procedure. The formula-tion, implementation and preliminary results of the approach arepresented.

ELEMENT-FREEGALERKINMETHODIn the EFG method, the discrete system of equations is con-

structed via discretization of the domain by a set of nodes instead ofa complex mesh. However, in order to implement the Galerkin pro-cedure (Xuan, 2002), it is necessary to compute the integrals overthe solution domain; this is done by defining the support of thebasis functions using either a set of quadrature points or a back-ground mesh.

Let u(x) be a continuous function to be approximated by Uh(x).The EFG method utilizes a moving least squares approximation,which relies on three components: a weight function; a polynomialbasis; and a set of position-dependent coefficients.The weight func-tion is defined over the domain of influence of a node that plays animportant role in the performance of the EFG method in terms ofaccuracy of solution, complexity of computation (coding) and rateof convergence. The shape of the domain of influence is arbitrary,and typically, a circular or rectangular domain of influence is usedin two dimensions, as shown in Figure 2.

,.

(a)

Q

(b)

Figure 2 - Domain of influence: (a) circular; (b) square.

The selection of the weight function within the domain of in-fluence is not restrictive, but it must conform to the followingconditions:. The weight function is positive valued.. The weight function should be relatively large for the node Xjclose to X, and small for more distant Xj.In other words, it shoulddecrease in magnitude with the distance from x to Xj.. The weight function is continuous together with its derivativesup to the desired degree.

Typical weight functions include Gaussian, exponential andcubic splines. Figure 3 shows one possible choice of weight functionover a one-dimensional domain of influence.

In moving least squares approximation, the interpolant uh(x) isgiven by (Belytschko et al., 1996)

m

Uh(x)= L Pj(x)aj(x)=pT(x)a(x)}=o

(1)

wherem + 1 is the number of terms in the basis functionp}{x)are monomial basis functionsaj(x)are coefficients that depend on position x.

. e .. .x

.4

Figure 3 - The weight function over one-dimensionaldomain ofinfluence.

When x and x are different,Lancaster and Salkauskas (1981)expressthe local approximation by

(2) Uh(X,X)= f Pj(x)aj(x)=pT (x)a(x)}=o

where

x is the approximation pointx is a particular nodePj(x)are monomial basis functionsaj(x)are coefficients that depend on position x.

For example, in two dimensions, Uh(X) can be expressed in terms ofeither a linear or a quadratic basis as

(3) Uh(x, y) = ao(x, y)+ ~ (x, y)x+ a2(x, y)y

(linear basis)

(4)Uh(x, y) = ao(x, y)+ ~ (x, y )x+ a2 (x, y)y+

a3 (x, Y)X2 +a4 (x, y)xy+as (x,y)l

(quadratic basis)

The coefficients aj(x)are determined by minimizing the weight-ed error between local approximation and the nodal values Uj,thatis, by minimizing the following quadratic form:

(5)

n 2

J= LW(X-Xj) [Uh(X,Xj)-Uj ]}=1

= f W (x-x .)[ f p;(x. )a; (X)-U. ]

2

j=l }i=O} }

where

w(x - Xj) is a weight function with compact supportn is the number of nodes in the neighborhood of x where the

weight function does not vanish.

In matrix notation, Equation 5 can be rewritten as

(6) J= (pa-uf W(x)(Pa-u)

whereUT = (U1,U2,...,Un)are the unknownsp = [p;(Xj)]mxnW(x) = diag[w(x - X1),W(X - X2),.. .,W(X - Xn)].

The minimization of Jwith respect to a(x) leads to

(7) B(x)a(x)-C(x)u = 0

whereB(x)= pIW(x)pC(x)= pIW(x).

It follows that

(8) a(x) =B-1(x)C(x)u

MaterialsEvaluation/July2008 741

!--..

Y ---I.

' . .\

\. . ,'- ."'\ . I J. . .

-----/'. --Q'- - I

Substituting Equation 8 into Equation 2, and letting x = x, themoving least squares approximation can be written as

(9) Uh(X)= L<1>(x)u;=1 } }

where the shape functions <1>;are given by

(10)<1>;(x) =j~Pj (X)[ B-1 (x)c(x) 1; =pTB-1C;

The shape functions in Equation 10are used in a Galerkin proce-dure to construct the discrete system of equations. In the EFGmethod, each local integral involves a higher order of terms, andhence yields a more accurate solution in comparison with a linearinterpolation in conventional FEM.Additional details of implemen-tation and validation results of the EFG modeling method for one-,two- and three-dimensional problems in electromagnetics are pro-

.vided by Liu et al. (2006).

INVERSION BASEDON STATESPACESEARCHThe discontinuity updating scheme in the inversion procedure

uses a state space search and is presented in this section for two-dimensional problems. The two-dimensional model can be appliedto characterize the cross section of the crack in terms of width anddepth. Three major components of this approach are:. state space definition of the problem geometry and discontinu-ity representation. search procedure using tree representation of state space. definition of cost function.

State Space Representation of Discontinuity GeometryThe schematic of a typical two-dimensional problem geometry

is shown in Figure 4. Figure 4a is the true three-dimensional geom-etry of an infinite conducting plate with an infinitely long disconti-nuity. The time-harmonic current in the y-direction is applied usinga thin foil as the excitation source. The magnetic flux density associ-ated with the induced eddy currents is detected using magnetore-sistive sensors and constitutes the probe signal. Since the geometryextends to infinity in the y-direction, the geometry can be simplifiedto the two-dimensional problem shown in Figure 4b.

(a)

z

~x

x-z Plane

(b)

Figure4- TIreEFG model geometry: (a) three-dimensional view;(b) cross-section view.

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The discontinuity geometry is parameterized in terms of a statespace (Udpa, 1991;Udpa and Lord, 1989).In order to derive a statespace representation, the cross section of the discontinuity is dis-cretized in terms of M discrete cells along the length and N layersalong the depth into the test specimen, as shown in Figure 5 with M= 7 and N = 4. Each cell in the grid is of value 1 if there is no discon-tinuity and value 0 if the cell is in the discontinuity region. A state Xis an M x N matrix with element 1 or O.The set of all possible states(2MN)is called the state space. Figure 5 shows three possible states(discontinuity profiles), where Figures 5a and 5b are surface discon-tinuities, while Figure 5c is a subsurface discontinuity.

(a)

(b)

(c)

Figure 5 - Cross-sectionof discontinuities,representedby the states(discontinuity profiles): (a) surface discontinuity; (b) surfacediscontinuity variation; (c) subsurface discontinuity.

An initial state is a rectangular, one layer deep discontinuity pro-file that is estimated for the locations of the peaks in the measuredsignal. An objective state is the discontinuity profile that minimizesthe cost function. The search procedure is performed layer by layer.The best discontinuity profile at each layer is obtained by minimiz-ing the cost function, which is commonly defined as

(11)K 2

F = L IBn- BmnIn=l

where

Bnis the imaginary part of the model predicted normalcomponent of magnetic flux density

Bmnis the corresponding experimental signal measured usingthe magnetoresistive sensors

K is the number of data points in the two signals.

J

The cost function is computed for all the possible states of the cur-rent layer, and the state, corresponding to the lowest cost function,is selected in each layer.

Tree Representation of State SpaceIn order to derive the solution to the inverse problem, we search

for the best-fit discontinuity state that minimizes the difference be-tween the signals corresponding to true and reconstructed disconti-nuities. The exhaustive search procedure is summarized using atree structure, which is shown in Figure 6.The search is constrainedby two assumptions, namely: that the discontinuity grows vertical-ly downwards; and that the discontinuity grows narrower or staysthe same width.

d(O) Cost Function CO

Figure 6 - The tree structure.

The initial state d(O) is chosen as the root node in the tree struc-ture. In each layer, the nodes are expanded to obtain other statesthat satisfy the two assumptions, and the cost function of each nodeor state is computed using the forward model. The node corre-sponding to the minimum cost function is selected and retained forexpansion to the next layer. In Figure 6, the red nodes represent theselected minimum cost node in each layer.

The tree expansion continues until a leaf node is reached. AEthetree grows to the successive layer, if the minimum cost function be-gins to increase, the search stops and the selected node in the cur-rent layer, which corresponds to the minimum of the cost function,is labeled as the leaf node. Once a leaf node is obtained, the searchprocedure stops and the objective state is reached. In Figure 6, thecost function of the blue node is seen to be greater than the mini-mum cost function of the parent node in the previous layer. The redparent node inside the square corresponds to the desired disconti-nuity profile.

Tree Search ProcedureTheoverallsearchprocedureis divided into the threesteps:

. Step1- Estimation of the initial state d(O):A simple peak detection algorithm determines the discontinuity

location. The peak-to-peak distance in the input signal is used to es-timate the discontinuity length in the initial layer, as illustrated inFigure 7.

Figure 7a represents the input signal, where the peak-to-peakseparation of the signal estimates the initial state as a rectangulardiscontinuity of 6 mm length and one cell 0.33 mm deep. The loca-tion of the discontinuity could be on the top or bottom surface, asshown in Figures 7b and 7c. The cost functions corresponding tothese two discontinuity geometries are simulated using the EFGmodel, and the state with the lower cost function is chosen as initialstate d(O).

(a)

8

6

4

2

0

-2

-4

-6

(b).au..-10 10.a -2 0 2-6 -4 4 6 8

8

6

4

2

0

-2

-4

-6

(c)-8u..

-10 10.a -6 -4 -2 0 2 4 6 8

Figure 7 - A peak-ta-peak detection algorithm determines thediscontinuity location: (a) input signal; (b) initial guess of top surfacediscontinuity; (c) initial guess of bottom surface discontinuity.

. Step2- Tree search:The state space search procedure outlined above is performed.

The allowed discontinuity shapes in each layer are generated andmodeled, and the cost function associated with each node is com-puted. The node / discontinuity corresponding to the minimumcost function is retained. Figures 8a through 8f depict the node ex-pansions in each layer until the desired objective state is reached.. Step3- Check for stopping criteria:

If the minimum cost function in a layer is larger than the costfunction of the parent node, the search stops. The geometry withthe global minimum cost function is the desired discontinuity pro-file or the solution to the inverse problem.

Materials Evaluation/July 2008 743

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(a)

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(d) (e)

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(c)

(1)

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Figure 8 - Candidate discontinuity shapes used in the search in layer 2: (a)-{f) node expansions in each layer until desired objective state is reached.

...-2 ...... .

..

-6

oS

(a) -10 oS-6 .. -2

-2 .. ..

..

-6

oS

(d) .10 oS -6 .. -2

-2

..

-6

oS

(g) .10 oS-6 .. -2

-2 -2

..

-6

10.8

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..

-6

10

oS

(c) .10 oS 10-6 -,

. . .....

-6 .. -2

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. ....

.. .2 10

-2

10

Figure 9 - Top surface: (a) true discontinuity profile; (b) initial guess of a top surface discontinuity; (c) initial guess of a bottom surface

discontinuity; (d) reconstructed result. Bottom surface: (e) true discontinuity profile; (f) initial guess of a top surface discontinuity; (g) initial guess of

a bottom surface discontinuity; (h) reconstructed result.

744 Materials Evaluation/July 2008

..

-6

10-6L

(e) -10 oS .2-6 ..

.2 . .~

..

-6

10oS

(h)'l0 oS.. .2-6

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10 (1) -10 oS -6

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-6 -6

-8

(a) .10 -8 -6 .. .2

-8

e 10 (b) ,'0 -8 -6 .. .2

.2' ; .2

..

-6

-8

e 10 (c) .10 -8 -6 .. .2e 10

.. ..

-6 -6

(d) ~'0 -8 -6 .. .2 e 10 (e) -810 -8 -6 .. .2

.2

..

-6

. ~ (f) ~'O -8 -6 .. .2e 10 e 10

Figure 10 - Topsurface: (a) true discontinuity profile; (b) reconstructed result with previous cost function; (c) reconstructed result with new costfunction. Bottom surface: (d) true discontinuity profile; (e) reconstructed result with previous cost function; (f) reconstructed result with new costfunction.

Results Using Simulation DataThe algorithm was implemented on two discontinuity profiles

using simulation model data in place of experimental measure-ments. A linear excitation current of 3 kHz frequency was appliedalong the Y axis, as shown in Figure 4a. In Figure 9, the current foilis shown in dark gray and the light gray region represents a sectionof the aluminum plate. The background dots are the discretizationnodes of the background mesh in the EFG method. As before, theunits are in millimeters.

The cost function is square error as defined in Equation 11.Thesignal shown in Figure 7a is obtained for the true discontinuity pro-file in Figure 9a, which is a top surface discontinuity profile. In Step I,the cost function of the top surface discontinuity (Figure 9b) is 0.1708,while that of the bottom surface discontinuity (Figure 9c) is 0.3198.This leads to the choice of the top surface discontinuity as d(O). Thefinal reconstructed discontinuity profile is shown in Figure 9d.

Similar results for the bottom surface discontinuity are alsoshown. Figure ge gives the true discontinuity profile. In this case thecost function of the top surface discontinuity (Figure 9f) is 0.3766while that of the bottom surface discontinuity (Figure 9g) is 0.1770.Hence the initial discontinuity profile is determined as a bottomsurface discontinuity and the final reconstructed discontinuity pro-file is shown in Figure 9h.

Choice of Cost FunctionThe objective of inverse problems is to estimate the "best-fit"

profile of the detected discontinuity, based on the minimization ofthe cost function. Hence, different cost functions may lead to differ-ent resultant profiles and should be chosen with care.

The cost function acts as the evaluation function to describe theerror between the model prediction and input signal, which is ob-tained from the true discontinuity geometry. The most commonlyused cost function is the L2norm defined in Equation 11, which isused in obtaining the results in Figure 9. An alternative cost func-tion that can be employed is the L. (or Chebychev) norm defined inEquation 12:

(12) F=max(JBn -Bmnl), n=I,2,...K

This cost function is defined as the maximum difference betweenthe input signal and model prediction, among all the data points.

Two discontinuity profiles of large depth (shown in Figures lOaand lOd) were considered next. The reconstructed top surface dis-continuity profile obtained with the L2cost function is shown inFigure lOb, and results obtained with the L. cost function in Equa-tion 16 are shown in Figure lOc. The reconstructed discontinuityprofiles are significantly different in the two cases with two differ-ent cost functions. Similar results for a bottom layer discontinuityare shown in Figures lOe and lOf, with the two cost functions de-fined in Equations 11and 12.It is seen that for discontinuity profilesof large depth, the L. cost function gives more accurate reconstruct-ed results.

CONCLUSIONA model based inversion technique in conjunction with a tree

search algorithm has been presented. The approach has beendemonstrated for a two-dimensional eddy current problem withfoil excitation and using magnetoresistive pick-up sensors. TheEFG forward model is more advantageous in that it is more accu-rate and efficient than traditional finite element models used in pre-vious research (Liu et al., 2006). The state space search method isimplemented as the discontinuity updating scheme. Both surfaceand subsurface discontinuities can be reconstructed through thetree search technique. The dependence of the reconstructed discon-tinuity on the choice of the cost function to be minimized has beendemonstrated.

Implementation of the technique on experimental signals is cur-rently underway. The performance and robustness of the inversiontechnique will then be evaluated on field signals from aircraft test-ing. The study of the noise level contained in the measurement datais of interest; to address this problem, the use of a new cost functionsuch as the signal similarity is also under investigation.

Materials Evaluation/July2008 745

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