Submitted January 2006

An Element-free Galerkin Model for SimulatingMagneto-optic Imaging for Aerospace

Applicationsby x. Liu,*Y.Deng,+z. Zeng,+L. Udpat and J.S. Knopp:!:

ABSTRACTThe finite element technique has been employed intensively in model-

ing magneto-optical imaging of multilayer airframe structures and fasten-ers. However, the inclusion of a fastener hole with a tight crack results in avery dense finite element mesh. This paper introduces the use of theelement-free Galerkin technique in simulating magneto-optic imaging.The element-free Galerkin technique relies on only a cloud of nodes anddoes not require a complex mesh to describe the domain. The efficacy of theelement-free Galerkin technique is demonstrated by comparing the modelwith those obtained using traditional finite element modeling, and validat-ed with experimental images for three-dimensional magneto-optic imagingproblems.

Keywords: Element-free Galerkin technique, finite element technique,magneto-optic imaging, nondestructive testing, multilayer aircraftgeometry.

INTRODUCTION

Magneto-optic imaging (Fitzpatrick et al., 1993)is one of the re-cent nondestructive testing (NDT) techniques to detect surface andsubsurface cracks and corrosion in aircraft skin. magneto-opticimaging uses eddy current excitation and polarized light in con-junction with a magneto-optic sensor to form visual images that re-flect the structures of the sample being tested. In order to optimizethe design and operational parameters of the magneto-optic imag-ing system, a model using the linear finite element technique (finiteelement modeling) utilizing the A-V formulation was developed(Chen, 2000; Xuan et al., 2001). The reliance of finite elementmodeling on a mesh leads to some obvious disadvantages, how-ever (Belytschko et al., 1996). Finite element modeling needs adenser mesh for discontinuities such as air gaps, corrosion andtight cracks, which makes the computation more expensive andless accurate.

Recently, a new class of numerical methods known as the"meshless methods" (Belytschko et al., 1994;Belytschko et aL,1996;Dolbow and Belytschko, 1998)has been developed, where the un-known function is approximated entirely in terms of "local" func-tions defined at a set of nodes. No elements or characterization ofthe interrelationship of nodes is needed to construct the discreteequations. With the implementation of a moving least squares ap-proximation and a Galerkin formulation, the element-free Galerkintechnique has been successfully applied to problems in fracture me-chanics (Belytschko et al., 1994;Belytschko et al., 1996;Dolbow andBelytschko, 1998) and static and quasistatic electromagnetic fieldcomputation (Xuan et aL, 2004). This paper describes the formula-tion and implementation of the element-free Galerkin technique.Results of application for the simulation of magneto-optic testing ofaircraft skin structures are also presented.

* Department of Electricaland Computer Engineering,MichiganStateUniversity, East Lansing, MI 48824; e-mail <liuxin2@msu.edu>.

t Department of Electrical and Computer Engineering, Michigan StateUniversity, East Lansing, MI 48824.

* US Air Force Research Laboratory, Wright-Patterson Air Force Base, OH45433.

MODELING

Let QI and Q2be partitions of the solution domain, where QI de-notes the conducting region and Q2represents the surrounding freespace. Let A be the magnetic vector potential both in QI and Qz, andVbe the electric scalar potential in QI. J.Land crare the permeabilityand conductivity of the media, respectively. The governing equa-tions for the magneto-optic imaging problem are derived fromMaxwell's equations and expressed as (Xuan et al., 2001):

Vx!VXA+ jrocrA+crVV =0 inQlJ.L

(1)

(2) V.(jrocr A+crVV)=O inQl

(3) VX!VXA=Js inQ2J.L

where

Js = the assigned source current density.

Expanding the potentials in terms of shape functions associatedwith the nodes, we get

(4) A = I( AxjN/lx + AyjNjay + AzjNjaz)J

(5) V =IVjNjJ

where

AXj,Ayj and Azj are the three components of the vector potentialat node j

Vj is the scalar potentialNj is the shape function discussed in the following sectionax,ayand az are the Cartesian unit vectors.

Appropriate Dirichlet boundary conditions are applied at theexternal boundaries and In = cr(-jroA- VV) . n equal to zero is ap-plied on interfaces to obtain a unique solution. The Galerkin formu-lation (Xuan, 1997)is used together with an explicit coulomb gage(Chen, 2000) on Equations 1 to 3 to obtain a system of linear alge-braic equations:

(6) GU=Q

where

G is a complex and sparse matrixU is the vector of unknowns, including the electric scalar

potential and the three components of the magnetic vectorpotential at each node

Q is the load vector incorporating the current source.

Materials Evaluation/October2006 1009

Only nonzero entries of G need to be stored in the iterative solu-tion procedure. The solution of Equation 6 is then used to computevarious physically measurable quantities such as flux density B andinduced eddy current density in the area of interest.

ELEMENT-FREEGALERKINMETHODIn the element-free Galerkin technique, a set of nodes is used to

construct the discrete equations. However, to implement theGalerkin procedure, it is necessary to compute the integrals overthe solution domain; this is done by defining the support of thebasis functions using a set of quadrature points on a backgroundmesh.

The element-free Galerkin technique utilizes a moving leastsquares approximation. The central idea of the moving leastsquares approximation is that the approximation can be achievedby a "moving" process, which relies on three components: a weightfunction; a polynomial basis; and a set of position-dependent coef-ficients. The weight function is nonzero only over a small subdo-main around a particular node, which is termed the domain of in-fluence of that node.

Moving Least Squares Approximation

In moving least s3,uares (Dolbow and Belytschko, 1998), thelocal approximation U (x,x) is expressed as the inner product of avector of the polynomial basis p(x), and a vector of the position-dependent coefficients a(x)

(7) Uh(x, x) = ~pj (x)a; (x) = pT (x)a( x);=0

wherex is the approximation pointx is a particular nodem + 1 is the number of terms in the basis function.

The coefficients aj(x) are determined by minimizing the differ-ence between the local approximation and the nodal value Uj,thatis, by minimizing the following weighted discrete quadratic errorform:

(8)J=! w(x-Xj)[ Uh(X,Xj)-Uj r= I.W (x-x.

)[~p; (x.)a;(x)-u.]

2

j=l J ;=0 J J

Here, W (x - Xj)is a weight function with compact support, n isthe number of nodes in the domain of influence of x where theweight function does not vanish. Equation 7 in matrix notation isgiven by

(9) J=(pa-ut W(x)(Pa-u)

whereUT = (U1,Uz, ..., Un)are the unknownsP = [Pi (Xj)]mxn

(10) W(x)=diag[ W(X-X1)' W(X-X2)' ..., W(X-Xn)]

The minimization of Jwith respect to a(x) leads to a linear sys-tem

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

whereB(x)= p1W(x)pC(x) = p1W(x)

1010 MaterialsEvaluation/October2006

So a(x) can be solved as

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

Substituting Equation 12 into the local approximation Equation 7and letting x = x, complete the moving least squares approximationas

(13) Uh(X)= I.<P.(x)uj=l J J

where

the shape functions <Pjare given by

(14) <Pj(X)=~Pi(x)[B-1(X)C(X)].. =pTB-1Cj1=0 9

Note that the shape function does not satisfy the Kroneckerdelta criterion, that is, <Pj(Xk)'# Ojk; therefore, uh(Xj)'# Uj,which makesit difficult to impose boundary conditions. Techniques to solve thisproblem include using either a Lagrange multiplier or couplingwith standard finite elements at the boundary.

The shape function is inherently of a high order, which makesthe element-free Galerkin technique more accurate than linear finiteelement modeling.

Weight FunctionThe shape of the domain of influence of each node is chosen ar-

bitrarily. A circular or rectangular domain is typically used. In ourimplementation, a rectangular domain is used and the correspond-ing weight function at each point is derived as (Dolbow and Be-lytschko, 1998)

w(x-Xj) = Wx'Wy ,wz = w(rJ.w(ry ).w(rz)(15)

This paper uses Cubic splines weight function, which can bewritten in 1D as

(16) w{r;)=

2-- 4r2+4r33 I 1

4 2 4 3--4r.+4r. --r.3 I I 3 I

0

1forr;:::;;-2

fl.

or -<rj:::;;l,I=X,y,z2

for r; > 1

where

(17) IX-Xjl !Y-Yjl IZ-Zjlrx=-,ry=-,rz=-

dmx drrry dmz

and

(18) dmx = dmax ,cx' drrry= dmax 'Cy' dmz = dmax ,cz

Here, dmaxis a scaling factor, and (cx,Cy,cz) is the distance be-tween node Xjand its neighbor. Note that dmaxis chosen such thatmatrix B in Equation 12 is non-singular.

Discontinuity ApproximationMultiply-connected regions are typically involved in NDT prob-

lems. This leads to discontinuities of the tangential component ofcurrent density at the interface between different materials, whichcan be realized by introducing the discontinuity into the shapefunction and its derivatives at the interface. Since the continuity ofshape functions is inherited from the continuity of the weight func-tion, it is necessary to introduce discontinuity into the weight func-tion. This is realized by applying the visibility criterion (Belytschkoet al., 1996).

As shown in Figure 1, if there is no material discontinuity, thedomain of influence of node Xjis the total area of the square. How-ever, in the presence of a material discontinuity, the domain of in-fluence of node Xjshrinks to the area covered by the dashed hori-zontalline, and the weight function vanishes outside of that area.This procedure directly results in the discontinuity of the weightfunction, which in turn introduces the discontinuity of the shapefunction and its derivatives (Xuan et al., 2004).

1x.

]

.

2

Figure 1 - Domain of influence of the node adjacent to the materialdiscontinuity.

Expressing the potentials as a linear combination of shape func-tions in three-dimensional using Equation 14

(19)3N.. N

A = L. O.A- and Y = L. Y.<I>.

j=l J J j=l J J

where

OJ = <l>jax' <Pjay or <PjaZ

Substituting the above expansions into the integral equationsusing the Galerkin formulation on the governing Equations 1 to 3and applying the Lagrange multiplier technique, the matrix equa-tion is obtained and solved using a standard iterative techniquesuch as transpose-free quasiminimal residual (Freund, 1992).

8mm

(b) (c)

h=1nvn~I~ '-'mm

Air gap

RESULTSA typical 3D magneto-optic imaging inspection geometry used

in both finite element modeling and the element-free Galerkinmethods is shown in Figure 2.

The geometry consists of two layers of aluminum plates of 3 mmthickness and of infinite width and length. An infinitesimally smallair gap exists between the two layers. An infinite inductive foil of1 mm thickness, carrying a linear sinusoidal excitation current densi-ty of amplitude Iff'AI m2 and frequency 3 kHz is placed above theconducting multilayer plate at a distance of 1 mm. The solution re-gionis of dimensions [-12,12]mm x [-12,12]mm x [-8,8] mm.Afas-tener hole of 6 mm diameter is introduced, extending through thetwo layers, and a second layer tight crack of 5 mm radial length isalso modeled. The number of nodes used for discretization in the el-ement-free Galerkin technique is 18x 18x 12and in the finite elementmodeling is 28 x 28 x 24 and is illustrated in Figure 3.

"'0""'''' .....

0.01

0.015

0.005

0

-0.01

(a) .-O.Q1 .0.005-0.015 0.01

00000000000

0.01, , 0 0 0 0 0

000000000000000

0000000000000000000

0000000000000000000

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0000000000000000000

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00000000000

00000000000

-0.005 0000000000000000000

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(b) .0.01 .0.005 0.005 0.01 0.015-0.015

Figure 3 - 2D top view of: (a) finite element modeling mesh;(b) element-free Galerkin discretization of the solution domain.

0

(a) -8 mm

G-II1

L

1h=1 mm

Figure 2 - Magneto-optical imaging simulation geometry (both for finite element modeling and element-free Galerkin): (a) 3D view; (b) 2D topview; (c) side view of the multilayer plate with a 6 mm diameter fastener hole.

MaterialsEvaluation/October2006 1011

.,,10 .,,,.,

(a) .001

(d) .o01

.,,10

3.5

25

15

0.5

0.006 (c) 0010 .o01 .0.005 0005001

.0.005

001

0.005

.o005

.om

(e).om

Figure 4 - The normal component of the magnetic flux density in the discontinuity-free fastener hole: (a) finite element modeling, 3D view; (b) finiteelement modeling, 2D top view; (c) finite element modeling, 2D side view; (d) element-free Galerkin, 3D view; (e) element-free Galerkin, 2D top view;(f) element-free Galerkin, 2D side view.

Model Geometry 1: Fastener Hole without CrackSimulation results obtained using finite element modeling

and the element-free Galerkin technique for a discontinuity-freefastener hole are shown in Figure 4. The normal component ofthe magnetic flux density is plotted at the magneto-optic sensorlayer placed above the induction foil. A comparison of the com-putational time and mesh parameters are summarized in Table 1.

Table 1 Summary of simulation resultsfor a fastener hole free ofdiscontinuities

(a)

Figure 5 shows a comparison of the model predictions after thresh-olding and the experimental magneto-optic image.

Model Geometry 2: Fastener Hole with CrackSimulation results obtained using finite element modeling and

the element-free Galerkin technique for a fastener hole with a 5 mmradial crack are shown in Figure 6. The normal component of themagnetic flux density is plotted at the magneto-optic sensor layerplaced above the induction foiLA comparison of the computationaltime and mesh parameters are summarized in Table 2. Figure 7shows a comparison of the model predictions after thresholding.

CONCLUSIONThis paper presents a description of the formulation and imple-

mentation of the element-free Galerkin technique for electromag-netic NDT applications. The simulation results are compared withthose obtained using the well-established finite element models.Initial results show that the element-free Galerkin technique can besuccessfully applied to magneto-optic imaging testing problems,and offers a significant advantage over the linear finite elementmodeling.

88 II(b) (c)

Figure 5 - Binary results obtained by thresholding simulation results in Figure 4: (a) finite element modeling; (b) element-free Galerkin;(c) experimental image.

1012 Materials Evaluation/October2006

001

0.006

0.006

0.004

0.002

0

.oe",

.oe'"

.o006

.o006

(b) .om.0.01 .0.005

.' 104

3.5

3

2.5

2

15

0.5

0.005 001 (f) 0" .10 .5 0 5 " "

Finite Element Element-freeGalerkin

Peak value of

magnetic flux 3.422 mT 3.593 mT

density (34.223 G) (35.927 G)Computation time 3h O.5hNumber of

nodes 28 x 28 x 24 = 18816 18 x 18 x 12 = 3888

Discretization complex (non-uniform simple (uniformlycomplexity elements) distributed nodes)

:ii:~~::; DOl

0.005

"';...

~005

~OI

(b)(a)~OI oms D." 0.01'.0.01 ~.005 oms DOl

""""',....,."

~OIh"""

""":1;;' ~ms

0.005

(e)(d)om

0.005 ...,0.01 oms ~.005 ~DI

Figure 6 - The normal component of the 11Ulgneticflux density in the fastener hole with a 5 mm crack: (a) finite element modeling, 3D view;(b) finite element modeling, 2D top view; (c) finite element modeling, 2D side view; (d) element-free Galerkin, 3D view; (e) element-free Galerkin,2D top view; (f) element-free Galerkin, 2D side view.

finite element modeling with respect to computation time, num-ber of unknowns and mesh complexity. With the visibility criteri-on, we can set the discretization of element-free Galerkin to beperfectly symmetric and uniform when modeling a fastener hole,even in the presence of an asymmetric tight crack, while finite ele-ment mesh needs a more complex and wasteful mesh for model-ing the combination of fastener hole and tight crack. Furthermore,the interpolation in the element-free Galerkin technique is inher-ently of high order and hence is more accurate. More extensivetesting of the model for other geometries and excitation currentsare in progress.

Table 2 Summary of simulation results for a fastener hole with a5 mm radial crack

REFERENCES

Belytschko, T., Y.Y.Lu and 1. Gu, "Element-free Galerkin Methods," Interna-tional Journal of Numerical Engineering, VoL 37, 1994, pp. 229-256.

Belytschko, T., Y. Krongauz, D. Organ, M. Fleming and P. Krysl, "MeshIessMethods: An Overview and Recent Developments," Computer Methods inApplied Mechanics and Engineering, VoL 139, 1996,pp. 3-47.

Chen, c., "Finite Element Modeling of MOl for NDE Applications," Mas-ter's Thesis, Iowa State University, 2000.

Dolbow, J. and T. Belytschko, "An Introduction to Programming the Mesh-less Element Free Galerkin Method," Archives in Computational Mechanics,VoL 5, 1998,pp. 207-241.

Fitzpatrick, G.L., D.K. Thome, R.L. Skaugset, E.Y.C. Shih and w.c.L. Shih,"Magneto-optic/Eddy Current Imaging of Aging Aircraft: A New NDITechnique," Materials Evaluation, VoL 51,1993, pp. 1402-1407.

Freund, R.w., "Transpose-free Quasi-minimal Residual Methods for Non-hermitian Linear Systems," Advances in Computer Metlwds for Partial Dif-ferential Equations, Corfu, Greece, IMACS, 1992,pp. 2Q8-264.

Xuan, 1., "Finite Element and MeshIess Methods in NDT Applications,"Ph.D. Dissertation, Iowa State University, 1997.

Xuan, 1., B. Shanker, 1. Udpa, W. Shih and G.L. Fitzpatrick, "Finite Ele-ment Model for MOl Applications Using A-V Formulation," Review ofProgress in Quantitative Nondestructive Evaluation, D.O. Thompson andD.E. Chimenti, eds., VoL 20A, Melville, New York, AlP, 2001, pp. 385-391.

Xuan, 1., Z. Zeng, B. Shanker and 1. Udpa, "Element-free Galerkin Methodfor Static and Quasi-static Electromagnetic Field Computation," IEEETransactionson Magnetics, VoL 40, 2004, pp. 12-20.

Figure 7 - Binaryresultsobtainedby thresholdingsimulation resultsin Figure6: (a)finite elementmodeling;(b)element-freeGalerkin.

The numerical results predicted by the finite element modelingand element-free Galerkin techniques produce similar results andcompare well with the experimental image (without crack). How-ever, the element-free Galerkin technique is more beneficial than

Materials Evaluation/October 2006 1013

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",,4

3.'

3

2.'

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os(f)

,p.,, ., .o.oos

Finite Element Element-freeGalerkin

Peak value of

magnetic flux 3.547 mT 3.778 mT

density (35.467G) (37.779 G)Computation time 3h O.shNumber of

nodes 28 x 28 x 24 = 18816 18 x 18 x 12 = 3888Discretization complex (non-uniform simple (uniformly

complexity elements) distributed nodes)

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