Advances in Engineering Software 40 (2009) 1097–1103

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Advances in Engineering Software

journal homepage: www.elsevier .com/locate /advengsoft

Adaptive finite element mesh triangulation using self-organizing neural networks

H. Jilani a, A. Bahreininejad b,*, M.T. Ahmadi a

a Civil Engineering Department, Tarbiat Modares University, Tehran, Iranb Industrial Engineering Department, Tarbiat Modares University, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 January 2009Accepted 28 June 2009Available online 18 July 2009

Keywords:Finite elementMesh generationNeural networksTriangulationSelf-organizing map

0965-9978/$ - see front matter Crown Copyright � 2doi:10.1016/j.advengsoft.2009.06.008

* Corresponding author. Tel.: +98 21 82883247.E-mail address: bahreininejad@modares.ac.ir (A. B

The finite element method is a computationally intensive method. Effective use of the method requiressetting up the computational framework in an appropriate manner, which typically requires expertise.The computational cost of generating the mesh may be much lower, comparable, or in some cases higherthan the cost associated with the numeric solver of the partial differential equations, depending on theapplication and the specific numeric scheme at hand.

The aim of this paper is to present a mesh generation approach using the application of self-organizingartificial neural networks through adaptive finite element computations. The problem domain is initiallyconstructed using the self-organizing neural networks. This domain is used as the background meshwhich forms the input for finite element analysis and from which adaptive parameters are calculatedthrough adaptivity analysis. Subsequently, self-organizing neural network is used again to adjust thelocation of randomly selected mesh nodes as is the coordinates of all nodes within a certain neighbor-hood of the chosen node. The adjustment is a movement of the selected nodes toward a specific inputpoint on the mesh. Thus, based on the results obtained from the adaptivity analysis, the movement ofnodal points adjusts the element sizes in a way that the concentration of elements will occur in theregions of high stresses. The methods and experiments developed here are for two-dimensional triangu-lar elements but seem naturally extendible to quadrilateral elements.

Crown Copyright � 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Almost any finite element (FE) analysis initially requires the do-main under consideration to be discretized into an appropriate FEmesh. This process is a tedious task to undertake manually and thequality of the resulting mesh depends upon the skills of the engi-neer. A poor quality FE mesh, where there are large differences be-tween the stresses of adjacent elements, will ultimately reduce theaccuracy of the analysis results. Adaptive mesh generation can beused to produce FE meshes having acceptable quality and accuracy.The objective of adaptivity and error estimation is to remove thehuman factors in mesh design and automate the FE procedure[1–5].

For any adaptive FE technique to be implemented, the availabil-ity of a mesh generator which works in accordance with the infor-mation received from the adaptive calculations of the posteriorierror estimator using FE solution is necessary [2]. A limit on theoverall error for the FE problem is fixed and the procedure is re-peated until a solution mesh is obtained with the error over the do-main being within acceptable limits.

009 Published by Elsevier Ltd. All r

ahreininejad).

Subsequent to the adaptive computations, mesh generationschemes may be used to increase the population of elements ofthe coarse mesh based on adaptivity results. Another way to createmeshes based on adaptivity analysis is to populate a mesh with therequired number of elements and subsequently reconfigure theelement sizes in order to comply with the FE and adaptivity anal-ysis conducted on the initial mesh.

FE meshes must have certain properties in order to be accept-able for computation. In this paper structural mechanics relatedFE has been considered. The following guidelines are consideredstandard [2]:

1. The mesh should be finer in regions where the solution isbelieved to be changing rapidly or to have large gradients. Thussmaller elements should be used near singularity points such asreentrant corners or cracks, near holes, near small features ofthe boundary, near the location of rapidly changing boundarydata, at and near in-homogeneities, etc.

2. All elements should be well proportioned. The aspect ratio ofthe element (namely, the ratio between its largest and smallestdimensions) should be close to unity. Square elements are thebest quadrilaterals, but even an aspect ratio of 1.5 or 2 isacceptable.

ights reserved.

1098 H. Jilani et al. / Advances in Engineering Software 40 (2009) 1097–1103

3. All interior angles of the element must be significantly smallerthan 180�. For example, a quadrilateral with three of its verticeslying on a nearly straight line is usually unacceptable.

4. Transition from large elements to small elements must be madegradually. The ratio between the sizes of two neighboring ele-ments may be 1.5 or 2 but should not be much greater than this.

Manevitz et al. [6] conducted mesh generation using a densityfunction to generate finer elements in the regions where the solu-tion is believed to be changing rapidly or to have large gradients.This density function was not based on adaptivity analysis.

In this paper the concept of self-organizing neural networks isused to generate an initial mesh. The same approach is used onceadaptive FE has been carried out on the initial mesh in order toreconfigure the size and location of the elements of the initial meshto comply with the adaptivity analysis.

The following sections describe the concepts of self-organizingneural network, finite element and adaptivity methods used in thispaper, in addition to the application of self-organizing neural net-works to the mesh generation and the adjustment of FE domains.Examples are presented to illustrate the performance of the pro-posed approach and finally concluding remarks are given.

2. Self-organizing neural networks

A self-organizing neural network, as described by Kohonen [7,8]is a system of neurons linked by a topology. Such a network canthen learn to adjust its weights’ parameters based on the inputin such a way as to automatically create a map of responsive neu-rons that topologically resembles the input data. Such networksare often referred to as self-organizing maps (SOM). A SOM con-sists of neurons organized on an array or two-dimensional grid.The number of neurons may vary from a few dozen up to severalthousand. Each neuron is represented by an n-dimensional weightvector, m = [m1, . . ., mn], where n is equal to the dimension of theinput vectors. The neurons are connected to adjacent neurons bya neighborhood relation, which dictates the topology, or structure,of the map. The SOM is trained iteratively. In each training step,one sample vector x from the input data-set is chosen randomly,and the distance between it and all the weight vectors of theSOM is calculated using some distance measure, e.g. Euclidean dis-tance. The neuron (unit) c whose weight vector is closest to the in-put vector x is called the Best-Matching Unit (BMU):

Fig. 1. Updating the BMU and its neighbors towards the input sample marked withx. Solid lines refer to the initial state and the dashed lines refer to the state afterupdating is made.

kx�mck ¼minifkx�mikg ð1Þ

where || || denotes the distance measure.After finding the BMU, the weight vectors of the SOM are up-

dated so that the BMU is moved closer to the input vector in theinput space. The topological neighbors of the BMU are also treatedin a similar way. This adaptation procedure stretches the BMU andits topological neighbors toward the sample vector as shown inFig. 1. The SOM updating rule for the weight vector of the unit iis given as:

miðt þ 1Þ ¼ miðtÞ þ hciðtÞ½xðtÞ �miðtÞ� ð2Þ

where t denotes time (time is virtual and corresponds to iteration).The x(t) is the input vector randomly drawn from the input data-setat time t and hci(t) is the neighborhood kernel (or learning rate)around the winner unit c at time t. The neighborhood kernel is anon-increasing function of time and of the distance of unit i fromthe winner unit c. It defines the region of influence that the inputsample has on the SOM [8]. In this paper the neighborhood kernelis selected based on trial and error which lies between 0 and 1.

3. Finite element and adaptivity

In the case of FE idealization based upon simple constant straintriangular elements the accuracy of the results depends on thetopology of the mesh chosen for FE analysis [2,6]. Adaptive reme-shing may be used to determine an efficient mesh by taking intoaccount the domain error. An efficient mesh is defined as the onein which the error is equally distributed over the domain [9].

Through adaptive remeshing the domain error is reduced aswell as uniformly distributed over the domain until the domain er-ror becomes less than a pre-defined error value [1,2,5].

For a constant strain triangular element, the elemental stressesobtained from FE analysis are generally defined as the stress resul-tants at the centroid of the element which satisfy the equilibriumand compatibility requirements. This element provides no furtherinformation on the distribution of the stresses within the elementwhich consequently leads to stress discontinuities at the boundarywith adjacent elements. This feature is most dramatically apparentin areas of high stresses within a mesh where a large number ofsmall elements are required to accurately model the domain. Byadaptively remeshing a domain, a denser mesh is automaticallyproduced in areas of high error. There are two methods of increas-ing the accuracy of an analysis adaptively:

1. h-Refinement: Where the element size is reduced.2. p-Refinement: Where the order of polynomial displacement

function is increased.

In this research only h-refinement method has been considered.The calculations of h-refinement approach are as follows.

3.1. Nodal averaging

Simple nodal averaging may be used to calculate the errorswithin the solution. The stresses of each element adjacent to eachnode in the mesh are summed and averaged. The stresses at each ofthe three nodes on a triangular element are then averaged to givethe value of smoothed stresses in each element as follows:

ð�rnÞi ¼1nc

Xnc

j¼1

j�rjj; i ¼ 1; . . . ; nn ð3Þ

ðrÞi ¼13

X3

j¼1

ð�rnÞj; i ¼ 1; . . . ; ne ð4Þ

H. Jilani et al. / Advances in Engineering Software 40 (2009) 1097–1103 1099

where �rn is the vector consisting of the averaged nodal stresses, ncis the number of elements common to a node, ne is the number ofelements in the mesh, nn is the number of nodes in the mesh, �r isthe vector of element stresses frxx;ryy;rxygT and r is the vector ofsmoothed element stresses.

3.2. h-Refinement

The standard linear elastic FE problem may be approximated to:

Ku� f ¼ 0 ð5Þ

where

K ¼Z

XBT DBdX ð6Þ

for which f is the vector of applied loads, u is the vector of nodal dis-placements, X is the problem domain, B is the strain displacementmatrix, D is the stress–strain matrix and K is the stiffness matrix.

The results of FE analysis will yield approximate solutions forthe displacement and stress. Therefore, the error in the analysisis the difference between the exact and calculated solutions. Thedisplacement error is given by:

e ¼ u� u ð7Þ

and the error in stresses is given by:

er ¼ �r� r ð8Þ

where u is the exact displacement and u is the approximated value.The error in the strain may be determined from the error in the

stresses [2].

ee ¼ D�1er ð9Þ

The error energy norm in the specific case of elasticity is definedin Refs. [1,2] as:

kek ¼Z

XðBeÞT DðBeÞdX

� �12

ð10Þ

Substituting Eq. (9) in (10) the following expressions for the en-ergy norm will be obtained.

kek ¼Z

XeTrD�1erdX

� �12

ð11Þ

The total energy norm is given as:

kuk ¼Z

XeT D�1edX

� �12

ð12Þ

where e ¼ �rD�1 [1].All these norms have been defined for the whole domain. In

practice, the norms for each individual element of the mesh arecalculated and summed as follows:

kek ¼Xne

i¼1

keki2

!12

ð13Þ

An adaptivity control parameter g is defined to quantify thepercentage error which is given as:

g ¼ kekkuk � 100 ð14Þ

If �g is the limit on the error, then while g > �g; the mesh elementrefinement parameter is defined as:

ni ¼keki

emð15Þ

where

em ¼ �gkuk2

ne

!12

ð16Þ

The mesh is refined according to:

hnew ¼hi

nið17Þ

until g 6 �g, where hi and hnew are the previous and new elementsizes determined from the adaptive analysis [2].

4. FE meshing using self-organizing neural networks

When applying the FE method to a given domain, one has to di-vide the domain into a finite number of non-overlapping subdo-mains (elements). (In two dimensions, the elements are usuallytriangles or quadrilaterals.) One also has to define a finite numberof nodes, which are the vertices of the elements, and possibly otherpoints as well. The collections of elements and nodes (and the con-nections among them), constitutes the FE mesh, whose quality isan essential ingredient in achieving accurate and reliable numericresults for all FE codes.

The density of the mesh affects the accuracy of the FE results. Afiner mesh would give more accurate solutions but also wouldnecessitate a larger computational effort. Thus the actual densityof the mesh used in a certain computation is a compromise be-tween accuracy and cost. The main parameter that controls thedensity of the mesh is called the mesh parameter; this is roughlythe size of the largest element in the mesh. Of course, the densityof the mesh should not necessarily be uniform. The mesh may befiner in some regions and coarser in others [6].

A self-organizing neural network, as described earlier, is a sys-tem of neurons (units) linked by a topology. Such a network canthen learn to adjust its weight parameters, based on the input, insuch a way as to automatically create a map of responsive neuronsthat topologically resembles the input data. The presented methodis independent of the specific topology chosen for the mesh. Theessence of the implementation method in this paper is the Koho-nen self-organizing neural network algorithm. This algorithm al-lows a network to choose its weights to fix its topologicalelements in weight space in such a way as to mimic as closely aspossible the arrangement of sample input data. In other words,the neural network becomes a representative map of the sampledata information. This is utilized in order to arrange for the place-ment of the FE mesh by identifying the mesh nodes with neuralnodes in addition to identifying the weight space with the physicalspace of the domain. This happens by randomly choosing samplepoints of the mesh domain as input to the self-organizing neuralnetwork.

Thus the coordination of elements of the mesh is carried outautomatically by the Kohonen algorithm with the only input nec-essary being sample points of the domain chosen randomly to re-flect the desired density function. The mesh then self-organizes tomake the best possible representation of the domain by the meshelements.

The aim of the proposed method is to reconfigure the size of theelements of a generated mesh after FE and adaptivity computa-tions are conducted This is carried out using the SOM network con-cepts in addition to adaptivity parameter (hnew), so that bychanging the locations of nodal coordinates, while preserving themesh topology, the value of hi is less than hnew for each element.

4.1. Creating the initial mesh using SOM method

A region is defined initially which represents the two-dimen-sional domain corresponding to the boundaries of the FE mesh to

1100 H. Jilani et al. / Advances in Engineering Software 40 (2009) 1097–1103

be generated. The geometry of this region is based on the region’sboundary coordinates. An example of such region is shown inFig. 2. Furthermore, the locations of loads and restrained pointsare also specified to be used for FE computations later. We call thisregion the physical domain (PD). Fig. 3 represents the PD for whichthe shaded area represents the initial mesh.

To generate elements inside the physical domain, a computa-tional domain (CD) is considered for which a regular mesh consist-ing of the number of desired triangular elements is constructed.Fig. 4 shows the computational domain (CD) and the regular meshinside it.

The aim is to map the regular mesh within the CD onto the PDon a one-to-one basis. As shown in Fig. 4, the CD consists of r axisand s axis (limited to �1 and +1), and the regular mesh lies within�a, +a, �b, +b on the r and s, respectively. The regular mesh con-sists of n and m number of nodes on horizontal and vertical direc-tions. The length of each element on the horizontal direction isgiven as:

di ¼a� ð�aÞ

n� 1¼ 2a

n� 10 6 i 6 n� 1 ð18Þ

Fig. 2. An initial region for SOM-based mesh generation.

Fig. 3. The physical domain with known coordinates.

Fig. 4. The computational domain containing a regular mesh.

and in the vertical direction is given as:

dj ¼b� ð�bÞ

n� 1¼ 2b

n� 10 6 j 6 n� 1 ð19Þ

Thus, the coordinates of the elements of the regular mesh in ther direction are obtained using:

ri ¼ �aþ ði � diÞ 0 6 i 6 n� 1 ð20Þ

and in the s direction are obtained using:

si ¼ �bþ ðj � djÞ 0 6 i; j 6 m� 1 ð21Þ

In order to map the CD coordinates onto the PD, a one-to-onerelation is required. For the xi and yi coordinates of the PD on thex axis and y axis, the following relation may be defined for the xi

coordinate:

xi ¼ a1 þ a2 � ri þ a3 � si þ a4 � ri � si ð22Þ

where a represents the mapping ratio in the horizontal direction.Hence:

x1

x2

x3

x4

26664

37775 ¼

1 r1 s1 r1s1

1 r2 s2 r2s2

1 r3 s3 r3s3

1 r4 s4 r4s4

26664

37775 ¼

a1

a2

a3

a4

26664

37775 ð23Þ

Eq. (23) may be represented as:

�X ¼ P � �a ) �a ¼ P�1 � �X ð24Þ

Using the information for �X (from the four nodes specified forthe PD) and P (r and s values of the CD), the mapping ratio valuescan be obtained.

The same procedure may be carried out for the mapping ratiovalues of the y axis which is given as:

yi ¼ b1 þ b2 � ri þ b3 � si þ b4 � ri � si ð25Þ

where b represents the mapping ratio in the vertical direction.Hence:

�Y ¼ P � �b ) �b ¼ P�1 � �Y ð26Þ

Once the one-to-one mapping mechanism is available and theinitial mesh within the PD is constructed (the shaded area withinthe PD in Fig. 3), the next stage is to use the SOM concepts to

Fig. 5. One-to-one mapping of the computational domain onto the physical domain.

H. Jilani et al. / Advances in Engineering Software 40 (2009) 1097–1103 1101

expand the generated nodes so that the initial mesh with the de-sired number of elements covers the whole PD.

A SOM network consisting of nodes (or neurons) representingthe generated nodes inside the CD is developed. Each networknode is associated with a coordinate location in CD (in neural net-work terminology, the weight space of the neuron is associatedwith the Cartesian coordinates).

The input to the network is a sequence of points (coordinates)in the CD, chosen randomly. For each input, a point in the networkis selected. The location of the chosen node is adjusted, as is thecoordinates of all nodes within the chosen point neighborhood.The adjustment is a movement of the neighborhood nodes towardthe input point. Then, by using the one-to-one mapping relationsdescribed earlier, the nodes in the PD will be adjusted. The move-ment of the winning node coincides with the movement of otherneighboring nodes toward the selected random point. The winnernode determination is based on the Euclidean distance given as:

kxchosen; ychosen � xwinner; ywinnerk 6 kxchosen; ychosen � xi; yik ð27Þ

where i = 1, . . ., n � 1 on the horizontal axis and i = 1, . . ., m � 1 onthe vertical axis. The weight changes to the SOM network are car-ried out according to Eq. (2).

The random point selection procedure is continued until themesh reaches a desired topological state. However, to reduce thecomputations, duplications in the use of random point selectionis avoided. Fig. 5 represents the mapping criteria from the CD tothe PD.

In order to distribute the points appropriately on the boundary(but after the network has reached the boundary), a one-dimen-sional Kohonen algorithm is used. Thus, while the two-dimen-sional Kohonen algorithm is executed, random points on theboundary are also selected for the one-dimensional Kohonenalgorithm.

4.2. FE and adaptivity computations

The generated mesh is then used for FE and adaptivity analysis.The locations of loads and restrained points have their correspond-ing Cartesian points on the generated mesh. Finite element analy-sis is carried out on the generated mesh considering the loadingand restrained conditions. Adaptivity analysis is carried out usingthe FE analysis results [2].

Mesh reconfiguration is then carried out using the adaptivity re-sults. This is fulfilled through the elements’ size parameters hi andelements’ adaptivity size parameters hnew.

4.3. Mesh reconfiguration

The reconfiguration procedure is carried out as follows.Initially all nodes throughout the mesh are categorized into the

following three types:

� Nodes within the mesh which are able to move freely withinthe mesh.

� Nodes on the mesh boundary which can move freely on theboundary.

� Nodes which are fixed and are not allowed to move.

A search is conducted throughout the mesh to find the nodaland element neighborhood criteria. Thus, the maximum numberof nodes in the neighborhood of a single node in the mesh is found.The nodal neighborhood matrix is constructed using the nodalnumbers. If a node has fewer numbers of neighbors than the max-imum, a value of �1 is inserted into the matrix for identificationpurposes. The element neighborhood matrix is also created usingthe nodal neighborhood matrix and the maximum number of ele-ments connecting to a single element. The elements of this matrixcorrespond to the element numbers in the mesh. If an element hasfewer numbers of neighbors than the maximum, a value of �1 isinserted into the matrix for identification purposes. The aim is tocategorize the elements in the mesh into layers corresponding tothe element undergoing size minimization. Fig. 6 shows the layercategorization of the mesh for which the layers around the darkcentral element are identified and categorized.

Another search is also carried out to validate whether the valueof hi > hnew for each element in the mesh is valid. If such conditionis reached for an element, this element undergoes sizeminimization.

In order to conduct the element size minimization, a SOM net-work is formed for the element undergoing size minimization. Thisnetwork has three neurons each corresponding to the elementnodes. The adjustment of weights in this network is same for thethree neurons and is calculated as:

wkðt þ 1Þ ¼ wkðtÞ þ w � a � ðxðtÞ �wkðtÞÞ ð28Þ

where the amount of the movement on mesh nodes correspondingto the neighborhood of the chosen element is determined by aparameter a (learning rate or neighborhood kernel which is be-tween 0 and 1) that decreases dynamically as the algorithm pro-ceeds in time and w is a parameter effecting a changes where wis between 0 and 1. Since the value of a is fixed for each layer of ele-

Fig. 6. Layer categorization of the mesh domain.

Fig. 9. Initial mesh domain having 2376 elements.

1102 H. Jilani et al. / Advances in Engineering Software 40 (2009) 1097–1103

ments, w is used to reduce a. The reason for using w is to control theelement size changes on different layers. The value ofa � (x(t) �wk(t)) is reduced according to w to avoid the elementsin surrounding layers being pulled excessively towards the elementunder size minimization which results in irregular element shapes(i.e. narrow geometry elements). Eq. (28) corresponds to Eq. (2)for SOM.

The input points to SOM network are divided into three types;the points within the mesh boundary, the points on the meshboundary, and the fixed points (e.g. loading and constrainedpoints). Such points are identified to avoid the movement of fixedpoints in addition to the movement of boundary points to withinthe mesh region.

5. Examples

The first example is a simply supported beam loaded on thecenter. The number of elements in the mesh is 2242. Fig. 7 showsthe domain after the initial mesh generation using SOM. Fig. 8shows the domain after reconfiguration of elements using FE,adaptivity and SOM methods.

The second example represents a dam loaded on the top lefthand side and restrained at the bottom edges. The number of ele-ments in the domain is 2376. Figs. 9 and 10 show the initial andreconfigured mesh domain after FE, adaptivity and SOM methodswere applied.

The third example is shown in Fig. 11. This domain underwentadaptive mesh generation using SOM method and the method gi-

Fig. 10. Final reconfigured mesh domain.

Fig. 7. Initial mesh domain having 2242 elements.

Fig. 8. Final mesh after reconfiguration. Fig. 11. Initial domain.

Fig. 12. Adaptive unstructured mesh generation using the method in [2] with 1048elements.

Fig. 13. Adaptive mesh generation using SOM method with 968 elements.

H. Jilani et al. / Advances in Engineering Software 40 (2009) 1097–1103 1103

ven in [2]. Fig. 12 represents the adaptive unstructured mesh usingthe method given in Ref. [2] with 1048 elements, for which the er-ror ratio is 31.45%. Fig. 13 shows the adaptive mesh using the pro-posed SOM method with 968 elements, for which the error ratio is34.5%. The proposed SOM method has 80 elements less than themethod in [2]. While, the error ratios are close, the SOM methodgenerates less number of elements than the method given in [2].

The method given in [2] is unstructured for which additional ele-ments may have been produced during the adaptive mesh genera-tion. Furthermore, the computational CPU times of the twomethods are relatively close, while the SOM consumes slightoverhead.

6. Conclusions

This paper presented the use of self-organizing map (SOM)neural networks based on Kohonen algorithm for generating finiteelement meshes. The initial mesh is generated using the SOM con-cepts based on two different domain criteria, physical domain andcomputational domain. A mesh is generated inside a computa-tional domain and is mapped on a one-to-one basis to a physicaldomain with a different mesh topology. Using SOM concepts ran-dom points are selected within the computational domain andconsequently the nodes of neighboring elements are moved to-ward the selected point. The mesh is then reconfigured and byusing a one-to-one mapping relationship, the mesh inside thephysical domain is reconfigured.

Consequently, the mesh is subjected to finite element and adap-tivity analysis based on the mesh data, load and restraining condi-tions. The adaptivity information is then used to reconfigure themesh elements through the movement of nodal points in a waythat the concentration of elements will occur in the regions of highstresses. The methods and experiments developed here are fortwo-dimensional triangular elements but seem naturally extend-ible to quadrilateral elements.

The computational cost of generating meshes increases with thenumber of elements within the domain. However, the method ishighly parallelizable and the computational cost may be decreasedusing parallel or distributed computing environment.

This research dealt with convex problems. However, further re-search on non-convex domains, quadrilateral, and mixed triangu-lar-quadrilateral elements, using other Euclidean distancemeasurement, and more advanced SOM neighborhood kernel(learning rate) functions may aid in discovering the benefits ofneural networks mesh generation. Furthermore, the hybrid ap-proach such as neuro-genetic algorithm or neuro-PSO (particleswarm optimization) methods for generating adaptive FE meshesmay worth the challenge.

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