Research ArticleDevelopment ofHigh-Order Infinite ElementMethod forBendingAnalysis of MindlinndashReissner Plates

D S Liu1 Y W Chen 1 and C J Lu2

1Department of Mechanical Engineering and Advanced Institute of Manufacturing with High-tech InnovationsNational Chung Cheng University Chiayi Taiwan2Department of Intelligent Manufacturing Engineering Guangdong-Taiwan College of Industrial Science amp TechnologyDongguan University of Technology Dongguan Guangdong China

Correspondence should be addressed to Y W Chen ywc0314gmailcom

Received 11 June 2020 Revised 7 September 2020 Accepted 15 September 2020 Published 1 October 2020

Academic Editor Nhon Nguyen )anh

Copyright copy 2020D S Liu et al)is is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An approach is presented for solving plate bending problems using a high-order infinite element method (IEM) based onMindlinndashReissner plate theory In the proposed approach the computational domain is partitioned into multiple layers ofgeometrically similar virtual elements which use only the data of the boundary nodes Based on the similarity a reduction processis developed to eliminate virtual elements and overcome the problem that the conventional reduction process cannot be directlyapplied Several examples of plate bending problems with complicated geometries are reported to illustrate the applicability of theproposed approach and the results are compared with those obtained using ABAQUS software Finally the bending behavior of arectangular plate with a central crack is analyzed to demonstrate that the stress intensity factor (SIF) obtained using the high-orderPIEM converges faster and closer than low-order PIEM to the analytical solution

1 Introduction

)e finite element method (FEM) is the most commonlyused numerical approach to accurately predict the static anddynamic behaviors of plate structures )e FEM is flexible inidentifying solutions to engineering problems that involvecomplex geometric shapes different material compositionsand different load forms therefore it has become the mostextensively implemented numerical method in commercialanalysis and simulation software in the market Neverthe-less the FEM necessitates first establishing an analysis gridcontaining elements and nodes a process that is timeconsuming and labor intensive Furthermore to obtainaccurate analysis results a relatively high number of ele-ments and nodes must be established in the calculationdomain and this process exerts a considerable burden oncomputer memory and reduces the calculation speedMoreover direct application of the FEM to the Mind-linndashReissner theory of beams and plates may experience thesame numerical error known as transverse shear locking

that is frequently encountered in FEM analyses Nguyen-Xuan et al [1 2] introduced edge-based and node-basedsmoothed stabilized discrete shear gap method (ES-DSGNS-DSG) in conjunction with the high-order shear defor-mation theory (HSDT) to investigate statics and free vi-bration behavior of plates the numerical examplesillustrated that both methods are free of shear locking andthe results are extremely efficient and accurate

In the past years meshless methods have been used [3 4]as alternatives to the FEM Suchmethods require only spatialnodes obviating the necessity of establishing an analysisgrid )erefore the disadvantages of the FEM can beovercome and the analysis time and labor required forextensive modeling tasks can be reduced Recently naturallystabilized nodal integration (NSNI) mesh-free formulationhas been extensively developed by )ai et al [5 6] andsuccessfully used in many complex plate structures such aslaminated composite sandwich plate and multilayerfunctionally graded graphene platelets reinforced compositeplates Numerical results show the current approach is

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9142193 13 pageshttpsdoiorg10115520209142193

promising and highly accurate Isogeometric analysis (IGA)is a recently introduced technique in the fields of numericalanalysis IGA was first proposed by Hughes et al [7] as anovel technology to bridge computer-aided design (CAD)and finite element analysis (FEA) Its essential idea is toadopt the same basis functions that are used in geometricdesign such as B-splines and nonuniform rational B-splines(NURBS) for the FEM simulations )e combined conceptof IGA allows for improved convergence and smoothnessproperties of the FEM solutions and faster overall simula-tions )us IGA has been successfully applied to solvedemanding problems as geometrically nonlinear analysis[8] buckling and free vibration analysis problem for lam-inated composite plates [9] and crack growth analysis inthin-shell structures by isogeometric mesh-free couplingapproach with a local adaptive mesh refinement scheme nearthe crack tip [10 11]

An alternative numerical method called infinite elementmethod (IEM) is a meshless method based on the FEM Inthis method the special similarity between elements can beused to easily create lots of elements as required and backsubstitution can be applied to degenerate an infinite numberof elements into a multinode super element )erefore theIEM can effectively prevent the problems of considerablememory usage and low computing efficiency and speed )epresented method is equally well suited for the usual reg-ularity closed domain and other types of singularitiesFurthermore it can be easily combined with FEM )atcher[12 13] has combined the concept of the FEM and similarsplitting to create many tiny elements near a singularitypoint to approximate Laplacersquos equation near a boundarysingularity Moreover to resolve the problem of structuralcracks Ying and Han [14 15] have produced many similartriangular elements near a crack tip and combined them intoa single element )e calculation results and theoreticalsolution regarding the stress intensity factor (SIF) werecomparable To solve two-dimensional (2D) and three-di-mensional (3D) crack problems Go et al [16 17] have usedthe similarity of quadrilateral elements to generate so-calledsuper elements by using iterative methods Liu et al [18 19]have combined the IEM with the FEM to solve static linearproblems and have continuously extended equations from2D to 3D Liu et al [20] further derived a high-order IEMequation for analyzing various 2D elastic static problemsthey compared their results with those of the traditional low-order IEM and with analytical solutions provided in theliterature )eir findings revealed that the results obtainedusing their method were more accurate than those obtainedusing the low-order IEM and were in good agreement withthe analytical solutions Furthermore Liu et al [21] com-bined the IEM with MindlinndashReissner plate theory and aclosed mode of the IEM to analyze the effects of the sizeposition and shape of a circular hole on the flexural stiffnessof a thin plate

MindlinndashReissner plate theory can be applied to ap-propriately reduce 3D problems to 2D problems and can beused to increase computing efficiency and reduce memoryusage Currently this theory is extensively used by scholarsTo increase the accuracy and speed of numerical analyses

several scholars have focused on the development of higherorder thin plate elements [22 23] High-order IEM andMindlinndashReissner plate theory are the available methodsrespectively However the conventional reduction processcannot be directly applied when these two theories arecombined Accordingly a new reduction process has beendeveloped to eliminate virtual elements in the IEM domainso that the IE range is condensed and transformed to form asuper element with the master nodes on the boundary onlyTo demonstrate the effectiveness of the proposedmethod wecompared the results with that obtained using ABAQUSsoftware Finally the analysis results were compared withthose obtained using the traditional low-order IEM

2 MindlinndashReissner Plate Theory

MindlinndashReissner plate theory is an extension of Kirch-hoffndashLove plate theory which considers shear deformationsthrough the thickness of a plate When MindlinndashReissnertheory is applied the following assumptions are used (a) thethickness of the plate remains unchanged during defor-mation (b) the normal stress through the thickness can beignored and (c) the normal line of the thickness is per-pendicular to the neutral axis line after deformation

On the basis of the aforementioned assumptions acomplete 3D solid mechanics problem can be reduced to a2D problem )erefore in-plane displacements can beexpressed in equations (1) and (2) and the transverse dis-placement can be expressed as indicated in equation (3)

u minuszθx(x y) minuszzw

zxminus cxz1113888 1113889 (1)

v minuszθy(x y) minuszzw

zyminus cyz1113888 1113889 (2)

w w(x y) (3)

where x and y are the in-plane axes located in the midplaneof the plate and z is the in-plane axis located along thedirection of plate thickness (Figure 1) θx and θy are therotations of the midplane about the y and x axes respec-tively and c is the angle caused by transverse shear de-formation Executing a transformation from physical tonatural coordinates yields the rotation and transverse dis-placements as follows

θx 1113944n

i1Hi(ζ η) θx( 1113857i

θy 1113944n

i1Hi(ζ η) θy1113872 1113873

i

w 1113936n

i1Hi(ζ η)wi

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

where Hi represents the n-node plate finite element shapefunction and (ζ η) represents the natural coordinates )enine-node-plate finite element stiffness matrix can be de-rived using MindlinndashReissner theory and by transforming

2 Mathematical Problems in Engineering

physical coordinates to natural coordinates )e associatedplate stiffness is expressed in equation (5) where [KB] and[KS] denote the bending stiffness and shear stiffness re-spectively )e plate material is considered linear elasticisotropic and homogenous )e resultant equation of eachelement can be expressed in equation (12)

[K] KB1113858 1113859 + KS1113858 1113859 (5)

where

KB1113858 1113859 h3

1211139461

minus111139461

minus1BB1113858 1113859

TDB1113858 1113859 BB1113858 1113859det[J]dζ dη

KS1113858 1113859 κh 11139461

minus111139461

minus1BS1113858 1113859

TDS1113858 1113859 BS1113858 1113859det[J]dζ dη

(6)

where h is the plate thickness κ is the shear energy cor-rection factor (usually 56) and [J] is the Jacobian matrix

[J]

zx

zζzy

zζ

zx

zηzy

zη

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

[BB] and [BS] comprise shape functions as presented inequations (8) and (9) respectively In addition [DB] and[DS] are related to the material properties of the model aspresented in equations (10) and (11) respectively

BB1113858 1113859

zH1

zx0 0

zH2

zx0 0 middot middot middot

zH9

zx0 0

0zH1

zy0 0

zH2

zy0 middot middot middot 0

zH9

zy0

zH1

zy

zH1

zx0

zH2

zy

zH2

zx0 middot middot middot

zH9

zy

zH9

zx0

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

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

(8)

BS1113858 1113859

minusH1 0zH1

zxminusH2 0

zH2

zxmiddot middot middot minusH9 0

zH9

zx

0 minusH1zH1

zy0 minusH2

zH2

zymiddot middot middot 0 minusH9

zH9

zy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)

DB1113858 1113859 E

1 minus ]2

1 ] 0

] 1 0

0 01 minus ]2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

DS1113858 1113859 E

2(1 minus ])

1 00 1

1113890 1113891 (11)

[K]

θx

θy

w

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

Mx

My

fz

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (12)

z w

x u

dx

y v

θx

θy

dy

Mx

Mxy

MyMxy

h

fz

Figure 1 Free-body diagram of a plate element

Mathematical Problems in Engineering 3

3 High-Order PIEM

31 Similarity Characteristic Figure 2 presents the basicconcept of the infinite element (IE) model In this model thecomputational domain is partitioned into multiple layers ofgeometrically similar elements For element I the localnodes i are numbered 1 2 and 9 If the global origin Oand ξ are considered the center of the similarity and theproportionality ratio respectively then element II can becreated )e global coordinates of elements I and II arerelated as presented in equation (13) According to equa-tions (13) and (7) the determinants of the Jacobian matricesof elements I and II are related as expressed in equation (13)Similarly according to equations (13) and (8) the relationbetween [BB] of element I and [BB] of element II can bepresented in equation (15)

xIIi y

IIi1113872 1113873 ξx

Ii ξy

Ii1113872 1113873 (13)

det[J]II

ξ2det[J]I (14)

BB1113858 1113859II

1ξ

BB1113858 1113859I (15)

)erefore as shown in equation (16) the bendingstiffness matrix [KB] of the first and second element layers isrelated

KB1113858 1113859II

h3

1211139461

minus111139461

minus1BB1113858 1113859

IITDB1113858 1113859

IIBB1113858 1113859

IIdet[J]IIdζ dη

h3

1211139461

minus111139461

minus1

1ξ

BB1113858 1113859IT

DB1113858 1113859I1ξ

BB1113858 1113859Iξ2det[J]

Idζ dη

h3

1211139461

minus111139461

minus1BB1113858 1113859

ITDB1113858 1113859

IBB1113858 1113859

Idet[J]Idζ dη

KB1113858 1113859I

(16)

To adapt the conventional IEM to MindlinndashReissnerplate problems the shear stiffness of the first element layer

[BS] can be partitioned into two submatrices namely [BlowastS ]

and [BlowastlowastS ]

BS1113858 1113859 BlowastS1113858 1113859 + B

lowastlowastS1113858 1113859 (17)

where

Blowasts1113858 1113859

0 0zH1

zx0 0

zH2

zxmiddot middot middot 0 0

zH9

zx

0 0zH1

zy0 0

zH2

zymiddot middot middot 0 0

zH9

zy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Blowasts1113858 1113859

minusH1 0 0 minusH2 0 0 middot middot middot minusH9 0 0

0 minusH1 0 0 minusH2 0 middot middot middot 0 minusH9 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(18)

Substituting equation (17) into equation (9) yields thefollowing

KS1113858 1113859 κh 11139461

minus111139461

minus1BlowastS1113858 1113859 + B

lowast lowastS1113858 1113859( 1113857

TDS1113858 1113859 B

lowastS1113858 1113859 + B

lowast lowastS1113858 1113859( 1113857det[J]dζ dη (19)

Let

KlowastS1113858 1113859 κh 1113946

1

minus111139461

minus1BlowastS1113858 1113859

TDS1113858 1113859 B

lowastS1113858 1113859det[J]dζ dη (20)

Klowast lowastS1113858 1113859 κh 1113946

1

minus111139461

minus1BlowastS1113858 1113859

TDS1113858 1113859 B

lowast lowastS1113858 1113859 + B

lowast lowastS1113858 1113859

TDS1113858 1113859 B

lowastS1113858 11138591113872 1113873det[J]dζ dη (21)

Klowast lowast lowastS1113858 1113859 κh 1113946

1

minus111139461

minus1Blowast lowastS1113858 1113859

TDS1113858 1113859 B

lowast lowastS1113858 1113859det[J]dζ dη (22)

I

II

X

Y

O

(xiII yiII)

(xiI yiI)

1

2

3

4

56

7

8

9

Figure 2 Geometrically similar 2D elements in IE formulation

4 Mathematical Problems in Engineering

)us equation (19) becomes

KS1113858 1113859 KlowastS1113858 1113859 + K

lowastlowastS1113858 1113859 + K

lowastlowastlowastS1113858 1113859 (23)

According to the geometric similarity the relationshipbetween the first and second element layers in terms of theshear stiffness matrix can be expressed as follows

KS1113858 1113859II

KlowastS1113858 1113859

II+ KlowastlowastS1113858 1113859

II+ KlowastlowastlowastS1113858 1113859

II

KlowastS1113858 1113859

I+ ξ K

lowastlowastS1113858 1113859

I+ ξ2 K

lowastlowastlowastS1113858 1113859

I

(24)

Substituting equations (16) and (24) into equation (5)yields the plate stiffness matrix as follows

[K]I

KB1113858 1113859I

+ KlowastS1113858 1113859

I+ Klowast lowastS1113858 1113859

I+ Klowast lowast lowastS1113858 1113859

I

[K]II

KB1113858 1113859I

+ KlowastS1113858 1113859

I+ ξ K

lowast lowastS1113858 1113859

I+ ξ2 K

lowast lowast lowastS1113858 1113859

I

⋮[K]

s KB1113858 1113859

I+ KlowastS1113858 1113859

I+ ξsminus 1

Klowast lowastS1113858 1113859

I+ ξ2(sminus 1)

Klowast lowast lowastS1113858 1113859

I

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)

32 Combined Stiffness of High-Order PIEM According toequation (25) the nine-node elements I II and s can bemapped using the same square-shaped master elementSpecifically these elements can be designated as similarelements when the coordinate of an element is similar to thatof other elements )e matrices of the first element layer canbe expressed as follows

KB1113858 1113859I

Ka minusBT

minusAT

minusBmK minusC

T

minusA minusC Kb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

KlowastS1113858 1113859

I

Klowasta minusB

lowastTminusAlowastT

minusBlowast

Klowastm minus C

lowastT

minusAlowast

minusClowast

Klowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Klowast lowastS1113858 1113859

I

Klowast lowasta minusB

lowastlowastTminusAlowastlowastT

minusBlowastlowast

Klowast lowastm minusC

lowastlowastT

minusAlowastlowast

minusClowastlowast

Klowast lowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Klowast lowast lowastS1113858 1113859

I

Klowast lowast lowasta minusB

lowastlowastlowastTminusAlowastlowastlowastT

minusBlowastlowastlowast

Klowast lowast lowastm minusC

lowastlowastlowastT

minusAlowastlowastlowast

minusClowastlowastlowast

Klowast lowast lowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

When equation (26) is substituted into equation (12) andthe result is expanded the equations of the s element layersin the computational domain can be derived as follows

Ka1 minusBT1 minusA

T1

minusB1 Km1 minusCT1

minusA1 minusC1 Kb1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δ0δm1

δ2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

f0

0

f2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

⋮

(27)

Kai minusBTi minusA

Ti

minusBi Kmi minusCTi

minusAi minusCi Kbi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δiminus1

δmi

δi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

minusfiminus1

0fi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

⋮(28)

Kas minusBTs minusA

Ts

minusBs Kms minusCTs

minusAs minusCs Kbs

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δsminus1

δms

δs

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

minusfsminus1

0fs

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (29)

where

Kai Ka + Klowasta + ξiminus 1

Klowast lowasta + ξ2(iminus 1)

Klowast lowast lowasta

Kbi Kb + Klowastb + ξiminus 1

Klowast lowastb + ξ2(iminus 1)

Klowast lowast lowastb

Kmi Km + Klowastm + ξiminus 1

Klowast lowastm + ξ2(iminus 1)

Klowast lowast lowastm

Ai A + Alowast

+ ξiminus 1Alowastlowast

+ ξ2(iminus 1)Alowastlowastlowast

Bi B + Blowast

+ ξiminus 1Blowastlowast

+ ξ2(iminus 1)Blowastlowastlowast

Ci C + Clowast

+ ξiminus 1Clowastlowast

+ ξ2(iminus 1)Clowastlowastlowast

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

In equations (27)ndash(29) δi is the nodal displacementvector associated with the ith node layer and fi is thecorresponding nodal force vector Combining the equationsfrom the first element layer to the sth element layer andassuming that no internal force is applied to the ith nodelayer (ie fs 0) can yield the following expression

Ka1 minusBT1 minusA

T1 0 0 0 0 0 0 0

minusB1 Km1 minusCT1 0 0 0 0 0 0 0

minusA1 minusC1 Q1 minusBT2 minusA

T2 0 0 0 0 0

0 0 minusB2 Km2 minusCT2 0 0 0 0 0

0 0 minusA2 minusC2 Q2 minusBT3 minusA

T3 0 0 0

⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱0 0 0 0 0 minusBsminus1 Km(sminus1) minusC

Tsminus1 0 0

0 0 0 0 0 minusAsminus1 minusCsminus1 Qsminus1 minusBTs minusA

Ts

0 0 0 0 0 0 0 minusBs Kms minusCTs

0 0 0 0 0 0 0 minusAs minusCs Kbs

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

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

δ0δm1

δ1δm2

δ2⋮

δm(sminus1)

δsminus1

δms

δs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

f0

0

0

0

0

⋮

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(31)

Mathematical Problems in Engineering 5

where Qi Ka(i+1) + Kbi If Ms Kbs in the last row ofequation (31) then

δs Mminus1s

sAδsminus1 + Csδms( 1113857 (32)

Substituting equation (32) into the second-to-last row ofequation (31) yields

δms Kms minus CTs M

minus1s Cs1113872 1113873

minus 1 sB + C

Ts M

minus1s As1113872 1113873δ4

Mminus1msNsδsminus1

(33)

Similarly substituting equations (32) and (33) into thesecond-to-last row of equation (31) yields

δsminus1 Qsminus 1 minus ATs M

minus1s As minus N

Ts M

minus1msNs1113872 1113873

minus 1Asminus1δsminus2 + Csminus1δms( 1113857

Mminus1sminus1 Asminus1δsminus2 + Csminus1δm(sminus1)1113872 1113873

(34)

According to equations (32)ndash(34) the following iterationformulas can be inferred

Mmi Kmi minus CTi M

minus1i Ci i ms m(s minus 1) m1

(35)

Ni iB + C

Ti M

minus1i Ai i s s minus 1 1

(36)

Mi Qiminus1 minus ATi M

minus1i Ai minus N

Ti M

minus1miNi i s minus 1 s minus 2 1

(37)

δmi Mminus1miNiδiminus1 i s s minus 1 1 (38)

δi Mminus1i Aiδiminus1 + Ciδmi( 1113857 i s s minus 1 1 (39)

Because Ms is equal to Kbs we can iterate Msminus1 Msminus2 and M1 by using equation (37) According to equation(39) δ1 Mminus1

1 (A1δ0 + C1δm1) Substituting δ1 into the firstrow of equation (31) yields the following fundamental IEMformula

Ka1 minus AT1 M

minus12 A1 minus N

T1 M

minus11 N11113872 1113873δ0 Kzδ0 f0 (40)

where Kz (Ka1 minus AT1 Mminus1

2 A1 minus NT1 Mminus1

1 N1) is the com-bined stiffness matrix which preserves the inherent sym-metry characteristic of the global stiffness matrix used in theconventional finite element procedure Using equations (38)and (39) we can condense all inner layer elements andtransform them into a single super element with masternodes at the outer boundary

Ying [24] proved that Kz converges toward a certainconstant quantity as the number of element layers ap-proaches infinity that is

lims⟶infin

K(s)z Kz (41)

where s denotes the number of the defined element layersHowever equation (41) cannot be directly applied to thenumerical formulation because the infinity element layers

are not countable in a physical sense )erefore Liu [23]proposed a convergence method for observing the diagonaltrace term K

(s)Z )e desired accuracy criterion can be

expressed as follows

ε K

(i+1)Z minus K

(i)Z

K(i+1)Z

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868times 100le 10minus 6

(42)

When this criterion is satisfied the iterative program isterminated and the critical number of element layers (scr) isdetermined as equal to the terminated iterative value (i) scr isthe minimum number of element layers required for con-vergence this implies that sufficient elements are available tocover the entire domain )e proportionality ratio ξ isanother important factor in the convergence study A higherξ indicates that a higher number of element layers scr isrequired Specifically given a sufficiently high s value thestiffness K

(s)Z is approximately equal to the combined stiff-

ness KZ

4 Case Studies

41 Circular Plate Subject to a Concentrated LoadConsider for example a simply supported circular platesubjected to a concentrated load P of 1 lbf at its centroid(Figure 3(a)) )e material and geometric parameters are asfollows Youngrsquos modulus E 3times106 psi Poissonrsquos ratio] 03 plate radius R 10 in and thickness h 02 in )eanalytical maximum deflection was provided by a previousstudy [25]

wmax (3 + ])PR

2

16π(1 + ])D (43)

where

D Eh

3

12 1 minus ]21113872 1113873 (44)

)e solution procedure of the high-order PIEM entailsthe assumption that the outer boundary comprises 30uniformly distributed master nodes in addition the pro-portionality ratio ξ is set to 064 (Figure 3(b)) On the basisof the convergence criterion (equation (42)) the number ofvirtual element layers s required is 19 Table 1 illustrates theconvergence process Given the geometric symmetry andload only a quarter of the entire strip under the providedload and boundary conditions must be considered Forcomparison we determine the maximum deflection usingABAQUS software (S4R 394 elements) )e results ob-tained using ABAQUS are in good agreement with thoseobtained using the proposed method as presented inTable 2

42 Square Plate Subject to a Concentrated LoadConsider a simply supported square plate subjected to acenter unit point load (Figure 4(a)) )e material andgeometric parameters are listed as follows Youngrsquosmodulus E 3times106 psi Poissonrsquos ratio ] 03 dimension

6 Mathematical Problems in Engineering

a 80 in and thickness h 08 in )e analytical solutionfor this problem was provided by a previous study [25]where the deflection at the plate centroid can be expressedas follows

wmax αPa

2

D (45)

In equation (45) the coefficient α (00116) is a functionof the dimension ratio a b and D is the flexural rigidity ofthe plate )e solution procedure of the high-order PIEMinvolves the assumption that 40 nodes are uniformly dis-tributed and deployed at the boundary moreover theproportionality ratio ξ is set to 056 (Figure 4(b)) Given theproportionality ratio (056) the number of element layers srequired to achieve convergence is 33 Because of the geo-metric symmetry and load only a quarter of the completestrip under the given load and boundary conditions mustbe considered For comparison we also determine themaximum deflection using ABAQUS software (S4R 400elements) )e results obtained using ABAQUS are in goodagreement with the results obtained using the proposedhigh-order PIEM (Table 3) nevertheless the results ob-tained using the high-order PIEM are closer to the analyticalsolution

43 Rectangular Plate Subject to Bending MomentsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 5(a)) Assume that two of the opposite edges aresimply supported and that the other two edges are free suchthat the applied bending moment (M 100N-mm) vanishesalong the two simply supported edges Assume that the platehas a Youngrsquos modulus of 200GPa and a Poissonrsquos ratio of03 Figure 5(b) presents the corresponding virtual meshpattern obtained by the high-order PIEM before the mesh isdegenerated to form a single super element )e high-orderPIEM solution procedure involves the assumption that theouter boundary comprises 60 uniformly distributed masternodes furthermore the proportionality ratio ξ is set to 081On the basis of the convergence criterion (equation (42))the number of virtual element layers s required is 31 Table 4presents a comparison of the deflection profile of edge (AB)of the plate (Figure 5(a)) obtained using the proposed high-order PIEM scheme with the profile obtained using ABA-QUS )e results obtained from the high-order PIEM are ingood agreement with those obtained from ABAQUS (rel-ative deviationlt 06) (Table 4)

44 Cantilever Plate Subject to Concentrated LoadsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 6(a)) Assume that one of the edges is clamped andthat the other three edges are free such that the concentratedloads (P 5N) are applied at the end Moreover considerthat the plate has a Youngrsquos modulus of 200GPa and aPoissonrsquos ratio of 03 Figure 6(b) presents the correspondingmesh pattern obtained from the high-order PIEM before the

Table 1 Convergence of the required virtual element layer

Number of element layers 15 16 17 18 19Accuracy criterion ε 207times10minus5 781times 10minus6 296times10minus6 112times10minus6 422times10minus7

P = 1

R

(a)

10

9

8

7

6

5

4

3

2

1

00 2 4 6 8 10

(b)

Figure 3 Circular plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 2 Maximum deflection of the circular plate subjected to apoint load at the centroid

Methods Analytical ABAQUS High-orderPIEM

Maximum deflection wmax minus000230 minus000231 minus000231Relative deviation mdash 049 049

Mathematical Problems in Engineering 7

P

a a

(a)

40

35

30

25

20

15

10

5

04035302520151050

(b)

Figure 4 Square plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 3 Maximum deflection of the square plate subjected to a point load at the centroid

Methods Analytical ABAQUS High-order PIEMMaximum deflection wmax minus0000528 minus0000531 minus0000527Relative deviation mdash 056 048

MM

A B

CD

E

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 5 Simply supported rectangular plate subjected to bending moments (a) Analysis model and (b) high-order PIEM

Table 4 Comparisons between proposed approach and ABAQUS of maximum deflection along AB shown in Figure 5

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 0450 0452 051minus30 0793 0795 028minus20 1037 1039 017minus10 1183 1184 0120 1231 1233 01110 1183 1184 01220 1037 1039 01730 0793 0795 02840 0450 0452 05150 0000 0000 000

8 Mathematical Problems in Engineering

mesh is degenerated to form a single super element )esolution procedure of the high-order PIEM involves theassumption that the outer boundary comprises 60 uniformlydistributed master nodes moreover the proportionalityratio ξ is set to 081 Given the proportionality ratio ξ (081)the number of element layers s required is 31 Table 5 il-lustrates a comparison of the deflection profile of edge (AB)of the plate (Figure 6(a)) obtained using the proposed high-order PIEM with the profile obtained using ABAQUS )etwo profiles are in good agreement (relativedeviationlt 02) (Table 5)

45 L-Shaped Plate Subject to a Concentrated LoadConsider an L-shaped plate with the dimensions36mm times 36mm times 18mm (L1 times L2 timesW) (Figure 7(a)) andthe thickness is 05mm )e material and geometric pa-rameters are listed as follows Youngrsquos modulus E 200000and Poissonrsquos ratio ] 03 Assume that two of the oppositeedges are clamped and the concentrated loads (P 1 N)are applied at the point B Figure 7(b) presents the cor-responding mesh pattern obtained from the high-orderPIEM before the mesh is degenerated to form a singlesuper element )e solution procedure of the high-orderPIEM involves the assumption that the outer boundarycomprises 48 uniformly distributed master nodes Giventhe proportionality ratio (081) the number of elementlayers s required to achieve convergence is 34 Table 6illustrates a comparison of the deflection profile of edge(BC) of the L-shaped plate obtained using the proposedhigh-order PIEM with the profile obtained using ABA-QUS )e two profiles are in good agreement (relativedeviation lt 03)

46 Multihole Plate Subject to Bending MomentsConsider a multihole plate with the dimensions144mmtimes 36mmtimes 1mm (LtimesWtimes thickness) (Figure 8) andthe circle holes have a radius R 09mm )e material andgeometric parameters are listed as follows Youngrsquos modulusE 200000 and Poissonrsquos ratio ] 03 Assume that two ofthe opposite edges are simply supported and that the othertwo edges are free such that the applied bending moment

(M 100N-mm) vanishes along the two simply supportededges Figure 9(a) presents the corresponding mesh patternof one quarter of the complete strip )e model consists offour subdomains each of which is the IE model Given theproportionality ratio (081) the number of element layers srequired to achieve convergence is 34)e one quarter of thecomplete strip can be a cell and the complete model bycombining four cells (Figure 9(b)) Table 7 illustrates acomparison of the deflection profile of edge (AB) of themultihole plate obtained using the proposed high-orderPIEM with the profile obtained using ABAQUS )e twoprofiles are in good agreement (relative deviationlt 025))is example not only demonstrates the feasibility ofcombining IE subdomains but also copying

47 Cracked Plate Subject to Bending Moments Finallyconsider a central crack on a rectangular plate (Figure 10(a))As boundary conditions assume that the two edges parallelto the crack are simply supported and momentM is appliedto the edges the other two edges are free )e properties ofthe plate are as follows bh 2 ba 2 cb 2 M 1Youngrsquos modulus E 210000 and Poissonrsquos ratio ] 03)ehigh- and low-order PIEM algorithms can be used to in-vestigate the stress intensity factor (SIF) which can be

P

P

A

B

C

D

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 6 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 5 Comparisons between proposed approach and ABAQUSof maximum deflection along AB shown in Figure 6

CoordinateDeflection(mm) Relative deviation

()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 minus0311 minus0310 012minus30 minus1449 minus1451 012minus20 minus3378 minus3383 014minus10 minus6000 minus6009 0150 minus9212 minus9225 01410 minus12914 minus12932 01420 minus17008 minus17030 01330 minus21399 minus21425 01240 minus25991 minus26020 01150 minus30689 minus30722 011

Mathematical Problems in Engineering 9

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

physical coordinates to natural coordinates )e associatedplate stiffness is expressed in equation (5) where [KB] and[KS] denote the bending stiffness and shear stiffness re-spectively )e plate material is considered linear elasticisotropic and homogenous )e resultant equation of eachelement can be expressed in equation (12)

[K] KB1113858 1113859 + KS1113858 1113859 (5)

where

KB1113858 1113859 h3

1211139461

minus111139461

minus1BB1113858 1113859

TDB1113858 1113859 BB1113858 1113859det[J]dζ dη

KS1113858 1113859 κh 11139461

minus111139461

minus1BS1113858 1113859

TDS1113858 1113859 BS1113858 1113859det[J]dζ dη

(6)

where h is the plate thickness κ is the shear energy cor-rection factor (usually 56) and [J] is the Jacobian matrix

[J]

zx

zζzy

zζ

zx

zηzy

zη

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

[BB] and [BS] comprise shape functions as presented inequations (8) and (9) respectively In addition [DB] and[DS] are related to the material properties of the model aspresented in equations (10) and (11) respectively

BB1113858 1113859

zH1

zx0 0

zH2

zx0 0 middot middot middot

zH9

zx0 0

0zH1

zy0 0

zH2

zy0 middot middot middot 0

zH9

zy0

zH1

zy

zH1

zx0

zH2

zy

zH2

zx0 middot middot middot

zH9

zy

zH9

zx0

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

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

(8)

BS1113858 1113859

minusH1 0zH1

zxminusH2 0

zH2

zxmiddot middot middot minusH9 0

zH9

zx

0 minusH1zH1

zy0 minusH2

zH2

zymiddot middot middot 0 minusH9

zH9

zy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)

DB1113858 1113859 E

1 minus ]2

1 ] 0

] 1 0

0 01 minus ]2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

DS1113858 1113859 E

2(1 minus ])

1 00 1

1113890 1113891 (11)

[K]

θx

θy

w

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

Mx

My

fz

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (12)

z w

x u

dx

y v

θx

θy

dy

Mx

Mxy

MyMxy

h

fz

Figure 1 Free-body diagram of a plate element

Mathematical Problems in Engineering 3

3 High-Order PIEM

31 Similarity Characteristic Figure 2 presents the basicconcept of the infinite element (IE) model In this model thecomputational domain is partitioned into multiple layers ofgeometrically similar elements For element I the localnodes i are numbered 1 2 and 9 If the global origin Oand ξ are considered the center of the similarity and theproportionality ratio respectively then element II can becreated )e global coordinates of elements I and II arerelated as presented in equation (13) According to equa-tions (13) and (7) the determinants of the Jacobian matricesof elements I and II are related as expressed in equation (13)Similarly according to equations (13) and (8) the relationbetween [BB] of element I and [BB] of element II can bepresented in equation (15)

xIIi y

IIi1113872 1113873 ξx

Ii ξy

Ii1113872 1113873 (13)

det[J]II

ξ2det[J]I (14)

BB1113858 1113859II

1ξ

BB1113858 1113859I (15)

)erefore as shown in equation (16) the bendingstiffness matrix [KB] of the first and second element layers isrelated

KB1113858 1113859II

h3

1211139461

minus111139461

minus1BB1113858 1113859

IITDB1113858 1113859

IIBB1113858 1113859

IIdet[J]IIdζ dη

h3

1211139461

minus111139461

minus1

1ξ

BB1113858 1113859IT

DB1113858 1113859I1ξ

BB1113858 1113859Iξ2det[J]

Idζ dη

h3

1211139461

minus111139461

minus1BB1113858 1113859

ITDB1113858 1113859

IBB1113858 1113859

Idet[J]Idζ dη

KB1113858 1113859I

(16)

To adapt the conventional IEM to MindlinndashReissnerplate problems the shear stiffness of the first element layer

[BS] can be partitioned into two submatrices namely [BlowastS ]

and [BlowastlowastS ]

BS1113858 1113859 BlowastS1113858 1113859 + B

lowastlowastS1113858 1113859 (17)

where

Blowasts1113858 1113859

0 0zH1

zx0 0

zH2

zxmiddot middot middot 0 0

zH9

zx

0 0zH1

zy0 0

zH2

zymiddot middot middot 0 0

zH9

zy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Blowasts1113858 1113859

minusH1 0 0 minusH2 0 0 middot middot middot minusH9 0 0

0 minusH1 0 0 minusH2 0 middot middot middot 0 minusH9 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(18)

Substituting equation (17) into equation (9) yields thefollowing

KS1113858 1113859 κh 11139461

minus111139461

minus1BlowastS1113858 1113859 + B

lowast lowastS1113858 1113859( 1113857

TDS1113858 1113859 B

lowastS1113858 1113859 + B

lowast lowastS1113858 1113859( 1113857det[J]dζ dη (19)

Let

KlowastS1113858 1113859 κh 1113946

1

minus111139461

minus1BlowastS1113858 1113859

TDS1113858 1113859 B

lowastS1113858 1113859det[J]dζ dη (20)

Klowast lowastS1113858 1113859 κh 1113946

1

minus111139461

minus1BlowastS1113858 1113859

TDS1113858 1113859 B

lowast lowastS1113858 1113859 + B

lowast lowastS1113858 1113859

TDS1113858 1113859 B

lowastS1113858 11138591113872 1113873det[J]dζ dη (21)

Klowast lowast lowastS1113858 1113859 κh 1113946

1

minus111139461

minus1Blowast lowastS1113858 1113859

TDS1113858 1113859 B

lowast lowastS1113858 1113859det[J]dζ dη (22)

I

II

X

Y

O

(xiII yiII)

(xiI yiI)

1

2

3

4

56

7

8

9

Figure 2 Geometrically similar 2D elements in IE formulation

4 Mathematical Problems in Engineering

)us equation (19) becomes

KS1113858 1113859 KlowastS1113858 1113859 + K

lowastlowastS1113858 1113859 + K

lowastlowastlowastS1113858 1113859 (23)

According to the geometric similarity the relationshipbetween the first and second element layers in terms of theshear stiffness matrix can be expressed as follows

KS1113858 1113859II

KlowastS1113858 1113859

II+ KlowastlowastS1113858 1113859

II+ KlowastlowastlowastS1113858 1113859

II

KlowastS1113858 1113859

I+ ξ K

lowastlowastS1113858 1113859

I+ ξ2 K

lowastlowastlowastS1113858 1113859

I

(24)

Substituting equations (16) and (24) into equation (5)yields the plate stiffness matrix as follows

[K]I

KB1113858 1113859I

+ KlowastS1113858 1113859

I+ Klowast lowastS1113858 1113859

I+ Klowast lowast lowastS1113858 1113859

I

[K]II

KB1113858 1113859I

+ KlowastS1113858 1113859

I+ ξ K

lowast lowastS1113858 1113859

I+ ξ2 K

lowast lowast lowastS1113858 1113859

I

⋮[K]

s KB1113858 1113859

I+ KlowastS1113858 1113859

I+ ξsminus 1

Klowast lowastS1113858 1113859

I+ ξ2(sminus 1)

Klowast lowast lowastS1113858 1113859

I

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)

32 Combined Stiffness of High-Order PIEM According toequation (25) the nine-node elements I II and s can bemapped using the same square-shaped master elementSpecifically these elements can be designated as similarelements when the coordinate of an element is similar to thatof other elements )e matrices of the first element layer canbe expressed as follows

KB1113858 1113859I

Ka minusBT

minusAT

minusBmK minusC

T

minusA minusC Kb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

KlowastS1113858 1113859

I

Klowasta minusB

lowastTminusAlowastT

minusBlowast

Klowastm minus C

lowastT

minusAlowast

minusClowast

Klowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Klowast lowastS1113858 1113859

I

Klowast lowasta minusB

lowastlowastTminusAlowastlowastT

minusBlowastlowast

Klowast lowastm minusC

lowastlowastT

minusAlowastlowast

minusClowastlowast

Klowast lowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Klowast lowast lowastS1113858 1113859

I

Klowast lowast lowasta minusB

lowastlowastlowastTminusAlowastlowastlowastT

minusBlowastlowastlowast

Klowast lowast lowastm minusC

lowastlowastlowastT

minusAlowastlowastlowast

minusClowastlowastlowast

Klowast lowast lowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

When equation (26) is substituted into equation (12) andthe result is expanded the equations of the s element layersin the computational domain can be derived as follows

Ka1 minusBT1 minusA

T1

minusB1 Km1 minusCT1

minusA1 minusC1 Kb1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δ0δm1

δ2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

f0

0

f2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

⋮

(27)

Kai minusBTi minusA

Ti

minusBi Kmi minusCTi

minusAi minusCi Kbi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δiminus1

δmi

δi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

minusfiminus1

0fi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

⋮(28)

Kas minusBTs minusA

Ts

minusBs Kms minusCTs

minusAs minusCs Kbs

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δsminus1

δms

δs

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

minusfsminus1

0fs

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (29)

where

Kai Ka + Klowasta + ξiminus 1

Klowast lowasta + ξ2(iminus 1)

Klowast lowast lowasta

Kbi Kb + Klowastb + ξiminus 1

Klowast lowastb + ξ2(iminus 1)

Klowast lowast lowastb

Kmi Km + Klowastm + ξiminus 1

Klowast lowastm + ξ2(iminus 1)

Klowast lowast lowastm

Ai A + Alowast

+ ξiminus 1Alowastlowast

+ ξ2(iminus 1)Alowastlowastlowast

Bi B + Blowast

+ ξiminus 1Blowastlowast

+ ξ2(iminus 1)Blowastlowastlowast

Ci C + Clowast

+ ξiminus 1Clowastlowast

+ ξ2(iminus 1)Clowastlowastlowast

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

In equations (27)ndash(29) δi is the nodal displacementvector associated with the ith node layer and fi is thecorresponding nodal force vector Combining the equationsfrom the first element layer to the sth element layer andassuming that no internal force is applied to the ith nodelayer (ie fs 0) can yield the following expression

Ka1 minusBT1 minusA

T1 0 0 0 0 0 0 0

minusB1 Km1 minusCT1 0 0 0 0 0 0 0

minusA1 minusC1 Q1 minusBT2 minusA

T2 0 0 0 0 0

0 0 minusB2 Km2 minusCT2 0 0 0 0 0

0 0 minusA2 minusC2 Q2 minusBT3 minusA

T3 0 0 0

⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱0 0 0 0 0 minusBsminus1 Km(sminus1) minusC

Tsminus1 0 0

0 0 0 0 0 minusAsminus1 minusCsminus1 Qsminus1 minusBTs minusA

Ts

0 0 0 0 0 0 0 minusBs Kms minusCTs

0 0 0 0 0 0 0 minusAs minusCs Kbs

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

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

δ0δm1

δ1δm2

δ2⋮

δm(sminus1)

δsminus1

δms

δs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

f0

0

0

0

0

⋮

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(31)

Mathematical Problems in Engineering 5

where Qi Ka(i+1) + Kbi If Ms Kbs in the last row ofequation (31) then

δs Mminus1s

sAδsminus1 + Csδms( 1113857 (32)

Substituting equation (32) into the second-to-last row ofequation (31) yields

δms Kms minus CTs M

minus1s Cs1113872 1113873

minus 1 sB + C

Ts M

minus1s As1113872 1113873δ4

Mminus1msNsδsminus1

(33)

Similarly substituting equations (32) and (33) into thesecond-to-last row of equation (31) yields

δsminus1 Qsminus 1 minus ATs M

minus1s As minus N

Ts M

minus1msNs1113872 1113873

minus 1Asminus1δsminus2 + Csminus1δms( 1113857

Mminus1sminus1 Asminus1δsminus2 + Csminus1δm(sminus1)1113872 1113873

(34)

According to equations (32)ndash(34) the following iterationformulas can be inferred

Mmi Kmi minus CTi M

minus1i Ci i ms m(s minus 1) m1

(35)

Ni iB + C

Ti M

minus1i Ai i s s minus 1 1

(36)

Mi Qiminus1 minus ATi M

minus1i Ai minus N

Ti M

minus1miNi i s minus 1 s minus 2 1

(37)

δmi Mminus1miNiδiminus1 i s s minus 1 1 (38)

δi Mminus1i Aiδiminus1 + Ciδmi( 1113857 i s s minus 1 1 (39)

Because Ms is equal to Kbs we can iterate Msminus1 Msminus2 and M1 by using equation (37) According to equation(39) δ1 Mminus1

1 (A1δ0 + C1δm1) Substituting δ1 into the firstrow of equation (31) yields the following fundamental IEMformula

Ka1 minus AT1 M

minus12 A1 minus N

T1 M

minus11 N11113872 1113873δ0 Kzδ0 f0 (40)

where Kz (Ka1 minus AT1 Mminus1

2 A1 minus NT1 Mminus1

1 N1) is the com-bined stiffness matrix which preserves the inherent sym-metry characteristic of the global stiffness matrix used in theconventional finite element procedure Using equations (38)and (39) we can condense all inner layer elements andtransform them into a single super element with masternodes at the outer boundary

Ying [24] proved that Kz converges toward a certainconstant quantity as the number of element layers ap-proaches infinity that is

lims⟶infin

K(s)z Kz (41)

where s denotes the number of the defined element layersHowever equation (41) cannot be directly applied to thenumerical formulation because the infinity element layers

are not countable in a physical sense )erefore Liu [23]proposed a convergence method for observing the diagonaltrace term K

(s)Z )e desired accuracy criterion can be

expressed as follows

ε K

(i+1)Z minus K

(i)Z

K(i+1)Z

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868times 100le 10minus 6

(42)

When this criterion is satisfied the iterative program isterminated and the critical number of element layers (scr) isdetermined as equal to the terminated iterative value (i) scr isthe minimum number of element layers required for con-vergence this implies that sufficient elements are available tocover the entire domain )e proportionality ratio ξ isanother important factor in the convergence study A higherξ indicates that a higher number of element layers scr isrequired Specifically given a sufficiently high s value thestiffness K

(s)Z is approximately equal to the combined stiff-

ness KZ

4 Case Studies

41 Circular Plate Subject to a Concentrated LoadConsider for example a simply supported circular platesubjected to a concentrated load P of 1 lbf at its centroid(Figure 3(a)) )e material and geometric parameters are asfollows Youngrsquos modulus E 3times106 psi Poissonrsquos ratio] 03 plate radius R 10 in and thickness h 02 in )eanalytical maximum deflection was provided by a previousstudy [25]

wmax (3 + ])PR

2

16π(1 + ])D (43)

where

D Eh

3

12 1 minus ]21113872 1113873 (44)

)e solution procedure of the high-order PIEM entailsthe assumption that the outer boundary comprises 30uniformly distributed master nodes in addition the pro-portionality ratio ξ is set to 064 (Figure 3(b)) On the basisof the convergence criterion (equation (42)) the number ofvirtual element layers s required is 19 Table 1 illustrates theconvergence process Given the geometric symmetry andload only a quarter of the entire strip under the providedload and boundary conditions must be considered Forcomparison we determine the maximum deflection usingABAQUS software (S4R 394 elements) )e results ob-tained using ABAQUS are in good agreement with thoseobtained using the proposed method as presented inTable 2

42 Square Plate Subject to a Concentrated LoadConsider a simply supported square plate subjected to acenter unit point load (Figure 4(a)) )e material andgeometric parameters are listed as follows Youngrsquosmodulus E 3times106 psi Poissonrsquos ratio ] 03 dimension

6 Mathematical Problems in Engineering

a 80 in and thickness h 08 in )e analytical solutionfor this problem was provided by a previous study [25]where the deflection at the plate centroid can be expressedas follows

wmax αPa

2

D (45)

In equation (45) the coefficient α (00116) is a functionof the dimension ratio a b and D is the flexural rigidity ofthe plate )e solution procedure of the high-order PIEMinvolves the assumption that 40 nodes are uniformly dis-tributed and deployed at the boundary moreover theproportionality ratio ξ is set to 056 (Figure 4(b)) Given theproportionality ratio (056) the number of element layers srequired to achieve convergence is 33 Because of the geo-metric symmetry and load only a quarter of the completestrip under the given load and boundary conditions mustbe considered For comparison we also determine themaximum deflection using ABAQUS software (S4R 400elements) )e results obtained using ABAQUS are in goodagreement with the results obtained using the proposedhigh-order PIEM (Table 3) nevertheless the results ob-tained using the high-order PIEM are closer to the analyticalsolution

43 Rectangular Plate Subject to Bending MomentsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 5(a)) Assume that two of the opposite edges aresimply supported and that the other two edges are free suchthat the applied bending moment (M 100N-mm) vanishesalong the two simply supported edges Assume that the platehas a Youngrsquos modulus of 200GPa and a Poissonrsquos ratio of03 Figure 5(b) presents the corresponding virtual meshpattern obtained by the high-order PIEM before the mesh isdegenerated to form a single super element )e high-orderPIEM solution procedure involves the assumption that theouter boundary comprises 60 uniformly distributed masternodes furthermore the proportionality ratio ξ is set to 081On the basis of the convergence criterion (equation (42))the number of virtual element layers s required is 31 Table 4presents a comparison of the deflection profile of edge (AB)of the plate (Figure 5(a)) obtained using the proposed high-order PIEM scheme with the profile obtained using ABA-QUS )e results obtained from the high-order PIEM are ingood agreement with those obtained from ABAQUS (rel-ative deviationlt 06) (Table 4)

44 Cantilever Plate Subject to Concentrated LoadsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 6(a)) Assume that one of the edges is clamped andthat the other three edges are free such that the concentratedloads (P 5N) are applied at the end Moreover considerthat the plate has a Youngrsquos modulus of 200GPa and aPoissonrsquos ratio of 03 Figure 6(b) presents the correspondingmesh pattern obtained from the high-order PIEM before the

Table 1 Convergence of the required virtual element layer

Number of element layers 15 16 17 18 19Accuracy criterion ε 207times10minus5 781times 10minus6 296times10minus6 112times10minus6 422times10minus7

P = 1

R

(a)

10

9

8

7

6

5

4

3

2

1

00 2 4 6 8 10

(b)

Figure 3 Circular plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 2 Maximum deflection of the circular plate subjected to apoint load at the centroid

Methods Analytical ABAQUS High-orderPIEM

Maximum deflection wmax minus000230 minus000231 minus000231Relative deviation mdash 049 049

Mathematical Problems in Engineering 7

P

a a

(a)

40

35

30

25

20

15

10

5

04035302520151050

(b)

Figure 4 Square plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 3 Maximum deflection of the square plate subjected to a point load at the centroid

Methods Analytical ABAQUS High-order PIEMMaximum deflection wmax minus0000528 minus0000531 minus0000527Relative deviation mdash 056 048

MM

A B

CD

E

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 5 Simply supported rectangular plate subjected to bending moments (a) Analysis model and (b) high-order PIEM

Table 4 Comparisons between proposed approach and ABAQUS of maximum deflection along AB shown in Figure 5

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 0450 0452 051minus30 0793 0795 028minus20 1037 1039 017minus10 1183 1184 0120 1231 1233 01110 1183 1184 01220 1037 1039 01730 0793 0795 02840 0450 0452 05150 0000 0000 000

8 Mathematical Problems in Engineering

mesh is degenerated to form a single super element )esolution procedure of the high-order PIEM involves theassumption that the outer boundary comprises 60 uniformlydistributed master nodes moreover the proportionalityratio ξ is set to 081 Given the proportionality ratio ξ (081)the number of element layers s required is 31 Table 5 il-lustrates a comparison of the deflection profile of edge (AB)of the plate (Figure 6(a)) obtained using the proposed high-order PIEM with the profile obtained using ABAQUS )etwo profiles are in good agreement (relativedeviationlt 02) (Table 5)

45 L-Shaped Plate Subject to a Concentrated LoadConsider an L-shaped plate with the dimensions36mm times 36mm times 18mm (L1 times L2 timesW) (Figure 7(a)) andthe thickness is 05mm )e material and geometric pa-rameters are listed as follows Youngrsquos modulus E 200000and Poissonrsquos ratio ] 03 Assume that two of the oppositeedges are clamped and the concentrated loads (P 1 N)are applied at the point B Figure 7(b) presents the cor-responding mesh pattern obtained from the high-orderPIEM before the mesh is degenerated to form a singlesuper element )e solution procedure of the high-orderPIEM involves the assumption that the outer boundarycomprises 48 uniformly distributed master nodes Giventhe proportionality ratio (081) the number of elementlayers s required to achieve convergence is 34 Table 6illustrates a comparison of the deflection profile of edge(BC) of the L-shaped plate obtained using the proposedhigh-order PIEM with the profile obtained using ABA-QUS )e two profiles are in good agreement (relativedeviation lt 03)

46 Multihole Plate Subject to Bending MomentsConsider a multihole plate with the dimensions144mmtimes 36mmtimes 1mm (LtimesWtimes thickness) (Figure 8) andthe circle holes have a radius R 09mm )e material andgeometric parameters are listed as follows Youngrsquos modulusE 200000 and Poissonrsquos ratio ] 03 Assume that two ofthe opposite edges are simply supported and that the othertwo edges are free such that the applied bending moment

(M 100N-mm) vanishes along the two simply supportededges Figure 9(a) presents the corresponding mesh patternof one quarter of the complete strip )e model consists offour subdomains each of which is the IE model Given theproportionality ratio (081) the number of element layers srequired to achieve convergence is 34)e one quarter of thecomplete strip can be a cell and the complete model bycombining four cells (Figure 9(b)) Table 7 illustrates acomparison of the deflection profile of edge (AB) of themultihole plate obtained using the proposed high-orderPIEM with the profile obtained using ABAQUS )e twoprofiles are in good agreement (relative deviationlt 025))is example not only demonstrates the feasibility ofcombining IE subdomains but also copying

47 Cracked Plate Subject to Bending Moments Finallyconsider a central crack on a rectangular plate (Figure 10(a))As boundary conditions assume that the two edges parallelto the crack are simply supported and momentM is appliedto the edges the other two edges are free )e properties ofthe plate are as follows bh 2 ba 2 cb 2 M 1Youngrsquos modulus E 210000 and Poissonrsquos ratio ] 03)ehigh- and low-order PIEM algorithms can be used to in-vestigate the stress intensity factor (SIF) which can be

P

P

A

B

C

D

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 6 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 5 Comparisons between proposed approach and ABAQUSof maximum deflection along AB shown in Figure 6

CoordinateDeflection(mm) Relative deviation

()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 minus0311 minus0310 012minus30 minus1449 minus1451 012minus20 minus3378 minus3383 014minus10 minus6000 minus6009 0150 minus9212 minus9225 01410 minus12914 minus12932 01420 minus17008 minus17030 01330 minus21399 minus21425 01240 minus25991 minus26020 01150 minus30689 minus30722 011

Mathematical Problems in Engineering 9

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

3 High-Order PIEM

31 Similarity Characteristic Figure 2 presents the basicconcept of the infinite element (IE) model In this model thecomputational domain is partitioned into multiple layers ofgeometrically similar elements For element I the localnodes i are numbered 1 2 and 9 If the global origin Oand ξ are considered the center of the similarity and theproportionality ratio respectively then element II can becreated )e global coordinates of elements I and II arerelated as presented in equation (13) According to equa-tions (13) and (7) the determinants of the Jacobian matricesof elements I and II are related as expressed in equation (13)Similarly according to equations (13) and (8) the relationbetween [BB] of element I and [BB] of element II can bepresented in equation (15)

xIIi y

IIi1113872 1113873 ξx

Ii ξy

Ii1113872 1113873 (13)

det[J]II

ξ2det[J]I (14)

BB1113858 1113859II

1ξ

BB1113858 1113859I (15)

)erefore as shown in equation (16) the bendingstiffness matrix [KB] of the first and second element layers isrelated

KB1113858 1113859II

h3

1211139461

minus111139461

minus1BB1113858 1113859

IITDB1113858 1113859

IIBB1113858 1113859

IIdet[J]IIdζ dη

h3

1211139461

minus111139461

minus1

1ξ

BB1113858 1113859IT

DB1113858 1113859I1ξ

BB1113858 1113859Iξ2det[J]

Idζ dη

h3

1211139461

minus111139461

minus1BB1113858 1113859

ITDB1113858 1113859

IBB1113858 1113859

Idet[J]Idζ dη

KB1113858 1113859I

(16)

To adapt the conventional IEM to MindlinndashReissnerplate problems the shear stiffness of the first element layer

[BS] can be partitioned into two submatrices namely [BlowastS ]

and [BlowastlowastS ]

BS1113858 1113859 BlowastS1113858 1113859 + B

lowastlowastS1113858 1113859 (17)

where

Blowasts1113858 1113859

0 0zH1

zx0 0

zH2

zxmiddot middot middot 0 0

zH9

zx

0 0zH1

zy0 0

zH2

zymiddot middot middot 0 0

zH9

zy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Blowasts1113858 1113859

minusH1 0 0 minusH2 0 0 middot middot middot minusH9 0 0

0 minusH1 0 0 minusH2 0 middot middot middot 0 minusH9 0⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

(18)

Substituting equation (17) into equation (9) yields thefollowing

KS1113858 1113859 κh 11139461

minus111139461

minus1BlowastS1113858 1113859 + B

lowast lowastS1113858 1113859( 1113857

TDS1113858 1113859 B

lowastS1113858 1113859 + B

lowast lowastS1113858 1113859( 1113857det[J]dζ dη (19)

Let

KlowastS1113858 1113859 κh 1113946

1

minus111139461

minus1BlowastS1113858 1113859

TDS1113858 1113859 B

lowastS1113858 1113859det[J]dζ dη (20)

Klowast lowastS1113858 1113859 κh 1113946

1

minus111139461

minus1BlowastS1113858 1113859

TDS1113858 1113859 B

lowast lowastS1113858 1113859 + B

lowast lowastS1113858 1113859

TDS1113858 1113859 B

lowastS1113858 11138591113872 1113873det[J]dζ dη (21)

Klowast lowast lowastS1113858 1113859 κh 1113946

1

minus111139461

minus1Blowast lowastS1113858 1113859

TDS1113858 1113859 B

lowast lowastS1113858 1113859det[J]dζ dη (22)

I

II

X

Y

O

(xiII yiII)

(xiI yiI)

1

2

3

4

56

7

8

9

Figure 2 Geometrically similar 2D elements in IE formulation

4 Mathematical Problems in Engineering

)us equation (19) becomes

KS1113858 1113859 KlowastS1113858 1113859 + K

lowastlowastS1113858 1113859 + K

lowastlowastlowastS1113858 1113859 (23)

According to the geometric similarity the relationshipbetween the first and second element layers in terms of theshear stiffness matrix can be expressed as follows

KS1113858 1113859II

KlowastS1113858 1113859

II+ KlowastlowastS1113858 1113859

II+ KlowastlowastlowastS1113858 1113859

II

KlowastS1113858 1113859

I+ ξ K

lowastlowastS1113858 1113859

I+ ξ2 K

lowastlowastlowastS1113858 1113859

I

(24)

Substituting equations (16) and (24) into equation (5)yields the plate stiffness matrix as follows

[K]I

KB1113858 1113859I

+ KlowastS1113858 1113859

I+ Klowast lowastS1113858 1113859

I+ Klowast lowast lowastS1113858 1113859

I

[K]II

KB1113858 1113859I

+ KlowastS1113858 1113859

I+ ξ K

lowast lowastS1113858 1113859

I+ ξ2 K

lowast lowast lowastS1113858 1113859

I

⋮[K]

s KB1113858 1113859

I+ KlowastS1113858 1113859

I+ ξsminus 1

Klowast lowastS1113858 1113859

I+ ξ2(sminus 1)

Klowast lowast lowastS1113858 1113859

I

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)

32 Combined Stiffness of High-Order PIEM According toequation (25) the nine-node elements I II and s can bemapped using the same square-shaped master elementSpecifically these elements can be designated as similarelements when the coordinate of an element is similar to thatof other elements )e matrices of the first element layer canbe expressed as follows

KB1113858 1113859I

Ka minusBT

minusAT

minusBmK minusC

T

minusA minusC Kb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

KlowastS1113858 1113859

I

Klowasta minusB

lowastTminusAlowastT

minusBlowast

Klowastm minus C

lowastT

minusAlowast

minusClowast

Klowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Klowast lowastS1113858 1113859

I

Klowast lowasta minusB

lowastlowastTminusAlowastlowastT

minusBlowastlowast

Klowast lowastm minusC

lowastlowastT

minusAlowastlowast

minusClowastlowast

Klowast lowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Klowast lowast lowastS1113858 1113859

I

Klowast lowast lowasta minusB

lowastlowastlowastTminusAlowastlowastlowastT

minusBlowastlowastlowast

Klowast lowast lowastm minusC

lowastlowastlowastT

minusAlowastlowastlowast

minusClowastlowastlowast

Klowast lowast lowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

When equation (26) is substituted into equation (12) andthe result is expanded the equations of the s element layersin the computational domain can be derived as follows

Ka1 minusBT1 minusA

T1

minusB1 Km1 minusCT1

minusA1 minusC1 Kb1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δ0δm1

δ2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

f0

0

f2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

⋮

(27)

Kai minusBTi minusA

Ti

minusBi Kmi minusCTi

minusAi minusCi Kbi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δiminus1

δmi

δi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

minusfiminus1

0fi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

⋮(28)

Kas minusBTs minusA

Ts

minusBs Kms minusCTs

minusAs minusCs Kbs

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δsminus1

δms

δs

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

minusfsminus1

0fs

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (29)

where

Kai Ka + Klowasta + ξiminus 1

Klowast lowasta + ξ2(iminus 1)

Klowast lowast lowasta

Kbi Kb + Klowastb + ξiminus 1

Klowast lowastb + ξ2(iminus 1)

Klowast lowast lowastb

Kmi Km + Klowastm + ξiminus 1

Klowast lowastm + ξ2(iminus 1)

Klowast lowast lowastm

Ai A + Alowast

+ ξiminus 1Alowastlowast

+ ξ2(iminus 1)Alowastlowastlowast

Bi B + Blowast

+ ξiminus 1Blowastlowast

+ ξ2(iminus 1)Blowastlowastlowast

Ci C + Clowast

+ ξiminus 1Clowastlowast

+ ξ2(iminus 1)Clowastlowastlowast

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

In equations (27)ndash(29) δi is the nodal displacementvector associated with the ith node layer and fi is thecorresponding nodal force vector Combining the equationsfrom the first element layer to the sth element layer andassuming that no internal force is applied to the ith nodelayer (ie fs 0) can yield the following expression

Ka1 minusBT1 minusA

T1 0 0 0 0 0 0 0

minusB1 Km1 minusCT1 0 0 0 0 0 0 0

minusA1 minusC1 Q1 minusBT2 minusA

T2 0 0 0 0 0

0 0 minusB2 Km2 minusCT2 0 0 0 0 0

0 0 minusA2 minusC2 Q2 minusBT3 minusA

T3 0 0 0

⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱0 0 0 0 0 minusBsminus1 Km(sminus1) minusC

Tsminus1 0 0

0 0 0 0 0 minusAsminus1 minusCsminus1 Qsminus1 minusBTs minusA

Ts

0 0 0 0 0 0 0 minusBs Kms minusCTs

0 0 0 0 0 0 0 minusAs minusCs Kbs

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

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

δ0δm1

δ1δm2

δ2⋮

δm(sminus1)

δsminus1

δms

δs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

f0

0

0

0

0

⋮

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(31)

Mathematical Problems in Engineering 5

where Qi Ka(i+1) + Kbi If Ms Kbs in the last row ofequation (31) then

δs Mminus1s

sAδsminus1 + Csδms( 1113857 (32)

Substituting equation (32) into the second-to-last row ofequation (31) yields

δms Kms minus CTs M

minus1s Cs1113872 1113873

minus 1 sB + C

Ts M

minus1s As1113872 1113873δ4

Mminus1msNsδsminus1

(33)

Similarly substituting equations (32) and (33) into thesecond-to-last row of equation (31) yields

δsminus1 Qsminus 1 minus ATs M

minus1s As minus N

Ts M

minus1msNs1113872 1113873

minus 1Asminus1δsminus2 + Csminus1δms( 1113857

Mminus1sminus1 Asminus1δsminus2 + Csminus1δm(sminus1)1113872 1113873

(34)

According to equations (32)ndash(34) the following iterationformulas can be inferred

Mmi Kmi minus CTi M

minus1i Ci i ms m(s minus 1) m1

(35)

Ni iB + C

Ti M

minus1i Ai i s s minus 1 1

(36)

Mi Qiminus1 minus ATi M

minus1i Ai minus N

Ti M

minus1miNi i s minus 1 s minus 2 1

(37)

δmi Mminus1miNiδiminus1 i s s minus 1 1 (38)

δi Mminus1i Aiδiminus1 + Ciδmi( 1113857 i s s minus 1 1 (39)

Because Ms is equal to Kbs we can iterate Msminus1 Msminus2 and M1 by using equation (37) According to equation(39) δ1 Mminus1

1 (A1δ0 + C1δm1) Substituting δ1 into the firstrow of equation (31) yields the following fundamental IEMformula

Ka1 minus AT1 M

minus12 A1 minus N

T1 M

minus11 N11113872 1113873δ0 Kzδ0 f0 (40)

where Kz (Ka1 minus AT1 Mminus1

2 A1 minus NT1 Mminus1

1 N1) is the com-bined stiffness matrix which preserves the inherent sym-metry characteristic of the global stiffness matrix used in theconventional finite element procedure Using equations (38)and (39) we can condense all inner layer elements andtransform them into a single super element with masternodes at the outer boundary

Ying [24] proved that Kz converges toward a certainconstant quantity as the number of element layers ap-proaches infinity that is

lims⟶infin

K(s)z Kz (41)

where s denotes the number of the defined element layersHowever equation (41) cannot be directly applied to thenumerical formulation because the infinity element layers

are not countable in a physical sense )erefore Liu [23]proposed a convergence method for observing the diagonaltrace term K

(s)Z )e desired accuracy criterion can be

expressed as follows

ε K

(i+1)Z minus K

(i)Z

K(i+1)Z

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868times 100le 10minus 6

(42)

When this criterion is satisfied the iterative program isterminated and the critical number of element layers (scr) isdetermined as equal to the terminated iterative value (i) scr isthe minimum number of element layers required for con-vergence this implies that sufficient elements are available tocover the entire domain )e proportionality ratio ξ isanother important factor in the convergence study A higherξ indicates that a higher number of element layers scr isrequired Specifically given a sufficiently high s value thestiffness K

(s)Z is approximately equal to the combined stiff-

ness KZ

4 Case Studies

41 Circular Plate Subject to a Concentrated LoadConsider for example a simply supported circular platesubjected to a concentrated load P of 1 lbf at its centroid(Figure 3(a)) )e material and geometric parameters are asfollows Youngrsquos modulus E 3times106 psi Poissonrsquos ratio] 03 plate radius R 10 in and thickness h 02 in )eanalytical maximum deflection was provided by a previousstudy [25]

wmax (3 + ])PR

2

16π(1 + ])D (43)

where

D Eh

3

12 1 minus ]21113872 1113873 (44)

)e solution procedure of the high-order PIEM entailsthe assumption that the outer boundary comprises 30uniformly distributed master nodes in addition the pro-portionality ratio ξ is set to 064 (Figure 3(b)) On the basisof the convergence criterion (equation (42)) the number ofvirtual element layers s required is 19 Table 1 illustrates theconvergence process Given the geometric symmetry andload only a quarter of the entire strip under the providedload and boundary conditions must be considered Forcomparison we determine the maximum deflection usingABAQUS software (S4R 394 elements) )e results ob-tained using ABAQUS are in good agreement with thoseobtained using the proposed method as presented inTable 2

42 Square Plate Subject to a Concentrated LoadConsider a simply supported square plate subjected to acenter unit point load (Figure 4(a)) )e material andgeometric parameters are listed as follows Youngrsquosmodulus E 3times106 psi Poissonrsquos ratio ] 03 dimension

6 Mathematical Problems in Engineering

a 80 in and thickness h 08 in )e analytical solutionfor this problem was provided by a previous study [25]where the deflection at the plate centroid can be expressedas follows

wmax αPa

2

D (45)

In equation (45) the coefficient α (00116) is a functionof the dimension ratio a b and D is the flexural rigidity ofthe plate )e solution procedure of the high-order PIEMinvolves the assumption that 40 nodes are uniformly dis-tributed and deployed at the boundary moreover theproportionality ratio ξ is set to 056 (Figure 4(b)) Given theproportionality ratio (056) the number of element layers srequired to achieve convergence is 33 Because of the geo-metric symmetry and load only a quarter of the completestrip under the given load and boundary conditions mustbe considered For comparison we also determine themaximum deflection using ABAQUS software (S4R 400elements) )e results obtained using ABAQUS are in goodagreement with the results obtained using the proposedhigh-order PIEM (Table 3) nevertheless the results ob-tained using the high-order PIEM are closer to the analyticalsolution

43 Rectangular Plate Subject to Bending MomentsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 5(a)) Assume that two of the opposite edges aresimply supported and that the other two edges are free suchthat the applied bending moment (M 100N-mm) vanishesalong the two simply supported edges Assume that the platehas a Youngrsquos modulus of 200GPa and a Poissonrsquos ratio of03 Figure 5(b) presents the corresponding virtual meshpattern obtained by the high-order PIEM before the mesh isdegenerated to form a single super element )e high-orderPIEM solution procedure involves the assumption that theouter boundary comprises 60 uniformly distributed masternodes furthermore the proportionality ratio ξ is set to 081On the basis of the convergence criterion (equation (42))the number of virtual element layers s required is 31 Table 4presents a comparison of the deflection profile of edge (AB)of the plate (Figure 5(a)) obtained using the proposed high-order PIEM scheme with the profile obtained using ABA-QUS )e results obtained from the high-order PIEM are ingood agreement with those obtained from ABAQUS (rel-ative deviationlt 06) (Table 4)

44 Cantilever Plate Subject to Concentrated LoadsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 6(a)) Assume that one of the edges is clamped andthat the other three edges are free such that the concentratedloads (P 5N) are applied at the end Moreover considerthat the plate has a Youngrsquos modulus of 200GPa and aPoissonrsquos ratio of 03 Figure 6(b) presents the correspondingmesh pattern obtained from the high-order PIEM before the

Table 1 Convergence of the required virtual element layer

Number of element layers 15 16 17 18 19Accuracy criterion ε 207times10minus5 781times 10minus6 296times10minus6 112times10minus6 422times10minus7

P = 1

R

(a)

10

9

8

7

6

5

4

3

2

1

00 2 4 6 8 10

(b)

Figure 3 Circular plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 2 Maximum deflection of the circular plate subjected to apoint load at the centroid

Methods Analytical ABAQUS High-orderPIEM

Maximum deflection wmax minus000230 minus000231 minus000231Relative deviation mdash 049 049

Mathematical Problems in Engineering 7

P

a a

(a)

40

35

30

25

20

15

10

5

04035302520151050

(b)

Figure 4 Square plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 3 Maximum deflection of the square plate subjected to a point load at the centroid

Methods Analytical ABAQUS High-order PIEMMaximum deflection wmax minus0000528 minus0000531 minus0000527Relative deviation mdash 056 048

MM

A B

CD

E

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 5 Simply supported rectangular plate subjected to bending moments (a) Analysis model and (b) high-order PIEM

Table 4 Comparisons between proposed approach and ABAQUS of maximum deflection along AB shown in Figure 5

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 0450 0452 051minus30 0793 0795 028minus20 1037 1039 017minus10 1183 1184 0120 1231 1233 01110 1183 1184 01220 1037 1039 01730 0793 0795 02840 0450 0452 05150 0000 0000 000

8 Mathematical Problems in Engineering

mesh is degenerated to form a single super element )esolution procedure of the high-order PIEM involves theassumption that the outer boundary comprises 60 uniformlydistributed master nodes moreover the proportionalityratio ξ is set to 081 Given the proportionality ratio ξ (081)the number of element layers s required is 31 Table 5 il-lustrates a comparison of the deflection profile of edge (AB)of the plate (Figure 6(a)) obtained using the proposed high-order PIEM with the profile obtained using ABAQUS )etwo profiles are in good agreement (relativedeviationlt 02) (Table 5)

45 L-Shaped Plate Subject to a Concentrated LoadConsider an L-shaped plate with the dimensions36mm times 36mm times 18mm (L1 times L2 timesW) (Figure 7(a)) andthe thickness is 05mm )e material and geometric pa-rameters are listed as follows Youngrsquos modulus E 200000and Poissonrsquos ratio ] 03 Assume that two of the oppositeedges are clamped and the concentrated loads (P 1 N)are applied at the point B Figure 7(b) presents the cor-responding mesh pattern obtained from the high-orderPIEM before the mesh is degenerated to form a singlesuper element )e solution procedure of the high-orderPIEM involves the assumption that the outer boundarycomprises 48 uniformly distributed master nodes Giventhe proportionality ratio (081) the number of elementlayers s required to achieve convergence is 34 Table 6illustrates a comparison of the deflection profile of edge(BC) of the L-shaped plate obtained using the proposedhigh-order PIEM with the profile obtained using ABA-QUS )e two profiles are in good agreement (relativedeviation lt 03)

46 Multihole Plate Subject to Bending MomentsConsider a multihole plate with the dimensions144mmtimes 36mmtimes 1mm (LtimesWtimes thickness) (Figure 8) andthe circle holes have a radius R 09mm )e material andgeometric parameters are listed as follows Youngrsquos modulusE 200000 and Poissonrsquos ratio ] 03 Assume that two ofthe opposite edges are simply supported and that the othertwo edges are free such that the applied bending moment

(M 100N-mm) vanishes along the two simply supportededges Figure 9(a) presents the corresponding mesh patternof one quarter of the complete strip )e model consists offour subdomains each of which is the IE model Given theproportionality ratio (081) the number of element layers srequired to achieve convergence is 34)e one quarter of thecomplete strip can be a cell and the complete model bycombining four cells (Figure 9(b)) Table 7 illustrates acomparison of the deflection profile of edge (AB) of themultihole plate obtained using the proposed high-orderPIEM with the profile obtained using ABAQUS )e twoprofiles are in good agreement (relative deviationlt 025))is example not only demonstrates the feasibility ofcombining IE subdomains but also copying

47 Cracked Plate Subject to Bending Moments Finallyconsider a central crack on a rectangular plate (Figure 10(a))As boundary conditions assume that the two edges parallelto the crack are simply supported and momentM is appliedto the edges the other two edges are free )e properties ofthe plate are as follows bh 2 ba 2 cb 2 M 1Youngrsquos modulus E 210000 and Poissonrsquos ratio ] 03)ehigh- and low-order PIEM algorithms can be used to in-vestigate the stress intensity factor (SIF) which can be

P

P

A

B

C

D

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 6 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 5 Comparisons between proposed approach and ABAQUSof maximum deflection along AB shown in Figure 6

CoordinateDeflection(mm) Relative deviation

()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 minus0311 minus0310 012minus30 minus1449 minus1451 012minus20 minus3378 minus3383 014minus10 minus6000 minus6009 0150 minus9212 minus9225 01410 minus12914 minus12932 01420 minus17008 minus17030 01330 minus21399 minus21425 01240 minus25991 minus26020 01150 minus30689 minus30722 011

Mathematical Problems in Engineering 9

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

)us equation (19) becomes

KS1113858 1113859 KlowastS1113858 1113859 + K

lowastlowastS1113858 1113859 + K

lowastlowastlowastS1113858 1113859 (23)

According to the geometric similarity the relationshipbetween the first and second element layers in terms of theshear stiffness matrix can be expressed as follows

KS1113858 1113859II

KlowastS1113858 1113859

II+ KlowastlowastS1113858 1113859

II+ KlowastlowastlowastS1113858 1113859

II

KlowastS1113858 1113859

I+ ξ K

lowastlowastS1113858 1113859

I+ ξ2 K

lowastlowastlowastS1113858 1113859

I

(24)

Substituting equations (16) and (24) into equation (5)yields the plate stiffness matrix as follows

[K]I

KB1113858 1113859I

+ KlowastS1113858 1113859

I+ Klowast lowastS1113858 1113859

I+ Klowast lowast lowastS1113858 1113859

I

[K]II

KB1113858 1113859I

+ KlowastS1113858 1113859

I+ ξ K

lowast lowastS1113858 1113859

I+ ξ2 K

lowast lowast lowastS1113858 1113859

I

⋮[K]

s KB1113858 1113859

I+ KlowastS1113858 1113859

I+ ξsminus 1

Klowast lowastS1113858 1113859

I+ ξ2(sminus 1)

Klowast lowast lowastS1113858 1113859

I

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(25)

32 Combined Stiffness of High-Order PIEM According toequation (25) the nine-node elements I II and s can bemapped using the same square-shaped master elementSpecifically these elements can be designated as similarelements when the coordinate of an element is similar to thatof other elements )e matrices of the first element layer canbe expressed as follows

KB1113858 1113859I

Ka minusBT

minusAT

minusBmK minusC

T

minusA minusC Kb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

KlowastS1113858 1113859

I

Klowasta minusB

lowastTminusAlowastT

minusBlowast

Klowastm minus C

lowastT

minusAlowast

minusClowast

Klowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Klowast lowastS1113858 1113859

I

Klowast lowasta minusB

lowastlowastTminusAlowastlowastT

minusBlowastlowast

Klowast lowastm minusC

lowastlowastT

minusAlowastlowast

minusClowastlowast

Klowast lowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Klowast lowast lowastS1113858 1113859

I

Klowast lowast lowasta minusB

lowastlowastlowastTminusAlowastlowastlowastT

minusBlowastlowastlowast

Klowast lowast lowastm minusC

lowastlowastlowastT

minusAlowastlowastlowast

minusClowastlowastlowast

Klowast lowast lowastb

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

When equation (26) is substituted into equation (12) andthe result is expanded the equations of the s element layersin the computational domain can be derived as follows

Ka1 minusBT1 minusA

T1

minusB1 Km1 minusCT1

minusA1 minusC1 Kb1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δ0δm1

δ2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

f0

0

f2

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

⋮

(27)

Kai minusBTi minusA

Ti

minusBi Kmi minusCTi

minusAi minusCi Kbi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δiminus1

δmi

δi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

minusfiminus1

0fi

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

⋮(28)

Kas minusBTs minusA

Ts

minusBs Kms minusCTs

minusAs minusCs Kbs

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

δsminus1

δms

δs

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭

minusfsminus1

0fs

⎧⎪⎪⎨

⎪⎪⎩

⎫⎪⎪⎬

⎪⎪⎭ (29)

where

Kai Ka + Klowasta + ξiminus 1

Klowast lowasta + ξ2(iminus 1)

Klowast lowast lowasta

Kbi Kb + Klowastb + ξiminus 1

Klowast lowastb + ξ2(iminus 1)

Klowast lowast lowastb

Kmi Km + Klowastm + ξiminus 1

Klowast lowastm + ξ2(iminus 1)

Klowast lowast lowastm

Ai A + Alowast

+ ξiminus 1Alowastlowast

+ ξ2(iminus 1)Alowastlowastlowast

Bi B + Blowast

+ ξiminus 1Blowastlowast

+ ξ2(iminus 1)Blowastlowastlowast

Ci C + Clowast

+ ξiminus 1Clowastlowast

+ ξ2(iminus 1)Clowastlowastlowast

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(30)

In equations (27)ndash(29) δi is the nodal displacementvector associated with the ith node layer and fi is thecorresponding nodal force vector Combining the equationsfrom the first element layer to the sth element layer andassuming that no internal force is applied to the ith nodelayer (ie fs 0) can yield the following expression

Ka1 minusBT1 minusA

T1 0 0 0 0 0 0 0

minusB1 Km1 minusCT1 0 0 0 0 0 0 0

minusA1 minusC1 Q1 minusBT2 minusA

T2 0 0 0 0 0

0 0 minusB2 Km2 minusCT2 0 0 0 0 0

0 0 minusA2 minusC2 Q2 minusBT3 minusA

T3 0 0 0

⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱0 0 0 0 0 minusBsminus1 Km(sminus1) minusC

Tsminus1 0 0

0 0 0 0 0 minusAsminus1 minusCsminus1 Qsminus1 minusBTs minusA

Ts

0 0 0 0 0 0 0 minusBs Kms minusCTs

0 0 0 0 0 0 0 minusAs minusCs Kbs

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

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

δ0δm1

δ1δm2

δ2⋮

δm(sminus1)

δsminus1

δms

δs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

f0

0

0

0

0

⋮

0

0

0

0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(31)

Mathematical Problems in Engineering 5

where Qi Ka(i+1) + Kbi If Ms Kbs in the last row ofequation (31) then

δs Mminus1s

sAδsminus1 + Csδms( 1113857 (32)

Substituting equation (32) into the second-to-last row ofequation (31) yields

δms Kms minus CTs M

minus1s Cs1113872 1113873

minus 1 sB + C

Ts M

minus1s As1113872 1113873δ4

Mminus1msNsδsminus1

(33)

Similarly substituting equations (32) and (33) into thesecond-to-last row of equation (31) yields

δsminus1 Qsminus 1 minus ATs M

minus1s As minus N

Ts M

minus1msNs1113872 1113873

minus 1Asminus1δsminus2 + Csminus1δms( 1113857

Mminus1sminus1 Asminus1δsminus2 + Csminus1δm(sminus1)1113872 1113873

(34)

According to equations (32)ndash(34) the following iterationformulas can be inferred

Mmi Kmi minus CTi M

minus1i Ci i ms m(s minus 1) m1

(35)

Ni iB + C

Ti M

minus1i Ai i s s minus 1 1

(36)

Mi Qiminus1 minus ATi M

minus1i Ai minus N

Ti M

minus1miNi i s minus 1 s minus 2 1

(37)

δmi Mminus1miNiδiminus1 i s s minus 1 1 (38)

δi Mminus1i Aiδiminus1 + Ciδmi( 1113857 i s s minus 1 1 (39)

Because Ms is equal to Kbs we can iterate Msminus1 Msminus2 and M1 by using equation (37) According to equation(39) δ1 Mminus1

1 (A1δ0 + C1δm1) Substituting δ1 into the firstrow of equation (31) yields the following fundamental IEMformula

Ka1 minus AT1 M

minus12 A1 minus N

T1 M

minus11 N11113872 1113873δ0 Kzδ0 f0 (40)

where Kz (Ka1 minus AT1 Mminus1

2 A1 minus NT1 Mminus1

1 N1) is the com-bined stiffness matrix which preserves the inherent sym-metry characteristic of the global stiffness matrix used in theconventional finite element procedure Using equations (38)and (39) we can condense all inner layer elements andtransform them into a single super element with masternodes at the outer boundary

Ying [24] proved that Kz converges toward a certainconstant quantity as the number of element layers ap-proaches infinity that is

lims⟶infin

K(s)z Kz (41)

where s denotes the number of the defined element layersHowever equation (41) cannot be directly applied to thenumerical formulation because the infinity element layers

are not countable in a physical sense )erefore Liu [23]proposed a convergence method for observing the diagonaltrace term K

(s)Z )e desired accuracy criterion can be

expressed as follows

ε K

(i+1)Z minus K

(i)Z

K(i+1)Z

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868times 100le 10minus 6

(42)

When this criterion is satisfied the iterative program isterminated and the critical number of element layers (scr) isdetermined as equal to the terminated iterative value (i) scr isthe minimum number of element layers required for con-vergence this implies that sufficient elements are available tocover the entire domain )e proportionality ratio ξ isanother important factor in the convergence study A higherξ indicates that a higher number of element layers scr isrequired Specifically given a sufficiently high s value thestiffness K

(s)Z is approximately equal to the combined stiff-

ness KZ

4 Case Studies

41 Circular Plate Subject to a Concentrated LoadConsider for example a simply supported circular platesubjected to a concentrated load P of 1 lbf at its centroid(Figure 3(a)) )e material and geometric parameters are asfollows Youngrsquos modulus E 3times106 psi Poissonrsquos ratio] 03 plate radius R 10 in and thickness h 02 in )eanalytical maximum deflection was provided by a previousstudy [25]

wmax (3 + ])PR

2

16π(1 + ])D (43)

where

D Eh

3

12 1 minus ]21113872 1113873 (44)

)e solution procedure of the high-order PIEM entailsthe assumption that the outer boundary comprises 30uniformly distributed master nodes in addition the pro-portionality ratio ξ is set to 064 (Figure 3(b)) On the basisof the convergence criterion (equation (42)) the number ofvirtual element layers s required is 19 Table 1 illustrates theconvergence process Given the geometric symmetry andload only a quarter of the entire strip under the providedload and boundary conditions must be considered Forcomparison we determine the maximum deflection usingABAQUS software (S4R 394 elements) )e results ob-tained using ABAQUS are in good agreement with thoseobtained using the proposed method as presented inTable 2

42 Square Plate Subject to a Concentrated LoadConsider a simply supported square plate subjected to acenter unit point load (Figure 4(a)) )e material andgeometric parameters are listed as follows Youngrsquosmodulus E 3times106 psi Poissonrsquos ratio ] 03 dimension

6 Mathematical Problems in Engineering

a 80 in and thickness h 08 in )e analytical solutionfor this problem was provided by a previous study [25]where the deflection at the plate centroid can be expressedas follows

wmax αPa

2

D (45)

In equation (45) the coefficient α (00116) is a functionof the dimension ratio a b and D is the flexural rigidity ofthe plate )e solution procedure of the high-order PIEMinvolves the assumption that 40 nodes are uniformly dis-tributed and deployed at the boundary moreover theproportionality ratio ξ is set to 056 (Figure 4(b)) Given theproportionality ratio (056) the number of element layers srequired to achieve convergence is 33 Because of the geo-metric symmetry and load only a quarter of the completestrip under the given load and boundary conditions mustbe considered For comparison we also determine themaximum deflection using ABAQUS software (S4R 400elements) )e results obtained using ABAQUS are in goodagreement with the results obtained using the proposedhigh-order PIEM (Table 3) nevertheless the results ob-tained using the high-order PIEM are closer to the analyticalsolution

43 Rectangular Plate Subject to Bending MomentsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 5(a)) Assume that two of the opposite edges aresimply supported and that the other two edges are free suchthat the applied bending moment (M 100N-mm) vanishesalong the two simply supported edges Assume that the platehas a Youngrsquos modulus of 200GPa and a Poissonrsquos ratio of03 Figure 5(b) presents the corresponding virtual meshpattern obtained by the high-order PIEM before the mesh isdegenerated to form a single super element )e high-orderPIEM solution procedure involves the assumption that theouter boundary comprises 60 uniformly distributed masternodes furthermore the proportionality ratio ξ is set to 081On the basis of the convergence criterion (equation (42))the number of virtual element layers s required is 31 Table 4presents a comparison of the deflection profile of edge (AB)of the plate (Figure 5(a)) obtained using the proposed high-order PIEM scheme with the profile obtained using ABA-QUS )e results obtained from the high-order PIEM are ingood agreement with those obtained from ABAQUS (rel-ative deviationlt 06) (Table 4)

44 Cantilever Plate Subject to Concentrated LoadsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 6(a)) Assume that one of the edges is clamped andthat the other three edges are free such that the concentratedloads (P 5N) are applied at the end Moreover considerthat the plate has a Youngrsquos modulus of 200GPa and aPoissonrsquos ratio of 03 Figure 6(b) presents the correspondingmesh pattern obtained from the high-order PIEM before the

Table 1 Convergence of the required virtual element layer

Number of element layers 15 16 17 18 19Accuracy criterion ε 207times10minus5 781times 10minus6 296times10minus6 112times10minus6 422times10minus7

P = 1

R

(a)

10

9

8

7

6

5

4

3

2

1

00 2 4 6 8 10

(b)

Figure 3 Circular plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 2 Maximum deflection of the circular plate subjected to apoint load at the centroid

Methods Analytical ABAQUS High-orderPIEM

Maximum deflection wmax minus000230 minus000231 minus000231Relative deviation mdash 049 049

Mathematical Problems in Engineering 7

P

a a

(a)

40

35

30

25

20

15

10

5

04035302520151050

(b)

Figure 4 Square plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 3 Maximum deflection of the square plate subjected to a point load at the centroid

Methods Analytical ABAQUS High-order PIEMMaximum deflection wmax minus0000528 minus0000531 minus0000527Relative deviation mdash 056 048

MM

A B

CD

E

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 5 Simply supported rectangular plate subjected to bending moments (a) Analysis model and (b) high-order PIEM

Table 4 Comparisons between proposed approach and ABAQUS of maximum deflection along AB shown in Figure 5

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 0450 0452 051minus30 0793 0795 028minus20 1037 1039 017minus10 1183 1184 0120 1231 1233 01110 1183 1184 01220 1037 1039 01730 0793 0795 02840 0450 0452 05150 0000 0000 000

8 Mathematical Problems in Engineering

mesh is degenerated to form a single super element )esolution procedure of the high-order PIEM involves theassumption that the outer boundary comprises 60 uniformlydistributed master nodes moreover the proportionalityratio ξ is set to 081 Given the proportionality ratio ξ (081)the number of element layers s required is 31 Table 5 il-lustrates a comparison of the deflection profile of edge (AB)of the plate (Figure 6(a)) obtained using the proposed high-order PIEM with the profile obtained using ABAQUS )etwo profiles are in good agreement (relativedeviationlt 02) (Table 5)

45 L-Shaped Plate Subject to a Concentrated LoadConsider an L-shaped plate with the dimensions36mm times 36mm times 18mm (L1 times L2 timesW) (Figure 7(a)) andthe thickness is 05mm )e material and geometric pa-rameters are listed as follows Youngrsquos modulus E 200000and Poissonrsquos ratio ] 03 Assume that two of the oppositeedges are clamped and the concentrated loads (P 1 N)are applied at the point B Figure 7(b) presents the cor-responding mesh pattern obtained from the high-orderPIEM before the mesh is degenerated to form a singlesuper element )e solution procedure of the high-orderPIEM involves the assumption that the outer boundarycomprises 48 uniformly distributed master nodes Giventhe proportionality ratio (081) the number of elementlayers s required to achieve convergence is 34 Table 6illustrates a comparison of the deflection profile of edge(BC) of the L-shaped plate obtained using the proposedhigh-order PIEM with the profile obtained using ABA-QUS )e two profiles are in good agreement (relativedeviation lt 03)

46 Multihole Plate Subject to Bending MomentsConsider a multihole plate with the dimensions144mmtimes 36mmtimes 1mm (LtimesWtimes thickness) (Figure 8) andthe circle holes have a radius R 09mm )e material andgeometric parameters are listed as follows Youngrsquos modulusE 200000 and Poissonrsquos ratio ] 03 Assume that two ofthe opposite edges are simply supported and that the othertwo edges are free such that the applied bending moment

(M 100N-mm) vanishes along the two simply supportededges Figure 9(a) presents the corresponding mesh patternof one quarter of the complete strip )e model consists offour subdomains each of which is the IE model Given theproportionality ratio (081) the number of element layers srequired to achieve convergence is 34)e one quarter of thecomplete strip can be a cell and the complete model bycombining four cells (Figure 9(b)) Table 7 illustrates acomparison of the deflection profile of edge (AB) of themultihole plate obtained using the proposed high-orderPIEM with the profile obtained using ABAQUS )e twoprofiles are in good agreement (relative deviationlt 025))is example not only demonstrates the feasibility ofcombining IE subdomains but also copying

47 Cracked Plate Subject to Bending Moments Finallyconsider a central crack on a rectangular plate (Figure 10(a))As boundary conditions assume that the two edges parallelto the crack are simply supported and momentM is appliedto the edges the other two edges are free )e properties ofthe plate are as follows bh 2 ba 2 cb 2 M 1Youngrsquos modulus E 210000 and Poissonrsquos ratio ] 03)ehigh- and low-order PIEM algorithms can be used to in-vestigate the stress intensity factor (SIF) which can be

P

P

A

B

C

D

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 6 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 5 Comparisons between proposed approach and ABAQUSof maximum deflection along AB shown in Figure 6

CoordinateDeflection(mm) Relative deviation

()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 minus0311 minus0310 012minus30 minus1449 minus1451 012minus20 minus3378 minus3383 014minus10 minus6000 minus6009 0150 minus9212 minus9225 01410 minus12914 minus12932 01420 minus17008 minus17030 01330 minus21399 minus21425 01240 minus25991 minus26020 01150 minus30689 minus30722 011

Mathematical Problems in Engineering 9

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

where Qi Ka(i+1) + Kbi If Ms Kbs in the last row ofequation (31) then

δs Mminus1s

sAδsminus1 + Csδms( 1113857 (32)

Substituting equation (32) into the second-to-last row ofequation (31) yields

δms Kms minus CTs M

minus1s Cs1113872 1113873

minus 1 sB + C

Ts M

minus1s As1113872 1113873δ4

Mminus1msNsδsminus1

(33)

Similarly substituting equations (32) and (33) into thesecond-to-last row of equation (31) yields

δsminus1 Qsminus 1 minus ATs M

minus1s As minus N

Ts M

minus1msNs1113872 1113873

minus 1Asminus1δsminus2 + Csminus1δms( 1113857

Mminus1sminus1 Asminus1δsminus2 + Csminus1δm(sminus1)1113872 1113873

(34)

According to equations (32)ndash(34) the following iterationformulas can be inferred

Mmi Kmi minus CTi M

minus1i Ci i ms m(s minus 1) m1

(35)

Ni iB + C

Ti M

minus1i Ai i s s minus 1 1

(36)

Mi Qiminus1 minus ATi M

minus1i Ai minus N

Ti M

minus1miNi i s minus 1 s minus 2 1

(37)

δmi Mminus1miNiδiminus1 i s s minus 1 1 (38)

δi Mminus1i Aiδiminus1 + Ciδmi( 1113857 i s s minus 1 1 (39)

Because Ms is equal to Kbs we can iterate Msminus1 Msminus2 and M1 by using equation (37) According to equation(39) δ1 Mminus1

1 (A1δ0 + C1δm1) Substituting δ1 into the firstrow of equation (31) yields the following fundamental IEMformula

Ka1 minus AT1 M

minus12 A1 minus N

T1 M

minus11 N11113872 1113873δ0 Kzδ0 f0 (40)

where Kz (Ka1 minus AT1 Mminus1

2 A1 minus NT1 Mminus1

1 N1) is the com-bined stiffness matrix which preserves the inherent sym-metry characteristic of the global stiffness matrix used in theconventional finite element procedure Using equations (38)and (39) we can condense all inner layer elements andtransform them into a single super element with masternodes at the outer boundary

Ying [24] proved that Kz converges toward a certainconstant quantity as the number of element layers ap-proaches infinity that is

lims⟶infin

K(s)z Kz (41)

where s denotes the number of the defined element layersHowever equation (41) cannot be directly applied to thenumerical formulation because the infinity element layers

are not countable in a physical sense )erefore Liu [23]proposed a convergence method for observing the diagonaltrace term K

(s)Z )e desired accuracy criterion can be

expressed as follows

ε K

(i+1)Z minus K

(i)Z

K(i+1)Z

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868times 100le 10minus 6

(42)

When this criterion is satisfied the iterative program isterminated and the critical number of element layers (scr) isdetermined as equal to the terminated iterative value (i) scr isthe minimum number of element layers required for con-vergence this implies that sufficient elements are available tocover the entire domain )e proportionality ratio ξ isanother important factor in the convergence study A higherξ indicates that a higher number of element layers scr isrequired Specifically given a sufficiently high s value thestiffness K

(s)Z is approximately equal to the combined stiff-

ness KZ

4 Case Studies

41 Circular Plate Subject to a Concentrated LoadConsider for example a simply supported circular platesubjected to a concentrated load P of 1 lbf at its centroid(Figure 3(a)) )e material and geometric parameters are asfollows Youngrsquos modulus E 3times106 psi Poissonrsquos ratio] 03 plate radius R 10 in and thickness h 02 in )eanalytical maximum deflection was provided by a previousstudy [25]

wmax (3 + ])PR

2

16π(1 + ])D (43)

where

D Eh

3

12 1 minus ]21113872 1113873 (44)

)e solution procedure of the high-order PIEM entailsthe assumption that the outer boundary comprises 30uniformly distributed master nodes in addition the pro-portionality ratio ξ is set to 064 (Figure 3(b)) On the basisof the convergence criterion (equation (42)) the number ofvirtual element layers s required is 19 Table 1 illustrates theconvergence process Given the geometric symmetry andload only a quarter of the entire strip under the providedload and boundary conditions must be considered Forcomparison we determine the maximum deflection usingABAQUS software (S4R 394 elements) )e results ob-tained using ABAQUS are in good agreement with thoseobtained using the proposed method as presented inTable 2

42 Square Plate Subject to a Concentrated LoadConsider a simply supported square plate subjected to acenter unit point load (Figure 4(a)) )e material andgeometric parameters are listed as follows Youngrsquosmodulus E 3times106 psi Poissonrsquos ratio ] 03 dimension

6 Mathematical Problems in Engineering

a 80 in and thickness h 08 in )e analytical solutionfor this problem was provided by a previous study [25]where the deflection at the plate centroid can be expressedas follows

wmax αPa

2

D (45)

In equation (45) the coefficient α (00116) is a functionof the dimension ratio a b and D is the flexural rigidity ofthe plate )e solution procedure of the high-order PIEMinvolves the assumption that 40 nodes are uniformly dis-tributed and deployed at the boundary moreover theproportionality ratio ξ is set to 056 (Figure 4(b)) Given theproportionality ratio (056) the number of element layers srequired to achieve convergence is 33 Because of the geo-metric symmetry and load only a quarter of the completestrip under the given load and boundary conditions mustbe considered For comparison we also determine themaximum deflection using ABAQUS software (S4R 400elements) )e results obtained using ABAQUS are in goodagreement with the results obtained using the proposedhigh-order PIEM (Table 3) nevertheless the results ob-tained using the high-order PIEM are closer to the analyticalsolution

43 Rectangular Plate Subject to Bending MomentsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 5(a)) Assume that two of the opposite edges aresimply supported and that the other two edges are free suchthat the applied bending moment (M 100N-mm) vanishesalong the two simply supported edges Assume that the platehas a Youngrsquos modulus of 200GPa and a Poissonrsquos ratio of03 Figure 5(b) presents the corresponding virtual meshpattern obtained by the high-order PIEM before the mesh isdegenerated to form a single super element )e high-orderPIEM solution procedure involves the assumption that theouter boundary comprises 60 uniformly distributed masternodes furthermore the proportionality ratio ξ is set to 081On the basis of the convergence criterion (equation (42))the number of virtual element layers s required is 31 Table 4presents a comparison of the deflection profile of edge (AB)of the plate (Figure 5(a)) obtained using the proposed high-order PIEM scheme with the profile obtained using ABA-QUS )e results obtained from the high-order PIEM are ingood agreement with those obtained from ABAQUS (rel-ative deviationlt 06) (Table 4)

44 Cantilever Plate Subject to Concentrated LoadsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 6(a)) Assume that one of the edges is clamped andthat the other three edges are free such that the concentratedloads (P 5N) are applied at the end Moreover considerthat the plate has a Youngrsquos modulus of 200GPa and aPoissonrsquos ratio of 03 Figure 6(b) presents the correspondingmesh pattern obtained from the high-order PIEM before the

Table 1 Convergence of the required virtual element layer

Number of element layers 15 16 17 18 19Accuracy criterion ε 207times10minus5 781times 10minus6 296times10minus6 112times10minus6 422times10minus7

P = 1

R

(a)

10

9

8

7

6

5

4

3

2

1

00 2 4 6 8 10

(b)

Figure 3 Circular plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 2 Maximum deflection of the circular plate subjected to apoint load at the centroid

Methods Analytical ABAQUS High-orderPIEM

Maximum deflection wmax minus000230 minus000231 minus000231Relative deviation mdash 049 049

Mathematical Problems in Engineering 7

P

a a

(a)

40

35

30

25

20

15

10

5

04035302520151050

(b)

Figure 4 Square plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 3 Maximum deflection of the square plate subjected to a point load at the centroid

Methods Analytical ABAQUS High-order PIEMMaximum deflection wmax minus0000528 minus0000531 minus0000527Relative deviation mdash 056 048

MM

A B

CD

E

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 5 Simply supported rectangular plate subjected to bending moments (a) Analysis model and (b) high-order PIEM

Table 4 Comparisons between proposed approach and ABAQUS of maximum deflection along AB shown in Figure 5

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 0450 0452 051minus30 0793 0795 028minus20 1037 1039 017minus10 1183 1184 0120 1231 1233 01110 1183 1184 01220 1037 1039 01730 0793 0795 02840 0450 0452 05150 0000 0000 000

8 Mathematical Problems in Engineering

mesh is degenerated to form a single super element )esolution procedure of the high-order PIEM involves theassumption that the outer boundary comprises 60 uniformlydistributed master nodes moreover the proportionalityratio ξ is set to 081 Given the proportionality ratio ξ (081)the number of element layers s required is 31 Table 5 il-lustrates a comparison of the deflection profile of edge (AB)of the plate (Figure 6(a)) obtained using the proposed high-order PIEM with the profile obtained using ABAQUS )etwo profiles are in good agreement (relativedeviationlt 02) (Table 5)

45 L-Shaped Plate Subject to a Concentrated LoadConsider an L-shaped plate with the dimensions36mm times 36mm times 18mm (L1 times L2 timesW) (Figure 7(a)) andthe thickness is 05mm )e material and geometric pa-rameters are listed as follows Youngrsquos modulus E 200000and Poissonrsquos ratio ] 03 Assume that two of the oppositeedges are clamped and the concentrated loads (P 1 N)are applied at the point B Figure 7(b) presents the cor-responding mesh pattern obtained from the high-orderPIEM before the mesh is degenerated to form a singlesuper element )e solution procedure of the high-orderPIEM involves the assumption that the outer boundarycomprises 48 uniformly distributed master nodes Giventhe proportionality ratio (081) the number of elementlayers s required to achieve convergence is 34 Table 6illustrates a comparison of the deflection profile of edge(BC) of the L-shaped plate obtained using the proposedhigh-order PIEM with the profile obtained using ABA-QUS )e two profiles are in good agreement (relativedeviation lt 03)

46 Multihole Plate Subject to Bending MomentsConsider a multihole plate with the dimensions144mmtimes 36mmtimes 1mm (LtimesWtimes thickness) (Figure 8) andthe circle holes have a radius R 09mm )e material andgeometric parameters are listed as follows Youngrsquos modulusE 200000 and Poissonrsquos ratio ] 03 Assume that two ofthe opposite edges are simply supported and that the othertwo edges are free such that the applied bending moment

(M 100N-mm) vanishes along the two simply supportededges Figure 9(a) presents the corresponding mesh patternof one quarter of the complete strip )e model consists offour subdomains each of which is the IE model Given theproportionality ratio (081) the number of element layers srequired to achieve convergence is 34)e one quarter of thecomplete strip can be a cell and the complete model bycombining four cells (Figure 9(b)) Table 7 illustrates acomparison of the deflection profile of edge (AB) of themultihole plate obtained using the proposed high-orderPIEM with the profile obtained using ABAQUS )e twoprofiles are in good agreement (relative deviationlt 025))is example not only demonstrates the feasibility ofcombining IE subdomains but also copying

47 Cracked Plate Subject to Bending Moments Finallyconsider a central crack on a rectangular plate (Figure 10(a))As boundary conditions assume that the two edges parallelto the crack are simply supported and momentM is appliedto the edges the other two edges are free )e properties ofthe plate are as follows bh 2 ba 2 cb 2 M 1Youngrsquos modulus E 210000 and Poissonrsquos ratio ] 03)ehigh- and low-order PIEM algorithms can be used to in-vestigate the stress intensity factor (SIF) which can be

P

P

A

B

C

D

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 6 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 5 Comparisons between proposed approach and ABAQUSof maximum deflection along AB shown in Figure 6

CoordinateDeflection(mm) Relative deviation

()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 minus0311 minus0310 012minus30 minus1449 minus1451 012minus20 minus3378 minus3383 014minus10 minus6000 minus6009 0150 minus9212 minus9225 01410 minus12914 minus12932 01420 minus17008 minus17030 01330 minus21399 minus21425 01240 minus25991 minus26020 01150 minus30689 minus30722 011

Mathematical Problems in Engineering 9

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

a 80 in and thickness h 08 in )e analytical solutionfor this problem was provided by a previous study [25]where the deflection at the plate centroid can be expressedas follows

wmax αPa

2

D (45)

In equation (45) the coefficient α (00116) is a functionof the dimension ratio a b and D is the flexural rigidity ofthe plate )e solution procedure of the high-order PIEMinvolves the assumption that 40 nodes are uniformly dis-tributed and deployed at the boundary moreover theproportionality ratio ξ is set to 056 (Figure 4(b)) Given theproportionality ratio (056) the number of element layers srequired to achieve convergence is 33 Because of the geo-metric symmetry and load only a quarter of the completestrip under the given load and boundary conditions mustbe considered For comparison we also determine themaximum deflection using ABAQUS software (S4R 400elements) )e results obtained using ABAQUS are in goodagreement with the results obtained using the proposedhigh-order PIEM (Table 3) nevertheless the results ob-tained using the high-order PIEM are closer to the analyticalsolution

43 Rectangular Plate Subject to Bending MomentsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 5(a)) Assume that two of the opposite edges aresimply supported and that the other two edges are free suchthat the applied bending moment (M 100N-mm) vanishesalong the two simply supported edges Assume that the platehas a Youngrsquos modulus of 200GPa and a Poissonrsquos ratio of03 Figure 5(b) presents the corresponding virtual meshpattern obtained by the high-order PIEM before the mesh isdegenerated to form a single super element )e high-orderPIEM solution procedure involves the assumption that theouter boundary comprises 60 uniformly distributed masternodes furthermore the proportionality ratio ξ is set to 081On the basis of the convergence criterion (equation (42))the number of virtual element layers s required is 31 Table 4presents a comparison of the deflection profile of edge (AB)of the plate (Figure 5(a)) obtained using the proposed high-order PIEM scheme with the profile obtained using ABA-QUS )e results obtained from the high-order PIEM are ingood agreement with those obtained from ABAQUS (rel-ative deviationlt 06) (Table 4)

44 Cantilever Plate Subject to Concentrated LoadsConsider a rectangular plate with the dimensions100mmtimes 50mmtimes 05mm (lengthtimeswidthtimes thickness)(Figure 6(a)) Assume that one of the edges is clamped andthat the other three edges are free such that the concentratedloads (P 5N) are applied at the end Moreover considerthat the plate has a Youngrsquos modulus of 200GPa and aPoissonrsquos ratio of 03 Figure 6(b) presents the correspondingmesh pattern obtained from the high-order PIEM before the

Table 1 Convergence of the required virtual element layer

Number of element layers 15 16 17 18 19Accuracy criterion ε 207times10minus5 781times 10minus6 296times10minus6 112times10minus6 422times10minus7

P = 1

R

(a)

10

9

8

7

6

5

4

3

2

1

00 2 4 6 8 10

(b)

Figure 3 Circular plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 2 Maximum deflection of the circular plate subjected to apoint load at the centroid

Methods Analytical ABAQUS High-orderPIEM

Maximum deflection wmax minus000230 minus000231 minus000231Relative deviation mdash 049 049

Mathematical Problems in Engineering 7

P

a a

(a)

40

35

30

25

20

15

10

5

04035302520151050

(b)

Figure 4 Square plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 3 Maximum deflection of the square plate subjected to a point load at the centroid

Methods Analytical ABAQUS High-order PIEMMaximum deflection wmax minus0000528 minus0000531 minus0000527Relative deviation mdash 056 048

MM

A B

CD

E

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 5 Simply supported rectangular plate subjected to bending moments (a) Analysis model and (b) high-order PIEM

Table 4 Comparisons between proposed approach and ABAQUS of maximum deflection along AB shown in Figure 5

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 0450 0452 051minus30 0793 0795 028minus20 1037 1039 017minus10 1183 1184 0120 1231 1233 01110 1183 1184 01220 1037 1039 01730 0793 0795 02840 0450 0452 05150 0000 0000 000

8 Mathematical Problems in Engineering

mesh is degenerated to form a single super element )esolution procedure of the high-order PIEM involves theassumption that the outer boundary comprises 60 uniformlydistributed master nodes moreover the proportionalityratio ξ is set to 081 Given the proportionality ratio ξ (081)the number of element layers s required is 31 Table 5 il-lustrates a comparison of the deflection profile of edge (AB)of the plate (Figure 6(a)) obtained using the proposed high-order PIEM with the profile obtained using ABAQUS )etwo profiles are in good agreement (relativedeviationlt 02) (Table 5)

45 L-Shaped Plate Subject to a Concentrated LoadConsider an L-shaped plate with the dimensions36mm times 36mm times 18mm (L1 times L2 timesW) (Figure 7(a)) andthe thickness is 05mm )e material and geometric pa-rameters are listed as follows Youngrsquos modulus E 200000and Poissonrsquos ratio ] 03 Assume that two of the oppositeedges are clamped and the concentrated loads (P 1 N)are applied at the point B Figure 7(b) presents the cor-responding mesh pattern obtained from the high-orderPIEM before the mesh is degenerated to form a singlesuper element )e solution procedure of the high-orderPIEM involves the assumption that the outer boundarycomprises 48 uniformly distributed master nodes Giventhe proportionality ratio (081) the number of elementlayers s required to achieve convergence is 34 Table 6illustrates a comparison of the deflection profile of edge(BC) of the L-shaped plate obtained using the proposedhigh-order PIEM with the profile obtained using ABA-QUS )e two profiles are in good agreement (relativedeviation lt 03)

46 Multihole Plate Subject to Bending MomentsConsider a multihole plate with the dimensions144mmtimes 36mmtimes 1mm (LtimesWtimes thickness) (Figure 8) andthe circle holes have a radius R 09mm )e material andgeometric parameters are listed as follows Youngrsquos modulusE 200000 and Poissonrsquos ratio ] 03 Assume that two ofthe opposite edges are simply supported and that the othertwo edges are free such that the applied bending moment

(M 100N-mm) vanishes along the two simply supportededges Figure 9(a) presents the corresponding mesh patternof one quarter of the complete strip )e model consists offour subdomains each of which is the IE model Given theproportionality ratio (081) the number of element layers srequired to achieve convergence is 34)e one quarter of thecomplete strip can be a cell and the complete model bycombining four cells (Figure 9(b)) Table 7 illustrates acomparison of the deflection profile of edge (AB) of themultihole plate obtained using the proposed high-orderPIEM with the profile obtained using ABAQUS )e twoprofiles are in good agreement (relative deviationlt 025))is example not only demonstrates the feasibility ofcombining IE subdomains but also copying

47 Cracked Plate Subject to Bending Moments Finallyconsider a central crack on a rectangular plate (Figure 10(a))As boundary conditions assume that the two edges parallelto the crack are simply supported and momentM is appliedto the edges the other two edges are free )e properties ofthe plate are as follows bh 2 ba 2 cb 2 M 1Youngrsquos modulus E 210000 and Poissonrsquos ratio ] 03)ehigh- and low-order PIEM algorithms can be used to in-vestigate the stress intensity factor (SIF) which can be

P

P

A

B

C

D

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 6 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 5 Comparisons between proposed approach and ABAQUSof maximum deflection along AB shown in Figure 6

CoordinateDeflection(mm) Relative deviation

()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 minus0311 minus0310 012minus30 minus1449 minus1451 012minus20 minus3378 minus3383 014minus10 minus6000 minus6009 0150 minus9212 minus9225 01410 minus12914 minus12932 01420 minus17008 minus17030 01330 minus21399 minus21425 01240 minus25991 minus26020 01150 minus30689 minus30722 011

Mathematical Problems in Engineering 9

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

P

a a

(a)

40

35

30

25

20

15

10

5

04035302520151050

(b)

Figure 4 Square plate subjected to a point load at the centroid (a) Analysis model and (b) high-order PIEM

Table 3 Maximum deflection of the square plate subjected to a point load at the centroid

Methods Analytical ABAQUS High-order PIEMMaximum deflection wmax minus0000528 minus0000531 minus0000527Relative deviation mdash 056 048

MM

A B

CD

E

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 5 Simply supported rectangular plate subjected to bending moments (a) Analysis model and (b) high-order PIEM

Table 4 Comparisons between proposed approach and ABAQUS of maximum deflection along AB shown in Figure 5

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 0450 0452 051minus30 0793 0795 028minus20 1037 1039 017minus10 1183 1184 0120 1231 1233 01110 1183 1184 01220 1037 1039 01730 0793 0795 02840 0450 0452 05150 0000 0000 000

8 Mathematical Problems in Engineering

mesh is degenerated to form a single super element )esolution procedure of the high-order PIEM involves theassumption that the outer boundary comprises 60 uniformlydistributed master nodes moreover the proportionalityratio ξ is set to 081 Given the proportionality ratio ξ (081)the number of element layers s required is 31 Table 5 il-lustrates a comparison of the deflection profile of edge (AB)of the plate (Figure 6(a)) obtained using the proposed high-order PIEM with the profile obtained using ABAQUS )etwo profiles are in good agreement (relativedeviationlt 02) (Table 5)

45 L-Shaped Plate Subject to a Concentrated LoadConsider an L-shaped plate with the dimensions36mm times 36mm times 18mm (L1 times L2 timesW) (Figure 7(a)) andthe thickness is 05mm )e material and geometric pa-rameters are listed as follows Youngrsquos modulus E 200000and Poissonrsquos ratio ] 03 Assume that two of the oppositeedges are clamped and the concentrated loads (P 1 N)are applied at the point B Figure 7(b) presents the cor-responding mesh pattern obtained from the high-orderPIEM before the mesh is degenerated to form a singlesuper element )e solution procedure of the high-orderPIEM involves the assumption that the outer boundarycomprises 48 uniformly distributed master nodes Giventhe proportionality ratio (081) the number of elementlayers s required to achieve convergence is 34 Table 6illustrates a comparison of the deflection profile of edge(BC) of the L-shaped plate obtained using the proposedhigh-order PIEM with the profile obtained using ABA-QUS )e two profiles are in good agreement (relativedeviation lt 03)

46 Multihole Plate Subject to Bending MomentsConsider a multihole plate with the dimensions144mmtimes 36mmtimes 1mm (LtimesWtimes thickness) (Figure 8) andthe circle holes have a radius R 09mm )e material andgeometric parameters are listed as follows Youngrsquos modulusE 200000 and Poissonrsquos ratio ] 03 Assume that two ofthe opposite edges are simply supported and that the othertwo edges are free such that the applied bending moment

(M 100N-mm) vanishes along the two simply supportededges Figure 9(a) presents the corresponding mesh patternof one quarter of the complete strip )e model consists offour subdomains each of which is the IE model Given theproportionality ratio (081) the number of element layers srequired to achieve convergence is 34)e one quarter of thecomplete strip can be a cell and the complete model bycombining four cells (Figure 9(b)) Table 7 illustrates acomparison of the deflection profile of edge (AB) of themultihole plate obtained using the proposed high-orderPIEM with the profile obtained using ABAQUS )e twoprofiles are in good agreement (relative deviationlt 025))is example not only demonstrates the feasibility ofcombining IE subdomains but also copying

47 Cracked Plate Subject to Bending Moments Finallyconsider a central crack on a rectangular plate (Figure 10(a))As boundary conditions assume that the two edges parallelto the crack are simply supported and momentM is appliedto the edges the other two edges are free )e properties ofthe plate are as follows bh 2 ba 2 cb 2 M 1Youngrsquos modulus E 210000 and Poissonrsquos ratio ] 03)ehigh- and low-order PIEM algorithms can be used to in-vestigate the stress intensity factor (SIF) which can be

P

P

A

B

C

D

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 6 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 5 Comparisons between proposed approach and ABAQUSof maximum deflection along AB shown in Figure 6

CoordinateDeflection(mm) Relative deviation

()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 minus0311 minus0310 012minus30 minus1449 minus1451 012minus20 minus3378 minus3383 014minus10 minus6000 minus6009 0150 minus9212 minus9225 01410 minus12914 minus12932 01420 minus17008 minus17030 01330 minus21399 minus21425 01240 minus25991 minus26020 01150 minus30689 minus30722 011

Mathematical Problems in Engineering 9

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

mesh is degenerated to form a single super element )esolution procedure of the high-order PIEM involves theassumption that the outer boundary comprises 60 uniformlydistributed master nodes moreover the proportionalityratio ξ is set to 081 Given the proportionality ratio ξ (081)the number of element layers s required is 31 Table 5 il-lustrates a comparison of the deflection profile of edge (AB)of the plate (Figure 6(a)) obtained using the proposed high-order PIEM with the profile obtained using ABAQUS )etwo profiles are in good agreement (relativedeviationlt 02) (Table 5)

45 L-Shaped Plate Subject to a Concentrated LoadConsider an L-shaped plate with the dimensions36mm times 36mm times 18mm (L1 times L2 timesW) (Figure 7(a)) andthe thickness is 05mm )e material and geometric pa-rameters are listed as follows Youngrsquos modulus E 200000and Poissonrsquos ratio ] 03 Assume that two of the oppositeedges are clamped and the concentrated loads (P 1 N)are applied at the point B Figure 7(b) presents the cor-responding mesh pattern obtained from the high-orderPIEM before the mesh is degenerated to form a singlesuper element )e solution procedure of the high-orderPIEM involves the assumption that the outer boundarycomprises 48 uniformly distributed master nodes Giventhe proportionality ratio (081) the number of elementlayers s required to achieve convergence is 34 Table 6illustrates a comparison of the deflection profile of edge(BC) of the L-shaped plate obtained using the proposedhigh-order PIEM with the profile obtained using ABA-QUS )e two profiles are in good agreement (relativedeviation lt 03)

46 Multihole Plate Subject to Bending MomentsConsider a multihole plate with the dimensions144mmtimes 36mmtimes 1mm (LtimesWtimes thickness) (Figure 8) andthe circle holes have a radius R 09mm )e material andgeometric parameters are listed as follows Youngrsquos modulusE 200000 and Poissonrsquos ratio ] 03 Assume that two ofthe opposite edges are simply supported and that the othertwo edges are free such that the applied bending moment

(M 100N-mm) vanishes along the two simply supportededges Figure 9(a) presents the corresponding mesh patternof one quarter of the complete strip )e model consists offour subdomains each of which is the IE model Given theproportionality ratio (081) the number of element layers srequired to achieve convergence is 34)e one quarter of thecomplete strip can be a cell and the complete model bycombining four cells (Figure 9(b)) Table 7 illustrates acomparison of the deflection profile of edge (AB) of themultihole plate obtained using the proposed high-orderPIEM with the profile obtained using ABAQUS )e twoprofiles are in good agreement (relative deviationlt 025))is example not only demonstrates the feasibility ofcombining IE subdomains but also copying

47 Cracked Plate Subject to Bending Moments Finallyconsider a central crack on a rectangular plate (Figure 10(a))As boundary conditions assume that the two edges parallelto the crack are simply supported and momentM is appliedto the edges the other two edges are free )e properties ofthe plate are as follows bh 2 ba 2 cb 2 M 1Youngrsquos modulus E 210000 and Poissonrsquos ratio ] 03)ehigh- and low-order PIEM algorithms can be used to in-vestigate the stress intensity factor (SIF) which can be

P

P

A

B

C

D

(a)

20

10

0

ndash10

ndash20

ndash50 ndash40 ndash30 ndash20 ndash10 0 10 20 30 40 50

(b)

Figure 6 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 5 Comparisons between proposed approach and ABAQUSof maximum deflection along AB shown in Figure 6

CoordinateDeflection(mm) Relative deviation

()High-order PIEM ABAQUS

minus50 0000 0000 000minus40 minus0311 minus0310 012minus30 minus1449 minus1451 012minus20 minus3378 minus3383 014minus10 minus6000 minus6009 0150 minus9212 minus9225 01410 minus12914 minus12932 01420 minus17008 minus17030 01330 minus21399 minus21425 01240 minus25991 minus26020 01150 minus30689 minus30722 011

Mathematical Problems in Engineering 9

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

A

F

WE D

C

BL1

L2P

(a)

ndash5

0

5

10

15

20

25

ndash5 0 5 10 15 20 25

(b)

Figure 7 Cantilever rectangular plate subjected to concentrated loads (a) Analysis model and (b) high-order PIEM

Table 6 Comparisons between proposed approach and ABAQUS of maximum deflection along BC shown in Figure 7

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus9 minus01812 minus01808 026minus3 minus01388 minus01385 0203 minus00987 minus00985 0159 minus00622 minus00622 01115 minus00313 minus00313 00621 minus00088 minus00089 02827 00000 00000 000

M MR

L

W

A B

CD

Figure 8 Cantilever rectangular plate subjected to concentrated loads

15

10

5

0

ndash5

ndash10

ndash15

151050ndash5ndash10ndash15

(a)

15

10

5

0

ndash5

ndash10

ndash15

0 20 40 60 80 100 120 140

(b)

Figure 9 Mesh pattern of a multihole plate (a) Quarter model and (b) complete model

10 Mathematical Problems in Engineering

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

evaluated through crack surface displacement extrapolationand can be expressed as follows [26]

SIF Eh

3

48lim

r⟶0

π2r

1113970

θy(r)11138681113868111386811138681113868 π minusθy(r)

11138681113868111386811138681113868 minusπ1113876 1113877 (46)

where θy is the rotations of themidplane about the y axis andr is the distance of the nodal point from the crack tip asshown in Figure 11 where the stresses near the tip of thecrack are modeled using a polar coordinate framework (r θ)mounted at the crack tip

)e virtual mesh patterns obtained from the high- andlow-order PIEM models are displayed in Figures 10(b) and

10(c) respectively Because of the geometric symmetry andload only one-half of the complete model must be con-sidered )e outer boundary comprises 101 distributedmaster nodes)is example uses an open loop model that isno elements are generated between the first and last masternode thereby creating a crack Figure 12 presents theconvergence of the high- and low-order PIEM solutions interms of the required virtual element layers and SIF )ederived results indicate that for more accurate results thenumber of element layers and proportionality ratio shouldbe set to 100 and 087 respectively with respect to the cracktip Table 8 presents the results obtained using the variousmethods indicating that the results are in good agreement

Free Free

2a

2b

2c

Mx

Simply supported

MSimply supported

Thickness h

y

(a)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(b)

Crack

2

15

1

05

0

1050

ndash05

ndash1

ndash05

ndash2

(c)

Figure 10 Central crack in the rectangular plate (a) Analysis (b) high-order and (c) low-order models

Table 7 Comparisons between the proposed approach and ABAQUS of maximum deflection along AB shown in Figure 8

CoordinateDeflection (mm)

Relative deviation ()High-order PIEM ABAQUS

minus72 00000 00000 000minus54 02883 02877 021minus36 04813 04805 016minus18 06029 06020 0160 06388 06379 01518 06029 06020 01636 04813 04805 01654 02883 02877 02172 00000 00000 000

Mathematical Problems in Engineering 11

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

with those obtained using the proposed method never-theless the results obtained using the high-order PIEM arecloser to the analytical solution

5 Conclusions

We present a high-order PIEM based on MindlinndashReissnerplate theory A new reduction process has been developedto eliminate virtual elements in the IEM domain so that theIE range is condensed and transformed to form a superelement with the master nodes on the boundary only andovercome the problem that the conventional reductionprocess cannot be directly applied Several numerical

examples with complex geometries including L-shaped andmultihole plates subject to bending moments have beenstudied the numerical results are compared with the resultsobtained using ABAQUS software and the comparisonproves the effectiveness of the proposed scheme Finallythe bending behavior of a rectangular plate with a centralcrack is analyzed to demonstrate that the stress intensityfactor (SIF) obtained using the high-order PIEM areconverge faster and closer to the analytical solution )enumerical results demonstrate that the high-order PIEMprovides an accurate and computationally efficient methodfor analyzing the plate bending problems

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

40 45 50 55 6035 8075 9590 1008565 70Number of layers

08

082

084

086

088

09

092

094

096

098

1

SIF

High-order PIEMLow-order PIEM

Figure 12 Convergence of high- and low-order PIEM solutions for SIF for a central crack on a plate

Table 8 SIF for a central crack on a plate subjected to bendingmoments

Methods Ref [26] High-orderPIEM

Low-orderPIEM

SIF 09094 08895 08825Relative deviation mdash 219 296

y

Crack

r

x

0

σxx

τyx

τxy

σyy

θ

Figure 11 Modeling of stresses near tip of crack in the elastic body

12 Mathematical Problems in Engineering

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13

Acknowledgments

)is study was partially supported by the Advanced Instituteof Manufacturing with High-Tech Innovations (AIM-HI)from the Featured Areas Research Center Program withinthe framework of the Higher Education Sprout Project bythe Ministry of Education (MOE) in Taiwan )is researchwas also supported by ROC MOST Foundation Contractnos MOST109-2634-F-194-004 and MOST109-2634-F-194-001

References

[1] H Nguyen-Xuan G R Liu C )ai-Hoang and T Nguyen-)oi ldquoAn edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysisof Reissner-Mindlin platesrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 9ndash12 pp 471ndash4892010

[2] C H )ai L V Tran D T Tran T Nguyen-)oi andH Nguyen-Xuan ldquoAnalysis of laminated composite platesusing higher-order shear deformation plate theory and node-based smoothed discrete shear gap methodrdquo Applied Math-ematical Modelling vol 36 no 11 pp 5657ndash5677 2012

[3] T Belytschko Y Y Lu and L Gu ldquoElement-free galerkinmethodsrdquo International Journal for Numerical Methods inEngineering vol 37 no 2 pp 229ndash256 1994

[4] M Labibzadeh ldquoVoronoi based discrete least squaresmeshless method for assessment of stress field in elasticcracked domainsrdquo Journal of Mechanics vol 32 no 3pp 267ndash276 2016

[5] C H)ai A J M Ferreira and H Nguyen-Xuan ldquoNaturallystabilized nodal integration meshfree formulations for anal-ysis of laminated composite and sandwich platesrdquo CompositeStructures vol 178 pp 260ndash276 2017

[6] C H )ai and P Phung-Van ldquoA meshfree approach usingnaturally atabilized nodal integration for multilayer FGGPLRC complicated plate structuresrdquo Engineering Analysiswith Boundary Elements vol 117 pp 346ndash358 2020

[7] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquo Computer Methods in Applied Mechanicsand Engineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[8] T T Yu S Yin T Q Bui and S Hirose ldquoA simple FSDT-based isogeometric analysis for geometrically nonlinearanalysis of functionally graded platesrdquo Finite Elements inAnalysis and Design vol 96 pp 1ndash10 2015

[9] T Yu S Yin T Q Bui S Xia S Tanaka and S HiroseldquoNURBS-based isogeometric analysis of buckling and freevibration problems for laminated composites plates withcomplicated cutouts using a new simple FSDT theory andlevel set methodrdquo Din-Walled Structures vol 101 pp 141ndash156 2016

[10] N Nguyen-)anh W Li and K Zhou ldquoStatic and free-vi-bration analyses of cracks in thin-shell structures based on anisogeometric-meshfree coupling approachrdquo ComputationalMechanics vol 62 pp 1287ndash1309 2018

[11] W Li N Nguyen-)anh J Huang and K Zhou ldquoAdaptiveanalysis of crack propagation in thin-shell structures via anisogeometric-meshfree moving least-squares approachrdquoComputer Methods in Applied Mechanics and Engineeringvol 358 2020

[12] R W )atcher ldquoSingularities in the solution of Laplacersquosequation in two dimensionsrdquo IMA Journal of AppliedMathematics vol 16 no 3 pp 303ndash319 1975

[13] R W )atcher ldquoOn the finite element method for un-bounded regionsrdquo SIAM Journal on Numerical Analysisvol 15 no 3 pp 466ndash477 1978

[14] L A Ying ldquo)e infinite similar element method for calcu-lating stress intensity factorsrdquo Scientia Sinica vol 21pp 19ndash43 1978

[15] H D Han and L A Ying ldquoAn iterative method in the finiteelementrdquo Mathematica Numerica Sinica vol 1 pp 91ndash991979

[16] C G Go and Y S Lin ldquoInfinitely small element for theproblem of stress singularityrdquo Computers amp Structuresvol 37 pp 547ndash551 1991

[17] C G Go and C C Guang ldquoOn the use of an infinitely smallelement for the three-dimensional problem of stress singu-larityrdquo Computers amp Structures vol 45 no 1 pp 25ndash30 1992

[18] D S Liu and D Y Chiou ldquoA coupled IEMFEM approach forsolving elastic problems with multiple cracksrdquo InternationalJournal of Solids and Structures vol 40 no 8 pp 1973ndash19932003

[19] D S Liu D Y Chiou and C H Lin ldquo3D IEM formulationwith an IEMFEM coupling scheme for solving elastostaticproblemsrdquo Advances in Engineering Software vol 34 no 6pp 309ndash320 2003

[20] D-S Liu K-L Cheng and Z-W Zhuang ldquoDevelopment ofhigh-order infinite element method for stress analysis ofelastic bodies with singularitiesrdquo Journal of Solid Mechanicsand Materials Engineering vol 4 no 8 pp 1131ndash1146 2010

[21] D S Liu C Y Tu and C L Chung ldquoCoupled PIEMFEMalgorithm based onMindlin-Reissner plate theory for bendinganalysis of plates with through-thickness holerdquo CMESComputer Modeling in Engineering amp Sciences vol 92pp 573ndash594 2013

[22] H Nguyen-Xuan T Rabczuk S Bordas and J F DebongnieldquoA smoothed finite element method for plate AnalysisrdquoComputer Methods in Applied Mechanics and Engineeringvol 197 no 13-16 pp 1184ndash1203 2008

[23] G R Liu X Y Cui andG Y Li ldquoAnalysis of mindlin-reissnerplates using cell-based smoothed radial point interpolationmethodrdquo International Journal of Applied Mechanics vol 2no 3 pp 653ndash680 2010

[24] L A Ying Infinite Element Method Peking University PressBeijing China 1995

[25] S P Timoshenko and S Woinowsky-KriegerDeory of Platesand Shells McGraw-Hill New York NY USA Second edi-tion 1959

[26] T Dirgantara and M H Aliabadi ldquoStress intensity factors forcracks in thin platesrdquo Engineering Fracture Mechanics vol 69no 13 pp 1465ndash1486 2002

Mathematical Problems in Engineering 13