Numerical manifold space of Hermitian form and application …jgxy.bjut.edu.cn/picture/article/36/1a/58/83c9ba574362a9...Numerical manifold space of Hermitian form and application

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2013; 95:721–739Published online 28 May 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4515

Numerical manifold space of Hermitian form and application toKirchhoff’s thin plate problems

Hong Zheng1,*,†, Zhijun Liu2 and Xiurun Ge2

1Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology,Beijing 100124, China

2State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, ChineseAcademy of Sciences, Wuhan 430071, China

SUMMARY

For second-order problems, where the behavior is described by second-order partial differential equations,the numerical manifold method (NMM) has gained great success. Because of difficulties in the construc-tion of the H2-regular Lagrangian partition of unity subordinate to the finite element cover; however, fewapplications of the NMM have been found to fourth-order problems such as Kirchhoff’s thin plate problems.Parallel to the finite element methods, this study constructs the numerical manifold space of the Hermitianform to solve fourth-order problems. From the minimum potential principle, meanwhile, the mixed primalformulation and the penalized formulation fitted to the NMM for Kirchhoff’s thin plate problems are derived.The typical examples indicate that by the proposed procedures, even those earliest developed elements inthe finite element history, such as Zienkiewicz’s plate element, regain their vigor. Copyright © 2013 JohnWiley & Sons, Ltd.

Received 5 September 2012; Revised 19 March 2013; Accepted 31 March 2013

KEY WORDS: numerical manifold method; mathematical cover; physical cover; Kirchhoff’s thin platetheory; Zienkiewicz’s plate element

1. INTRODUCTION

The numerical manifold method (NMM) [1], or the finite cover method called by Terada et al.[2], has been paid more and more attention during the past decade due to its great ability to treatcomplicated problems in geotechnical engineering.

Advantageous over the FEM, the NMM can perform the p-adaptive analysis without limitation.By using higher order polynomials as the patch functions, Chen et al. [3] and Jiang et al. [4]derived the general scheme of the NMM, which, however, will lead to the rank deficiency of theglobal stiffness matrix. Employing the topological information inherent in the finite element cover,An et al. [5] proposed an algorithm for predicting the rank deficiency of the global stiffness matrix.Using the finite element cover, Lin [6] reformulated the NMM with more clarity in the frameworkof the partition of unity and explored the merits and limitations of the method. Having solved anumber of typical examples including Cook’s beam, and made comparisons with the FEM, Teradaet al. [7] concluded that the performance deterioration due to the distortion of the physical elementin a mathematical element of the finite cover method is less serious than that due to element distor-tion in the FEM. Having taken the linear polynomials for the patch functions in terms of physicallymeaning nodal unknowns, Tian and Yagawa [8] developed two-dimensional and three-dimensionalsimplex elements and associated the physical patch with a so-called generalized node. In simulation

*Correspondence to: H. Zheng, State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rockand Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China.

†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

722 H. ZHENG, Z. LIU AND X. GE

of soil consolidation, Zhang and Zhou [9] interpolated displacement and pore pressure indepen-dently, leading to a numerically stable scheme for the coupling problems, particularly in the nearlyincompressible case.

The major advantage of the NMM is to solve in a unified way the problems involving continu-ous and discontinuous deformation. Many researchers, including Wu and Wong [10], Kurumataniand Terada [11], and many others, investigated the initiation and propagation of cracks in geoma-terials. An et al. [12] made a comparison between the NMM and the eXtended Finite ElementMethod (XFEM) in simulating discontinuities. The most difficult in the discontinuous deformationanalysis (DDA), in the authors’ viewpoint, is to treat impact and contact between blocks, becausethe open–close iteration developed in the DDA cannot always assure the convergence of the pro-cess, particularly in the three-dimensional cases [13]. To overcome the dependence of the DDA onthe penalty parameters, Zheng and Jiang [14] introduced the nonlinear complementarity algorithmsin the discontinuous deformation analysis, significantly enhancing the numerical properties of theDDA procedure.

Because the NMM is the generalization of the classical FEM, in principle all techniques devel-oped in the FEM and objects treated by the FEM can be carried over to the NMM. However, itseems to the authors that the developments and applications of the NMM are almost limited to thesecond-order problems. Even in their very comprehensive review, for example, Ma et al. [15] didnot refer to the applications of the NMM to the structural problems. Therefore, this study aims toextend the applications of the NMM to the problems of higher order and is organized as follows.

In Section 2, from a viewpoint different from the literature, we first recapitulate the NMM in thecontext of a general finite cover; then in the framework of the finite element cover, we emphasize thedifficulties in constructing the Lagrangian partition of unity that satisfy both global C1 continuityand reproduction of linear polynomials, leading to the necessity of the Hermitian form of the NMMspace. Section 3 concisely states the basic boundary value problem of Kirchhoff’s plate bending. InSection 4, we propose two variational formulations of Kirchhoff’s plate problem fitted to the NMM.The discretization forms corresponding to the variational formulations are discussed in Section 5. InSection 6, we analyze a rectangular plate and an elliptic plate to confirm the proposed procedures.The element adopted therein is Zienkiewicz’s plate element of the first version [16]. Earliest devel-oped in the finite element history, Zienkiewicz’s plate element was an incompatible plate element,but it can pass the patch test only for triangular meshes created by three sets of equally spacedstraight lines. Many researchers, such as Felippa and Bergan [17], Samulelsson [18], and Chen andCheung [19] have made efforts to amend Zienkiewicz’s plate element so that the resulting elementpasses the patch test in all mesh configurations. Experiences tell us that none of them works wellfor all situations. With the procedures in this study, however, we do not need to amend the elementinterpolation scheme but to extend the classical variational formulations such that they are fittedto the NMM. Thus, the NMM offers alternative ways to rescue those incompatible elements earlydeveloped, such as Zienkiewicz’s plate element, Wilson’s Q6 element (for second-order problems),and many others, which behave very well with regular meshes but poorly with meshes of low qual-ity. We know that a high quality mesh should have nearly equal angles for each element as far aspossible. Section 7 concludes this study.

2. ORTHODOX STATEMENTS OF NUMERICAL MANIFOLD METHOD ANDITS EXTENSION

The NMM introduces two sets of covers referred to as the mathematical cover (MC) and the physicalcover (PC), which, albeit a little elusive, facilitates the unified solution of continuous and discontin-uous deformation. Independent of the problem domain, the MC must completely cover the problemdomain, and the MC configuration specifies the solution accuracy. The PC is formed by cutting theMC with the problem domain components, including the boundary segments, the material interfacesand the discontinuities. Having been formed, the PC might move as the problem domain moves. Thisis why the NMM is able to simulate conveniently large movement and the damage process involvingthe generation of new discontinuities.

Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2013; 95:721–739DOI: 10.1002/nme

NUMERICAL MANIFOLD SPACE OF HERMITIAN FORM 723

2.1. Orthodox statements of the numerical manifold method based on the general cover

Shown schematically in Figure 1 is the problem domain �, which is labeled by thick line segmentsand contains a bifurcated crack. Now, we use M1, a big circle; M2, a small circle; and M3, a rect-angle, to cover �. The collection of ¹Miº, i D 1, 2, 3, constitutes the MC, in which each Mi isreferred to as a local cover. Now, we use the components of� to cutM1, obtaining the two physicalpatches: P1�1 and P1�2 in Figure 2. Here, Pi�j represents the j th physical patch from Mi . Pro-ceeding such as this, we obtain the two physical patches P2�1 and P2�2 from M2 in Figure 3, andthe two physical patches P3�1 and P3�2 from M3 in Figure 4. We mention that the physical patchP2�2 (Figure 3) contains a crack terminating inside the patch. We call such a patch a singular patch.For the singular patch, we should better adopt a singular patch function, defined subsequently, that

Figure 1. Problem domain and mathematical cover.

P1-2

P1-1

Figure 2. Two physical patches from M1.





Figure 5. Manifold elements from the PC.

represents the asymptotic behavior of the solution. Similar to P2�2, the physical patch P3�1 shownin Figure 4 is also singular. The collection of

®Pi�j

¯constitutes the PC.

For the convenience of statements, now we use single indices to number all physical patches Pi�jand represent them by Pi , i D 1, 2, : : :,n, with n the number of all Pi�j . A nonempty region sharedby several patches is called a manifold element, which is the basic unit in numerical integration ofvariational formulations. Figure 5 illustrates all the manifold elements from all six patches in theformer text. For example, the manifold elementE1 in Figure 5, is the intersection of P1�1 (Figure 2)and P3�1 (Figure 4) but excluding P2�1 (Figure 3), or expressed as

E1 D P1�1 \P3�1 �P2�1,

in terms of the language of set theory. Likewise, another manifold element E2 in Figure 5 isexpressed as

E2 D P1�2 \P2�2 �P3�2.

Associated with eachPi is the weight function,wi .r/, with r the position vector. The NMM requiresthat the collection of all ¹wi .r/º be the partition of unity (PU) subordinate to ¹Piº, indicating that

wi .r/D 0, if r … Pi , (1.1)

06 wi .r/6 1, if r 2 Pi , (1.2)

nXiD1

wi .r/D 1, if r 2� (1.3)

In the partition of unity method [20], the requirement for wi .r/ > 0 is discarded, which is dif-ferent a little from the standard statements of the partition of unity theorem, for example, Reference[21]. The purpose for the partition of unity method to discard this, in the authors’ opinion, is to



Figure 6. Independence of physical covers at different moments. PC, physical cover.

include the experiences gained in the finite element analysis where the shape functions of a nonlin-ear isoparametric element such as the two-dimensional isoparametric element of eight nodes can benegative in part of the element, for example, reference [22].

Incidentally, in the literature, all physical patches Pi�j , j D 1, 2, : : :, generated from the samelocal cover, Mi , are said to share the same weight function, wi .r/. Here, we argue that this is trueonly for small deformation. In the case of large movement, all physical patches Pi might move asthe solids of interest move. Here is an example. We consider a block’s movement along a ramp.Figure 6(a) displays the initial PC. At this moment, all physical patches Pi�j generated from thesame local cover Mi share the same weight function wi .r/. Once the block slides some distance asshown in Figure 6(b), however, these patches Pi�j have their own weight functions.

Now, we introduce a local finite-dimensional function space Vi defined only on the patch Pi . Viis referred to as the patch space. The functions in Vi , called the patch functions, should represent theasymptotic behavior of the solution on Pi as far as possible. In case that the asymptotic behavior ofthe solution is not available, the functions in Vi can be designated polynomials, where constants aremost frequently chosen.

The global function space V , called the NMM space in the variational formulation, is obtainedby pasting all patch spaces Vi together with the weight functions wi .r/, reading

V �X

wiVi �°X

wivi

ˇ̌̌vi 2 Vi

±(2)

To this point, therefore, the regularity of V is determined by less regular one of wi and vi .

2.2. Hermitian form of numerical manifold method space on finite element cover

The way to generate the MC in the NMM is diverse. If the moving least squares method [23] orthe reproducing kernel method [24] is used to construct the trial and test functions, for example, thelocal covers in the MC can be the supports of the weight functions. In this case, the weight functionscan be arbitrarily smooth.

Although the mesh-free methods have gained great progress, it still takes a long route for them toreach the same status as the FEM in the aspects of application. Therefore, most of applications anddevelopments of the NMM select finite element meshes to form their mathematical covers. If a finiteelement mesh is used as the MC, one node, not always inside the problem domain, corresponds toone local cover that consists of all elements sharing this node, shown in Figure 7(a), and a manifoldelement is just an ordinary finite element that might be cut partially by the domain boundary. Theweight function of the patch is formed by collecting all element shape functions at this node, whoseimage is like a tent and illustrated in Figure 7(b).

For second-order problems, it is enough for the weight functions in the PU to be globally C0

continuous but piecewise C1 continuous, or more precisely, H1 regular. The finite element literaturehas had mature techniques for constructing such H1-regular weight functions (shape functions). Inthe literature, we refer to the approximation to the solution u

uh.r/DXi

uiwi .r/ (3)



Figure 7. One node-associated patch and the weight function. (a) Finite element cover and a patch and(b) weight function of a patch.

Figure 8. A C1 weight function on a regular hexagon. (a) Image of weight function and (b) contours ofweight function.

as the Lagrangian interpolant, where ui D uh.r i /, with r i , position vector of node i . In this case,one node has only one degree of freedom, ui .

For fourth-order problems, the primal variational formulation requires uh.r/ to be globally C1

continuous but piecewise C2 continuous, or more precisely, H2 regular. Nevertheless, the finite ele-ment literature has no off-the-shelf procedures for constructing H2 regular ¹wi .r/º such that theLagrangian interpolant uh of u (Equation (3)) is also H2 regular.

As an aside, in principle the weight functions wi .r/ in the PU can be made C1 for any PC.Moreover, the proof of the partition of unity theorem in the literature, such as [21], is also construc-tive. By using the construction procedure in the proof [21], we indeed construct such a C1 weightfunction wi .r/ on a regular hexagon patch consisting of six equilateral triangle elements, whoseimage is like a cap, illustrated in Figure 8(a).

As long as the weight functions wi .r/ in the interpolant (3) are H2 regular, in principle we do notcommit any variational crime if starting from a primal variational formulation. Unfortunately, theweight functions wi .r/ constructed in this way, as in Figure 8, only satisfy Equations (1.1) – (1.3),implying that they reproduce rigid body translations, but they do not reproduce monomials x and y,that is to say that either

x DXi

xiwi .r/

or

y DXi

yiwi .r/



does not hold anymore. The resulting discretization model will be over-stiff and the conver-gence will be very slow. For fourth-order problems, therefore, we should discard the Lagrangianinterpolant and turn to the Hermitian interpolant. In this way, we have the approximation to thesolution u

uh.r/DXi

�uiNi .r/C u

ixRi .r/C u

iySi .r/

�(4)

if a triangular mesh is adopted where each node has three degrees of freedom, ui D uh.r i /,

uix [email protected] /@x

, and uiy [email protected] /@y

. Here, Ni .r/, Ri .r/, and Si .r/ are shape functions of node i ,satisfying

Ni .rj /D ıij ,@Ni .rj /

@xD@Ni .rj /

@yD 0, (5.1)

Ri .rj /D 0,@Ri .rj /

@xD ıij ,

@Ri .rj /

@yD 0, (5.2)

Si .rj /D 0,@Si .rj /

@xD 0,

@Si .rj /

@yD ıij . (5.3)

A number of procedures for constructing a triangular plate element of nine parameters are availablein the finite element literature, some of which are explicit such as Zienkiewicz’s plate element to bediscussed subsequently, whereas others are implicit such as [25].

3. REVIEW ON KIRCHHOFF’S THIN PLATE BENDING THEORY

A basic assumption in Kirchhoff’s thin plate bending theory, for example, Reference [26], is thata straight line normal to the middle plane before deformation remains straight and normal to themiddle plane after deformation.

3.1. Basic unknowns and equations

The deflection w.x,y/ of the middle plane (´ D 0) – the lateral deformation of the plate, is theunique basic variable and decides everything of the plate. Given w.x,y/, the deformation of theplate can be determined by

u.x,y, ´/D�´@w

@x, v.x,y, ´/D�´

@w

@y, w.x,y,´/D w .x,y/ (6)

under the coordinate system as shown in Figure 9.

Figure 9. Coordinate system and internal forces.



The magnitude of stresses in the plate is in order of .�x , �y , �xy/ >> .�x´, �y´/ >> �´. Theformer five stress components are related to the five internal forces by

�x D12´

h3Mx , �y D

12´

h3My , �xy D

12´

h3Mxy

�x´ D3

2h

�1�

4´2

h2

�Qx , �y´ D

3

2h

�1�

4´2

h2

�Qy

(7)

Here,Mx andMy are referred to as the bending moments normal to x-planes and y-planes, respec-tively, and Mxy the twisting moment. Qx and Qy are called the lateral shear forces acting onx-planes and y-planes, respectively. All are resultant moments and resultant forces of the relevantstresses on the unit width along the thickness h, that is,

Mx D

Z h2

�h2

�x´d´, My D

Z h2

�h2

�y´d´, Mxy D

Z h2

�h2

�xy´d´ (8)

with the unit of [N�m/m], and

Qx D

Z h2

�h2

�x´d´, Qy D

Z h2

�h2

�y´d´ (9)

with the unit of [N/m].The positive values of the bending momentsMx andMy , and the lateral shear forcesQx andQy

are illustrated in Figure 9, with the exact descriptions: positive Mx and My lead to positive �x and�y on the side of ´ > 0, positive Mxy to positive �xy on the side of ´ > 0, and positive Qx and Qy

to positive �x´ and �y´.Qx and Qy generate no strains, but depend on the three moments,

Qx D@Mx

@xC@Mxy

@y, Qy D

@Mxy

@xC@My

@y(10)

Parallel to �x , �y , and �xy in two-dimensional cases, as a result, Mx , My , and Mxy are called thegeneralized stresses.

The equilibrium equation in terms of Mx , My , and Mxy reads

@2Mx

@x2C 2

@2Mxy

@x@yC@2My

@y2C Np D 0 (11)

with Np � Np.x,y/ the lateral traction on ´D�h=2. Hereafter, a quantity under the bar ‘–’ indicatesit is known.

Dual to Mx , My , and Mxy are the generalized strains

�x D�@2w

@x2, �y D�

@2w

@y2, �xy D�

@2w

@x@y(12)

with the first two being the curvatures and the third, the twisting curvature.With the triples .Mx ,My ,Mxy/ and .�x , �y , �xy/, our concerns will be focused on the middle

plane. For example, the strain energy density U of the plate can be regarded as the strain energy ofunit area of the plane, U

��x , �y , �xy

�, having the unit of [N�m/m2 D N/m]. The variation of U in

terms of�ı�x , ı�y , ı�xy

�is defined as

ıU DMxı�x CMyı�y C 2Mxyı�xy (13)

For elastic linear solids, U should be a homogeneous positive definite function of��x , �y , �xy

�. In

the case of isotropic solids, the expression of U is

U D1

2Dh��x C �y

�2C 2.1� �/

��2xy � �x�y

�i(14)



or

U D1

2�TD� (14.1)

Here, �T D��x �y �xy

�and

D DD

24 1 � 0

� 1 0

0 0 1� �

35 (15)

where D

D DEh3

12.1� �2/(16)

is the flexural rigidity of the plate, E is the modulus of elasticity, and � is Poisson’s ratio.Accordingly, Hooke’s law for the plate is

M DD� (17)

withMT D .Mx ,My ,Mxy/.Substitution of Equations (12) and (17) into (11) leads to the equilibrium equation

D�2w D Np (18)

in terms of w, with �2 ��@2

@x2C @2

@y2

2.

3.2. Typical boundary conditions

The planar domain of the middle plate is denoted by �. The local frame on the boundary isconstituted by n and s and displayed in Figure 10.

Let

nD .nx , ny/T (19)

be the exterior unit normal vector; then, the unit tangential vector is defined as

sD .�ny , nx/T. (20)

The transformations of first-order derivatives read

@

@nD nx

@

@xC ny

@

@y,

@

@sD�ny

@

@xC nx

@

@y. (21)

Figure 10. Definition of local frame.



The transformations of the generalized strains and stresses are

n D xn2x C 2 xynxny C yn

2y ,

s D xn2y � 2 xynxny C yn

2x ,

ns D� xnxny C xy�n2x � n

2y

�C ynxny

(22)

where the symbol is being used to represent either � or M .The transformation of the lateral shear force is

Qn DQxnx CQyny . (23)

In general, we divide the boundary of�, into three disconnected parts, that is, @�D C1[C2[C3.C1 is said to be clamped, if

on C1 W w D Nw,@w

@nD N�n. (24)

Here, Nw and N�n are functions of arc length s. C2 is said to be simply supported, if

on C2 W w D Nw,Mn D NMn (25)

with NMn a function of arc length s. C3 is free, if

on C3 WMn D NMn,@Mns

@sCQn D Nq (26)

with Nq a linearly distributed load.

4. VARIATIONAL FORMULATIONS FITTED TO NUMERICAL MANIFOLD METHOD

The minimum potential principle for Kirchhoff’s plate problems says that if w D Nw and @w@nD N�n

on C1, and w D Nw on C2, all of which constitute the essential boundary conditions of the principle,then the true w is such that

ı0.w/D 0 and 0.w/Dmin

with

0.w/D

“�

.U � Npw/dxdy �

ZC3

NqwdsC

ZCM

NMn

@w

@nds (27)

with CM D C2 [C3.In the NMM based on the finite element cover, however, the nodes are allowed not to match with

the domain boundary, implying that the essential boundary conditions stated in the former cannot beimposed directly. Using the Lagrange multiplier method and the penalty method, we subsequentlypropose the mixed primal form and the penalized form fitted to the NMM.

4.1. The mixed primal variational formulation

By using the Lagrange multiplier method to remove the essential boundary conditions and thenidentifying the Lagrange multipliers, we arrive at a functional with regard to the three independentvariables w, q, and Mn,

.w, q,Mn/D

“�

.U � Npw/dxdy �

ZCw

q .w � Nw/ds �

ZC3

Nqwds

C

ZC1

Mn

�@w

@n� N�n

�dsC

ZCM

NMn

@w

@nds

(28)



with Cw D C1 [ C2. The functional .w, q,Mn/ has the property that ı D 0 is equivalent to thefollowing equations:

Mn D NMn, on CM I

QnC@Mns

@sD q, on Cw I

QnC@Mns

@sD Nq, on C3I

@w

@nD N�n, on C1, and w D Nw, on Cw I

@2Mx

@x2C 2

@2Mxy

@x@yC@2My

@y2C Np D 0, in �.

Here, Mx , My , and Mxy are deemed w’s functions in terms of Hooke’s law (17).While formulating the finite element schemes, we use the expression of ı.w, q,Mn/ as follows:

ı.w, q,Mn/D

“�

ıUdxdy �

“�

Npıwdxdy �

ZCw

qıwds �

ZC3

Nqıwds

C

ZC1

Mn

@ıw

@ndsC

ZCM

NMn

@ıw

@nds

�

ZCw

wıqdsC

ZCw

Nwıqds

C

ZC1

@w

@nıMnds �

ZC1

N�nıMnds

(29)

4.2. The penalized variational formulation

We further emphasize that w, q, and Mn in Equation (29) are independently interpolated. Thismight lead to the issue of what kinds of interpolation to q and Mn should be taken such that theinf-sup condition [27] is satisfied. Usually, the schemes of interpolation to q and Mn depend on theplate element type. Inadequate interpolation schemes would incur the rank deficiency of the result-ing augmented matrix. To evade this problem, we might also start from the penalized formulation,reading

p.w/D

“�

.U � Npw/dxdy C

ZCw

1

2Kw .w � Nw/

2 ds �

ZC3

Nqwds

C

ZC1

1

2KM

�@w

@n� N�n

�2dsC

ZCM

NMn

@w

@nds

(30)

with Kw and KM user-specified penalty parameters. Because we always use those meshes of highquality to approximate w, we are also justified to derive the NMM scheme from p .

5. DISCRETIZATION VERSIONS OF VARIATIONAL FORMULATIONS

Now, we use plate elements of the simplest nodal variables to approximate the deflection w

w D

npXiD1

�Niw

i CRiwix C Siw

iy

�, (31)

where np is the number of nodes in the PC; wi D w�xi ,yi

�, wix D

@w.xi ,yi /@x

, and wiy [email protected] ,yi /

@y

are the variables of node i . To be more compact in the derivation, we rewrite it as



w D

npXiD1

N iwi , (32)

with N i D�Ni Ri Si

�and wi D

�wi wix wiy

�T. Accordingly,

�D

npXiD1

Biwi (33)

with Bi a 3� 3 matrix

Bi D�

26664

@2N i@x2

@2N i@y2

@2N i@x@y

37775 (34)

5.1. The mixed primal formulation

We write the interpolation to Mn as

Mn D

nmXiD1

MiMin, on C1 (35)

with M in D Mn.x

i ,yi /, Mi is interpolant to Mn, and nm is the number of the nodes deployed onC1 for interpolating Mn.

Similarly, the interpolation to q is

q D

nqXiD1

Qiqi , on Cw (36)

with qi D q.xi ,yi /, Qi is interpolant to q, and nqis the number of the nodes set on Cw forinterpolating q.

Considering that q and Mn on the boundary are dual to w and @w@n

, respectively, if interpolationproperties (Equations (5.1) – (5.3)) are satisfied, it is appropriate for both Mi and Qi to take theDirac delta function, that is to say,

Mi .r/DQi .r/D ı.r � r i / (37)

All numerical tests we have made, without exception, justified this choice.Substituting Equations (32) – (36) into Equation (29), we have the mixed formulation of

ı.w, q,Mn/D 0 ,

24 Kww Kwq Kwm

Kqw 0 0

Kmw 0 0

350@ w

q

m

1AD

0@ f wf qf m

1A (38)



with

Kww DXe

K eww.3np � 3np/,K

eww Œi , j D

“�eBTi DBjdxdy.3� 3/I

f w D�f iw

�.3np � 1/,

f iw D

“�eNpNT

i dxdy C

ZC3

NqNTi ds �

ZCM

NMn

@NTi

@nds.3� 1/I

Kwq D�K ijwq

�.3np � nq/,K

ijwq D�

ZCw

NTi Qjds.3� 1/I

f q D�f iq�.nq � 1/, f

jq D�

ZCw

Qj NwdsI

Kwm D�K ijwm

�.3np � nm/,K

ijwm D

ZCw

Mj

@NTi

@nds.3� 1/I

f m D�f im�.nm � 1/,f

jm D

ZC1

QjN�nds.

where (3�1), and so on, denote the dimensions of the matrices direct ahead.In the mixed primal formulation (MPF), there are three kinds of nodes involved: 1) w nodes are

simply those nodes of the finite element mesh in use. Each w node has three degrees of freedomrepresenting w, wx , and wy ; 2) q nodes are deployed along the boundary Cw . Each q node has onedegree of freedom, q; and 3) m nodes are set along the boundary C1. Each m node has one degreeof freedom, Mn. The three kinds of nodes are coded independently.

We mention that because all boundary conditions are incorporated into the MPF and they allbecome the natural boundary conditions of the MPF, we do not have to impose the boundaryconditions onto system (38) prior to solving it.

5.2. The penalized formulation

In general, the block (1, 1) – Kww in Equation (38) has a rank deficiency of three, causing sometroubles for the solution. Fortunately, some effective algorithms dedicated to this type of equations,direct or iterative, are available. For example, Zheng and Li [28] proposed a stable and efficientdirect solution.

Alternatively, we can also start from ıp.w/ D 0 (Equation (30)) leading to the penalizedformulation, in which only w nodes are present. In this way, we have the more compact form

KwD f (39)

where

K DXe

K e , K eŒi , j DK eww Œi , j C

ZCw

KwNTi N jdsC

ZC1

KM@NT

i

@n

@N j

@nds,

f D .f i /.3np � 1/,fi D f iw C

ZCw

NwKwNTi dsC

ZC1

N�nKM@NT

i

@nds

Likewise, we do not have to impose boundary conditions on system (39) prior to solution.

6. NUMERICAL EXAMPLES BASED ON ZIENKIEWICZ’S PLATE ELEMENT

There have been many excellent plate elements available that can serve as the analysis in this study.To demonstrate the power of the NMM in treating structural problems, however, we particularlyselect Zienkiewicz’s plate element of the earliest version [16] in analyzing the examples later.As stated in the introduction, Zienkiewicz’s element was the thin plate element that was earliestdeveloped in the finite element history. Due to its simplest scheme, it played an important role



in applications during 1960s. But later on, it was found that this incompatible element can pass thepatch test only for triangular meshes created by three sets of equally spaced straight lines, limiting itsapplication to thin plates of simple shape. Many researchers have then made great efforts to amendZienkiewicz’s plate element so that the resulting element passes the patch test in all configurations,see [26] for the retrospect.

Regardless of how complex the problem domains are in shape, in the NMM, we always adoptmathematical meshes of highest quality to interpolate the field variables. With such regular meshes,Zienkiewicz’s plate element certainly has no difficulty in the patch test.

Zienkiewicz’s plate element [16] has a set of explicit shape functions as follows:

Ni D �2i .3� 2�i /C 2�1�2�3 (40.1)

Ri D�x1 � x3

�ORi C

�x2 � x3

�OSi (40.2)

Si D�y1 � y3

�ORi C

�y2 � y3

�OSi (40.3)

with

OR1 D �21.�1 � 1/� �1�2�3, OR2 D �1�

22C

1

2�1�2�3, OR3 D �1�

23C

1

2�1�2�3I

OS1 D �21�2C

1

2�1�2�3, OS2 D �

22.�2 � 1/� �1�2�3, OS3 D �

23�2C

1

2�1�2�3.

For all the plates in the following examples, we assume they are loaded by an even distributedtraction Np D 1.0 with the flexural rigidity D D 1 (Equation (16)). All the boundary conditions arehomogeneous.

6.1. A simply supported square plate

The edge of this plate has a unit length. Because all boundary segments are simply supported, theyare all type C2. Accordingly, only q nodes on the edge are involved. We adopt two groups of meshes.Each group has three meshes with nearly equal element size (ES). But we match the second mesh ineach group with the boundary, whereas the first and the third not. The ES of the boundary-matchingmesh is made between that of the two boundary-mismatching meshes. The meshes of two groupsare shown in Figures 11 and 12, respectively.

Figure 11. Group 1’s meshes of different element size (ES). (a) ES D 0.30, (b) ES D 0.25, and(c) ESD 0.20.



Figure 12. Group 2’s meshes of different element size (ES). (a) ES D 0.15, (b) ES D 0.125, and (c) ES D0.11.

Table I. Results of example 1 from the two formulations.

Element size of group 1

0.30n 0.25m 0.20n

Method Dc10�3 Err% Dc10

�3 Err % Dc10�3 Err%

MPF 4.373 7.65 4.373 7.65 4.286 5.52PF 4.559 12.24 4.373 7.65 4.286 5.52

Element size of group 2

0.150n 0.125m 0.110n

Method Dc10�3 Err % Dc10

�3 Err % Dc10�3 Err %

MPF 4.156 2.31 4.143 1.99 4.124 1.53PF 4.253 4.70 4.153 2.23 4.224 4.00

MPF, mixed primal form; PF, penalized form; Dc , deflection of centralpoint; n, boundary mismatching mesh; m, boundary matching mesh.

Table I lists deflection and error percentage of the central point from the two formulations withthe different meshes. Here, the error definition is

Err DDc �De

De� 100%,

with Dc , calculated deflection of the central point and De D 4.062 � 10�3, exact deflection of thecentral point.

We mention that the results from the penalized formulation (PF) are generated with the penaltiesKw DKM D 10

6, varying little for the range of 104 � 109.From the results, we draw the conclusions as follows:

1) The accuracy of the MPF depends on the mesh density, rather than whether the mesh matcheswith the boundary or not. This does not always apply to the PF, however. The PF seems toprefer the boundary matching mesh among the meshes with nearly equal ESs.

2) The accuracy of the primal mixed formulation is a little better than that of the PF.

6.2. An elliptic plate with the clamped edge

Now that the plate is not a parallelogram, we cannot design a regular mesh to match the boundary.This example is hence suited for illustrating the ability of the NMM to rescue those incompatibleelements that are convergent only for regular meshes, such as Zienkiewicz’s plate element. Figure 13displays two meshes of different ES.



Figure 13. Two meshes of different element size (ES). (a) ESD 0.100 and (b) ESD 0.069.

Table II. Results of example 2 from the two formulations.

MPF PF

Element size Dc.10�3/ Err(%) Dc.10

�3/ Err(%)

0.100 2.282 7.72 2.282 7.720.069 2.191 3.40 2.191 3.40

MPF, mixed primal formulation; PF, penalized formulation.

Horizontal ordinate

Def

lect

ion

(0.0

01) Analytical

NMM (0.1)

Shell63 (0.1)NMM (0.069)

Shell63 (0.069)

Figure 14. Comparison of deflection along the major axis.

The problem has an analytic solution as follows:

w DNp�x2

a2C y2

b2� 1

2

8D�3a4C 2

a2b2C 3

b4

With the semi-major axis length a D 1 and the semi-minor axis length b D 0.5, the example hasan exact deflection of the central point, Dc D 2.119 � 10�3. Table II lists the results from the twoformulations with two meshes shown in Figure 13. Because the boundary belongs to type C1, bothq and m nodes on the boundary are included in the system.

We also use elements of type Shell63 in ANSYS 12.0 to analyze this plate. ANSYS is a com-mercial program based on FEM, developed by ANSYS Inc., located at Pittsburgh, PA, USA. TheShell63 is also a plate element with nine parameters. Figures 14 and 15 illustrate deflection andmoment along the major axis, respectively. Figures 16 and 17 show deflection and moment alongthe minor axis, respectively. The mesh configurations used by ANSYS are displayed in Figure 18,with the same mesh density as those used by the NMM.

To this point, all the results appear to suggest that Zienkiewicz’s plate element overestimates thedeflection, that is to say, it approximates the analytic solution from above. This is opposite to the



Horizontal ordinate

My

Analytical

NMM (0.1)

Shell63 (0.1)

NMM (0.069)

Shell63 (0.069)

Figure 15. Comparison of moment My along the major axis.

Figure 16. Comparison of deflection along the minor axis.

Figure 17. Comparison of moment Mx along the minor axis.

compatible elements. Meanwhile, although Zienkiewicz’s plate element is the first developed ele-ment and the requirement on the mesh configuration is most stringent, it still generates satisfactoryresults in the setting of the NMM.



Figure 18. Mesh configurations used by ANSYS. (a) ES� 0.10 and (b) ES� 0.069.

7. CONCLUSIONS

The ability of the NMM to treat complicated problems is significantly enhanced because of intro-duction of two sets of covers. Separating the PC from the MC, the NMM combines the advantagesof the finite difference method and the FEM.

The enhancements of the NMM to the FEM are not limited in large movement and damage sim-ulation. If the MC is generated from the finite element mesh of high quality, for any complicateddomains the interpolation based on the partition of unity naturally enjoys high accuracy. Using thisproperty, we can rescue those incompatible elements developed in the finite element history withno need to amend the interpolation schemes in the element. These incompatible elements exhibitexcellent numerical properties with regular meshes, such as Taylor’s Q6 elements for second-orderproblems, Zienkiewicz’s element for fourth-order problems, and many others.

The idea of two covers in the NMM can be easily extended to the structural problems if the NMMspace takes the Hermitian form.

For Kirchhoff’s thin plate problems, the MPF and the PF have nearly the same numerical proper-ties. The solution accuracy given by the former is dependent on the mesh density rather than whetherthe mesh matches with the problem domain boundary. This is believed another excellent propertyinherent in the NMM.

ACKNOWLEDGEMENTS

This study is supported by the National Basic Research Program of China (973 Program), under the GrantNo. 2011CB013505; and the National Natural Science Foundation of China, under the grant numbers:50779031, 50925933.

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