Research Article Shape-Free Finite Element Method: Another Way … · 2019. 7. 31. · formulae of...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 491626, 14 pageshttp://dx.doi.org/10.1155/2013/491626

Research ArticleShape-Free Finite Element Method: Another Way betweenMesh and Mesh-Free Methods

Song Cen,1,2,3 Ming-Jue Zhou,1 and Yan Shang1

1 Department of Engineering Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China2High Performance Computing Center, School of Aerospace, Tsinghua University, Beijing 100084, China3 Key Laboratory of Applied Mechanics, School of Aerospace, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Song Cen; [email protected]

Received 25 June 2013; Accepted 18 July 2013

Academic Editor: Zhiqiang Hu

Copyright © 2013 Song Cen et al.This is an open access article distributed under theCreative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Performances of the conventional finite elements are closely related to the mesh quality. Once distorted elements are used, theaccuracy of the numerical results may be very poor, or even the calculations have to stop due to various numerical problems.Recently, the author and his colleagues developed two kinds of finite element methods, named hybrid stress-function (HSF) andimproved unsymmetric methods, respectively. The resulting plane element models possess excellent precision in both regular andseverely distorted meshes and even perform very well under the situations in which other elements cannot work. So, they are calledshape-free finite elements since their performances are independent to element shapes. These methods may open new ways fordeveloping novel high-performance finite elements. Here, the thoughts, theories, and formulae of above shape-free finite elementmethods were introduced, and the possibilities and difficulties for further developments were also discussed.

1. Introduction

As the cornerstone of computational mechanics and mathe-matics, the finite elementmethod (FEM) has been consideredas one of the greatest academic achievements in the lastcentury [1, 2]. Over the past 60 years, with the progress intechniques of computers, the FEM has obtained remarkabledevelopments in theories and applications, and became oneof themain tools for computations andnumerical simulationsof science and engineering. However, it should be pointed outthat, the FEM is a kind of numerical piecewise interpolationmethods, so that its performance often depends on the meshquality. For example, the accuracy ofmost conventional finiteelements may drop dramatically once the mesh is distorted[3, 4], and this phenomenon will lead to incorrect results orinterruption of computations due to numerical difficulties.Since the mesh distortions are unavoidable for complexgeometry and large deformation problems, the sensitivityproblem to mesh distortion has been regarded as one of theseverest defects existing in the FEM.

In order to circumvent above trouble caused by mesh,numerous researchers began to develop so-called meshless

or mesh-free approaches to replace the FEM, such as theelement-free Galerkin method [5], the reproducing kernelparticle method [6], the meshless local Petrov-Galerkin(MLPG) and the local boundary integral equation (LBIE)methods [7], the boundary node method [8], the hybridboundary node method [9], the least-squares collocationmeshless method [10], the PU-based meshless Shepard inter-polation method [11]. These methods can produce excellentresults without meshing (only distributed points are neededfor most cases), so that they are free of many numerical prob-lems aroused by the FEMmesh. However, since higher orderinterpolations and more complicated techniques are oftenused, these methods usually need much more computationcosts, which is not acceptable for applications of large-scaleengineering problems, especially for nonlinear problems.Therefore, at present, besides several special problems, themeshless or mesh-free methods have not completely takenthe place of the FEM yet.

On the other hand, during the history of the FEMitself, numerous efforts have been also made for improvingperformance and robustness of the traditional finite elementmodels, such as the hybrid stress method proposed by

2 Mathematical Problems in Engineering

Pian et al. [12–14], the incompatible displacement modesproposed by Wilson et al. [15] and Taylor et al. [16], theenhanced strainmethod proposed by Simo and Rifai [17], thestabilization method proposed by Belytschko and Bacharch[18], the selectively reduced integration scheme proposedby Hughes [19], the assumed strain formulations proposedby MacNeal [20] and Piltner and Taylor [21], the quasi-conforming element method proposed by Tang et al. [22],the generalized conforming method proposed by Long andHuang [23], the Alpha finite element method (𝛼FEM) [24]and the smoothed finite element method (S-FEM) [25, 26]proposed by Liu et al., the new spline finite element method[27] proposed by Chen et al., the FE-mesh-free elementmethod proposed byRajendran et al. [28, 29], the newnaturalcoordinate methods proposed by Long et al. [30–37], andthe Hamilton hybrid tress element method proposed by Cenet al. [38]. These works made great contributions on thefinite element method. However, the sensitivity problem tomesh distortion has never been overcome from the outset.For example, once a quadrilateral element almost degeneratesinto a triangle or becomes a concave quadrangle, the accuracywill be lost greatly, or even the computation has to be stoppeddue to the numerical difficulties.

Recently, two mesh distortion immune techniques wereproposed by the author and his coworkers. The first one isthe hybrid stress-function (HSF) element method [39–42],which can be treated as the improved version for the initialhybrid stress element method proposed by Pian [12]. It startsfrom the principle of minimum complementary energy andemploys the fundamental analytical solutions of the Airystress function as the trial functions (analytical trial functionmethod). The second one is the improved unsymmetricelement method [43], which can solve the interpolationfailure and rotational frame dependence problems existingin the original unsymmetric element method proposed byRajendran et al. [44–46] and can produce more accurateresults. It is based on the virtual work principle and used twodifferent strain matrices in the final formulae: one is from theconventional isoparametric element, and the other is derivedfrom the fundamental analytical solutions of elasticity.

The element models constructed by above methods pos-sess excellent precision in both regular and severely distortedmeshes, and even perform very well under the situationsin which other elements cannot work (e.g., a quadrilateralelement degenerates into triangular or concave quadrangularshapes). So, they are called shape-free finite elements sincetheir performances are independent to the element shapes. Itis very interesting that effective ways for completely avoidingsensitivity problem to mesh distortion may be found.

In the following sections, the thoughts, theories, andformulae of the shape-free finite element methods will beintroduced, and the possibilities and difficulties for furtherdevelopments were also discussed.

2. Fundamental Analytical Solutions forTrial Functions

The fundamental analytical solutions of the mechanics playimportant roles in both the hybrid stress-function (HSF)

and the improved unsymmetric element methods. For planeproblem without body forces (or with only constant bodyforces), the analytical solutions 𝜙

𝑖of the Airy stress function

𝜙 should satisfy the Beltrami-Michell equation (the compati-bility equation expressed in terms of stress function). For theisotropic case, the equation is written as

∇4𝜙𝑖=

𝜕4𝜙𝑖

𝜕𝑥4+ 2

𝜕4𝜙𝑖

𝜕𝑥2𝜕𝑦2+

𝜕4𝜙𝑖

𝜕𝑦4= 0, (1)

and for the anisotropic case, the equation becomes

𝐶11

𝜕4𝜙𝑖

𝜕𝑦4+ 𝐶22

𝜕4𝜙𝑖

𝜕𝑥4+ (2𝐶

12+ 𝐶66)

𝜕4𝜙𝑖

𝜕𝑥2𝜕𝑦2

− 2𝐶16

𝜕4𝜙𝑖

𝜕𝑥𝜕𝑦3− 2𝐶26

𝜕4𝜙𝑖

𝜕𝑥3𝜕𝑦

= 0,

(2)

where 𝐶𝑖𝑗= 𝐶𝑗𝑖are the reduced elastic compliances and have

been defined in [41].The fundamental analytical solutions𝜙

𝑖can be used as the

trial function in finite element method or other numericalmethods. In order to choose appropriate number of thesolutions for the construction of new element model, twoprinciples must be followed: (i) the fundamental analyticalsolutions should be selected in turn from the lowest-orderto higher-order; and (ii) the resulting stress or displacementfields should possess completeness in Cartesian coordinates.In plane elasticity, there are 3 solutions corresponding tothe rigid-body displacement state, 3 for the constant stressstate (linear displacement state), and 4 for each other higherorder stress or displacement state. As an example, the first18 solutions 𝜙

𝑖(𝑖 = 1 ∼ 18) and resulting stresses and

displacements for isotropic case are given in Table 1.

3. Shape-Free FEM I: Plane HybridStress-Function (HSF) Element

3.1. The Construction Procedure for 2D Hybrid Stress-FunctionElements. For a plane finite element model, its complemen-tary energy functional can be written in the following matrixform [12, 39–42]:

Π𝐶= Π∗

𝐶+ 𝑉∗

𝐶=

1

2

∬

𝐴𝑒

𝜎TC𝜎𝑡 d𝐴 − ∫

Γ𝑒

TTu𝑡 d𝑠, (3)

Π∗

𝐶=

1

2

∬

𝐴𝑒

𝜎TC𝜎𝑡 d𝐴, 𝑉

∗

𝐶= −∫

Γ𝑒

TTu𝑡 d𝑠, (4)

𝜎 =

{

{

{

𝜎𝑥

𝜎𝑦

𝜏𝑥𝑦

}

}

}

, T = {

𝑇𝑥

𝑇𝑦

} , u = {

𝑢

V} , (5)

where Π∗𝐶is the complementary energy within the element;

𝑉∗

𝐶is the complementary energy along the kinematic bound-

aries (here, all element boundaries are treated as the kine-matic boundaries because the boundary displacements willbe prescribed in (6);C is the elasticity matrix of compliances,it possesses different forms for isotropic and anisotropiccases, or for plane strain and plane stress states; t is the

Mathematical Problems in Engineering 3

Table1:Eighteen

fund

amentalanalyticalsolutio

nsof

theA

irystr

essfun

ctionandresulting

stressa

nddisplacementsolutions

forp

lane

stressprob

lems∗.

i𝜙𝑖

𝜎𝑥𝑖

𝜎𝑦𝑖

𝜏𝑥𝑦𝑖

𝑈𝑖

𝑉𝑖

11

00

01/E

02

x0

00

01/E

3y

00

0y/E

−x/E

4x2

02

0−2𝜇

x/E

2y/E

5xy

00

−1

−(1+𝜇)y/E

−(1+𝜇)x/E

6y2

20

02x/E

−2𝜇

y/E

7x3

06x

0−3(𝜇x2

+y2)/E

6xy/E

8x2y

02y

−2x

−2𝜇

xy/E

[y2 −

(2+𝜇)x

2 ]/E

9xy

22x

0−2y

[𝑥2−(2+𝜇)𝑦2]/𝐸

−2𝜇

xy/E

10y3

6y0

06xy/E

−3(𝜇y2

+x2)/E

11x3y

06xy

−3x

2−(3𝜇x2y+

y3)/E

[3𝑥𝑦2−(2+𝜇)𝑥3]/𝐸

12xy

36xy

0−3y

2[3x2y−

(2+𝜇)y

3 ]/E

−(3𝜇x2y+

y3)/E

13𝑥4−𝑦4

−12y2

12x2

0−4(3xy2+𝜇x3)/E

4(3x

2 y+𝜇y3)/E

146𝑥2𝑦2−𝑥4−𝑦4

12(𝑥2−𝑦2)

−12(𝑥2−𝑦2)

−24xy

4(1+

𝜇)(𝑥3−3𝑥𝑦2)/E

4(1+

𝜇)(𝑦3−3𝑥2𝑦)/E

15𝑥3𝑦2−𝑥𝑦4

2x(𝑥2−6𝑦2)

6xy2

−2y(3𝑥2−2𝑦2)

[𝑥4−6(2

+𝜇)x

2 y2+(3

+2𝜇

)y4 ]/2E

[2(1+2𝜇

)𝑥𝑦3−2𝜇x3y]/E

165𝑥3𝑦2−𝑥5

10x3

−10x(2𝑥2−3𝑦2)

−30x2y

2.5[(1+2𝜇

)𝑥4−6𝜇x2y2−y4]/E

10xy[𝑦2−(2+𝜇)𝑥2]/E

17𝑥2𝑦3−𝑥4𝑦

6x2 y

−2y(6𝑥2−𝑦2)

2x(2𝑥2−3𝑦2)

[2(1+2𝜇

)x3 𝑦

−2𝜇xy

3 ]/E

[𝑦4−6(2

+𝜇)x

2 y2+(3+2𝜇)x

4 ]/2E

185𝑥2𝑦3−𝑦5

10y(3𝑥2−2𝑦2)

10y3

−30xy

210xy[x

2 −(2

+𝜇)y

2 ]/E

2.5[(1+2𝜇

)𝑦4−6𝜇x2y2−x4]/E

∗Note:𝐸→𝐸/(1−𝜇2)and𝜇→𝜇/(1−𝜇)forp

lane

strainprob

lem.E

and𝜇areY

oung’smod

ulus

andPo

isson’sratio

,respectively

.


thickness of the element; 𝜎, the element stress vector; T, thetraction force vector along the element boundaries; u, thedisplacement vector along element boundaries, which can beinterpolated by the element nodal displacement vector q𝑒:

u = N|Γq𝑒, (6)

where matrix N|Γis the interpolation function matrix for

element boundary displacements.The stress vector 𝜎 in (5) can be derivable from the Airy

stress function 𝜙, that is,

𝜎 =

{

{

{

𝜎𝑥

𝜎𝑦

𝜏𝑥𝑦

}

}

}

=

{{{{{{{{{{{

{{{{{{{{{{{

{

𝜕2𝜙

𝜕𝑦2

𝜕2𝜙

𝜕𝑥2

−

𝜕2𝜙

𝜕𝑥𝜕𝑦

}}}}}}}}}}}

}}}}}}}}}}}

}

=R (𝜙) . (7)

And the traction force vector T in (5) can be written as

T = {

𝑇𝑥

𝑇𝑦

} = [

𝑙 0 𝑚

0 𝑚 𝑙]

{

{

{

𝜎𝑥

𝜎𝑦

𝜏𝑥𝑦

}

}

}

= LR (𝜙) with L = [

𝑙 0 𝑚

0 𝑚 𝑙] ,

(8)

where 𝑙 and𝑚 are the direction cosines of the outer normal 𝑛of the element boundaries.

Substituting (7) and (8) into (3) yields

Π𝐶= Π∗

𝐶+ 𝑉∗

𝐶=

1

2

∬

𝐴𝑒

R(𝜙)TCR (𝜙) 𝑡 d𝐴

− ∫

Γ

[LR (𝜙)]

Tu𝑡 d𝑠.

(9)

Let

𝜙 =

𝑁

∑

𝑖=1

𝜙𝑖+3𝛽𝑖= 𝜑𝛽, (10)

with

𝜑 = [𝜙4𝜙5𝜙6⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 𝜙

𝑁+3] ,

𝛽 = [𝛽1𝛽2𝛽3⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 𝛽

𝑁]T,

(11)

where 𝑁 is the number of the fundamental analytical solu-tions used for stress function𝜙 in (10);𝜙

𝑖(𝑖 = 4 ∼ 𝑁+3) are𝑁

fundamental analytical solutions (in Cartesian coordinates)of the Airy stress function 𝜙, which must be selected fromcomplete second-order terms (𝑥2,𝑦2,𝑥𝑦) to complete higher-order terms (see Table 1); 𝛽

𝑖(𝑖 = 1 ∼ 𝑁) are 𝑁 unknown

constants. Obviously, such trial functions will directly lead tomore reasonable stress fields satisfying both equilibrium andcompatibility conditions.

Substitution of (6) and (10) into (9) yields

Π𝐶= Π∗

𝐶+ 𝑉∗

𝐶=

1

2

𝛽TM𝛽 − 𝛽THq𝑒, (12)

M = ∬

𝐴𝑒

STCS𝑡 d𝐴, H = ∫

Γ

STLTN𝑡 d𝑠 (13)

with

S =

[

[

[

[

[

[

[

[

[

[

[

[

𝜕2𝜙4

𝜕𝑦2

𝜕2𝜙5

𝜕𝑦2

⋅ ⋅ ⋅

𝜕2𝜙𝑁+3

𝜕𝑦2

𝜕2𝜙4

𝜕𝑥2

𝜕2𝜙5

𝜕𝑥2

⋅ ⋅ ⋅

𝜕2𝜙𝑁+3

𝜕𝑥2

−

𝜕2𝜙4

𝜕𝑥𝜕𝑦

−

𝜕2𝜙5

𝜕𝑥𝜕𝑦

⋅ ⋅ ⋅ −

𝜕2𝜙𝑁+3

𝜕𝑥𝜕𝑦

]

]

]

]

]

]

]

]

]

]

]

]

=[

[

𝜎𝑥4

𝜎𝑥5

𝜎𝑥6

⋅ ⋅ ⋅ 𝜎𝑥(𝑁+3)

𝜎𝑦4

𝜎𝑦5

𝜎𝑦6

⋅ ⋅ ⋅ 𝜎𝑦(𝑁+3)

𝜏𝑥𝑦4

𝜏𝑥𝑦5

𝜏𝑥𝑦6

⋅ ⋅ ⋅ 𝜏𝑥𝑦(𝑁+3)

]

]

,

(14)

where 𝜎𝑥𝑖, 𝜎𝑦𝑖and 𝜏𝑥𝑦𝑖

(𝑖 = 4 ∼ 𝑁 + 3) are the correspondingstresses of 𝜙

𝑖(𝑖 = 4 ∼ 𝑁 + 3) (see Table 1), and all are

expressed in Cartesian coordinates.According to the principle of minimum complementary

energy, we require

𝜕Π𝐶

𝜕𝛽

= 0. (15)

Then, the unknown constant vector 𝛽 can be expressed interms of the nodal displacement vector q𝑒:

𝛽 = M−1Hq𝑒. (16)

Substitution of (16) into the Π∗𝐶in (12) yields

Π∗

𝐶=

1

2

q𝑒TK∗q𝑒,

K∗ = (M−1H)TH = HTM−1H.

(17)

From the viewpoint of the element definition given in [12],matrixK∗mentioned earlier can be considered as the stiffnessmatrix of the hybrid stress-function element, and therefore, itcan readily be incorporated into the standard finite elementprogram framework.

Once the element nodal displacement vector q𝑒 is solved,the element stresses can be given by

𝜎 = SM−1Hq𝑒. (18)

3.2. A Typical HSF Element HSF-Q8-15𝛽 [40]. Up to present,several plane HSF element models for both isotropic andanisotropic materials, including 8-node and 12-node quadri-lateral elements [39–41], have been successfully developed.Asan example, the 8-node HSF element using 15𝛽, denoted byHSF-Q8-15𝛽 [40], is introduced here.


𝜉

𝜂

1(−1, −1) 2(1, −1)

4(−1, 1)

5(0, −1)

7(0, 1)

8(−1, 0) 6(1, 0)

3(1, 1)

(a)

4

1

2

3

5

68

7

u2

�4

u1

�1

u3

�3

u4

u5

�5

u6

�6

u7

�7

u8

�8

�2

(b)

x

y

4

12

3

5

6

7

8

(c)

(d) (e)Figure 1: 8-node quadrilateral plane element with arbitrary shapes.

Consider an 8-node quadrilateral element shown inFigure 1, and any edge of the element can be either straightor curved, and (𝜉, 𝜂) are the usual isoparametric coordinates.

Differing from the usual models, the element shapes can beeither convex or concave. The element nodal displacementvector q𝑒 is given by

q𝑒 = [𝑢1

V1

𝑢2

V2

𝑢3

V3

𝑢4

V4

𝑢5

V5

𝑢6

V6

𝑢7

V7

𝑢8

V8]T. (19)

Assume that the displacements along each element edgeare interpolated by the nodal displacements of each edge.Therefore, 𝑢, V and corresponding matrixN (see (6) and (12))of each element edge can be given as follows:

u12= {

𝑢

V}

12

= N|𝜂=−1q𝑒, u

23= {

𝑢

V}

23

= N|𝜉=1q𝑒,

u34= {

𝑢

V}

34

= N|𝜂=1q𝑒, u

41= {

𝑢

V}

41

= N|𝜉=−1q𝑒,

(20)

with

N = [

𝑁0

10 𝑁

0

20 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 𝑁

0

80

0 𝑁0

10 𝑁

0

2⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 𝑁

0

8

] , (21)

in which 𝑁0

𝑖(𝜉, 𝜂) (𝑖 = 1 ∼ 8) are the shape functions of the

plane 8-node isoparametric serendipity elements

𝑁0

𝑖=

{{{{{{{{{

{{{{{{{{{

{

−

1

4

(1 + 𝜉𝑖𝜉)(1 + 𝜂

𝑖𝜂)(1 − 𝜉

𝑖𝜉 − 𝜂𝑖𝜂) (𝑖 = 1, 2, 3, 4)

1

2

(1 − 𝜉2) (1 + 𝜂

𝑖𝜂) (𝑖 = 5, 7)

1

2

(1 − 𝜂2) (1 + 𝜉

𝑖𝜉) (𝑖 = 6, 8) ,

(22)

where (𝜉𝑖, 𝜂𝑖) are the isoparametric coordinates of node 𝑖.

Fifteen analytical solutions 𝜙𝑖(𝑖 = 4 ∼ 18) for the stress

function 𝜙 (see (10)), which have been listed in Table 1, aretaken as the trial functions. That is to say, fifteen unknownconstants 𝛽

𝑖are introduced. Then, the matrix S in (14) is a

3×15matrix. It can be seen that the stress fields possess third-order completeness in both 𝑥 and 𝑦. The resulting elementmodel is denoted as HSF-Q8-15𝛽.

In order to evaluate the matrices M and H by Gaus-sian numerical integration procedure, the Cartesian coor-dinates 𝑥 and 𝑦 within an element should be expressedin terms of local coordinates (isoparametric coordinates).Let

𝑥 =

8

∑

𝑖=1

𝑁0

𝑖(𝜉, 𝜂) 𝑥

𝑖, 𝑦 =

8

∑

𝑖=1

𝑁0

𝑖(𝜉, 𝜂) 𝑦

𝑖, (23)

where (𝑥𝑖, 𝑦𝑖) (𝑖 = 1 ∼ 8) are the Cartesian coordinates of the

node 𝑖.

Example 1 (pure bending for a cantilever beam (Figure 2)).As shown in Figure 2, a cantilever beam under plane stresscondition is subjected to a constant bending moment 𝑀.This problem was earlier used by Lee and Bathe [4] to assessthe distortion sensitivity of the quadratic plane elements.


M = 20c2:distributed as

fx =240y

c− 120x

(0, 0)

(0, c)

y

E = 107, 𝜇 = 0.3, t = 1.0M = 20c2

(50, 5)

(85, 3)

Mesh 4

Mesh 5

(51, 5.5) (55, 7)(30, 4)(66, 3)

Mesh 6

(0, 10)

(0, 0)

(100, 10)

(100, 0)

(0, 10)

(0, 0)

(100, 10)

(100, 0)(75, 10)

(25, 0)

Mesh 1

Mesh 2

(33, 6) (68, 7)

(35, 4) (66, 3)Mesh 3

(L, c)

(L, 0)

Figure 2: Pure bending problem for a cantilever beam.

The theoretical solutions for this problem are given by[47]:

𝜎𝑥=

240

𝑐

𝑦 − 120, 𝜎𝑦= 0, 𝜏

𝑥𝑦= 0,

𝑢 = −

120

𝐸

𝑥 +

240

𝑐𝐸

𝑥𝑦, V =36

𝐸

𝑦 −

120

𝑐𝐸

𝑥2−

36

𝑐𝐸

𝑦2.

(24)

Six regular and distorted mesh divisions are employed (seeFigure 2), inwhichmeshes 4, 5, and 6 are so severely distortedthat some quadrilateral elements degenerate into triangles orconcave quadrangles. Numerical results, obtained by presentelement HSF-Q8-15𝛽, 8-node isoparametric serendipity ele-ment Q8, 9-node isoparametric Lagrangian element Q9L, arelisted in Table 2. It can be seen that exact solutions can alwaysbe obtained by HS-F elements, no matter the meshes aredistorted or not, and nomatter the element shapes are convexor concave.

Actually, element HSF-Q8-15𝛽 can produce the exactsolutions for constant stress/strain problems, no matter theelement edges are straight or curved, and no matter theshapes of the elements are convex or concave. And so longas all element edges keep straight, element HSF-Q8-15𝛽can also produce the exact solutions for linear stress/strainproblems, no matter the shapes of the elements are convex orconcave. For higher-order problems, HSF-Q8-15𝛽 elementspossess much better convergence than the correspondingisoparametric elements.

3.3. A 4-Node HSF Quadrilateral Plane Membrane Elementwith Drilling Degrees of Freedom [42]. Consider a 4-nodequadrilateral element with drilling degrees of freedom shownin Figure 3. Differing from the usual displacement-based

Table 2: Results at selected locations for the pure bending problem(Figure 2).

Q8 Q9L HSF-Q8-15𝛽 Exact

Mesh 1𝜎𝑥(0, 10) 120.000 120.000 120.000 120.0

𝜎𝑥(0, 0) −120.000 −120.000 −120.000 −120.0

v(100, 0) × 103 −12.000 −12.000 −12.000 −12.0

Mesh 2𝜎𝑥(0, 10) 56.447 120.000 120.000 120.0

𝜎𝑥(0, 0) −74.863 −120.000 −120.000 −120.0

v(100, 0) × 103 −2.328 −12.000 −12.000 −12.0

Mesh 3

𝜎𝑥(0+, 10) 13.665 120.000 120.000 120.0

𝜎𝑥(0, 10

−) 5.262 120.000 120.000 120.0

𝜎𝑥(0, 0+) −5.665 −120.000 −120.000 −120.0

𝜎𝑥(0+, 0) −14.299 −120.000 −120.000 −120.0

v(100, 0) × 103 −0.477 −12.000 −12.000 −12.0

Mesh 4𝜎𝑥(0, 10) — — 120.000 120.0

𝜎𝑥(0, 0) — — −120.000 −120.0

v(100, 0) × 103 — — −12.000 −12.0

Mesh 5𝜎𝑥(0, 10) — — 120.000 120.0

𝜎𝑥(0, 0) — — −120.000 −120.0

v(100, 0) × 103 — — −12.000 −12.0

Mesh 6

𝜎𝑥(0+, 10) — — 120.000 120.0

𝜎𝑥(0, 10

−) — — 120.000 120.0

𝜎𝑥(0, 0+) — — −120.000 −120.0

𝜎𝑥(0+, 0) — — −120.000 −120.0

v(100, 0) × 103 — — −12.000 −12.0

models, the element shape is allowed to be either convex orconcave.The element nodal displacement vector q𝑒 is definedas


x

y

𝜉

𝜂

1(−1, −1) 2(1, −1)

3(1, 1)4(−1, 1)

u2

�4

u1

�1

u3

�3

u4

�2

𝜔4

𝜔3

𝜔1 𝜔212

34

Figure 3: A 4-node HSF quadrilateral plane membrane element with drilling degrees of freedom.

q𝑒 = [𝑢1

V1

𝜃𝑧1

𝑢2

V2

𝜃𝑧2

𝑢3

V3

𝜃𝑧3

𝑢4

V4

𝜃𝑧4]T, (25)

where 𝜃𝑧𝑖(𝑖 = 1 ∼ 4) are not the physical rotations of

the element nodes. Instead, the definitions of the drillingdegrees of freedom given by Allman [48] are employed. So,the element boundary displacements can be written as

{

𝑢

V}

𝑖𝑗

=

[

[

[

[

𝑁𝑖

0 −

(𝑦𝑗− 𝑦𝑖)

2

𝑁𝑖𝑁𝑗𝑁𝑗

0

(𝑦𝑗− 𝑦𝑖)

2

𝑁𝑖𝑁𝑗

0 𝑁𝑖

(𝑥𝑗− 𝑥𝑖)

2

𝑁𝑖𝑁𝑗

0 𝑁𝑗−

(𝑥𝑗− 𝑥𝑖)

2

𝑁𝑖𝑁𝑗

]

]

]

]

×

{{{{{{{

{{{{{{{

{

𝑢𝑖

V𝑖

𝜃𝑧𝑖

𝑢𝑗

V𝑗

𝜃𝑧𝑗

}}}}}}}

}}}}}}}

}

, (

𝑖 = 1, 2, 3, 4

𝑗 = 2, 3, 4, 1) ,

𝑁1=

1

2

(1 − 𝑠) , 𝑁2=

1

2

(1 + 𝑠) , −1 ≤ 𝑠 ≤ 1.

(26)

For matrix S defined in (14), let 𝑁 = 7, that is, the firstseven analytical solutions 𝜙

𝑖(𝑖 = 1 ∼ 7) for the stress function

𝜙, which have been listed in Table 1, are taken as the trialfunctions. Then, the matrix S is a 3 × 7 matrix. It can be seenthat the stress fields possess linear completeness in both𝑥 and𝑦. The resulting element model is denoted as HSF-Q4𝜃-7𝛽.

Example 2 (a wedge subjected to a uniformly distributedload (Figure 4)). As shown in Figure 4, a cantilever wedgeis subjected to a uniformly distributed load 𝑞. Because ofits triangular shape, the wedge can not be modeled withoutthe use of triangular and/or distorted quadrilateral elements.Since the present quadrilateral element HSF-Q4𝜃-7𝛽 canstill perform when its shape degenerates into triangle, it cantherefore be readily used to model such wedge problem.Numerical results and the percentage errors of the radialstresses at selected points are listed in Table 3. The present

element HSF-Q4𝜃-7𝛽 performs very well for such high-orderbending problem.

Other examples can be found in [42]. Actually, elementHSF-Q4𝜃-7𝛽 may be the best model among all the 4-nodequadrilateral plane membrane elements with drilling degreeof freedom.

4. Shape-Free FEM II: Improved UnsymmetricElement Method [43]

4.1. Brief Reviews on Original 8-Node Unsymmetric ElementUS-QUAD8 [44]. Based on the virtual work principle [44],the final element stiffness matrix (unsymmetric) can bewritten as [44–46]

K𝑒 = ∬

𝐴𝑒

BTDB𝑡 d𝐴 = ∬

1

−1

B∗T

|J|DB |J| 𝑡 d𝜉 d𝜂

= ∬

1

−1

B∗TDB𝑡 d𝜉 d𝜂,

(27)

where |J| is the Jacobian determinant; D is the elasticitymatrix;B is the element strainmatrix for conventional 8-nodeplane isoparametric element Q8:

B =

1

|J|B∗ = 1

|J|

×[

[

𝑁1,𝑥

0 𝑁2,𝑥

0 𝑁3,𝑥

0 ⋅ ⋅ ⋅ 𝑁8,𝑥

0

0 𝑁1,𝑦

0 𝑁2,𝑦

0 𝑁3,𝑦

⋅ ⋅ ⋅ 0 𝑁8,𝑦

𝑁1,𝑦

𝑁1,𝑥

𝑁2,𝑦

𝑁2,𝑥

𝑁3,𝑦

𝑁3,𝑥

⋅ ⋅ ⋅ 𝑁8,𝑦

𝑁8,𝑥

]

]

,

(28)

in which 𝑁𝑖(𝜉, 𝜂) (𝑖 = 1 ∼ 8) have been given by (22); B is

given by

B = [

𝑀1,𝑥

0 𝑀2,𝑥

0 𝑀3,𝑥

0 ⋅ ⋅ ⋅ 𝑀8,𝑥

0

0 𝑀1,𝑦

0 𝑀2,𝑦

0 𝑀3,𝑦

⋅ ⋅ ⋅ 0 𝑀8,𝑦

𝑀1,𝑦

𝑀1,𝑥

𝑀2,𝑦

𝑀2,𝑥

𝑀3,𝑦

𝑀3,𝑥

⋅ ⋅ ⋅ 𝑀8,𝑦

𝑀8,𝑥

] ,

(29)


A (0, 5)

E = 10000.0

𝜇 = 1.0/3.0

t = 1.0

B(1, 5)

10

2

q = 0.1

x

y

r

B

A

Typical mesh 1 × 6

10/24 10/24

The tip division of the mesh 2 × 12

10/48 10/48

The tip division of the mesh 4 × 24

𝜃

𝛼

Corner nodes of a quadrilateralFigure 4: A wedge subjected to a uniformly distributed load.

where 𝑀𝑖(𝑥, 𝑦) (𝑖 = 1 ∼ 8) are Cartesian shape functions,

given by

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

1 1 1 1 1 1 1 1

𝑥1

𝑥2

𝑥3

𝑥4

𝑥5

𝑥6

𝑥7

𝑥8

𝑦1

𝑦2

𝑦3

𝑦4

𝑦5

𝑦6

𝑦7

𝑦8

𝑥2

1𝑥2

2𝑥2

3𝑥2

4𝑥2

5𝑥2

6𝑥2

7𝑥2

8

𝑥1𝑦1𝑥2𝑦2𝑥3𝑦3𝑥4𝑦4𝑥5𝑦5𝑥6𝑦6𝑥7𝑦7𝑥8𝑦8

𝑦2

1𝑦2

2𝑦2

3𝑦2

4𝑦2

5𝑦2

6𝑦2

7𝑦2

8

𝑥2

1𝑦1𝑥2

2𝑦2𝑥2

3𝑦3𝑥2

4𝑦4𝑥2

5𝑦5𝑥2

6𝑦6𝑥2

7𝑦7𝑥2

8𝑦8

𝑥1𝑦2

1𝑥2𝑦2

2𝑥3𝑦2

3𝑥4𝑦2

4𝑥5𝑦2

5𝑥6𝑦2

6𝑥7𝑦2

7𝑥8𝑦2

8

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

×

{{{{{{{{{{

{{{{{{{{{{

{

𝑀1

𝑀2

𝑀3

𝑀4

𝑀5

𝑀6

𝑀7

𝑀8

}}}}}}}}}}

}}}}}}}}}}

}

=

{{{{{{{{{{{

{{{{{{{{{{{

{

1

𝑥

𝑦

𝑥2

𝑥𝑦

𝑦2

𝑥2𝑦

𝑥𝑦2

}}}}}}}}}}}

}}}}}}}}}}}

}

,

(30)

in which (𝑥𝑖, 𝑦𝑖) are the Cartesian coordinates of node 𝑖.

The resulting element model is denoted as US-QUAD8.It can keep producing exact solutions for constant andlinear stress/strain problems by using both regular andseverely distorted meshes. However, once the element isdistorted from quadrilateral to a certain shape, such as atriangle, the first matrix on the left hand of (30) will besingular, that is, interpolation failure will occur. Further-more, this element exhibits rotational frame dependence inhigher-order problems since the displacement fields con-structed by 𝑀

𝑖(𝑥, 𝑦) (𝑖 = 1 ∼ 8) are not compl-

ete.

4.2. The Shape-Free 8-Node Unsymmetric Element US-ATFQ8[43]. In [43], the above unsymmetric element method wasimproved by analytical trial function method. The elementdisplacement fields in terms of Cartesian coordinates areassumed as

u𝑒 = {

𝑢

V} = [

U UV V]{

��

𝛽

} = {

𝑢𝛼+ 𝑢𝛽

V𝛼+ V𝛽

} , (31)

in which

U = [𝑈1

𝑈2

𝑈3

𝑈4

𝑈5

𝑈6

𝑈7

𝑈8

𝑈9

𝑈10

𝑈11

𝑈12

𝑈13

𝑈14]

V = [𝑉1

𝑉2

𝑉3

𝑉4

𝑉5

𝑉6

𝑉7

𝑉8

𝑉9

𝑉10

𝑉11

𝑉12

𝑉13

𝑉14] ,

U = [𝑈15

𝑈16

𝑈17

𝑈18] ,

V = [𝑉15

𝑉16

𝑉17

𝑉18] ,

�� = [𝛼1

𝛼2

𝛼3

𝛼4

𝛼5

𝛼6

𝛼7

𝛼8

𝛼9

𝛼10

𝛼11

𝛼12

𝛼13

𝛼14]T,

𝛽 = [𝛽1

𝛽2𝛽3𝛽4]

T,

(32)


Table 3: HSF-Q4𝜃-7𝛽 results of radial stress at selected points for a wedge subjected to a uniformly distributed load (Figure 4).

Mesh 1 × 6 2 × 12 4 × 24 Exact [48]𝜎𝑟at point A (0, 5) 7.6835 (1.38%) 7.5894 (0.13%) 7.5806 (0.02%) 7.5792

𝜎𝑟at point B (1, 5) −7.7920 (1.47%) −7.7088 (0.39%) −7.6832 (0.05%) −7.6792

where 𝑈𝑖and 𝑉

𝑖(𝑖 = 1 ∼ 18) are the displacement solutions

derived from the fundamental analytical solutions of the Airystress function, and have been given inTable 1;𝛼

𝑖(𝑖 = 1 ∼ 14)

are 14 unknown parameters related to 𝑢𝛼and V𝛼; 𝛽𝑖(𝑖 = 1 ∼

4) are 4 unknown parameters related to 𝑢𝛽and V𝛽;

{

𝑢𝛼

V𝛼

} = {

U��V��} , (33)

are assumed displacements which possess third-order com-pleteness in 𝑥 and y; and

{

𝑢𝛽

V𝛽

} = {

U𝛽V𝛽

} , (34)

are assumed fourth-order displacement fields.For the fourth-order displacement fields 𝑢

𝛽and V

𝛽, two

additional relaxed point bubble conditions are introduced8

∑

𝑖=5

𝑢𝛽(𝑥𝑖, 𝑦𝑖) =

8

∑

𝑖=5

[𝑈15(𝑥𝑖, 𝑦𝑖) 𝛽1+ 𝑈16(𝑥𝑖, 𝑦𝑖) 𝛽2

+ 𝑈17(𝑥𝑖, 𝑦𝑖) 𝛽3+ 𝑈18(𝑥𝑖, 𝑦𝑖) 𝛽4] = 0,

8

∑

𝑖=5

V𝛽(𝑥𝑖, 𝑦𝑖) =

8

∑

𝑖=5

[𝑉15(𝑥𝑖, 𝑦𝑖) 𝛽1+ 𝑉16(𝑥𝑖, 𝑦𝑖) 𝛽2

+ 𝑉17(𝑥𝑖, 𝑦𝑖) 𝛽3+ 𝑉18(𝑥𝑖, 𝑦𝑖) 𝛽4] = 0.

(35)

So,𝛽2and𝛽4can be expressed in terms of𝛽

1and𝛽3as follows

{

𝛽2

𝛽4

} = −[

𝑎2𝑎4

𝑏2𝑏4

]

−1

[

𝑎1𝑎3

𝑏1𝑏3

]{

𝛽1

𝛽3

} = [

𝑟11

𝑟12

𝑟21

𝑟22

]{

𝛽1

𝛽3

} ,

(36)

in which

𝑎1= 0.5

8

∑

𝑖=5

𝑥4

𝑖− (6 + 3𝜇)

8

∑

𝑖=5

𝑥2

𝑖𝑦2

𝑖+ (1.5 + 𝜇)

8

∑

𝑖=5

𝑦4

𝑖,

𝑎2= (2.5 + 5𝜇)

8

∑

𝑖=5

𝑥4

𝑖− 15𝜇

8

∑

𝑖=5

𝑥2

𝑖𝑦2

𝑖− 2.5

8

∑

𝑖=5

𝑦4

𝑖,

𝑎3= (2 + 4𝜇)

8

∑

𝑖=5

𝑥3

𝑖𝑦𝑖− 2𝜇

8

∑

𝑖=5

𝑥𝑖𝑦3

𝑖,

𝑎4= 10

8

∑

𝑖=5

𝑥3

𝑖𝑦𝑖− (20 + 10𝜇)

8

∑

𝑖=5

𝑥𝑖𝑦3

𝑖,

𝑏1= −2𝜇

8

∑

𝑖=5

𝑥3

𝑖𝑦𝑖+ (2 + 4𝜇)

8

∑

𝑖=5

𝑥𝑖𝑦3

𝑖,

𝑏2= − (20 + 10𝜇)

8

∑

𝑖=5

𝑥3

𝑖𝑦𝑖+ 10

8

∑

𝑖=5

𝑥𝑖𝑦3

𝑖,

𝑏3= (1.5 + 𝜇)

8

∑

𝑖=5

𝑥4

𝑖− (6 + 3𝜇)

8

∑

𝑖=5

𝑥2

𝑖𝑦2

𝑖+ 0.5

8

∑

𝑖=5

𝑦4

𝑖,

𝑏4= −2.5

8

∑

𝑖=5

𝑥4

𝑖− 15𝜇

8

∑

𝑖=5

𝑥2

𝑖𝑦2

𝑖+ (2.5 + 5𝜇)

8

∑

𝑖=5

𝑦4

𝑖.

(37)

Let

𝛽1= 𝛼15

𝛽3= 𝛼16.

(38)

Then, the displacement fields in (31) can be rewritten as

u𝑒 = {

𝑢

V} = [

U UV V

]

{

{

{

��

{

𝛼15

𝛼16

}

}

}

}

= P𝛼, (39)

with

U = [(𝑈15+ 𝑟11𝑈16+ 𝑟21𝑈18) (𝑈

17+ 𝑟12𝑈16+ 𝑟22𝑈18)]

V = [(𝑉15+ 𝑟11𝑉16+ 𝑟21𝑉18) (𝑉

17+ 𝑟12𝑉16+ 𝑟22𝑉18)] ,

P = [

U UV V

] ,

𝛼 = [𝛼1

𝛼2

𝛼3

⋅ ⋅ ⋅ 𝛼14

𝛼15

𝛼16]T.

(40)

Substitution of coordinates of eight nodes into (39) yields

d𝛼 = q𝑒, (41)

in which

d =

[

[

[

[

[

P(𝑥1, 𝑦1)

P(𝑥2, 𝑦2)

...P(𝑥8, 𝑦8)

]

]

]

]

]16 × 16

, (42)


(0, 0)

(0, c)

x

y

distributed as

P = 20c2/L

P = 20c2/L

distributed as

fy =120y

L−120y2

cL

M = 20c2

fx =240y

c− 120

E = 107, 𝜇 = 0.3, t = 1.0

(100, 10)

Mesh 11

(0, 10)

(0, 0)

(50, 10)

(75, 0) (100, 0)

(100, 10)(0, 10)

Mesh 11a

(0, 0) (50, 0)

(50, 10)(25, 10) (75, 10)

(32.5, 5) (82.5, 0)

(100, 0)

Mesh 2

(75, 10)

(25, 0)

Mesh 3(33, 6) (68, 7)

(35, 4) (66, 3)

(100, 10)(0, 10)

Mesh 1

(0, 0) (50, 0)

(50, 10)

(100, 0)

(100, 10)(0, 10)

Mesh 4(50, 5)

(0, 0) (100, 0)

Mesh 5 (85, 3)

Mesh 6(66, 3)

(30, 4) (55, 7)(51, 5.5)

Mesh 9

(0, 0)

(0, 20) (10, 20)

(10, 0)

(0, 10)

Mesh 7

(0, 0)

(20, 10)

(20, 0)

Mesh 8

(10, 10)

(10, 0)

x

x(y) = 10 − 8(y/10)(1 − y/10) Mesh 10

(0, 10) (10, 10)

y

y(x) = 10 − 8(x/10)(1 − x/10)

(L, c)

(L, 0)

Figure 5: Linear bending problem for a cantilever beam.

where q𝑒 is the element nodal displacement vector and hasbeen given by (19). Then, 𝛼

𝑖(𝑖 = 1 ∼ 16) can be solved by

𝛼 = d−1

q𝑒. (43)

It should be noted that so long as the element nodes are not incoincidence with each other, the matrix d is hardly singularfor various element shapes.

The displacement fields in terms of Cartesian coordinatesof the element US-QUAD8 can be replaced by (39).Thus, theelement stiffness matrix in (27) can be rewritten as

K𝑒 = ∬

𝐴𝑒

BTSd−1

𝑡 d𝐴 = ∬

1

−1

B∗T

|J|Sd−1

|J| 𝑡 d𝜉 d𝜂

= ∬

1

−1

B∗TS𝑡 d𝜉 d𝜂 ⋅ d−1

,

(44)

where B and B∗ are given by (28), and

S = [

[

0 0 0 𝜎𝑥4

𝜎𝑥5

⋅ ⋅ ⋅ 𝜎𝑥14

𝜎𝑥15

+ 𝑟11𝜎𝑥16

+ 𝑟21𝜎𝑥18

𝜎𝑥17

+ 𝑟12𝜎𝑥16

+ 𝑟22𝜎𝑥18

0 0 0 𝜎𝑦4

𝜎𝑦5

⋅ ⋅ ⋅ 𝜎𝑦14

𝜎𝑦15

+ 𝑟11𝜎𝑦16

+ 𝑟21𝜎𝑦18

𝜎𝑦17

+ 𝑟12𝜎𝑦16

+ 𝑟22𝜎𝑦18

0 0 0 𝜏𝑥𝑦4

𝜏𝑥𝑦5

⋅ ⋅ ⋅ 𝜏𝑥𝑦14

𝜏𝑥𝑦15

+ 𝑟11𝜏𝑥𝑦16


𝜏𝑥𝑦17



]

]3 × 16

, (45)


Table 4: Results at selected locations for the linear bending problem (Figure 5).

Q8 (3 × 3) HSF-Q8-15𝛽 (5 × 5) US-QUAD8 (3 × 3) US-ATFQ8(5 × 5 or 3 × 3) Exact

𝜎𝑥(0, 10) −66.871 −120.00 −66.871 −120.000 −120.0

Mesh 1 𝜎𝑥(0, 0) 66.871 120.00 66.871 120.000 120.0

v(100, 0) × 103 6.282 8.558 6.282 8.0460 8.046𝜎𝑥(0, 10) −35.248 −113.63 −74.317 −120.000 −120.0

Mesh 2 𝜎𝑥(0, 0) 67.040 119.63 71.781 120.000 120.0

v(100, 0) × 103 1.611 7.104 6.415 8.0460 8.046𝜎𝑥(0+,10) −11.307 −68.645 −66.349 −120.000 −120.0

𝜎𝑥(0, 10

−) −55.699 −66.980 −67.974 −120.000 −120.0

Mesh 3 𝜎𝑥(0, 0+) 55.305 66.146 67.995 120.000 120.0

𝜎𝑥(0+, 0) 11.073 69.328 66.328 120.000 120.0

v(100, 0) × 103 4.520 6.350 6.292 8.0460 8.046𝜎𝑥(0, 10) −72.471 −120.00 −120.0

Mesh 4 𝜎𝑥(0, 0) Failed 76.265 Failed 120.00 120.0

v(100, 0) × 103 6.663 8.0460 8.046𝜎𝑥(0, 10) −77.515 −58.158 −120.000 −120.0

Mesh 5 𝜎𝑥(0, 0) Failed 66.306 59.158 120.000 120.0

v(100, 0) × 103 6.729 5.471 8.0460 8.046𝜎𝑥(0+, 10) −70.346 −42.671 −120.000 −120.0

𝜎𝑥(0, 10

−) −66.835 −44.636 −120.000 −120.0

Mesh 6 𝜎𝑥(0, 0+) Failed 66.877 40.974 120.000 120.0

𝜎𝑥(0+, 0) 68.512 41.048 120.000 120.0

v(100, 0) × 103 6.373 3.752 8.0460 8.046𝜎𝑥(0, 10) −105.69 −120.00 −105.69 −120.000 −120.0

Mesh 7 𝜎𝑥(0, 0) 105.69 120.00 105.69 120.000 120.0

v(20, 0) × 104 3.485 3.975 3.485 3.6600 3.66𝜎𝑥(0, 10) −115.25 −119.02 −118.99 −120.000 −120.0

Mesh 8 𝜎𝑥(0, 0) 115.25 119.02 118.99 120.000 120.0

v(20, 0) × 104 3.3423 3.796 3.5570 3.6600 3.66𝜎𝑥(0, 20) −118.68 −120.00 −118.68 −120.000 −120.0

Mesh 9 𝜎𝑥(0, 0) 118.68 120.00 118.68 120.000 120.0

v(10, 0) × 105 11.1963 15.750 11.1963 13.200 13.2𝜎𝑥(0, 20) −97.066 −121.61 −118.94 −120.00b/−119.65a −120.0

Mesh 10 𝜎𝑥(0, 0) 138.21 124.61 117.74 120.00b/119.36a 120.0

v(10, 0) × 105 12.8162 13.738 12.9157 13.200b/13.190a 13.2𝜎𝑥(0, 10) −24.732 −36.104 −120.000 −120.0

Mesh 11 𝜎𝑥(0, 0) Failed 20.048 36.230 120.000 120.0

v(100, 0) × 103 1.517 3.1470 8.0460 8.046a3 × 3 integration scheme; b5 × 5 integration scheme.

where𝜎𝑥𝑖,𝜎𝑦𝑖and 𝜏𝑥𝑦𝑖

(𝑖 = 4 ∼ 18) have been given in Table 1.And the element stresses can be evaluated by

𝜎 =

{

{

{

𝜎𝑥

𝜎𝑦

𝜏𝑥𝑦

}

}

}

= Sd−1

q𝑒. (46)

The resulting element model is denoted as US-ATFQ8.It can avoid the interpolation failure and rotational frame

dependence problems existing in the original unsymmetricelement US-QUAD8, and possesses much better accuracy forhigher-order problems.

Example 3 (linear bending for a cantilever beam (Figure 5)).As shown in Figure 5, a cantilever beam under plane stresscondition is subjected to a linear bending moment causedby a shear force 𝑃 at the free end. This problem is often


used to assess the distortion sensitivity of the cubic planeelements [4]. The theoretical solutions for this problem aregiven by [47]. The meshes used for this example are alsogiven in Figure 5. Table 4 lists results using full or suggestedintegration schemes. It is astonishing that the present elementUS-ATFQ8 can almost produce the exact solutions under allmeshes.This phenomenon has not been found before for anyother 8-node plane elements.

The element US-ATFQ8 can nearly produce the exactsolutions for constant, pure bending and linear bendingproblems, nomatter the element edges are straight or curved,and no matter the shapes of the elements are convex orconcave. This important advantage cannot be achieved byother 8-node, 16-DOF elements. Compared with the orig-inal unsymmetric 8-node plane element US-QUAD8, thepresent element possesses obvious advantages: (i) elementUS-ATFQ8 can still work well when interpolation failuremodes for US-QUAD8 occur; (ii) element US-ATFQ8 nat-urally eliminates the rotation dependency which exists inelement US-QUAD8 for higher-order problems; and (iii)element US-ATFQ8 can provide more accurate results thanthose obtained by element US-QUAD8 in most complicatedproblems.Therefore, element US-ATFQ8 is a truly shape-freefinite element model.

5. Discussions and Concluding Remarks

Since there is no inverse of the Jacobianmatrix existing in theexpressions of the element stiffness matrix, most sensitivityproblems to mesh distortion can be automatically avoided.And the use of the fundamental analytical solutions ofelasticity significantly improved the element accuracy. Havewe already found effective ways for developing new finiteelements whose performances are independent to elementshapes? If it was true, the users of FEM would not worryabout the mesh quality in practical analyses, and it was notnecessary for researchers to make more efforts on meshgeneration techniques.

However, the answer is not optimistic. We have toencountermany additional difficulties when generalize abovemethods for developing other models.

When one develops 3D 20-node hexahedral element,in order to construct a complete fourth-order displacementfields (similar with (31) for 2D case), 75 fundamental ana-lytical solutions must be considered, which is too com-plicated and not economical for practical applications. Onthe other hand, when one wants to develop lower-orderelement models (such as plane 4-node quadrilateral elementwith 8 DOFs, or 3D 8-node hexahedral element with 24DOFs) directly using the above method, the number of theeffective fundamental analytical solutions will be limitedby the number of the element DOFs, so that there is nonoteworthy improvements on accuracy.

In HSF element method, in order to guarantee its robustperformance, the assumed exact displacements of the ele-ment boundaries may be required. However, it is difficult toobtain such displacements in 3D problems.

Furthermore, how to select the appropriate forms ofthe fundamental analytical solutions [49], how to construct

shape-free plate and shell elements, how to improve thecomputational efficiency of the elements with unsymmetricstiffness matrices, how to deal with the problems in whichthere are no fundamental analytical solutions (such as non-linear problems), and so forth, are all important issues thatwe have to face to.

Obviously, new techniques must be developed for solvingabove problems. But at the same time, do not forget that someexisting achievements, such as the new natural coordinatemethods [30–38], generalized conforming method [2, 23],assumed strainmethod [17], could play important roles in theconstructions of the new shape-free finite elements, especiallyfor low-order high-performance models.

So, for establish systematic shape-free finite elementmethod, there is still a long way to go.

Acknowledgments

The authors would like to acknowledge the financial sup-port of the National Natural Science Foundation of China(11272181), the Specialized Research Fund for the DoctoralProgram of Higher Education of China (20120002110080),and the National Basic Research Program of China (Projectno. 2010CB832701).

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