DSpace at VNU: A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis

Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366

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Comput. Methods Appl. Mech. Engrg.
journal homepage: www.elsevier.com/locate/cma

A moving Kriging interpolation-based element-free Galerkin method
for structural dynamic analysis
Tinh Quoc Bui a,b,⇑, Minh Ngoc Nguyen c, Chuanzeng Zhang a
a

Chair of Structural Mechanics, Department of Civil Engineering, University of Siegen, Paul-Bontz-Strasse 9-11, D-57076, Siegen, Germany
Department of Computational Mechanics, Faculty of Mathematics and Computer Science, University of Natural Science-National University of Ho Chi Minh City, Viet Nam
c
Institute of Computational Engineering, Department of Civil Engineering, Ruhr University Bochum, Germany
b

a r t i c l e

i n f o

Article history:
Received 25 June 2010
Received in revised form 16 December 2010
Accepted 21 December 2010
Available online 25 December 2010
Keywords:
Dynamic analysis
Vibration
Meshfree method

Moving Kriging interpolation

a b s t r a c t
In this paper, a meshfree method based on the moving Kriging interpolation is further developed for free
and forced vibration analyses of two-dimensional solids. The shape function and its derivatives are essentially established through the moving Kriging interpolation technique. Following this technique, by possessing the Kronecker delta property the method evidently makes it in a simple form and efficient in
imposing the essential boundary conditions. The governing elastodynamic equations are transformed
into a standard weak formulation. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard implicit Newmark time integration scheme. Numerical examples
illustrating the applicability and effectiveness of the proposed method are presented and discussed in
details. As a consequence, it is found that the method is very efficient and accurate for dynamic analysis
compared with those of other conventional methods.
Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction
The analysis of structural dynamics problems is of great importance in the field of structural mechanics and computational
mechanics. Generally, the dynamic analysis needs more efforts in
modeling because of acting of many different conditions of complicated external loadings than the static one. To find an exact solution to the class of dynamic problems usually is a hard way and
in principle it could be reachable only with a simple loading condition and geometrical configuration. Due to many requirements of
engineering applications in reality, such a task of finding a solution
analytically is generally difficult and often impossible. Therefore,
numerical computational methods emerge as an alternative way
in finding an approximate solution. The finite element method
(FEM), e.g. see [1,2], formed into that issue and becomes the most
popular numerical tool for dealing with these problems. The necessity of such numerical computational methods is nowadays
unavoidable.
In the past two decades, the so-called meshfree or meshless
methods, e.g. see [3–5], have emerged alternatively, where a set
of scattered ‘‘nodes’’ in the domain is used instead of a set of ‘‘ele⇑ Corresponding author at: Chair of Structural Mechanics, Department of Civil
Engineering, University of Siegen, Paul-Bonatz Strasse 9-11, D-57076, Siegen,
Germany. Tel.: +49 2717402836; fax: +49 2717404074.
E-mail addresses: ,

(T.Q. Bui).
0045-7825/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.cma.2010.12.017

ments’’ or ‘‘mesh’’ as in the FEM. No meshing is generally required
in meshfree methods. Note that the meshing here means different
from the concept of background cells which are usually needed for
performing the domain integrations. There is another concept of
‘‘truly’’ meshfree or meshless methods, in which no meshing at
all including the background cells for the domain integrations is required, e.g. see [5–7]. In particular, the last author has developed
the meshless local Petrov-Galekin (MLPG) method for analysis of
static, dynamic and crack problems of nonhomogeneous, orthotropic, functionally graded materials as well as Reissner–Mindlin and
laminated plates [8–13]. Recently, Belytschko et al. [14] proposed
and promoted by Moes et al. [15] an effective method by substantially adding an enrichment function into the traditional finite element approximation function; the extended finite element method
(X-FEM), which aims at modeling of the discontinuity.
The present work belongs to the meshfree scheme, and a novel
meshfree method based on a combination of the classical elementfree Galerkin (EFG) method [3] and the moving Kriging (MK) interpolation is further developed for analysis of structural dynamics
problems. Previously, the present method has been developed by
the first author for static analysis [16] and recently [17] for free
vibration analysis of Kirchhoff plates. The MK interpolation-based
meshfree method was first introduced by Gu [18] and its application to solid and structural mechanics problems is still young
and more potential. Gu [18] successfully demonstrated its applicability for solving a simple problem of steady-state heat conduction.
Dai et al. [19] reported a comparison between the radial point


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T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366

interpolation method (RPIM) and the Kriging interpolations for

elasticity. Lam et al. [20] introduced an alternative approach,
a Local Kriging (LoKriging) method to two-dimensional solid
mechanics problems, where a local weak-form of the governing
partial differential equations was appplied. Li et al. [21] further
developed the LoKriging method for structural dynamics analysis.
Furthermore, Tongsuk et al. [22,23] and Sayakoummane et al.
[24] recently illustrated the applicability of the method to investigations of solid mechanics problems and shell structures,
respectively.
Imposing essential boundary conditions is a key issue in meshfree methods because of the lack of the Kronecker delta property
and, therefore, the imposition of prescribed values is not as
straightforward as in the FEM. Thus, many special techniques
have been proposed to avoid such difficulty by various ways e.g.
Lagrange multipliers [3], penalty method [4], or coupling with
the FEM [24–26], etc. Due to the possession of the Kronecker delta
property, the present method is hence capable of getting rid of
such drawback of enforcing the essential boundary conditions.
Note here that a majority of meshfree methods has been developed by displacement-based approaches and, in contrast, Duflot
et al. [27] and the first author have also implemented an equilibrium-based meshfree method for elastostatic problems where a
stress-based approach is taken into consideration, see [28,29].
With respect to the linear structural dynamics analysis in twodimension, a variety of studies has been reported so far. Gu et al.
[30] successfully used the meshless local Petrov–Galerkin (MLPG)
method for free and forced vibration analyses for solids, while in
a similar manner Hua Li et al. [21] developed the LoKriging, Dai
and Liu [31] proposed the smoothed finite element method (SFEM),
Gu and Liu [32] presented a meshfree weak-strong form (MWS)
approach, and recently, Liu et al. [33] and Nguyen-Thanh et al.
[34] implemented the edge-based smoothed finite element method (ES-FEM) and an alternative alpha finite element method (AaFEM), respectively. As mentioned above, the proposed method has
a significant advantage in the treatment of the boundary conditions, which is easier than the classical EFG. This present work
essentially makes use of that good feature to structural dynamics
analysis. At the standing point of view and to the best knowledge

of the authors, such a task has not yet been carried out while this
work is being reported.
The paper is organized as follows. The moving Kriging shape
function is introduced in the second section. The governing equations and their discretization of elastodynamic problems will then
be presented in Section 3. In Section 4, numerical examples for free
and forced vibration analyses are investigated and discussed in details. Finally, some conclusions from this study are given in
Section 5.
2. Moving Kriging shape function
Essentially, the MK interpolation technique is similar to the MLS
approximation. In order to approximate the distribution function
u(xi) within a sub-domain Xx # X, this function can be interpolated based on all nodal values xi(i 2 [1, nc]) within the sub-domain,
with n being the total number of the nodes in Xx. The MK interpolation uh(x), "x 2 Xx is frequently defined as follows [16,18,22,23].

uh ðxÞ ¼ ½pT ðxÞA þ rT ðxÞBŠuðxÞ

ð1Þ

or in a shorter form of
h

u ðxÞ ¼

n
X

/I ðxÞuI ¼ UðxÞu

ð2Þ

ÁÁÁ

uðxn Þ ŠT and /I(x) is the MK shape

m
X

pj ðxÞAjI þ

j

n
X

r k ðxÞBkI

ð3Þ

k

The matrixes A and B are determined by

A ¼ ðPT RÀ1 PÞÀ1 PT RÀ1

ð4Þ

À1

B ¼ R ðI À PAÞ

ð5Þ

where, I is an unit matrix and the vector p(x) is the polynomial with
m basis functions

pðxÞ ¼ f p1 ðxÞ p2 ðxÞ Á Á Á pm ðxÞ gT

ð6Þ

The matrix P has a size n  m and represents the collected values of
the polynomial basis functions (6) as

2

3
p1 ðx1 Þ p2 ðx1 Þ Á Á Á pm ðx1 Þ
6 .
..
..
.. 7
6 ..
.
.
. 7
7
P¼6
6
7
4 p1 ðx2 Þ p2 ðx2 Þ Á Á Á pm ðx2 Þ 5

ð7Þ

p1 ðxn Þ p2 ðxn Þ Á Á Á pm ðxn Þ
and r(x) in Eq. (1) is

rðxÞ ¼ f Rðx1 ; xÞ Rðx2 ; xÞ Á Á Á Rðxn ; xÞ gT

ð8Þ

where R(xi,xj) is the correlation function between any pair of the n
nodes xi and xj, and it is belong to the covariance of the field value
u(x): R(xi,xj) = cov[u(xi)u(xj)] and R(xi, x) = cov[u(xi)u(x)]. The correlation matrix R[R(xi, xj)]nÂn is explicitly given by

2

1

6 Rðx2 ; x1 Þ
6
R½Rðxi ; xj ފ ¼ 6
..
6
4
.

Rðx1 ; x2 Þ Á Á Á Rðx1 ; xn Þ
1
..
.

3

Á Á Á Rðx2 ; xn Þ 7
7
7
..
..
7
5
.
.

Rðxn ; x1 Þ Rðxn ; x2 Þ Á Á Á

ð9Þ

1

Many different correlation functions can be used for R but the
Gaussian function with a correlation parameter h is often and
widely used to best fit the model
Àhr2ij

Rðxi ; xj Þ ¼ e

ð10Þ

where rij = kxi À xjk, and h > 0 is a correlation parameter. As studied
in the previous work by the author [16], the correlation parameter
has a significant effect on the solution. In this work, the quadratic
bassis functions p T ðxÞ ¼ ½ 1 x y x2 y2 xy Š are used for all

numerical computations. Furthermore, the MK shape function in
one-dimension and its first-order derivatives used in the dynamic
analysis are presented in Fig. 1.
One of the most important features in meshfree methods is the
concept of the influence domain where an influence domain radius
is defined to determine the number of scattered nodes within an
interpolated domain of interest. In fact, no exact rules can be derived appropriately to all types of nodal distributions. The accuracy
of the method depends on the number of nodes inside the support
domain of the interest point. Therefore, the size of the support domain should be chosen by analysts somehow to ensure the convergence of the considered problems. It might also be found in the
same manner as in [4,5]. Often, the following formula is employed
to compute the size of the support domain.

dm ¼ adc

I

where u ¼ ½ uðx1 Þ uðx2 Þ
function and defined by

UðxÞ ¼ /I ðxÞ ¼

ð11Þ

where dc is a characteristic length regarding the nodal spacing close
to the point of interest, while a stands for a scaling factor. Other features related to the method can be found in [16,18,22,23] for more
details.


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with u0 and v0 being the initial displacements and velocities at the
initial time t0, respectively, and nj standing for the unit outward
normal to the boundary C = Cu [ Ct. By using the principle of virtual work, the variational formulation of the initial-boundary value
problems of Eq. (12) involving the inertial and damping forces can
be written as [1,2,31]

1

0.8

Z

φ(x)

0.6

deT rdX À

Z

X

€ À cuŠd
_ XÀ
duT ½b À qu

Z

du T tdC ¼ 0

In the meshfree method, the approximation (2) is utilized to calculate the displacements uh(x) for a typical point x. The discretized
form of Eq. (14) using the meshfree procedure based on the approximation (2) can be written as

0.4

0.2

€ þ Cu_ þ Ku ¼ f
Mu
0
0

0.2

0.4

0.6

0.8

1

x

b

8
6

4

φ’(x)

2

ð15Þ

where u is known as the vector of the general nodal displacements,
M,C,K and f stand for the matrixes of mass, damping and stiffness
and force vector, respectively. They are defined as follows

Z
MIJ ¼
UTI qUJ dX
X
Z
UTI cUJ dX
CIJ ¼
X
Z
BTI DBJ dX
KIJ ¼
Z X
Z
UTI bI dX þ
UTI tI dC
fI ¼

ð16Þ

ð17Þ
ð18Þ
ð19Þ

Ct

X

where c in Eq. (17) is the damping coefficient, U is the MK shape
function defined in Eq. (3), the elastic matrix D and the displacement gradient matrix B in Eq. (18) are given, respectively, by

0
−2

2
3
1 m
0
E 6
7
D¼
0
4m 1
5ðplane stressÞ
1 À m2
0 0 ð1 À mÞ=2

−4
−6
−8

ð14Þ

Ct

X

2
0

0.2

0.4

0.6

0.8

1

Fig. 1. The MK shape function (a) and its first-order derivative (b).

/I;x

6
BI ¼ 4 0
/I;y

0

ð20Þ

3

/I;y 7
5

ð21Þ

/I;x

3.2. Free vibration analysis
3. Meshfree elastodynamic formulation

For the free vibration analysis, the damping and the external
forces are not taken into account in the system. Then, Eq. (15)
can be reduced to a system of homogeneous equations as [1]

3.1. Discrete governing equations
Let us consider a deformable body occupying a planar linear
elastic domain X in a two-dimensional configuration bounded by
C subjected to the body force bi acting on the domain. The strong
form of the initial-boundary value problems for small displacement elastodynamics with damping can be written in the form

qu€i þ cu_ i ¼ rij;j þ bi in X

ð12Þ

where q stands for the mass density, c is the damping coefficient, üi
and u_ i are accelerations and velocities, and rij specifies the stress

tensor corresponding to the displacement field ui, respectively.
The corresponding boundary conditions are given as

i
ui ¼ u

on the essential boundary Cu
t i ¼ rij nj ¼ t i on the natural boundary Ct

ð13aÞ
ð13bÞ

€ þ Ku ¼ 0
Mu

ð22Þ

A general solution of such a homogeneous equation system can be
written as

 expðixtÞ
u¼u

ð23Þ

 is the eigenvector
where i is the imaginary unit, t indicates time, u
and x is natural frequency or eigenfrequency. Substitution of Eq.
(23) into Eq. (22) leads to the following eigenvalue equation for
the natural frequency x

¼0
ðK À x2 MÞu

ð24Þ

The natural frequencies and their corresponding mode shapes of a
structure are often referred to as the dynamic characteristics of
the structure.

 i and ti are the prescribed displacements and tractions,
with u
respectively, and the initial conditions are defined by

3.3. Forced vibration analysis

uðx; t 0 Þ ¼ u0 ðxÞ;
_
uðx;
t 0 Þ ¼ v 0 ðxÞ;

For the forced vibration analysis, the approximation function
Eq. (2) is a function of both space and time. For the displacements,

x2X

ð13cÞ

x2X

ð13dÞ


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velocities and accelerations at time t + Dt, the dynamic equilibrium
equations or equations of motion presented in Eq. (15) are also
considered at time t + Dt as follows

1000

There are many different methods available to solve the secondorder time dependent problems such as Houbolt, Wilson, Newmark,
Crank–Nicholson, etc. [2,31]. In this study, the Newmark time integration scheme is adopted to solve the equations of motion expressed in Eq. (25) at time step t + Dt. The Newmark scheme can
be given in the form [1,2]

€ tþDt ¼
u



1
1
1
€t
À1 u
ðutþDt À ut Þ À
u_ t À
bDt

b Dt
2b

€ tþDt ŠDt
u_ tþDt ¼ u_ t þ ½ð1 À cÞut þ cu

ð26bÞ

4. Numerical results
In order to demonstrate the efficiency and the applicability of
the present method to analysis of structural dynamics problems,
some typical numerical examples are considered for free and
forced vibrations and their dynamic responses are reported
correspondingly.
4.1. Free vibration analysis
4.1.1. Cantilever beam
A cantilever beam as shown in Fig. 2 is first considered as a
benchmark example. To do so, the non-dimensional parameters
in the computation have the length L = 48 and height D = 12. The
beam is assumed to have a unit thickness so that plane stress condition is valid. Young’s modulus E = 3.0 Â 107, Poisson’s ratio
m = 0.3, and mass density q = 1.0 [16] are used.
As confirmed by static analysis in the previous work [16], two
important parameters involving the correlation coefficient h and
the scaling factor a related to the interpolation function expressed
in Eqs. (10) and (11), respectively; have certain influences on the
numerical solutions. Thus, they are of importance to the present
method and may also have effects on the dynamic analysis in the
present work. This implies that the choice of these two parameters
must be in carefulness, and the choice might be different from the
static analysis. Fig. 3 shows the computed results of the natural

frequency of the beam compared to those of available reference

y

x

L=48
Fig. 2. The geometry of the cantilever beam.

800

600

400

ð26aÞ

By substituting both Eqs. (26a) and (26b) into Eq. (25) one can obtain the dynamic responses at time t + Dt. Since the Newmark time
integration scheme is an implicit method, the initial conditions of
€ 0 Þ are thus assumed to be known and
the state at t ¼ t 0 ðu0 ; u_ 0 ; u
€ 1 Þ is needed to be determined
the new state at t 1 ¼ t 0 þ Dtðu1 ; u_ 1 ; u
correspondingly. In addition, the choice of c = 0.5 and b = 0.25,
unconditionally guarantees the stability of the Newmark scheme
with c P 0.5 and b P 0.25(c + 0.5)2.

D=12

LoKriging [21]

FEM [21]
θ=0.004
θ=0.2
θ=1
θ=5
θ=30
θ=1000

ð25Þ
Natural frequency

€ tþDt þ Cu_ tþDt þ KutþDt ¼ f tþDt
Mu

1200

200

0

1

2

3

4

5
6

Mode No.

7

8

9

10

Fig. 3. Natural frequency versus the correlation parameter h for the cantilever beam
(a = 2.8).

solutions, where the correlation parameter is varied in an interval
of 0.004 6 h 6 1000 whilst a = 2.8 is fixed. A regular set of 189 scattered nodes is taken in this example and its distribution will be
seen later. A comparison of the obtained results of the present
method to that of the LoKriging [21] and the FEM (4850 DOFs)
[21] is given in Table 1 below. It is found that a good agreement
can be reached if 0.004 6 h < 5 is chosen, it fails with
0 6 h < 0.004, and other h values are though possible but the error
increases and a bad result is unavoidable. Additionally, the corresponding percentage error is then estimated and presented in
Fig. 4. Note that the FEM (4850 DOFs) derived from [21] is used
as a reference solution for the verification purpose. Similarly, the
influence of the scaling factor a to the selected quantity of scattered nodes within the influence domain is also studied in the
same manner. The results are plotted in Fig. 5. The correlation coefficient h = 0.2 is kept unchanged in the computation. Definitely, a
smaller error is obtained with a scaling factor 2.4 6 a 6 3.0.
Table 1 listing the first ten frequencies shows a comparison of
natural frequencies among LoKriging [21], FEM (4850 DOFs) [21]
and the present method, in which two scattered nodes of coarse
and fine node distributions with 55 and 189 are considered for

the cantilever beam associated with the chosen parameters of
h = 0.2 and a = 3.0. An excellent agreement with other solutions
can be found. Furthermore, the first twenty eigenmodes of the cantilever beam are also provided in Fig. 6. The applicability of the
method to irregularly scattered nodes is also given in Table 2,
which shows a very good result compared to that of the FEM [21].
To analyze the influences of the density of nodal distributions
and the convergence of the natural frequencies versus the nodal
densities, five regular nodal distributions with 5 Â 5; 11 Â 5;
15 Â 9; 21 Â 9 and 21 Â 16 are additionally applied to the beam
problem and four of them are illustrated in Fig. 7. The corresponding results of the non-dimensional frequencies are calculated individually for each set of scattered nodes and presented in Table 3 in
comparison with those obtained by LoKriging [21] and the FEM
[21]. It shows a very good convergence of the frequencies to the
reference solutions even with a coarse set of 55 nodes.
4.1.2. A shear wall with four openings
The next numerical example dealing with a shear wall with four
openings as shown in Fig. 8 is considered. This example has been
solved using several different computational methods such as
BEM [35], MLPG [30], SFEM [31], ES-FEM [33], AaFEM [34], etc.


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T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366

Table 1
Comparison of the natural frequencies for different node distributions for the cantilever beam.
Mode

FEM [21](4850 DOFa)

1
2
3
4
5
6
7
8
9
10

27.72
140.86
179.71
323.89
523.43
536.57
730.04
881.28
899.69
1000.22

a

55 Nodes

189 Nodes

MWS [32]

LoKriging [21]

Present method

MWS [32] (regular)

LoKriging [21]

Present method

26.7693
141.3830
179.7013
327.0243
527.3999
539.0598
730.1131
886.5635
896.9009
1004.7952

28.16
142.94
178.90
329.08
529.77
535.58
733.34
882.40
902.75

1001.55

27.952
143.943
179.874
334.562
537.394
548.201
776.301
884.231
929.177
1046.214

27.8370
141.1300
179.9077
323.8497
522.3307
537.1464
727.2628
881.5703
896.1059
997.7824

27.76
140.46
178.81
323.83
523.96
534.12

731.11
877.89
899.46
999.39

27.781
142.525
179.781
331.385
538.608
542.063
763.999
888.505
921.521
1028.855

DOF-Degree of freedom.

1st (27.7403Hz)

5
10
0
−10

0

Error (%)

10

0
−10

−10
θ=0.004
θ=0.08
θ=0.2
θ=1.0
θ=5.0
θ=10.0
θ=30.0
θ=1000

−15

−20

−25

1

2

3

4

5
6
Mode No.

7

8

9

10

α=2.4
α=2.8
α=3.0
α=3.2
α=4.0
α=6.0
α=10.0

8

0
50
7th (747.897Hz)

0
50
8th (887.0532Hz)
10
0
−10

0
50
0
50
0
50
10th (1010.478Hz) 11th (1085.3417Hz) 12th (1181.1089Hz)
10
0
−10

10
0
−10

0
50
0
50
0
50
0
50
13th (1251.4518Hz) 14th (1261.0004Hz) 15th (1315.9708Hz) 16th (1350.3932Hz)

14

10

4th (326.8239Hz)

10
0
−10

10
0
−10

10
0
−10

10
0
−10

10
0
−10

10
0
−10

10
0
−10

0
50

0
50
0
50
0
50
17th (1428.2654Hz) 18th (1452.5852Hz) 19th (1469.6634Hz) 20th (1519.1815Hz)
10
0
−10

12

10
0
−10

10
0
−10

10
0
−10

3rd (179.8042Hz)

0
50
6th (538.1866Hz)

0
50
9th (904.1602Hz)

Fig. 4. Influence of correlation parameter h on the natural frequency (a = 2.8).

Error (%)

10
0
−10

0
50
5th (532.2126Hz)

−5

−30

2nd (141.2899Hz)

10
0
−10
0

50

10
0
−10
0

50

10
0
−10
0

50

0

50

Fig. 6. The first twenty eigenmodes of the cantilever beam by the present method.

Table 2
A comparison of natural frequencies of the cantilever beam for both regular and
irregular node distributions (h = 0.2, a = 2.8).

6
Mode

4
2
0

−2

1

2

3

4

5
6
Mode No.

7

8

9

10

Fig. 5. Influence of scaling factor a on the natural frequency of the beam (h = 0.2).

The geometrical parameters of the shear wall can be found in Fig. 8
and other relevant material parameters are taken exactly the same
as in [30,31] with E = 10000, m = 0.2, t = 1.0 and q = 1.0 in a plane
stress state. The first eight natural frequencies are given in Table 4.

1

2
3
4
5
6
7
8
9
10

FEM [21]
(4850 DOFs)

27.72
140.86
179.71
323.89
523.43
536.57
730.04
881.28
899.69
1000.22

Present method
200 Nodes

325 Nodes

Irregular

Regular

Irregular

Regular

27.788
142.050
179.755
328.248
533.935
537.464
748.069
884.706
913.835
1018.345

27.752
139.152
178.403
322.164
515.272
535.144
728.552
881.552
922.056
995.822

27.711

138.977
178.957
319.114
530.419
529.225
729.687
879.388
907.737
1009.411

27.720
139.515
179.300
318.097
518.802
530.712
742.449
889.150
913.706
1080.830

Two sets of 261 and 556 scattered nodes are used, as well as h = 0.2
and a = 3.0 are specified in the computation. As a consequence, it is
found that the present solutions are in a good agreement with the


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T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366

a

20

6
4

1.8

18

2
0

3.0

16

−2
−4

14

1.8

−6
0

b

10

20

30

40

50

12

6

3.0

4

10
2

1.8

0

8

−2
−4

3.0

6

−6
0

10

20

30

40

50

1.8

4

c

6
4

21

2

3.0

2
0

0

−2

0

3.0

2

4

3.0

6

8

4.8

10

12

−4

Fig. 8. A shear wall with four openings with 556 scattered nodes.

−6
0

d

10

20

30

40

50

6
4
2
0
−2
−4
−6
0

10

20

30

40

50

Fig. 7. Various regular nodal distributions: 5 Â 5 (a), 11 Â 5 (b), 21 Â 9(c) and
21 Â 16 (d).

chosen. The beam is considered to be in plane stress condition with
parameters E = 3 Â 107, m = 0.3, mass density q = 1.0, and the thickness t = 1.0, respectively. A regular set of 189 scattered nodes is
used for all implementations of the forced vibration analysis. Three
main kinds of dynamic loadings depicted in Fig. 11 as harmonic
loading, Heaviside step loading, and transient loading with a finite
decreasing time are analyzed associated with a traction at the free
end of the beam by P = 1000 Â g(t), where g(t) is the time-dependent function. The implicit Newmark time integration scheme is
applied. The vertical displacement or deflection at point A as depicted in Fig. 10 is computed, and the detailed results obtained
by the present method are then compared either to those of the
ANSYS FEM software or other available solutions.

ones obtained by BEM, FEM and MLPG. Additionally, the first
twelve eigenmodes are also presented in Fig. 9 for the shear wall.

4.2.1. Harmonic loading
The loading in this case is shown in Fig. 10(a) with the loading
function g(t) given by

4.2. Forced vibration analysis

gðtÞ ¼ sin xf t

Regarding the analysis of the forced vibration, a benchmark
cantilever beam in two-dimensional setting shown in Fig. 10 is

in which xf is the forced frequency of the traction loading P, and
xf = 27 is used in the computation in this example. Fig. 12 shows

ð27Þ

Table 3
Convergence of the natural frequencies with various nodal densities of the cantilever beam.
Mode

5Â5

11 Â 5

15 Â 9

21 Â 9

21 Â 16

LoKriging [21]

FEM [21] (4850DOF)

1
2
3

4
5
6
7
8
9
10

31.384
149.236
162.053
314.614
326.115
365.320
551.751
1112.647
1209.457
1396.263

27.952
143.943
179.874
334.562
537.394
548.201
776.301
894.231
929.177
1046.214

27.648
142.474
179.653
331.419
538.284
545.519
765.665
889.478
925.343
1032.822

27.781
142.525
179.781
331.385
538.608
542.063
763.999
888.505
921.521
1028.855

27.725
141.770
179.433
324.355
524.267
537.306
738.306
884.180

899.173
1002.132

27.76
140.46
178.81
323.83
523.96
534.12
731.11
877.89
899.46
999.39

27.72
140.86
179.71
323.89
523.43
536.57
730.04
881.28
899.69
1000.22


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T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366

Table 4
Comparison of the first eight natural frequencies of a shear wall with four openings
among different methods.
Mode

MLPG[30]

FEM[30]

Brebbia et al.[35]

1
2
3
4
5
6
7
8

2.069
7.154
7.742
12.163
15.587
18.731
20.573
23.081

2.073

7.096
7.625
11.938
15.341
18.345
19.876
22.210

2.079
7.181
7.644
11.833
15.947
18.644
20.268
22.765

1st (2.1176rad/s)

Present method
261 Nodes

556 Nodes

2.180
7.385
7.631
13.008
16.088
18.475

20.059
22.438

2.117
7.199
7.616
12.317
15.633
18.014
19.841
22.241

2nd (7.1992rad/s)

3rd (7.6161rad/s)

20
10

10

10

0
−10

0

10

20

0
−10

4th (12.3174rad/s)
20

10

10
0

10

20

20

0

10

20

0

10

20

0
−10

0

10

20

0
−10

11th (25.3297rad/s)
20

10

10

10

0

10

20

0

10

20

0
−10

0

10

20

0

10

20

12th (26.3196rad/s)

20

−10

20

9th (23.2189rad/s)

20

0

10

10

0
−10

10th (23.553rad/s)

0

6th (18.0144rad/s)

8th (22.2409rad/s)

10

0
−10

0
−10

10

0
−10

7th (19.8407rad/s)
10

10

5th (15.6329rad/s)

20

0
−10

0

0

−10

0

10

20

Fig. 9. The first twelve eigenmodes for the shear wall with four openings by the
present method.

Fig. 10. A cantilever beam subjected to a tip uniform traction.

the computed results of the vertical displacement uy of point A by
different time-steps. Obviously, the results obtained including for
large time-steps are very stable compared with that of the FEM.
Additionally, Fig. 13 illustrates the influence of the correlation
parameter h on the displacement uy of point A, and it shows that
an acceptable result can be yielded even though the h value takes
up to 500. Various different time-steps are then applied for this

loading case and the results of the computed displacement uy at
point A are presented in Fig. 14. It can be found that all considered
time-steps could give stable results and they have a good agreement with the one obtained by the FEM except for Dt = 5 Â 10À2s.
This implies that the accuracy of the present method will be decreased if the time-step is taken too large, which is known as the
numerical damping effect.
Similar to the free vibration analysis, the effect of the densities
of the nodal distributions on the dynamic response is also investigated here numerically and presented in Fig. 15 for five different
nodal distributions, which are the same as used in the free vibration analysis of the cantilever beam. Compared with the FEM solution, it shows that a large error may occur when a very coarse set of
5 Â 5 nodes is taken whereas all other nodal densities can yield
good agreements even with a coarse set of 11 Â 5 nodes.
Many time-steps are further studied to check the stability of the
method involving damping effect by comparing the displacement
uy obtained at point A versus the forced frequency of the dynamic
loading xf. The numerical results till to 20s are plotted in Fig. 16. In
this figure, the computed responses with xf = 18 on the left and
with xf = 27 on the right are given. A damping coefficient c = 0.4
is selected for the two cases and the time-step is taken as
Dt = 5 Â 10À3s. In fact, this example has also been investigated by
Li et al. in [21], and the same conclusion is found here by comparing Fig. 16(a) and (b). That is, the amplitude in the case of xf = 18 is
smaller than about six times of that with xf = 27, because xf = 27
is close to the first natural frequency of the beam and a resonance
is hence occurred. The present results are very stable compared

with the results in [21,30]. Without damping, i.e. c = 0, the present
method can also give stable dynamic response with many timesteps as shown in Fig. 17.
When using a time integration technique to elastodynamic
analysis, dissipation error representing the amplitude decay is
known as a critical issue in measuring of the accuracy of the method. As concluded in [1,37,38], the standard Newmark method with
c = 1/2 leads to no numerical dissipation, whereas with values of
c > 1/2 it gives rise to numerical dissipation. In the present study,
the valuec = 1/2 in conjunction with b = 1/4 is chosen to eliminate
the numerical dissipation for the case of constant average acceleration. By using these two values i.e.c = 1/2, b = 1/4, the method is
always stable. To verify the increase of the numerical dissipation
for values of c > 1/2, the harmonic loading condition of the cantilever beam for several specific values of c > 1/2 such as 0.5; 0.8; 1.5;
2.0 and 10.0 are considered, respectively. The corresponding results are given in Fig. 18 for a typical peak in comparison with
the one obtained by the FEM (ANSYS). It is evident that the amplitudes decay gradually when the c parameter is increasing, especially a very large error is observed with c = 10.0.
The dispersion error is related to the nodal distributions (or
mesh density in FEM) and the time-step used in the time integration technique. Generally speaking, a straightforward way of
reducing the dispersion error is to use a finer mesh in FEM and
smaller time-step size [38]. This issue is investigated numerically
and the corresponding results are presented in Figs. 14 and 15
for various time-step sizes and for various nodal distributions,
respectively. Consequently, it can be concluded that a sufficiently
small time-step and a sufficiently small nodal density can yield
very good agreement between the results obtained by the present
method and the FEM, whereas a large dispersion error can be found
for a large time-step as well as a coarse nodal distribution. It is
worth noting that such a refinement in time-step or nodal distribution always results in more computational effort and thus a more
efficient technique for reducing the dispersion errors is desirable.
More details on dissipation and dispersion errors arising in
structural dynamics by using numerical time-integration techniques can be found in [1,37–44].

1361

T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366

Fig. 11. Schematic diagram of dynamic loadings: (a) harmonic loading, (b) Heaviside step loading and (c) transient loading with a finite decreasing time.

0.2

0.025
−3

FEM(ANSYS) (Δt=1x10 )

0.15

Δt=1x10−3

0.02

Δt=1x10−4

0.1

−2

Δt=1x10

Displacement uy

Displacement u

y

0.015
0.01
0.005
0

0.05
0
−0.05
−3

FEM(ANSYS) (Δt=1x10 )

−0.1

−3

Δt=1x10

−0.005

Δt=1x10−4

−0.15

−2

Δt=1x10

−0.01

Δt=5x10−2

−0.2
−0.015

0

0.05

0.1
Time t

0.15

0

0.2

Fig. 12. Displacement uy at point A using Newmark time integration scheme
(c = 0.5, b = 0.25) and the scaling and correlation parameters a = 2.8 and h = 0.2 for
time-harmonic loading.

1.5

2

0.2

FEM(ANSYS)
θ = 0.1
θ=1
θ=5
θ = 10
θ = 50
θ = 100
θ = 500
θ = 1000

0.015
0.01
0.005

0.15
0.1
Displacement uy

0.02

y

1
Time t

Fig. 14. Displacement uy at point A with various time-steps using Newmark
method (c = 0.5, b = 0.25) and a = 2.8 and h = 0.2 for time-harmonic loading.

0.025

Displacement u

0.5

0

0.05
0
−0.05

−0.005

−0.1

−0.01

−0.15

−0.015

FEM (ANSYS)
5x5
11x5
15x9
21x9
21x16

−0.2
0

0.05

0.1
Time t

0.15

0.2

Fig. 13. Influence of the correlation parameter h on the displacement uy at point A
using Newmark method (c = 0.5, b = 0.25) and a = 2.8 for time-harmonic loading.

4.2.2. Heaviside step loadings
In this section, three different types of dynamic Heaviside step
loadings are examined. The scaling factor and the correlation
parameter are taken as a = 2.8 and h = 0.2 in all the computations.

0

0.5

1
Time t

1.5

2

Fig. 15. Displacement uy at point A with various nodal densities using Newmark
method (c = 0.5, b = 0.25), Dt = 1 Â 10À3 and a = 2.8 and h = 0.2 for time-harmonic

loading.

4.2.2.1. Heaviside step loading with an infinite duration. When the
loading function is specified as

gðtÞ ¼ HðtÞ

ð28Þ


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T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366

0.025

FEM (ANSYS)
γ =0.5
γ =0.8
γ =1.0
γ =1.5
γ =2.0
γ =10.0

0.1

−3

Δt=5x10 , ω = 18

0.02

0.08

0.015

Displacement u

Displacement u

y

y

0.01
0.005
0
−0.005

0.06

0.04

0.02
−0.01
−0.015

0

−0.02

−0.02
−0.025

b

0

5

10
Time t

15

20

0.85

0.9

0.95

1

Time t
Fig. 18. Dissipation verification at point A without damping (c = 0) for c = 0.5; 0.8;
1.0; 1.5; 2.0 and 10.0 (b = 0.25) using Newmark method for time-harmonic loading

with D t = 1 Â 10À3.

0.25
Δt=5x10−3, ω = 27

0.2

0.8

0.15

0

0.05

−0.002

0

−0.004

−0.05

−0.006
Displacement u

y

Displacement u

y

0.1

−0.1
−0.15
−0.2
−0.25

−0.01
−0.012

0

5

10
Time t

15

20

−0.014

Fig. 16. Response at point A for different loading frequency using Newmark method
(c = 0.5,b = 0.25), Dt = 5 Â 10À3 and c = 0.4 for time-harmonic loading.x = 18 (a)
and x = 27(b).

−3

Δt=4x10 , c=0

−0.016
−0.018

0

0.5

1
Time t

1.5

2

Fig. 19. Response at point A under Heaviside step loading with an infinite duration
and without damping.

0.3

0.2

Displacement uy

−0.008

0.1

0

−0.1

−0.2

−0.3
0

5

10
Time t

15

20

Fig. 17. Response at point A without damping (c = 0) using Newmark method
(c = 0.5, b = 0.25) and Dt = 5 Â 10À3 for time-harmonic loading.

as depicted in Fig. 11(b), the dynamic loading is often referred to as
impact loading and this type of dynamic analysis is often defined as
dynamic relaxation [21,32]. The dynamic relaxation implies that the

loading will keep unchanged once a constant loading is suddenly
applied to the structure. The static analytical solution at point A is
uy,exact = À0.0089 [36]. The results with and without damping specified by c = 0 and c = 0.4, respectively, are plotted in Figs. 19 and 20.
It is evidently found that the response will become a steady state
harmonic vibration with the static deformation of the beam as

the equilibrium position if the damping is neglected, and the response converges to the static deformation once a damping is introduced. Both obtained results are obviously very stable and have an
excellent agreement with those computed by different methods
such as MWS [32] and LoKriging [21].
In addition, the results for a damping with c = 0.4 are listed in
Table 5 for several time-steps at t % 50s. Fig. 20 conforms that
the response converges to the static solution uy = À0.00888811.
The computed percentage errors compared to the exact solution
for both the LoKriging [21] and the present methods are also estimated in Fig. 21. This result implies that the present method gives
a remarkable convergence with a smaller error about 0.1% compared to that of about 0.6% obtained by the LoKriging [21]. As a
consequence, it is hence demonstrated that the present method
works very well and accurate for the forced vibration analysis.


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T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366
0

0.9
Δt=4x10 ,c=0.4

−0.002

0.8

−0.004

0.7

−0.006

0.6

y

−0.008

Error (%)

Displacement u

Present
LoKriging

−3

−0.01
−0.012

u = −0.008888112425049
y

0.5
0.4
0.3

−0.014

0.2
−0.016

0.1
−0.018

0

5

10

15

20
Time t

25

30

35

40

Fig. 20. Response at point A under Heaviside step loading with an infinite duration
and with damping.

0
47

47.5

Time (s)

Displacement uy
LoKriging [21]

Present method

11750
11875
12000
12125
12250
12375
12500

0.470000E+02
0.475000E+02
0.480000E+02
0.485000E+02
0.490000E+02
0.495000E+02
0.500000E+02

À0.00883283
À0.00883255
À0.00883264
À0.00882592
À0.00883220
À0.00884123

À0.00884174

À0.008888400762486
À0.008889026243629
À0.008890959168876
À0.008885829057817
À0.008888068307071
À0.008889391269139
À0.008888348924774

48.5
Time t

49

49.5

50

Fig. 21. A comparison of the percentage errors between the LoKriging [21] and the
present methods with damping.

Table 5
Computed results at several time-steps (about t % 50s) under dynamic Heaviside step
loading with an infinite duration.
Number of
time-steps

48

g(t)

1.0

0

t=0.5s

t(s)

Fig. 22. The dynamic Heaviside step loading with a finite duration.
Exact: uy = À0.0089 [36].

4.2.2.2. Heaviside step loading with a finite duration. The dynamic
Heaviside rectangular step loading with a finite duration has a similar form as the dynamic impact loading considered previously but
suddenly vanished at time t = 0.5 s as depicted in Fig. 22. This can
be considered as a special case of the dynamic relaxation, and the
loading function is given by

The results computed for both c = 0 and c = 0.4 are compared to that
of the FEM (ANSYS) and given in Figs. 23 and 24, respectively. Several different time-steps are chosen for c = 0, and the results show
again that the accuracy of the present method decreases for large
time-steps, while in other cases the results fit well with the FEM’s
solution. Alternatively, these results with and without damping
can be compared with the results of Gu et al. [30] and a good agreement could be found. It is worth noting that after 0.5 s the response
oscillates around zero as an equilibrium position because of the
vanishing loading on the system.
4.2.2.3. Heaviside step loading with a finite rise time. Furthermore,
we consider the dynamic Heaviside step loading with a finite rise
time as depicted in Fig. 25. In this case, the loading function is defined by



t
t
gðtÞ ¼ HðtÞ À
À 1 Hðt À t 0 Þ
t0
t0

0.005
y

ð29Þ

0.01

Displacement u

gðtÞ ¼ HðtÞ À Hðt À 0:5Þ

0.015

0

−0.005

−0.01
−3

FEM (ANSYS)(Δt=5x10 )
Δt=1x10−4

−0.015

Δt=1x10−3
Δt=5x10−2

−0.02

0

0.5

1
Time t

1.5

2

Fig. 23. Transient displacement uy at point A without damping under Heaviside
step loading with a finite duration.

and 27 that the displacement response always converges to the static deformation given by the static analytical solution.

ð30Þ

The corresponding numerical results are presented in Figs. 26 and
27, respectively. As can be seen from the results shown in Figs. 20

4.2.3. Transient loading with a finite decreasing time
For the transient loading with a finite decreasing time as depicted in Fig. 11c, the loading function is determined by


1364

T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366
0.015

0
−0.001

−3

Δt=5x10 ,c=0.4

0.01

−3

Δt=1x10 ,c=0.4

−0.002
g(t)
0.005
y

Displacement u

Displacement u

y

−0.003

0

−0.005

1.0

−0.004
−0.005
−0.006

0

t

t(s)

0

−0.007

−0.01

−0.008
−0.015

−0.009
−0.02

0

5

10
Time t

15

−0.01

20

Fig. 24. Transient displacement uy at point A with damping for many time-steps
under Heaviside step loading with a finite duration.

0

5

10
Time t

15

20

Fig. 27. Response at point A of the beam under Heaviside step loading with a finite
rise time and with damping.

0.01

g(t)

0.005
1.0

0

t0

Displacement u

y

0

t(s)

Fig. 25. The dynamic Heaviside step loading with a finite rise time.

−0.01

0

−0.015
Δt=4x10−3,c=0

−3

Δt=1x10 ,c=0

−0.001
−0.002

−0.02

g(t)

−0.003

1.0

0

0.5

1

1.5

2
Time t

2.5

3

3.5

4

Fig. 28. Transient displacement uy at point A without damping under transient
loading with a finite decrease time. This result can be compared to that of the MSW
method in [32].

y

Displacement u

−0.005

−0.004
−0.005
0

t

t(s)

0

−0.006
0.01

−0.007
−0.008

0.005

−0.009
1

2

3

4

5

Time t
Fig. 26. Response at point A of the beam under Heaviside step loading with a finite
rise time and without damping.

y

0

0

Displacement u

−0.01

−0.005

−0.01

gðtÞ ¼ ð1 À tÞ½HðtÞ À Hðt À 1ފ

ð31Þ
−0.015

The corresponding results with and without damping are provided
in Figs. 28–30 by using different time-steps, respectively. The different time-steps are selected in such a way that the corresponding
numerical results can be compared with others. Because the input
values of the problem are set up the same as that of [21,32], thus
the responses obtained in Figs. 28–30 can be directly compared
with the results presented in [21,32]. Here again, a very good agree-

−3

Δt=5x10 ,c=0
−0.02

0

5

10

15

Time t

Fig. 29. Transient response uy at point A without damping under transient loading

with finite decreasing time. This result can be compared with that of the LoKrging
method in [21].


T.Q. Bui et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1354–1366

References

0.01

0.005

Displacement u

y

0

−0.005

−0.01

−0.015

−0.02

1365

−3

Δt=5x10 ,c=0.5

0

5

10

15

Time t

Fig. 30. Transient displacement uy at point A with damping under transient loading
with a finite decreasing time. This result can be compared to that of both the MSW
[32] and LoKriging [21] methods.

ment can be found. Furthermore, it can be observed in Figs. 28 and
29 that the amplitude decreases as the time increases from 0 to
1.0 s. The response oscillates in a steady state way after 1.0 s because damping affect is ignored in the system. On the other hand,
a damping in the system results an amplitude decrease to zero as
the time increases, as illustrated in Fig. 30. Very stable results for
the forced vibration analysis are obviously achieved by the present
meshfree method.
5. Conclusions
Free and forced vibration analyses of two-dimensional solid
mechanics problems are presented in this paper. Numerical examples are investigated in details for many different dynamic loading
cases using the Newmark time integration scheme. It demonstrates
a successful application of the present moving Kriging interpolation-based element-free Galerkin method for structural dynamics
problems. By making use of the good feature of the possession of
the Kronecker delta property, the method is effective in enforcing

the essential boundary conditions, and the procedure is straightforward as in the FEM.
As a result, the combination between the novel MK interpolation technique and the standard EFG method is an attractive method. Numerical results presented in this paper show an excellent
agreement with other reference solutions. The present moving
Kriging meshfree method is efficient, accurate and stable for solving the structural dynamics problems. Therefore, the proposed
meshfree method has a good potential and is promising to be extended to non-linear problems, crack problems and so forth, especially for numerical modeling of multifield coupled problems in
smart materials such as piezoelectric, magnetoelectroelastic media, etc. Furthermore, studies on the effects of the size of the domain
of influence on the numerical solutions of higher modes in free
vibration analysis by using an adaptive procedure as proposed in
[45] are useful in the future research. Also, a detailed investigation
of dispersion and dissipation errors for various time-integration
schemes in the proposed method for elastodynamic problems
would be interesting.
Acknowledgements
The support of the German Research Foundation (DFG) under
the Project No. ZH15/11-1 is gratefully acknowledged.

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