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

Contents lists available at ScienceDirect

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

(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

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


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


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

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

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


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


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


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

ð13dÞ


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

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

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]

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

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]


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

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.

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)

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)

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
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

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


10

20

30

40

50

12

6

3.0

4

10
2

1.8

0

8

−2
−4


3.0

2
0

0

−2

0

3.0

2

4

3.0

6

8

4.8

10

12

−4

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

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


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


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

0
−10

4th (12.3174rad/s)
20

10

10
0

10

20

20

0

10

20

0

10

20

10

20

0
−10

0

10

20

0

10

20

12th (26.3196rad/s)

20

−10

20

9th (23.2189rad/s)

20


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.


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].




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

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


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

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

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



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


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


0.1
−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

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





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



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


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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


−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



−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


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


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


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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