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

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

A moving Kriging interpolation-based element-free Galerkin method

for structural dynamic analysis

Tinh Quoc Buia,b,⇑, Minh Ngoc Nguyenc, Chuanzeng Zhanga

a

Chair of Structural Mechanics, Department of Civil Engineering, University of Siegen, Paul-Bontz-Strasse 9-11, D-57076, Siegen, Germany

b

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

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 essen-tially established through the moving Kriging interpolation technique Following this technique, by pos-sessing 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 equa-tions, 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

 2010 Elsevier B.V All rights reserved

1 Introduction

The analysis of structural dynamics problems is of great

impor-tance 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

compli-cated external loadings than the static one To find an exact

solu-tion to the class of dynamic problems usually is a hard way and

in principle it could be reachable only with a simple loading

condi-tion and geometrical configuracondi-tion 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

neces-sity 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-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 re-quired, 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, orthotro-pic, 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 substan-tially adding an enrichment function into the traditional finite ele-ment approximation function; the extended finite eleele-ment 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 element-free Galerkin (EFG) method[3]and the moving Kriging (MK) inter-polation 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 applica-tion to solid and structural mechanics problems is still young and more potential Gu[18]successfully demonstrated its applica-bility for solving a simple problem of steady-state heat conduction Dai et al.[19] reported a comparison between the radial point 0045-7825/$ - see front matter  2010 Elsevier B.V All rights reserved.

⇑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: bui-quoc@bauwesen.uni-siegen.de , tinh.buiquoc@gmail.com

(T.Q Bui).

Contents lists available atScienceDirect

Comput Methods Appl Mech Engrg.

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c m a

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

investi-gations of solid mechanics problems and shell structures,

respectively

Imposing essential boundary conditions is a key issue in

mesh-free 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

devel-oped by displacement-based approaches and, in contrast, Duflot

et al.[27]and the first author have also implemented an

equilib-rium-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

two-dimension, 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

meth-od (ES-FEM) and an alternative alpha finite element methmeth-od (A

a-FEM), respectively As mentioned above, the proposed method has

a significant advantage in the treatment of the boundary

condi-tions, 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

equa-tions and their discretization of elastodynamic problems will then

be presented in Section3 In Section4, numerical examples for free

and forced vibration analyses are investigated and discussed in

de-tails Finally, some conclusions from this study are given in

Section5

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-domainXx# X, this function can be

interpo-lated based on all nodal values xi(i 2 [1, nc]) within the sub-domain,

with n being the total number of the nodes inXx The MK

interpo-lation uh(x), "x 2Xxis frequently defined as follows[16,18,22,23]

uh

or in a shorter form of

uh

ðxÞ ¼Xn

I

where u ¼ ½ uðx1Þ uðx2Þ    uðxnÞ T and /I(x) is the MK shape

function and defined by

UðxÞ ¼ /IðxÞ ¼Xm

j

pjðxÞAjIþXn

k

The matrixes A and B are determined by

A ¼ ðPTR1PÞ1PTR1

ð4Þ

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

m basis functions pðxÞ ¼ f p1ðxÞ p2ðxÞ    pmðxÞ gT ð6Þ The matrix P has a size n  m and represents the collected values of the polynomial basis functions(6)as

P ¼

p1ðx1Þ p2ðx1Þ    pmðx1Þ

p1ðx2Þ p2ðx2Þ    pmðx2Þ

p1ðxnÞ p2ðxnÞ    pmðxnÞ

2 6 6 4

3 7 7

and r(x) in Eq.(1)is rðxÞ ¼ f Rðx1;xÞ Rðx2;xÞ    Rðxn;xÞ gT ð8Þ where R(xi,xj) is the correlation function between any pair of the n nodes xiand 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 corre-lation matrix R[R(xi, xj)]nnis explicitly given by

R½Rðxi;xjÞ ¼

1 Rðx1;x2Þ    Rðx1;xnÞ Rðx2;x1Þ 1    Rðx2;xnÞ

Rðxn;x1Þ Rðxn;x2Þ    1

2 6 6 4

3 7 7

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

where rij= kxi 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 pTðxÞ ¼ ½ 1 x y x2 y2 xy  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 inFig 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 de-rived 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 do-main should be chosen by analysts somehow to ensure the conver-gence 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

where dcis a characteristic length regarding the nodal spacing close

to the point of interest, whileastands for a scaling factor Other fea-tures related to the method can be found in[16,18,22,23]for more details

3 Meshfree elastodynamic formulation

3.1 Discrete governing equations

Let us consider a deformable body occupying a planar linear

elastic domainXin a two-dimensional configuration bounded by

Csubjected to the body force biacting on the domain The strong

form of the initial-boundary value problems for small

displace-ment elastodynamics with damping can be written in the form

whereqstands for the mass density, c is the damping coefficient, üi

and _uiare accelerations and velocities, andrijspecifies the stress

tensor corresponding to the displacement field ui, respectively

The corresponding boundary conditions are given as

ui¼ ui on the essential boundaryCu ð13aÞ

ti¼rijnj¼ ti on the natural boundaryCt ð13bÞ

with ui and ti are the prescribed displacements and tractions,

respectively, and the initial conditions are defined by

with u0and v0being the initial displacements and velocities at the initial time t0, respectively, and njstanding for the unit outward normal to the boundaryC=Cu [Ct By using the principle of vir-tual 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]

Z X

deTrdX

Z X

duT½b qu  c _ud€ X

Z

C t

duTtdC¼ 0 ð14Þ

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

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

MIJ¼ Z X

CIJ¼ Z X

KIJ¼ Z X

BT

fI¼ Z X

UTIbIdXþ

Z

C t

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

1 m2

0 0 ð1 mÞ=2

2 6

3 7

BI¼

/I;x 0

0 /I;y /I;y /I;x

2 6

3

3.2 Free vibration analysis 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]

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

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

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

3.3 Forced vibration analysis For the forced vibration analysis, the approximation function

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

0

0.2

0.4

0.6

0.8

1

x

−8

−6

−4

−2

0

2

4

6

8

b

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

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

M€utþDtþ C _utþDtþ KutþDt¼ ftþDt ð25Þ

There are many different methods available to solve the

second-order time dependent problems such as Houbolt, Wilson, Newmark,

Crank–Nicholson, etc.[2,31] In this study, the Newmark time

inte-gration scheme is adopted to solve the equations of motion

ex-pressed in Eq.(25)at time step t +Dt The Newmark scheme can

be given in the form[1,2]

€

utþDt¼ 1

bDtðutþDt utÞ 

1

bDt_ut

1 2b 1

€

_

utþDt¼ _utþ ½ð1 cÞutþcu€tþDtDt ð26bÞ

By substituting both Eqs.(26a) and (26b)into Eq.(25)one can

ob-tain the dynamic responses at time t +Dt Since the Newmark time

integration scheme is an implicit method, the initial conditions of

the state at t ¼ t0ðu0; _u0; €u0Þ are thus assumed to be known and

the new state at t1¼ t0þDtðu1; _u1; €u1Þ is needed to be determined

correspondingly In addition, the choice of c= 0.5 and b = 0.25,

unconditionally guarantees the stability of the Newmark scheme

withcP0.5 and b P 0.25(c+ 0.5)2

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 inFig 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

con-dition is valid Young’s modulus E = 3.0  107, Poisson’s ratio

m= 0.3, and mass densityq= 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 factorarelated 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 3shows the computed results of the natural

frequency of the beam compared to those of available reference

solutions, where the correlation parameter is varied in an interval

of 0.004 6 h 6 1000 whilsta= 2.8 is fixed A regular set of 189 scat-tered 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 inTable 1below 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 corre-sponding 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 factorato the selected quantity of scat-tered nodes within the influence domain is also studied in the same manner The results are plotted inFig 5 The correlation coef-ficient h = 0.2 is kept unchanged in the computation Definitely, a smaller error is obtained with a scaling factor 2.4 6a63.0 Table 1listing 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 can-tilever 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  5; 11  5;

15  9; 21  9 and 21  16 are additionally applied to the beam problem and four of them are illustrated inFig 7 The correspond-ing results of the non-dimensional frequencies are calculated indi-vidually for each set of scattered nodes and presented inTable 3in 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 inFig 8is 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

D=12

L=48

y

x

Fig 2 The geometry of the cantilever beam.

0 200 400 600 800 1000 1200

Mode No.

LoKriging [21]

FEM [21]

θ=0.004 θ=0.2 θ=1 θ=5 θ=30 θ=1000

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

The geometrical parameters of the shear wall can be found inFig 8

and other relevant material parameters are taken exactly the same

as in[30,31]with E = 10000,m= 0.2, t = 1.0 andq= 1.0 in a plane

stress state The first eight natural frequencies are given inTable 4

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

Table 1

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

Mode FEM [21] (4850 DOF a

MWS [32] LoKriging [21] Present method MWS [32] (regular) LoKriging [21] Present method

a

DOF-Degree of freedom.

−30

−25

−20

−15

−10

−5

0

5

Mode No.

Error (%) θ=0.004

θ=0.08

θ=0.2

θ=1.0

θ=5.0

θ=10.0

θ=30.0

θ=1000

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

−2

0

2

4

6

8

10

12

14

Mode No

α=2.4

α=2.8

α=3.0

α=3.2

α=4.0

α=6.0

α=10.0

Fig 5 Influence of scaling factoraon the natural frequency of the beam (h = 0.2).

−10 0 10 1st (27.7403Hz)

−10 0 10 2nd (141.2899Hz)

−10 0 10 3rd (179.8042Hz)

−10 0 10 4th (326.8239Hz)

−10 0 10 5th (532.2126Hz)

−10 0 10 6th (538.1866Hz)

−10 0 10 7th (747.897Hz)

−10 0 10 8th (887.0532Hz)

−10 0 10 9th (904.1602Hz)

−10 0 10 10th (1010.478Hz)

−10 0 10 11th (1085.3417Hz)

−10 0 10 12th (1181.1089Hz)

−10 0 10 13th (1251.4518Hz)

−10 0 10 14th (1261.0004Hz)

−10 0 10 15th (1315.9708Hz)

−10 0 10 16th (1350.3932Hz)

−10 0 10 17th (1428.2654Hz)

−10 0 10 18th (1452.5852Hz)

−10 0 10 19th (1469.6634Hz)

−10 0 10 20th (1519.1815Hz)

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).

Mode FEM [21]

(4850 DOFs)

Present method

Irregular Regular Irregular Regular

10 1000.22 1018.345 995.822 1009.411 1080.830

ones obtained by BEM, FEM and MLPG Additionally, the first

twelve eigenmodes are also presented inFig 9for the shear wall

4.2 Forced vibration analysis

Regarding the analysis of the forced vibration, a benchmark

cantilever beam in two-dimensional setting shown inFig 10 is

chosen The beam is considered to be in plane stress condition with parameters E = 3  107,m= 0.3, mass densityq= 1.0, and the thick-ness 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 inFig 11as 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  g(t), where g(t) is the time-depen-dent function The implicit Newmark time integration scheme is applied The vertical displacement or deflection at point A as de-picted inFig 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

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

in whichxfis the forced frequency of the traction loading P, and

xf= 27 is used in the computation in this example.Fig 12shows

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

a

b

c

d

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

21  16 (d).

Table 3

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

0 2 4 6 8 10 12 14 16 18 20

21

3.0 3.0 3.0 3.0

1.8 1.8 1.8 1.8

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

the computed results of the vertical displacement uyof 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 uyof 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 uyat point A are presented inFig 14 It can be found that all considered time-steps could give stable results and they have a good agree-ment with the one obtained by the FEM except forDt = 5  102s This implies that the accuracy of the present method will be de-creased 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 investi-gated here numerically and presented inFig 15for five different nodal distributions, which are the same as used in the free vibra-tion analysis of the cantilever beam Compared with the FEM solu-tion, it shows that a large error may occur when a very coarse set of

5  5 nodes is taken whereas all other nodal densities can yield good agreements even with a coarse set of 11  5 nodes

Many time-steps are further studied to check the stability of the method involving damping effect by comparing the displacement

uyobtained at point A versus the forced frequency of the dynamic loadingxf The numerical results till to 20s are plotted inFig 16 In this figure, the computed responses withxf= 18 on the left and withxf= 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  103s In fact, this example has also been investigated by

Li et al in[21], and the same conclusion is found here by compar-ingFig 16(a) and (b) That is, the amplitude in the case ofxf= 18 is smaller than about six times of that withxf= 27, becausexf= 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 time-steps as shown inFig 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

meth-od 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 acceler-ation 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 ofc> 1/2, the harmonic loading condition of the cantile-ver beam for secantile-veral specific values ofc > 1/2 such as 0.5; 0.8; 1.5; 2.0 and 10.0 are considered, respectively The corresponding re-sults are given inFig 18for a typical peak in comparison with the one obtained by the FEM (ANSYS) It is evident that the ampli-tudes decay gradually when thecparameter is increasing, espe-cially a very large error is observed withc= 10.0

The dispersion error is related to the nodal distributions (or mesh density in FEM) and the time-step used in the time integra-tion 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 inFigs 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 distribu-tion always results in more computadistribu-tional 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 tech-niques can be found in[1,37–44]

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] Present method

261 Nodes 556 Nodes

0

10

1st (2.1176rad/s)

0 10 20 2nd (7.1992rad/s)

0 10 3rd (7.6161rad/s)

0

10

20

4th (12.3174rad/s)

0 10 20 5th (15.6329rad/s)

0 10 6th (18.0144rad/s)

0

10

7th (19.8407rad/s)

0 10 8th (22.2409rad/s)

0 10 9th (23.2189rad/s)

0

10

20

10th (23.553rad/s)

0 10 20 11th (25.3297rad/s)

0 10 20 12th (26.3196rad/s)

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.

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 asa= 2.8 and h = 0.2 in all the computations

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

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

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time t

FEM(ANSYS) (Δt=1x10 −3 )

Δt=1x10 −3

Δt=1x10 −4

Δt=1x10 −2

Fig 12 Displacement u y at point A using Newmark time integration scheme

(c= 0.5, b = 0.25) and the scaling and correlation parametersa= 2.8 and h = 0.2 for

time-harmonic loading.

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time t

FEM(ANSYS)

θ = 0.1

θ = 1

θ = 5

θ = 10

θ = 50

θ = 100

θ = 500

θ = 1000

Fig 13 Influence of the correlation parameter h on the displacement u y at point A

using Newmark method (c = 0.5, b = 0.25) anda= 2.8 for time-harmonic loading.

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

Time t

FEM(ANSYS) ( Δt=1x10 −3 ) Δt=1x10 −3

Δt=1x10 −4

Δt=1x10 −2

Δt=5x10 −2

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

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

Time t

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

Fig 15 Displacement u y at point A with various nodal densities using Newmark method (c= 0.5, b = 0.25), Dt = 1  10 3

anda= 2.8 and h = 0.2 for time-harmonic loading.

as depicted inFig 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= 0.0089[36] The results with and without damping spec-ified by c = 0 and c = 0.4, respectively, are plotted inFigs 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 re-sponse converges to the static deformation once a damping is intro-duced 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 5for several time-steps at t  50s Fig 20 conforms that the response converges to the static solution uy= 0.00888811 The computed percentage errors compared to the exact solution for both the LoKriging[21]and the present methods are also esti-mated inFig 21 This result implies that the present method gives

a remarkable convergence with a smaller error about 0.1% com-pared 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

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

Time t

Δt=5x10 −3 , ω = 18

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time t

Δt=5x10 −3 , ω = 27

a

b

Fig 16 Response at point A for different loading frequency using Newmark method

(c= 0.5,b = 0.25), Dt = 5  10 3 and c = 0.4 for time-harmonic loading.x= 18 (a)

andx= 27(b).

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time t

Fig 17 Response at point A without damping (c = 0) using Newmark method

(c= 0.5, b = 0.25) and Dt = 5  10 3

for time-harmonic loading.

−0.02 0 0.02 0.04 0.06 0.08 0.1

Time t

FEM (ANSYS)

γ =0.5

γ =0.8

γ =1.0

γ =1.5

γ =2.0

γ =10.0

Fig 18 Dissipation verification at point A without damping (c = 0) forc= 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  10 3

−0.018

−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002 0

Time t

Δt=4x10 −3 , c=0

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

4.2.2.2 Heaviside step loading with a finite duration The dynamic

Heaviside rectangular step loading with a finite duration has a

sim-ilar form as the dynamic impact loading considered previously but

suddenly vanished at time t = 0.5 s as depicted inFig 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 inFigs 23 and 24, respectively

Sev-eral 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

agree-ment 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 inFig 25 In this case, the loading function is

de-fined by

gðtÞ ¼ t

t0

HðtÞ  t

t0

 1

The corresponding numerical results are presented inFigs 26 and

27, respectively As can be seen from the results shown inFigs 20

and 27that the displacement response always converges to the sta-tic deformation given by the stasta-tic analysta-tical solution

4.2.3 Transient loading with a finite decreasing time For the transient loading with a finite decreasing time as de-picted inFig 11c, the loading function is determined by

−0.018

−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

Time t

Δt=4x10 −3

,c=0.4

uy= −0.008888112425049

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

and 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

Time (s) Displacement u y

LoKriging [21] Present method

11750 0.470000E+02 0.00883283 0.008888400762486

11875 0.475000E+02 0.00883255 0.008889026243629

12000 0.480000E+02 0.00883264 0.008890959168876

12125 0.485000E+02 0.00882592 0.008885829057817

12250 0.490000E+02 0.00883220 0.008888068307071

12375 0.495000E+02 0.00884123 0.008889391269139

12500 0.500000E+02 0.00884174 0.008888348924774

Exact: u y = 0.0089 [36]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time t

Present LoKriging

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

g(t)

t(s) t=0.5s

1.0

0

Fig 22 The dynamic Heaviside step loading with a finite duration.

−0.02

−0.015

−0.01

−0.005 0 0.005 0.01 0.015

Time t

FEM (ANSYS)(Δt=5x10 −3

) Δt=1x10 −4

Δt=1x10 −3

Δt=5x10 −2

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