DSpace at VNU: FREE AND FORCED VIBRATION ANALYSIS USING THE n-SIDED POLYGONAL CELL-BASED SMOOTHED FINITE ELEMENT METHOD (nCS-FEM)

5 Hochiminh City, Vietnam † Division of Computational Mechanics Ton Duc Thang University 98 Ngo Tat To St., War 19 Binh Thanh Dist., Hochiminh City, Vietnam § Institute of Structural Mec

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DOI: 10.1142/S0219876213400082

FREE AND FORCED VIBRATION ANALYSIS USING THE

n-SIDED POLYGONAL CELL-BASED SMOOTHED FINITE

T NGUYEN-THOI∗,†,,∗∗, P PHUNG-VAN†, T RABCZUK§,

H NGUYEN-XUAN∗,†and C LE-VAN¶

∗ Department of Mechanics, Faculty of Mathematics &

Computer Science, University of Science, VNU-HCM

227 Nguyen Van Cu, Dist 5 Hochiminh City, Vietnam

† Division of Computational Mechanics Ton Duc Thang University

98 Ngo Tat To St., War 19 Binh Thanh Dist., Hochiminh City, Vietnam

§ Institute of Structural Mechanics Bauhaus-University Weimar Marienstrasse 15, 99423, Weimar

¶ Department of Civil Engineering, International University

VNU-HCM, Vietnam

thoitrung76@gmail.com

Received 14 March 2011 Accepted 10 June 2011 Published 18 January 2013

An-sided polygonal cell-based smoothed ﬁnite element method (nCS-FEM) was recently

proposed to analyze the elastic solid mechanics problems, in which the problem domain can be discretized by a set of polygons with an arbitrary number of sides In this paper, the nCS-FEM is further extended to the free and forced vibration analyses of

two-dimensional (2D) dynamic problems A simple lump mass matrix is proposed and hence the complicated integrations related to computing the consistent mass matrix can be avoided in thenCS-FEM Several numerical examples are investigated and the results

found of thenCS-FEM agree well with exact solutions and with those of others FEM Keywords: Numerical methods; ﬁnite element method (FEM); cell-based smoothed ﬁnite

element method (CS-FEM); polygonal element;n-sided polygonal cell-based smoothed

ﬁnite element method (nCS-FEM).

1 Introduction

In the front of development of novel numerical methods, by incorporating the strain

smoothing technique of meshfree methods [Chen et al (2001)] into the FEM, Liu

∗∗Corresponding author.

Int J Comput Methods 2013.10 Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 05/30/13 For personal use only.

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and Nguyen-Thoi et al (2010) have formulated a series of smoothed FEM (S-FEM) models named as cell-based S-FEM (CS-FEM) [Bordas et al (2009); Cui et al (2008); Dai and Liu (2007); Dai et al (2007); Liu et al (2007); Liu et al (2007); Liu

et al (2009); Nguyen-Thanh et al (2008); Nguyen-Thoi et al (2007); Nguyen-Xuan

et al (2008a, 2008b); Nguyen-Xuan et al (2008); Nguyen-Xuan and Nguyen-Thoi (2009); Nguyen-Van et al (2008)], node-based S-FEM (NS-FEM) [Nguyen-Xuan

et al (2010); Liu et al (2009); Liu et al (2010); Thoi et al (2009); Nguyen-Thoi et al (2009); Nguyen-Nguyen-Thoi et al (2010)], edge-based S-FEM (ES-FEM) [Liu

et al (2009); Nguyen-Thoi et al (2009); Nguyen-Thoi et al (2009); Nguyen-Xuan

et al (2009); Nguyen-Xuan et al (2009); Tran et al (2010)], face-based (FS-FEM) [Nguyen-Thoi et al (2009); Nguyen-Thoi et al (2009)], and alpha-FEM [Liu et al.

(2008)] that use linear interpolations In these S-FEM models, the ﬁnite element mesh is used similarly as in the standard FEM However, these S-FEM models evaluate the weak form based on smoothing domains created from the entities of the element mesh such as cells/elements, or nodes, or edges or faces These smoothing domains can be located inside the elements (CS-FEM) or cover parts of adjacent elements (NS-FEM, ES-FEM, and FS-FEM) These smoothing domains are linear independent and hence ensure stability and convergence of the S-FEM models They

cover parts of adjacent elements, and therefore the number of supporting nodes in

smoothing domains is larger than that in elements This leads to the bandwidth

of stiﬀness matrix in the S-FEM models to increase and the computational cost is hence higher than those of the FEM However, also due to contributing of more supporting nodes in the smoothing domains, the S-FEM models often produce the solution that is much more accurate than that of the FEM Therefore in general, when the eﬃciency of computation (computation time for the same accuracy) in terms of the error estimator versus computational cost is considered, the S-FEM

models are more eﬃcient than the counterpart FEM models [Liu et al (2009); Nguyen-Thoi et al (2009); Nguyen-Thoi et al (2009)] It is clear that these S-FEM models have the features of both models [Rabczuk et al (2006)]: meshfree and FEM.

The element mesh is still used but the smoothed gradients bring the information

beyond the concept of only one element in the FEM: they bring in the information

from the neighboring elements

In these S-FEM models, only the CS-FEM applies the strain smoothing tech-nique for elements seperately, while the others S-FEM models apply this techtech-nique for two or more adjacent elements As a result, the others S-FEM models are closer

to meshfree methods [Liu and Nguyen Thoi (2010); Liu et al (2010); Rabczuk et al.

(2004); Rabczuk and Belytschko (2005)], while the CS-FEM model is closer to the standard FEM and simpler than other S-FEM models The CS-FEM is ﬁrst

devel-oped for the quadrilateral elements [Liu et al (2007); Liu et al (2007); Liu et al (2009); Nguyen-Xuan et al (2008a)] to analyze elastic solid mechanics problems [Liu et al (2007)], and free and forced vibration analysis [Dai and Liu (2007)] The CS-FEM is then further developed for n-sided polygonal elements to analyze the

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elastic solid mechanics problems [Dai et al (2007)] In Dai et al [2007], the stability condition of the nCS-FEM is examined and some criteria are provided to avoid the presence of spurious zero-energy modes An approach to constructing nCS-FEM

shape functions are also suggested with emphasis on a novel and simple averaging method A selective integration scheme is recommended to overcome volumetric locking for nearly incompressible materials

In this paper, the nCS-FEM is further extended to the free and forced vibration

analyses of two-dimensional (2D) dynamic problems A simple lump mass matrix is proposed and hence the complicated integrations related to computing the

consis-tent mass matrix can be avoided in the nCS-FEM Several numerical examples are investigated and the results found of the nCS-FEM agree well with exact solutions

and with those of others FEM

The paper is outlined as follows In Sec 2, the brieﬁng on the nCS-FEM is

presented including the proposal of lump mass matrix Some numerical examples are presented and examined in Sec 3 and some concluding remarks are made in the Sec 4

2 Brief of Dynamic Analysis of the nCS-FEM

2.1. Brief of the ﬁnite element method (FEM )

The discrete equations of the FEM are generated from the Galerkin weak form and the integration is performed on the basis of element as follows

Ω (∇ s δu) TD(∇ s u)dΩ −

Ωδu T(b− ρ¨u − c ˙u)dΩ −

Γt

δu T¯tdΓ = 0, (1)

where b is the vector of external body forces, D is a symmetric positive deﬁnite (SPD) matrix of material constants, ¯ t is the prescribed traction vector on the

nat-ural boundary Γt , u is trial functions, δu is test functions, ρ is the mass density, c

is the damping coeﬃcient and∇ su is the symmetric gradient of the displacement ﬁeld [Bathe et al (1996); Liu and Quek (2003); Hughes (1987); Zienkiewicz and

Taylor (2000)]

The FEM uses the following trial and test functions

uh(x) =N n

I=1

NI(x)dI; δu h(x) =

N n

I=1

where N n is the total number of nodes of the problem domain, dI is the nodal

displacement vector and NI (x) is the shape function matrix of Ith node.

By substituting the approximations, uh and δu h, into the weak form and invok-ing the arbitrariness of virtual nodal displacements, Eq (1) yields the standard discretized system of algebraic equation:

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where M is the mass matrix; C is the damping matrix; KFEMis the stiﬀness matrix

and f is the element force vector that are assembled with entries of

MIJ =

Ω

NT

CIJ =

Ω

NT

KFEM

ΩB

T

fI =

Ω

NT

I (x)bdΩ +

Γt

NT

with the strain gradient matrix deﬁned as

2.2. Smoothed stiﬀness matrix in the nCS-FEM

In the nCS-FEM [Dai et al (2007)], the problem domain is also discretized into

N e polygonal elements of arbitrary number of sides such that Ω = N e

e=1Ωe and

Ωi ∩ Ω j = However, we will replace the stiﬀness matrix KFEM in Eq (3) by

the smoothed stiﬀness matrix ˜K by using strain smoothing technique [Chen et al.

(2001)] based on the triangular smoothing cells created inside the polygonal

ele-ments In such a cell-based integration process, each n-sided polygonal element Ω e

is divided into n triangular smoothing cells Ω (k) e such that Ωe = n

k=1Ω(k) e and

Ω(i)

e ∩ Ω (j) e =, i = j, by connecting n nodes of the element to the central point of

the element as shown in Fig 1

: field nodes : added node to form the smoothing cells

6 5

n

O

2

=6

Fig 1 Division of a six-sided polygonal element into six sub-triangles by connectingn ﬁeld nodes

with the central pointO.

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Applying the cell-based smoothing operation, the compatible strains ε = ∇ su

used in Eq (1) is used to create a smoothed strain on the triangular smoothing cell

Ω(k)

e :

˜

ε (k) e =

Ω(k)

e

ε(x)Φ (k) (x)dΩ =

Ω(k)

e

∇ su(x)Φ(k) (x)dΩ, (9)

where Φ(k)(x) is a given smoothing function that satisﬁes at least unity property

Ω(k) e

Using the following constant smoothing function

Φ(k)(x) =

1/A (k) e x∈ Ω (k) e

0 x / ∈ Ω (k) e

(11)

where A (k) e is the area of the triangular smoothing cell Ω(k)

e , and applying a diver-gence theorem, one can obtain the smoothed strain

˜

ε (k)

1

A (k) e

Γ(k) e

n(k)

where Γ(k)

e is the boundary of the smoothing cell Ω(k)

e , and n(k)

e (x) is the outward

normal vector matrix on the boundary Γ(k)

e and has the following form for 2D problems

n(k)

e (x) =







n (k) ex 0

0 n (k) ey

n (k) ey n (k) ex





In the nCS-FEM, the trial function u h(x) is the same as in Eq (2) of the FEM

and therefore the force vector f in the nCS-FEM is calculated in the same way as

in the FEM

Substituting Eq (2) into Eq (12), the smoothed strain on the smoothing cell

Ω(k)

e can be written in the following matrix form of n nodal displacements of n-sided

polygonal element Ωe

˜

ε (k)

n

i=1

˜

B(k)

where ˜ B(k)

i is termed as the smoothed strain gradient matrix on the smoothing cell

Ω(k)

e ,

˜

B(k)







˜

b (k) ix 0

0 ˜b (k) iy

˜

b (k) iy ˜b (k) ix





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and it is calculated numerically using

˜

b (k) ih = 1

A (k) e

Γ(k) e

n (k) h (x)N i (k) (x)dΓ, (h = x, y). (16) When a linear compatible displacement ﬁeld along the boundary Γ(k)

e is used, one

Gaussian point is suﬃcient for line integration along each segment Γ(k)

ej of boundary

Γ(k)

e of Ω(k)

e , the above equation can be further simpliﬁed to its algebraic form

˜

b (k) ih = 1

A (k) e

M

j=1

n (k) hj N i(xGP

where M is the total number of the boundary segments of Γ (k) e , xGP

j is the midpoint

(Gaussian point) of the boundary segment of Γ(k)

ej , whose length and outward unit normal are denoted as l (k) j and n (k) hj , respectively

Equation (17) implies that only shape function values at some particular points along segments of boundary Γ(k)

ej are needed and no derivatives of the shape function

are required This gives tremendous freedom in shape function construction In this

paper, the simple averaging method [Dai et al (2007); Liu and Nguyen Thoi (2010)] for constructing nCS-FEM shape functions is used.

The smoothed stiﬀness matrix ˜ K of the system is then assembled by a similar

process as in the FEM

˜

K = N e

A

e=1

˜

whereA is assembled operator and ˜ Keis the stiﬀness matrix of the element Ωeand

is calculated by

˜

Ke= An

k=1

˜

K(k)

n

A

k=1

Ω(k)

e

˜

BT

ID ˜ BJ dΩ = An

k=1

˜

BT

ID ˜ BJ A (k) e (19)

2.3. Lump mass matrix in the nCS-FEM

In dynamic analysis using the nCS-FEM, we can use the usual consistent mass

matrix defined in Eq (4) to compute However, this computational process will be rather difficult and cumbersome due to the sub-division of polygonal elements into smoothing cells, and also due to the sub-division of the shape function of polygonal elements into linear piecewise shape functions on smoothing cells [Liu and Nguyen Thoi (2010)] In order to avoid such difficulty and to increase the computational

eﬃciency, in this paper, we propose the well-known lumped mass matrix for the n-sided polygonal elements Ω esuch as

M = N n

A

p=1Mp=

N n

A

where Mp is the lump mass matrix of pth node; I pis the identity matrix of size 2×2;

N n is the total number of nodes of the problem domain; A p is the area surrounding

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Fig 2 AreaA pof ﬁeld nodep in a mesh of n-sided polygonal elements.

the pth node and is created by connecting sequentially the mid-edge-point to the central points of the surrounding n-sided polygonal elements of the node pth as shown in Fig 2; ρ and t are the mass density and the thickness of the element,

respectively

Note that the diagonal form of lumped mass matrix gives the superiority in terms

of computational eﬃciency over the consistent mass matrix in solving transient dynamics problems [Liu and Quek (2003)]

2.4. Shape functions of the nCS-FEM

The general shape functions of n-sided polygonal elements in the nCS-FEM was

presented by Liu and Nguyen-Thoi [2010] However, in actual computation of the

nCS-FEM, it is not necessary to use such shape functions to compute the smoothed

stiﬀness matrix ˜ K Instead, as shown in sub-section 2.2, we only need to evaluate

the shape function values at Gauss points along boundary segments of

triangu-lar smoothing cells to compute ˜ K This computational process is very simple and

performed in three steps as follows:

• Step 1: For each n-sided polygonal element, write explicitly the available shape

function values at the ﬁeld nodes

• Step 2: Evaluate the shape function values at the central point of the n-sided polygonal element by averaging the shape function values of n ﬁeld nodes at

Step 1

• Step 3: Evaluate the shape function values at Gauss points along boundary

seg-ments of triangular smoothing cells by linear interpolation from the available

shape function values of n ﬁeld nodes and central point.

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Fig 3 Positions of Gauss points at mid-segment-points on segments of six triangular smoothing cells in an six-sided polygonal element.

Figure 3 and Table 1 presents explicitly the shape function values at diﬀerent points of a six-sided polygonal element divided into six triangular smoothing cells The number of support ﬁeld nodes for this six-sided element is 6 (from #1 to #6)

We have a total of 12 segments (1–2, 2–3, 3–4, 4–5, 5–6, 6–1, 1–O, 2–O, 3–O, 4–

O, 5–O, 6–O) Each segment needs only one Gauss point (due again to the linear interpolation) Therefore, there are a total of 12 Gauss points (from g1, to g12) to

be used for all the smoothing cells, and the shape function values at all these 12 Gauss points can be tabulated in Table 1 by simple inspection

It should be mentioned that the purpose of introducing of central points such

as point O in Fig 3 is to facilitate the evaluation of the values of shape functions

at some discrete points inside and on the segments of the interested element There

is no extra degrees of freedom are associated with these added points In other words, these points carry no additional independent ﬁeld variable Therefore, the

total degrees of freedom (DOFs) of a nCS-FEM model will be exactly the same as

the standard FEM using the same set of nodes

2.5. Dynamic analyses of the nCS-FEM

By replacing the stiﬀness matrix K of the FEM in Eq (3) by the smoothed stiﬀness matrix ˜K in Eq (18) of the nCS-FEM, the dynamic discretized system of equations

in the nCS-FEM is expressed as a set of diﬀerential equations with respect to time

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Table 1 Values of shape functions at diﬀerent points within ann-sided polygonal element (Fig 3).

Node Node Node Node Node Node

g1 7/12 1/12 1/12 1/12 1/12 1/12 Gauss point (mid-segment point of Γ(k)

ej )

g2 1/2 1/2 0 0 0 0 Gauss point (mid-segment point of Γ(k)

ej )

g4 0 1/2 1/2 0 0 0 Gauss point (mid-segment point of Γ(k)

ej )

g6 0 0 1/2 1/2 0 0 Gauss point (mid-segment point of Γ(k)

ej )

g8 0 0 0 1/2 1/2 0 Gauss point (mid-segment point of Γ(k)

ej )

g10 0 0 0 0 1/2 1/2 Gauss point (mid-segment point of Γ(k)

ej )

g12 1/2 0 0 0 0 1/2 Gauss point (mid-segment point of Γ(k)

ej )

For simplicity, the Rayleigh damping is used, and the damping matrix C is

assumed to be a linear combination of M and ˜ K,

where α and β are the Rayleigh damping coeﬃcients.

Many existing standard schemes can be used to solve the second-order time dependent problems, such as the Newmark method, Crank-Nicholson method, etc [Smith and Griﬃths (1998)] In this paper, the Newmark method is used When the

current state at t = t0is known as (d0, ˙ d0, ¨ d0), we aim to ﬁnd a new state (d1, ˙ d1,

¨

d1) at t1= t0+ θ∆t where 0.5 ≤ θ ≤ 1, using the following formulations:

α + 1 θ∆t M + (β + θ∆t) ˜K

d1= θ∆tf1+ (1− θ)∆tf0

+

α + 1 θ∆t Md0+

1

θM ˙ d0+ [β − (1 − θ)∆t] ˜Kd0 (23)

˙

d1= 1

θ∆t(d1− d0)−1− θ

θ

˙

¨

d1= 1

θ∆t( ˙ d1− ˙d0)−1− θ

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Without the damping and forcing terms, Eq (21) is reduced to a homogenous diﬀerential equation:

A general solution of such a homogenous equation can be written as

where t indicates time, D is the amplitude of the sinusoidal displacements and ω

is the angular frequency On its substitution into Eq (26), the natural frequency ω

can be found by solving the following eigenvalue equation

3 Numerical Examples

In this section some examples will be analyzed to demonstrate the eﬀectiveness

and accuracy of the nCS-FEM The results of the nCS-FEM will be compared

with analytic solutions and with those of FEM using triangular elements (FEM-T3), FEM using quadrilateral elements (FEM-Q4) and FEM using 8-node elements (FEM-Q8)

3.1. Free vibration analysis of a rectangular cantilever beam

In this example, a rectangular cantilever beam is studied The parameters used are

length L = 100 mm, height H = 10 mm, thickness t = 1.0 mm, Young’s modulus

E = 2.1 × 104kgf/mm2, Poisson’s ratio ν = 0.3, and mass density ρ = 8.0 ×

10−10kgf s2/mm4 A plane stress problem is considered This problem has also been investigated in Dai and Liu [2007] Using the Euler–Bernoulli beam theory

we obtain the fundamental frequency f1 = 0.08276 × 104Hz that can serve as a reference Three types of meshes of triangular, quadrilateral and polygonal elements are used for comparison purpose as shown in Fig 4 Because the exact solution is not available, numerical results using the FEM-Q4 with a very ﬁne mesh (100× 10)

for the same problem are computed and used as the reference solutions

Table 2 lists the ﬁrst nine natural frequencies of the beam, and the ﬁrst nine

free vibration modes using nCS-FEM are plotted in Fig 5 It is observed that the nCS-FEM does not have any spurious nonzero energy and all the modes obtained

corresponds to physical modes In addition, the convergence of the ﬁrst natural

frequency of the beam using FEM-T3, FEM-Q4, and nCS-FEM is shown in Fig 6.

It is shown that the nCS-FEM converges to the reference solution much faster than

FEM-T3 and FEM-T4

Figure 7 plots the ﬁrst nine natural frequencies of the beam using FEM-T3,

FEM-Q4, and nCS-FEM compared to those of reference solution Again, it is shown that the results of nCS-FEM are closer to the reference solution.

for the same problem are computed and used as the reference solutions

Table lists the ﬁrst nine natural frequencies of the beam, and the ﬁrst nine

free vibration modes using. .. nCS-FEM The results of the nCS-FEM will be compared

with analytic solutions and with those of FEM using triangular elements (FEM-T3), FEM using quadrilateral elements (FEM-Q4) and FEM using. .. been investigated in Dai and Liu [2007] Using the Euler–Bernoulli beam theory

we obtain the fundamental frequency f1 = 0.08276 × 104Hz that can serve

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