A Smoothed Finite Element Method for the Static and ...

Sep. 2010, Volume 4, No. 9 (Serial No.34) Journal of Civil Engineering and Architecture, ISSN 1934-7359, USA

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells Nhon Nguyen-Thanh1, Chien Thai-Hoang2, Hung Nguyen-Xuan2, 3 and Timon Rabczuk1 1. Institute of Structural Mechanics, Bauhaus-University Weimar, Weimar D-99423, Germany 2. Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 3. Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science-VNU HCM, Vietnam Abstract: A four-node quadrilateral shell element with smoothed membrane-bending based on Mindlin-Reissner theory is proposed. The element is a combination of a plate bending and membrane element. It is based on mixed interpolation where the bending and membrane stiffness matrices are calculated on the boundaries of the smoothing cells while the shear terms are approximated by independent interpolation functions in natural coordinates. The proposed element is robust, computationally inexpensive and free of locking. Since the integration is done on the element boundaries for the bending and membrane terms, the element is more accurate than the MITC4 element for distorted meshes. This will be demonstrated for several numerical examples. Key words: Shell element, MITC4 elements, smoothed finite elements method (SFEM).

1. Introduction The static and free vibration analysis of shell structures plays an important role in engineering applications as shells are widely used as structural components. Due to limitations of analytical methods [1-4] for practical applications, numerical methods have become the most widely used tool for designing shell structures. One of the most popular numerical approaches for analyzing vibration characteristics of shells is the Finite Element Method (FEM). Although the FEM provides a general and systematic technique for constructing basis functions, a number of difficulties still exist in the development of shell elements based on shear deformation theories. One is the shear locking phenomenon for low order displacement models based on Mindlin-Reissner theory [5, 6] as the shell thickness decreases. Membrane locking also occurs for shell elements and curved geometries. In order to avoid this drawback, various improvements and numerical techniques have Corresponding author: Nhon Nguyen-Thanh, research fields: computational mechanics, isogeometric analysis, E-mail: [email protected].

been developed, e.g., reduced and selective integration elements [7, 8], mixed formulation/hybrid elements [9], the Assumed Natural Strain (ANS) method [10-12] and Enhanced Assumed Strain (EAS) method [13, 14]. Many improved shell elements have been developed [15-18] and can be found in the textbooks [8, 19]. Recently, this smoothing technique was incorporated into the FEM, leading to the smoothed finite element method (SFEM) proposed by Liu et al. [20]. It was shown by numerical examples that the SFEM is very robust, accurate and computational inexpensive [21-22]. As we will show by several numerical examples, the proposed shell element is especially useful for distorted elements. The paper is organized as follows: In the next section, we will state the formulation shell element formulation. Section 3 describes the smoothing technique in order to evaluate the bending and membrane stiffness. Section 4 discusses several numerical examples that are compared to analytical solutions and other elements from the literature. Finally, we close our paper with some concluding remarks.

14

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

2. Formulations for Quadrilateral Shell Element A typical Mindlin-Reissner shell with notations shown in Fig. 1 is considered here. For an initially flat isotropic thick shell, the membrane deformations are accounted for since they are uncoupled from the bending and shear deformations. Hence, the basic assumptions for the displacement behavior [23] are:

u ( x , y , z ) = u 0 ( x , y ) + z β x ( x, y ) v ( x, y , z ) = v 0 ( x , y ) + z β y ( x , y )

(1)

w ( x, y , z ) = w ( x, y )

By means of the spatial discretization procedure in the FEM, the displacement and strains within any element can be written as ⎡ Ni 0 0 0 0 0⎤ ⎢0 N 0 0 0 0 ⎥⎥ i ⎢ np ⎢ 0 0 Ni 0 0 0⎥ uh = ∑ ⎢ ⎥q i (4) 0 0 0 Ni 0⎥ i =1 ⎢ 0 ⎢ 0 0 0 Ni 0 0⎥ ⎢ ⎥ ⎣ 0 0 0 0 0 0⎦ T where qi = {ui vi wi θ xi θ yi θ zi } is the nodal displacement vector

0

0

0

εm =

where u , v , w are displacement components in the x, y, z directions (local coordinate system), respectively. β x and β y are the rotations of the normal to the

⎡ ⎤ ∂β x ⎢ ⎥ ∂x ⎢ ⎥ ⎢ ∂β y ⎥ κ=⎢ − ⎥ ∂y ⎢ ⎥ ⎢ ∂β ∂β y ⎥ ⎢ x − ⎥ ∂ x ⎦⎥ ⎣⎢ ∂ y

and the transverse shear strain vector is ⎧ ∂w ⎫ + βx ⎪ ⎧γ xz ⎫ ⎪⎪ ∂x ⎪ γ =⎨ ⎬=⎨ ⎬ ⎩γ yz ⎭ ⎪ ∂w − β y ⎪ ⎪⎩ ∂y ⎪⎭

(2)

(3)

B is q i ; κ =

i

⎡ Ni, x ⎢ B =⎢ 0 ⎢ Ni, y ⎣ m i

∑

B bi q i

(5)

i

0 Ni, y Ni, x

0 0 0 0⎤ ⎥ 0 0 0 0⎥ ; 0 0 0 0 ⎥⎦

⎡0 0 0 0 ⎢ B = ⎢0 0 0 − Ni, y ⎢0 0 0 − Ni, x ⎣ b i

and curvature strains κ are

⎡ ∂u 0 ⎤ ⎢ ⎥ ∂x ⎢ ⎥ ⎢ ⎥ ∂v 0 εm = ⎢ ⎥; ∂y ⎢ ⎥ ⎢ ∂u 0 ∂v 0 ⎥ + ⎢ ⎥ ∂ x ⎦⎥ ⎣⎢ ∂ y

∑

in which

undeformed mid-surface in the xz and yz planes, ∂w ∂w and β y = . respectively, β x = ∂x ∂y m

B im q i ; γ =

i

0

The membrane ε defined as

∑

Ni,x 0 Ni, y

0⎤ ⎥ 0⎥ ; 0 ⎥⎦ 0⎤ 0 ⎥⎦

(6)

0 Ni ⎡0 0 Ni , x Bis = ⎢ ⎣0 0 Ni , y − Ni 0 As known in Refs. [24, 25], the use of reduced integration on the shear term ks can avoid shear locking as the thickness of the shell tends to zero. However, these elements fail the patch test and exhibit an instability due to rank deficiency [22]. In order to improve these elements, we use independent interpolation fields in the natural coordinate system for the approximation of the shear strains [10].

γ ⎡γ x ⎤ −1 ⎡ ξ ⎤ ⎢γ ⎥ = J ⎢γ ⎥ ⎣ η⎦ ⎣ y⎦

(7)

where

1 2 (8) 1 A C γ η = [(1 − ξ )γ η + (1 + ξ )γ η ] 2 where J is the Jacobian matrix and the mid-side nodes A, B, C, D are shown in Fig. 1. In case of bending around the η axis, it is useful to place the sampling points at positions ξ =0 where the

γ ξ = [(1 − η )γ ξB + (1 + η )γ ξD ];

Fig. 1

Quadrilateral shell element.

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

parasitic transverse shear strains vanish. We recall, that γ ξ linearly varies in ξ direction. In order to retain a

γ ξ in η direction, we choose two sampling points, at ξ = 0 , η = 1 and at ξ = 0 , η = −1 (points A and C). For the transverse shear strains γ η we proceed in a similar way (points B and

linear variation of

A C D). Presenting γ η , γ η based on the discretized fields

uh, we obtain the shear matrix:

⎡0 0 Ni,ξ −bi12 Ni,ξ bi11Ni,ξ 0⎤ Bis = J−1 ⎢ ⎥ 22 21 ⎣⎢0 0 Ni,η −bi Ni,η bi Ni,η 0⎥⎦ where

bi11 = ξi x,Mξ ; bi12 = ξi y,Mξ ; bi21 = ηi x,Lη ; bi22 = ηi y,Lη with

(9)

and

(i, M , L) ∈ {(1, B, A); (2, B, C ); (3, D, C ); (4, D, A)} The formulation for the free vibration of a Mindlin-Reissner shell can be written in matrix form as

me q + k e q = 0

(11)

me = ∫ e N T mNd Ω

(12)

where Ω

⎡t ⎢0 ⎢ ⎢0 ⎢ m = ρ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢0 ⎣

Dm =

Db =

0⎤ 0 0 0 0 ⎥⎥ t 0 0 0⎥ ⎥ t3 0 0 0⎥ ⎥ 12 ⎥ 3 t 0 0 0⎥ ⎥ 12 0 0 0 0 ⎥⎦

0 0 t 0 0 0 0

(13)

(14)

⎡ ⎤ ⎢1 ν 0 ⎥ ⎢ ⎥ 0 ⎥ ⎢ν 1 ⎢ 1 −ν ⎥ ⎢0 0 ⎥ ⎣ 2 ⎦

(15)

kEt ⎡1 0 ⎤ 2 (1 + ν ) ⎢⎣0 1 ⎥⎦

(16)

2

Et 3 12 (1 −ν )

Ds =

0

⎡ ⎤ ⎢1 ν 0 ⎥ ⎢ ⎥ 0 ⎥ ⎢ν 1 ⎢ 1 −ν ⎥ ⎢0 0 ⎥ 2 ⎦ ⎣

Et

(1 −ν )

0

2

The transformation between global coordinates and local coordinates is required to generate the local element stiffness matrix in the local coordinate system.

(17)

where Tl is the transformation matrix as given in Ref. [8]. Finally, the element stiffness K mass M in the global coordinate system, can be written as (18) K = T k T ; M = T m T T

(10)

15

T

e

e

where ⎡Tl ⎢0 ⎢ ⎢0 T=⎢ ⎢0 ⎢0 ⎢ ⎣⎢ 0

0

0

0

0

Tl

0

0

0

0 0

Tl 0

0 Tl

0 0

0 0

0 0

0 0

Tl 0

0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ Tl ⎦⎥

(19)

3. A Mixed Interpolation and a Smoothed Method for Four-Node Quadrilateral Shell Element The strain smoothing method was proposed by J. S. Chen et al. [26]. A strain smoothing stabilization is created to compute the nodal strain as the divergence of a spatial average of the strain field. This strain smoothing avoids evaluating derivatives of mesh-free shape functions at nodes and thus eliminates defective modes. The motivation of this work is to develop the strain smoothing approach for the FEM. The method developed here can be seen as a stabilized conforming nodal integration method, as in Galerkin mesh-free methods applied to the finite element method. The smooth strain field at an arbitrary point xC is written as

ε ij ( x C ) = ∫ ε ij ( x ) Φ ( x − xC ) d Ω Ωh

(20)

where Φ is a smoothing function that satisfies the following properties

Φ ≥ 0 and

∫

Ωh

Φd Ω = 1

(21)

16

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

For simplicity, Φ is assumed to be a step function defined by

⎧1/ AC , x ∈ ΩC Φ ( x − xC ) = ⎨ ⎩ 0, x ∉ ΩC

(22)

where AC is the area of the smoothing cell;

ΩC ⊂ Ω e ⊂ Ω h , as shown in Fig. 2. Substituting Eq. (22) into Eq. (20) and applying the divergence theorem, we obtain 1 2 AC

ε ij ( xC ) =

∫

ΩC

⎛ ∂ui ∂u j ⎞ + ⎜⎜ ⎟⎟d Ω ⎝ ∂x j ∂xi ⎠

(23) 1 + Γ u n u n d ( i j j j) 2 AC ∫ΓC Next, we consider an arbitrary smoothing cell ΩC =

Fig. 2 cells.

Example of finite element meshes and smoothing

nb

illustrated Fig. 2 with boundary Γ = Γb , where ΓbC ∪ C C b=1

is the boundary segment of ΩC and nb is the total number of edges of each smoothing cell. The relationship between the strain field and the nodal displacement is rewritten as ⎧κ ⎫ ε = ⎨ m ⎬ w ith κ = Β bC q ; ⎩ε ⎭

ε m = Β Cm q

(24)

The smoothed element membrane and bending stiffness matrix is obtained by k be =

∫ (B

b

)T D b B b d Ω =

Ωe

k em =

∫ (B

Ω

e

m

)T D m B m d Ω =

nc

∑ (B C =1

) T D b B bC AC

(25)

) T D m B Cm AC

(26)

b C

nc

∑ (B C =1

m C

where nc is the number of smoothing cells of the element, see Fig. 3. The integrands are constant over each ΩC and the non-local strain displacement matrix reads

⎛0 0 0 0 Ni nx 0⎞ 1 ⎜ ⎟ B (xC ) = ∫ ⎜ 0 0 0 −Ni ny 0 0⎟dΓ AC ΓC ⎜ ⎟ ⎝ 0 0 0 −Ni nx Niny 0⎠ b Ci

⎛ Nn 0 0 0 0 0⎞ i x m 1 ⎜ ⎟ BCi ( xC ) = ∫ ⎜ 0 Nn i y 0 0 0 0⎟dΓ Γ AC C ⎜ ⎟ i x 0 0 0 0⎠ ⎝ Niny Nn

(27)

(28)

From Eq. (28) we can use Gauss points for line integration along each segment ΓbC . If the shape

Fig. 3 Division of an element into smoothing cells (nc) and the value of the shape function along the boundaries of cells: k-Subcell stands for the shape function of the MISCk element, k = 1, 2, 4.

functions are linear on each segment of a cell’s boundary, one Gauss point is sufficient for an exact integration: Ni ( xGb ) nx 0 ⎞ ⎟ 0 0 ⎟lbC (29) ⎟ Ni ( xGb ) ny 0 ⎟⎠ 0 0 0 0 0⎞ ⎟ Ni xGm ny 0 0 0 0⎟lbC (30) ⎟ Ni xGm nx 0 0 0 0⎟⎠

⎛0 0 0 0 ⎜ 1 b BCi = ∑⎜ 0 0 0 −Ni ( xGb ) ny AC 1 ⎜ ⎜ 0 0 0 −N ( xG ) n i b x ⎝ nb

⎛ Ni ( xGm ) nx ⎜ 1 BCmi = ∑⎜ 0 AC 1 ⎜ ⎜ N ( xG ) n ⎝ i m y nb

( ) ( )

where xG and lbC are the midpoint (Gauss point) and the length of Γ bC , respectively, and nb is the total number

17

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

of edges of each smoothing cell. The smoothed membrane and curvatures lead to high flexibility such as arbitrary polygonal elements, and a slight reduction in computational cost. The element is subdivided into nc non-overlapping sub-domains also called smoothing cells. Fig. 3 illustrates different smoothing cells for nc = 1, 2, and 4 corresponding to 1-subcell, 2-subcell, and 4-subcell methods. The membrane and curvature are smoothed over each sub-cell. The values of the shape functions are indicated at the corner nodes in (Fig. 3) in the format (N1, N2, N3, N4). The values of the shape functions at the integration nodes are determined based on the linear interpolation of shape functions along boundaries of the element or the smoothing cells. Hence, the element stiffness matrix and geometrical stiffness matrix write: e

e

e

k = kb + km

(31)

where k be =

∫ (B

Ω

e

) D b B b dΩ =

b T

nc

∑ (B C =1

b C

) T D b B bC AC

(32)

nc

k em = ∫ (Bm )T DmBmdΩ = ∑(BCm )T DmBCm AC Ωe

C =1

(33)

It is seen that the element membrane-bending and geometrical stiffness matrices are now constructed based on the smoothing operator on each smoothing cell of the element while the shear term kse is derived from an interpolation independent from that of the shear strains, in the natural coordinates [11]. The shear contribution is therefore calculated as usual using Gauss quadrature and, in this paper, we use 2 point rule. The transformation of the element stiffness matrix and geometrical stiffness matrix from the local to the global coordinate system is given by T (34) ⎡⎣K ⎤⎦ = [ T]24×24 ⎡⎣k e ⎤⎦ [T]24×24 24×24 24×24 The final formulation of the free vibration of shells with the smoothed version reads: (35) Mq + Kq = 0 A general solution of such an equation can be written

q = q exp(iωt ) . Upon substitution of this general

solution into Eq. (35) the frequency ω can be found by solving (36) ( K − ω 2M ) q = 0 where K is the global smoothed stiffness matrix, M is the global mass matrix, vector q contains the vibration mode shapes, ω is the natural frequency.

4. Numerical Results 4.1 Static Analysis We name our element MISTk (Mixed Interpolation with Smoothing Technique with k ∈ {1, 2, 4}) related to number of smoothing cells as given by Fig. 3. For several numerical examples, we will now compare the MISTk elements to the widely used MITC4 elements. One major advantage of our element is that it is especially accurate for distorted meshes. To obtain mesh distortion that occurs naturally under phenomena such as shear bending or cracking, the coordinates of the initially regularly (structured):

x′ = x + α rc Δx y′ = y + α rc Δy

(37)

where rc is a random number between–1.0 and 1.0, α ∈ [0, 0.5] is used to control the amount of distortion of the elements and Δx , Δy are initial regular element sizes in the x–and y–directions, respectively. 4.1.1 Pinched Cylinder with Diaphragm Consider a cylindrical shell with rigid end diaphragm subjected to a point load at the center of the cylindrical surface. Due to its symmetry, only one eighth of the cylinder shown in Fig. 4 is modeled. The expected deflection under a concentrated load is 1.8248×10–5 [27]. The problem is described with N × N MITC4 or MISTk elements in regular and irregular configurations. The meshes used are shown in Figs. 4, 5 and 6 illustrate the convergence of the displacement at the center point and the strain energy, respectively, for the MITC4 element and our MISTk elements for regular meshes. Our element is slightly more accurate than the MITC4 element for structured meshes. This problem

18

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

300 300

250 250

200

z

z

200

150 100

150 100

50

50

0 0

0 0

50 100 200

300

300

0

y

Table 1 mesh.

Analytical solu. MITC4 MIST1 MIST2 MIST4

4×4 8×8 12×12 16×16 20×20 24×24

0.7

0.6

0.5

0.4

Table 2 4

6

8

10

12

14 16 Index mesh N

18

20

22

24

Fig. 5 The convergence of deflection under the load for a regular mesh. −6

x 10

2.2

MITC4 MIST1 MIST2 MIST4

2

1.2

1

Fig. 6

10

12

14 16 Index mesh N

18

20

0.37 0.74 0.93 -

0.373 0.747 0.935 -

0.4751 0.8094 0.9159 0.9574 0.9774 0.9889

0.4418 0.7878 0.9022 0.9483 0.9709 0.984

0.3875 0.7554 0.882 0.9347 0.9612 0.9767

The strain energy for a regular mesh. MITC4

4×4 8×8 12×12 16×16 20×20 24×24

8.47E-07 1.70E-06 1.99E-06 2.12E-06 2.18E-06 2.22E-06

Present elements MIST1

MITS2

MIST4

1.08E-06 1.85E-06 2.09E-06 2.18E-06 2.23E-06 2.26E-06

1.01E-06 1.80E-06 2.06E-06 2.16E-06 2.21E-06 2.24E-06

8.84E-07 1.72E-06 2.01E-06 2.13E-06 2.19E-06 2.23E-06

Mesh

MIST2 MITC4 α = 0.5 α = 0.1 α = 0.2 α = 0.3 α = 0.4 α = 0.5

4×4

0.3539 0.437

8×8

0.695

12×12

0.7402 0.8941 0.8938 0.8945 0.8959 0.893

16×16

0.8488 0.9397 0.9394 0.9344 0.9402 0.935

1.4

8

0.399 0.763 0.935 -

Mesh

1.6

6

0.3712 0.7434 0.874 0.9292 0.9573 0.9737

Present elements MIST1 MIST2 MIST3

Table 3 Normal displacement under the load for an irregular mesh.

1.8

4

y

Normal displacement under the load for a regular

Mesh MITC4 Mixed QPH SRI

0.8

0

x

υ = 0.3; E = 3×10 ) and mesh its (regular and irregular mesh).

0.9

Normalized deflection w

50

7

1

The energy value E

100

250

50

x

0.8

200 150

200

100

250

2.4

250 150

150

200

0.3

300

100

250 150

Fig. 4 Pinched cylinder (P = 1; R = 300; L = 600; t = 3;

50

300

22

0.4342 0.4331 0.4261 0.4398

0.7777 0.7786 0.7839 0.7803 0.786

24

20×20

0.896

The convergence of strain energy for regular mesh.

24×24

0.8718 0.9746 0.9739 0.9764 0.9755 0.9672

was studied repeatedly in the literature [28-30]. In Table 1, we have compared the normalized displacement at the center point of our element to the MITC4 element. The strain energy is summarized in Table 2. The advantage of our element becomes more

0.9614 0.9631 0.9586 0.9628 0.9601

relevant for distorted meshes, see Tables 3 and 4 and Figs. 7 and 8. For the same reasons as outlined in the previous section, the MISTk elements are significantly more accurate as compared to the MITC4 element with increasing mesh distortion.

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

Table 4

The strain energy for an irregular mesh. MIST2

Mesh

MITC4 α = 0.5

α = 0.1

α = 0.2

α = 0.3

α = 0.4

α = 0.5

4×4

8.15E-07

1.01E-06

1.00E-06

9.97E-07

9.81E-07

1.01E-06

8×8

1.60E-06

1.79E-06

1.79E-06

1.81E-06

1.80E-06

1.81E-06

12×12

1.70E-06

2.06E-06

2.06E-06

2.06E-06

2.06E-06

2.06E-06

16×16

1.95E-06

2.16E-06

2.16E-06

2.15E-06

2.17E-06

2.15E-06

20×20

2.06E-06

2.21E-06

2.22E-06

2.21E-06

2.22E-06

2.21E-06

24×24

2.01E-06

2.24E-06

2.24E-06

2.25E-06

2.25E-06

2.23E-06

Table 5 The reference values for the total strain energy E and vertical displacement w at point B x=L/2, y=0. t/L Strain energy E (N.m) Displacement w (m)

1

Normalized deflection w

0.9

1/1000 1/10000

0.8 Analytical solu. MITC4(α=0.5) MIST2(α=0.1) MIST2(α=0.2) MIST2(α=0.3) MIST2(α=0.4) MIST2(α=0.5)

0.7

0.6

0.5

0.4

0.3

4

6

Fig. 7

8

10

12

14 16 Index mesh N

18

20

22

24

The convergence of deflection for irregular meshes. −6

2.4

x 10

2.2

2

The energy value E

19

MITC4(α=0.5) MIST2(α=0.1) MIST2(α=0.2) MIST2(α=0.3) MIST2(α=0.4) MIST2(α=0.5)

1.8

1.6

1.2

1

4

6

8

10

12

14 16 Index mesh N

18

20

−6.3941E-3 −5.2988E-1

there is no analytical solution, and reference values for the total strain energy E and vertical displacement w present in Table 5, previously obtained by K. J. Bathe [31]. Figs.10 and 11 illustrate the convergence of deflection at point B and strain energy error for a regular mesh with ratio t/L=1000, t/L=1/10000, respectively. We note that the MISTk elements are always more accurate compared to the elements compared with. The results for the distorted meshes are illustrated in Figs. 12 and 13. The convergence of deflection and strain energy for an irregular mesh (t/L = 1/10000) are shown in Figs. 14 and 15. 4.2 Free Vibration Analysis

1.4

0.8

1.10E-02 8.99E-02

22

24

Fig. 8 The convergence of strain energy for irregular meshes.

4.1.2 Partly Clamped Hyperbolic Parabolic We consider the partly clamped hyperbolic parabolic shell structure, loaded by self-weight and clamped along one side. The geometric, material and load data are given in Fig. 9, and only one half of the surface needs to be considered in the analysis. For this problem

4.2.1 A Cylindrical Shell Panel In this example, a clamped cylindrical shell panel is analyzed. The geometry of the shell is illustrated in Fig. 16 and its mesh. The following parameters are used in the analysis: length L = 7.62 cm, radius R = 76.2 cm, thickness t = 0.033 cm, elastic modulus E = 6.8948 ×1010 N/m2, Poisson ratio ν =0.33 and mass density ρ = 2657.3 kg/m3. The central subtended angle of the section is θ = 7.64o. This problem was studied repeatedly in the literature: experimentally by Nath [32] numerically by the extended Rayleigh-Ritz method (ERR) in Ref. [33], using triangular finite elements (FET) in Ref. [34], analytically by a higher order theory

20

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05 z

0.05 z

0

0

−0.05

−0.05

−0.1

−0.1 −0.15

−0.15 0

−0.2

0

−0.2 −0.1

−0.1

−0.2

−0.25 −0.5

−0.2

−0.25 −0.5

−0.3

−0.3

−0.4

0

−0.4

0

−0.5

0.5

0.5

y

−0.5 y

x

x

Fig. 9 Partly clamped hyperbolic parabolic (L = 1 m, E = 2×1011 N/m2, ν = 0.3, ρ = 8000 kg/m3, z = x2 - y2, x ∈ [-0.5, 0.5], y ∈ [-0.5, 0.5]) and mesh its (regular and irregular mesh). −3

−3

The convergence of deflection at the point B (t/L=1/1000)

x 10

x 10

6.4

6.2

6.2 Ref. solu MITC4 MIST1 MIST2 MIST4

5.8

6

5.8

Displacement w (cm)

6

Displacement w (m)

The convergence of displacement for a irregular mesh (t/L=1/1000)

6.4

5.6

5.4

Ref solu. MITC4(α=0.5) MIST2(α=0.1) MIST2(α=0.2) MIST2(α=0.3) MIST2(α=0.4) MIST2(α=0.5)

5.6

5.4

5.2 5.2 5 5 4.8 4.8 4.6 4.6

10

15

20

25 30 Index mesh N

35

40

45

10

Fig. 10 The convergence of deflection of point B for a regular mesh (t/L = 1/1000).

15

20

25 30 Index mesh N

35

40

45

Fig. 12 The convergence of deflection of point B for an irregular mesh (t/L = 1/1000).

The convergence of energy (t/L=1/1000)

The convergence of energy for a irregular mesh (t/L=1/1000)

0.35 MITC4 MIST1 MIST2 MIST4

0.3

MITC4(α=0.5) MIST2(α=0.1) MIST2(α=0.2) MIST2(α=0.3) MIST2(α=0.4) MIST2(α=0.5)

0.3

0.25

1−E /E

0.2 h

h

1−E /E

0.25

0.15

0.2

0.15 0.1

0.1

0.05

0

5

10

15

20

25 30 Index mesh N

35

40

45

50

0.05

10

15

20

25 30 Index mesh N

35

40

45

Fig. 11 Convergence in strain energy for a regular mesh (t/L=1/1000).

Fig. 13 Convergence in strain energy for an irregular mesh (t/L=1/1000).

in Ref. [35] and with a nine-node assumed natural degenerated shell element in Ref. [36]. To analyze the effectiveness of the present method for distorted

meshes, we calculate frequencies using 8×8, 12×12, 16×16, 20×20 meshes for both regular and distorted elements.

21

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells The convergence of deflection at the point B (t/L=1/10000)

The convergence of energy (t/L=1/10000) 0.5 MITC4 MIST1 MIST2 MIST4

0.45

0.5

0.45

0.35

0.3

1 − Eh / E

Displacement w (m)

0.4

Ref. solu MITC4 MIST1 MIST2 MIST4

0.4

0.25

0.2

0.15

0.35

0.1

0.05

0.3

0

10

15

20

25 30 Index mesh N

35

40

45

5

10

15

20

25 30 Index mesh N

35

40

45

50

Fig. 15 Convergence in strain energy for an irregular mesh (t/L = 1/10000).

Fig. 14 The convergence of deflection of point B for an irregular mesh (t/L = 1/10000). −3

x 10

−3

x 10

0

0

−1

−1

−2

−2

−3

z

−3

z

−4

−4

−5

−5

−6

−6

−7 0 0.01

−7 0

0.02

0.02 0

0.03

0

0.04 0.06

0.06 0.08

10

0.06

0.06

0.08

0.08

0.07

Table 6

0.04

0.04

y

Fig. 16

0.02

0.04

0.02

0.05

0.1

0.1 0.12

0.08

0.12

y

x

2

x

3

Cylindrical shell panel: E = 6.8948×10 N/m , ν = 0.33, ρ = 2657.3 kg/m . First eight frequencies of clamped cylindrical shell panel for a regular mesh.

Modes

MITC4

MIST1

MIST2

MIST4

Ref.solu.: Olson [34] Petyt [33] Lim [35] Lee [36] Nath [32]

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

899.34 849.72 833.42 826.09 888.13 844.77 830.62 824.29 892.05 846.41 831.52 824.86 896.76 848.54 832.74 825.65

993.14 951.66 934.6 926.21 980.04 945.76 931.24 924.04 985.36 947.99 932.47 924.82 990.01 950.23 933.78 925.68

1439.03 1315.32 1271.52 1253.18 1399.66 1306.53 1266.68 1250.11 1416.7 1309.88 1268.51 1251.27 1429.84 1313.19 1270.34 1252.42

1476.1 1365.06 1340.2 1328.74 1444.49 1347.22 1330.04 1322.18 1460.68 1354.52 1334.07 1324.73 1470.99 1360.8 1337.74 1327.13

1487.88 1384.48 1350.3 1334.82 1454.61 1365.03 1339.15 1327.6 1462.96 1371.65 1342.77 1329.88 1478.56 1380.07 1347.68 1333.09

1897.88 1683.56 1617.52 1588.41 1818.21 1650.67 1599.1 1576.59 1847.83 1661.89 1605.12 1580.37 1880.34 1676.05 1613.18 1585.58

2496.93 2009.36 1848.14 1780.96 2436.97 1996.11 1841.04 1776.5 2463.57 2001.18 1843.68 1778.13 2483.34 2006.15 1846.4 1779.86

2571.05 2188.12 2097.67 2058.23 2490.22 2159.51 2081.1 2047.46 2521.81 2171.5 2087.84 2051.77 2554.94 2181.42 2093.71 2055.62

869.56 890 870 878.253 814

957.56 973 958 966.97 940

1287.56 1311 1288 1300.51 1260

1363.21 1371 1364 1377.21 1306

1440.26 1454 1440 1453.5 1452

1755.59 1775 1753 1768.54 1802

1779.63 1816 1779 1797.46 1735

2056.08 2068 2055 2077.21 2100

22

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells MODE 3, FREQUENCY = 1250.1096 [Hz]

MODE 2, FREQUENCY = 924.0485 [Hz]

MODE 1, FREQUENCY = 824.2904 [Hz]

0.02

0

−0.02

z −0.02

0.04 0.02 z

0

z

0.02

0

−0.04

−0.02

−0.04

0.1

0.06

0.1

0.06

0.04

0.04

0.06 0.04

0.02 0

0.08 0.04

0.06 0.04

0.02 y

x

0

0

MODE 4, FREQUENCY = 1338.0376 [Hz]

0.02 y

x

0

−0.02

MODE 6, FREQUENCY = 1576.5934 [Hz]

0.02 0 −0.02

−0.02

−0.04

−0.04

−0.04

0.1

0.06

0.1

0.06

0.08 0.06

0.08 0.04

0.06 0.04

0.02

0.02 0

0.1

0.06

0.08 0.04

0.04

0.02

0.06 0.04

0.02

0.02 y

x

0

x

0

z

z

0

z

0.02

0

0.04

0

MODE 5, FREQUENCY = 1366.7376 [Hz]

0.02

y

0.06 0.04

0.02

0.02

0.02 y

0.1

0.06

0.08

0.08

0

0.02 y

x

0

MODE 7, FREQUENCY = 1776.5038 [Hz]

0

0

x

MODE 8, FREQUENCY = 2047.4617 [Hz]

0.02

0.02

0

z

z

0

−0.02

−0.02

−0.04

−0.04

0.1

0.06

0.1

0.06

0.08 0.04

0.06

0.08 0.04

0.04

0.02

0.06 0.04

0.02

0.02 y

0

0

0.02

x

y

0

0

x

Fig. 17 Mode shapes of a clamped cylindrical panel for a regular mesh.

The first eight frequencies of the clamped cylindrical shell panel are shown in Table 6 for regular. The frequencies obtained using the MISTk element are lower than those obtained using the MITC4 element, which is consistent with the fact that strain smoothing leads to softer responses in the case of bilinear interpolants (see, e.g., Refs. [37, 38]). Fig.17 illustrate eight shape modes of free vibration of the clamped cylindrical shell panel with regular meshes. 4.2.2 Hemispherical Panel CCFF Let us consider a hemispherical panel as shown in Fig. 18 with radius R=1 m, thickness t = 0.1 m, ϕ0 = 30o; ϕ1 = 90o, ψ = 120o. The material parameters are:

Fig. 18 Hemispherical panel CCFF: R = 1 m, h = 0.1 m,

ϕ0 = 30o, ϕ1 = 90o, ψ

= 120o.

Young’s modulus E = 2.1×1011 Pa; Poisson’s ratio ν =0.3; mass density ρ =7800 kg/m3. The results of both MITC4 and MISTk are compared with the Generalized Differential Quadrature (GDQ) method of

23

A Smoothed Finite Element Method for the Static and Free Vibration Analysis of Shells

MODE 2, FREQUENCY = 462.2669 [Hz]

MODE 1, FREQUENCY = 330.3421 [Hz]

MODE 3, FREQUENCY = 718.4209 [Hz]

1

1 0.8

1

0.8

0.8

0.6

0.6

0.4

z

z

z

0.6 0.4

0.4

0.2

0.2

0.2

0

0

−0.2

0 −0.5 0

0

0

0.5

0.5

0.5 1

1

1

0.6

0.8

0.2

0.4

0

−0.2

−0.4

0 0.5 1

x

1

x

y

y

0.4

0.6

0.8

0.2

0

−0.2

−0.4

x

y

MODE 6, FREQUENCY = 1308.8918 [Hz] MODE 5, FREQUENCY = 1064.6284 [Hz] MODE 4, FREQUENCY = 904.0717 [Hz]

1 1

1

0.8

0.8

0.6

0.6

0.8

z

z

z

0.6

0.2

0.2

0.2

0

0

0 0

0 0.5 1

1

y

Fig. 19

0.4 0.4

0.4

0.5 x

0

0 0.5 1

1

y

0.5

0

x

0.5 1 y

0.8

0.6

0.4

0.2

0

−0.2 −0.4

x

Mode shapes of a hemispherical panel CCFF.

Ref. [39] and with results obtained using commercial software packages such as Abaqus, Ansys, Nastran, Straus. It is observed that the solutions of the MISTk element are closer to the reference values than the MITC4 element. The first six eigen modes of hemispherical panels are given in Fig. 19.

as non-linear material and geometric non-linearity, problems where large mesh-distortion play a major role. Providing an association of boundary integration with partition of unity methods in the extended finite element method [40-49] may be an interesting subject for improving discontinuous approximations.

5. Conclusions

References

In this paper, a smoothed finite element method (SFEM) was further developed for static and free vibration analysis of shell structures. The present method is derived from the linear combination of the gradient smoothing technique and an independent interpolation of assumed natural strains as given in the MITC4 element. The major advantage of the method, emanating from the fact that the membrane and bending stiffness matrix are evaluated on element boundaries instead of on their interiors is that the proposed formulation gives very accurate and convergent results for distorted meshes. In addition to the above points, the author believes that the strain smoothing technique herein is seamlessly extendable to complex shell problems such

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