An alternative alpha finite element method

An alternative alpha finite element method (AαFEM) free and forced vibration analysis of solids using triangular meshes N. Nguyen-Thanh

∗

T. Rabczuk

†

H. Nguyen-Xuan

‡

St´ephane Bordas

§

March 23, 2009

Abstract This paper presents a novel approach to modified a strain field of linear triangular FEM model with an adjustable parameter α. Using meshes with the same aspect ratio, a preferable approach has been proposed to obtain a nearly exact solution in strain energy for linear problems. In this paper, it is extended to the free and forced analysis of two-dimensional dynamic for solid mechanics problems without increasing the computational cost. The numerical results show in AαFEM are more accurate results and higher convergence rates as compared to that of standard FEM when the same sets of nodes are used.

1

Introduction

The analysis of natural frequencies and forced responde has been played a very important role in the design of structures in mechanical, civil and aerospace engineering applications [17, 43, 27]. A thorough study of the dynamic behaviors of these structures is essential in assessing their full potential. Therefore, it is necessary to develop appropriate models capable of accurately predicting their dynamic characteristics. In practical applications, the lowerorder linear triangular element is most preferred due to its simplicity, efficiency, less demand on the smoothness of the solution, and easy for adaptive mesh refinements for solutions of desired accuracy. On the other front of the development, Liu [30] has formulated the linear conforming point interpolation method (LC-PIM) using PIM shape functions constructed using a set of nodes in a local support domain that can overlap [21, 40] and the generalized strain smoothing technique [18] that are extended from the strain smoothing technique [7].Because the nodebased smoothing operation is used in the LC-PIM, it is also termed as a node-based smoothed point interpolation method (NS-PIM). Introducing the strain smoothing operation into the finite elements, the element-based smoothed finite element method (SFEM) [20,22] has also been formulated. The theoretical base of SFEM was then established and proven in detail in [22]. The SFEM has also been developed for general n-sided polygonal elements (nSFEM) [10], dynamic analyses [9], plate and shell analyses [36,35,34]. Based on the idea of the NSPIM and the SFEM, a node-based smoothed finite element method (NS-FEM) [25] for 2D solid mechanics problems has been developed. Furthermore, both the NS-PIM and NS-FEM can provide an upper bound [25,29] to exact solution in the strain energy ∗ Institute of Structural Mechanics (ISM), Bauhaus of University, MarienstraBe 15, 99423 Weimar, Germany email: [email protected] † Institute of Structural Mechanics (ISM), Bauhaus of University, MarienstraBe 15, 99423 Weimar, Germany email:[email protected] ‡ Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore 117576, Singapore email: [email protected] § Department of Civil Engineering, University of Glasgow, Rankine building, G12 8LT email: [email protected] , Tel. +44 (0) 141 330 5204, Fax. +44 (0) 141 330 4557 http://www.civil.gla.ac.uk/~ bordas

1

for elasticity problems with non-zero external forces [11,12]. However, it is also observed that the NS-PIM and NS-FEM can lead to spurious non-zero energy modes for dynamic problems, due to an overly-soft behavior that is in contrary to the overlystiff phenomenon of the compatible FEM (T3) [19, 18]. The finite element methods with free parameters have been well known via previous contributions in [4, 14, 13]. An alpha finite element method (α) [8] of quadrilateral elements was formulated to obtain ”exact” or best possible solution for a given problem by scaling the gradient of strains in the natural coordinates and Jacobian matrices with a scaling factor alpha. The method is not variationally consistent but stable and convergent. The AαFEM can produce ”nearly exact” solutions in the strain energy for all overestimation problems, and the ”exact” possible solution to underestimation problems. An AαFEM using triangular elements for exact solution to mechanics problems has also proposed [31]. In this paper, an alternative alpha finite element method (AαFEM) using triangular meshes is proposed. A strain field is carefully constructed by combining the compatible strains and the averaged nodal strains with an adjustable factor α. A novel variationally consistent Galerkin-like weak form for the AαFEM is derived from the Hellinger-Reissner variational principle. Due to the particular way of the strain field constructed, the new Galerkin-like weak form is as simple as the Galerkin weak form and the resultant stiffness matrix is symmetric. We prove theoretically and show numerically that the AαFEM is much more accurate than the original FEM-T3 and even more accurate than the FEM-Q4 when the same sets of nodes are used. The AαFEM can produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an α. The paper is arranged as following: In next section we will briefly describe discrete governing equations. In section 3 an assumed strain field based on linear triangular element (T3) is introduced. In section 4 some theoretical properties of the AαFEM. Next, in section 5 forced and for free vibration analyses are presented and proven . In section 6 presents and discusses numerical results. Finally, we close our paper with some concluding remarks.

2

Discrete governing equations

Consider a deformable body occupying domain Ω in motion, subjected to body forces b, external applied ¯ on Γu . It undergoes arbitrary traction on boundary Γt and displacement boundary conditions u = u virtual displacements δd accordingly give rise to compatible virtual strains δε and internal displacements δu. If the inertial and damping forces are also considered in the dynamic equilibrium equations, the principle of virtual work requires that Z Z Z ˙ dΩ − δεT DεdΩ − δuT [b − ρ¨ u − cu] δuT dΓ = 0 (1) Ω

Ω

Γt

where D is a matrix of material constants that is symmetric positive definite. The virtual displacements and the compatible strains ε = ∇s u within any element can be written as δuh =

np X

NI δdI , uh =

I=1

δεh =

X

np X

NI dI

(2)

BI dI

(3)

I=1

BI δdI , εh =

I

X I

T where np is the total number of nodes in the mesh, dI = uI vI is the nodal displacement vector, B is the the standard gradient matrix.   NI ′ x 0 NI ′ y  BI = ∇s NI (x) =  0 (4) NI ′ y NI ′ x 2

The energy assembly process gives Z Z Z T T T T ˙ dΩ − δd B DεdΩ − δd N [b − ρ¨ u − cu] Ω

Ω

δdT NT dΓ = 0

Since the expressions hold for any arbitrary virtual displacements δd, we obtain Z Z Z ˙ dΩ − BT DεdΩ − NT [b − ρ¨ u − cu] NT dΓ = 0 Ω

(5)

Γt

Ω

(6)

Γt

The resultant discrete governing equation can be written as ¨ + Cd˙ + Kd = f Md In which the vector of general nodal displacements is d and Z K= BT DBT dΩ

(7)

(8)

Ω

f=

Z

NT bdΩ +

Ω

M=

Z

NT dΓ

(9)

Γt

Z

NT ρNdΩ

(10)

Z

NT cNdΩ

(11)

Ω

C=

Ω

The formulation given above is the same as that of standard FEM. When triangular elements are used, the shape functions used in the AαFEM is also the same as that in FEM.

3

Construction of an assumed strain field

The problem domain Ω is fist divided into a set of triangular mesh with N nodes and Ne elements. We then divide, in a overlay fashion, the domain into a set of smoothing domains Ωk , k =1,2,..N, by connecting node k to centroid of the surrounding triangles as shown in Fig. 1. Ω is further subdivided into Ωk,i as M S shown in Fig. 2 such as Ωk = Ωk,i , Ωk,i ∩ Ωk,j = ∅ , i 6= j. i=1

ˆ = (uhx , uhy ) of the elasticity problem can be then expressed as The approximation of the displacement u ˆ (x) = u

np X

ˆi Ni (x) d

(12)

i=1

ˆ i is the vector of nodal displacements and where np is the total number of nodes in the mesh, d Ni (x) 0 Ni (x) = 0 Ni (x)

(13)

is the matrix of the FEM shape functions for node i created based on the elements. ˜i,j at any point using the assumed displacement field based on First, we obtain the compatible strain ε triangular elements: ˜i,j = ∇s u ˆ k,i (x) ε 3

(14)

Figure 1: Triangular elements and smoothing cells associated to nodes

Figure 2: Smoothing cell and triangular sub-domains associated with node k

4

where ∇s is a differential operator matrix given by ∇s =

"

∂ ∂x

0 ∂ ∂y

0

#T

∂ ∂y ∂ ∂x

(15)

˜i,j is constant in Ωi,j and different from element to element. Such Because the displacement is linear, ε a piecewise constant strain field obviously does not represent well the exact strain field, and should be somehow modified or corrected. To make a proper correction, a smoothed strain for node k (see Fig. 2) is introduced as follows: Z 1 ¯k = ˜i,j (x)dΩ ε ε (16) Ak Ωk where Ak is the area of smoothing domain Ωk . The strains εP , εI at points P, I are then assigned as √

˜ ) ˜ − α 3 6 (¯ ε −ε εP = εI = ε √ k,i √ k k,i α√6 ˜k,i − 3 (¯ ˜k,i ) εP = α 6¯ εk + 1 − α 6 ε εk − ε

(17)

ˆ at any points within a sub-triangular domain Ωk,i is where α is an adjustable factor. The strain field ε now re-constructed as [28]: √ √ ˆ (x) = L1 (x) εk + L2 (x) εp + L3 (x) εI = L1 (x) α 6¯ ˜k,i − ε εk + 1 − α 6 ε √ √ 6 6 ˜k,i − 3 α (¯ ˜k,i ) + L3 (x) ε ˜k,i − 3 α (¯ ˜k,i ) +L2 (x) ε εk − ε εk − ε

√

6 εk 3 α (¯

˜k,i ) −ε

(18)

where L1, L2, L3 is the area coordinates for the sub-triangular Ωk,i , which are partitions of unity. Eq. (18) can be simplified as √ √ 6 ˆ ˜ ˜ ˜k,i ) ε (x) = (L1 + L2 + L3 ) εk,i + α 6L1 (x) (¯ εk − εk,i ) − (L1 + L2 + L3 ) α (¯ εk − ε (19) 3 which can be simplified as ˆ (x) = ε ˜k,i + αεad ε k,i

(20)

√

(21)

˜k,i is constant in Ωk,i and where ε εad k,i (x) =

1 ˜k,i ) L1 (x) − 6 (¯ εk − ε 3

which is the ”adjusted” strain. It is a linear function representing the variation of the re-constructed strain field in . It is clear that we have successfully constructed a linear strain field in without adding any degrees of freedoms. We now need to examine the ”legality” of the constructed strain field. Using the formula [43, 27] Z p!q!r! Lp1 Lp2 Lr3 dA = 2Ak,i (22) (p + q + r + 2)! Ak,i it is easy to prove that Z

Ωk,i

εad k,i dΩ

=

√

˜k,i ) 6 (¯ εk − ε

5

Z

Ωk,i

1 L1 − 3

dΩ = 0

(23)

which is termed as zero-sum property of the correction strain, which is similar to orthogonal condition that used in the stabilization formulation [2, 3, 15]. The zero-sum property results in the following total zero-sum of the additional strain over the entire problem domain: Z

ad

ε dΩ =

Ωk,i

ˆdΩ = ε

Ω

Z

˜dΩ + ε Ω

Z

εad dΩ

(24)

Ωk,i

k=1 i=1

Therefore, we shall have Z

N X M Z X

N X M Z X

εad dΩ =

Ω

˜k,i dΩ ε

(25)

Ωk,i

k=1 i=1

which implies that the strain εk,i does not effect on the constant stress state that is needed to satisfy a patch test [37, 38], and hence ensures the convergence. The zero-sum Eq. (23) as shown later is also important to simplify the formulation. ˆ using the constant It is clear now that the key idea of this work is to re-construct carefully strain field ε ˜ of the FEM and the node-based smoothed strains ε ¯ of the NS-FEM, so that AαFEM compatible strains ε can always pass the standard patch tests ensuring the convergence. In addition, we introduce an α to regularize the variation of the strain field. Based on the findings in [24] that the gradient of strain field can be ”freely” scaled as long as the zero-sum property is maintained, the AαFEM will converge for any α ∈ ℜ.

4

Weak form for modified strain field

The following Galerkin-like weakform [26]: "Z # Z Z N X M X ad T ad ˜k,i + αεk,i D ε ˜k,i − αεk,i dΩ − δ ε δuT bdΩ − k=1 i=1

Ωk,i

Ω

δuT dΓ = 0

(26)

Γt

With the constructed strain field given by Eq. (20) is variationally consistent for elasticity problems. The new Galerkin-like weakform Eq. (26) is as simple as the Galerkin weak formulation: the bi-linear form is still symmetric, and hence the AαFEM should keep all the good properties of the standard Galerkin weak form. The Galerkin-like weakform provides a ”legal” means to re-construct the strain field. In the case of no reconstruction is made, we have εad k,i = 0 , and the Galerkin-like becomes the standard Galerkin weakform. Substituting the approximation Eq. (12) into Eq. (26) and using the arbitrary property of variation, we obtain ˆ AαF EM d ˆ = ˆf K α

(27)

ˆ AαF EM is the element stiffness matrix with the scaled gradient strains, and ˆf is the element force where K α vector given by T M R N M R N P P ad T ˆ AαF EM = P P B DBad K B DB dΩ − α k,i α k,i k,i dΩ k,i Ωk,i Ωk,i k=1 i=1 F EM−T 3

=K

2

EM KAαF ad

k=1 i=1

−α Z Z ˆf = NT (x) bdΩ + Ω

Γt

6

NT (x) ¯tdΓ

(28)

(29)

ˆ AαF EM is derived from the where KF EM−T 3 is the global stiffness matrix of the standard FEM (T3). K ad correction strain, and hence it is termed as correction stiffness matrix that help to reduce the well-known overly-stiffness of the standard FEM model. In Eq. (28), √ 1 ad ¯ Bk,i = 6 Bk − Bk,i L1(x) − (30) 3 ˆ AαF EM can be rewritten explicitly as and K ad

T N M R ad ˆ AαF EM = P P K DBad ad k,i dΩ Ωk,i Bk,i k=1 i=1 N P M P

=6 =

¯k = where B

1 Ak

M R P

i=1

Ωk,i

1 3

k=1 i=1 N P M P

¯ k − Bk,i B

k=1 i=1

, Bk,i dΩ =

1 Ak

T

¯ k − Bk,i B M P

T

¯ k − Bk,i D B

R

Ωk,i

L1 −

¯ k − Bk,i Ak,i D B

1 2 3

dΩ

(31)

Ak,i Bk,i is the nodal strain displacement matrix of node k,

i=1

Bk,i is the strain displacement matrix of sub-triangular domain i connecting to vertex k see in Fig. 2. ˆ AαF EM counts for the strain gap Eq. (21) between the It is clear that correction stiffness matrix K ad compatible (element) strains of FEM and the smoothed nodal strains of the NS-FEM. Note that the present formulation Eq. (28) is always stable for any finite parameters α. Hence α can be manipulated without affecting the convergence of the model [23], and it plays a crucial role to ensure the accuracy of ˆ AαF EM through α can, however, change the convergent rate of the model. the model. Manipulating K ad Eq. (28) may be rewritten in the following form : ˆ AαF EM = K ˆ AαF EM − α2 K ˆ AαF EM K α α=0 ad

(32)

ˆ (x) is the same as that given in Eq. (12) and therefore the Finally, we note that the trial function u force vector f , mass matrix M and damping matrix C in the AαFEM are calculated in the same way as in the FEM. In other words, the AαFEM changes only the stiffness matrix.

5

Forced and for free vibration analyses

From Eq. (32), the discretized equation system in the AαFEM can be expressed as ¨ + Cd˙ + K ˆ AαF EM d = f Md α

(33)

For simplicity, the Rayleigh damping is used, and the damping matrix C is assumed to be a linear ˆ AαF EM combination of M and K α The Newmark method which is a generation of the linear acceleration method. This latter method assumes that the acceleration varies linearly within the interval (t, t+δt). The formulations of the Newmark method is: ¨=d ¨t + 1 d ¨ t+∆t − d ¨t τ d ∆t h i ¨ t + δdt+∆t ∆t d˙ t+∆t = d˙ t + (1 − δ) d dt+∆t

˙ = dt + d∆t +

1 ¨ ¨ − β dt + β dt+∆t ∆t2 2 7

(34) (35) (36)

The response at t + δt is obtained by evaluating the equation of motion at time t + δt. The Newmark method is unconditionally stable provided 1 2 (δ + 0.5) 4 In this paper, α = 0.5 and β = 0.25 are used for the simplification. δ ≥ 0.5 and β ≥

6 6.1

(37)

Numerical results Cantilever beam to a parabolic traction at the free end

A cantilever beam with length L and height D and a unit thickness is studied as a benchmark here, which is subjected to a parabolic traction at the free end as shown in Fig. 3. The analytical solution is available and can be found in a textbook by Timoshenko and Goodier [39]. h i 2 y ux = 6PEI (6L − 3x) x + (2 + ν¯) y 2 − D4 ¯ i h (38) y D2 x 2 2 + (3L − x) x uy = − 6PEI 3¯ ν y (L − x) + (4 + 5¯ ν ) ¯ 4 where the moment of inertia I for a beam with rectangular cross section and unit thickness is given by 3 I=D 12 and ν f or plan stress ¯= E 2 E , ν¯ = (39) E 1−ν ν/(1 − ν) f or plan strain The stresses corresponding to the displacements Eq. (38) are σxx (x, y) =

P (L − x) y I

;

σyy (x, y) = 0

;

τx,y (x, y) = −

P 2I

D2 − y2 4

(40)

The related parameters are taken as E = 3.0 × 107 kP a, ν = 0.34, D = 12m, L = 48m and P = 1000N . The meshes using triangular is used as shown in Fig. 4. The compare results of displacements and energy of the AαFEM with the FEM-T3, FEM-Q4, NS-FEM and ES-FEM in Fig. 5 and Fig. 6. It is shown that the FEM-T3 is very stiff while the NS-FEM is very soft compared to the exact solution. The AαFEM is stiffer than the NS-FEM and softer than the FEM-T3, and is very close the exact solution. Compared with all methods, the AαFEM is best and even better than the FEM-Q4. From Fig. 7 and Fig. 8, it is observed that all the computed convergence displacement and energy using the AαFEM are in good agreement with the analytical solutions.

8

Figure 3: A cantilever beam (E = 3.0 × 107 kP a, ν = 0.3, D= 12m, L = 48m, P = 1000N)

y

5 0 −5

0

20

40

60

80

100

x

Figure 4: Domain discretization using triangular elements of the cantilever beam

9

−3

2

x 10

Analytical solu. FEM−T3 FEM−Q4 NS−FEM ES−FEM AαFEM

Vertical displacement v

0

−2

−4

−6

−8

−10

0

5

10

15

20

25 x (y=0)

30

35

40

45

50

Figure 5: Vertical displacement at central line (y = 0) using triangular elements of the cantilever beam

6 Exact energy FEM−T3 FEM−Q4 NS−FEM ES−FEM AαFEM

5.5

Strain energy

5

4.5

4

3.5

3

2.5

10

15

20 Mesh index

25

30

Figure 6: Convergence of the strain energy in a cantilever subjected to a parabolic traction at the free end

10

2

10

T3 ES−FEM−T3 NS−FEM−T3 Q4 AαFEM

1

10

d

log (e )

10

0

10

−1

10

−2

10

0

10

log10(h)

Figure 7: Convergence and the estimated rate in the displacement norm of the cantilever beam

2

10

T3 ES−FEM−T3 NS−FEM−T3 Q4 AαFEM

1

10

e

log (e )

10

0

10

−1

10

−2

10

0

10

log10(h)

Figure 8: Convergence and the estimated rate in the energy norm of the cantilever beam

11

6.2

Infinite plate with a circular hole

Fig. 6.2 represents a square 2D solid with a central circular hole of radius a = 1m , subjected to a unidirectional tensile load of σ = 1.0N/m at infinity in the x-direction. Due to its symmetry, only the upper right quadrant of the plate is modeled. Plane strain condition is assumed and E = 1.0 × 103 N/m2 , Poisson ratio ν = 0.3. Symmetric conditions are imposed on the left and bottom edges, and the inner boundary of the hole is traction free. The exact solution of the stress for the corresponding infinite solid is [39] 2 4 σ11 = 1 − ar2 32 cos 2θ + cos 4θ + 3a 2r 4 cos 4θ 4 2 σ22 = − ar2 12 cos 2θ − cos 4θ − 3a 2r44 cos 4θ 2 τ12 = − ar2 12 sin 2θ + sin 4θ + 3a 2r 4 sin 4θ

(41)

where (ρ, θ) are the polar coordinates and θ is measured counterclockwise from the positive x-axis. Traction boundary conditions are imposed on the right (x = 5.0) and top (y = 5.0) edges based on the exact solution Eq. (41). The displacement components corresponding to the stresses are h i a r a a3 (κ + 1) cos θ + 2 ((1 + κ) cos θ + cos 3θ) − 2 cos 3θ u1 = 8µ 3 r r ha i (42) a r a a3 u1 = 8µ (κ − 1) sin θ + 2 ((1 − κ) sin θ + sin 3θ) − 2 sin 3θ a r r3

where µ = E/(2(1 + ν)) , κ is defined in terms of Poisson’s ratio by κ = 3 − 4ν for plane strain cases. An illustration of 128 triangular elements is given in Fig. 9. In Fig. 10 and Fig. 11 illustrate the results for displacements along bottom and left boundaries of the methods using the coarse mesh with 128 triangular elements. It is shown that the AαFEM is a strong competitor of the ES-FEM, whereas the FEM-T3 and NS-FEM are less accurate. Note that the displacements of the AαFEM at are even better than those of the FEM-Q4. In Fig. 12 and Fig. 13 exhibit the comparison between the computed stresses using the AαFEM and analytical values. It is observed that the AαFEM solutions are in a good agreement with exact solution and display smooth solutions without using any post-process. In Fig. 15 plot the convergence rates for the error in displacement and energy using . It is observed that (1) the present AαFEM is more accurate than FEM-T3 and even more accurate than FEM-Q4; (2) a convergent solution is obtained for the AαFEM in energy norm; (3) the AαFEM stands out clearly in energy norm measure.

12

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

1

2

3

4

5

6

Figure 9: Domain discretization using 128 triangular elements for the quarter model of the infinite plate with a circular hole

6.3

Free vibration analysis of a cantilever beam

The free vibration analysis presented for a 2-D cantilever beam as shown in Fig. 3. The computation are length L = 100mm, height D = 10mm, thickness t = 1.0mm, Young’s module E = 2.1 × 104 kgf /mm4 , Poisson ratio ν = 0.3, and mass density ρ = 8.0 × 10−10kgf s2 /mm4 . A plane stress problem is considered. The first eight eigen-modes of the beam are show in Fig. 16 and results are listed in Table 1. It is observed that AαFEM convergent much faster than the FEM-T3, FEM-Q4 especially for a very coarse mesh (20 × 2 elements). For 40 × 4 AαFEM achieves nearly comparable accuracy as the corresponding reference solution. In Table 2 are estimate error with EulerBernoulli beam theory, we can see that the AαFEM errors are slightly smaller than other methods.

6.4

Free vibration analysis of tapered cantilever plate with central circular hole

Consider the tapered cantilever plate with a central circular hole, shown in Fig. 17. The side length L = 10 and the radius R = 1.5. The following material property parameters are used in the analysis: with mass density ρ = 1, Youngs modulus E = 1 and Poissons ratio ν = 0.3, and plane stress condition is assumed. The results are listed in Table 3 and the eight modes show in Fig. 18. We can see that method estimate error the natural frequency of the tapered cantilever are much smaller than FEM-T3 as well as reference solution. Furthermore, this shows that the AαFEM using triangular elements can be applied to the vibration analysis with excellent accuracy.

6.5

Free vibration analysis of a shear wall

A shear wall with four openings (see Fig. 19) is analyzed, which has been solved using BEM by Brebbia [6]. The bottom edge is fully clamped. Plane stress case is considered with E = 10 × 103 N/m2 ,ν = 0.2 ,t = 1.0m,ρ = 1.0N/m3 . The natural frequencies of the first 8 modes are calculated show in Fig. 20 and results show in Table 4. In Table 4 lists the first eight natural frequencies and the first eight modes using the AαFEM are demonstrated in Fig. 20. The natural frequencies obtained using the AαFEM is larger than those of the 13

−3

x 10

Exact solu. FEM−Q4 FEM−T3 NS−FEM ES−FEM AαFEM

Horizontal displacement ux

5

4.5

4

3.5

3

2.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x (y=0)

Figure 10: Horizontal displacement of the infinite plate with a hole along the bottom boundary

−3

−0.6

x 10

Exact solu. FEM−Q4 FEM−T3 NS−FEM ES−FEM AαFEM

−0.8

Vertical displacement uy

−1

−1.2

−1.4

−1.6

−1.8

−2

−2.2

−2.4

1

1.5

2

2.5

3

3.5

4

4.5

5

y (x=0)

Figure 11: Vertical displacement of the infinite plate with a hole along the left boundary

14

3

Analytical solution AαFEM

2.8 2.6

Stress σx

2.4 2.2 2 1.8 1.6 1.4 1.2 1

1

1.5

2

2.5

3

3.5

4

4.5

5

y(x=0)

Figure 12: Distribution of stress along the left boundary (x = 0) of the infinite plate with a hole subjected to unidirectional tension

0.4

Analytical solution AαFEM

0.35

0.3

Stress σy

0.25

0.2

0.15

0.1

0.05

0

1

1.5

2

2.5

3

3.5

4

4.5

5

y(x=0)

Figure 13: Distribution of stress along the bottom boundary (y = 0) of the infinite plate with a hole subjected to unidirectional tension

15

Exact energy FEM−T3 FEM−Q4 NS−FEM ES−FEM AαFEM

0.0119

0.0118

Strain energy

0.0118

0.0118

0.0118

0.0118

0.0117

0.0117

10

15

20 Mesh index

25

30

Figure 14: Solution bounds of energy for infinite plate with a circular hole

1

10

FEM−T3 ES−FEM NS−FEM FEM−Q4 AαFEM

0

log10(ed)

10

−1

10

−2

10

−0.9

10

−0.8

10

−0.7

10

−0.6

10

−0.5

10

−0.4

10

−0.3

10

log10(h)

Figure 15: Convergence and the estimated rate in the displacement norm of the infinite plate with a circular hole

16

Table 1: First eight natural frequencies of a cantilever beam No.elements

FEM-T3

FEM-Q4

ES-FEM

MLPG [16]

NBNM [32]

AαFEM

(20 × 2)

1119 6643 12852 17306 31173 38686 47342 64769

924 5561 13484 14732 27003 40502 41636 58075

853 5078 12828 13246 23783 35784 38298 48533

824.44 5070.32 12894.73 13188.12 24044.43 36596.15 38723.90 50389.01

844.19 5051.21 12827.60 13258.21 23992.82 36432.15 38436.43 49937.19

867.80 5228.45 12833.66 13858.33 25366.81 38473.60 38977.74 54022.01

907 5431 12834 14286 25949 38511 39612 54647

879 5267 13467 13863 25193 38456 40370 53043

827 4950 12826 13006 23554 35778 38408 49029

824.44 5070.32 12894.73 13188.12 24044.43 36596.15 38723.90 50389.01

844.19 5051.21 12827.60 13258.21 23992.82 36432.15 38436.43 49937.19

827.62 4968.00 12826.57 13099.22 23836.27 36409.61 38453.30 50215.89

80 triangular elements

(40 × 4) 320 triangular elements

Table 2: Natural frequencies of a cantilever beam Mode

FEM-T3

FEM-Q4

ES-FEM

MLPG [16]

Nagashima [32]

AαFEM

Euler beam

1 Error with Euler beam(percent)

907 9.594

824 5.841

827 -0.072

824 -0.435

844.19 1.959

827.62 -0.0036

827.65 -

2 Error with Euler beam(percent)

5431 4.709

4944 4.681

4950 -4.565

5070 -2.251

5051.21 -2.683

4968.00 -4.2178

5186.77 -

17

MODE 1, FREQUENCY = 822.0314 [Hz]








18 Figure 16: First eight modes of a cantilever beam by the AαFEM

Table 3: First eight frequencies (rd/s) of the tapered cantilever plate Frequency (rad/s)

FEM-T3

Zhao [42]

AαFEM

Exact solution [41]

Mode 1 Error(percent)

0.0773 3.9

0.0833 11.9624

0.0731 -1.75

0.0744 -


0.1699 3.09

0.1661 0.7888

0.1638 -0.61

0.1648 -


0.2223 8.18

0.2334 13.5766

0.2117 3.02

0.2055 -


0.3055 3.95

0.3302 12.3511

0.2779 -5.44

0.2939 -


0.3531 8.88

0.3525 8.6957

0.3209 -1.05

0.3243 -


0.4763 8.27

0.4234 -3.7509

0.4441 0.95

0.4399 -


0.4821 7.56

0.4624 3.1682

0.4499 0.38

0.4482 -


0.5421 9.01

0.4805 -3.3782

0.5148 3.52

0.4973 -

19

Figure 17: Tapered cantilever plate with central circular hole (E = 1.0, ν = 0.3, t = 1.0,ρ = 1.0) FEM-T3 that is known overly-stiff. The AαFEM results are generally closest to the reference solution, and they converges faster even than the FEM-T3 with the same number of nodes used.

6.6

Free vibration analysis of a connecting rod

A free vibration analysis of a connecting rod is performed as shown in Figure Fig. 29. Plane stress problem is considered with E = 1.0 × 1010 N/m2 , ν = 0.3, ρ = 7.8 × 103kg/m3 . The nodes on the left inner circumference are fixed in two directions. The AαFEM results are better than FEM-T3 show in Table 5 and Fig. 22. In addition, the AαFEM using triangular elements can be applied to the vibration analysis with excellent accuracy.

6.7

Free vibration analysis of a machine part

The eigenvalue analysis for a machine part designed by CAD, shown in Fig. 23. For the plane stress analysis was performed for an in-plane load with E = 2.1 × 104kgf /mm2, ν = 0.3, ρ = 8.0 × 10−10kgf s2 /mm4 , t = 1.0mm. The natural frequencies of the first eight modes are calculated show in Fig. 24 and results show in Table 6.

6.8

Forced vibration analysis of a cantilever beam

A benchmark cantilever beam is investigated using the Newmark method. It is subjected to a tip harmonic loading f (t) = 1000¯ g(t) in y-direction.Plane strain problem is considered with non dimensional numerical parameters as L = 48, D = 12, t = 1, E = 3×7 , ν = 0.3. The time step ∆t = 1 × 10−3 is used for time integration. From the dynamic responses in Fig. 27 it is seen that the amplitude of the AαFEM seems closer to that of the FEM-T3.

20

MODE 1, FREQUENCY = 0.073057 [rad/s]








21 Figure 18: First eight modes of the tapered cantilever plate by the AαFEM

Figure 19: A shear wall with four openings (E = 10 × 103 N/m2 , ν = 0.2, t = 1.0m,ρ = 1.0N/m3 )

Table 4: First eight frequencies (rd/s) of a shear wall No.elements



FEM-T3

FEM(ABAQUS) [16]

Brebbia [6]

MLPG [16]

AαFEM

2.146 7.342 7.653 12.601 16.079 18.865 20.525 22.793

2.073 7.096 7.625 11.938 15.341 18.345 19.876 22.210

2.079 7.181 7.644 11.833 15.947 18.644 20.268 22.765

2.069 7.154 7.742 12.163 15.587 18.731 20.573 23.081

2.0270 7.0054 7.6085 11.6181 15.1223 18.1770 19.6959 22.0139

2.0634 7.0926 7.6206 11.8926 15.3559 18.3444 19.8988 22.2439

2.073 7.096 7.625 11.938 15.341 18.345 19.876 22.210

2.079 7.181 7.644 11.833 15.947 18.644 20.268 22.765

2.069 7.154 7.742 12.163 15.587 18.731 20.573 23.081

2.0105 6.9554 7.6011 11.4767 14.9788 18.0807 19.5951 21.8834

22

MODE 1, FREQUENCY = 2.027 [rd/s]








23 Figure 20: First eight frequencies of the shear wall by the AαFEM

Figure 21: Geometric model of an automobile connecting rod(E = 1.0 × 1010 N/m2 , ν = 0.3, ρ = 7.8 × 103 kg/m3 )

Table 5: First eight natural frequencies (Hz) of a Connecting rod No.elements



FEM-T3

FEM-Q4 [1]

FEM-Q8 [43]

ES-FEM

AαFEM

5.454 23.466 49.803 55.516 99.715 116.243 147.192 166.975

5.1369 22.050 49.299 52.232 93.609 108.59 134.64 159.45

5.1222 21.840 49.115 51.395 91.787 106.153 130.146 156.142

5.1368 22.0595 49.3809 52.0420 92.7176 109.5887 132.6795 158.2376

5.0865 21.8690 49.2016 51.6044 92.3010 107.9777 132.3226 158.0875

5.209 22.291 49.360 52.592 94.154 109.474 135.321 160.165

5.124 21.909 49.211 51.657 92.390 107.51 131.48 157.51

5.1222 21.840 49.115 51.395 91.787 106.153 130.146 156.142

5.1246 21.8805 49.1726 51.5181 91.9305 106.8473 130.5546 156.3497

5.1082 21.8191 49.1125 51.3580 91.7107 106.3088 130.2115 156.1473

24









25 Figure 22: First eight modes of a connecting rod by the AαFEM

Figure 23: A machine part (E = 2.1 × 104 kgf /mm2, ν = 0.3, ρ = 8.0 × 10−10 kgf s2 /mm4 , t = 1.0mm)

26









27

Figure 24: First eight modes a machine part by the AαFEM

Table 6: First eight natural frequencies (Hz) of a machine part No.nodes



FEM-T3

FEM [33]

NBNM [32]

AαFEM

954.208 1684.134 4602.111 10383.386 11498.753 17898.438 22907.050 24794.438

909.09 1639.45 4434.00 9944.57 11209.39 17522.44 -

906.50 1640.04 4426.57 9932.99 11226.98 17516.87 -

899.5509 1613.7050 4372.3597 9831.4070 11192.9315 17337.4634 22376.2442 24057.6648

961.745 1692.864 4611.028 10363.961 11539.766 17961.157 22941.550 24878.957

909.09 1639.45 4434.00 9944.57 11209.39 17522.44 -

906.50 1640.04 4426.57 9932.99 11226.98 17516.87 -

899.3090 1614.9336 4377.1788 9833.8197 11182.8249 17349.3836 22370.2123 24034.7601

Figure 25: A cantilever beam (E = 3 × 107 , ν = 0.3, t = 1.0,ρ = 1)

28

0.025 −3

FEM−T3(∆ t=1x10 ) 0.02

−3

FEM−Q4(∆ t=1x10 ) AαFEM(∆ t=1x10−4)

0.015

−3

AαFEM(∆ t=1x10 ) −2

AαFEM(∆ t=1x10 )

Displacement u

y

0.01 0.005 0 −0.005 −0.01 −0.015 −0.02

0

0.05

0.1 Time t

0.15

0.2

Figure 26: Displacement uy at point A using Newmark method (δ = 0.5, β = 0.25, g(t) = sinωt)

0.3

0.2

Displacement u

y

0.1

0

−0.1 −3

FEM−T3(∆ t=1x10 ) −3

−0.2

FEM−Q4(∆ t=1x10 ) −4

AαFEM(∆ t=1x10 ) −3

AαFEM(∆ t=1x10 )

−0.3

−2

AαFEM(∆ t=5x10 ) −0.4

0

0.5

1 Time t

1.5

2

Figure 27: Displacement uy at point using Newmark method (δ = 0.5, β = 0.25, g(t) = sinωt)

29

0.02 FEM−T3 FEM−Q4

0.015

−3

AαFEM(∆ t=1x10 ,c=0) 0.01

Displacement u

y

0.005 0 −0.005 −0.01 −0.015 −0.02 −0.025

0

0.5

1 Time t

1.5

2

Figure 28: Transient displacement uy at point A using Newmark method (δ = 0.5 and β = 0.25)

0.01 −3

AαFEM(∆ t=5x10 ,c=0.4) 0.005

Displacement u

y

0

−0.005

−0.01

−0.015

−0.02

−0.025

0

5

10 Time t

15

20

Figure 29: Transient displacement uy at point A using Newmark method (δ = 0.5 and β = 0.25)

30

7

Conclusions

The method applied to static, free and force vibrations analysis of two dimensional solids and structural have seen presented in this paper. This paper presents carefully a designed procedure to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter α. The method has the following attractive features: (1) The numerical results of the AαFEM using triangular elements are even more accurate than the FEM using triangle and quadrilateral elements with the same number of nodes; (2) In the natural frequency analysis it is found that the AαFEM gives much more accurate results and higher convergence rate than the corresponding FEM-Q4; (3) The AαFEM is easy to implement into a finite element program using triangular meshes that can be generated with ease for complicated problem domains. In addition, it is promising to extend the present method for the 3D problems and the plate and shell problems by combining the AαFEM with DSG method [5] to get rid of shear locking and to improve the accuracy of solutions. It is also promising to maintain accuracy in a local region and to improve the rough solution of strong discontinuities in fracture structures by coupling the AαFEM with the extended finite element method XFEM .

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