A Comparative Study of Least-Squares and the Weak- Form Galerkin Finite Element …journals.iau.ir/article_514360_26f9d2cf13309ce7b8237772c... · 2020-03-31 · Form Galerkin Finite

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Journal of Solid Mechanics Vol. 2, No. 2 (2010) pp. 101-114

A Comparative Study of Least-Squares and the Weak-Form Galerkin Finite Element Models for the Nonlinear Analysis of Timoshenko Beams

W. Kim1, J.N. Reddy2,* 1Department of Mechanical Engineering, Korea Army Academy at Yeong cheon, Yeong cheon, 770-849 South Korea 2Department of Mechanical Engineering, Texas A&M University, College Station, TX 77840, USA

Received 19 June 2010; accepted 17 July 2010

ABSTRACT In this paper, a comparison of weak-form Galerkin and least-squares finite element models of Timoshenko beam theory with the von Kármán strains is presented. Computational characteristics of the two models and the influence of the polynomial orders used on the relative accuracies of the two models are discussed. The degree of approximation functions used varied from linear to the 5th order. In the linear analysis, numerical results of beam bending under different types of boundary conditions are presented along with exact solutions to investigate the degree of shear locking in the newly developed mixed finite element models. In the nonlinear analysis, convergences of nonlinear finite element solutions of newly developed mixed finite element models are presented along with those of existing traditional model to compare the performance.

© 2010 IAU, Arak Branch. All rights reserved.

Keywords: Finite element models; Least-squares model; Weak-form Galerkin model; Geometric nonlinearity; Timoshenko beam theory

1 INTRODUCTION

OR several decades, finite element models based on the weak-form Galerkin or Ritz formulation [1] have dominated both commercial finite element software and academic research. An advantage of the weak-form

Galerkin model as applied to structural mechanics problems is that it is the most natural approach resulting from the application of the principle of virtual displacements. For problems outside of solid and structural mechanics, such principles are not available. It is possible, however, to develop the so-called weak forms or weighted-residual integral statements from the governing differential equations of any physical problem. The weak forms, in most cases, are not equivalent to any principle or minimization of error introduced in the approximation of the variables. Thus, the weak forms of problems for which there is no underlying physical or mathematical principle, are merely a means to computing solutions of the integral statements represented by the weak forms [2]. Consequently, it is possible that such solutions may degenerate for certain combinations of physical characteristics of the problem and finite element approximations used. It is well known that the weak formulations lower the differentiability requirements on the variables of the formulation. The traditional displacement based (i.e., principle of virtual displacements) finite element models can provide high level of accuracy only for the generalized displacements, increasing errors in the post-computed variables involving the derivatives of the generalized displacements. To remedy this, the so-called mixed formulations [3] are adopted. The mixed models include secondary variables as independent degrees of freedom and they yield increased accuracy of the secondary variables, sometimes at the expense of the primary variables and increased computational effort. Thus weak-form, mixed, Galerkin finite element models can provide higher level of accuracy for secondary variables included in the model. But to establish

‒‒‒‒‒‒ * Corresponding author. Tel.: 979 862 2417; fax: 979 845 3081. E-mail address: [email protected] (J.N. Reddy).

F

102

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A Comparative Study of Least-Squares and the Weak-Form Galerkin Finite Element Models for … 103

© 2010 IAU, Arak Branch

20 0 00.5 ( ), 0.5 ( )xz x xx xw u w z ¢ ¢ ¢ ¢= + = + + (2)

The constitutive relations are

, 2xx xx xz xzE G = = (3) where E is Young’s modulus and G is shear modulus. The stress resultants are

20 0d d [ 0.5 ( ) ]xx xx

AN y z EA u w ¢ ¢= = +ò

0 d d ( )xz s xA

Q y z K GA wx

¢= = +ò

d dxx xx xA

M z y z EI ¢= =ò

(4)

where the Ks (5/6) is shear correction factor, the Nxx is axial force, the Vx is the transverse shear force, and Mxx is bending moment at any cross section of the beam. The equilibrium equations of the TBT are

0( ) 0, ( ) ( ) 0, 0xx x xx xx xN f x Q w N q x M Q¢ ¢ ¢ ¢+ = + + = - = (5) where f(x) and q(x) are distributed axial force and transverse loads, respectively. The resultant forces and bending moment can be included in the finite element models to develop mixed finite element models, which will be discussed in the next section.

3 FINITE ELEMENT MODELS

Traditionally, the three equilibrium equations in (5) are written in terms of the generalized displacements (u0, w0, x) to develop the displacement finite element model of the TBT. Here, we discuss weighted-residual finite element models of these equations. Since we approximate the variables of the formulation, the differential equations in (4) and (5) are not satisfied exactly everywhere in the domain. Thus, there is error introduced into the differential equations, called residuals. All approximation methods try to minimize the residuals in some suitable sense. For example, consider the three equations in Eq. (5), expressed in terms of the displacements. The residuals in the three equations are

1 0 02{ [ 0.5 ( ) ]} ( ) 0R EA u w f x¢ ¢=- + - ¹

22 0 0 0 0{ ( ) [ 0.5 ( ) ]} ( ) 0s xR K GA w w EA u w q x ¢ ¢ ¢ ¢ ¢=- + + + - ¹

3 0 ( ) 0x s xR EI K GA w ¢¢ ¢=- + + ¹

(6)

The resulting weighted-residual or weak-form model will contain only the displacement variables. On the other

hand, one may also consider Eqs. (4) and (5) to be six independent equations and treat three displacements (u0, w0, x) and there stress resultants (Nxx, Vx, Mxx) as independent variables. The resulting finite element model is termed a mixed model of the TBT because it contains displacements as well as stress resultants:

1 ( ) 0,xxR N f x¢= + ¹

2 0( ) ( ) 0,x xxR Q w N q x¢ ¢= + + ¹

3 0,xx xR M Q¢= - ¹ 1

4 0( ) 0,( ) xs xR Q wK GA - ¢= - + ¹ 1

5 0 0 0( 0.5 ) 0,( ) xxR N u w wEA - ¢ ¢ ¢= - + ¹ 1

6 ( ) 0.xx xR EI M - ¢= - ¹ (7)

104 W. Kim and J.N. Reddy


An advantage of mixed formulations is that they yield increased accuracy of the resultants and they can be computed at the nodes as opposed to the Gauss points in displacement finite element models. As a special case of the weighted -residual method, one may construct least-squares finite element models. In Eq. (7), one can find that the differentiability requirements on u0, w0 and x are lowered by inclusion of the stress resultants Nxx, Vx and Mxx.

3.1 Traditional displacement based Galerkin weak -form model

In this section, a traditional displacement based Galerkin model is reviewed to provide better comparison of the characteristics and behavior of the newly developed mixed models herein. The traditional displacement based model includes only u0, w0 and x as nodal variables, yields the smallest stiffness matrix. The weak formulation of the traditional displacement model is based on the statements [1, 3]

[ ]

[ ]

[ ]

20 0 0 1 00

20 0 0 0 0 0 2 00

0 3 00

[ [ 0.5( ) ] ( ) ] d

{ ( ) [ 0.5(

(

) ] ( ) } d

[ )] d

e e

e e

e e

L L

o o

L L

s x o o

L L

x x s x x x

EA u u w f x u x u Q

K GA w w w EA w u w q x w x w Q

EI K GA w x Q

¢ ¢ ¢+ - -

¢ ¢ ¢ ¢ ¢ ¢+ + + - -

¢ ¢ ¢+ + -

ò

ò

ò

δ δ δ

δ δ δ δ

δ δ δ

(8)

where Q1Nxx, 2 0x xx xQ Q w N V¢= + = , and Q3Mxx at the boundaries. In the traditional displacement models,

boundary conditions can be imposed exactly only on the displacements and in integral sense on the stress resultants. However, mixed models allow exact imposition of boundary conditions even on the stress resultants that appear in the model. The displacements appearing in Eq. (8) can be approximated by

0 1

l

j jju u

=@å

0 1

m

j jjw w

=@å

1

n

x j jj

=@å

(9)

and substitution into Eq. (8) yields the following finite element equations:

11 12 13 1

21 22 23 2

31 32 33 3

[ ] [ ] [ ] { } { }

[ ({ })]{ } { } or [ ] [ ] [ ] { } { }

[ ] [ ] [ ] { } { }

j

j

j

T T T TK

é ù ì üì ü ï ïï ï ï ïï ïê ú ï ïï ïï ï ï ïê ú =í ý í ýê úï ï ï ïê úï ï ï ïï ï ï ïê úï ï ï ïî þû î þ

=

ë

K K K F

U U F K K K F

K K K F

w

u

(10)

where

11 120

0 0

23 22 20

0 0

21 33 320

0 0

( ) d , (0.5 ) d

( ) d , ( 0.5 ( ) ) d

( ) d , ( ) d ,

L L

ij i j ij i j

L L

ij s i j ij s i j i j

L L

ij i j ij i j s i j ij j

K EA x K EAw x

K K GA x K K GA EA w x

K EAw x K EI K GA x K K

¢ ¢ ¢ ¢ ¢= =

¢ ¢ ¢ ¢ ¢ ¢= = +

¢ ¢ ¢ ¢ ¢= = + =

ò ò

ò ò

ò ò 23

1 2 31 2 20 0 00

0

[ ( ) ] d , [ ( ) ] d and [ ] [ ] [ ]e e e

i

LLL L L

i i i i i i i if f x x Q f q x x Q f Q = + = + =ò ò



3.2 Mixed Galerkin model

For a mixed finite element model, the principle of minimum potential energy and the Lagrange multiplier method can be employed simultaneously to minimize the total potential energy and include the stress resultant-displacement relations as constrains through the Lagrange multiplier method. For the problem at hand, the total potential energy functional of the TBT beam can be expressed as

0 00

[0.5( ) ] deL

c xx xx xz xzL fu qw x = + - -ò (11)

where the Lc is the potential energy functional. Then minimizing Lc subject to the constraints can be expressed as

3

0 00

1

1 2 1 2 1 20 0

0

1 11 0 0 0 2 0

0 0

, , , , , d

[0.5( ) 0.5(

( )

[( ) ] (

) 0.5( ) ] d

( 0.5 ) d () ) d

e

e

e e

L

l x x x x c k kk

L

x xx s x

L L

xx s x x

L u w N Q M L G x

EA N EI M K GA Q fu qw x

EA N u w w x K GA Q w x

=

- - -

- -

æ ö÷ç= + ÷ç ÷çè ø

= + + - -

é ù¢ ¢ ¢ ¢+ - + + - +ê úë û

å ò

ò

ò ò

13

0( )[ ] d

eL

xx xEI M x - ¢+ -ò (12)

where Ll is the mixed functional and i are the Lagrange multipliers. The condition for the minimum of Ll is Ll0, where the is variational operator. Then we can obtain the following relations, which can be used to develop the mixed Galerkin finite element model. Since Ll contains 6 variables and their first derivatives only, coefficients of variations of the variables can be computed from the Euler-Lagrange equations [2] as follows:

10

0 0

2 1 00

0 0

2 30

1

: ( ) d 0

: [ ( ) ] d 0

: ( ) d 0

: [( ) ( )

e

e

e

Ll l

o

Ll l

o

Ll l

xx x

lxx xx

xx

L Lu f x

u x u

L Lw q w x

w x w

L Lx

x

LN EA N EA

N

- -

æ ö¶ ¶¶ ÷ç ¢÷- = - - =ç ÷ç ÷ç ¢¶ ¶ ¶è øæ ö¶ ¶¶ ÷ç ¢ ¢ ¢÷- = - - - =ç ÷ç ÷ç ¢¶ ¶ ¶è øæ ö¶ ¶¶ ÷ç ¢÷- = - =ç ÷ç ÷ç ¢¶ ¶ ¶è ø

¶= -

¶

ò

ò

ò

δ

δ

δ

δ 11

0

1 12

0

1 13

0

] d 0

: [( ) ( ) ] d 0

: [( ) ( ) ] d 0

e

e

e

L

Ll

x s x sx

Ll

xx xxxx

x

LQ K GA Q K GA x

Q

LM EI M EI x

M

- -

- -

=

¶= - =

¶

¶= - =

¶

ò

ò

ò

δ

δ

(13)

The Lagrange multipliers can be chosen to be 1Nx, 2Qx and 3Mx satisfying all conditions given in Eq. (7). By using the Lagrange multiplier method, all governing equations of the TBT beam can be fully recovered. Thus, we can develop mixed Galerkin finite element model with properly chosen weight functions for the residuals. We have

0 0 0

0

1 10 0 0 0

( ) { ( ) [ ] ( )

(

, , , , , ( )

( 0.5) ) ( ( )

d 0

)

( )}

eL

l x x x x o x o x x x x x

xx xx x s x x

xx x x

L u w N Q M u f N w q Q N w Q M

N EA N u w w Q K GA Q w

M EI M x

- -

¢ ¢ ¢ ¢= - - + - - + + -

é ù é ù¢ ¢ ¢ ¢+ - + + - +ê ú ê úë û ë û¢+ - =

òδ δ δ δ

δ δ

δ

(14)



Note that following relation [3] can be used to lower differentiability requirement for the w0 simplifying force vector terms in the matrix form of finite element equations.

0 , x x xx x x xxV Q w N Q V w N¢ ¢= + = - (15)

In the least squares method, which will be discussed in the next section, use of the relation Eq. (15) will allow us to have only first-order differential equation, and to construct simple element-wise force vectors that contain linear terms only. In Eq. (15), Vx is the shear resultant on the undeformed edge, and Qx is the shear force on the deformed edge. We write

0 0

0

0

1 10 0 0 0 0 0

1

, , , , ,

( ) ( )

( 0.5 ) ( )[( ) ( )]

( )

{ [ ] [ ] ( )

[( ) ] ( )

[( d 0) ]}

e

l x x x x

L

o xx o x x xx x xx

xx xx x xx s x xx x

xx xx x

L u w N V M

u N f x w V q x M V w N

N EA N u w w V w N K GA V w N w

M xEI M

- -

-

¢ ¢ ¢ ¢= + + + + - +

¢ ¢ ¢ ¢ ¢ ¢+ - + + - - - +

¢+ - =

ò

δ

δ δ δ

δ δ δ

δ

(16)

Then the variables and their variations in Eq. (16) can be approximated as linear combinations of nodal values and known interpolation functions, as

01 1 1 1 1

1 1

, , , , ,

,

p ql m n

o j j j j x j j xx j j j jj j j j j

q r

x i i xx j ji j

u w N

V M

= = = = =

= =

@ @ @ @ @

@ @

å å å å å

å å

u w N V

V M

(17) where , , , , u w N Q and M are nodal unknowns of the finite element model, the i and the j are ith and jth are

Lagrange type interpolation functions. By substituting all the approximations given in Eq. (17) into the function Ll of Eq. (16) and collecting coefficients of the variation of the nodal unknowns (i.e. , , , , u w N Q and M ),

the following finite element equations can be obtained:

[ ]

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ]({ })

[ ] [{ } {

] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [

} or

] [

G G G G

K K K K K K

K K K K K K

K K K K K KU

K K K K K K

K K K K K K

K K K K K K

=K U F

1

2

3

4

5

66 6

{ } { }{ } { }{ } { }

{ } { }{ } { }{ }] { }

j

j

j

j

j

j

F

F

F

F

F

F

ì üé ù ì üï ï ï ïï ï ï ïê úï ï ï ïï ïê ú ï ïï ï ï ïê úï ï ï ïï ï ï ïê úï ï ï ïï ïê ú ï ïï ï=í ý í ýê úï ï ï ïê úï ï ï ïï ï ï ïê úï ï ï ïï ï ï ïê úï ï ï ïê úï ï ï ïï ï ï ïê úï ï ï ïë û î þï ïî þ

u

w

N

V

M

(18)

where KG is element stiffness matrix of the mixed Galerkin model, UG is the column vector of unknowns, and FG is the force vector. Definitions of all nonzero stiffness and force coefficients are given below.

14 25 36

0 0 0

35 34 44 10

0 0 0

41 42 43 340

0 0

44

( ) d , ( ) d , ( ) d ,

( ) d , ( ) d , [( ) ] d ,

( ) d , ( 0.5 )

(

d ,

[

,

L L L

ij i j ij i j ij i j

L L L

ij i j ij i j ij i j

L L

ij i j ij i j ij ji

ij s

K x K x K x

K x K w x K EA x

K x K w x K K

K K GA

-

¢ ¢ ¢= = =

¢= - = =

¢ ¢ ¢= - = - =

=

ò ò ò

ò ò ò

ò ò1 2 45 1

0 00 0

] d , [ ( ) ] d) ,L L

i j ij s i jw x K K GA w x - -¢ ¢= -ò ò



52 53 54 450

00

55 1 66 1 63

0 0 0

1 2

0 0

( ) d , ( ) d ,

[( ) ] d , [( ) ] d , ( ) d

[ ( ) ] d and [ ( ) ] d

LL

ij i j i j ij i j ij ji

L L L

ij s i j ij i j ij i j

L L

i i i i

K w x K x K K

K K GA x K EI x K x

f f x x f q x x

- -

¢ ¢ ¢= - = - =

¢= = = -

= - = -

ò ò

ò ò ò

ò ò

(19)

One advantage of this mixed model is the use of equal interpolation functions for all variables. In the displacement model, which was reviewed in Subsection 3.1, each variable should be approximated with consistent interpolation or reduced integrations must be used to prevent shear and membrane locking. However, increase of the number of unknowns is inevitable in mixed models.

3.3 Mixed least-squares model

In the least-squares method, the sum of the squares of the residuals in the governing equations is minimized. Using relations given in Eq. (15), the residuals in Eq. (7) can be expressed as

1

2

3 0

14 0 0

15 0 0 0

16

( )

( ) 0

( ) 0

0

( ) ( ) 0

( ) ( 0.5 ) 0

( ) 0

xx

x

xx x xx

s x xx x

xx

xx x

R N f x

R V q x

R M V w N

R K GA V w N w

R EA N u w w

R EI M

-

-

-

¢= + ¹

¢= + ¹

¢ ¢= - + ¹

¢ ¢= - - + ¹

¢ ¢ ¢= - + ¹

¢= - ¹

(20)

Unlike weak-form Galerkin model, the least-squares finite element model can be developed directly from Eq. (20) without consideration of the weight functions because they are naturally defined [5-8]. But before considering the development of the mixed least-squares finite element model, we need to scale the residuals so that all of them have the same magnitudes [5]. For example, the units of R1 and R2 are forced per length while unit of the R3 is force. Residual R6 has unit of force times length while residuals R4 and the R5 have no dimension. When the residuals are of different magnitude, the convergence of each residual to zero is dictated by their relative magnitudes (residuals with large magnitudes converge faster than those with less magnitude). To make all terms in the least-squares functional to be dimensionally the same, the following weights are chosen [9]:

3 21 0 0 2 30 0 0 0 04 5 6 / / /, , , 1,L EA L EI L EI L = = = = = = (21)

where L0, A0 and I0 are characteristic length, characteristic cross-sectional area, and characteristic second moment of inertia about the y-axis, respectively. In the present study, L0, A0 and I0 are chosen to be 50.0 in, 1.0 in2 and 1/12 in4, respectively. Then, the least-squares functional becomes

1 22 2 2 2 2 2

1 32 3 4 5 6 60

4 5( , , , , , ) ( ) deL

o o x x x xI u w N V M R R R R R R x = + + + + +ò (22)

To obtain symmetric coefficient matrix of in the nonlinear mixed least-squares finite element model,

linearization of nonlinear terms in is important. The linearization can be explained by separating the residuals including nonlinear terms (i.e. R3, R4 and R5) and linear terms (i.e. R1, R2 and R6). Then, the least-squares functional I can be rewritten as

2

1

1( , , , , , ) [ ] d

2

x

o o x x x xx

I u w N V M x = +ò (23)

where



2 2 2 2 2 21 21 2 6 3 4 56 3 4 5( ), ( )R R R R R R = + + = + +

We can minimize the above least-squares equation, which includes nonlinear terms, by taking first variation of it

and setting it to zero:

2

1

2

1

2

1

2 2 23 3 3 4 4 4 5 5 5

2 ' 13 0 0 0

2 10 0 0 4 0 0 0

0 ( , , , , , ) ( ) d

( ) d

{ ( )

( 0.5 ) ( )

( )( )[

] [ ( )][(

x

o o x x x xx

x

x

x

xx x xx xx xx x xx xxx

xx s

I u w N V M N x

R R R R R R x

M V w N w N M V w N EA N

u w w EA N u w w K

-

-

= = +

= + + +

¢ ¢ ¢ ¢= + - + + - + -

¢ ¢ ¢ ¢ ¢ ¢+ + - +

ò

ò

ò

10 0) ( )) ](x xx xGA V w N w- ¢ ¢- - +

2 15 0 0 0([( ) ( ] d }) )s x xx xx xK GA V w N w N w x - ¢ ¢ ¢+ - - - + (24)

But if we linearize I (i.e., treat 0w¢ as known from the previous iteration) before taking its variation, then we have

2

1

2

1

23 0 0

2 1 14 0 0 0 0 0 0

2 1 15 0 0

, , , , , ( ) d

{

( ) ( 0.5 ) ( ) ( 0

( )

( )( )

[ ][ ]

[( ) ( ) ][ (

.

)

5 )

( ()

x

o o x x x xx

x

xx x xx xx x xxx

xx xx

s x xx x s x

I u w N V M N x

M V w N M V w N

EA N u w w EA N u w w

K GA V w N w K GA V

- -

- -

= +

¢ ¢ ¢ ¢= + - + - +

¢ ¢ ¢ ¢ ¢ ¢+ - + - +

¢ ¢ ¢+ - - + -

ò

ò

0 0) ]( ) d}xx xw N w x ¢- +

(25)

where quantities with bars indicate that they are linearized values. Then 0I = is not the same condition as 0I = unless

( )( )

2 2 13 0 0 4 0 0 0 0 0

12 ' 15 0 0

0 [ ]

[( ) ( ) ]

.5 ( ) ( 0.5 )

( ) 0

xx xx x xx xx

s o xx s x xx x

w N M V w N w w EA N u w w

K GA w N K GA V w N w

-

- -

¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢- + + - +

¢ ¢- - - + =

(26)

Since the sufficient condition for the minimum of I is 2 0I > ,

2

1

2 23 0 0 0

2 1 14 0 0 0 0 0 0

2 1 15 0 0 0 0

( )( )

[ ][

{

( ) ]

[( ) ( ) ][

( ) ( ) ( 0.5 )

( )

(

( ) ( )

x

xx x xx xx xx x xxx

xx xx

s x xx xx x s x xx

x

I M V w N w N M V w N


K GA V w N w N w K GA V w N

w

- -

- -

¢ ¢ ¢ ¢ ¢= + - + + - +

¢ ¢ ¢ ¢ ¢ ¢+ - + - +

¢ ¢ ¢ ¢+ - - - + -

¢- +

ò

0 ) } ] d 0x >

(27)

Thus, in order for 2 0,I > we must have

( )

2 2 13 0 0 4 0 0 0 0 0

12 15 0 0 0

0.5 [( ) ( 0.5( )

[( ) ( ) ( ]

)]

) 0

xx xx x xx xx

s xx s x xx x

w N M V w N w w EA N u w w

K GA w N K GA V w N w

-

- -

¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢- + + - +

¢ ¢ ¢- - - + ³

(28) or Eq. (26) should hold. The linearization of the least-squares function before taking its first variation results in

0I = and 2 0,I > which does not alter the minimizing conditions of 0I = and 2 0I > as given in Eq. (26). Now we can develop symmetric bilinear form of the mixed finite element model from the first variation of the linearized I



2 2 2 '1 2 3 0 0

2 1 14 0 0 0 0

0

0 0

2 1 15 0 0

0 { ( )( )

[ ]

( ) ( )

( ) ( 0.5 ) ( ) ( 0.5 )

( ) (

[ ]

[ ) (( ) ][ ()

e

xx xx x x xx x xx xx x xx

xx xx

s x xx x x

L

s

N N f x V V q x M V w N M V w N


K GA V w N w K GA V

- -

- -

é ù é ù¢ ¢ ¢ ¢ ¢ ¢ ¢+ + + + - + - +ë û ë û

¢ ¢ ¢ ¢ ¢ ¢+ - + - +

¢ ¢ ¢+ - - +

=

-

ò

0 0

2 1 16

) ]( )

( ) ( ) } d .[ ][ ]

xx x

xx x xx x

w N w

ME M xI EI

- - -

¢- +

¢ ¢+ -

(29)

Note that all bars are omitted for consistency with the Galerkin mixed model. Then the variables involved in Eq. (29) can be approximated and replaced by using Eq. (17) which are the same functions used in the Galerkin model. By using matrix notation, above equations can be rewritten simply as

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 6

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ]({ })

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [

[ ]{ } { } or

] [ ] [ ] [

L L L L

K K K K K K

K K K K K K

K K K K K K

K K K K K K

K K K K K K

K K K K K

=K U U F

1

2

3

4

5

5 66 6

{ } { }{ } { }{ } { }

{ } { }{ } { }{ }] [ ] { ]

j

j

j

j

j

j

F

F

F

F

F

K F

ì üé ù ì üï ï ï ïï ï ï ïê ú ï ï ï ïï ïê ú ï ïï ï ï ïê ú ï ï ï ïï ï ï ïê ú ï ï ï ïï ïê ú ï ïï ï=í ý í ýê ú ï ï ï ïê ú ï ï ï ïï ï ï ïê ú ï ï ï ïï ï ï ïê ú ï ï ï ïê ú ï ï ï ïï ï ï ïê ú ï ï ï ïë û î þï ïî þ

u

w

N

V

M

(30)

where KL is element stiffness matrix of the mixed least-squares model, UL is the vector of unknowns, and FL is the force vector. The nonzero coefficients of Eq. (30) are defined as

11 2 12 2 14 2 14 4 0 4

0 00d , 0.5 d , ( ) ( ) ( ) ( d ,)

L L

ij i j ij i

L

j ij i jK x K w x K EA x -¢ ¢ ¢ ¢ ¢ ¢= = -=ò ò ò

21 12 22 2 2 2 23 24 0 5 5

0 0( ) (, ) ( ) ( [0.25 ] d , d) ,

L L

ij ji ij i j i j ij i jK K K w x K x ¢ ¢ ¢ ¢ ¢ ¢= = + =ò ò

24 2 1 2 1 25 2 14 0 5 0 5

0 0( ) ([ 0.5 ( ) ( )] d , ( )( ,) d)

L L

ij i j s i j ij s i jK EA w K GA w x K K GA x - - -¢ ¢ ¢ ¢ ¢= - + = -ò ò

32 23 33 2 2 34 2 15 6 5 0

0 0, [ ] d , (( ) ( ) ( ) ) d ,

L L

ij ji ij i j i j ij s i jK K K x K K GA w x -¢ ¢ ¢= = + =ò ò

35 2 1 36 2 15 6

0 0 d , ( ) ( ) ( ) ( ) d ,

L L

ij s i j ij i jK K GA x K EI x - - ¢= - = -ò ò

41 14 42 2 2 1 43 344 0 5 0

0(, [ 0.5 ( ) ( )] d ,) ,

L

ij ji ij i j s i j ij jiK K K w K GA w x K K -¢ ¢ ¢ ¢= = - + =ò

44 2 2 2 2 2 2 2 21 3 0 4 5 0

0( ) ( ) ( )[ ( ( ) ( )) ] ,) d( ( )

L

ij i j i j i j s i jK w EA K GA w x - -¢ ¢ ¢ ¢= + + +ò

45 2 2 2 46 23 0 5 0 3 0

0 0[ ( ) ( )] d , ( ) d ,( )

L L

ij i j s i j ij i jK w K GA w x K w x -¢ ¢ ¢ ¢= - - =ò ò

55 2 2 2 2 56 23 2 5 3

00

( ) ( ) ( ) ([ ] d , ( )) d ,L

L

ij i j i j s i j ij i jK K GA x K x -¢ ¢ ¢= + + = -ò ò

52 25 53 35 54 45, , ,ij ji ij ji ij jiK K K K K K= = =

63 36 64 46 65 56 66 2 2 23 6

0, , , [ ( ) ( ) ] d ,( )

L

ij ji ij ji ij ji ij i j i jK K K K K K K EI x -¢ ¢= = = = +ò

4 21

0

[ ( ) ] dL

i if f x x ¢= -ò and 5 22

0[ ( ) ] d

L

i if q x x ¢= -ò (31)

Comparing Eqs. (30) and (31) with Eqs. (18) and (19), it can be seen that the KL is symmetric while the KG is

not. In Eqs. (30) and (31), force vector contains only linear terms, by replacing Qx with 0 .x xxV w N¢+

110

4 NUME4.1 Descrip

A steel beused, witharchive pro

30E =

Three d

C-C (clamallow mod

The C-elements, athe beam h

have Nxx0polynomia

x and the

displaceme

Fig.2 Geometry

Fig. 3 Description

W. Kim and

ERICAL RESption of proble

am with the gh uniform apploper nonlinear

60 10 psi, ´ =

different typesmped-clamped),deling half of th

C and P-P bouand the H-H bhas no horizon

0. In the traditial orders betwe

e 0w¢ in sK GA

ent based mode

of beam and cho

0 (0) 0w =

(0) 0xxM =

(0) 0xxN =

0 (0) 0w =

0 (0) 0u =

(0) 0x =

0 (0) 0w =

0 (0) 0u =

(0) 0xxM =

of symmetry bo

J.N. Reddy

ULTS em

eometry shownlied distributed convergence.

0.25, 5sK= =

of boundary c, were considehe beam. undary conditioboundary condintal support un

ional displacemeen the 0u¢ and

0( )xA w ¢+ ma

els, performanc

osen coordinate

a. A h

b. A c

c. A p

oundary conditio

n in Fig. 2 wad load q(x), wh

5 / 6, ( ) 0f x =

conditions shoered to evalua

ons can be usedition can be usnder the H-H

ment model, thd the 2

0( )w¢ ter

ay cause shear

ce of beam ele

system for the n

hinged-hinged(H

clamped- clampe

pined- pined(P-P

ons.

as chosen for thich was set t

, ( ) 1.0 ~q x =

wn in Fig. 3, iate the perform

d to test the nosed to determinboundary cond

his condition isrms. Also, the

locking [3, 10

ment is mostly

numerical analys

H-H) beam

ed(C-C) beam

P) beam

the study. The to vary from 1

~1 0.0 lb/in

i.e. H-H (hingmance of the e

onlinear behavine membrane dition, 0 0.u¢ +

s barely satisfi inconsistency

0] which resul

y depends on th

sis [source from

©

properties giv1.0 to 10.0 lb/

ed-hinged), P-elements. All b

ior of newly delocking is exp

20.5( ) 0w¢ = sh

ied because of in polynomia

lts in inaccurat

he relations of

3].

0 ( / 2)u L =

( / 2)x L =

( / 2)xV L =

0 ( / 2)u L =

( / 2)x L =

( / 2)xV L =

0 ( / 2)u L =

( / 2)x L =

( / 2)xV L =

© 2010 IAU, Arak

ven in Eq. (32)/in with ten st

-P (Pined-Pineboundary cond

eveloped beamperienced [10].hould be satisf

the inconsistel orders betwe

te linear soluti

the displaceme

0

0

0

0

0

0

0

0

0

Branch

) were teps to

(32)

d) and ditions

m finite Since fied to

ency in een the

ion. In

ents



and their derivatives, thus they experience sever locking compared with mixed models. Some techniques like reduced integration and use of consistency approximations of the displacement field may be employed, but it is not easy (but possible) to implement them into computers. On the other hand, the locking can be weakened or even eliminated by the inclusion of resultants which can be done by mixed formulations, because certain conditions which cause locking are not solely depend on the relations of displacement but also on the resultant in mixed models. The numerical results that show the effects of mixed formulation are presented in the next sections.

4.2 Linear analysis

In this section linear solutions of the newly developed mixed models are compared with known exact solutions and solutions of traditional displacement based model. By taking all nonlinear terms to be zero, we can eliminate geometric nonlinearity in the beam element to study linear behavior of beam element. Linear study is important because finding nonlinear solutions can be started from linear solutions in most of structural problems. Linear study of the TBT beam bending includes interesting phenomena like shear locking. For shear locking, accuracy level of the solutions can be possibly calculated by using known exact solutions. As mentioned in previous section, displacement based models experience severe shear locking even with the use of higher interpolations. But present models developed with mixed formulations showed less locking, especially for the mixed Galerkin model, if higher order of interpolation functions were used. The solutions of the newly developed models are presented in the Table. 1, to investigate shear locking. With the results presented in the Table 1, we can find that accuracy level of the solution does depend on methods and formulations adopted. The mixed Galerkin model showed the best accuracy in finite element solutions obtained under both C-C and P-P boundary conditions, while traditional displacement models showed the most inaccurate results. With the results presented in the Table 1, errors which mean degree of shear locking are compared in the Fig. 3.

The degree of shear locking is measured using the definition

Degree of shear locking exact sl

exact

w w

w

-= (33)

Table.1 Linear results of beam obtained with 4-element mesh under C-C boundary condition, q(x) =1.0 lb/in, with full integration Interpolation order

Center deflection of the beam (w0), in C-C boundary condition(exact: .104291667) P-P boundary condition(exact: .520958333) Mixed Galerkin

Mixed Least-squares

Traditional displacement

Mixed Galerkin

Mixed Least-squares


1st 0.112972222 0.096004271 0.001964678 0.526745370 0.497652359 0.009691326 2nd 0.105593750 0.104283529 0.098352041 0.522000000 0.520938133 0.515034294 3rd 0.104291667 0.104291651 0.104296375 0.520958335 0.520958540 0.521078238 4th 0.104291667 0.104291612 0.104273233 0.520958333 0.520958178 0.520491516 5th 0.104291667 0.104291670 0.104270593 0.520958335 0.520959431 0.520417581 6th 0.104291668 0.104291730 0.104363783 0.520958337 0.520958056 0.522800451 Table 2 Linear results of beam obtained with 1-element mesh under C-C boundary condition, q(x) =1.0 lb/in, with full integration Interpolation order

Center deflection of the beam (w0), in C-C boundary condition(exact: .104291667) P-P boundary condition(exact: .520958333) Mixed Galerkin

Mixed Least-squares


Mixed Galerkin

Mixed Least-squares

Traditional Displacement

1st 0.139013889 0.072895696 0.000498209 0.544106482 0.433308428 0.002364252 2nd 0.109500000 0.104161909 0.078860218 0.525125000 0.520637115 0.495521997 3rd 0.104291667 0.104291651 0.104291468 0.520958334 0.520958051 0.520951347 4th 0.104291667 0.104291668 0.104290570 0.520958333 0.520958341 0.520926382 5th 0.104291665 0.104291663 0.104283217 0.520958326 0.520958009 0.520744177 6th 0.104291671 0.104291639 0.104299478 0.520958357 0.520957624 0.521167093



a. C-C boundary condition b. H-H boundary condition

Fig. 4 Decay of error verses the interpolation order with 4-element uniform mesh, full integration.

Table 3 Converged center deflection w0 (iterations taken) of beam obtained with 4-element mesh, forth order interpolation, under C-C boundary condition Load Mixed Galerkin Mixed Least-squares Traditional Displacement

(full integration) Traditional Displacement (reduced integration)

1.0 0.1035(3) 0.1035(3) 0.1026(2) 0.1034(3) 2.0 0.2025(4) 0.2025(4) 0.2008(4) 0.2023(4) 3.0 0.2942(5) 0.2941(5) 0.2919(4) 0.2939(4) 4.0 0.3777(5) 0.3775(6) 0.3749(5) 0.3774(5) 5.0 0.4532(6) 0.4530(6) 0.4499(6) 0.4530(6) 6.0 0.5217(7) 0.5215(7) 0.5177(6) 0.5216(6) 7.0 0.5842(7) 0.5838(7) 0.5799(7) 0.5841(7) 8.0 0.6411(8) 0.6407(8) 0.6368(8) 0.6414(8) 9.0 0.6941(9) 0.6933(8) 0.6890(8) 0.6943(9) 10.0 0.7427(10) 0.7426(9) 0.7376(10) 0.7433(10) where wsl is center vertical deflection of beam which is obtained by linear finite element analysis with full integration, and wexact is mathematically exact solution. Since wsl is obtained with full integration, it will contain certain degree of shear locking caused by inconsistent approximation of the variables. Thus, degree of shear locking can be determined from Eq. (33).

4.3 Nonlinear analysis

Under C-C boundary condition, each model showed similar convergence, as shown in the Fig. 4a, while the converged solutions of the current mixed Galerkin and the least-squares models are more accurate than those of the displacement model (see Fig. 4b). The converged solutions of the displacement model close to the solutions predicted by the current mixed models when proper reduced integration techniques or consistent approximations of variables are used.

Converged solutions are presented in Table 3. The direct iterative method was used to get the solutions. Two mixed models showed good result with full integrations, while the traditional displacement model showed some degree of locking. Normally, locking of element is sever in finite element solution of lower interpolations, while it is likely to disappear in higher interpolations. Two mixed models showed closest converged solutions, while the

‐0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 2 3 4 5 6 7

degree of shear locking

interpolation order

Mixed Galerkin

Mixed least‐squares

Displacement based

‐0.001

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012

1 2 3 4 5 6 7

degree of shear locking

interpolation order

Mixed Galerkin


Displacement based



displacement based model did not as presented in the Fig. 5b. As mentioned before, membrane locking [3] is caused by the use of inconsistent order of interpolation of the terms like 2 1

0 0 .0.5( ) ( ) xxu w EA N-¢ ¢+ = In particular, when the

beam undergoes no extensional deformation, it should produce 2 10 0 00.5( ) ( ) .xxu w EA N-¢ ¢+ = @

(a) Load verses center defevtion under C-C boundary condition

(b) Nomalixed difference of converged solutions respect to that of the mixed Galerkin model

Fig. 5 Nonlinear behavior of beams under C-C boundary condition, with 4 cubic elements.

(a) Load verses center defevtion (b) Esitimated degree of menbrane locking

Fig. 6 Nonlinear behavior of beams under H-H boundary condition, with 4 cubic elements.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0

vertical deflection of center w

o

incremental load qo

Displacement based

Mixed Galerkin


0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0norm

alized

difference between solutions

incremental load qo

Displacement based


0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0

vertical deflection of center w

o

incremental load qo


Mixed Galerkin

Displacement based

‐0.010

‐0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

1 2 3 4 5 6 7 8 9 10 11

degree of mem

brane locking

interpolation order


Mixed Galerkin

Displacement based



This is satisfied only when the axial displacement and transverse displacements are interpolated such a way that the membrane strain has the possibility of becoming zero. Thus, the beam element may behave in a manner physically unrealistic. Traditionally, this phenomenon is overcome using reduced integration of the nonlinear stiffness coefficients. In the case of the mixed model presented herein, membrane locking completely disappears. As in Eq. (33), the degree of membrane locking can be measured as

Degree of membrane locking exact ml

exact

w w

w

-= (34)

where wml is converged center vertical deflection of the beam obtained with full integration, and wexact is mathematically exact solution of the same. Judging from the nonlinear results obtained, both models showed good nonlinear convergence while the mixed Galerkin model showed least degree of membrane locking.

5 CONCLUSIONS

Developing procedures of nonlinear beam bending finite element models were presented with two different methods. All element wise coefficient matrices and force vectors are also presented. Mixed formulation provided superior accuracy in linear solutions and better performance in nonlinear analysis with the use of same order interpolations and full integrations. Two types of locking phenomena were discussed and current mixed models showed less locking compared with traditional displacement based model.

ACKNOWLEDGEMENTS

The research reported here was supported by a subcontract from the University of Kansas of an ARO grant (W911NF-09-1-0548 KUCR No. FED65623). The support is gratefully acknowledged. The authors are also pleased to acknowledge many discussions on beam LSFEM with Mr. Gregory Payette (doctoral student of Professor Reddy).

REFERENCES

[1] Reddy J.N., 2006, An Introduction to the Finite Element Method, Third Edition, McGraw-Hill, New York. [2] Reddy J.N., 2002, Energy Principles and Variational Methods in Applied Mechanics, Second Edition, John Wiley &

Sons, New York. [3] Reddy J.N., 2004, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Oxford, UK. [4] Wang C.M., Reddy J.N., Lee K.H., 2000, Shear Deformable Beams and Plates: Relationships with Classical Solutions,

Elsevier, New York. [5] Pontaza J.P., Reddy J.N., 2004, Mixed plate bending elements based on least-squares formulation, International

Journal for Numerical Methods in Engineering 60: 891-922. [6] Pontaza J.P., 2005, Least-squares variational principles and the finite element method: Theory, formulations, and

models for solids and fluid mechanics, Finite Elements in Analysis and Design 41: 703-728. [7] Bochev P.B., Gunzburger M.D., 2009, Least-squares Finite Element Methods, Springer, New York. [8] Jiang B.-N., 1998, The least-squares Finite Element Method, Springer, New York. [9] Jou J., Yang S.-Y., 2000, Least-squares finite element approximations to the Timoshenko beam problem, Applied

Mathematics and Computation 115: 63-75. [10] Reddy, J.N., 1997, On locking-free shear deformable beam finite elements, Computer Methods in Applied Mechanics

and Engineering 149: 113-132.

Documents

A Comparative Study of Least-Squares and the Weak- Form Galerkin Finite Element …journals.iau.ir/article_514360_26f9d2cf13309ce7b8237772c... · 2020-03-31 · Form Galerkin Finite