### Least-squares ®nite element approximations

### to the Timoshenko beam problem

### Jang Jou

a,b,*_{, Suh-Yuh Yang}

c,1
a_{Department of Applied Mathematics, National Chiao Tung University, 1001, Ta Hsueh Road,}

Hsinchu 30050, Taiwan, ROC

b_{Department of Statistics, Ming Chuan University, Taipei 11120, Taiwan, ROC}
c_{Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung 84008, Taiwan, ROC}

Abstract

In this paper a least-squares ®nite element method for the Timoshenko beam problem is proposed and analyzed. The method is shown to be convergent and stable without requiring extra smoothness of the exact solutions. For suciently regular exact

solutions, the method achieves optimal order of convergence in the H1_{-norm for all the}

unknowns (displacement, rotation, shear, moment), uniformly in the small parameter which is generally proportional to the ratio of thickness to length. Thus the locking phenomenon disappears as the parameter tends to zero. A sharp a posteriori error

es-timator which is exact in the energy norm and equivalent in the H1_{-norm is also brie¯y}

discussed. Ó 2000 Published by Elsevier Science Inc. All rights reserved.

AMS classi®cation: 65N15; 65N30

Keywords: Timoshenko beam problem; Least-squares; Finite element method; Locking phenom-enon; A posteriori error estimator

1. Introduction

The locking phenomenon is mainly concerned in the ®nite element analysis for parameter-dependent problems [2,5]. In this paper we shall propose and analyze a ®nite element method based on the least-squares principles for

www.elsevier.com/locate/amc

*_{Corresponding author. E-mail: jangjou@math.nctu.edu.tw}

1_{The work of this author was supported in part by NSC-grant 87-2811-M-007-0017, Taiwan,}

ROC.

0096-3003/00/$ - see front matter Ó 2000 Published by Elsevier Science Inc. All rights reserved. PII: S0096-3003(99)00139-3

approximating the solution of the Timoshenko beam problem in its ®rst-order
system formulation. The method avoids the locking problem as the thickness of
the beam tends to zero, and achieves optimal order error estimates for all the
unknowns (displacement, rotation, shear, moment) in the H1_{-norm.}

Finite element analysis of the Timoshenko beam problem has been fre-quently used as a starting point for a better understanding of the much more complex problem of constructing accurate ®nite element approximations to the Reissner±Mindlin plate problem. It is well-known that some bad behaviors may occur such as the locking phenomenon when we solve these problems with standard Galerkin ®nite element methods [2,19,26,27]. That is, the convergence results of the standard ®nite element approximations deteriorate as the small parameters (which depend on the thickness of the beam and the plate for the Timoshenko beam model and the Reissner±Mindlin plate model, respectively) tend to zero. The most widely used eective approach to overcome the di-culty is based on the reduced integration technique. A complete analysis for the Timoshenko beam problem by using this technique has been addressed in [2]. Recently, a modi®ed reduced integration method with linear ®nite elements is proposed and analyzed by Cheng et al. [19]. The method presented in [19] uses the reduced integration technique to compute the term involving the small parameter and adds a bubble function space to the rotation to increase the solution accuracy. This method can also be applied to solve the circular arch problem and the Reissner±Mindlin plate problem. Another way to eliminate the locking phenomenon is to use the p and h ÿ p versions of the ®nite element method for which optimal error estimates are established in [25].

In the present investigation we provide an alternate way to avoid the
diculty by exploiting the least-squares principles on a ®rst-order system
formulation of the Timoshenko beam problem. We ®rst introduce a quadratic
least-squares functional Q over a function space V consisting of functions
which satisfy the boundary conditions of the problem. The functional is de®ned
to be the sum of the squared L2_{-norms of the residuals in the dierential}

equations. Then the exact solution must be the unique zero minimizer of the functional Q over V. Therefore, the least-squares ®nite element approximate solution is de®ned to be the minimizer of Q over a ®nite-dimensional subspace Vp

hof V. Mathematical analyses show this approach can eliminate the locking

phenomenon.

Over the past decade, the use of least-squares principles in connection with ®nite element techniques has been extensively applied to the approximations in many dierent ®elds such as ¯uid dynamics [9,12,13,16±18], elasticity [14,29,30], electromagnetism [15,24], and semiconductor device physics [8]. The approach oers certain advantages, especially for large-scale computations. It leads to minimization problems rather than saddle point problems led by the mixed ®nite element approach. A single continuous piecewise polynomial space can be used for the approximation of all unknowns, and accurate

approxi-mations of all unknowns can be simultaneously obtained. The resulting alge-braic system is symmetric and positive de®nite. In addition, the value of the least-squares functional of the approximate solution provides a practical and sharp a posteriori error estimator at no additional cost [21,23,28]. This feature is very important in adaptive computations.

The layout of the remainder of the paper is as follows. In Section 2, we
introduce the ®rst-order system formulation for the Timoshenko beam
prob-lem and we then derive important a priori estimates for the system. The
least-squares ®nite element method is given in Section 3. Then the method is proved
to be convergent and stable in Section 4, where optimal error estimates in the
H1_{-norm are also established. Finally, in Section 5, we brie¯y discuss the sharp}

a posteriori error estimator for the least-squares approach. 2. The Timoshenko beam problem

In this paper we shall consider the in-plane bending of a clamped uniform beam of length L, cross-section A, moment of inertia I, Young's modulus E, and shear modulus G, subjected to a distributed load p x, and a distributed moment m x, with x 2 0; L representing the independent variable. According to the Timoshenko beam theory, this problem is governed by the following system of ordinary dierential equations of ®rst-order (cf. [27]):

ÿdQ_{dx} p in
0; L;
2:1

ÿdM_{dx} ÿ Q m in
0; L;
2:2

ÿ_{jGA}Q dw_{dx}ÿ h 0 in
0; L;
2:3

ÿM_{EI}dh_{dx} 0 in
0; L;
2:4

supplemented with the boundary conditions:

w 0 w L 0; 2:5

h 0 h L 0; 2:6

where Q x is the shear force; M x the bending moment; w x the transverse displacement; h x the cross-section rotation; and j the shear correction factor.

To explicate the dependence of this problem on a small parameter,
e2_{} EI

we introduce the following change of variables:
x x
L;
2:8
u1w_{L}; u2 h;
2:9
r1QL
2
EI ; r2
ML
EI ;
2:10
f1pL
3
EI ; f2
mL2
EI ;
2:11

which reduces the original problem to ®nding u x u1 x; u2 x and r x

r1
x; r2
x, x 2
0; 1 satisfying
ÿ r0
1 f1 in
0; 1;
2:12
ÿ r0
2ÿ r1 f2 in
0; 1;
2:13
ÿ e2_{r}
1 u01ÿ u2 0 in
0; 1;
2:14
ÿ r2 u02 0 in
0; 1;
2:15

with the boundary conditions

u1 0 u1 1 0; 2:16

u2 0 u2 1 0; 2:17

where the prime superscript denotes dierentiation with respect to the nondi-mensional variable x. We observe that the nondinondi-mensional problem (2.12)± (2.17) depends explicitly on a parameter e. In general, e is a small parameter proportional to the ratio of the thickness to length. For instance, in the rect-angular cross-section case [26],

e2_{} E
12jG
T
L
2
;

where T represents the thickness of the beam. Thus in most realistic applica-tions, e.g., for thin beams, 0 < e 1, and the construction of accurate ®nite element approximations is delicate.

We need some function spaces throughout this paper. The classical Sobolev
spaces Hr_{
0; 1 with their associated norms k k}

r are employed [11,20]. As

usual, we denote by
; _{0}and k k_{0}the conventional inner product and norm
on the Hilbert space L2_{
0; 1 of square-integrable functions. The space H}1_{
0; 1}

of functions which together with their ®rst derivatives are square-integrable, is
a Hilbert space with the inner product
; _{1}, where

u; v_{1}
Z 1

0 uv u

The norm generated by this inner product is denoted by k k_{1}; that is,
kvk_{1}
v; v1=2

1 . We denote by H01 0; 1 the subspace of H1 0; 1 consisting of

functions which vanish at the ends of the interval. For the product space
H1_{
0; 1}4_{, the corresponding inner product and norm are also denoted by}

; _{1}and k k_{1}, respectively, when there is no chance of confusion.

We shall use C with or without subscripts in this paper to denote a generic positive constant, possibly dierent at dierent occurrences, that is independent of the parameter e and the mesh parameter h introduced in the next section.

The following regularity result plays an important role in the theoretical analysis later.

Theorem 2.1. Let v; x v1; v2; x1; x2 2 H01 0; 12 H1 0; 12. Then there

exist two positive constants C1and C2both independent of the parameter e such

that C1 kv1k21 kv2k21 kx1k21 kx2k21 6 Q v; x; 0 2:18 and Q v; x; 0 6 C2 kv1k21 kv2k21 kx1k21 kx2k21 ; 2:19 where Q v; x; 0 : kx0 1k20 kx02 x1k20 ke2x1ÿ v01 v2k20 kx2ÿ v02k20:

Proof. Upper bound (2.19) is straightforward from the triangle and Cauchy± Schwarz inequalities. To prove the lower bound (2.18), we ®rst de®ne

g1: ÿ x01; 2:20

g2: ÿ x02ÿ x1; 2:21

g3: ÿ e2x1 v01ÿ v2; 2:22

g4: ÿ x2 v02; 2:23

in 0; 1. Then from the ®rst Eq. (2.20), we ®nd x1 x ÿG1 x c1;

where

G1 x :

Z _{x}

0 g1 tdt; x 2 0; 1; c1: x1 0:

Integrating the second Eq. (2.21), we have x2 x ÿG2 x Z x 0 G1 tdt ÿ c1x c2; where G2 x : Z x 0 g2 tdt; x 2 0; 1; c2: x2 0:

Solving the last Eq. (2.23) together with the boundary condition v2
0 0, we
obtain
v2
x ÿ
Z _{x}
0 G2
sds
Z _{x}
0
Z _{s}
0 G1
tdt ds ÿ
1
2c1x2 c2x G4
x;
where
G4
x :
Z _{x}
0 g4
tdt; x; s 2 0; 1:

Finally, from the third Eq. (2.22) and the boundary condition v1 0 0, we

®nd for x; y; s 2 0; 1
v1
x ÿe2
Z _{x}
0 G1
tdt e
2_{c}
1x G3
x ÿ
Z _{x}
0
Z _{y}
0 G2
sdsdy
Z _{x}
0
Z _{y}
0
Z _{s}
0 G1
tdt dsdy ÿ
1
6c1x3
1
2c2x2
Z _{x}
0 G4
ydy;
where
G3
x :
Z _{x}
0 g3
tdt; x 2 0; 1:

Utilizing the boundary conditions v2 1 0 and v1 1 0, we get the following

2 2 linear system of equations in the unknowns c1 and c2:

ÿ1_{2}c1 c2
Z _{1}
0 G2
sds ÿ
Z _{1}
0
Z _{s}
0 G1
tdt ds ÿ G4
1
e2_{ÿ}1
6c1
1
2c2 e2
Z 1
0 G1
tdt ÿ G3
1
Z 1
0
Z y
0 G2
sdsdy
ÿ
Z _{1}
0
Z _{y}
0
Z _{s}
0 G1
tdt dsdy ÿ
Z _{1}
0 G4
ydy:

Solving this linear system, we can ®nd the following estimates for c1 and c2,

respectively, jc1j 6 C X4 i1 kgik0; jc2j 6 C X4 i1 kgik0

for some constant C independent of the parameter e. Now it is easy to see that kvik216 CQ v; x; 0;

for i 1; 2. Thus the assertion (2.18) follows immediately and this completes the proof.

3. The least-squares ®nite element method

In this section we introduce the least-squares ®nite element method for problem (2.12)±(2.17). De®ne a function space V for our problem by

V H1

0 0; 1 H01 0; 1 H1 0; 1 H1 0; 1 3:1

and then de®ne a quadratic least-squares energy functional Q: V ! R by Q v; x; f k ÿ x0

1ÿ f1k20 k ÿ x02ÿ x1ÿ f2k20 k ÿ e2x1 v01ÿ v2k20

k ÿ x2 v02k20; 3:2

where v v1; v2, x x1; x2, and f f1; f2. Note that the quadratic

en-ergy functional Q
v; x; f is de®ned to be the sum of the squared L2_{-norms of}

the residuals in the dierential equations. Obviously, the exact solution u; r 2 V of problem (2.12)±(2.17) is the unique zero minimizer of the functional Q on V, that is,

Q u; r; f 0 min Q v; x; f: v; x 2 Vf g: 3:3 Applying the variational techniques, we can ®nd that (3.3) is equivalent to

B u; r; v; x F v; x 8 v; x 2 V; 3:4

where the bilinear form B
; and the linear form F
are de®ned,
respec-tively, by
B
v; x;
z; .
Z _{1}
0
x
0
1.01
x02 x1
.02 .1
ÿe2_{x}
1 v01ÿ v2
ÿe2.1 z01ÿ z2
ÿx2 v02
ÿ.2 z02dx;
3:5
F
v; x
Z _{1}
0
ÿ
ÿ x0
1f1
ÿ x02ÿ x1f2dx
3:6
for all
v; x,
z; . 2 V.

It is evidently that B
; is symmetric and continuous (bounded) on V V
and, for each given f 2 L2_{
0; 1}2_{, F
is also continuous (bounded) on V.}

C1 kv1k21 kv2k21 kx1k21 kx2k21 6B v;x; v;x Q v;x;0 6C2 kv1k21 kv2k21 kx1k21 kx2k21 3:7 for all v; x 2 V. Thus B ; is coercive on V V and

k
v; xk_{B} B
v; x;
v; x
1=2 _{8
v; x 2 V} _{
3:8}

de®ne a new norm on V which is equivalent to the H1_{-norm.}

For the purpose of discretization we shall use ®nite element spaces de®ned with reference to partitions of 0; 1. Let Thbe a partition of 0; 1 such that the

interval 0; 1 is partitioned into subintervals Ii xiÿ1; xi, i 1; . . . ; N, with

0 x0< x1< < xN 1. The mesh parameter h is de®ned by

h maxfjxiÿ xiÿ1j: i 1; . . . ; Ng. We denote by Pph, p P 1 integer, the space

of continuous functions on 0; 1 whose restrictions to Ii, i 1; ; N, are

polynomials of degree p, that is,
Pp_{h} v 2 C0_{
0; 1: vj}
Iiis a polynomial of degree p
:
De®ne
~
Pp_{h} Pp
h\ H01
0; 1:

Then we will seek the least-squares ®nite element approximations in the fol-lowing ®nite-dimensional subspace of V,

Vp

h ~Pph ~P p

h Pph Pph: 3:9

By the interpolation theory, the ®nite element space possesses the following
approximation property: for any
v; x 2 V \ Hp1_{
0; 1}4_{, there exists}

vh; xh 2 Vph such that

k v; x ÿ vh; xhk16 Chpk v; xkp1; 3:10

where the positive constant C is independent of v; x and the mesh parameter h.

The least-squares ®nite element method for problem (2.12)±(2.17) is then the following.

Find uh; rh 2 Vph such that

B uh; rh; vh; xh F vh; xh 8 vh; xh 2 Vph: 3:11

Applying the Lax±Milgram theorem [11,20], we know that the approximation problem (3.11) has a unique solution. Once a basis for the space Vp

his chosen,

the matrix associated with problem (3.11) can easily be shown to be symmetric and positive de®nite.

4. Stability, convergence and error estimates

Let
u; r 2 V be the exact solution of problem (2.12)±(2.17) with the given
function f 2 L2_{
0; 1}2_{. We ®rst prove the stability of the least-squares ®nite}

element solution uh; rh 2 Vph.

Theorem 4.1. The unique solution uh; rh 2 Vphof problem (3.11) is stable in the

H1_{-norm in the following sense:}

k uh; rhk16 Ckfk0; 4:1

where the positive constant C is independent of e and h. Proof. By (3.7), (3.8) and (3.11), we have

C1k uh; rhk216 k uh; rhk2B

B uh; rh; uh; rh

F uh; rh

6 kfk_{0}k
uh; rhkB

6 Ckfk_{0}k
uh; rhk1;

which implies (4.1). This completes the proof.

Estimate (4.1) indicates that the least-squares ®nite element solution uh; rh

is stable with respect to the H1_{-norm, that is, when we change the given data}

function f slightly in the L2_{-norm, the least-squares solution
u}

h; rh changes

only slightly in the H1_{-norm.}

We now introduce some function spaces which will be used to prove the next theorem. Let C1

0 0; 1 denote the linear space of in®nitely dierentiable

func-tions with compact support in
0; 1, and let C1_{0; 1 denote the restrictions of}

the functions in C1

0 R to 0; 1. Then it is obvious that the product space

S : C1

0 0; 12 C10; 12 4:2

is dense in V with respect to the H1_{-norm.}

Now utilizing the standard density argument [20], we can obtain the fol-lowing results for the convergence.

Theorem 4.2. The least-squares ®nite element solution uh; rh is convergent in the

H1_{-norm without requiring any extra regularity assumption on the exact solution}

u; r, that is, lim

h!0k u; r ÿ uh; rhk1 0: 4:3

Moreover, if the exact solution
u; r 2 V \ Hp1_{
0; 1}4_{, then we have the}

k u; r ÿ uh; rhk16 Chpk u; rkp1; 4:4

where C is a positive constant independent of e and h.

Proof. Subtracting the equation in (3.11) from the equation in (3.4), we get the following orthogonality relation:

B u; r ÿ uh; rh; vh; xh 0 8 vh; xh 2 Vph: 4:5

Using (3.7), (4.5) and the Cauchy±Schwarz inequality, we obtain
k
u; r ÿ
uh; rhk216_{C}1

1k u; r ÿ uh; rhk 2 B

_{C}1

1B u; r ÿ uh; rh; u; r ÿ uh; rh

_{C}1

1B u; r ÿ uh; rh; u; r ÿ vh; xh

6_{C}C

1k u; r ÿ uh; rhk1k u; r ÿ vh; xhk1

for all vh; xh 2 Vph. Thus,

k u; r ÿ uh; rhk16 Ck u; r ÿ vh; xhk1 8 vh; xh 2 Vph: 4:6

Now, since the subspace S V \ Hp1_{
0; 1}4 _{is dense in V with respect to}

the H1_{-norm, for any d > 0, there exists
~u; ~r 2 S independent of h such that}

k
u; r ÿ
~u; ~rk_{1}< d

2C; 4:7

where C is the same constant as in (4.6). For this ®xed suciently smooth
function
~u; ~r 2 S Hp1_{
0; 1}4_{, by the approximation property (3.10), we}

can ®nd ~uh; ~rh 2 Vph so that,

k ~u; ~r ÿ ~uh; ~rhk16 Chpk ~u; ~rkp1;

which implies, for suciently small h,

k
~u; ~r ÿ
~uh; ~rhk1<_{2C}d ;
4:8

where C is the same constant as in (4.6). Combining inequalities (4.7) and (4.8) with (4.6), we immediately obtain

0 6 k u; r ÿ uh; rhk16 Ck u; r ÿ ~uh; ~rhk1

6 C k
u; r ÿ
~u; ~rk_{1} k
~u; ~r ÿ
~uh; ~rhk1

< d;

We now assume that
u; r 2 V \ Hp1_{
0; 1}4_{. By (4.6) and the }

approxi-mation property (3.10) of the ®nite element space Vp

h, we can obtain (4.4)

immediately. This completes the proof.

Since the energy norm k k_{B}is equivalent to the H1_{-norm, from (4.3), we can}

conclude the following consequence.

Corollary 4.3. Let uh; rh u1h; u2h; r1h; r2h be the least-squares ®nite element

solution. Then
lim
h!0 k
ÿ
ÿ r0
1hÿ f1k0 k ÿ r02hÿ r1hÿ f2k0
k ÿ e2_{r}
1h u01hÿ u2hk0 k ÿ r2h u02hk0
0:
4:9

Remark 4.4. The error estimate (4.4) is optimal in the H1_{-norm with}

respect to the order of approximation of the ®nite element space Vph. Under

suitable assumptions (cf. [22]), it is quite possible to derive the optimal error
estimates in the L2_{-norm for all the unknowns by using the Aubin±Nitsche}

trick.

5. An equivalent a posteriori error estimator in the H1_{-norm}

The use of a posteriori error estimators has become an accepted tool for assessing and controlling computational errors in adaptive computations. One of the most important advantageous features of the least-squares ®nite element approach is that the square root of the value of the least-squares functional of the approximate solution provides a practical and sharp a posteriori error estimator at no additional cost [21,23,28]. This is quite dierent from the previous error estimators, see [1,3,4,6,7,10,31] for example.

In this section we shall brie¯y show that the simple estimator is indeed an
exact error estimator in the energy norm, k k_{B}. Thus, it is an equivalent error
estimator in the H1_{-norm. Let
u}

h; rh u1h; u2h; r1h; r2h be the least-squares

®nite element solution. For all subintervals Ii2 Th, let QIi uh; rh; f denote the value of quadratic least-squares functional of the approximate solution re-stricted on Ii, that is,

QIi
uh; rh; f
Z
Ii
ÿr0
1hÿ f12dx
Z
Ii
ÿr0
2hÿ r1hÿ f22dx
Z
Ii
ÿe2_{r}
1h u01hÿ u2h2dx
Z
Ii
ÿr2h u02h2dx

for all Ii2 Th. Then
QIi
uh; rh; f
Z
Ii
ÿr0
1hÿ
ÿr012dx
Z
Ii
ÿr0
2hÿ r1hÿ
ÿr02ÿ r12dx
Z
Ii
ÿe2_{r}
1h u01hÿ u2hÿ
ÿe2r1 u01ÿ u22dx
Z
Ii
ÿr2h u02hÿ
ÿr2 u022dx

k u; r ÿ uh; rhk2B;Ii;

where k k_{B;I}_{i} denotes the energy norm restricted on Ii. Therefore, the

com-putable value QIi uh; rh; f1=2 de®nes an ``exact'' error indicator of the sub-interval Iiwhich assesses the quality of the approximate solution uh; rh in that

subinterval Ii and indicates whether the element needs to be re®ned, dere®ned

or unchanged. Taking the summation, Q uh; rh; f 1=2 X Ii2Th QIi uh; rh; f !1=2 k u; r ÿ uh; rhkB;

we get an exact a posteriori error estimator Q uh; rh; f1=2, which can serve as

one of the major stopping criteria for an entire adaptive process. In general, the eectivity index de®ned by

H _{k
u; r ÿ
u}
Q
uh; rh; f1=2

h; rhkB

is then used to quantify the quality of the estimator and hence the quality of the approximate solution uh; rh. In our case, H exactly equals 1.

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