Locking and robustness in the nite element ... - Semantic Scholar

Numer. Math. 69: 509{522 (1995)

Numerische Mathematik

c Springer-Verlag 1995 Electronic Edition

Locking and robustness in the nite element method for circular arch problem

Zhimin Zhang?

Department of Mathematics, Texas Tech University, Lubbock, TX 79409, USA e-mail: [email protected] Received June 5, 1992 / Revised version received May 17, 1994

Summary. In this paper we discuss locking and robustness of the nite ele-

ment method for a model circular arch problem. It is shown that in the primal variable (i.e., the standard displacement formulation), the p-version is free from locking and uniformly robust with order p?k and hence exhibits optimal rate of convergence. On the other hand, the h-version shows locking of order h?2 , and is uniformly robust with order hp?2 for p > 2 which explains the fact that the quadratic element for some circular arch problems suers from locking for thin arches in computational experience. If mixed method is used, both the h-version and the p-version are free from locking. Furthermore, the mixed method even converges uniformly with an optimal rate for the stress. Mathematics Subject Classi cation (1991): 65N30, 73K05, 73V05

1. Introduction The development of robust nite element methods for the circular arch problem is of interest in many practical applications. For the standard nite element method, namely the h-version method, locking occurs in the thin arch limit [2]. Many eorts have been focused on overcoming the locking eects, for example, mixed methods which use mixed variational principles (see [8], [12], [15], and [20]{[22]), the Petrov-Galerkin method [11] in which the test function space diers from the trial function space, and reduced integration methods (see [12], [14], [16], and [18]{[22]). In [20], the equivalence of a mixed method with the selective reduced integration method was established for a family of circular arch models and a uniform optimal convergence rate was obtained for the mixed method. Another remedy for the locking eects is the p-version method, that is, using higher order polynomials. Although satisfactory computational results were observed in the engineering literature [7], there is at present no theoretical analysis in the literature regarding the p-version of the nite element method for circular arch problems. Also, the locking eect has not been characterized mathematically. These are the topics of the current paper. At this point we should mention ?

This work was partially supported by the National Science Foundation grant DMS-9023063

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that for a closely related problem, the Timoshenko beam, the investigation is quite complete (see [1], [9] and [10]). Recently, Babuska and Suri set up a general mathematical framework on locking eects and robustness in the nite element method for a certain class of parameter dependent problems [3], [4]. In the spirit of this framework, we are able to characterize locking and robustness in nite element methods for circular arch models. The major step is to verify that the so-called \condition ()" of [3] is satis ed by a particular circular arch model. This is achieved in Sect. 2. With the help of results in [3], in Sect. 3, we establish an optimal rate of convergence uniformly with respect to the arch thickness for the p-version and characterize the locking eect for the h-version for the circular arch model considered. We shall also discuss mixed method and the combination of the h-p version with the mixed method. Concluding this introduction, we list notation used in the paper. Let I = (0; 1) and L2(I ) be the space of square integrable functions on I . Denote as usual H k (I ) = fv 2 L2 (I )j v0 ;


; v(k) 2 L2(I )g; H01 (I ) = fv 2 H 1 (I ); v(0) = v(1) = 0g; and denote H ?1 (I ) as the dual space of H01 (I ). De ne

kvk0 =

Z 1 0

1=2

v(x)2 dx

0 11=2 X ; jvjk = kv(k) k0; kvkk = @ jvj2i A : ik

We consider vector spaces W = L2(I )2 ; V = H01 (I )3 ; V 0 = H ?1 (I )3 ; where H (I )3 = H (I )  H (I )  H (I ). We de ne on W the inner product (w; z) = which induces the norm

0

w (x)
z (x)dx;

kwk0 = kwkW = (w; w)1=2 ;

and de ne on V the inner product

(u; v)1 = which induces the norm

Z1

Z1 0

u0 (x)
v 0 (x)dx;

kukV = (u; u)11=2 : From the Poincare inequality, k
kV is equivalent to the Sobolev norm k
k1 on

V which is de ned by

kuk1 = ((u; u)1 + (u; u))1=2 : The Sobolev norms jujk , kukk on H k (I )n , n = 2; 3, are de ned similarly.

Throughout the paper C is used to denote a generic constant which is not necessarily same at each occurrence. Numerische Mathematik Electronic Edition { page numbers may dier from the printed version page 510 of Numer. Math. 69: 509{522 (1995)

Finite element method for circular arch problem Φ f

511

u

d

x

A

w

y R

Fig. 1.

2. Circular arch problems and their properties Let us consider a uniform linear elastic circular arch with thickness d. Its centerline has radius R and length L. See Fig. 1. The arch has Young's modulus E , shear modulus G, cross-sectional area A R and moment of inertia I = A y2 dA = O(Ad2 ), and is subjected to a distributed load (per unit length) f = (f1 ; f2; f3 ) and clamped at both ends. We assume the Mindlin-Reissner hypotheses, so the deformation of the arch is described by the rotation of the cross-section , the radial displacement w and the tangential displacement u of its center-line. We consider the following circular arch model which is expressed by its potential energy: Z L "  d w 1 du 2 1  EI ds + R2 ? R ds  (; w;  u) = 2 0  2  w du 2 +kGA  ? ddws ? Ru + EA R ? ds  (1) ?2f1 ? 2f2w ? 2f3 u ds; where s is the arch length coordinate, and k is the shear correction factor. This model was discussed in [20] where it was treated by a mixed nite element method. In the expression for the strain energy, the term with coecient EI is the bending energy; the term with coecient kGA is the shear energy; and the term with coecient EA is the membrane energy. Indeed, the above model is the degenerate form of the Naghdi's shell model, see [16] for the physical background of this model. Because of this reason, the investigation of the circular arch model can give us some insights for more general and more complicated mechanical structure - the shell model. If we let R ! 1 and impose the Mindlin's hypotheses u = 0 on the centerline, we then recover the potential energy for the Timoshenko beam: #  dw 2 Z L "  d 2 1      (2) 0 (; w) = 2 EI ds + kGA  ? ds ? 2f1 ? 2f2 w ds; 0 Numerische Mathematik Electronic Edition { page numbers may dier from the printed version page 511 of Numer. Math. 69: 509{522 (1995)

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which contains no membrane energy term. For the relationship of this circular arch model with other models, the reader is referred to [20]. We introduce the change of variables: s = Ls; (R(s); w(s); u(s)) = ((s); w(s); u(s)); (f1 (s); f2 (s); f3 (s)) = LEI 2 R2 (Rf1 (s); f2 (s); f3 (s)):

Multiplying (1) throughout by LR2=(EI ) yields the nondimensional form of the potential energy as Z 1 1 (0 + w ? u0 )2 + mt (  ? w0 ? u)2 + 1t ( w ? u0 )2 (; w; u) = 2 0 (3)



? 2f1 ? 2f2 w ? 2f3u ds;

where = L=R, t?1 = AR2 =I , m = kG=E , and v0 = dv=ds. Clearly, t = O(d2 ), and is the nondimensional measure of the arch depth which varies from 0 to 2. The corresponding variational problem is: (P t ) Given f = (f1 ; f2 ; f3 ) 2 V 0 , nd ut = (t ; wt ; ut) 2 V such that B t (ut ; v) = (f ; v); 8v = ( ; z; v) 2 V; where B t is a bilinear form de ned on V  V as, (4) B t (u; v) = a(u; v) + m (S u; S v) + 1 (Du; Dv);

t

t

with (5) a(u; v) = (0 + w ? u0 ; 0 + z ? v0 ); (6) S v = ? z 0 ? v; (7) Dv = z ? v0 : Problem (P t ) is equivalent to the following mixed variational problem: (M t ) Given f 2 V 0 and g 2 W 0 , nd (ut ;  t ) = (t ; wt ; ut ; t ; t ) 2 V  W such that (8) a(ut ; v) + b(v;  t ) = (f ; v) 8v 2 V; (9) b(ut ; y) ? t( t ; y) = (g; y) 8y 2 W; where p b(v; y) = m(S v; y1 ) + (Dv; y2 ); g = (0; 0): Let us consider a slightly more general case when g 6= (0; 0). Theorem 2.1. Let f 2 H k?1 (I )3 , g 2 H k (I )2 . Then there exists a unique sequence of solutions fut ;  t g 2 V  W to problem (M t ) for t 2 [0; 1] such that (10) kut kk+1 + k t kk  C (kf kk?1 + kgkk ); where C is a constant independent of f , g and t. Numerische Mathematik Electronic Edition { page numbers may dier from the printed version page 512 of Numer. Math. 69: 509{522 (1995)

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Proof. The following three properties have been proved in [20]. 1) a(
;
) is symmetric and positive semide nite. 2) There exists  > 0 such that for any z 2 N (b) = fv 2 V j b(v; y) = 0

8y 2 W g

a(z ; z)  kz k2V : 3) There exists > 0 such that for any w 2 W , there is a v 2 V with b(v; w)  kvkV kwkW : Since V and W are Hilbert spaces, by Theorem 5.1 of [1] (See also [5]), we conclude from 1), 2) and 3) that for each pair (f ; g) 2 V 0  W 0 , there exists a unique pair (ut ;  t ) 2 V  W that solves Problem (M t ), moreover, (11) kut kV + k t kW  C (kf kV + kgkW ); 0

0

or equivalently (12) kut k1 + k t k0  C (kf k?1 + kgk0 ): After performing integration by parts, we may obtain the strong form of (M t ), 8 m t ? (Dut )0 = f1 ; ?00t + p > 0t + pmt0 + t + Dut = f2 ; >  > < 00t ? pm t + t0 + (Dut )0 = f3 ;  t p p (S ) > mS ut ? tt = m( t ? wt0 + ut ) ? tt = g1 ; > Dut ? tt = wt ? u0t ? tt = g2 ; > : t (0) = wt(0) = ut(0) = t (1) = wt (1) = ut(1) = 0: Therefore we can express higher derivatives by lower derivatives by reorganizing the terms in (S t ) as 8 00 = pm t ? f1 ? (tt + g2)0 ; > pmt0 = f ? (0 +  + t + g ); > < 2 t 2 t0 t t  = f + f ; (S t )a 1 3 p t p ( + u ) ? t ? g ; > t t t 1 : mwut00 == wm t ? tt ? g2 : t Applying mathematical induction, the regularity of (10) is obtained by the \bootstrapping" method from (12) and successive dierentiation of (S t )a . ut Corollary 2.1. Let f 2 H k?1(tI )3 . Then there exists a unique sequence of solutions fut g 2 V to problem (P ) for t 2 [0; 1] such that (13) kut kk+1  C kf kk?1 ; where C is a constant independent of f and t. Now we go back to our special case where g = (0; 0) and we have  a(u ; v) + b(v;  ) = (f ; v) 8v 2 V; t t (M t ) b(ut ; y) ? t( t ; y) = 0 8y 2 W; when t > 0, and Numerische Mathematik Electronic Edition { page numbers may dier from the printed version page 513 of Numer. Math. 69: 509{522 (1995)

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 a(u ; v) + b(v;  ) =

(f ; v) 8v 2 V; 0 b(u0 ; y) = 0 8y 2 W; for t = 0. Let us have a further discussion on the problem (M 0 ). From b(u0 ; y) = 0

(M 0 )

0, we have

S u0 = 0 ? w00 ? u0 = 0; Du0 = w0 ? u00 = 0: Observe that b(u0 ;  0 ) = 0, and we have a(u0 ; u0 ) = (f ; u0 ): Applying S u0 = 0 and Du0 = 0 yields a(u0 ; u0 ) = =

Z1 0

(00 + Du0 )ds =

Z 11 0



w00 + u0 0

0

2

Z1 0

ds =

(00 )2 ds

Z 11 0



w00 + w0 0

2

ds;

which is the strain energy for the circular ring. We see that, as the limit case, (M 0 ) is the variational formulation for the circular ring model. Next theorem tells the relationship between solutions of the circular arch and the circular ring models. Theorem 2.2. Let f 2 H k?1 (I )3 and let (ut ;  t), (u0;  0 ) be solutions of (M t) 0 and (M ), respectively. Then kut ? u0 kk+1 + k t ?  0 kk  C 2 tkf kk?1 ; where C , independent of t, is the same constant as in Theorem 2.1. Proof. First, we apply Theorem 2.1 with g = (0; 0) to Problem (M 0 ), and we have (14) ku0 kk+1 + k 0 kk  C kf kk?1 : 0 Subtraction of (M ) from (M t ) yields  a(u ? u ; v) + b(v;  ?  ) = 0 8v 2 V; t 0 t 0 b(ut ? u0 ; y) ? t( t ?  0 ; y) = t( 0 ; y) 8y 2 W; Applying Theorem 2.1 with f = (0; 0; 0) and g = t 0 yields kut ? u0 kk+1 + k t ?  0 kk  Ctk 0 kk  C 2 tkf kk?1 : The last step is from (14). ut Corollary 2.2. Assume that f 2 H k?1 (I )3 . Let ut , (u0 ;  0 ) be solutions of t 0 (P ) and (M ), respectively. Then

kut ? u0 kk+1  C 2 tkf kk?1 ;

where C , independent of t, is the same constant as in Theorem 2.1.

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For the bilinear form B t , we have

(15) B t (v; v )  Ct?1=2 kvkV ; 8v 2 V; (16) B t (v; v)  C ?1 kvkV ; 8v 2 V; where constant C depends only on and m. The inequality (15) is obvious from the de nition. The proof of (16) may be found in [20]. From the Lax-Milgram Lemma [6], the existence and uniqueness of (P t ) is guaranteed by (16). De ne kvk2E;t = B t (v ; v); then k
kE;t is a norm (called energy norm) on V which is equivalent to the norm k
kV for any xed t. Now we de ne for k  0, 0  t  1, the spaces Ht;k = fu 2 H k+1 (I )3 ; u is the solution of (P t ) for some f 2 H k?1 (I )3 g; HkB = fu 2 H k+1 (I )3 ; kukk+1 + k kk + kkk  B g; B = Ht;k \ H B : Ht;k k B , there is a u0 2 H B Following [3], we introduce condition (): For any ut 2 Ht;k 0;k such that, kut ? u0 kk+1  Ct1=2 B:

Then Theorem 2.2 simply says that: For the problems considered, Condition () is satis ed with power t (instead of just t1=2 , as needed). Note that Ht;k = H k+1 (I )3 for t > 0. Also, using Corollary 2.2, for any u0 2 H0B;k , there exists a sequence ut 2 Ht;k such that

kut kE;t  C ku0 kk+1 ; kut kk+1  C ku0 kk+1 ; and kut ? u0 kV ! 0 as t ! 0. This shows that H0;k is precisely the limit set of Ht;k as t ! 0, in the sense of [3].

3. Locking and robustness of nite element methods Now we are going to approximate problem (P t ). Partition the interval I = [0; 1] into subintervals Ij = [sj?1 ; spj ] with 0 = s0 < s1 <


< sn = 1. Let h = maxj (sj ? sj?1 ). Denote by V?1 (
), the space of piecewise polynomials of order p which are not necessarily continuous at nodal points sj . De ne

V p (
) = V?p1 (
) \ C 0 (I ); V0p (
) = V p (
) \ H01 (I ); VN = V0p (
)3 ; WN = V?p1 (
)2 ; where N is the dimension. Obviously, VN  V . The standard nite element solution of (P t ) is de ned as following: (V t ) Find uNt 2 VN such that B t (uNt ; v) = (f ; v); 8v 2 VN : Numerische Mathematik Electronic Edition { page numbers may dier from the printed version page 515 of Numer. Math. 69: 509{522 (1995)

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The sequence fVN g de nes a rule that decreases the error by increasing N . We now de ne locking and robustness following [3]. We assume that fVN g is F0 -admissible, i.e., it leads to a certain xed rate F0 (N ) of convergence when functions in H k+1 (I )3 are approximated, in the following sense C ?1 F0 (N )  sup vinf (17) ku ? v kV  CF0 (N ): 2V u2HkB

N

Here F0 (N ) ! 0 as N ! 1 and C depends on B but is independent of N . Note that since for t > 0, Ht;k = H k+1 (I )3 , we may show by using (17), that for 1  t  t0 > 0, t0 xed, C ?1 F0 (N )  sup Et (ut ? uNt )  CF0 (N ); (18) B u2Ht;k

where C is independent of t (but depends on t0 ) and is not necessarily same as in (17). Here we choose measure Et as k
kV or the energy norm. Hence, F0 (N ) measures the rate of convergence without locking (i.e., when t  t0 ). To see if there is locking, we compare the observed rate of convergence as t ! 0 with F0 (N ). For t 2 (0; 1] and N , we de ne by L(t; N ) = sup Et (ut ? uNt )F0 (N )?1 ; B u2Ht;k

the locking ratio with respect to the space Ht;k and error measure Et for the problem (P t ). Then if (18) holds, we have the following de nitions. De nition 3.1. The nite element method (V t) is free from locking for the family of problems (P t ), t 2 (0; 1] with respect to the solution sets Ht;k and error measures Et i

"

#

lim sup sup L(t; N ) = M < 1:

N !1

t2(0;1]

It shows locking of order f (N ) i

#

"

0 < Nlim sup sup L(t; N )f (N )?1 = C < 1 !1 t2(0;1]

where f (N ) ! 1 as N ! 1. It shows locking of at least (respectively at most) order f (N ) if C > 0 (respectively C < 1). De nition t3.2. The nite element method (V t ) is robust for the family of problems (P ), t 2 (0; 1] with respect to the solution sets Ht;k and error measures Et i lim sup sup Et (ut ? uNt ) = 0: N !1 t u 2H t

B t;k

It is robust with uniform order g(N ) i sup sup Et (ut ? uNt )  g(N ) t u 2H t

B t;k

where g(N ) ! 0 as N ! 1. We quote the following theorems from [3]. Numerische Mathematik Electronic Edition { page numbers may dier from the printed version page 516 of Numer. Math. 69: 509{522 (1995)

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Theorem 3.1. (V t) is free from locking i it is robust with uniform order F0 (N ). Moreover, let f (N ) be such that f (N )F0 (N ) = g(N ) ! 0 as N ! 1. t

Then (V ) shows locking of order f (N ) i it is robust with uniform order g(N ). Theorem 3.2. Let fVN g be F0-admissible and condition () be satis ed. Assume that Et (v) = kvkV . Then (V t ) is free from locking i g(N ) = sup v2V ;Sinf ku ? vkV  CF0 (N ): v =Dv =0 u2H0B;k

N

It shows locking of order f (N ) i C ?1 F0 (N )f (N )  g(N )  CF0 (N )f (N ): In this paper we consider following classes of nite element methods. The h-version of the nite element method. Let the polynomial degree p be xed, and let the partition
be varied in a quasi-uniform manner, which means that there exists a constant  > 0 such that  min (s ? sj?1 ) > h holds j j uniformly for all partitions
. The convergence is expected when we let h ! 0. In this case we write VN = Vh and we have N = O(h?1 ). The p-version of the nite element method. Let the partition
be xed, and let the polynomial degree p be varied. The convergence is expected when p ! 1. In this case we write VN = Vp and we have N = O(p). The h-p version of the nite element method. This is a combination of the above two methods. Both h and p are varied, and convergence is expected when N ! 1 where N = O(p=h). We rst investigate the p-version method. Instead of quoting Theorem 3.2, we pursue a direct error estimate. As in [3], we have the following lemma. Lemma 3.1. Let ut, u0 and uNt be solutions of (P t), (M 0) and (V t), respectively, and let et = ut ? u0 . Then kut ? uNt kE;t  v 2V ;Sinf ku0 ? v1 kE;t + Ct?1=2 v inf (19) ke ? v k : v =Dv =0 2V t 2 V 1

1

N

1

2

N

where C is a constant which depends only on and m. Proof. Since uNt is determined by projection in a Hilbert space, kut ? uNt kE;t  vinf kut ? vkE;t 2V  v inf ku0 ? v1 kE;t + v inf ke ? v k 2V 2V t 2 E;t N

1

N

2

N

inf ku0 ? v1 kE;t + Ct?1=2 v 2inf ke ? v k : v 1 2V ;S v 1 =Dv 1 =0 2V t 2 V The last step is from (15), where C depends only on and m. ut Theorem 3.3. Let ut 2 V and upt 2 Vp be solutions of (P t ) and (V t), respectively. Then kut ? upt kE;t  Cp?k kf kk?1 [1 + O(t1=2 )]; where C is a constant independent of t.



N

N

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Proof. We estimate the right hand side of (19). We know that S u0 = Du0 = S v1 = Dv1 = 0; where u0 = (0 ; w0 ; u0 ) and v1 = ( ; z; v). By virtue of (6) and (7), (20) 0 ? w00 ? u0 = ? z 0 ? v = 0; (21) w0 ? u00 = z ? v0 = 0: From (20), (21), and the boundary condition, we have (22) w00 (0) = w00 (1) = u00 (0) = u00(1) = u000 (0) = u000 (1) = 0; (23) z 0(0) = z 0 (1) = v0 (0) = v0 (1) = v00 (0) = v00 (1) = 0; (24)  ? = 1 (w ? z )0 + u ? v = 1 (u ? v)00 + u ? v: 0



Then (25)

0

0

2

0

0

ku0 ? v1 kE;t = k(0 ? )0 k0  (1 + 12 )k(u0 ? v)000 k0 ; since k(u0 ? v)0 k0  k(u0 ? v)00 k0  k(u0 ? v)000 k0 under condition (22) and (23). By the standard approximation theory of the p-version [3],  1 inf ku ? v 1 kE;t  1 + 2 inf ju0 ? vj3 v 2V ;S v =Dv =0 0 1

(26) Note that

1

p

1

 Cp?(k+3?3) ju0 jk+3 = Cp?k ju0 jk+3 :

v2V0

p

ju0 jk+3 = jw0 jk+2 = 2 j0 ? u0 jk+1 ;

thus by the regularity result (Corollary 2.1), we have, (27) ju0 jk+3  C kf kk?1 : For the second term in (19), we apply the standard approximation theory and Corollary 2.2 to get inf ket ? v2 kV  Cp?k ket kk+1  Ctp?k kf kk?1 : (28) v 2V 2

p

The conclusion follows by substituting (26), (27) and (28) into (19). ut Corollary 3.1. Under the same conditions of Theorem 3.3, we have

kut ? upt kV  Cp?k kf kk?1 [1 + O(t1=2 )]; Proof. Observe that kut ? upt kV  kut ? upt kE;t . ut

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From the proof of Theorem 3.3 we see that when t is small enough, the dominant error appears due to the approximation of the limiting problem. In other words, the ability of the nite element space to approximate the limiting problem decides the quality of the method. The interesting fact is that the limiting problem gives extra regularity: w0 has one more order of smoothness and u0 has two more orders of smoothness. The p-version of the nite element method takes advantage of this phenomenon and achieves the optimal rate of the convergence uniformly with respect to t. By the standard p-version approximation theory we have, F0 (N ) = sup vinf ku ? vkV  O(p?k )jujk+1  O(p?k )kf kk?1 ; 2V u2HkB

N

when B = kf kk?1 . Comparing this with Theorem 3.3 and Corollary 3.1, according to De nition 3.1 and 3.2, we say that the p-version method is free of locking and robust with order p?k uniformly with respect to t in both the V -norm and the energy norm. Next, we consider the h-version of the nite element method. We have the following theorem. Theorem 3.4. Consider problem (P t). Assume that Et (v) = kvkV . Then the h-version method shows locking of order h?2 . It is robust with order hp?2 for k  p. Proof. By the standard approximation theory, we have, F0 (N ) = sup vinf (29) ku ? vkV  O(hp ); 2V u2HkB

h

since k  p. Next we estimate g(N ). For u = (; w; u) 2 H0B;k , we know that S u = Du = 0. By the samilar argument as in the proof of Theorem 3.3, for v = ( ; z; v) 2 Vh , S v = Dv = 0, we have

2

2

1

2 (u ? v)000

+ k(u ? v)k20 + 12 k(u ? v)00 k20 + k(u ? v)0 k20: 0

ku ? vk2V =

12 (u ? v)000 + (u ? v)

+ 12 k(u ? v)00 k20 + k(u ? v)0 k20 0 =

Here we have used the fact that (u ? v)000 and u ? v are orthogonal. This can be veri ed by integration by parts and boundary conditions (u?v)(0) = (u?v)(1) = (u ? v)0 (0) = (u ? v)0 (1) = 0 as following: ((u ? v)000 ; u ? v) = ?((u ? v)00 ; (u ? v)0 ) = ((u ? v)0 ; (u ? v)00 ) = ?(u ? v; (u ? v)000 ); Therefore, (30) cku ? vk23  ku ? vk2V  C ku ? vk23 : But by the standard spline theory [17], (31) inf ku ? vk3  juj3 ; for p = 1; 2; k  0; juj3 6= 0; v2V0 (
) p

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(32) inf ku ? vk3  O(hp?2 )jujp+1 ; for p  3; k  p ? 2; jujp+1 6= 0: v2V0 (
) p

Note here, as u 2 H0B;k , u 2 H k+3 (I ). Combining (30) with (31) and (32), we have   C; p = 1; 2; k  0; (33) sup v2V ;Sinf k u ? v kV p?2 ) p  3; k  p ? 2: = O ( h v = D v =0 u2H0 B ;k

N

Relating (29) with (33), the conclusion follows from Theorem 3.2. ut By virtue of (16), the conclusion in Theorem 3.4 is also true for the energy norm. Comparing with the result in [1] and [3] for the Timoshenko beam which says that the h-version method shows locking of order h?1 and is uniformly robust with order hp?1 for p > 1, we see that for the circular arch problem considered, the locking is more serious. This can be seen from comparing the potential energy of the circular arch model (1) and that of the Timoshenko beam model (2). We see that the interplay between w and u disappears for the Timoshenko beam which relaxes one constraint of the circular arch model in the limit t = 0. Hence, the locking is less serious for the Timoshenko beam model. In general, the method which works well for the circular arch model also works well for the Timoshenko beam model. This is dierent from the cylindrical shell model considered in [13] and its R ! 1 counterpart, the Reissner-Mindlin plate model. Namely, There are schemes that converge asymptotically when applied to a shell problem but lock when applied to a plate problem. See [13] for details. We consider now the following modi ed scheme of the problem (V t ): (Vt ) Find uNt 2 VN such that, Bt (ut ; v) = (f ; v); 8v 2 VN ; with (34) Bt (u; v) = a(u; v) + mt (S u; S v) + 1t (Du; Dv) ;

where (; ) indicates reduced integration scheme, i.e., applying the p-point Gauss quadrature rule (Note that the (p + 1)-point rule is the exact integration for the problem considered). De ne, by , the L2 -projection into WN . It has been proved in [20] that (S u; S v) = (S u; S v); (Du; Dv) = (Du; Dv); and (Vt ) is equivalent to the following mixed method, (Mht ) Find (uNt ;  Nt ) = (Nt ; wtN ; uNt ; tN ; tN ) 2 VN  WN such that

a(uNt ; v) + b(v;  Nt ) = (f ; v) 8v 2 VN ; (35) (36) b(uNt ; y) ? t( Nt ; y) = 0 8y 2 WN ; with (37) tN = mt ( Nt ? (wtN )0 ? uNt ); tN = 1t ( wtN ? (uNt )0 ): Furthermore,

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Finite element method for circular arch problem

521

(38) kut ? uNt kV + k t ?  Nt k0  v2V inf (kut ? vkV + k t ? k0 ): ;2W N

N

For more information regarding mixed methods, the reader is referred to [5] and the reference therein. By virtue of (38), the standard approximation theory, and the regularity result in Theorem 2.1, we have kut ? uNt kV + k t ?  Nt k0  Chmin(p;k) kf kk?1 : We see that the mixed method exhibits optimal convergence uniformly with respect to t not only for the rotation t and the displacements wt ; ut , but also for the shear stress and the membrane stress,  = m (  ? w0 ? u );  = 1 ( w ? u0 ): t

t

t

t

t

t

t

t

t

Motivated by this fact and the result of Theorem 3.3, we now consider a combined scheme - mixed h-p version in which we let both h and p vary in the variational formulation (Mht ) (or (Vt )). We need the following approximation properties of spaces VN and WN . Lemma 3.2. If w 2 H k (I ), k  0, there exists z 2 WN such that kw ? z k0  C (k)p?k hmin(k;p+1) jwjk : If u 2 H k+1 (I ) \ V , there exists v 2 VN such that ku ? vk1  C (k)p?k hmin(k;p) kukk+1: Based on this lemma, (38), and Theorem 2.1, we have the following: Theorem 3.5.t Let (ut;  t ) 2 V  W and (uNt ;  Nt ) 2 VN  WN be solutions of t (M ) and (Mh ), respectively. Then kut ? upt kV + k t ?  Nt k0  C (k)p?k hmin(k;p) kf kk?1 ; where C (k) is a constant depending only on k. We summarize our results. For the model circular arch problem, the standard nite element method, i.e., the h-version method to (V t ) suers locking; the pversion can be applied directly to the standard variational formulation (V t ) and has optimal rate of convergence uniformly with respect to the thickness of the arch; mixed method has further advantage, it converges in an optimal rate uniformly with respect to the arch thickness even for the stresses. Hence, the mixed h-p method is recommended. From a practical point of view (reducing the computational cost), we suggest reduced integration scheme (Vt ) with the stresses calculated by (37). Finally, we would like to point out that all results in this paper hold for most of the arch models discussed in [20]. Acknowledgement. The author would like to thank Professor Ivo Babuska for his encouragement on this area of investigation and helpful discussions about the literature. The author is very grateful to Professor Manil Suri and an anonymous referee for many valuable comments and suggestions which considerably improve the paper.

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Numerische Mathematik Electronic Edition { page numbers may dier from the printed version page 522 of Numer. Math. 69: 509{522 (1995)