Numerische Mathematik

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1 Numer. Math. 69: (1995) Numerische Mathematik c Springer-Verlag 1995 Locking and robustness in the finite element method for circular arch problem Zhimin Zhang Department of Mathematics, Texas Tech University, Lubbock, TX 7949, USA zhang@ttmath.ttu.edu Received June 5, 1992 / Revised version received May 17, 1994 Summary. In this paper we discuss locking and robustness of the finite element 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 h p 2 for p > 2 which explains the fact that the quadratic element for some circular arch problems suffers 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 Classification (1991): 65N3, 73K5, 73V5 1. Introduction The development of robust finite element methods for the circular arch problem is of interest in many practical applications. For the standard finite element method, namely the h-version method, locking occurs in the thin arch limit [2]. Many efforts have been focused on overcoming the locking effects, for example, mixed methods which use mixed variational principles (see [8], [12], [15], and [2] [22]), the Petrov- Galerkin method [11] in which the test function space differs from the trial function space, and reduced integration methods (see [12], [14], [16], and [18] [22]). In [2], 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 effects 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 finite element method for circular arch problems. This work was partially ported by the National Science Foundation grant DMS

2 51 Z. Zhang Also, the locking effect has not been characterized mathematically. These are the topics of the current paper. At this point we should mention that for a closely related problem, the Timoshenko beam, the investigation is quite complete (see [1], [9] and [1]). Recently, Babuška and Suri set up a general mathematical framework on locking effects and robustness in the finite 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 finite element methods for circular arch models. The major step is to verify that the so-called condition (α) of [3] is satisfied 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 effect 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 = (, 1) and L 2 (I) be the space of square integrable functions on I. Denote as usual H k (I) = {v L 2 (I) v,, v (k) L 2 (I)}, H 1 (I) = {v H 1 (I), v() = v(1) = }, and denote H 1 (I) as the dual space of H 1 (I). Define ( 1/2 1 v = v(x) dx) 2, v k = v (k), v k = v 2 i i k We consider vector spaces W = L 2 (I) 2, V = H 1 (I) 3, V = H 1 (I) 3, where H(I) 3 = H(I) H(I) H(I). We define on W the inner product which induces the norm (w, z) = and define on V the inner product which induces the norm 1 w(x) z(x)dx, w = w W = (w, w) 1/2 ; (u, v) 1 = 1 u (x) v (x)dx, u V = (u, u) 1/2 From the Poincaré inequality, V is equivalent to the Sobolev norm 1 on V which is defined by u 1 = ((u, u) 1 + (u, u)) 1/2. The Sobolev norms u k, u k on H k (I) n, n = 2, 3, are defined similarly. Throughout the paper C is used to denote a generic constant which is not necessarily same at each occurrence. 1. 1/2.

3 Finite element method for circular arch problem 511 f x Φ u d A y w R Fig Circular arch problems and their properties Let us consider a uniform linear elastic circular arch with thickness d. Its center-line has radius R and length L. See Fig. 1. The arch has Young s modulus E, shear modulus G, cross-sectional area A and moment of inertia I = A y2 da = O(Ad 2 ), and is subjected to a distributed load (per unit length) f = ( f 1, f 2, f 3 ) 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 φ, the radial displacement w and the tangential displacement ū of its center-line. We consider the following circular arch model which is expressed by its potential energy: (1) π( φ, w, ū) = 1 2 L [ EI ( d φ d s + w R 2 1 R ( +kga φ d w d s ū R 2f 1 φ 2f 2 w 2f 3 ū ] d s, ) 2 dū d s ) 2 ( w + EA R dū d s where s is the arch length coordinate, and k is the shear correction factor. This model was discussed in [2] where it was treated by a mixed finite element method. In the expression for the strain energy, the term with coefficient EI is the bending energy; the term with coefficient kga is the shear energy; and the term with coefficient 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 and impose the Mindlin s hypotheses ū = on the center-line, we then recover the potential energy for the Timoshenko beam: (2) π ( φ, w) = 1 2 L [ EI ( d φ d s ) 2 + kga ( φ d w d s ) 2 ) 2 2 f 1 φ 2 f 2 w which contains no membrane energy term. For the relationship of this circular arch model with other models, the reader is referred to [2]. ] d s,

4 512 Z. Zhang We introduce the change of variables: s = Ls, (R φ( s), w( s), ū( s)) = (φ(s), w(s), u(s)), ( f 1 ( s), f 2 ( s), f 3 ( s)) = EI L 2 R 2 (Rf 1(s), f 2 (s), f 3 (s)). Multiplying (1) throughout by LR 2 /(EI) yields the nondimensional form of the potential energy as π(φ, w, u) = 1 1 [ (φ + βw u ) 2 + m 2 t (βφ w βu) t (βw u ) 2 ] (3) 2f 1 φ 2f 2 w 2f 3 u ds, where β = L/R, t 1 = AR 2 /I, m = kg/e, and v = dv/ds. Clearly, t = O(d 2 ), and β is the nondimensional measure of the arch depth which varies from to 2π. The corresponding variational problem is: (P t ) Given f = (f 1, f 2, f 3 ) V, find u t = (φ t, w t, u t ) V such that B t (u t, v) = (f, v), v = (ψ, z, v) V, where B t is a bilinear form defined on V V as, (4) B t (u, v) = a(u, v) + m t (Su, Sv) + 1 (Du, Dv), t with (5) (6) (7) a(u, v) = (φ + βw u, ψ + βz v ), Sv = βψ z βv, Dv = βz v. Problem (P t ) is equivalent to the following mixed variational problem: (M t ) Given f V and g W, find (u t, ζ t ) = (φ t, w t, u t ; ξ t, η t ) V W such that (8) (9) a(u t, v) + b(v, ζ t ) = (f, v) v V, b(u t, y) t(ζ t, y) = (g, y) y W, where b(v, y) = m(sv, y 1 ) + (Dv, y 2 ), g = (, ). Let us consider a slightly more general case when g (, ). Theorem 2.1. Let f H k 1 (I) 3, g H k (I) 2. Then there exists a unique sequence of solutions {u t, ζ t } V W to problem (M t ) for t [, 1] such that (1) u t k+1 + ζ t k C( f k 1 + g k ), where C is a constant independent of f, g and t.

5 Finite element method for circular arch problem 513 Proof. The following three properties have been proved in [2]. 1) a(, ) is symmetric and positive semidefinite. 2) There exists α > such that for any z N (b) = {v V b(v, y) = y W } a(z, z) α z 2 V. 3) There exists γ > such that for any w W, there is a v V with b(v, w) γ v V w W. 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) V W, there exists a unique pair (u t, ζ t ) V W that solves Problem (M t ), moreover, (11) u t V + ζ t W C( f V + g W ), or equivalently (12) u t 1 + ζ t C( f 1 + g ). After performing integration by parts, we may obtain the strong form of (M t ), φ t + mβξ t (Du t ) = f 1, βφ t + mξ t + βη t + βdu t = f 2, (S t φ ) t mβξ t + η t + (Du t ) = f 3, msut tξ t = m(βφ t w t + βu t ) tξ t = g 1, Du t tη t = βw t u t tη t = g 2, φ t () = w t () = u t () = φ t (1) = w t (1) = u t (1) =. Therefore we can express higher derivatives by lower derivatives by reorganizing the terms in (S t ) as φ t = mβξ t f 1 (tη t + g 2 ), mξ t = f 2 β(φ (S t t + η t + tη t + g 2 ), ) a η t = f 1 + f 3, mw t = mβ(φ t + u t ) tξ t g 1, u t = βw t tη t g 2. Applying mathematical induction, the regularity of (1) is obtained by the bootstrapping method from (12) and successive differentiation of (S t ) a. Corollary 2.1. Let f H k 1 (I) 3. Then there exists a unique sequence of solutions {u t } V to problem (P t ) for t [, 1] such that (13) u t k+1 C f k 1, where C is a constant independent of f and t. Now we go back to our special case where g = (, ) and we have (M t ) { a(ut, v) + b(v, ζ t ) = (f, v) v V, b(u t, y) t(ζ t, y) = y W, when t >, and

6 514 Z. Zhang (M ) { a(u, v) + b(v, ζ ) = (f, v) v V, b(u, y) = y W, for t =. Let us have a further discussion on the problem (M ). From b(u, y) =, we have Su = βφ w βu =, Du = βw u =. Observe that b(u, ζ ) =, and we have Applying Su = and Du = yields a(u, u ) = (f, u ). a(u, u ) = = 1 1 (φ + Du )ds = 1 ( 1 β w + u ) 2 ds = (φ ) 2 ds 1 ( 1 β w + βw ) 2 ds, which is the strain energy for the circular ring. We see that, as the limit case, (M ) 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 H k 1 (I) 3 and let (u t, ζ t ), (u, ζ ) be solutions of (M t ) and (M ), respectively. Then u t u k+1 + ζ t ζ k C 2 t f k 1, where C, independent of t, is the same constant as in Theorem 2.1. Proof. First, we apply Theorem 2.1 with g = (, ) to Problem (M ), and we have (14) u k+1 + ζ k C f k 1. Subtraction of (M ) from (M t ) yields { a(ut u, v) + b(v, ζ t ζ ) = v V, b(u t u, y) t(ζ t ζ, y) = t(ζ, y) y W, Applying Theorem 2.1 with f = (,, ) and g = tζ yields u t u k+1 + ζ t ζ k Ct ζ k C 2 t f k 1. The last step is from (14). Corollary 2.2. Assume that f H k 1 (I) 3. Let u t, (u, ζ ) be solutions of (P t ) and (M ), respectively. Then u t u k+1 C 2 t f k 1, where C, independent of t, is the same constant as in Theorem 2.1.

7 Finite element method for circular arch problem 515 For the bilinear form B t, we have (15) B t (v, v) Ct 1/2 v V, v V, (16) B t (v, v) C 1 v V, v V, where constant C depends only on β and m. The inequality (15) is obvious from the definition. The proof of (16) may be found in [2]. From the Lax-Milgram Lemma [6], the existence and uniqueness of (P t ) is guaranteed by (16). Define v 2 E,t = B t (v, v), then E,t is a norm (called energy norm) on V which is equivalent to the norm V for any fixed t. Now we define for k, t 1, the spaces H t,k = {u H k+1 (I) 3, u is the solution of (P t ) for some f H k 1 (I) 3 }, H B k = {u H k+1 (I) 3, u k+1 + ξ k + η k B}, H B t,k = H t,k H B k. Following [3], we introduce condition (α): For any u t H B t,k, there is a u H B,k such that, u t u k+1 Ct 1/2 B. Then Theorem 2.2 simply says that: For the problems considered, Condition (α) is satisfied with power t (instead of just t 1/2, as needed). Note that H t,k = H k+1 (I) 3 for t >. Also, using Corollary 2.2, for any u H B,k, there exists a sequence u t H t,k such that u t E,t C u k+1, u t k+1 C u k+1, and u t u V as t. This shows that H,k is precisely the limit set of H t,k as t, in the sense of [3]. 3. Locking and robustness of finite element methods Now we are going to approximate problem (P t ). Partition the interval I = [, 1] into subintervals I j = [s j 1, s j ] with = s < s 1 < < s n = 1. Let h = max j (s j s j 1 ). Denote by V p 1 ( ), the space of piecewise polynomials of order p which are not necessarily continuous at nodal points s j. Define V p ( ) = V p 1 ( ) C (I), V p ( ) = V p ( ) H 1 (I), V N = V p ( )3, W N = V p 1 ( )2, where N is the dimension. Obviously, V N V. The standard finite element solution of (P t ) is defined as following: (V t ) Find u N t V N such that B t (u N t, v) = (f, v), v V N.

8 516 Z. Zhang The sequence {V N } defines a rule that decreases the error by increasing N. We now define locking and robustness following [3]. We assume that {V N } is F -admissible, i.e., it leads to a certain fixed rate F (N) of convergence when functions in H k+1 (I) 3 are approximated, in the following sense (17) C 1 F (N) u H B k inf u v V CF (N). v V N Here F (N) as N and C depends on B but is independent of N. Note that since for t >, H t,k = H k+1 (I) 3, we may show by using (17), that for 1 t t >, t fixed, (18) C 1 F (N) E t (u t u N t ) CF (N), u H B t,k where C is independent of t (but depends on t ) and is not necessarily same as in (17). Here we choose measure E t as V or the energy norm. Hence, F (N) measures the rate of convergence without locking (i.e., when t t ). To see if there is locking, we compare the observed rate of convergence as t with F (N). For t (, 1] and N, we define by L(t, N) = u H B t,k E t (u t u N t )F (N) 1, the locking ratio with respect to the space H t,k and error measure E t for the problem (P t ). Then if (18) holds, we have the following definitions. Definition 3.1. The finite element method (V t ) is free from locking for the family of problems (P t ), t (, 1] with respect to the solution sets H t,k and error measures E t iff [ ] lim N It shows locking of order f(n) iff [ < lim N L(t, N) t (,1] = M <. ] L(t, N)f(N) 1 = C < t (,1] where f(n) as N. It shows locking of at least (respectively at most) order f(n) if C > (respectively C < ). Definition 3.2. The finite element method (V t ) is robust for the family of problems (P t ), t (, 1] with respect to the solution sets H t,k and error measures E t iff lim N t u t H B t,k It is robust with uniform order g(n) iff t where g(n) as N. u t H B t,k We quote the following theorems from [3]. E t (u t u N t ) =. E t (u t u N t ) g(n)

9 Finite element method for circular arch problem 517 Theorem 3.1. (V t ) is free from locking iff it is robust with uniform order F (N). Moreover, let f(n) be such that f(n)f (N) = g(n) as N. Then (V t ) shows locking of order f(n) iff it is robust with uniform order g(n). Theorem 3.2. Let {V N } be F -admissible and condition (α) be satisfied. Assume that E t (v) = v V. Then (V t ) is free from locking iff g(n) = u H B,k It shows locking of order f(n) iff inf u v V CF (N). v V N,Sv=Dv= C 1 F (N)f(N) g(n) CF (N)f(N). In this paper we consider following classes of finite element methods. The h-version of the finite element method. Let the polynomial degree p be fixed, and let the partition be varied in a quasi-uniform manner, which means that there exists a constant τ > such that τ min(s j s j 1 ) > h holds uniformly for all j partitions. The convergence is expected when we let h. In this case we write V N = V h and we have N = O(h 1 ). The p-version of the finite element method. Let the partition be fixed, and let the polynomial degree p be varied. The convergence is expected when p. In this case we write V N = V p and we have N = O(p). The h-p version of the finite element method. This is a combination of the above two methods. Both h and p are varied, and convergence is expected when N where N = O(p/h). We first 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 u t, u and u N t and let e t = u t u. Then be solutions of (P t ), (M ) and (V t ), respectively, (19) u t u N t E,t inf v 1 V N,Sv u v 1 E,t + Ct 1/2 inf e 1=Dv 1= v t v 2 V. 2 V N where C is a constant which depends only on β and m. Proof. Since u N t is determined by projection in a Hilbert space, u t u N t E,t inf v V N u t v E,t inf u v v 1 E,t + inf e 1 V N v t v 2 E,t 2 V N inf v 1 V N,Sv u v 1 E,t + Ct 1/2 inf e 1=Dv 1= v t v 2 V. 2 V N The last step is from (15), where C depends only on β and m. Theorem 3.3. Let u t V and u p t V p be solutions of (P t ) and (V t ), respectively. Then u t u p t E,t Cp k f k 1 [1 + O(t 1/2 )], where C is a constant independent of t.

10 518 Z. Zhang Proof. We estimate the right hand side of (19). We know that Su = Du = Sv 1 = Dv 1 =, where u = (φ, w, u ) and v 1 = (ψ, z, v). By virtue of (6) and (7), (2) (21) βφ w βu = βψ z βv =, βw u = βz v =. From (2), (21), and the boundary condition, we have (22) (23) (24) Then (25) w () = w (1) = u () = u (1) = u () = u (1) =, z () = z (1) = v () = v (1) = v () = v (1) =, φ ψ = 1 β (w z) + u v = 1 β 2 (u v) + u v. u v 1 E,t = (φ ψ) (1 + 1 β 2 ) (u v), since (u v) (u v) (u v) under condition (22) and (23). By the standard approximation theory of the p-version [3], (26) Note that inf v u v 1 E,t (1 + 1β ) 1 V p,sv 1=Dv 1= 2 Cp (k+3 3) u k+3 = Cp k u k+3. u k+3 = β w k+2 = β 2 φ u k+1, thus by the regularity result (Corollary 2.1), we have, inf v V p u v 3 (27) u k+3 C f k 1. For the second term in (19), we apply the standard approximation theory and Corollary 2.2 to get (28) inf v 2 V p e t v 2 V Cp k e t k+1 Ctp k f k 1. The conclusion follows by substituting (26), (27) and (28) into (19). Corollary 3.1. Under the same conditions of Theorem 3.3, we have u t u p t V Cp k f k 1 [1 + O(t 1/2 )], Proof. Observe that u t u p t V u t u p t E,t.

11 Finite element method for circular arch problem 519 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 finite element space to approximate the limiting problem decides the quality of the method. The interesting fact is that the limiting problem gives extra regularity: w has one more order of smoothness and u has two more orders of smoothness. The p-version of the finite 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, F (N) = u H B k inf u v V O(p k ) u k+1 O(p k ) f k 1, v V N when B = f k 1. Comparing this with Theorem 3.3 and Corollary 3.1, according to Definition 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 finite element method. We have the following theorem. Theorem 3.4. Consider problem (P t ). Assume that E t (v) = v V. Then the h-version method shows locking of order h 2. It is robust with order h p 2 for k p. Proof. By the standard approximation theory, we have, (29) F (N) = u H B k inf u v V O(h p ), v V h since k p. Next we estimate g(n). For u = (φ, w, u) H,k B, we know that Su = Du =. By the samilar argument as in the proof of Theorem 3.3, for v = (ψ, z, v) V h, Sv = Dv =, we have 2 u v 2 V = 1 β 2 (u v) + (u v) + 1 β 2 (u v) 2 + (u v) 2 2 = 1 (u v) β2 + (u v) β 2 (u v) 2 + (u v) 2. Here we have used the fact that (u v) and u v are orthogonal. This can be verified by integration by parts and boundary conditions (u v)() = (u v)(1) = (u v) () = (u v) (1) = as following: ((u v), u v) = ((u v), (u v) ) = ((u v), (u v) ) = (u v, (u v) ), Therefore, (3) But by the standard spline theory [17], c u v 2 3 u v 2 V C u v 2 3. (31) (32) inf v V p ( ) u v 3 u 3, for p = 1, 2, k, u 3 ; inf v V p ( ) u v 3 O(h p 2 ) u p+1, for p 3, k p 2, u p+1.

12 52 Z. Zhang Note here, as u H B,k, u Hk+3 (I). Combining (3) with (31) and (32), we have (33) u H B,k inf u v V v V N,Sv=Dv= { C, p = 1, 2, k ; = O(h p 2 ) p 3, k p 2. Relating (29) with (33), the conclusion follows from Theorem 3.2. 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 h p 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 ū disappears for the Timoshenko beam which relaxes one constraint of the circular arch model in the limit t =. 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 different from the cylindrical shell model considered in [13] and its R 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 modified scheme of the problem (V t ): (V t ) Find u N t V N such that, B t (u t, v) = (f, v), v V N, with (34) B t (u, v) = a(u, v) + m t (Su, Sv) + 1 t (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). Define, by π, the L 2 -projection into W N. It has been proved in [2] that (Su, Sv) = (πsu, Sv), (Du, Dv) = (πdu, Dv), and (V t ) is equivalent to the following mixed method, (M t h ) Find (un t, ζ N t ) = (φ N t, w N t, u N t ; ξ N t, η N t ) V N W N such that (35) (36) with (37) a(u N t, v) + b(v, ζ N t ) = (f, v) v V N, b(u N t, y) t(ζ N t, y) = y W N, ξ N t = m t π(βφn t (w N t ) βu N t ), η N t = 1 t π(βwn t (u N t ) ). Furthermore, (38) u t u N t V + ζ t ζ N t inf ( u t v V + ζ t χ ). v V N,χ W N For more information regarding mixed methods, the reader is referred to [5] and the reference therein.

13 Finite element method for circular arch problem 521 By virtue of (38), the standard approximation theory, and the regularity result in Theorem 2.1, we have u t u N t V + ζ t ζ N t Ch min(p,k) f k 1. We see that the mixed method exhibits optimal convergence uniformly with respect to t not only for the rotation φ t and the displacements w t, u t, but also for the shear stress and the membrane stress, ξ t = m t π(βφ t w t βu t ), η t = 1 t π(βw t u 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 (Mh t t ) (or (V )). We need the following approximation properties of spaces V N and W N. Lemma 3.2. If w H k (I), k, there exists z W N such that w z C(k)p k h min(k,p+1) w k. If u H k+1 (I) V, there exists v V N such that u v 1 C(k)p k h min(k,p) u k+1. Based on this lemma, (38), and Theorem 2.1, we have the following: Theorem 3.5. Let (u t, ζ t ) V W and (u N t, ζ N t ) V N W N be solutions of (M t ) and (Mh t ), respectively. Then u t u p t V + ζ t ζ N t C(k)p k h min(k,p) f k 1, where C(k) is a constant depending only on k. We summarize our results. For the model circular arch problem, the standard finite element method, i.e., the h-version method to (V t ) suffers locking; the p-version 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 (V t ) 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 [2]. Acknowledgement. The author would like to thank Professor Ivo Babuška 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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