ANALYSIS OF A BILINEAR FINITE ELEMENT FOR SHALLOW SHELLS II: CONSISTENCY ERROR

Transkriptio

1 Helsinki University of Technology Institute of Mathematics Research Reports Teknillisen korkeakoulun matematiikan laitoksen tutkimusraporttisarja Espoo 2001 A433 ANALYSIS OF A BILINEAR FINITE ELEMENT FOR SHALLOW SHELLS II: CONSISTENCY ERROR Ville Havu Juhani Pitkäranta AB TEKNILLINEN KORKEAKOULU TEKNISKA HÖGSKOLAN HELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D HELSINKI

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3 Helsinki University of Technology Institute of Mathematics Research Reports Teknillisen korkeakoulun matematiikan laitoksen tutkimusraporttisarja Espoo 2001 A433 ANALYSIS OF A BILINEAR FINITE ELEMENT FOR SHALLOW SHELLS II: CONSISTENCY ERROR Ville Havu Juhani Pitkäranta Helsinki University of Technology Department of Engineering Physics and Mathematics Institute of Mathematics

4 Ville Havu and Juhani Pitkäranta: Analysis of a bilinear finite element for shallow shells II: Consistency error; Helsinki University of Technology Institute of Mathematics Research Reports A433 (2001). Abstract: We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane-dominated case, under severe assumptions on the domain, the finite element mesh and on the regularity of the solution, an error bound O(h + t 1 h 1+s ) can be obtained if the contribution of transverse shear is neglected. Here t is the thickness of the shell, h the mesh spacing, and s a smoothness parameter. In the bending-dominated case, the uniformly optimal bound O(h) is achievable but requires that membrane and transverse shear strains are oforder O(t 2 ) as t! 0. In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound O(h). AMS subject classifications: 65N30, 73K15 Keywords: finite elements, locking, shells Ville.Havu@hut.fi Juhani.Pitkaranta@hut.fi ISBN ISSN Espoo 2001 Helsinki University of Technology Department of Engineering Physics and Mathematics Institute of Mathematics P.O. Box 1100, HUT, Finland math@hut.fi

5 1 Introduction Approximation of deformation states arising in thin shells by low-order finite element methods is known to be a nontrivial task. Different locking modes degrade the convergence rate of the most basic formulations when approximating bending-dominated or inextensional deformations. However, it is equally well-known by now that a suitable variational crime can be used to retain the convergence properties in such cases. This can even be done up to an optimal order and smoothness requirements for certain shell geometries as it was shown in Part I of this paper [3], see also [6]. The real challenge begins when one aims to find a formulation that has a satisfactory behavior also in the membrane-dominated states of deformation. In this case one is inevitably led to consider the questions of consistency and stability of the formulation since the approximation properties will rarely be a problem in such a case, but lack of consistency or stability can yield a very large error component. Probably most low-order shell elements aimed to be general in nature contain the basic ideas of MITC4 by Bathe and Dvorkin [1]. In [4] it was shown that this formulation is in fact equivalent to a certain variational crime already considered in [6]. In this paper we extend our analysis of the MITCtype elements and address their consistency and stability properties. We show that at least under favorable circumstances, this kind of an element can indeed approximate well also membrane-dominated deformation. However, due to the lack of stability inthe membrane-dominated case, we can bound the consistency error in this case only non-uniformly with respect to the thickness t of the shell. We need also strong assumptions on the problem setup and on the finite element mesh as in the previous part [3]. Under such hypotheses and under certain additional hypotheses on the solution we show that the consistency error is at most of order O(h + t 1 h 1+s ) where h is the mesh spacing and s 0 is a parameter depending on the degree of smoothness of the exact solution. As s can be arbitrarily large in principle, one can have t 1 h s = O(1) for reasonable sequences of (t; h) if the solution is very smooth. In such a case the consistency error is O(h), which is the optimal order for bilinear elements. The conclusion from the Parts I-II of the paper is then that at least under extremely favorable circumstances, both bendingand membrane-dominated smooth deformations can be approximated with nearly optimal accuracy by the bilinear MITC4 element. To what extent this holds for more general problem setups, deformation states, and finite element meshes, isawideopen problem. Another topic to be considered in this paper is the asymptotic behavior of the consistency error in the case of an inextensional deformation. In [3] we considered the problem of finding a best finite element approximation of agiven inextensional deformation. At that point the question of consistency was deliberately left aside. However, in real computations one is inevitably faced with the fact that since the reduced inextensional space is not a subspace of the corresponding continuous space, the consistency error does not 3

6 tend to zero when the thickness t! 0, but to some finite value depending on h. Here we show that this error term is of the optimal order O(h). However, to obtain this result we need much stronger regularity assumptions on the exact solution than in the previous Part I [3] where we bounded the approximation error. Whether our analysis here is sharp is not clear at the moment. The plan of this paper is as follows. In section 2 we describe the problems to be considered and in section 3 we consider two slightly different FEM approximations to these. Section 4 is devoted to the consistency error in the non-asymptotic case (t >0) whereas section 5 deals with the asymptotic consistency error in the inextensional deformation state. In the following we denote by C a generic constant that may take a differentvalue in different usage. The constants may depend on the geometry parameters of the problem but are otherwise independent of the parameters, unless indicated explicitly. The Sobolev norm and seminorm are denoted by jj jj k and j j k respectively on the assumed rectangular domain. Further, jj jj L 2 = jj jj 0. 2 The shell problem We use basically the same shell model of Reissner-Naghdi type as in [3] but with two different scalings. Denoting by u =(u; v; w; ; ψ) the vector of three translations and two rotations we let the (scaled) total energy of the shell be given either by F M (u) = 1 2 t2 A b (u;u)+a m (u;u) Q(u) or by F B (u) = 1 2 Ab (u;u)+t 2 A m (u;u) Q(u) where t is the thickness of the shell and Q represents the load potential. Here the subscripts M and B refer to the natural scalings of the total energy in membrane and bending-dominated deformations, respectively. We assume that in both cases Q(u) defines a bounded linear functional on the corresponding energy space to be defined later. The bilinear forms A b (u;v) and A m (u;v) arising from the bending and membrane energies are given by A b (u;v)= Z Ω Φ ν(»11 +» 22 )(u)(» 11 +» 22 )(v)+(1 ν) 2 i;j=1» ij (u)» ij (v) Ψ dxdy 4

7 and A m (u;v)=6fl(1 ν) +12 Z Z Ω Ω fρ 1 (u)ρ 1 (v) +ρ 2 (u)ρ 2 (v)gdxdy fν(fi 11 + fi 22 )(u)(fi 11 + fi 22 )(v) +(1 ν) 2 i;j=1 fi ij (u)fi ij (v)gdxdy where overbars denote complex conjugation. Here ν is the Poisson ratio of the material, fl is a shear correction factor and» ij, fi ij and ρ i represent the bending, membrane and transverse shear strains respectively depending on u as and fi + aw fi + bw fi 12 = )+cw = fi 21 ρ = )=» 21 (2.1) ρ 2 : (2.2) The integration is taken over the midsurface Ω of the shell which we assume to occupy the rectangular region (0;L) (0;H) in the xy-coordinate space satisfying d 1» L=H» d for some constant d>0. We are considering the shell to be shallow and assume that the parameters a, b and c defining the geometry can be taken constants. We further note that if ab c 2 > 0 the shell is elliptic, if ab c 2 = 0 it is parabolic and if ab c 2 < 0 we have a hyperbolic shell. The above two energy formulations lead naturally to two differently scaled variational formulations, the membrane (M) and bending (B) cases: (M) Find u 2U M such that A M (u;v)=t 2 A b (u;v)+a m (u;v)=q(v) 8v 2U M : (2.3) (B) Find u 2U B such that A B (u;v)=a b (u;v)+t 2 A m (u;v)=q(v) 8v 2U B : (2.4) Here U M and U B are the membrane and bending energy spaces, respectively, which we take to be subspaces of [Hp 1 (Ω)]5 where Hp 1 (Ω) is the usual Sobolev space with periodic boundary conditions imposed at y =0;H. In U B no constraints are imposed at x = 0;L whereas in U M we assume the constraints u = v = w = = ψ = 0 at x =0;L. In Case (B) we must also remove the rigid displacements from U B so as to make (2.4) uniquely solvable. For the 5

8 convenience of our error analysis, we make here somewhat stronger assumptions than are needed for the well-posedness of (2.4): We introduce the set of pseudo-rigid displacements as Z = fv 2 [H 1 p (Ω)]5 j v = 5 i=1 C i e i g where e i is the ith Euclidean unit vector, assume that Q(v) = 0 for every v 2 Z, and let U B = Z? in [Hp 1 (Ω)]5. Finally we denote the energy norms p on U M and on U B, respectively, by jjj jjj M = AM ( ; ) and jjj jjj B = p AB ( ; ) =t 1 jjj jjj M. Letting t! 0 in (2.4) we obtain the inextensional formulation of the problem (B): Find u 0 2U 0 such that A b (u 0 ;v)=q(v) 8v 2U 0 (2.5) where U 0 = fv 2 U B ja m (v;v) = 0g ρ U B is the space of inextensional deformations. 3 The reduced-strain FE scheme We consider the bilinear MITC4 finite element formulation of the problems (2.3) (2.5). As in [3] we make strong assumptions on the finite element mesh so as to allow theuse of Fourier methods in the error analysis. Assume that Ω is divided into rectangular elements with node points (x k ;y n ), k =0;:::;N x, n =0;:::;N y and a constant mesh spacing h y in the y-direction and that the aspect ratios of the elements satisfy d 1» h k x=h y» d for some d>0where h k x = xk+1 x k. To this mesh we associate the standard space V h ρ Hp 1 (Ω) of continuous piecewise bilinear functions. We then define the FE spaces U M;h and U B;h, respectively, as subspaces of Vh 5 where the boundary or orthogonality conditions of Problems (M) and (B) are enforced. The finite element formulation of problems (2.3) (2.5) are then obtained by replacing U M, U B by U M;h, U B;h and by modifying the bilinear form A m numerically as Z A h m (u;v)=6fl(1 ν) f~ρ 1 (u)~ρ 1 (v) +~ρ 2 (u)~ρ 2 (v)gdxdy +12 Z Ω fν( fi ~ 11 + fi ~ 22 )(u)( fi ~ 11 + fi ~ 22 )(v) Ω +(1 ν) 2 i;j=1 ~fi ij (u) ~ fi ij (v)gdxdy where ~ fi ij = R ij fi ij, ~ρ i = R i ρ i with suitable reduction operators R ij and R i. As in [3] we choose these operators for fi ii and ρ i to be ~fi 11 =Π x hfi 11 ; ~ fi 22 =Π y h fi 22; ~ρ 1 =Π x hρ 1 ~ρ 2 =Π y h ρ 2 (3.1) 6

9 and Wy h where Π x h and Πy h are orthogonal L2 -projections onto spaces Wh x consisting of functions that are constant in x an piecewise linear in y or constant in y and piecewise linear in x respectively. For the term fi 12 we consider two different alternatives (E1) ~ fi 12 =Π xy h fi 12 (E2) ~ fi 12 = fi 12 + S 12 where Π xy h =Πx h Πy h is the orthogonal L2 -projection onto elementwise constant functions and for every element K S 12jK (Πx hw)(x h k x=2) (Πy h w)(y h y=2) + (Π xy h cw cw) is essentially the term introduced in [4]. Remark 3.1. The formulation (E1) was assumed in [3], [6]. This is a straightforward interpretation of the MITC4 finite element formulation, but as shown recently in [4], (E2) is actually a closer interpretation of MITC4. The two formulations are practically equivalent when approximating inextensional deformations but may differ in other deformation states, as noted in [4]. Our error analysis here can only detect a small difference when approximating smooth membrane-dominated deformations, see Theorem 4.4 ahead. The above definitions give rise to two different FE-schemes for solving (2.3) and (2.4): (M h ) Find u h 2U M;h such that A h M (u h;v)=t 2 A b (u h ;v)+a h m (u h;v)=q(v) 8v 2U M;h (3.2) (B h ) Find u h 2U B;h such that A h B (u h;v)=a b (u h ;v)+t 2 A h m (u h;v)=q(v) 8v 2U B;h (3.3) Upon passing to the limit t! 0 in (3.3) we obtain a finite element formulation of the asymptotic problem (2.5): Find u h 2U 0;h such that A b (u h ;v)=q(v) 8v 2U 0;h (3.4) where U 0;h = fv 2U B;h ja h m (v;v)=0g. To analyze the discretization errors e M = jjju u h jjj M;h and e B = jjju u h jjj M;h as originating from (3.2) and (3.3) when t > 0, we split e M and e B into two orthogonal components in both cases, namely the approximation errors e a;m (u) = min jjju vjjj M;h v2um;h e a;b (u) = min jjju vjjj B;h v2ub;h 7

10 and the consistency errors e c;m (u) = sup v2um;h (A M A h M )(u;v) jjjvjjj M;h (3.5) (A B A h B e c;b (u) = sup )(u;v) (3.6) v2ub;h jjjvjjj B;h where jjj jjj M;h = p A h M ( ; ), jjj jjj B;h = p A h B ( ; ). These definitions imply that e 2 M = e2 a;m + e2 c;m e 2 B = e2 a;b + e2 c;b: (For a detailed reasoning, see [5].) We note that standard finite element theory gives the bound e a;m» Chjjujj 2 and for e a;b we refer to [3]. Hence, the main task of this paper is to bound e c;m and e c;b. We aim to analyze these error terms with both proposed strain-reductions (E1) and (E2). The asymptotic formulations (2.5), (3.4) lead to a similar error decomposition. We have for u 0 2U 0 the asymptotic approximation error e 0 a (u 0 )= min jjju 0 vjjj B;h v2u 0;h which was under consideration in [3]. On the other hand, at t =0we have that A b (u 0 ;v)=q(v) 8v 2U 0 for the inextensional solution u 0 2U 0, and that A b (u h ;v)=q(v) 8v 2U 0;h (3.7) for the corresponding finite element solution u h. Let ~u h be the best finite element approximation to u 0 in U 0;h, i.e. A b (~u h ;v)=a b (u 0 ;v) 8v 2U 0;h : (3.8) By (3.7), (3.8) the asymptotic consistency error u h ~u h 2U 0;h satisfies and thus we can define A b (u h ~u h ;v)=q(v) A b (u 0 ;v) 8v 2U 0;h (3.9) e 0 c (u 0 ) + jjju h ~u h jjj h = Q(v) A b (u 0 ;v) sup : (3.10) v2u 0;h jjjvjjj B;h As in [3], the main tool of our analysis will be the Fourier transform where we write u(x; y) = 2Λ ' (y) =e i y ; ' (y)ffi (x) = 2Λ # (x; y); Λ=f = 2ßν H ; ν 2 Zg; 8

11 making use of the periodic boundary conditions at y = 0;H. For functions in the FE space we write analogously where v(x; y) = 2ΛN ~' (y) ~ ffi (x) = 2ΛN ~# (x; y) Λ N = f 2 Λ j ß» h y» ß when N y is odd, or ß< h y» ß when N y is eveng: Here ~' (y) is the interpolant of ' (y), so that we are in fact considering a discrete Fourier transform of v 2U h. In our forthcoming analysis the following results are also needed. Proposition 3.1 (Korn's inequality). Let V = fv =(v 1 ;v 2 ) 2 [H 1 p (Ω)]2 j v(0; ) =v(l; ) =0g or let V = fv =(v 1 ;v 2 ) 2 [H 1 p (Ω)]2 j Z v 1 dxdy = Z Ω Ω v 2 dxdy =0g: Then there exists a constant c>0 such that jjvjj 1» c jj2 L 2 + jj@v jj2 L 2 +2jj1 2 (@v )jj2 L 2 1=2 8v 2V: (3.11) Proof. See [2]. Proposition 3.2. Assume that v =(v 1 ;v 2 ) 2 [V h ] jj L» C 2 jjπ xy h + )jj L 2 + jj@v jj L 2 + jj@v jj L 2 : (3.12) Proof. See Theorem 6.1 in[5]. 4 The consistency error at t > 0 We start by giving a stability result for U M;h. Lemma 4.1. Let v 2U M;h. Then jjvjj 1» Ct 1 jjjvjjj M;h : Proof. Assume first the modification (E1). By (3.3) we have that for v = (u; v; w; ; ψ) 2U jj L 2 + jj L jj L 2» Ct 1 jjjvjjj M;h 9

12 and thus by the Korn inequality (3.11) jj jj 1 + jjψjj 1» Ct 1 jjjvjjj M;h : (4.1) Also the definitions of the membrane strains fi ij (2.1) jj L 2 + jj L 2 + jjπxy h + )jj L 2» C(jjjvjjj M;h + jjwjj L 2) and by (3.12) we have resulting @x jj L 2» C(jjΠxy h + )jj L 2 + jj L 2 + jj L jj L 2 + jj L jj L 2» C(jjjvjjj M;h + jjwjj L 2) where from again by the Korn inequality (3.11) jjujj 1 + jjvjj 1» C(jjjvjjj B;h + jjwjj L 2): (4.2) By (2.2), (3.1) we = ~ρ 1 Π h = ~ρ 2 Π y hψ so jj L 2 + jj L 2» C(jjjvjjj M;h + jj jj L 2 + jjψjj L 2)» Ct 1 jjjvjjj M;h (4.3) by (4.1). The claim follows from (4.1) (4.3) together with the Poincaré's inequality. Similar calculations imply the result also for the modification (E2). Lemma 4.2. Let v 2U B;h. Then jjvjj 1» Cjjjvjjj B;h : Proof. By the definition of U B;h the Korn inequality (3.11) holds for the pairs ( ; ψ) and (u; v), aswell as the Poincaré's inequality for w. The result follows as in Lemma 4.1. We derive next more specific stability results for the low-order discrete Fourier modes in the FE space. Lemma 4.3. Let ~ # = ~' ~ ffi = ~' (~u ; ~v ; ~w ; ~ ; ~ ψ ) 2U M;h. Then, if b 6= 0 we have for such that j jh» c<ß, jj ~' ~u jj 1 + jj ~' ~v jj 1 + jj ~' ~w jj L 2» Cj j 1 m jjj ~ # jjj M;h ; (4.4) where m = 1 in the elliptic case and m = 0 in the parabolic and hyperbolic cases. 10

13 Proof. Consider first the case (E1). The translation components ~u and ~v of # ~ satisfy the difference equation (cf. [3])» ~v (x ) ~u k+1 ~v (x ~u k )= 1 fi ~v km (x )+ 2 ~u k+1 ~v (x ~u k ) +h k x F ~ k (4.5) where fi k =2 hk x h y tan ( 1 2 h y), M = i 2c 1 b a 0 b (4.6) and ~F k = 1 cos ( 1 2 h y) Here 2 ~ f 12 (xk+1=2 ) c b ( ~ f 22 (xk+1 )+ ~ f 22 (xk )) cos ( 1 2 h y) ~ f 11 (xk+1=2 ) a 2b ( ~ f 22 (xk+1 )+ ~ f 22 (xk )) : (4.7) ~f 11 (xk+1=2 )=e in hy ( ~ fi 11 ( ~ # )) j(x k+1=2 ;yn ) ~f 22 (xk )=e i(n+1=2) hy ( ~ fi 22 ( ~ # )) j(x k;y n+1=2 ) ~f 12 (xk+1=2 )=e i(n+1=2) hy ( ~ fi 12 ( ~ # )) j(x k+1=2 ;y n+1=2 ): Due to the constraints at x =0;Lwe may without loss of generality consider only the exponentially decreasing solution of (4.5) starting from x =0. Then if j jh y» c < ß, the standard theory for A-stable difference schemes (see also [3]) gives us the bound jj ~v (x ~u k+1 )jj» e jj ~v ffj jxk+1 (0)jj ~u (4.8) + Z x k+1 0 e fij j(xk+1 t) jj ~ F (t)jje ffj jt dt where jj jj is the Euclidean norm of vectors in R 2 2 f ~F ~ 12 2c f ~ = b 22 ~f 11 a f ~ : b 22 and Here ff > fi > 0 in the elliptic case and ff = fi = 0 in the parabolic and hyperbolic cases. Since ~u (0) = ~v (0) = 0 we obtain when 6= 0 R jj ~v (Cj j (x ~u 1 x e 2fij jxk+1 k+1 jj k+1 )jj 2 0» R F ~ (t)jj 2 dt in the elliptic case x C k+1 jj F ~ 0 (t)jj 2 dt in the parabolic and hyperbolic case and consequently jj~v jj 2 L 2 (0;L) + jj~u jj 2 L 2 (0;L)» Cj j 2m jj ~ F jj 2 L 2 (0;L) : (4.9) 11

14 Also, (4.5) gives the relation ~v (x ~u 0 k+1=2 )= 1 tan ( 1 h h y)m y 2 from which it follows that» ~v ~u (x )+ k+1 ~v (x ~u k ) + F ~ k jj~v 0 jj 2 L 2 (0;L) + jj~u0 jj 2 L 2 (0;L)»Cj j2 (jj~v jj 2 L 2 (0;L) + jj~u jj 2 L 2 (0;L) ) Combining (4.9) and (4.10) gives + jj F ~ jj 2 L 2 (0;L) (4.10)»Cj j 2(1 m) jj F ~ jj 2 L : 2 (0;L) jj ~' ~u jj 1 + jj ~' ~v jj 1» Cj j 1 m jjj ~ # jjj M;h (4.11) since jj ~ F jj L 2» Cjjj ~ # jjj M;h. To consider ~w we note that (cf. [3]) and thus leading to ~w (x k )= 2i bh y tan ( 1 2 h y)~v (x k )+ 1 b cos ( 1 h ~f 22 2 y) (xk ) jj ~w jj 2 L ( 0;L)» C(j j2 jj~v jj 2 L 2 (0;L) + jj ~ f 22jj 2 L 2 (0;L) ) jj ~' ~w jj L 2 (0;L)» Cj j 1 m jjj ~ # jjj M;h : (4.12) The claim for 6= 0follows from (4.11) together with (4.12). When =0we have from (4.8) and from ~w 0 (x k )= 1 b ~ f 22 (x k ) that jj ~' 0 ~u 0 jj 1 + jj ~' 0 ~v 0 jj 1 + jj ~' 0 ~w 0 jj L 2» Cjjj ~ # 0 jjj M;h in any geometry. Similar calculations show that the claim holds also for the case (E2). Remark 4.1. The assumption b 6= 0is not superfluous. This can be seen by taking b =0, =0, a = 1, c =1=2 and choosing ~ ffi 0 (x 1 )=(0; 0; 2; 4=h; 0), then repeating the sequence ~ffi 0 (x j )=(h; h; 0; 8=h; 0); ~ffi 0 (x j+1 )=(0; 0; 2; 4=h; 0); ~ffi 0 (x j+2 )=( h; h; 0; 0; 0); ~ffi 0 (x j+3 )=(0; 0; 2; 4=h; 0) for j =2; 6; 10;::: and finally letting ~ffi 0 (x Nx 2 )=(h; h; 0; 8=h; 0); ~ffi 0 (x Nx 1 )=(0; 0; 2; 4=h; 0): For this particular choice we have that ~u jj L 2 the stability isweaker when b =0. ο min f 1 h ; h2 t gjjj~ # 0 jjj M;h so 12

15 With the help of the stability estimates given in Lemmas we can now bound the consistency error. Theorem 4.4. Assume that b 6= 0, and let m = 1 in the elliptic case and m = 0 in the parabolic and hyperbolic cases.. The consistency error e c;m defined in (3.5) satisfies e c;m» C 1 (u)h + C 2 (t; u)h 2 + C 3 (t; s; u)h 1+s + C 4 (t; u)h 2 ; s 0 provided that C 1 (u) =C jfi ij (u)j 2 m ij ( 0 for the case(e1) C 2 (u;t)= Ct 1 jwj 1 for the case(e2) ( Ct 1 jfi C 3 (t; s; u) =Ct 1 12 (u)j 1+s for the case(e1) jfi ii (u)j 1+s + Ct 1 (jfi i 12 (u)j s + jwj 1+s ) for the case (E2) C 4 (t; u) =Ct 1 ( i jρ i (u)j 1 ) are all finite. The consistency error e c;b defined in (3.6) satisfies provided that e c;b» C 1 (t; u)h + C 2 (t; u)h 2 C 1 (t; u) =Ct 2 ij jfi ij (u)j 1 C 2 (t; u) =Ct 2 i jρ i (u)j 1 are both finite. Remark 4.2. The transverse shear strains ρ i are typically very small at small t in smooth deformation states, so the error term of e c;m is very likely negligible in practice. In the bending-dominated case, e c;b depends strongly on fi ij and ρ i. For smooth deformations one could assume realistically that jfi ij (u)j 1 ο jρ i (u)j 1 ο t 2 as t! 0, in which case e c;b = O(h) uniformly in t. In practice, however, boundary layer effects probably cause the growth of e c;b, via constant C 1 (u) in particular. Proof. We P P consider first the membrane case and write u = # 2Λ 2 U M and v = ~# 2ΛN 2 U M;h. Then by the orthogonality of the discrete and 13

16 continuous modes (cf. [3]) (A M A h M )(u;v)=(a m A h m )(u;v)=(a m A h m )( # ~# ) =(A m A h m )( ~# )+(A m A h m )( ~# ) = j j» 0 # ; 2ΛN j j» 0 (A m A h m )(# ; ~ # )+» C ij + 2Λ 2ΛN # ; j j> 0 2ΛN j j> 0 (A m A h m )(# ;v) Λ (fiij (# );fi ij (# ~ )) ( fi ~ ij (# ); fi ~ ij (# ~ )) (4.13) j j» 0 Λ (fiii (# );fi jj (# ~ )) ( fi ~ ii (# ); fi ~ jj (# ~ )) j j» 0 ij;i6=j +Cj 0 j s 1 +Cj 0 j s 2 +Cj 0 j s 3 i j j> 0 j j s1 j(fi ii (# ) ~ fi ii (# );fi ii (v) ~ fi ii (v))j i6=j j j> 0 j j s3 j +C i j j s2 j(fi ii (# ) Πh fi ii (# );fi jj(v))j j j> 0 2Λ Z Ω xy (fi 12 (# )fi 12 (v) ~ fi 12 (# ) ~ fi 12 (v))dxdyj (ρ i (# ) ~ρ i (# );ρ i (v) ~ρ i (v)) = I + II + III + IV + V + VI where we have chosen 0 such that 0 h» c<ß. We note first that in I (fi ij (# );fi ij ( ~ # )) ( ~ fi ij (# ); ~ fi ij ( ~ # )) =(fi ij (# ) ~ fi ij (# );fi ij ( ~ # )) +( ~ fi ij (# );fi ij ( ~ # ) ~ fi ij ( ~ # )) (4.14) where the first term can be bounded by Lemma 4.3 as (fi ij (# ) ~ fi ij (# );fi ij ( ~ # ))» Chj 1 m fi ij (# )j 1 (jj~u jj 1 + jj~v jj 1 + jj ~w jj L 2)» Chjfi ij (# )j 2 m jjj ~ # jjj M;h (4.15) and the second term in (4.14) is zero in the case of the modification (E1) since R ij is an orthogonal L 2 -projection. For the modification (E2) the second term gives (S 12 ( ~ # ); ~ fi 12 (# )) =(S 12 ( ~ # );fi 12 (# )) + (S 12 ( ~ # );S 12 ( ~ # ))»Chjfi 12 (# )j 1 jj ~w jj L 2 + Ch 2 jw j 1 j ~w j 1»Chjfi 12 (# )j 1 jjj ~ # jjj M;h + Ch 2 t 1 jw j 1 jjj ~ # jjj M;h (4.16) by Lemmas 4.1 and 4.3. For the term II we can write (fi ii (# );fi jj ( ~ # )) ( ~ fi ii (# ); ~ fi jj ( ~ # )) =(fi ii (# ) ~ fi ii (# );fi jj ( ~ # )) +( ~ fi ii (# );fi jj ( ~ # ) ~ fi jj ( ~ # )) (4.17) 14

17 where the first term can be treated as in case of the term I. The second term in (4.17) can be written as ( ~ fi ii (# );fi jj ( ~ # ) ~ fi jj ( ~ # )) =(R ii fi ii (# ); (I R jj )fi jj ( ~ # )) =((I R jj )R ii fi ii (# );fi jj ( ~ # ))»Chj 1 m fi ii (# )j 1 (jj~u jj 1 + jj~v jj 1 + jj ~w jj L 2) (4.18)» Chjfi ii (# )j 2 m jjj ~ #jjj M;h again by Lemma 4.3. By (4.14) (4.18) we have the bounds I + II»C 1 h jfi ij (# )j 2 m jjj # ~ jjj M;h ij j j» 0 + C 2 h 2 t 1 jw j 1 jjj # ~ jjj M;h j j» 0 (4.19)»C 1 h ij jfi ij (u)j 2 m jjjvjjj M;h + C 2 h 2 t 1 jwj 1 jjjvjjj M;h where C 2 =0for the modification (E1). For the rest of the terms in (4.13) standard approximation theory gives III» Ch 2+s 1 t 1 i jfi ii (u)j 1+s1 jjjvjjj M;h IV» Ch 1+s 2 t 1 jfi ii (u)j 1+s2 jjjvjjj M;h i ( Ch 1+s 3 V» t 1 jfi 12 (u)j 1+s3 jjjvjjj M;h for the case (E1) Ch 1+s 3 t 1 (jfi 12 (u)j s3 + jwj 1+s3 )jjjvjjj M;h for the case (E2) (4.20) VI» Ch 2 t 1 i jρ i (u)j 1 jjjvjjj M;h by Lemmas 4.1 and 4.3. The claim for e c;m follows form (4.13), (4.19) and (4.20) when we take s 1 = s 2 = s 3 = s. For the case e c;b the claim follows by the same arguments when we note that (A B A h B )(u;v)=t 2 (A m A h m )(u;v) and use the stability result given in Lemma The asymptotic consistency error In this section we bound the asymptotic consistency error in an inextensional deformation state, as defined by (3.10). In [3] we showed that the approximation error in the inextensional state is of order O(h) under nearly optimal regularity assumptions on u 0. Here we find that the consistency error is likewise of order O(h) at t =0, but we need a very strong regularity assumption on u 0. 15

18 We also make the additional assumption that the load is given by Q(v) = Z Ω (q 1 u + q 2 v + q 3 w)dxdy for some suitable q i 2 L 2 p (Ω), i = 1; 2; 3 where L2 p (Ω) denotes the usual L2 - space with periodic boundary conditions imposed at y =0;H and define the Fourier components of the load by Q (v) = Z Ω (q 1 u + q 2 v + q 3 w)dxdy P where for each q i we write q i (x; y) = 2Λ q i (x; y) =P 2Λ ^q i (x)' (y). We define the (semi-) norms 1=2 jqj s = jq j 2 s 2Λ where jq j s = j j s (jjq 1 jj 2 L 2 + jjq 2 jj 2 L 2 + jjq 3 jj 2 L 2)1=2 and write frequently jqj 0 = jjqjj L 2, jq j 0 = jjq jj L 2 and jjqjj k =( P k j=0 jqj2 j )1=2. Theorem 5.1. Assume that b 6= 0, u 0 2 [Hp 5 (Ω)]5. Then the asymptotic consistency error e 0 c;b (u 0 ), as defined by (3.10), satisfies e 0 c;b (u 0 )» C(jju 0jj 5 + jjqjj 1 )h: Proof. Let v 2U 0;h and write v = P 2ΛN ~# (x; y) = P 2ΛN A ~' (y) ~ (x) = P P A 2ΛN ~' (y)(~u P (x); ~v (x); ~w (x); ~ (x); ψ ~ (x)), u 0 = 2Λ u 0 = 2Λ (u 0;v 0 ;w 0 ; 0 ;ψ 0 ) and Q(v) =P 2Λ Q (v). Then by the orthogonality of the Fourier modes [3] we have that A b (u 0 ;v) Q(v) =A b ( =A b ( = u 0; 2Λ 2ΛN u 0; j j» 0 2ΛN u 0; j j> 0 + A b ( ~# ) Q ( 2Λ ~# ) 2ΛN 2ΛN j j» 0 Q ( ~# ) j j» 0 (A b (u 0; ~ # ) Q ( ~ # )) + =I + II ~# ) 2ΛN j j> 0 Q ( for any 0 such that 0 h y» c<ß. Let us first bound the term II. Here we have that A b (u 0;v)» 1 0 j j> 0 ~# ) 2ΛN ~# ) j j> 0 (A b (u 0;v) Q (v)) j j> 0 A b (j ju 0;v)» Chjju 0 jj 2 jjjvjjj B;h (5.1) 16

19 for 0 = c, c sufficiently small and similarly h Q (v)» 1 0 j jq (v)» ChjQj 1 jjjvjjj B;h : (5.2) j j> 0 j j> 0 To bound the term I when b 6= 0we note that for any # = A ' 2U 0 we can write A b (u 0; ~ # ) Q ( ~ # )=A b (u 0; ~ # # ) Q ( ~ # # ) =A (A b (u 0; ~' ~ ' ) Q (~' ~ ' )): (5.3) Integration by parts in the first term in (5.3) gives A b (u 0; ~' ~ ' )= Z H 0 + fi fi fi fi Z Ω L 0 ff 1 (~' ~ ' )+ff 2 (~' ~ ψ ' ψ )dy ffi 1 (~' ~ ' )+ffi 2 (~' ~ ψ ' ψ )dxdy where 8>< so that >: ff 1 w 0 + w 2 ff =(1 2 w ffi 1 ffi w w 0 A b (u 0; ~' ~ ' )»jjff (L; )jj 1 L (0;H)jj ~' ~ 2 (L; ) ' (L; )jj L 2 (0;H) +jjffi (L; )jj 1 L (0;H)jj ~' ~ 2 ψ (L; ) ' ψ (L; )jj L 2 (0;H) +jjff (0; )jj 1 L (0;H)jj ~' ~ 2 (0; ) ' (0; )jj L 2 (0;H) +jjffi (0; )jj 1 L (0;H)jj ~' ~ 2 ψ (0; ) ' ψ (0; )jj L 2 (0;H) +jjff 2jj L 2jj ~' ~ ' jj L 2 + jjffi 2 jj L 2jj ~' ~ ψ ' ψ jj L 2» Cjju 0jj 3 jj ~' ~ (L; ) ' (L; )jj L 2 (0;H) (5.4) +jj ~' ~ ψ (L; ) ' ψ (L; )jj L 2 (0;H) +jj ~' ~ (0; ) ' (0; )jj L 2 (0;H) +jj ~' ~ ψ (0; ) ' ψ (0; )jj L 2 (0;H) +jj ~' ~ ' jj L 2 + jj ~' ~ ψ ' ψ jj L 2 : Also for Q in (5.3) we have the bound Q (~' ~ ' )»CjjQ jj L 2 jj ~' ~u ' u jj L 2 + jj ~' ~v ' v jj L 2 + jj ~' ~w ' w jj L 2 : (5.5) To continue we need the following approximation results. The proof will be postponed to the end of this section. 17

20 Lemma 5.2. For every such that j jh y» c<ßthere exists a ' 2U 0 such that jj ~' ~ (L; ) ' (L; )jj L 2 (0;H) + jj ~' ~ ψ (L; ) ' ψ (L; )jj L 2 (0;H) +jj ~' ~ (0; ) ' (0; )jj L 2 (0;H) + jj ~' ~ ψ (0; ) ' ψ (0; )jj L 2 (0;H) +jj ~' ~ ' jj L 2 + jj ~' ~ ψ ' ψ jj L 2 (5.6)» C(h 2 j j 5 m + h 2 4 )+Ch 2 jjj ~' ~ jjj B;h and jj ~' ~u ' u jj L 2+jj ~' ~v ' v jj L 2+jj ~' ~w ' w jj L 2» Ch 2 j j 4 3m=2 (5.7) where m = 1 in the elliptic case and m = 0 in the hyperbolic and parabolic cases. To complete the proof of Theorem 5.1 we note that since j j 3 m=2» Cjjj ~' ~ jjj B;h and j jh» 0 h» c<ßwe obtain from (5.4) with the help of Lemma 5.2 and from (5.5) A b (u 0; ~' ~ ' )» Chjj 2 u 0jj 3 jjj ~' ~ jjj B;h (5.8) Q (~' ~ ' )» ChjjQ jj L 2jjj ~' ~ jjj B;h (5.9) so that by (5.3), (5.8) and (5.9) (A b (u 0; # ~ ) Q (# ~ )) = A (A b (u 0; ~' ~ ) Q (~' ~ )) j j» 0 j j» 0» Ch (jj 2 u 0jj 3 + jjq jj L 2)jjjA ~' ~ jjj B;h (5.10) j j» 0» Ch(jju 0 jj 5 + jjqjj L 2)jjjvjjj B;h and Theorem 5.1 follows from the estimates (5.1), (5.2) and (5.10). Proof of Lemma 5.2. In [3] it was shown that for every discrete mode ~' ~ 2 U 0;h with j jh y» c<ßthere corresponds a continuous mode ' = ' (y)(u (x);v (x);w (x); (x);ψ (x)) 2U 0 satisfying u (0) = ~u (0) and v (0) = ~v (0) and such that ju (x k ) ~u (x k )j»ch 2 j j 3 m e fij jxk jv (x k ) ~v (x k )j»ch 2 j j 3 m e fij jxk jw (x k ) ~w (x k )j»ch 2 j j 4 m e fij jxk jψ (x k ) ~ ψ (x k )j»ch 2 j j 5 m e fij jxk 18

21 with fi > 0 in the elliptic case and fi = 0 in the hyperbolic and parabolic cases, so that jj ~' ~ ψ (0; ) ' ψ (0; )jj L 2 (0;H)»Ch 2 j j 5 m jj ~' ~ ψ (L; ) ' ψ (L; )jj L 2 (0;H)»Ch 2 j j 5 m jj ~' ~ ψ ' ψ jj L 2»Ch 2 j j 5 3m=2 : (5.11) and jj ~' ~u ' u jj L 2»Ch 2 j j 3 3m=2 jj ~' ~v ' v jj L 2»Ch 2 j j 3 3m=2 (5.12) jj ~' ~w ' w jj L 2»Ch 2 j j 4 3m=2 Also, by [3] we have that 1 2 (~ (x k+1 )+ ~ (x k )) = 2 bh 2 y tan 2 ( 1 2 h y) (~u (x k+1 )+~u (x k )) 2c b (~v (x k+1 )+~v (x k )) = 1 2 (~g(xk+1 )+~g(x k )) so that ~ (x k+1 )=~g(x k+1 )+( 1) k ( ~ (x 0 ) ~g(x 0 )) (5.13) ~ ~ (x k+1 ) ~ (x k (xk+1=2 )= h k x = ~g(xk+1 ) ~g(x k ) h k x + 2 h k x ( 1) k ( ~ (x 0 ) ~g(x 0 )): (5.14) Since (x k )= 2 b (2c b v (x k ) u (x k )) = g(x k ) (5.15) it follows from (5.13) (5.15) that ~ (x k+1 ) (x k+1 )=~g(x k+1 ) g(x k+1 ) and finally that j ~ (x k+1 ) (x k+1 )j»c + hk (xk+1=2 ) hk x ~g(x k+1 ) ~g(x k ) 2 h k x h 2 j j 5 m e fij jxk+1 + (xk+1=2 )j + h (xk+1=2 )j + (xk+1=2 )j : 19

22 For the values at the endpoints we get similarly ~ (x 0 ) (x 0 )=( 1) k ( ~ (x k+1 ) (x k+1 )+g(x k+1 ) ~g(x k+1 )) +~g(x 0 ) g(x 0 ) and ~ (x Nx ) (x Nx )=( 1) Nx ( ~ (x 0 ) (x 0 )+g(x 0 ) ~g(x 0 )) +~g(x Nx ) g(x Nx ): Thus, we have the following bounds jj ~' ~ (0; ) ' (0; )jj L 2 (0;H)»C(h h 2 j j 5 3m=2 ) + C(h + h 2 )jjj ~' ~ jjj B;h jj ~' ~ (L; ) ' (L; )jj L 2 (0;H)»C(h h 2 j j 5 3m=2 ) + C(h + h 2 )jjj ~' ~ jjj B;h (5.16) + C(h 2 j j 5 m + h 2 4 ) jj ~' ~ ' jj L 2»Ch 2 j j 5 3m=2 + C(h + h 2 )jjj ~' ~ jjj B;h : and Lemma 5.2 follows from (5.11), (5.12) and (5.16) since j jh y» c<ß. References [1] K.J. Bathe, E.N. Dvorkin, A formulation of general shell elements the use of mixed interpolation of tensorial components,int. J. Numer. Methods Engrg. 22 (1986) [2] S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, 1994) [3] V. Havu, J. Pitkäranta, Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations, Helsinki University of Technology Institute of Mathematics Research Reports A430 (2000). [4] M. Malinen, On geometrically incompatible bilinear shell elements and classical shell models I II, to appear. [5] J. Pitkäranta, The first locking-free plane-elastic finite element: historia mathematica, Helsinki University of Technology Institute of Mathematics Research Reports A411 (1999). [6] J. Pitkäranta, The problem of membrane locking in finite element analysis of cylindrical shells, Numer. Math. 61, (1992)

23 (continued from the back cover) A435 A434 A433 A432 A431 A430 A428 A427 A426 A425 A424 A423 A422 Timo Salin On Quenching with Logarithmic Singularity, May 2001 Ville Havu An Analysis of Asymptotic Consistency Error in a Parameter Dependent Model Problem, Mar 2001 Ville Havu and Juhani Pitkäranta Analysis of a bilinear finite element for shallow shells I: Consistency error, Jan 2001 Tuomas Hytönen Optimal Wrap-Around Network Simulation, Dec 2000 Ville Turunen Pseudodifferential calculus on compact Lie groups, Jun 2001 Ville Havu and Juhani Pitkäranta Analysis of a bilinear finite element for shallow shells I: Approximation of Inextensional Deformations, May 2000 Jarmo Malinen Discrete Time H 1 Algebraic Riccati Equations, Mar 2000 Marko Huhtanen A Hermitian Lanczos method for normal matrices, Jan 2000 Timo Eirola A Refined Polar Decomposition: A = UPD, Jan 2000 Hermann Brunner, Arvet Pedas and Gennadi Vainikko Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, Dec 1999 Olavi Nevanlinna Resolvent conditions and powers of operators, Dec 1999 Jukka Liukkonen Generalized Electromagnetic Scattering in a Complex Geometry, Nov 1999 Marko Huhtanen A matrix nearness problem related to iterative methods, Oct 1999

24 HELSINKI UNIVERSITY OF TECHNOLOGY INSTITUTE OF MATHEMATICS RESEARCH REPORTS The list of reports is continued inside. Electronical versions of the reports are available at A440 Ville Turunen Pseudodifferential calculus on compact Lie groups and homogeneous spaces, Sep 2001 A439 Jyrki Piila and Juhani Pitkäranta On corner irregularities that arise in hyperbolic shell membrane theory, Jul 2001 A438 Teijo Arponen The complete form of a differential algebraic equation, Jul 2001 A437 Viking Högnäs Nonnegative operators and the method of sums, Jun 2001 A436 Ville Turunen Pseudodifferential calculus on compact homogeneous spaces, Jun 2001 ISBN ISSN Espoo 2001