ERROR ESTIMATES FOR LOW-ORDER ISOPARAMETRIC QUADRILATERAL FINITE ELEMENTS FOR PLATES

Transcription

1 ERROR ESTIATES FOR LOW-ORDER ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES RICARDO G DURÁN, ERWIN HERNÁNDEZ, LUIS HERVELLA-NIETO, ELSA LIBERAN, AND RODOLFO RODRíGUEZ Abstract This paper deals with the numerical approximation of the bending of a plate modeled by Reissner-indlin equations It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods In particular, one of the most widely used procedures is based on the family of elements called ITC (mixed interpolation of tensorial components We consider two lowest-order methods of this family on quadrilateral meshes Under mild assumptions we obtain optimal H and L 2 error estimates for both methods These estimates are valid with constants independent of the plate thickness We also obtain error estimates for the approximation of the plate vibration problem Finally, we report some numerical experiments showing the very good behavior of the methods, even in some cases not covered by our theory ey words Reissner-indlin, ITC methods, isoparametric quadrilaterals AS subject classifications 65N30, 74S05, 7420 Introduction Reissner-indlin model is the most widely used for the analysis of thin or moderately thick elastic plates It is now very well understood that standard finite element methods applied to this model produce very unsatisfactory results due to the so-called locking phenomenon Therefore, some special method based on reduced integration or mixed interpolation has to be used Among them, the ITC methods introduced by Bathe and Dvorkin in [7] or variants of them are very likely the most used in practice A great number of papers dealing with the mathematical analysis of this kind of methods have been published (see for example [2, 6, 0, 2, 3, 8, 20, 23] In those papers, optimal order error estimates, valid uniformly on the plate thickness, have been obtained for several methods However, although one of the most commonly used elements in engineering applications are the isoparametric quadrilaterals (indeed, the original Bathe and Dvorkin s paper deals with these elements, no available result seems to exist for this case On the other hand, it has been recently noted that the extension to general quadrilaterals of convergence results valid for rectangular elements is not straightforward and, even more, the order of convergence can deteriorate when non-standard finite elements are used in distorted quadrilaterals, even if they satisfy the usual shape regularity assumption (see [3, 4] Departamento de atemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 428 Buenos Aires, Argentina Supported by ANPCyT under grant PICT and by CONICET under grant PIP 0660/98 ember of CONICET (Argentina Departamento de Ingeniería atemática, Universidad de Concepción, Casilla 60-C, Concepción, Chile Supported by FONDECYT (Chile Departamento de atemáticas, Universidade da Coruña, 507 A Coruña, Spain Partially supported by FONDAP in Applied athematics Comisión de Investigaciones Científicas de la Provincia de Buenos Aires and Departamento de atemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CC 72, 900 La Plata, Argentina GI 2 A, Departamento de Ingeniería atemática, Universidad de Concepción, Casilla 60-C, Concepción, Chile Partially supported by FONDAP in Applied athematics (Chile

2 2 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ The aim of this paper is to analyze two low-order methods based on quadrilateral meshes One is the original ITC4 introduced in [7], while the other one is an extension to the quadrilateral case of a method introduced in [2] for triangular elements (from now on the latter will be called DL4 We are interested not only in load problems but also in the determination of the free vibration modes of the plate For nested uniform meshes of rectangles, an optimal order error estimate in H norm has been proved in [6] for ITC4 However, the regularity assumptions on the exact solution required in that paper are not optimal These assumptions have been weakened in [2], but they are still not optimal Let us remark that to obtain approximation results for the plate vibration spectral problem, it is important to remove this extra regularity assumption On the other hand, for low-order elements as those considered here, an optimal error estimate in L 2 norm is difficult to obtain because of the consistency term arising in the error equation For triangular elements such estimate has been only recently proved in [3] However, the proof given in that paper can not be extended straightforwardly, even for the case of rectangular elements In this paper we prove optimal in order and regularity H and L 2 error estimates for both methods, ITC4 and DL4, under appropriate assumptions on the family of meshes As a consequence, following the arguments in [3], we obtain also optimal error estimates for the approximation of the corresponding plate vibration spectral problem In order to prove the H error estimate for ITC4 we require an additional assumption on the meshes (which is satisfied, for instance, by uniform refinements of any starting mesh Instead, no assumption other than the usual shape regularity is needed for DL4 On the other hand, a further assumption on the meshes is made to prove the L 2 error estimates: the meshes must be formed by higher order perturbations of parallelograms This restriction is related with approximation properties of the Raviart- Thomas elements which are used in our arguments and do not hold for general quadrilateral elements However, this assumption is only needed for extremely refined meshes Indeed, the L 2 estimate holds for any regular mesh as long as the mesh-size is comparable with the plate thickness oreover, we believe that this quasi-parallelogram assumption is of a technical character In fact, the numerical experiments reported here seem to show that it is not necessary The rest of the paper is organized as follows In section 2 we recall Reissner- indlin equations and introduce the two discrete methods We prove optimal order error estimates for both methods in H and L 2 norms in sections 3 and 4, respectively In section 5 we prove error estimates for the spectral plate vibration problem Finally, in section 6, we report some numerical experiments Throughout the paper we denote by C a positive constant not necessarily the same at each occurrence, but always independent of the mesh-size and the plate thickness 2 Statement of the problem 2 Reissner-indlin model Let Ω ( t 2, t 2 be the region occupied by an undeformed elastic plate of thickness t, where Ω is a convex polygonal domain of R 2 In order to describe the deformation of the plate, we consider the Reissner-indlin model, which is written in terms of the rotations β = (β, β 2 of the fibers initially normal to the plate mid-surface and the transverse displacement w The following equations describe the plate response to conveniently scaled transversal and shear loads f L 2 (Ω and θ L 2 (Ω 2, respectively (see, for instance, [9, 3]:

3 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 3 (2 Problem 2 Find (β, w H 0(Ω 2 H 0(Ω such that: a(β, η + (γ, v η = (f, v + t2 2 (θ, η (η, v H 0(Ω 2 H 0(Ω, γ = κ ( w β t2 In this expression, κ := Ek/2(+ν is the shear modulus, with E being the Young modulus, ν the Poisson ratio, and k a correction factor We have also introduced the shear stress γ and denoted by (, the standard L 2 inner product Finally, a is the H 0(Ω 2 elliptic bilinear form defined by a(β, η := E 2( ν 2 Ω 2 ( νε ij (βε ij (η + ν div β div η, i,j= with ε ij (β = 2 ( β i/ x j + β j / x i being the components of the linear strain tensor Let us remark that we have included in our formulation the shear load term t 2 2 (θ, η since it arises naturally when considering the free vibration plate problem In fact, it is simple to see that the free vibration modes of the plate are determined by t 3 a(β, η + κt ( w β ( v η = ω (t 2 ρ wv + t3 ρ β η Ω Ω 2 Ω (η, v H 0(Ω 2 H 0(Ω, where ω denotes the angular vibration frequency, β and w the rotation and transversal displacement amplitudes, respectively, and ρ the plate density (see [3] for further details Thus, rescaling the problem with λ := ρ ω 2 /t 2, we obtain the following, which is the spectral problem associated to Problem 2: Problem 22 Find λ R and 0 (β, w H 0(Ω 2 H 0(Ω such that: ] a(β, η + (γ, v η = λ [(w, v + t2 (β, η (η, v H 2 0(Ω 2 H 0(Ω, γ = κ ( w β t2 This paper deals with the finite element approximation of Problems 2 and 22 It is well known that both are well-posed (see [9] and [3] Furthermore, we will use the following regularity result for the solution of equations (2 (see [2]: (22 β 2,Ω + w 2,Ω + γ 0,Ω + t γ,ω C ( t 2 θ 0,Ω + f 0,Ω C (θ, f t, where, for any open subset O of Ω and any integer k, k,o denotes the standard norm of H k (O or H k (O 2, as corresponds, and (, t is the norm in L 2 (Ω 2 L 2 (Ω induced by the weighted inner product in the right hand side of the first equation in (2 (see [3] 22 Discrete problems In what follows we consider two lowest-degree methods on isoparametric quadrilateral meshes for the approximation of Problem 2: the so-called ITC4 (see [7] and an extension to quadrilaterals of a method introduced in [2] that we call DL4 Both methods are based on relaxing the shear terms in equation (2 by introducing an interpolation operator, called reduction operator

4 4 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ Let {T h } be a family of decompositions of Ω into convex quadrilaterals, satisfying the usual condition of regularity (see for instance [9]; ie, there exist constants σ > and 0 < ϱ < independent of h such that h σρ, cos ϑ i ϱ, i =, 2, 3, 4, T h, where h is the diameter of, ρ the diameter of the largest circle contained in, and ϑ i, i =, 2, 3, 4, the four angles of Let := [0, ] 2 be the reference element We denote by Q i,j ( the space of polynomials of degree less than or equal to i in the first variable and to j in the second one Also, we set Q k ( = Q k,k ( Let T h We denote by F a bilinear mapping of onto, with Jacobian matrix and determinant denoted by DF and J F, respectively The regularity assumptions above lead to ch 2 J F Ch 2, with c and C only depending on σ and ϱ (see [9] In particular, J F > 0 and, hence, F is a one-to-one map Let l i, i =, 2, 3, 4, be the edges of ; then l i = F ( l i, with l i being the edges of Let τi be a unit vector tangent to l i on the reference element; then τ i := DF τ i / DF τ i is a unit vector tangent to l i on (see Figure 2 y F τ τ 3 τ τ 4 τ 4 x Fig 2 Bilinear mapping onto an element T h Let N ( := { p : p Q 0, ( Q,0 ( }, and, from this space, we define through covariant transformation { } N ( := p : p F = DF t p, p N (

5 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 5 Let us remark that the mapping between N ( and N ( is a kind of Piola transformation for the rot operator, rot p := p / x 2 p 2 / x (the Piola transformation is defined for the div operator in, for example, [9] Then we have the following results which are easily established (see [23, 24]: (23 p τ i = p τ i, i =, 2, 3, 4, l i l i and (24 (rot p F = J F rot p in We define the lowest-order rotated Raviart-Thomas space (see [2, 24] Γ h := {ψ H 0 (rot, Ω : ψ N ( T h }, which will be used to approximate the shear stress γ We remark that, since Γ h H 0 (rot, Ω, the tangential component of a function in Γ h must be continuous along inter-element boundaries and vanish on Ω In fact, the integrals (23 of these tangential components are the degrees of freedom defining an element of Γ h We consider the interpolation operator (25 R : H (Ω 2 H 0 (rot, Ω Γ h, defined by (see [2] (26 Rψ τ l = ψ τ l edge l of T h, l l where, from now on, τ l denotes a unit vector tangent to l Clearly, the operator R satisfies ψ H (Ω 2, (27 rot(ψ Rψ = 0 T h Taking into account the rotation mentioned above, it is proved in Theorem III44 of [4] that (28 rot Rψ 0,Ω C ψ,ω and (29 ψ Rψ 0,Ω Ch ψ,ω To approximate the transverse displacements we will use the space of standard bilinear isoparametric elements W h := { v H } 0(Ω : v Q( T h, { where, T h, Q( := p L 2 ( : p F Q ( } The following lemma establishes some relations between the spaces Γ h and W h : Lemma 2 The following properties hold: W h = {µ Γ h : rot µ = 0}

6 6 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ and R( w = (w I w H 2 (Ω, where w I is the Lagrange interpolant of w on W h Proof For µ Γ h and T h, let µ N ( be such that µ F = DF t µ Then, according to (24, we have rot µ F = J F rot µ Hence, since J F > 0, rot µ = 0 if and only if rot µ = 0 On the other hand, note that if µ N (, then µ = (a+bŷ, c+d x, with a, b, c, d R, and rot µ = d b Therefore, rot µ = 0 if and only if µ = (a + dŷ, c + d x = v, for v = a x + cŷ + d xŷ Q ( Thus, rot µ = 0 if and only if µ = (DF t µ F = v, with v = v F Q( To prove the second property, since we have already proved that w I Γ h, it is enough to show that the degrees of freedom defining R( w and w I coincide Indeed, consider an edge l with end points A and B as in Figure 22 Then, R( w τ l = w τ l = w(b w(a = w I (B w I (A = w I τ l, l and we conclude the proof l l A τ B Fig 22 Geometry of The two methods that we analyze in this paper only differ in the space used to approximate the rotations Let us now specify them: ITC4: The spaces W h and Γ h are the ones defined above, whereas the space of standard isoparametric bilinear functions is used for the rotations; namely: H h := { η H 0(Ω 2 : η Q( 2 T h } DL4: While for this method W h and Γ h are the same as for ITC4, the space for the rotations is enriched by using a rotation of a space used for the approximation of the Stokes problem in [4] In fact, for each edge l i of, i =, 2, 3, 4, let p i be cubic functions vanishing on l j for j i Namely, p = xŷ( ŷ, p 2 = xŷ( x, p 3 = ŷ( x( ŷ, and p 4 = x( x( ŷ (see Figure 2 Then we define p i := ( p i F τ i and we set H 2 h := { η H 0(Ω 2 : η Q( 2 p, p 2, p 3, p 4 T h }

7 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 7 From now on we use H h to denote any of the two spaces Hh or H2 h In both methods we use R defined by (25-(26 as reduction operator Then, the discretization of Problem 2 can be written in both cases as follows: Problem 23 Find (β h, w h H h W h such that: (20 a(β h, η + (γ h, v Rη = (f, v + t2 2 (θ, η (η, v H h W h, γ h = κ t 2 ( w h Rβ h On the other hand, the discretization of Problem 22 is as follows: Problem 24 Find λ h R and 0 (β h, w h H h W h such that: [ ] a(β h, η + (γ h, v Rη = λ h (w h, v + t2 2 (β h, η (η, v H h W h, γ h = κ t 2 ( w h Rβ h Existence and uniqueness of solution for Problem 23 follow easily (see [2] Regarding Problem 24, it leads to a well posed generalized matrix eigenvalue problem, since the bilinear form in the right hand side of the first equation is an inner product 3 H error estimates To prove optimal error estimates in H norm we will use the abstract theory developed in [2] In particular, sufficient conditions to obtain such estimates have been settled in Theorem 3 of this reference By virtue of Lemma 2 this theorem reads in our case: Theorem 3 Let H h, W h, Γ h, and the operator R be defined as above Let (β, w, γ and (β h, w h, γ h be the solutions of equations (2 and (20, respectively If there exist β H h and an operator Π : H 0 (rot, Ω H (Ω 2 Γ h satisfying (3 β β,ω Ch β 2,Ω, (32 η Πη 0,Ω Ch η,ω η H (Ω 2 H 0 (rot, Ω, and (33 ( t 2 rot Πγ + R β = 0, κ then, the following error estimate holds true: β β h,ω + t γ γ h 0,Ω + w w h,ω Ch ( β 2,Ω + t γ,ω + γ 0,Ω Then, our next step is to construct an approximation β of β and an operator Π satisfying the hypotheses of the previous theorem for each one of the methods, ITC4 and DL4 3 ITC4 Several studies have been carried out for this method in, for example, [6], [2], and [7] Since the variational equations for plates have a certain similitude with those of the Stokes problem, the main results are based on properties already known for the latter An order h of convergence is obtained in those references only for uniform meshes of square elements oreover, more regularity of the

8 8 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ solutions is also required Although these results can be adapted for parallelogram meshes, they cannot be extended to general quadrilateral ones In what follows we obtain error estimates optimal in order and regularity for this method on somewhat more general meshes We assume specifically the following condition: Assumption 3 The mesh T h is a refinement of a coarser partition T 2h, obtained by joining the midpoints of each opposite edge in each T 2h (called macroelement In addition, T 2h is a similar refinement of a still coarser regular partition T 4h Let Q h := { q h L 2 } 0(Ω : q h = c, c R, T h, where L 2 0(Ω := { q L 2 (Ω : Ω q = 0} Note that, for parallelogram meshes, we have Q h = rot Γ h, but this does not hold for general quadrilateral meshes For each macro-element T 2h we introduce four functions q i, i =, 2, 3, 4, taking the values and according to the pattern of Figure 3 q q q q Fig 3 Bases for the macro-elements Let Q h4 := {q h Q h : q h = c q 4, c R, T 2h } and Q h be its L 2 (Ω orthogonal complement on Q h ; then, Q h := {q h Q h : q h q, q 2, q 3 T 2h } We associate to these spaces the subspace of Hh defined by { } H h := η h Hh : rot η h q h = 0 q h Q h4 Ω The following lemma provides the approximation β required by Theorem 3 oreover, this β H h, and this fact will be used below to define the operator Π required by the same theorem Lemma 32 Let β H 0(Ω Then, there exists β H h such that rot( β βq h = 0 q h Q h and the estimate (3 holds true Ω

9 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 9 Proof It follows from the results in section VI54 of [9] by changing div by rot and rotating 90 the fields, which in its turn are based on the results for isoparametric elements in [22] (see also [9] Our next step is to define the operator Π satisfying the requirements of Theorem 3 To do this, we will use a particular projector P onto rot Γ h We have already mentioned that, in general, Q h rot Γ h In fact, it is simple to show that (34 rot Γ h = { T h c J F χ : c R T h } L 2 0(Ω, where χ denotes the characteristic function of For each macro-element T 2h, we consider the bilinear mapping F as shown in Figure 32 Therefore, for any η h Γ h we have rot η h = J F 4 c i χ i, i= where i are the four elements in (see Figure 32 F Fig 32 Bilinear mapping on macro-elements We define P : L 2 0(Ω rot Γ h as follows: given p L 2 0(Ω, = with c i chosen such that P p q i = 4 i T 2h, P p = i= pq i, i =, 2, 3, and 4 i= c i J F χ i, P p q 4 = 0 Straightforward computations show that P is well defined by the equations above, and that they can be equivalently written (35 P p q h = pq h q h Q h and P p q h = 0 q h Q h4 Ω Ω Ω

10 0 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ The following properties of this operator will be used in the sequel: Lemma 33 The following estimates hold p L 2 (Ω: (36 (37 p P p 0,Ω C p 0,Ω, p P p,ω Ch p 0,Ω Proof To verify (36 it is enough to prove that P p 0,Ω C p 0,Ω From the definition of P we have ( 4 ( 4 ( P p 2 c i = P p χ i J i= F min J P p c i χ i F i= On the other hand, if we write 4 i= c iχ i in terms of the basis functions q i, we obtain 4 i= c iχ i = 4 i= d iq i, with d i related to c i by d c d 2 d 3 = c 2 2 c 3 d 4 c 4 Hence, d i 2 max j 4 c j, i =, 2, 3, 4 Therefore, from the definition of P we have ( 4 ( 4 P p c i χ i = P p d i q i = i= where we have used that i= p 0, ( 3 i= 3 ( d i i= d i q i 0, Cmax J F p 0, P p 0,, j ( P p 2 = c 2 j j pq i ( C /2 p 0, max c j j 4 JF 2 j max J F 2 c 2 j and that C j, j =, 2, 3, 4, with C only depending on σ and ϱ Now, using the inequalities above and noting that, for a quadrilateral regular mesh, max J F C min J F with a constant C independent of h, we obtain (36 To verify (37, let P : L 2 (Ω Q h be the orthogonal projection onto Q h Let v H 0(Ω be such that v,ω = By the definition of P, (36, and the fact that Q h contains the piecewise constants over T 2h, we have (p P p, v = (p P p, v P v p P p 0,Ω v P v 0,Ω C p 0,Ω v P v 0,Ω Ch p 0,Ω v,ω

11 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES Thus we conclude (37 Now we are in order to define an operator Π as required in Theorem 3: Lemma 34 Let (β, w, γ be the solution of equations (2 and β H h be as in Lemma 32 Then, there exists an operator Π : H 0 (rot, Ω H (Ω 2 Γ h such that (32 and (33 hold true Proof For η H 0 (rot, Ω H (Ω 2, let Πη := R(η Lη, where Lη := curl φ := ( φ/ x 2, φ/ x, with φ H (Ω being a solution of φ = rot Rη P (rot Rη in Ω, with homogeneous Neumann boundary conditions Note that this problem is compatible since its right hand side belongs to rot Γ h L 2 0(Ω Then, the standard estimates for the Neumann problem yield (38 Lη m+,ω rot Rη P (rot Rη m,ω, m =, 0 Also note that (39 rot Lη = φ = rot Rη P (rot Rη From the definition of the operator Π, we have η Πη 0,Ω η Rη 0,Ω + RLη 0,Ω The first term in the right hand side is bounded by (29, while for the second term we use again (29, (38, Lemma 33, and (28, to obtain RLη 0,Ω Lη RLη 0,Ω + Lη 0,Ω Ch Lη,Ω + Lη 0,Ω Ch rot Rη P (rot Rη 0,Ω + C rot Rη P (rot Rη,Ω Ch rot Rη 0,Ω Ch η,ω Thus, we conclude (32 To prove (33, note that (27 together with Lemma 32 yield [ ] rot R( β β q h = 0 q h Q h, Ω whereas, since β H h, from (27 and the definition of H h we have rot R β q h = rot β q h = 0 q h Q h4 Ω Ω Hence, since rot R β rot Γ h, from (35 we conclude that P (rot Rβ = rot R β Therefore, (30 rot R β = P (rot Rβ = t2 κ P (rot Rγ, because of the definition of γ in (2 and the fact that rot R( w vanishes, as a consequence of Lemma 2 On the other hand, note that (3 rot RLγ = rot Lγ

12 2 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ Indeed, rot RLγ and rot Lγ both belong to rot Γ h (the latter because of (39 Then, from the characterization (34 of this space, it is enough to verify that rot RLγ = rot Lγ T h, which in its turn is a consequence of (27 Therefore, from the definition of Π, (3, and (39, we obtain rot Πγ = rot R(γ Lγ = rot Rγ rot Lγ = P (rot Rγ, which together with (30 allow us to conclude (32 32 DL4 The convergence of this method follows immediately from that of ITC4 However, we have an alternative proof valid for any regular mesh without the need of Assumption 3 In this case, to define the approximation β of β and the operator Π satisfying the hypotheses of Theorem 3, we use some known results for the Stokes problem (see Girault and Raviart[4] Lemma 35 There exists β Hh 2 such that (3 holds true Furthermore, R β = Rβ Proof By using results from [4] (section 3, chapter II and taking into account a rotation of the space H(div, Ω, it follows that for β H 0(Ω 2 there exists β Hh 2 such that ( β β τ l = 0 l T h, and l β β m,ω Ch k m β k,ω, m = 0,, k =, 2 Then R( β β = 0 because of the definition of R, whereas (3 corresponds to the inequality above for k = 2 and m = Lemma 36 There exists an operator Π : H 0 (rot, Ω H (Ω 2 Γ h such that (33 and (32 hold Proof Because of the previous lemma we have R( β β = 0 On the other hand, rot R( w = 0, because of Lemma 2 Then, it is enough to take Π = R to obtain (33, whereas (32 follows from (29 33 ain result in H norm Now we are in position to establish the error estimates As above, in the case of ITC4, we consider meshes satisfying Assumption 3 Theorem 37 Given (θ, f L 2 (Ω 2 L 2 (Ω, let (β, w and (β h, w h be the solutions of Problems 2 and 23, respectively Then, there exists a constant C, independent of t and h, such that β β h,ω + w w h,ω Ch (θ, f t Proof It is a direct consequence of Lemmas 32, 34, 35, 36, Theorem 3, and the a priori estimate (22 4 L 2 error estimates Our next goal is to prove L 2 error estimates optimal in order and regularity To do this, we follow the techniques in [3] where a triangular element similar to DL4 is analyzed, although the arguments therein cannot be directly

13 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 3 applied to our case Let us remark that, in the case of ITC4, this result completes the analysis carried out in [0, 8] for higher order methods Our proofs are based on a standard Nitsche s duality argument However, since the methods are non-conforming, additional consistency terms also arise Then, higher order estimates must be proved for these terms too, which is the most delicate part of the paper First, we introduce the dual problem corresponding to equations (2 Let (ϕ, u H 0(Ω 2 H 0(Ω be the solution of (4 a(η, ϕ + ( v η, δ = (v, w w h + (η, β β h (η, v H 0(Ω 2 H 0(Ω, δ = κ ( u ϕ t2 By taking η = 0 in (4, we have (42 div δ = w h w An a priori estimate analogous to (22 yields for this problem: (43 ϕ 2,Ω + u 2,Ω + δ 0,Ω + t δ,ω C ( β β h 0,Ω + w w h 0,Ω The arguments in the proof of Lemma 34 in [3] can be used in our case leading to the following result: Lemma 4 Given (θ, f L 2 (Ω 2 L 2 (Ω, let (β, w, γ and (β h, w h, γ h be the solutions of equations (2 and (20, respectively Let (ϕ, u, δ be the solution of (4 Let ϕ H h be the vector field associated to ϕ by Lemmas 32 or 35, for ITC4 or DL4, respectively Then, there exists a constant C, independent of t and h, such that β β h 0,Ω + w w h 0,Ω Ch 2 (θ, f t + (β h Rβ h, δ + (γ, ϕ R ϕ β β h 0,Ω + w w h 0,Ω Our next step is to prove that the last term in the inequality above is O(h 2 too A similar result has been proved in [3] in the case of triangular meshes That proof relies on a technical result for the rotated Raviart-Thomas interpolant R (Lemma 33 of that reference It is easy to check that the arguments given there do not apply for quadrilateral elements Therefore, we need to introduce new arguments and this is the aim of the following lemma Lemma 42 Given ζ H(div, Ω and ψ H 0(Ω 2, there holds (ζ, ψ Rψ Ch 2 ( Rψ ψ 2, /2 div ζ 0,Ω + Ch rot(rψ ψ 0,Ω ζ 0,Ω Proof For T h, let s H ( be a solution of s = rot(rψ ψ in, with homogeneous Neumann boundary conditions By virtue of (27 we know that the above problem is compatible Hence, s satisfies (44 curl s m+, C rot(rψ ψ m,, m =, 0

14 4 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ The Laplace equation above can be equivalently written rot [curl s (Rψ ψ] = 0 and, hence, there exists r H ( (unique up to an additive constant such that (45 r = curl s (Rψ ψ oreover, from the homogeneous Neumann boundary condition satisfied by s, we have r τ l = Rψ τ l + ψ τ l for each edge l of Thus, if we define G L 2 (Ω 2 such that G = r, then G H 0 (rot, Ω and rot G = 0 Hence, there exists r H (Ω/R, such that G = r in Ω Furthermore, since G H 0 (rot, Ω, r can be chosen in H 0(Ω and the additive constants defining r on each T h can be fixed as to satisfy r = r Let A and B be as in Figure 22 Then, because of (26, we have r(b = r(a + r τ l = r(a + ( Rψ + ψ τ l = r(a l Thus r vanishes at all nodes of T h, since r Ω = 0 Hence, a standard scaling argument on each element yields r 0, Ch 2 r 2, (see for instance [] and, then, by using (45 and (44, we have ( (46 r 0, Ch 2 r, Ch 2 curl s, + Rψ ψ, Ch 2 [ rot(rψ ψ 0, + Rψ ψ, ] Ch 2 Rψ ψ, On the other hand, let (, be the usual inner product in L 2 ( and P the orthogonal projection onto the constant functions Because of (27, we have η H 0(Ω, ( ( rot(rψ ψ, η rot(rψ ψ, η P η Hence, η, = l rot(rψ ψ 0, η P η 0, η, η, rot(rψ ψ, Ch rot(rψ ψ 0, Now, let S L 2 (Ω 2 be such that S = curl s Therefore, because of (44 we have (47 S 2 0,Ω = curl s 2 0, rot(rψ ψ 2, T h T h Ch 2 rot(rψ ψ 2 0, Ch2 rot(rψ ψ 2 0,Ω T h Finally, from (45 we obtain (ζ, ψ Rψ = ζ r + ζ S div ζ 0,Ω r 0,Ω + ζ 0,Ω S 0,Ω, Ω Ω and the lemma follows by using (46 and (47

15 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 5 To obtain a bound of the consistency term in Lemma 4, there only remains to estimate the terms involving (Rψ ψ of the previous lemma To this aim, we use the analogue of Theorem 43 in [24] applied to our situation in the space H(rot, Ω, which reads: (48 ψ Rψ, C ( ψ, + h rot ψ, ψ 2, and (49 ( δ rot(ψ Rψ 0, C rot ψ h 0, + h rot ψ,, where δ is a measure of the deviation of the quadrilateral from a parallelogram, as defined in Figure 4 v v 2 v=v 2 v δ = v Fig 4 Geometrical definition of δ Note that for shape-regular meshes clearly δ /h C T h On the other hand, {T h } is said to be a family of asymptotically parallelogram meshes when there exists a constant C such that max Th (δ /h Ch for all the meshes Now we are in position to estimate the consistency term in Lemma 4: Lemma 43 Let β h, δ, γ and ϕ be as in Lemma 4 Then, there holds (β h Rβ h, δ + (γ, ϕ R ϕ β β h 0,Ω + w w h 0,Ω Proof First, we have Ch ( δ h + t max T h h (θ, f t (β h Rβ h, δ ( (β h β R(β h β, δ + (β Rβ, δ By using (29, Theorem 37, and (43, we obtain ( (β h β R(β h β, δ Ch βh β,ω δ 0,Ω Ch 2 (θ, f t δ 0,Ω ( Ch 2 (θ, f t β β h 0,Ω + w w h 0,Ω On the other hand, by the definition of γ in (2 and the estimate (22, we have rot β 0, = t2 κ rot γ 0, Ct (θ, f t Then, by using Lemma 42, (48, (49, the estimate above, (22, (42, and (43, we have

16 6 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ ( /2 (β Rβ, δ Ch 2 Rβ β, 2 div δ + Ch rot(rβ β 0,Ω 0,Ω δ 0,Ω ( Ch 2 δ β 2,Ω div δ 0,Ω + Ch max rot β T h h 0,Ω + h rot β,ω ( Ch 2 δ (θ, f t div δ 0,Ω + Ch h + t max (θ, f T h h t δ 0,Ω Ch ( δ h + t max T h h (θ, f t ( β β h 0,Ω + w w h 0,Ω δ 0,Ω The term (γ, ϕ R ϕ can be bounded almost identically, by using Lemma 32 for ITC4 or Lemma 35 for DL4 to estimate ϕ ϕ,ω and the fact that div γ = f in Ω, which follows by taking η = 0 in the first equation of (2 Therefore, we obtain ( δ ( (γ, ϕ R ϕ Ch h + t max (θ, f T h h t β β h 0,Ω + w w h 0,Ω, which allows us to conclude the proof Finally, we can establish an L 2 (Ω error estimate As above, in case of ITC4 elements, we consider meshes satisfying Assumption 3 Theorem 44 Given (θ, f L 2 (Ω 2 L 2 (Ω, let (β, w and (β h, w h be the solutions of Problems 2 and 23, respectively Then, there exists a constant C, independent of t and h, such that ( δ β β h 0,Ω + w w h 0,Ω Ch h + t max (θ, f T h h t Proof It is a direct consequence of Lemmas 4 and 43 Corollary 45 The following error estimate holds for any family of asymptotically parallelogram meshes: β β h 0,Ω + w w h 0,Ω Ch 2 (θ, f t Remark 4 The asymptotically parallelogram assumption on the meshes is not necessary as long as h > αt, for α fixed Indeed, according to Theorem 44, for general regular meshes with h > αt, we have β β h 0,Ω + w w h 0,Ω C α h 2 (θ, f t Note that the condition h > αt is fulfilled in practice for reasonably large values of α 5 The spectral problem The aim of this section is to study how the eigenvalues and eigenfunctions of Problem 24 approximate those of Problem 22 We do this in the framework of the abstract spectral approximation theory as stated, for instance, in the monograph by Babuška and Osborn[5] In order to use this theory, we

17 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 7 define operators T and T h associated to the continuous and discrete spectral problems, respectively We consider the operator T : L 2 (Ω 2 L 2 (Ω L 2 (Ω 2 L 2 (Ω, defined by T (θ, f := (β, w, where (β, w H 0(Ω 2 H 0(Ω is the solution of Problem 2 Note that T is compact, as a consequence of estimate (22 Since the operator is clearly self-adjoint with respect to (, t, then, apart from µ = 0, its spectrum consists of a sequence of finite multiplicity real eigenvalues converging to zero Note that λ is an eigenvalue of Problem 22 if and only if µ := /λ is an eigenvalue of T, with the same multiplicity and corresponding eigenfunctions As shown in [3], each eigenvalue µ of Problem 2 converges to some limit µ 0, when the thickness t 0 Indeed, µ 0 is an eigenvalue of the operator associated with the irchhoff model of the same plate (see Lemma 2 in [3] From now on, for simplicity, we assume that µ = /λ is an eigenvalue of T which converges to a simple eigenvalue µ 0, as t goes to zero (see section 2 in [3] for further discussions Now, analogously to the continuous case, we introduce the operator T h : L 2 (Ω 2 L 2 (Ω L 2 (Ω 2 L 2 (Ω, defined by T h (θ, f := (β h, w h, where (β h, w h H h W h is the solution of Problem 23 The operator T h is also self-adjoint with respect to (, t Clearly, the eigenvalues of T h are given by µ h := /λ h, with λ h being the strictly positive eigenvalues of Problem 24, and the corresponding eigenfunctions coincide As a consequence of Theorem 37, for each simple eigenvalue µ of T, there is exactly one eigenvalue µ h of T h converging to µ as h goes to zero (see for instance [6] The following theorem shows optimal t-independent error estimates Let us remark that the results of this theorem are valid for both methods, ITC4 and DL4, although, for the former, under Assumption 3 on the meshes as in the previous section Theorem 5 Let λ and λ h be simple eigenvalues of Problems 22 and 24, respectively, such that λ h λ as h 0 Let (β, w and (β h, w h be corresponding eigenfunctions normalized in the same manner Then, under the assumptions stated above, there exists C > 0 such that, for t and h small enough, there holds β β h,ω + w w h,ω Ch Furthermore, for any family of asymptotically parallelogram meshes, there hold and β β h 0,Ω + w w h 0,Ω Ch 2 λ λ h Ch 2 Proof The proof, which relies on Theorem 37 and Corollary 45, are essentially the same as those of Theorem 2, 22, and 23 in [3] 6 Numerical experiments In this section we report some numerical experiments carried out with both methods applied to the spectral problem 22

18 8 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ First, we have tested the two methods by using different meshes, not necessarily satisfying the assumptions in the theorems above We have considered a square clamped moderately thick plate of side-length L and thickness-to-span ratio t/l = 0 We report the results obtained with both types of elements using the following three families of meshes: T U h T A h T T h : It consists of uniform subdivisions of the domain into N N sub-squares, for N = 4, 8, 6, (see Figure 6 Clearly, these are parallelogram meshes satisfying Assumption 3 : It consists of uniform refinements of a non-uniform mesh obtained by splitting the square into four quadrilaterals Each refinement step is obtained by subdividing each quadrilateral into other four, by connecting the midpoints of the opposite edges Thus we obtain a family of N N asymptotically parallelogram shape regular meshes as shown in Figure 62 Furthermore, for N = 4, 8, 6,, these meshes satisfy Assumption 3 : It consists of partitions of the domain into N N congruent trapezoids, all similar to the trapezoid with vertices (0, 0, (/2, 0, (/2, 2/3 and (0, /3, as shown in Figure 63 Clearly, these are not asymptotically parallelogram meshes and they do not satisfy Assumption 3 N=4 N=8 N=6 Fig 6 Uniform square meshes T U h N=4 N=8 N=6 Fig 62 Asymptotically parallelogram meshes T A h N=4 N=8 N=6 Fig 63 Trapezoidal meshes T T h

19 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 9 Let us remark that the third family was used in [3, 4] to show that the order of convergence of some finite elements deteriorate on these meshes, in spite of the fact that they are shape regular We have computed approximations of the free vibration angular frequencies ω = t λ/ρ corresponding to the lowest-frequency vibration modes of the plate In order to compare the obtained results with those in [] we present the computed frequencies ω h mn in the following non-dimensional form: [ ] /2 2( + νρ ˆω mn := ωmnl h, E m and n being the numbers of half-waves occurring in the mode shapes in the x and y directions, respectively Tables 6 and 62 show the four lowest vibration frequencies computed by our method with successively refined meshes of each type, Th U, T h A, and T h T Each table includes also the values of the vibration frequencies obtained by extrapolating the computed ones as well as the estimated order of convergence Finally, it also includes in the last column the results reported in [] In every case we have used a Poisson ratio ν = 03 and a correction factor k = 0860 The reported non-dimensional frequencies are independent of the remaining geometrical and physical parameters, except for the thickness-to-span ratio Table 6 Scaled vibration frequencies ˆω mn computed with ITC4 esh ode N = 6 N = 32 N = 64 Extrap Order [] ˆω Th U ˆω ˆω ˆω ˆω Th A ˆω ˆω ˆω ˆω Th T ˆω ˆω ˆω Table 62 Scaled vibration frequencies ˆω mn computed with DL4 esh ode N = 6 N = 32 N = 64 Extrap Order [] ˆω Th U ˆω ˆω ˆω ˆω Th A ˆω ˆω ˆω ˆω Th T ˆω ˆω ˆω

20 20 DURÁN, HERNÁNDEZ, HERVELLA-NIETO, LIBERAN, AND RODRÍGUEZ It can be clearly seen that both methods converge for the three types of meshes with an optimal O(h 2 order Hence, none of the two particular assumptions on the meshes (Assumption 3 and to be asymptotically parallelogram seem to be actually necessary As a second test, we have made a numerical experiment to assess the stability of the methods as the thickness t goes to zero We have used a sequence of clamped plates with decreasing values of the thickness-to-span ratios: t/l = 0, 00, 000, 0000 All the other geometrical and physical parameters have been taken as in the previous test We have computed again approximations of the free vibration angular frequencies ω = t λ/ρ The quotients ω/t are known to converge to the corresponding vibration frequencies of an identical irchhoff plate (ie, to the frequencies obtained from the irchhoff model for the deflection of a similar zero-thickness ideal plate; see Lemma 2 from [3] Because of this, we present now the computed frequencies ω h mn scaled in the following manner: ω mn := ωmn h L t [ ] /2 2( + νρ E The obtained results have been qualitatively similar for both methods We only report those obtained with DL4, since the performance of ITC4 has been assessed in many other papers (see for instance [8], as well as [5] for the vibration problem We present in Table 63 the results for the lowest-frequency vibration mode, with the same meshes as in the previous test In each case, for each thickness-to-span ratio t/l, we have computed again an extrapolated more accurate value of the scaled vibration frequency and the estimated order of convergence Finally we have also estimated by extrapolation the limit values of the scaled frequencies ω mn as t goes to zero Table 63 Scaled vibration frequency ω computed with DL4 for different thickness-to-span ratios t/l esh t/l N = 6 N = 32 N = 64 Extrap Order Th U (extrap Th A (extrap Th T (extrap Note that the extrapolated values for each thickness-to-span ratio are almost identical for the three meshes oreover, although the estimated orders of convergence seem to deteriorate a bit as t/l goes to zero for the non-uniform meshes, the values obtained with these meshes are better than those computed with the uniform mesh

21 ISOPARAETRIC QUADRILATERAL FINITE ELEENTS FOR PLATES 2 (ie closer to the extrapolated ones, even for the coarser meshes Therefore, this test suggests that the method is locking-free for any kind of regular meshes Finally, we report in Table 64 the corresponding extrapolated values as t/l goes to zero for the four lowest scaled vibration frequencies It can be seen from this table that the results are essentially the same as for ω Furthermore, the computed orders of convergence are even closer to 2 Table 64 Extrapolated values as (t/l 0 of the scaled vibration frequencies ω mn computed with DL4 esh ode N = 6 N = 32 N = 64 Extrap Order ˆω Th U ˆω ˆω ˆω ˆω Th A ˆω ˆω ˆω ˆω Th T ˆω ˆω ˆω Further experiments with ITC4 have been reported in [5], including other boundary condition and the extension of this method to compute the vibration modes of Naghdi shells REFERENCES [] B S Al Janabi and E Hinton, A study of the free vibrations of square plates with various edge conditions, in Numerical ethods and Software for Dynamic Analysis of Plates and Shells, E Hinton, ed, Pineridge Press, Swansea, 987 [2] D N Arnold and R S Falk, A uniformly accurate finite element method for the Reissner- indlin plate, SIA J Numer Anal, 26 ( [3] D N Arnold, D Boffi, and R S Falk, Approximation by quadrilateral finite elements, ath Comp, 7 ( [4] D N Arnold, D Boffi, R S Falk, and L Gastaldi, Finite element approximation on quadrilateral meshes, Comm Numer ethods Engrg, 7 ( [5] I Babuška and J Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, Vol II, P G Ciarlet and J L Lions, eds, North Holland, Amsterdam, 99, pp [6] J Bathe and F Brezzi, On the convergence of a four-node plate bending element based on indlin/reissner plate theory and a mixed interpolation, in The athematics of Finite Elements and Applications V, J R Whiteman, ed, Academic Press, London, 985, pp [7] J Bathe and E N Dvorkin, A four-node plate bending element based on indlin/reissner plate theory and a mixed interpolation, Internat J Numer ethods Engrg, 2 ( [8] J Bathe, A Iosilevich, and D Chapelle, An evaluation of the ITC shell elements Comp Struct, 75 ( [9] F Brezzi and Fortin, ixed and Hybrid Finite Element ethods, Springer-Verlag, New York, 99 [0] F Brezzi, Fortin, and R Stenberg, Quasi-optimal error bounds for approximation of shear-stresses in indlin-reissner plate models, ath odels ethods Appl Sci, ( [] P G Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Vol II, P G Ciarlet and J L Lions, eds, North-Holland, Amsterdam, 99, pp 7 35 [2] R Durán and E Liberman, On mixed finite element methods for the Reissner-indlin plate model, ath Comp, 58 (

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