ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD FOR REISSNER-MINDLIN PLATES

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 14, Number 1, Pages c 2017 Institute for Scientific Computing and Information ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD FOR REISSNER-MINDLIN PLATES GUOZHU YU, XIAOPING XIE*, AND YUANHUI GUO Abstract. This paper analyzes an existing 4-node hybrid mixed-shear-projected quadrilateral element MiSP4, presented by Ayad, Dhatt and Batoz (Int. J. Numer. Meth. Engng 1998, 42: ) for Reissner-Mindlin plates, which behaves robustly in numerical benchmark tests. This method is based on Hellinger-Reissner variational principle, where continuous piecewise isoparametric bilinear interpolations, as well as the mixed shear interpolation/projection technique of MITC family, are used for the approximations of displacements, and piecewise-independent equilibrium modes are used for the approximations of bending moments/shear stresses. Due to local elimination of the parameters of moments/stresses, the computational cost of MiSP4 element is almost the same as that of the conforming bilinear quadrilateral displacement element. We show that the element is free from shear locking in the sense that the error bound in the derived a priori estimate is independent of the plate thickness. Key words. Reissner-Mindlin plate, mixed-shear-projected quadrilateral element, shear-locking free. 1. Introduction Due to avoidance of C 1 -continuity difficulty, the Reissner-Mindlin (R-M) plate model is today the dominating two-dimensional model used to calculate the bending of a thick/thin three-dimensional plate of thickness t. It s well-known that for values of t close to zero, the standard low-order finite element discretization of this model suffers from shear locking ([1, 17]). To overcome the shear locking difficulty and derive locking-free or robust plate bending elements that are valid for the analysis of thick and thin plates, significant efforts are devoted to the development of simple and efficient triangular and quadrilateral finite elements in the past few decades. The most common approach is to modify the variational formulation with some reduction operator so as to weaken the Kirchhoff constraint (see, e.g. [2]-[8], [10], [12], [14]-[16], [18]-[20] and the references therein). Among the existing elements, the family of finite elements named mixed interpolated tensorial components (MITC) by Bathe et. al [4, 5] is one of the most attractive representative. By virtue of an independent shear approximation and a discrete Mindlin technique along edges, MITC elements define the shear strains in terms of the edge tangential strains that are projected on the element degrees of freedom. As the lowest order quadrilateral MITC element, the 4-node plate element MITC4 is very likely the most used in practice. Using the same technique of shear interpolation as in the element MITC family, Ayad, Dhatt and Batoz [3] presented an improved formulation for obtaining locking-free quadrilateral element, which is called MiSP4 element. It is based on Hellinger-Reissner variational principle, including variables of displacements, shear Received by the editors on October 25, Mathematics Subject Classification. 35R35, 49J40, 60G40. *Corresponding author. 48

2 ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD 49 stresses and bending moments. For the approximations of displacements, MiSP4 element uses continuous piecewise isoparametric bilinear interpolations. For the approximations of bending moments/shear stresses, it uses piecewise-independent equilibrium modes. The numerical experiments are presented to show that the MiSP4 element can avoid locking phenomenon, and it also passes the patch test for a general quadrilateral. However, so far there is no uniform error analysis with respect to plate thickness. It should be pointed out that in a veryrecentpaper [8], the shear interpolation treatment was replaced by enhancing a shear-stress-enhanced condition, and the resultant 4-node hybrid finite element scheme was shown to be locking-free. The main goal of this work is to establish uniform convergence for quadrilateral MiSP4 element. The main tools of our analysis are the self-equilibrium relation, i.e. (21), which contributes to the uniform coercivity of the corresponding bilinear forms, and the properties of shear interpolation proved in [11] for MITC4 element (see Lemma 4.11). We arrange the rest of this paper as follows. In Section 2 we give weak formulation of the model. Section 3 introduces the finite element spaces for MiSP4 element. We derive in Section 4 uniform error estimates for MiSP4 element. Finally in Section 5 we provide some numerical results to verify the theoretical results. For convenience, throughout the paper we use the notation a b to represent that there exists a generic positive constant C, independent of the mesh parameter h and the plate thickness t, such that a Cb. We also abbreviate a b a as a b. 2. Weak problem The Reissner-Mindlin model for the bending of a clamped isotropic elastic plate in equilibrium reads as: Find (w,β) H 1 0 () H1 0 ()2 such that (1) (2) divdǫ(β) λt 2 (grad w β) = 0 in, λt 2 div(grad w β) = g in. Here R 2, assumed to be a convex polygon for simplicity, is the region occupied by the midsection of the plate with plate thickness t, w and β denote respectively the transverse displacement of the midplane and the rotation of the fibers normal to it, ǫ(β) is the symmetric part of the gradient of β, g is the transverse loading, D is the elastic module tensor defined by E DQ = 12(1 ν 2 ) [(1 ν)q+νtr(q)i] with Q a 2 2 symmetric matrix, λ = κe 2(1+ν) with E the Young s modulus, ν the Poisson s ratio, and κ = 5 6 the shear correction factor. Set M := L 2 () 2 2 sym, Γ := L 2 () 2, W := H 1 0(), Θ := H 1 0() 2. When introducing the shear stress vector γ = λt 2 (grad w β) and the bending moment tensor M = Dǫ(β), the model problem (1)-(2) changes into the following system: Find (M,γ,w,β) M Γ W Θ such that (3) (4) (5) (6) divm γ = 0 in, divγ +g = 0 in, M+Dǫ(β) = 0 in, γ λt 2 (grad w β) = 0 in.

3 50 G. YU, X. XIE, AND Y. GUO Thevariationalformulationofthis systemreads: Find(M,γ,w,β) M Γ W Θ such that (7) (8) where the bilinear forms a(m,γ;q,τ)+b(q,τ;w,β) = 0 (Q,τ) M Γ, b(m,γ;v,ζ) = gvdx (v,ζ) W Θ, a(, ;, ) : (L 2 () 2 2 sym L2 () 2 ) (L 2 () 2 2 sym L2 () 2 ) R, b(, ;, ) : (L 2 () 2 2 sym L 2 () 2 ) (H0 1() H1 0 ()2 ) R are defined by (9) a(m,γ;q,τ) := : D M 1 Qdx+ t2 γ τdx, λ (10) b(q,τ;v,ζ) := Q : ǫ(ζ)dx τ (grad v ζ)dx. To get further regularity of the solution (M,γ,w,β), we introduce a weak problem: Find (w,β,γ) W Θ Γ such that (11) ǫ(β) : Dǫ(ζ)dx+ γ (grad v ζ)dx = gvdx (v,ζ) W Θ, (12) τ (grad w β)dx t2 λ We recall the following result (see [2, 6]). γ τdx = 0 τ Γ. Lemma 2.1. The problem (11)-(12) admits a unique solution with (13) w 2 + β 2 + γ 0 +t γ 1 g 0. In addition, if is a smoothly bounded domain and g H 1 (), then it holds (14) w 3 g 1. With the above lemma, we obtain some further results [8]: Theorem 2.2. Let (w,β,γ) W Θ Γ be the solution of the problem (11)-(12). Then the following three conclusions (i)-(iii) hold. (i) The quadruple (M = Dǫ(β),γ,w,β) M Γ W Θ is the unique solution of the problem (7)-(8); (ii) If M H(div;) := {Q L 2 () 2 2 sym : divq L 2 () 2 }, then the equilibrium relation (3) holds; (iii) Provided that g L 2 (), it holds (15) w 2 + β 2 + M 1 + γ 0 +t γ 1 g Finite element formulation for MiSP4 method 3.1. Finite element formulation. This subsection is devoted to the finite element formulation of the MiSP4 method on quadrilateral meshes. Let T h be a regular family of finite element subdivisions of the polygonal domain. Let M h M, Γ h Γ, W h W, Θ h Θ be finite dimensional spaces for the bending moment, shear stress, transverse displacement, and rotation approximations. Then

4 ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD 51 the corresponding finite element scheme for the problem (7)-(8) reads as: Find (M h,γ h,w h,β h ) M h Γ h W h Θ h such that (16) (17) a(m h,γ h ;Q h,τ h )+ b(q h,τ h ;w h,β h ) = 0 (Q h,τ h ) M h Γ h, b(mh,γ h ;v h,ζ h ) = gv h dx (v h,ζ h ) W h Θ h, where (18) b(qh,τ h ;v h,ζ h ) := Q h : ǫ(ζ h )dx τ h (grad v h R h ζ h )dx, and the reduction operator R h : H 1 () 2 H 0 (rot,) Z h is defined by [11] (19) R h ψ t e = ψ t e edge e of T h, where e (20) H 0 (rot,) := {ψ L 2 () 2 : rotψ L 2 (),ψ t = 0}, e Z h is to be defined in (30) for MiSP4, and t e denotes a unit vector tangent to e. For MiSP4 element, we define (21) Γ h = div h M h, with (Q h,τ h ) = (Q h,div h Q h ) for Q h M h. Here div h denotes the divergence operator piecewise defined with respect to T h. From the definition of the space Γ h, we have an equivalent form of the discrete scheme (16)-(17): Find (M h,w h,β h ) M h W h Θ h such that (22) a(m h,div h M h ;Q h,div h Q h )+ b(q h,div h Q h ;w h,β h ) = 0 Q h M h, (23) b(mh,div h M h ;v h,ζ h ) = gv h dx (v h,ζ h ) W h Θ h Finite dimensional subspaces. Let T h be a conventional quadrilateral mesh of. We denote by h K the diameter of a quadrilateral K T h, and denote h := max K Th h K. Let Z i (x i,y i ), 1 i 4 be the four vertices of K, and T i be the sub-triangle of K with vertices Z i 1, Z i and Z i+1 (the index on Z i is modulo 4). Define ρ K = min 1 i 4 {diameter of circle inscribed in T i}. Throughout the paper, we assume that the partition T h satisfies the following shape-regularity hypothesis: There exists a constant > 2 independent of h such that for all K T h, (24) h K ρ K. Let ˆK = [ 1,1] [ 1,1] be the reference square with vertices Ẑi, 1 i 4. For a quadrilateral K T h, there exists a unique invertible mapping F K that maps ˆK onto K with F K (ξ,η) Q 2 1(ξ,η) and F K (Ẑi) = Z i, 1 i 4 (Figure 3.1). Here ξ, η [ 1, 1] are the local isoparametric coordinates. This isoparametric bilinear mapping (x,y) = F K (ξ,η) is given by 4 4 (25) x = x i N i (ξ,η), y = y i N i (ξ,η), where i=1 N 1 = 1 4 (1 ξ)(1 η), N 2 = 1 4 (1+ξ)(1 η), N 3 = 1 4 (1+ξ)(1+η), N 4 = 1 4 (1 ξ)(1+η). i=1

5 52 G. YU, X. XIE, AND Y. GUO Ẑ 4 η 1 Ẑ 3 Ẑ 1-1 Ẑ 2 F K -1 1 ξ Z 3 y Z 4 Z 2 Z 1 x Figure 3.1. The mapping F K We can rewrite (25) as (26) x = a 0 +a 1 ξ +a 2 η +a 12 ξη, y = b 0 +b 1 ξ +b 2 η +b 12 ξη, with a 0 b 0 a 1 b 1 a 2 b 2 a 12 b 12 = 1 4 ( x ξ y ξ x η y η x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 The Jacobi matrix and the Jacobian of the transformation F K are respectively given by ) ( ) a1 +a DF K (ξ,η) = = 12 η a 2 +a 12 ξ, b 1 +b 12 η b 2 +b 12 ξ where J K = det(df K ) = J 0 +J 1 ξ +J 2 η,. J 0 = a 1 b 2 a 2 b 1, J 1 = a 1 b 12 a 12 b 1, J 2 = a 12 b 2 a 2 b 12. Remark 3.1. Notice that when K is a parallelogram, we have a 12 = b 12 = 0, and F K is reduced to an affine mapping. Especially, when K is a rectangle, we further have a 2 = b 1 = 0. For element MiSP4, the continuous isoparametric bilinear interpolations are used for the transverse displacement and rotation approximations, i.e. the transverse displacement space W h and rotation space Θ h are chosen as (27) W h := {v h H 1 0() C( ) : v h K F K Q 1 ( ˆK) K T h }, (28) Θ h := {ζ h (H 1 0 () C( )) 2 : ζ h K F K Q 1 ( ˆK) 2 K T h }. Here Q 1 ( ˆK) denotes the set of bilinear polynomials on ˆK. For the approximation of bending moment tensor, we define (29) M h := {Q h L 2 () 2 2 sym : (Q h K F K ) i,j Q 1 ( ˆK) K T h,i,j = 1,2}. We take the space Z h as (30) { { ( )} Z h := ψ h H 0 (rot,) : ψ h K F K = span DF t 1 η 0 0 K ξ K T h }.

6 ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD Error analysis for MiSP4 This section is denoted to the error estimates for the MiSP4 element. The corresponding subspaces in this section are defined as in subsection 3.2. We first give the following properties for the operator R h. Lemma 4.1. [13, Theorem III.4.4] The operator R h satisfies η R h η 0 h η 1, η H 1 () 2 H 0 (rot,). Lemma 4.2. [11, Lemma 2.1] The following property holds grad W h = {ψ h Z h,rot ψ h = 0}. With the property in Lemma 4.2 and the definition of R h in (19), we obvious haver h (grad v h ) = grad v h, v h W h, and sothe expression(18) canbe rewritten as (31) b(qh,τ h ;v h,ζ h ) = Q h : ǫ(ζ h )dx τ h R h (grad v h ζ h )dx. In general, for any Q (H 1 ()) 2 2 sym + M h, v (H 2 () H0 1()) + W h, ζ H0() 1 2 +Θ h, we define (32) b(q,divh Q;v,ζ) := Q : ǫ(ζ)dx div h Q R h (grad v ζ)dx, and define two mesh-dependent semi-norms for the finite dimensional spaces: (33) Q h,1 := Q 0 +(t+h) div h Q 0, (34) (v,ζ) h,2 := ǫ(ζ) 0 +(t+h) 1 R h (grad v ζ) 0. It is easy to see that h,1 is a norm on (H 1 ()) 2 2 sym + M h, and (, ) h,2 is a norm on W h Θ h. With this definition of mesh-dependent semi-norms, we can easily check the continuity results in Lemma 4.3. While the corresponding coercivity results are deduced in Lemma Lemma 4.5 is a preparation for Lemma 4.6. Lemma 4.3. For any M,Q (H 1 ()) 2 2 sym + M h, v (H 2 () H 1 0 ()) + W h, ζ H 1 0 ()2 +Θ h, it holds uniformly the continuity condition (35) a(m,div h M;Q,div h Q) M h,1 Q h,1, (36) b(q,divh Q;v,ζ) Q h,1 (v,ζ) h,2. Lemma 4.4. It holds uniformly the discrete coercivity condition (37) a(q h,div h Q h ;Q h,div h Q h ) Q h h,1, Q h M h. Proof. The proof immediately follows from the inverse inequality div h Q h 0 h 1 Q h 0. Lemma 4.5. The following two conclusions hold: (1) For any given ζ h Θ h, there exists Q 1 h M h, such that (38) (Q 1 h,ǫ(ζ h )) = Q 1 h 2 0 ǫ(ζ h ) 2 0, and div h Q 1 h = 0; (2)For any given v h W h, ζ h Θ h, there exists Q 2 h M h, such that (39) (div h Q 2 h,r h(grad v h ζ h )) = (t 2 +h 2 ) div h Q 2 h t 2 +h 2 R h(grad v h ζ h ) 2 0,

7 54 G. YU, X. XIE, AND Y. GUO and (40) div h Q 2 h 0 h 1 Q 2 h 0. Proof. The proof is similar to that in [8]. (1) Given ζ h Θ h, choose Q 1 h as the 5-parameter PS element in [8]. The proof for (38) can be found in [8, Lemma 4.4]. (2) Given v h W h, ζ h Θ h, for any K T h, R h (grad v h ζ h ) K can be expressed as = 1 J K R h (grad v h ζ h ) K ( b2 +b 12 ξ (b 1 +b 12 η) (a 2 +a 12 ξ) a 1 +a 12 η )( 1 η ξ ) c 1 c 2 c 3 c 4, here (c 1,c 2,c 3,c 4 ) T depends on v h,ζ h. Some calculations show R h (grad v h ζ h ) 2 0,K [ 4 = (b 2 c 1 b 1 c 3 ) J K (ξ 1,η 1 ) 3 (b 2c 2 b 12 c 3 ) (b 12c 1 b 1 c 4 ) (b 12c 2 b 12 c 4 ) 2 +(a 2 c 1 a 1 c 3 ) (a 2c 2 a 12 c 3 ) (a 12c 1 a 1 c 4 ) (a 12c 2 a 12 c 4 ) 2 ] C 1 [ = (b2 c 1 b 1 c 3 ) 2 +(b 2 c 2 b 12 c 3 ) 2 +(a 2 c 1 a 1 c 3 ) 2 +(a 12 c 1 a 1 c 4 ) 2]. J K (ξ 1,η 1 ) Take Q h K = c 1ξ +c 3 η +c 2 ξη c 1 ξ +c 3 η +c 4 ξη 0, then we have div h Q h K = 1 ( (b2 c 1 b 1 c 3 )+(b 12 c 1 b 1 c 2 )ξ +(b 2 c 2 b 12 c 3 )η J K (a 2 c 1 a 1 c 3 ) (a 12 c 1 a 1 c 4 )ξ +(a 2 c 4 a 12 c 3 )η ) and div h Q h 2 4 0,K = J K (ξ 2,η 2 ) [ (b 2 c 1 b 1 c 3 ) (b 2c 2 b 12 c 3 ) (b 12c 1 b 1 c 4 ) 2 +(a 2 c 1 a 1 c 3 ) (a 2c 4 a 12 c 3 ) (a 12c 1 a 1 c 4 ) 2 ] C 2 [ = (b2 c 1 b 1 c 3 ) 2 +(b 2 c 2 b 12 c 3 ) 2 J K (ξ 2,η 2 ) +(a 2 c 1 a 1 c 3 ) 2 +(a 12 c 1 a 1 c 4 ) 2].

8 ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD 55 On the other hand, it holds div h Q h R h (grad v h ζ h )dx K [ 4 = (b 2 c 1 b 1 c 3 ) J K (ξ 3,η 3 ) 3 (b 2c 2 b 12 c 3 ) (b 12c 1 b 1 c 4 ) 2 +(a 2 c 1 a 1 c 3 ) (a 2c 4 a 12 c 3 ) ] 3 (a 12c 1 a 1 c 4 ) 2 C 3 [ = (b2 c 1 b 1 c 3 ) 2 +(b 2 c 2 b 12 c 3 ) 2 +(a 2 c 1 a 1 c 3 ) 2 +(a 12 c 1 a 1 c 4 ) 2]. J K (ξ 3,η 3 ) Let C 0 = C3 J K(ξ 3,η 3) t 2 +h, and choose Q 2 2 h K = C 0 Q h K, i.e. div h Q 2 h K = C 0 div h Q h K, then a summation over all elements in T h completes the proof for (39). The result (40) follows from the construction of Q 2 h. C 2 J K(ξ 2,η 2) Lemma 4.6. It holds the inf-sup condition 1 (41) sup Q h M h b(qh,div h Q h ;v h,ζ h ) Q h h,1 (v h,ζ h ) h,2, (v h,ζ h ) W h Θ h. Proof. For ζ h Θ h, from (38) there exists a positive constant C 4 and Q 1 h M h, such that (42) (Q 1 h,ǫ(ζ h)) = Q 1 h 2 0 = C 4 ǫ(ζ h ) 2 0, and div hq 1 h = 0. Forv h W h, ζ h Θ h, from(39)foranypositiveconstantc 5 thereexistsq 2 h M h, such that (43) (div h Q 2 h,r h (grad v h ζ h )) = C 5 (t 2 +h 2 ) div h Q 2 h 2 0 = C 1 5 (t2 +h 2 ) 1 R h (grad v h ζ h ) 2 0, and there exists a positive constant C 6 independent of h and t, such that (44) div h Q 2 h 2 0 = C 6h 2 Q 2 h 2 0. Let Q h = Q 1 h +Q2 h, then we have b(qh,div h Q h ;v h,ζ h ) =(Q 1 h +Q 2 h,ǫ(ζ h )) (div h Q 1 h +div h Q 2 h,r h (grad v h ζ h )) =(Q 1 h,ǫ(ζ h ))+(Q 2 h,ǫ(ζ h )) (div h Q 2 h,r h (grad v h ζ h )) Q 1 h 2 0 Q 2 h 0 ǫ(ζ h ) 0 +C 5 (t 2 +h 2 ) div h Q 2 h 2 0 Q 1 h 2 0 C 4 2 ǫ(ζ h) Q 2 2C h 2 0 +C 5 (t 2 +h 2 ) div h Q 2 h Q 1 h 2 0 C 4 2 ǫ(ζ h) 2 0 h2 2C 4 C 6 div h Q 2 h 2 0 +C 5(t 2 +h 2 ) div h Q 2 h 2 0 C 4 2 Q1 h C 5 2 (t2 +h 2 ) div h Q 2 h 2 0 (by taking C 5 = 1 ǫ(ζ h ) 2 0 +(t 2 +h 2 ) 1 R h (grad v h ζ h ) 2 0 C 4C 6 ) Q 1 h +Q 2 h 2 0 +(t 2 +h 2 ) div h Q 1 h +div h Q 2 h 2 0 = Q h 2 0 +(t 2 +h 2 ) div h Q h 2 0. This immediately indicates the desired result. With the above continuity and coercivity results, we can obtain the following error estimates for MiSP4 element by following the standard error analysis.

9 56 G. YU, X. XIE, AND Y. GUO Theorem 4.7. Given g L 2 (), let (M,γ = divm,w,β) M Γ W Θ be the solution of the problem (7)-(8). Then the discretization problem (22)-(23) admits a unique solution (M h,w h,β h ) M h W h Θ h such that M M h h,1 + (w w h,β β h ) h,2 inf Q h M h M Q h h,1 + inf (w v h,β ζ h ) h,2 +ht γ 1 +h γ 0. Proof. Since a(m,γ;q h,div h Q h )+ b(q h,div h Q h ;w,β) (div h Q h,grad w β R h (grad w β)) = 0, Q h M h, a(m h,div h M h ;Q h,div h Q h )+ b(q h,div h Q h ;w h,β h ) = 0, Q h M h, then for all Q h M h, it holds a(m M h,γ div h M h ;Q h,div h Q h )+ b(q h,div h Q h ;w w h,β β h ) Denote (div h Q h,grad w β R h (grad w β)) = 0. Z h (g) = {Q h M h : b(q h,div h Q h ;v h,ζ h ) = (g,v h ), (v h,ζ h ) W h Θ h }. Let Q h be any element of Z h (g). Since Q h M h Z h (0), then Q h M h 2 h,1 a( Q h M h,div h ( Q h M h ); Q h M h,div h ( Q h M h )) = a( Q h M,div h Qh γ; Q h M h,div h ( Q h M h )) +a(m M h,γ div h M h ; Q h M h,div h ( Q h M h )) = a( Q h M,div h ( Q h M); Q h M h,div h ( Q h M h )) b( Q h M h,div h ( Q h M h );w w h,β β h ) +(div h ( Q h M h ),grad w β R h (grad w β)) = a( Q h M,div h ( Q h M); Q h M h,div h ( Q h M h )) b( Q h M h,div h ( Q h M h );w v h,β ζ h ) + t2 λ (div h( Q h M h ),γ R h γ) Q h M h h,1 ( Q h M h,1 + (w v h,β ζ h ) h,2 +ht γ 1 ). So we have Q h M h h,1 Q h M h,1 + (w v h,β ζ h ) h,2 +ht γ 1. Then, by using the triangle inequality, we get (45) M M h h,1 Q h M h,1 + inf Q h Z h (g) inf (w v h,β ζ h ) h,2 +ht γ 1. For any Q h M h, there exists Q h M h, such that, for all (v h,ζ h ) W h Θ h, b( Qh,div h Qh ;v h,ζ h ) = b(m Q h,div h (M Q h );v h,ζ h ) (γ,grad v h ζ h R h (grad v h ζ h ))

10 ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD 57 and Q h h,1 b( Qh,div h Qh ;v h,ζ h ) sup (v h,ζ h ) h,2 { b(m Qh,div h (M Q h );v h,ζ h ) sup (v h,ζ h ) h,2 (γ,grad v h ζ h R h (grad v h ζ h )) (v h,ζ h ) h,2 b(m Qh,div h (M Q h );v h,ζ h )+(γ,ζ h R h ζ h ) = sup (v h,ζ h ) h,2 M Q h h,1 +h γ 0. Choose Q h = Q h +Q h, then Q h Z h (g). Thus we get M Q h h,1 = M Q h Q h h,1 M Q h h,1 + Q h h,1 This estimate and (45) imply (46) (47) M Q h h,1 +h γ 0. M M h h,1 inf Q h M h M Q h h,1 + inf (w v h,β ζ h ) h,2 +ht γ 1 +h γ 0. On the other hand, from the coercivity and continuity properties we get (v h w h,ζ h β h ) h,2 b(qh,div h Q h ;v h w h,ζ h β h ) sup Q h M h Q h h,1 { a(m Mh,div h (M M h );Q h,div h Q h ) = sup Q h M h Q h h,1 b(q h,div h Q h ;w v h,β ζ h ) Q h h,1 + (div hq h,grad w β R h (grad w β)) Q h h,1 M M h h,1 + (w v h,β ζ h ) h,2 +ht γ 1. This inequality and (46) imply } } (48) (w w h,β β h ) h,2 inf M Q h h,1 Q h M h + inf (w v h,β ζ h ) h,2 +ht γ 1 +h γ 0. A combination of (46) and (48) completes the proof. Next we consider the approximation properties of finite element spaces. Lemma 4.8 gives the error estimates for space M h, and Lemma 4.12 is for space W h Θ h. We need to notice here the key for Lemma 4.12 is the property of the operator R h described in Lemma Finally the convergence theorem, i.e. Theorem 4.13, follows from these lemmas.

11 58 G. YU, X. XIE, AND Y. GUO Lemma 4.8. Given g L 2 (), let (M,γ = divm,w,β) M Γ W Θ be the solution of the problem (7)-(8). It holds inf Q h M h M Q h h,1 h( M 1 + γ 0 +t γ 1 ). Proof. FortheexactsolutionM,firstletQ 1 h beitspiecewiseconstantl2 projection, then M Q 1 h 0 h M 1. For the exact solution γ, secondly choose Q 2 h satisfying: (1) div h Q 2 h is the piecewise constant L2 projection of γ, then γ div h Q 2 h 0 h γ 1, div h Q 2 h 0 γ 0 ; (2) Q 2 h 0 h div h Q 2 h 0, then Q 2 h 0 h γ 0. Take Q h = Q 1 h +Q2 h, then we get the desired result M Q h h,1 M Q 1 h 0 + Q 2 h 0 +(h+t) γ div h Q 2 h 0 h M 1 +h γ 0 +h γ div h Q 2 h 0 +t γ div h Q 2 h 0 h M 1 +h γ 0 +h γ 0 +th γ 1 h( M 1 + γ 0 +t γ 1 ). Remark 4.9. We note that with the same technique as in Lemma 4.8, the condition t h in [8, Lemma 3.2] and in [8, Theorem 4.3] can be removed. Assumption [11] The mesh T h is a refinement of a coarser partition T 2h, obtained by jointing the midpoints of each opposite edge in each K 2h T 2h (called macroelement). In addition, T 2h is a similar refinement of a still coarser regular partition T 4h. Lemma [11, Lemma 3.2, 3.4] Given g L 2 (), let (M,γ = divm,w,β) M Γ W Θ be the solution of the problem (7)-(8). Then under Assumption 4.10, there exist (ŵ, ˆβ) W h Θ h and operator Π : H 1 () 2 H 0 (rot,) Z h satisfying (49) β ˆβ 1 h β 2, (50) R h (grad ŵ ˆβ) = Π(grad w β), and (51) η Πη 0 h η 1, η H 1 () 2 H 0 (rot,). Lemma Given g L 2 (), let (M,γ = divm,w,β) M Γ W Θ be the solution of the problem (7)-(8). Then under Assumption 4.10, it holds (52) inf (w v h,β ζ h ) h,2 h β 2 + ht2 t+h γ 1.

12 ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD 59 Proof. Choose (v h,ζ h ) = (ŵ, ˆβ), with (ŵ, ˆβ) W h Θ h as in Lemma 4.11, then we can get inf (w v h,β ζ h ) h,2 = inf ǫ(β) ǫ(ζ h ) (v h,ζ h ) W h Θ t+h R h(grad w β) R h (grad v h ζ h ) 0 h ǫ(β) ǫ(ˆβ) t+h R h(grad w β) R h (grad ŵ ˆβ) 0 = ǫ(β) ǫ(ˆβ) t+h R h(grad w β) Π(grad w β) 0 ǫ(β) ǫ(ˆβ) t+h R h(grad w β) (grad w β) t+h (grad w β) Π(grad w β) 0 h β 2 + ht2 t+h λt 2 (grad w β) 1 h β 2 + ht2 t+h γ 1. Theorem Given g L 2 (), let (M,γ = divm,w,β) M Γ W Θ and (M h,γ h,w h,β h ) M h Γ h W h Θ h be the solutions of the problems (7)-(8) and (16)-(17) respectively. Then under Assumption 4.10, it holds the error estimate (53) M M h h,1 + (w w h,β β h ) h,2 h( M 1 + β 2 + γ 0 +t γ 1 ) h g 0. Furthermore, it holds (54) M M h 0 +(t+h) γ γ h 0 + w w h 1 + β β h 1 h( M 1 + w 2 + β 2 +t γ 1 + γ 0 ) h g 0. Proof. The estimate (53) follows from the Theorem 4.7, Lemma 4.8 and Lemma For the second estimate, we only need to estimate w w h 1. In fact, grad w grad w h 0 = grad w R h grad w+r h (grad w grad w h β +β h )+R h (β β h ) 0 grad w R h grad w 0 + R h (grad w grad w h β +β h ) 0 + R h (β β h ) 0 h( w 2 + M 1 + β 2 +t γ 1 + γ 0 ). 5. Numerical results We compute a square plate with analytical solution to show the convergence. This example is taken from [9, 15]. The domain is the unit square (0,1) 2, the material parameters are taken as E = 1.0, ν = 0.3 and κ = 5 6. The exact solution is: the first component of the rotation β 1 = 100y 3 (y 1) 3 x 2 (x 1) 2 (2x 1), the second component of the rotation β 2 = 100x 3 (x 1) 3 y 2 (y 1) 2 (2y 1), and the displacement w = 100( 1 3 x3 (x 1) 3 y 3 (y 1) 3 2t2 5(1 ν) [y3 (y 1) 3 x(x 1)(5x 2 5x+ 1)+x 3 (x 1) 3 y(y 1)(5y 2 5y+1)]). Therefore,thetransverseloadg = 200E 1 ν (x 3 (x 2 1) 3 (5y 2 5y+1)+y 3 (y 1) 3 (5x 2 5x+1)+x(x 1)y(y 1)(5x 2 5x+1)(5y 2 5y+1)). For the plate thickness t, we consider four cases: t = 1.0,0.1,0.001,1e 8. The results for MiSP4 method under the uniform meshes (Figure 5.1) are reported in Table 5.1. These results are conformable to the error estimates in Theorem We note that the error analysis for MiSP4 element requires the partitions of domain to satisfy Assumption However, numerical results in Table 5.2 show that

13 60 G. YU, X. XIE, AND Y. GUO Figure 5.1. Uniform mesh Table 5.1. Results of error on uniform mesh with MiSP4 t rate 1 w w h β β h M M h γ γ h (t+h) γ γ h w w h β β h M M h γ γ h (t+h) γ γ h w w h β β h M M h γ γ h (t+h) γ γ h e-8 w w h β β h M M h γ γ h (t+h) γ γ h this assumption seems not to be absolutely necessary for the uniform convergence, as is similar to the MITC4 element [11]. Here the used partitions (Figure 5.2) do not satisfy Assumption Figure 5.2. Quadrilateral mesh

14 ANALYSIS OF A MIXED-SHEAR-PROJECTED QUADRILATERAL ELEMENT METHOD 61 Table 5.2. Results of error on quadrilateral mesh with MiSP4 t rate 1 w w h β β h M M h γ γ h (t+h) γ γ h w w h β β h M M h γ γ h (t+h) γ γ h w w h β β h M M h γ γ h (t+h) γ γ h e-8 w w h β β h M M h γ γ h (t+h) γ γ h Acknowledgments The work of the first author was partly supported by National Natural Science Foundation of China ( and ) and the Fundamental Research Funds for the Central Universities ( CX053). The work of the second author was partly supported by National Natural Science Foundation of China ( ) and Major Research Plan of National Natural Science Foundation of China ( ). References [1] D.N. Arnold. Discretization by finite elements of a model parameter dependent problem. Numerische Mathematik, 37(3) (1981) [2] D.N. Arnold and R.S. Falk. A uniformly accurate finite element method for the Reissner- Mindlin plate. SIAM Journal on Numerical Analysis, 26(6) (1989) [3] R. Ayad, G. Dhatt, and J.L. Batoz. A new hybrid-mixed variational approach for Reissner Mindlin plates. The MiSP model. International Journal for Numerical Methods in Engineering, 42(7) (1998) [4] K.J. Bathe, F. Brezzi, and S.W. Cho. The mitc7 and mitc9 plate bending elements. Computers & Structures, 32(3) (1989) [5] K.J. Bathe and E.N. Dvorkin. A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. International Journal for Numerical Methods in Engineering, 21(2) (1985) [6] F. Brezzi and M. Fortin. Numerical approximation of Mindlin-Reissner plates. Mathematics of Computation, 47(175) (186) [7] F. Brezzi, M. Fortin, and R. Stenberg. Error analysis of mixed-interpolated elements for Reissner-Mindlin plates. Mathematical Models and Methods in Applied Sciences, 1(2) (1991)

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