A locking-free model for Reissner-Mindlin plates:

Analysis and isogeometric implementation via

NURBS and triangular NURPS

L. Beirão da Veiga ∗ T.J.R. Hughes † J. Kiendl ‡

C. Lovadina § J. Niiranen ¶ A. Reali ‖ H. Speleers ∗∗††

January 13, 2015

Abstract

We study a reformulated version of Reissner-Mindlin plate theory inwhich rotation variables are eliminated in favor of transverse shear strains.Upon discretization, this theory has the advantage that the “shear lock-ing” phenomenon is completely precluded, independent of the basis func-tions used for displacement and shear strains. Any combination works, butdue to the appearance of second derivatives in the strain energy expres-sion, smooth basis functions are required. These are provided by Isogeo-metric Analysis, in particular, NURBS of various degrees and quadratictriangular NURPS. We present a mathematical analysis of the formula-tion proving convergence and error estimates for all physically interestingquantities, and provide numerical results that corroborate the theory.

∗Department of Mathematics, University of Milan, Via Saldini 50, 20133 Milan, Italy.Email: lourenco.beirao@unimi.it†Institute for Computational Engineering and Sciences, University of Texas

at Austin, 201East 24th Street, Stop C0200 Austin, TX 78712-1229, USA.Email: hughes@ices.utexas.edu‡Department of Civil Engineering and Architecture, University of Pavia, Via Ferrata 3,

27100, Pavia, Italy. Email: josef.kiendl@unipv.it§Department of Mathematics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy.

Email: carlo.lovadina@unipv.it¶Department of Civil and Structural Engineering, Aalto University, PO Box 12100, 00076

AALTO, Finland. Email: jarkko.niiranen@aalto.fi‖Department of Civil Engineering and Architecture, University of Pavia, Via Ferrata 3,

27100, Pavia, Italy. Email: alessandro.reali@unipv.it∗∗Department of Computer Science, University of Leuven, Celestijnenlaan 200A, 3001 Hev-

erlee (Leuven), Belgium. Email: hendrik.speleers@cs.kuleuven.be††Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scien-

tifica, 00133 Rome, Italy. Email: speleers@mat.uniroma2.it

1

1 Introduction

Finite element thin plate bending analysis, based on the Poisson-Kirchhoff the-ory, began with the MS thesis of Papenfuss at the University of Washington in1959 [24]. This four-node rectangular element employed C1-continuous inter-polation functions, but was deficient in the sense that its basis functions werenot complete through quadratic polynomials. Throughout the 1960s the devel-opment of quadrilateral and triangular thin plate elements was a focus of finiteelement research. The problem was surprisingly difficult. Eventually there weretechnical successes but the elements were complicated, hard to implement, anddifficult to use in practice. For an early history, we refer to Felippa [16]. Tocircumvent the difficulties, in the 1970s attention was redirected to the “thick”plate bending theory of Reissner-Mindlin. In this case only C0 basis functionswere required for the displacement and rotations, so the continuity and com-pleteness requirements were easily satisfied with standard isoparametric basisfunctions, but new difficulties arose associated with “shear locking” in the thinplate limit in which the transverse shear strains must vanish. Nevertheless,through a number of clever ideas, some tricks and some fundamental, successfulelements for many applications became available. The simplicity and efficiencyof these elements led to immediate incorporation in industrial and commer-cial structural analysis software programs, and that has been the situation eversince. Research in the subject then became somewhat stagnant for a number ofyears, but recently things changed.

Isogeometric Analysis was proposed by Hughes, Cottrell & Bazilevs in 2005[18]. The motivation for its development was to simplify and render more effi-cient the design-through-analysis process. It is often said that the developmentof suitable Finite Element Analysis (FEA) models from Computer Aided Design(CAD) files occupies over 80% of overall analysis time [12]. Some design/analysisengineers claim that this is an underestimate and the situation is actually worseperhaps 90%, or even more. Whatever the precise percentage, it is clear that theinterface between design and analysis is broken, and it is the stated aim of Isoge-ometric Analysis to repair it. The way this has been approached in IsogeometricAnalysis is to reconstitute analysis within the functions utilized in engineeringCAD, such as, for example, NURBS and T-splines, making it possible, at thevery least, to perform analysis within the representations provided by design,eliminating redundant data structures and unnecessary geometric approxima-tions. This has been the primary focus of Isogeometric Analysis, but in theprocess of its development new analysis opportunities have also presented them-selves. One emanates from a basic property of the functions utilized in CADthey are smooth, usually at least C1-continuous, more often C2-continuous, andthey do not require derivative degrees-of-freedom, one of the paradigmatic de-ficiencies of the early thin plate bending elements. CAD functions also haveother advantageous properties, but we will not go into these here. It turns outthat smoothness alone has created new opportunities in plate bending elementresearch.

The first and most obvious to be exploited by Isogeometric Analysis is re-

2

newed interest in the long abandoned Poisson-Kirchhoff theory, as anticipated in[18]. With convenient, smooth basis functions provided by Isogeometric Analy-sis, the role of Poisson-Kirchhoff theory is being reconsidered. A prime advan-tage is that no rotational degrees-of-freedom are required. This immediatelyreduces the size of equation systems and the computational effort necessaryto solve problems. In finite deformation analysis there is a further advantage.This stems from the fact that finite rotations involve a product-group structure,SO(3), a significant complication. It is completely obviated by rotationless thinbending elements.

Another innovation that occurred, due to Long, Bornemann & Cirak [21]and Echter, Oesterle & Bischoff [15] in the context of shells, was again moti-vated by the existence of smooth basis functions. It was observed that if onecould deal with second derivatives appearing in strain energy expressions, thena change of variables from rotations in Reissner-Mindlin plate theory to trans-verse shear strains would eliminate transverse shear locking, independent of thebasis functions employed. Why wasn’t this amazing formulation utilized previ-ously for the development of plate and shell elements? The answer seems to bethere simply were not convenient smooth basis functions available within thefinite element paradigm. Shear locking has been the fundamental obstacle tothe design of effective plate elements based on Reissner-Mindlin theory. It isremarkable that the discovery of a complete and general solution has occurrednearly 50 years after the widespread adoption of the theory as a framework forthe derivation of plate and shell elements.

Given the above, the question arises as to what other opportunities might beprovided by clever changes of variables? One answer has been presented in thework of Kiendl et al. [19] who have shown that smooth basis functions with onlytranslational displacement degrees-of-freedom can also be employed successfullyfor “thick” bending elements. The price to pay in the formulation of [19] is thatsquares of third derivatives appear in the strain energy expression, but theseare no problem for C2 continuous Isogeometric Analysis basis functions.

All these new ideas have created a renaissance in the development of methodsto solve problems involving thin and thick bending elements. It is clear thatthese and other related concepts will generate considerable interest in the comingyears.

In this paper we undertake the mathematical analysis of a class of methodsfor Reissner-Mindlin plate theory based on the change of variables introducedin [15, 21]. The dependent variables are then the transverse displacement ofthe plate and the transverse shear strain vector. The change of variables re-sults in squares of second derivatives of the displacement in the strain energy,which yield to spline discretizations of C1, or higher, continuity. It is apparentthat the shear strain vector is not as physically appealing or implementation-ally convenient as the rotation vector. However, by utilizing weakly enforcedrotation boundary conditions, by way of Nitsche’s method [23], these issues arecircumvented. Nitsche’s method also provides other analytical benefits in that italleviates “boundary locking”, a potential problem encountered in the presenttheory for clamped boundary conditions. In addition to Cp−1 NURBS dis-

3

cretizations, we consider quadratic C1 triangular NURPS, that is, non-uniformrational Powell-Sabin splines. We note in passing that Isogeometric Analysishas previously been applied to the standard displacement-rotation forms of theReissner-Mindlin plate and shell theories [6, 8].

An outline of the remainder of the paper follows: In Section 2 we presentthe Reissner-Mindlin plate model, first in terms of displacement and rotationvariables and then in an equivalent form in terms of displacement and transverseshear strain variables. We introduce the discrete form of the theory and summa-rize stability and convergence results that apply to displacements, shear strains,rotations, bending moments, and transverse shear force resultants. In Section 3we describe and perform numerical tests with NURBS spaces. In Section 4 wedo likewise with quadratic NURPS, and in Section 5 we present concluding re-marks. The technical details of proofs are postponed until Appendix A so asnot to interrupt the flow of the main ideas and results in the body of the paper.

2 The model and discretization

We start by considering the classical Reissner-Mindlin model for plates. Let Ωbe a bounded and piecewise regular domain in R2 representing the midsurfaceof the plate. We subdivide the boundary Γ of Ω in three disjoint parts (suchthat each is either void or the union of a finite sum of connected components ofpositive length),

Γ = Γc ∪ Γs ∪ Γf .

The plate is assumed to be clamped on Γc, simply supported on Γs and freeon Γf , with Γc,Γs defined to preclude rigid body motions. We consider forsimplicity only homogeneous boundary conditions. The variational space ofadmissible solutions is given by

X̃ ={

(w,θ) ∈ H1(Ω)× [H1(Ω)]2 : w = 0 on Γc ∪ Γs, θ = 0 on Γc}.

Following the Reissner-Mindlin model, see for instance [4] and [17], the platebending problem reads as{

Find (w,θ) ∈ X̃, such that

a(θ,η) + µkt−2(θ −∇w,η −∇v) = (f, v), ∀(v,η) ∈ X̃,(1)

where µ is the shear modulus and k is the so-called shear correction factor.In the above model, t represents the plate thickness, w the deflection, θ therotation of the normal fibers and f the applied scaled normal load. Moreover,(·, ·) stands for the standard scalar product in L2(Ω) and the bilinear form a(·, ·)is defined by

a(θ,η) = (Cε(θ), ε(η)), (2)

with C the positive definite tensor of bending moduli and ε(·) the symmetricgradient operator.

4

Provided the boundary conditions satisfy the conditions above, the bilinearform appearing in problem (1) is coercive, in the following sense (see Proposi-tion A.1 in Appendix A). There exists a positive constant α depending only onthe material constants and the domain Ω such that

a(η,η) + µkt−2(η −∇v,η −∇v)

≥ α(||η||2H1(Ω) + t

−2||η −∇v||2L2(Ω) + ||v||2H1(Ω)

), ∀(v,η) ∈ X̃.

(3)

The above result states the coercivity of the bilinear form on the product spaceX̃. It is well known that the discretization of the Reissner-Mindlin model posesdifficulties, due to the possibility of the locking phenomenon when the thicknesst

where we recall that we now have µk = 1 for notational simplicity.Problem (6) is clearly equivalent to (1) up to the above change of variables.

In particular, the coercivity property (3) translates immediately into

a(∇v + τ ,∇v + τ ) + t−2(τ , τ ) ≥ α|||τ , v|||2, ∀(v, τ ) ∈ X, (7)

for the same positive constant α.

2.2 Discretization of the model

We introduce a pair of finite dimensional spaces for deflections and shear strains:

Wh ⊂ H2(Ω), Ξh ⊂ [H1(Ω)]2.

We assume that the two spaces above are generated either by some finite elementor isogeometric technology, and are therefore associated with a (physical) meshΩh. In the following, we will indicate by K ∈ Ωh a typical element of themesh, and denote by hK its diameter. We denote by h the maximum mesh size.Moreover, we will indicate by Eh the set of all (possibly curved) edges of themesh, by e ∈ Eh the generic edge, and by he its length. As usual we assumethat the boundary parts Γc,Γs,Γf are unions of mesh edges. Furthermore, wewill assume that the set of elements K in the family {Ωh}h is uniformly shaperegular in the classical sense.

We then consider the discrete space with (partial) boundary conditions

Xh ={

(vh, τh) ∈Wh × Ξh : vh = 0 on Γc ∪ Γs}

;

see also Remark 2.3 below. The rotation boundary condition on Γc will beenforced with a penalized formulation in the spirit of Nitsche’s method [23],through the introduction of the following modified bilinear form. Let

ah(∇wh + γh,∇vh + τh) = a(∇wh + γh,∇vh + τh)

−∫

Γc

(Cε(∇wh + γh)ne

)· (∇vh + τh)

−∫

Γc

(Cε(∇vh + τh)ne

)· (∇wh + γh)

+ β tr(C)∑

e∈Eh∩Γc

c(e)−1∫e

(∇wh + γh)(∇vh + τh),

(8)

for all (wh,γh), (vh, τh) in Xh and for β > 0 a stabilization parameter. Above,c(e) is a characteristic quantity depending on the side e.

Remark 2.2. For a shape regular family of meshes, one can choose c(e) = heor c(e) = (AreaKe)

1/2, where Ke is the element containing e. This latter choicehas been employed in the numerical tests of Section 3, while the former has beenused in Section 4 . However, when a side e belongs to an element with a largeaspect ratio, a different choice could be preferable. For example, significantly

6

thin elements may be used in the presence of boundary layers, and one maychoose c(e) = h⊥e , where h

⊥e is the element size in the direction perpendicular

to the boundary.

We are now able to present the proposed discretization of the model (6):{Find (wh,γh) ∈ Xh, such thatah(∇wh + γh,∇vh + τh) + t−2(γh, τh) = (f, vh), ∀(vh, τh) ∈ Xh.

(9)

Remark 2.3. It is not wise to use a direct discretization of the space X byenforcing all the boundary conditions on Wh×Ξh. Indeed, the clamped conditionon rotations

∇wh + γh = 0 on Γc, ∀wh ∈Wh, γh ∈ Ξh,is very difficult to implement, and can be a source of boundary locking unlessthe two spaces Wh,Ξh are chosen in a very careful way. To illustrate why a poorapproximation might occur on Γc, let us consider the following example.

Let Ω = (0, 1) × (0, 1) and Γc = [0, 1] × {1}. Select Wh = Sp,pp−1,p−1(Ω) andΞh = S

p−1,p−1p−2,p−2(Ω)× S

p−1,p−1p−2,p−2(Ω), where S

p,qr,s (Ω) denotes the space of B-splines

of degree p and regularity Cr with respect to the x direction, and of degree q andregularity Cs with respect to the y direction.

Imposing ∇wh + γh = 0 on Γc implies in particular∂wh∂y

(x, 1) = −γ2,h(x, 1) ∀x ∈ [0, 1],

where γ2,h is the second component of the vector field γh. Since∂wh∂y (x, 1) ∈

Spp−1(0, 1) and γ2,h(x, 1) ∈ Sp−1p−2(0, 1), it follows that

γ2,h(x, 1) ∈ Sp−1p−2(0, 1) ∩ Spp−1(0, 1) = S

p−1p−1(0, 1).

Hence γ2,h(x, 1) is necessarily a global polynomial of degree at most p − 1 onΓc, regardless of the mesh. This means that on Γc γ2,h cannot converge to γ2,second component of γ, as the mesh size tends to zero, in general.

Instead, as we will prove in the next section, the method proposed above isfree of locking for any choice of the discrete spaces Wh,Ξh.

2.3 Stability and convergence results

In the present section we show the stability and convergence properties of theproposed method. All the proofs can be found in Appendix A.

In what follows, we set c(e) = he in (8), which is a suitable choice for shaperegular meshes, see Remark 2.2. We start by introducing the following discretenorm

|||vh, τh|||2h = |||vh, τh|||2 +∑

e∈Eh∩Γc

h−1e ||∇vh + τh||2L2(e), (10)

for all (vh, τh) ∈ Xh.In the theoretical analysis of the method we will make use of the following

assumptions on the solution regularity and space approximation properties.

7

A1) We assume that the solution w ∈ H2+s(Ω) and γ ∈ H1+s(Ω) for somes > 1/2.

A2) We assume that the following standard inverse estimates hold

|vh|H1(K) ≤ Ch−1K ||vh||L2(K), |τh|H1(K) ≤ Ch−1K ||τh||L2(K),

for all vh ∈Wh and τh ∈ Ξh with C independent of h.

A3) We assume the following approximation properties for Xh. Let s = 2, 3.For all (v, τ ) ∈ (Hs(Ω) × [H1(Ω)]2) ∩X there exists (vh, τh) ∈ Xh suchthat

||τ − τh||Hj(Ω) ≤ Ch1−j ||τ ||H1(Ω), j = 0, 1,||v − vh||Hj(Ω) ≤ Chs−j ||v||Hs(Ω), j = 0, . . . , s,

with C independent of h.

Moreover, we will make use of the following natural assumption, in order toavoid rigid body motions:

A4) We assume that Γc ∪ Γs has positive length and that eitheri) Γc has positive length, orii) Γs is not contained in a straight line.

We now introduce a coercivity lemma stating in particular the invertibilityof the linear system associated with (9).

Lemma 2.1. Let hypotheses A2 and A4 hold. There exist two positive constantsβ0, α

′ such that, for all β ≥ β0, we have

ah(∇vh+τh,∇vh+τh)+t−2(τh, τh) ≥ α′|||τh, vh|||2h, ∀(vh, τh) ∈ Xh. (11)

The constant α′ only depends on the material parameters and the domain Ω,while the constant β0 depends only on the shape regularity constant of Ωh.

Let A1 hold. By an integration by parts, it is immediately verified that thescheme (9) is consistent, in the sense that

ah(∇w + γ,∇vh + τh) + t−2(γ, τh) = (f, vh), ∀(vh, τh) ∈ Xh, (12)

where (w,γ) is the solution of problem (6) and the left-hand side makes sense dueto the regularity assumption A1. Note that condition A1 could be significantlyrelaxed by interpreting the integrals on Γc appearing in (8) in the sense ofdualities.

By combining the coercivity in Lemma 2.1 with the consistency property(12), the following convergence result follows.

8

Proposition 2.1. Let A1 and A2 hold. Let (w,γ) be the solution of problem(6) and (wh,γh) ∈ Xh be the solution of problem (9). Then, if β ≥ β0, we have

|||w − wh,γ − γh|||2h

≤ C inf(vh,τh)∈Xh

2∑j=0

( ∑K∈Ωh

h2(j−1)K |w − vh|

2Hj+1(K)

+∑K∈Ωh

h2(j−1)K |γ − τh|

2Hj(K) + t

−2||γ − τh||2L2(Ω)),

(13)

with C depending only on the material parameters and the domain Ω.

We now state a convergence result concerning bending moments and shearforces, the quantities of interest in engineering applications. To this aim, wefirst define the scaled bending moments and the scaled shear forces as follows:

M = −Cε(∇w + γ), Q = t−2µκγ = t−2γ, (14)

where (w,γ) is the solution to problem (6). The above quantities are knownto converge to non-vanishing limits, as t → 0 (see, e.g., [3] and [9]). Onceproblem (9) has been solved, we can define the scaled discrete bending momentsand the scaled discrete shear moments as

Mh = −Cε(∇wh + γh), Qh = t−2γh. (15)

We have the following estimates.

Proposition 2.2. Let A1 and A2 hold. Let (w,γ) be the solution of problem(6) and (wh,γh) ∈ Xh be the solution of problem (9). Then, if β ≥ β0, we have

||M −Mh||L2(Ω) + t ||Q−Qh||L2(Ω) ≤ C|||w − wh,γ − γh|||h. (16)

Moreover, let the mesh family {Ωh}h be quasi-uniform. Then, we have

h ||Q−Qh||L2(Ω) ≤ C(|||w − wh,γ − γh|||h + h inf

sh∈Ξh||Q− sh||L2(Ω)

), (17)

||Q−Qh||H−1(Ω) ≤ C(|||w − wh,γ − γh|||h + h inf

sh∈Ξh||Q− sh||L2(Ω)

). (18)

In addition, we can formulate the following improved result regarding theerror for the rotations in the L2-norm and the error for the deflections in theH1-norm. An analogous result (possibly with a smaller improvement in terms ofh, t) also holds if the additional hypotheses in the proposition are not satisfied;we do not detail here this more general case.

Proposition 2.3. Let the same assumptions and notation of Proposition 2.1hold. Moreover, let assumption A3 hold, Γc = Γ and let the domain Ω be

9

either regular, or piecewise regular and convex. Then, the following improvedapproximation result holds

||θ − θh||L2(Ω) ≤ C(h+ t) |||w − wh,γ − γh|||h, (19)||w − wh||H1(Ω) ≤ C(h+ t) |||w − wh,γ − γh|||h + ||γ − γh||L2(Ω), (20)

where θh = ∇wh +γh and the constant C depends only on the material param-eters and the domain Ω.

Note that the last term appearing in (13) is not a source of locking sincet−1γ = tQ is a quantity that is known to be uniformly bounded in the correctSobolev norms (see, e.g., [3] and [9]). The constants appearing in Propositions2.1 and 2.3 are independent of t, and so the results shown state that the proposedmethod is locking-free regardless of the discrete spaces adopted. This is a veryinteresting property that is missing in the standard methods for the Reissner-Mindlin problem. The accuracy of the discrete solution (9) will only dependon the approximation properties of the adopted discrete spaces, and will not behindered by small values of the plate thickness. We also remark that the normsfor the scaled shear forces appearing in the left-hand side of (17) and (18), areindeed the usual norms for which a convergence result can be established (see,e.g., [9] and [10]).

In the following two sections we will present a pair of particular choices forWh,Ξh within the framework of

• standard tensor-product NURBS-based isogeometric analysis (Section 3);

• triangular NURPS-based isogeometric analysis (Section 4).

For such choices we can apply Propositions 2.1, 2.3 in order to obtain the ex-pected convergence rates in terms of h. For example, in the case of standardtensor-product NURBS, combining Proposition 2.1 with the approximation es-timates in [5, 7] we obtain the following convergence result.

Corollary 2.1. Let the same assumptions and notation of Proposition 2.1 hold.Let standard isoparametric tensor-product NURBS of polynomial degree p for allvariables be used for the space Xh. Then, provided that the solution of problem(6) is sufficiently regular for the right-hand side to make sense, the followingerror estimates holds for all 2 ≤ s ≤ p:

|||w − wh,γ − γh|||h ≤ Chs−1(||w||Hs+1(Ω) + ||θ||Hs(Ω) + t||γ||Hs−1(Ω)

), (21)

||M −Mh||L2(Ω)+t ||Q−Qh||L2(Ω)≤ Chs−1

(||w||Hs+1(Ω) + ||θ||Hs(Ω) + t||γ||Hs−1(Ω)

).

(22)

Moreover, let the mesh family {Ωh}h be quasi-uniform. Then,

h ||Q−Qh||L2(Ω) + ||Q−Qh||H−1(Ω)≤ Chs−1

(||w||Hs+1(Ω) + ||θ||Hs(Ω) + t||γ||Hs−1(Ω) + ||γ||Hs−2(Ω)

).

(23)

The constant C depends only on p, the material parameters and the domain Ω.

10

The above corollary can also be combined with Proposition 2.3 to obtainimproved error estimates for the L2-norm of the rotations and the H1-norm ofthe deflections. Also recalling definition (5), one can easily obtain a convergencerate of (h+ t)hs−1 for such quantities. Finally, note that, like for all high ordermethods, the regularity requirement in Corollary 2.1 may be too demanding dueto the presence of layers in the solution. This situation is typically dealt withby making use of refined meshes, an example of which is shown in the numericaltests.

3 Isogeometric discretization with NURBS

In this section, NURBS-based isogeometric analysis is used to perform numericalvalidations of the presented theory. We begin with a brief summary of B-splinesand NURBS (Non-Uniform Rational B-Splines).

3.1 B-splines and NURBS

B-splines are piecewise polynomials defined by the polynomial degree p and aknot vector [ξ1, ξ2, . . . , ξn+p+1], where n is the number of basis functions. Theknot vector is a set of parametric coordinates ξi, called knots, which divide theparametric space into intervals called knot spans. A knot can also be repeated,in this case it is called a multiple knot. At a single knot the B-splines are Cp−1-continuous, and at a multiple knot of multiplicity k the continuity is reduced toCp−k.

The B-spline basis functions of degree p are defined by the following recursionformula. For p = 0,

Ni,0(x) =

{1, ξi ≤ x < ξi+1,0, otherwise.

For p ≥ 1,

Ni,p(x) =x− ξiξi+p − ξi

Ni,p−1(x) +ξi+p+1 − xξi+p+1 − ξi+1

Ni+1,p−1(x).

A bivariate NURBS function Rp,qi,j is defined as the weighted tensor-product ofthe B-spline functions Ni,p and Mj,q with polynomial degrees p and q,

Rp,qi,j (x, y) =Ni,p(x)Mj,q(y)ωi,j

n∑l=1

m∑r=1

Nl,p(x)Mr,q(y)ωl,r

,

where ωi,j are called control weights. Following the isogeometric concept, NURBSare employed to both represent the geometry and to approximate the solution,i.e. the isoparametric concept is invoked. Accordingly, the unknown variables

11

w and γ are approximated by

wh(x, y) =

nw∑i=1

mw∑j=1

Rpw,qwi,j (x, y)ŵi,j , γh(x, y) =

nγ∑i=1

mγ∑j=1

Rpγ ,qγi,j (x, y)γ̂i,j ,

with nw, mw the numbers of basis functions in the two parametric directions forwh, and nγ , mγ the numbers for γh. The test functions v and τ are discretizedaccordingly.

Since the rotations θ are not discretized in this approach, rotational bound-ary conditions cannot be imposed in a standard way by applying them directlyon the respective degrees of freedom at the boundary. Instead, such boundaryconditions are enforced by the modified bilinear form introduced in equation(8). Displacement boundary conditions are enforced in a standard way throughthe displacement degrees of freedom on the boundary.

3.2 Numerical tests

In this section, the proposed method is tested on different numerical examples.We always employ the same polynomial order and the highest regularity for allthe unknowns, but different choices can be made. In order to demonstrate thelocking-free behavior of this method we consider both thick and thin plates.Furthermore, we investigate an example which exhibits boundary layers. As anerror measure, an approximated L2-norm error for a variable u is computed as

||uex − uh||L2||uex||L2

=

√∑(xi,yi)∈G(uex − uh)

2∑(xi,yi)∈G u

2ex

,

where G is a 101 × 101 uniform grid in the parameter domain [0, 1]2 mappedonto the physical domain.

3.2.1 Square plate with clamped boundary conditions

The first example consists of a unit square plate [0, 1]2 with an analytical solutionas described in [11]. The plate is clamped on all four sides, and subject to aload given by

f(x, y) =E

12(1− ν2)[12y(y − 1)(5x2 − 5x+ 1)(2y2(y − 1)2

+ x(x− 1)(5y2 − 5y + 1))+ 12x(x− 1)(5y2 − 5y + 1)(2x2(x− 1)2

+ y(y − 1)(5x2 − 5x+ 1))].

12

The analytical solution for the displacement w is given by

w(x, y) =1

3x3(x− 1)3y3(y − 1)3

− 2t2

5(1− ν)[y3(y − 1)3x(x− 1)(5x2 − 5x+ 1)

+ x3(x− 1)3y(y − 1)(5y2 − 5y + 1)].

We perform an h-refinement study using equal polynomial degrees p = 2, 3, 4, 5for wh and γh, for the case of a thick plate with t = 10

−1 and a thin plate witht = 10−3. The material parameters are taken to be E = 106 and ν = 0.3.Figure 1(a) shows the convergence plots for the thick plate, whereas Figure 1(b)those for the thin plate. Dashed lines indicate the reference order of convergence.As can be seen, the convergence rates for all polynomial orders are at least oforder p.

In addition, we study the convergence for bending moments and shear forcessince these are of prime interest in the engineering design of plates. Bendingmoments m and shear forces q are obtained as

m = −t3Cε(∇w + γ), q = kµtγ.

The exact solution for bending moments and shear forces is

mxx = −Kb2(y3(y − 1)3(x− x2)(5x2 − 5x+ 1)

+ ν(x3(x− 1)3(y − y2)(5y2 − 5y + 1))),

myy = −Kb2(ν(y3(y − 1)3(x− x2)(5x2 − 5x+ 1))

+ x3(x− 1)3(y − y2)(5y2 − 5y + 1)),

mxy = myx = −Kb(1− ν)3y2(y − 1)2(2y − 1)x2(x− 1)2(2x− 1),

qx = −Kb2(y3(y − 1)3(20x3 − 30x2 + 12x− 1)

+ 3y(y − 1)(5y2 − 5y + 1)x2(x− 1)2(2x− 1)),

qy = −Kb2(x3(x− 1)3(20y3 − 30y2 + 12y − 1)

+ 3x(x− 1)(5x2 − 5x+ 1)y2(y − 1)2(2y − 1)),

where Kb =Et3

12(1− ν2)is the plate bending stiffness. For the error measure, the

Euclidean norms of m and q are used, m =

√2∑i=1

2∑j=1

m2ij and q =

√2∑i=1

q2i . The

13

1.4 1.6 1.8 2−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 /

||w

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#dof)−1

c2*(#dof)−3/2

c3*(#dof)−2

c4*(#dof)−5/2

(a)

1.4 1.6 1.8 2−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 /

||w

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#dof)−1

c2*(#dof)−3/2

c3*(#dof)−2

c4*(#dof)−5/2

(b)

Figure 1: Square plate with clamped boundary conditions. L2-norm approxi-mation error of displacements with tensor-product B-splines for (a) t = 10−1

and (b) t = 10−3.

1.4 1.6 1.8 2−7

−6

−5

−4

−3

−2

−1

0

log10

(#dof1/2

)

log

10(

||m

ex −

mh||

L2 / ||m

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#cp)−1/2

c2*(#cp)−1

c3*(#cp)−3/2

c4*(#cp)−2

(a)

1.4 1.6 1.8 2−7

−6

−5

−4

−3

−2

−1

0

log10

(#dof1/2

)

log

10(

||m

ex −

mh||

L2 / ||m

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#cp)−1/2

c2*(#cp)−1

c3*(#cp)−3/2

c4*(#cp)−2

(b)

Figure 2: Square plate with clamped boundary conditions. L2-norm approxi-mation error of bending moments with tensor-product B-splines for (a) t = 10−1

and (b) t = 10−3.

convergence plots for bending moments are presented in Figure 2, and those forshear forces in Figure 3. Recalling that m = t3M (resp. mh = t

3Mh) andq = t3Q (resp. qh = t

3Qh), we notice that the convergence rates for the relativeerrors displayed in Figures 2 and 3 are in accordance with the theoretical resultsof Corollary 2.1 (see estimates (22) and (23)). In particular, we remark thatFigure 3(b) displays an O(1) convergence rate for the L2-norm of the shear forceerrors, when p = 2, in agreement with estimate (23), for s = 2. In other words,there is no convergence.

14

1.4 1.6 1.8 2−8

−7

−6

−5

−4

−3

−2

−1

log10

(#dof1/2

)

log

10(

||q

ex −

qh||

L2 / ||q

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#cp)−1/2

c2*(#cp)−1

c3*(#cp)−3/2

c4*(#cp)−2

(a)

1.4 1.6 1.8 2−7

−6

−5

−4

−3

−2

−1

0

log10

(#dof1/2

)

log

10(

||q

ex −

qh||

L2 / ||q

ex||

L2 )

p=2

p=3

p=4

p=5

c2*(#cp)−1/2

c3*(#cp)−1

c4*(#cp)−2

(b)

Figure 3: Square plate with clamped boundary conditions. L2-norm approxi-mation error of shear forces with tensor-product B-splines for (a) t = 10−1 and(b) t = 10−3.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

x

y

Figure 4: Quarter of annulus plate. Geometry setup.

3.2.2 Quarter of an annulus with clamped and simply supportedboundary conditions

The second test consists of a quarter of an annulus with an inner diameter of 1.0and outer diameter of 2.5, as shown in Figure 4. The plate thickness is t = 0.01and the material parameters are E = 106 and ν = 0.3. The plate is loaded witha uniform load f(x, y) = 1 and two boundary conditions are considered: (a)all edges are clamped, (b) all edges are simply supported. For both cases, thisexample exhibits boundary layers. Therefore, we adopt a refinement strategyin order to better capture the boundary layers. Given that the knot vectorsrange from 0 to 1, we introduce as a first step additional knots at 0.1 and 0.9 in

15

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

x

y

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

x

yFigure 5: Quarter of annulus with boundary refinement. (a) Initial model, (b)boundary refined mesh.

1.2 1.4 1.6 1.8 2−8

−7

−6

−5

−4

−3

−2

−1

0

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 / ||w

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#dof)−1

c2*(#dof)−3/2

c3*(#dof)−2

c4*(#dof)−5/2

(a)

1.4 1.6 1.8 2−9

−8

−7

−6

−5

−4

−3

−2

−1

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 / ||w

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#cp)−1

c2*(#cp)−3/2

c3*(#cp)−2

c4*(#cp)−5/2

(b)

Figure 6: Quarter of annulus with clamped boundary conditions. L2-norm ap-proximation error of displacements with tensor-product NURBS and (a) bound-ary refinement, (b) uniform refinement.

both directions, see Figure 5. Then, we perform uniform refinement of the givenknot spans. In the following, we perform convergence studies with and withoutthe boundary refinement strategy. The refinement is performed such that thetotal number of degrees of freedom is comparable in both cases. Since analyticalsolutions are not available for these problems, we use as reference the solutionsobtained on a very fine mesh (an “overkill” solution) with 100 × 100 quinticelements and compute the L2-norm approximation errors for the displacement.

Figure 6 shows the results for the clamped case, (a) with boundary refine-ment and (b) with uniform refinement, while in Figure 7 the results for the

16

1.2 1.4 1.6 1.8 2−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 /

||w

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#dof)−1

c2*(#dof)−3/2

c3*(#dof)−2

c4*(#dof)−5/2

(a)

1.4 1.6 1.8 2−4

−3.8

−3.6

−3.4

−3.2

−3

−2.8

−2.6

−2.4

−2.2

−2

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 /

||w

ex||

L2 )

p=2

p=3

p=4

p=5

c1*(#dof)−1

(b)

Figure 7: Quarter of annulus with simply supported boundary conditions. L2-norm approximation error of displacements with tensor-product NURBS and(a) boundary refinement, (b) uniform refinement.

simply supported case are plotted, (a) with boundary refinement and (b) withuniform refinement. As expected, due to the presence of boundary layers, inboth the clamped and the simply supported case, the tests confirm that a suit-able boundary refined mesh is needed to achieve optimal orders of convergence.With a uniform refinement, suboptimal results are instead obtained; this is morepronounced in the simply supported case.

4 Isogeometric discretization with NURPS

In this section, we use a triangular NURPS-based isogeometric discretization toperform numerical validations. We first briefly summarize the construction andsome properties of quadratic B-splines over a Powell-Sabin (PS) refinement of atriangulation and their rational generalization, the so-called NURPS B-splines.Then, the discretized model is tested on the same examples as described in theprevious section.

4.1 Quadratic PS and NURPS B-splines

Let T be a triangulation of a polygonal (parametric) domain Ω̂ in R2, and letV i = (V

xi , V

yi ), i = 1, . . . , NV , be the vertices of T . A PS refinement T ∗ of T

is the refined triangulation obtained by subdividing each triangle of T into sixsubtriangles as follows (see also Figure 8).

• Select a split point Ci inside each triangle τi of T and connect it to thethree vertices of τi with straight lines.

• For each pair of triangles τi and τj with a common edge, connect the twopoints Ci and Cj . If τi is a boundary triangle, then also connect Ci to

17

Figure 8: A triangulation T and a Powell-Sabin refinement T ∗ of T .

an arbitrary point on each of the boundary edges.

These split points must be chosen so that any constructed line segment [Ci,Cj ]intersects the common edge of τi and τj . Such a choice is always possible: forinstance, one can take Ci as the incenter of τi, i.e. the center of the circleinscribed in τi. Usually, in practice, the barycenter of τi is also a valid choice,but not always.

The space of C1 piecewise quadratic polynomials on T ∗ is called the Powell-Sabin spline space [26] and is denoted by S12(T ∗). It is well known that thedimension of S12(T ∗) is equal to 3NV . Moreover, any element of S12(T ∗) isuniquely specified by its value and its gradient at the vertices of T , and can belocally constructed on each triangle of T once these values and gradients aregiven.

Dierckx [13] has developed a B-spline like basis {Bi,j , j = 1, 2, 3, i =1, . . . , NV } of the space S12(T ∗) such that

Bi,j(x, y) ≥ 0,NV∑i=1

3∑j=1

Bi,j(x, y) = 1, (x, y) ∈ Ω̂. (24)

The functions Bi,j will be referred to as Powell-Sabin (PS) B-splines. The PSB-splines Bi,j , j = 1, 2, 3, are constructed to have their support locally in the

molecule Ω̂i of vertex V i, which is the union of all triangles of T containing V i.It suffices to specify their values and gradients at any vertex of T . Due to thestructure of the support Ω̂i, we have

Bi,j(V k) = 0,∂

∂xBi,j(V k) = 0,

∂

∂yBi,j(V k) = 0,

for any vertex V k 6= V i. Moreover, we set

Bi,j(V i) = αi,j ,∂

∂xBi,j(V i) = βi,j ,

∂

∂yBi,j(V i) = γi,j .

18

Figure 9: Location of the PS points (black bullets), and a possible PS triangleassociated with the central vertex (shaded).

The triplets (αi,j , βi,j , γi,j) can be specified in a geometric way in order to satisfy(24). To this aim, for each vertex V i, i = 1, . . . , NV , we define three points

{Qi,j = (Qxi,j , Qyi,j), j = 1, 2, 3},

such that αi,1 αi,2 αi,3βi,1 βi,2 βi,3γi,1 γi,2 γi,3

Qxi,1 Qyi,1 1Qxi,2 Qyi,2 1Qxi,3 Q

yi,3 1

=V xi V yi 11 0 0

0 1 0

.The triangle with vertices {Qi,j , j = 1, 2, 3} will be referred to as the PS triangleassociated with the vertex V i and will be denoted by Ti. Finally, for each vertexV i we define its PS points as the vertex itself and the midpoints of all the edgesof the PS refinement T ∗ containing V i, see Figure 9. It has been proved in[13] that the functions Bi,j , j = 1, 2, 3, are non-negative if and only if the PStriangle Ti contains all the PS points associated with the vertex V i. From astability point of view, it is preferable to choose PS triangles with a small area.

Being equipped with a B-spline like basis, PS splines admit a straightforwardrational extension. A NURPS (Non-Uniform Rational PS) basis function isdefined as

Ri,j(x, y) =Bi,j(x, y)ωi,j

NV∑l=1

3∑r=1

Bl,r(x, y)ωl,r

,

where ωi,j are positive control weights. In our plate context, similar to thediscretization with NURBS, the unknown variables w and γ are approximatedby

wh =

NV w∑i=1

3∑j=1

Ri,j(x, y)ŵi,j , γh =

NV γ∑i=1

3∑j=1

Ri,j(x, y)γ̂i,j ,

19

where NV w is the number of vertices for wh and NV γ is the number of verticesfor γh. The test functions v and τ are discretized accordingly. Displacementboundary conditions are enforced in a standard way through the displacementdegrees of freedom on the boundary while rotation boundary conditions areenforced by the modified bilinear form introduced in equation (8).

PS and NURPS B-splines have already been successfully employed to solvepartial differential problems [30], in particular in the isogeometric environment[32, 31]. Certain spline spaces of higher degree and smoothness (regularity) havealso been defined on triangulations endowed with a PS refinement, and they canbe represented in a similar way as in the quadratic case. We refer to [27] for C2

quintics and to [28] for a family of splines with arbitrary smoothness. Moreover,the quadratic case has been extended to the multivariate setting in [29].

Unfortunately, they are lacking the same flexibility of any combination ofpolynomial degree and smoothness in contrast with the tensor-product B-splinecase. On the one hand, it is known how to construct stable spline spaces ontriangulations with a sufficiently high polynomial degree with respect to theglobal smoothness (see, e.g., [20]). In particular, one can quite easily do degree-elevation for the above mentioned existing spaces (i.e., raising the polynomialdegree and keeping the original smoothness). On the other hand, it is extremelychallenging to construct spline spaces on triangulations with a very high smooth-ness relatively to the degree (like the highest continuity Cp−1 for a degree p ≥ 2).Another interesting point of further investigation is the construction of splinespaces with mixed smoothness.

4.2 Numerical tests

In this section, we solve the same examples as illustrated in Section 3.2, usingquadratic PS or NURPS B-splines. In particular, the same error measure asdescribed before is adopted. Despite the fact that PS/NURPS splines can bedefined on arbitrary triangulations, we will only consider regular meshes in ourexamples, in order to be able to make a fair comparison with tensor-productsplines. Of course, in real applications one should exploit this feature anduse triangulations generated by an adaptive refinement strategy. For resultswith adaptive PS/NURPS approximations in isogeometric analysis, we refer to[32, 31].

4.2.1 Square plate with clamped boundary conditions

We perform the same test described in Section 3.2.1 using quadratic PS B-splines both for deflections and rotations defined on uniform triangulations.The coarsest triangulation is depicted in Figure 10 (left), and the approximationerror for the displacement is shown in Figure 11. The dashed line indicates thereference order of convergence. As can be seen, the convergence rate is of order 2.Figure 12 represents the approximation error for bending moments and shearforces. We remark that shear forces seem to converge like O(h) in the L2-normfor this case. However, other numerical tests (not reported here) exhibit an

20

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

Figure 10: Uniform triangulation and its mapping to a quarter of an annulus.

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−4

−3.5

−3

−2.5

−2

−1.5

−1

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 / ||w

ex||

L2 )

t=10−1

t=10−3

c1*(#dof)−1

Figure 11: Square plate with clamped boundary conditions. L2-norm approx-imation error of displacements for t = 10−1 and t = 10−3 with triangular PSsplines.

O(1) convergence rate, in agreement with the theoretical estimate (23), withs = 2.

4.2.2 Quarter of an annulus with clamped and simply supportedboundary conditions

We perform the same test described in Section 3.2.2 using NURPS B-splines.As before we consider a clamped and a simply supported case and in both caseswe perform boundary refinement and uniform refinement. The coarsest uniformmesh and its image are shown in Figure 10. The images of some of the boundaryrefined meshes are shown in Figure 13. Since there are no analytical solutionsavailable, we have taken as reference solutions the NURPS approximations ona fine mesh (an overkill solution): we have used a triangulation consisting of

21

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−2.5

−2

−1.5

−1

−0.5

0

0.5

log10

(#dof1/2

)

log

10(

||m

ex −

mh||

L2 / ||m

ex||

L2 )

m, t=10−1

m, t=10−3

c*(#dof)−1/2

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−2.5

−2

−1.5

−1

−0.5

0

0.5

log10

(#dof1/2

)

log

10(

||q

ex −

qh||

L2 / ||q

ex||

L2 )

q, t=10−1

q, t=10−3

c*(#dof)−1/2

Figure 12: Square plate with clamped boundary conditions. L2-norm approx-imation error of bending moments (left) and shear forces (right) for t = 10−1

and t = 10−3 with triangular PS splines.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

Figure 13: Some boundary refined meshes.

20000 triangles according to the two refinement schemes. Figure 14 shows theapproximation error for the displacement. The dashed line indicates the refer-ence order. As can be seen, boundary refinement yields improved results forboth cases. In particular, the following remark holds for the investigated rangeof degrees of freedom. For the simply supported case, the boundary refinementscheme achieves the correct convergence rate, whereas uniform refinement pro-duces a sub-optimal convergence rate. For the clamped case, both the boundaryrefinement scheme and the uniform refinement scheme give optimal convergencerates, but the former procedure exhibits a numerical better constant in the errorplots.

22

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−4

−3.5

−3

−2.5

−2

−1.5

−1

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 / ||w

ex||

L2 )

uniform

boundary

c1*(#dof)−1

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−4

−3.5

−3

−2.5

−2

−1.5

−1

log10

(#dof1/2

)

log

10(

||w

ex −

wh||

L2 / ||w

ex||

L2 )

uniform

boundary

c1*(#dof)−1

Figure 14: Quarter of annulus with clamped (left) and simply supported (right)boundary conditions. Uniform and boundary refined meshes are considered. L2-norm approximation error of displacements computed with triangular NURPS.

5 Conclusions

In this paper we mathematically and numerically investigated the reformulatedvariational formulation of Reissner-Mindlin plate theory in which the rotationvariables are eliminated in favor of the transverse shear strains. Boundary con-ditions on the rotations were enforced weakly by way of Nitsche’s method tomake the implementation easier and to overcome possible boundary lockingphenomena (see Remark 2.3). A distinct advantage of this theory is that shearlocking is precluded for any combination of trial functions for displacement andtransverse shear stains. However, second derivatives of the displacement ap-pear in the strain energy expression and these require basis functions of at leastC1 continuity. To deal with the smoothness requirements we employed Isogeo-metric Analysis, specifically various degree NURBS of maximal continuity, andquadratic triangular NURPS. The numerical results corroborated the theoreti-cal error estimates for displacement, bending moments and transvers shear forceresultants.

Acknowledgements

L. Beirão da Veiga, C. Lovadina and A. Reali were supported by the EuropeanCommission through the FP7 Factory of the Future project TERRIFIC (FoF-ICT-2011.7.4, Reference: 284981).T. J. R. Hughes was supported by grants from the Office of Naval Research(N00014- 08-1-0992), the National Science Foundation (CMMI-01101007) andSINTEF (UTA10- 000374), with the University of Texas at Austin.J. Kiendl and A. Reali were supported by the European Research Councilthrough the FP7 Ideas Starting Grant n. 259229 ISOBIO.Jarkko Niiranen was supported by Academy of Finland (decision number 270007).H. Speleers was supported by the Research Foundation Flanders and by the

23

MIUR ‘Futuro in Ricerca’ Programme through the project DREAMS.

A Proofs of the theoretical results

In the present section we prove all the theoretical results previously presentedin the body of the paper. In the following we will assume the obvious conditionthat 0 < t < diam(Ω), where diam(Ω) denotes the diameter of Ω. We will needthe following results.

• First Korn’s inequality (see [14]). There exists a positive constant C suchthat

||ε(η)||2L2(Ω) + ||η||2L2(Ω) ≥ C||η||

2H1(Ω), ∀η ∈ H1(Ω)2. (25)

• Second Korn’s inequality (see [14]). Suppose that |Γc| > 0. Then, thereexists a positive constant C such that

||ε(η)||2L2(Ω) ≥ C||η||2H1(Ω), ∀η ∈ H1(Ω)2, such that η|Γc = 0. (26)

• Agmon’s inequality (see [1, 2]). Let e be an edge of an element Ke. Then∃Ca(Ke) > 0 only depending on the shape of Ke such that

||ϕ||2L2(e) ≤ Ca(Ke)(h−1e ||ϕ||2L2(Ke) + he|ϕ|

2H1(Ke)

), ϕ ∈ H1(Ke). (27)

Clearly, (27) also holds for vector-valued and tensor-valued functions.

A.1 Coercivity of the continuous problem

Proposition A.1. Let assumption A4 hold. Then there exists a positive con-stant α depending only on the material constants and the domain Ω such that

a(η,η) + µkt−2(η −∇v,η −∇v)

≥ α(||η||2H1(Ω) + t

−2||η −∇v||2L2(Ω) + ||v||2H1(Ω)

), ∀(v,η) ∈ X̃.

(28)

Proof. It is easy to see that the hypotheses on Γc and Γs are sufficient to preventrigid body motions. We proceed by considering the two different cases.

i) Let Γc have positive length. Then, from the positive-definiteness of C andthe second Korn’s inequality, we get

a(η,η) = (Cε(η), ε(η)) ≥ C1||ε(η)||2L2(Ω) ≥ C2||η||2H1(Ω).

Therefore, estimate (28) follows from a little algebra and the Poincaré inequalityfor v.

ii) Let Γc have zero length. Then Γs is not contained in a straight line, and,since Γc ∪ Γs has positive length, it follows that Γs has positive length. It isenough to prove that one has

a(η,η) + ||η −∇v||2L2(Ω) ≥ C(||η||2H1(Ω) + ||v||

2H1(Ω)

), ∀(v,η) ∈ X̃. (29)

24

By contradiction, suppose that estimate (29) does not hold. Then, there exists

a sequence {(vk,ηk)} ∈ X̃ such that{a(ηk,ηk) + ||ηk −∇vk||2L2(Ω) → 0, for k → +∞;

||ηk||2H1(Ω) + ||vk||2H1(Ω) = 1.

(30)

Up to extracting a subsequence, the second equation of (30) shows that

ηk ⇀ η0 weakly in H1(Ω)2; vk ⇀ v0 weakly in H

1(Ω). (31)

By Rellich’s Theorem we infer that

ηk → η0 in L2(Ω)2; vk → v0 in L2(Ω). (32)

Therefore, recalling that C is positive-definite, from a(ηk,ηk) → 0 (cf. (30)),(32) and (25), we get that {ηk} is a Cauchy sequence in H1(Ω)2. Thus, we have

ηk → η0 in H1(Ω)2 and ε(η0) = 0. (33)

Moreover, since {ηk} is a Cauchy sequence in L2(Ω)2, from ||ηk−∇vk||2L2(Ω) → 0(cf. (30)), we have that also {∇vk} is a Cauchy sequence in L2(Ω)2. Therefore,from (31) and (32) we obtain that

vk → v0 in H1(Ω) and ∇v0 = η0.

Hence, from (33) we get ε(∇v0) = 0, which implies that v0 is an affine function.Since v0 = 0 on Γs and Γs is not contained in a straight line, it follows thatv0 = 0 in Ω. Therefore, η0 = 0 and we have proved that (ηk, vk) → (0, 0) inH1(Ω)2 ×H1(Ω), which contradicts the second equation of (30).

A.2 Stability and convergence analysis

In the present section we give the proofs of the results in Section 2.3. We needthe following Korn’s type inequality.

Lemma A.1. Suppose that Γc has positive length. Then, there exists a positiveconstant C such that

||ε(v)||2L2(Ω) + ||v||2L2(Γc)

≥ C||v||2H1(Ω), ∀v ∈ H1(Ω)2. (34)

Proof. By contradiction. If (34) does not hold, then there exists a sequence{vk} in H1(Ω)2 such that{

||ε(vk)||2L2(Ω) + ||vk||2L2(Γc)

→ 0, for k → +∞;

||vk||2H1(Ω) = 1.(35)

Up to extracting a subsequence, the second equation of (35) and Rellich’s the-orem show that there exists v0 ∈ H1(Ω)2 such that

25

{vk ⇀ v0 weakly in H

1(Ω)2;

vk → v0 strongly in L2(Ω)2.(36)

From (35) and (36) we get that {(vk, ε(vk))} is a Cauchy sequence in L2(Ω)2×L2(Ω)4s. Using the first Korn inequality (25) we deduce that {vk} is a Cauchysequence also in H1(Ω)2, and thus vk → v0, strongly in H1(Ω)2. Therefore,from the first equation of (35) we have

ε(v0) = 0 in Ω; v0|Γc = 0. (37)

Equation (37) easily implies v0 = 0. Therefore, vk → 0, which is in contradic-tion with ||vk||H1(Ω) = 1 (cf. (35)).

Proof of Lemma 2.1. We distinguish two cases.i) Γc has zero length. In this case we have

ah(∇vh + τh,∇vh + τh) = a(∇vh + τh,∇vh + τh),|||vh, τh|||h = |||vh, τh|||,

for every (vh, τh) ∈ Xh; see (8) and (10). Therefore, estimate (11) immediatelyfollows from estimate (7), since Γc with vanishing length implies Xh ⊂ X.

ii) Γc has positive length. First, for every (vh, τh) ∈ Xh we will show that(cf. (10)):

ah(∇vh + τh,∇vh + τh)

≥ C

(||∇vh + τh||2H1(Ω) +

∑e∈Eh∩Γc

h−1e ||∇vh + τh||2L2(e)

).

(38)

For notational simplicity, we set θh := ∇vh + τh. Then, recalling (8), we have

ah(θh,θh) = a(θh,θh)− 2∫

Γc

(Cε(θh)ne

)· θh + β tr(C)

∑e∈Eh∩Γc

h−1e

∫e

|θh|2

= a(θh,θh) +β

2tr(C)

∑e∈Eh∩Γc

h−1e

∫e

|θh|2

− 2∫

Γc

(Cε(θh)ne

)· θh +

β

2tr(C)

∑e∈Eh∩Γc

h−1e

∫e

|θh|2.

Applying (34) with v = θh, we obtain

ah(θh,θh) ≥ CK ||θh||2H1(Ω)

− 2∫

Γc

(Cε(θh)ne

)· θh +

β

2tr(C)

∑e∈Eh∩Γc

h−1e

∫e

|θh|2,(39)

26

for a suitable positive constant CK . For each edge e ∈ Γc ∩ Eh, let the symbolKe denote an element of Ωh such that e ∈ ∂K. We now have, by simple algebraand using (27):

− 2∫

Γc

(Cε(θh)ne

)· θh =

∑e∈Eh∩Γc

(−2∫e

(Cε(θh)ne

)· θh

)≥

∑e∈Eh∩Γc

(−2||Cε(θh)ne||L2(e)||θh||L2(e)

)≥

∑e∈Eh∩Γc

(−2CC||ε(θh)||L2(e)||θh||L2(e)

)≥ −

∑e∈Eh∩Γc

CC

(γhe||ε(θh)||2L2(e) +

1

γehe||θh||2L2(e)

)≥ −

∑e∈Eh∩Γc

(CCCa(Ke)γe

(||ε(θh)||2L2(Ke) + h

2e|ε(θh)|2H1(Ke)

)+

CCγ2he

||θh||2L2(e)),

(40)for positive constants {γe}e∈Γc∩Eh to be chosen. By using the inverse inequality

|ε(θh)|2H1(Ke) ≤ Cinv(Ke)h−2Ke||ε(θh)||2L2(Ke),

and setting

C(Ke) = CCCa(Ke)

(1 + Cinv(Ke)

h2eh2Ke

),

from (40) it follows that

−2∫

Γc

(Cε(θh)ne

)· θh ≥ −

∑e∈Eh∩Γc

(C(Ke)γe||ε(θh)||2L2(Ke) +

CCγehe

||θh||2L2(e)).

(41)Therefore, from (39) and (41) we get

ah(θh,θh) ≥ CK ||θh||2H1(Ω) −∑

e∈Eh∩Γc

C(Ke)γe||ε(θh)||2L2(Ke)

+∑

e∈Eh∩Γc

(β

2tr(C)− CC

γe

)h−1e

∫e

|θh|2 ≥(CK −

∑e∈Eh∩Γc

C(Ke)γe

)||θh||2H1(Ω) +

(β

2tr(C)− CC

γe

) ∑e∈Eh∩Γc

h−1e

∫e

|θh|2.

(42)Choosing

γe =CK2C(Ke)

−1 and β0 =γCK + 2CCγtr(C)

with γ = mine∈Eh∩Γc

γe,

from (42) we deduce that, for every β ≥ β0, we have

ah(θh,θh) ≥CK2

(||θh||2H1(Ω) +

∑e∈Eh∩Γc

h−1e

∫e

|θh|2).

27

Recalling that θh := ∇vh + τh, we get that (38) holds. Therefore, (11) followsfrom (38), (5), (10) and the Poincaré inequality applied to vh (recall that vh|Γc =0 and |Γc| > 0). Finally note that, due to the uniform shape regularity of theelements K in {Ωh}h, it is easy to check that the constant β0 is uniformlybounded from above independently of the mesh size h.

Proof of Proposition 2.1. In the following, C will denote a generic positiveconstant independent of h. Given any pair (vh, τh) in Xh, we denote by wE =wh − vh, γE = γh − τh and by wA = w − vh, γA = γ − τh. By applying firstthe coercivity Lemma 2.1 and then using the linearity of the bilinear forms andthe consistency condition (12), we get

α′|||γE , wE |||2h ≤ ah(∇wE + γE ,∇wE + γE) + t−2(γE ,γE)= ah(∇wA + γA,∇wE + γE) + t−2(γA,γE).

(43)

By definitions (8) and (2) and standard algebra we get from (43)

|||γE , wE |||2h ≤ C T1/2A · T

1/2E + t

−2||γA||L2(Ω)||γE ||L2(Ω), (44)

where the scalar terms are given by

TA = ||∇wA + γA||2H1(Ω) +∑

e∈Eh∩Γc

he||Cε(∇wA + γA)ne||2L2(e)

+∑

e∈Eh∩Γc

h−1e ||∇wA + γA||2L2(e),

and

TE = ||∇wE + γE ||2H1(Ω) +∑

e∈Eh∩Γc

he||Cε(∇wE + γE)ne||2L2(e)

+∑

e∈Eh∩Γc

h−1e ||∇wE + γE ||2L2(e).

Term TA can be bounded by using (27), as already done in (40). Without againshowing the details, and following the same notation for Ke introduced in (40),we get

TA ≤ C(||∇wA + γA||2H1(Ω) +

∑e∈Eh∩Γc

(h2e|wA|2H3(Ke) + |wA|

2H2(Ke)

+ h2e|γA|2H2(Ke) + |γA|2H1(Ke)

+ h−2e |wA|2H1(Ke) + h−2e ||γA||2L2(Ke)

)).

From the above bound, a triangle inequality, and the definition of wA, γA, weget

TA ≤ C2∑j=0

( ∑K∈Ωh

h2(j−1)K |w− vh|

2Hj+1(K) +

∑K∈Ωh

h2(j−1)K |γ − τh|

2Hj(K)

). (45)

28

We now bound TE . Again, using the Agmon inequality (27) and inverse esti-mates as done in (40), we get for all e ∈ Eh ∩ Γc:

he||Cε(∇wE + γE)ne||2L2(e) ≤ C(h2e|∇wE + γE |2H2(Ke)+

|∇wE + γE |2H1(Ke))≤ C|∇wE + γE |2H1(Ke).

Combining the above bound with the definition of TE and (10) yields

TE + t−2||γE ||2L2(Ω) ≤ C|||wE ,γE |||

2h. (46)

Now, recalling (44) and using (46) we easily get

|||γE , wE |||h ≤ C(T

1/2A + t

−1||γA||L2(Ω)).

Finally, recalling that the above inequality holds for all (vh, τh) ∈ Xh, thebound in (45) concludes the proof.

Proof of Proposition 2.2. We only sketch the proof. Estimate (16) immedi-ately follows from (5), by recalling (14) and (15). We prove (17) only for thecase of a simply supported plate. In this case, ah(·, ·) = a(·, ·), because Γc is theempty set. However, we notice that different boundary conditions can be dealtwith using a similar technique. Let sh and τh be given in Ξh. Using (6) and(9), we get

(Qh − sh, τh) = (Qh −Q, τh) + (Q− sh, τh)= a

(∇(w − wh) + (γ − γh), τh

)+ (Q− sh, τh).

Choosing ψh = h2(Qh − sh), and using the inverse inequality

|Qh − sh|H1(Ω) ≤ Ch−1||Qh − sh||L2(Ω),

we have

h2||Qh − sh||2L2(Ω) = h2a(∇(w − wh) + (γ − γh), (Qh − sh)

)+ h2(Q− sh,Qh − sh) ≤ C|||w − wh,γ − γh|||hh ||Qh − sh||L2(Ω)

+ h ||Q− sh||L2(Ω) h ||Qh − sh||L2(Ω).

Hence, we obtain

h ||Qh − sh||L2(Ω) ≤ C|||w − wh,γ − γh|||h + h ||Q− sh||L2(Ω).

Therefore, the triangle inequality gives

h ||Q−Qh||L2(Ω) ≤ C|||w − wh,γ − γh|||h + 2h ||Q− sh||L2(Ω),

from which we infer

h ||Q−Qh||L2(Ω) ≤ C|||w − wh,γ − γh|||h + 2h infsh∈Ξh

||Q− sh||L2(Ω),

29

i.e. (17).To prove (18), we use the so-called Pitkäranta-Verfürth trick (see [25] and [33]).

We start with noticing that

||Q−Qh||H−1(Ω) = supτ∈H10 (Ω)2

(Q−Qh, τ )|τ |H1(Ω)

. (47)

Then, fix τ ∈ H10 (Ω)2 and take τ I as its best approximation in Ξh with respectto the H1-norm. Then, we have

(Q−Qh, τ ) = (Q−Qh, τ − τ I) + (Q−Qh, τ I)≤ Ch ||Q−Qh||L2(Ω)|ψ|H1(Ω) + (Q−Qh, τ I).

(48)

Using (6) and (9), we get

(Q−Qh, τ I) = a(∇(wh − w) + (γh − γ), τ I

)≤ C|||w − wh,γ − γh|||h|τ I |H1(Ω)≤ C|||w − wh,γ − γh|||h|τ |H1(Ω).

(49)

Therefore, from (47), (48) and (49) we deduce

||Q−Qh||H−1(Ω) ≤ C(h ||Q−Qh||L2(Ω) + |||w − wh,γ − γh|||h

). (50)

Estimate (18) now follows from (50) and (17).

Proof of Proposition 2.3. We begin with introducing the following auxiliaryproblem:{

Find (w̃, γ̃) ∈ X, such thata(∇w̃ + γ̃,∇v + τ ) + t−2(γ̃, τ ) = (θ − θh,∇v + τ ), ∀(v, τ ) ∈ X.

(51)

Note that problem (51) is equivalent, up to the usual transformation (4), to aclassical Reissner-Mindlin problem with the load acting on the rotations. There-fore, also recalling the hypotheses above and the definition of γ, the followingregularity result holds [22]

||w̃1||H3(Ω) + t−1||w̃2||H2(Ω) + t−1||γ̃||H1(Ω) ≤ C||θ − θh||L2(Ω), (52)

with w̃ = w̃1 + w̃2 denoting a splitting of the deflections.The consistency of the discrete bilinear form, analogously as in (12), implies

thatah(∇w̃ + γ̃,∇vh + τh) + t−2(γ̃, τh) = (θ − θh,∇vh + τh), (53)

for all (vh, τh) ∈ Xh. Note that, whenever both entries are in X, the bilinearform ah(·, ·) is equal to the original form a(·, ·) since all the additional termsvanish. Therefore, from (51) we also have

ah(∇w̃ + γ̃,∇v + τ ) + t−2(γ̃, τ ) = (θ − θh,∇v + τ ), ∀(v, τ ) ∈ X. (54)

30

Combining (53) and (54) we can take (w − wh,γ − γh) as a test function, andobtain

||θ − θh||2L2(Ω) = ah(∇w̃ + γ̃,∇(w − wh) + γ − γh) + t−2(γ̃,γ − γh) (55)

where we recall the usual notation θ = ∇w + γ and θh = ∇wh + γh. Theremaining steps are standard in Aubin-Nitsche duality arguments and thereforeonly described briefly. By the symmetry of the bilinear form and using (9), (12),from (55) we get

||θ − θh||2L2(Ω) = ah(∇(w̃ − w̃I) + γ̃ − γ̃I ,∇(w − wh) + γ − γh)+ t−2(γ̃ − γ̃I ,γ − γh),

(56)

where (w̃I , γ̃I) ∈ Xh is an approximant of (w̃, γ̃).Due to the ||| · |||h-norm continuity of the bilinear form in (56), all we need

to show in order to prove (19) is that

|||w̃ − w̃I ,γ − γI |||h ≤ C (h+ t) ||θ − θh||L2(Ω), (57)

for some uniform constant C. The bound in (57) can be shown by making use ofthe regularity estimate (52) and the approximation properties A3 of the spaceXh; we do not show the details.

Finally, estimate (20) follows easily by the triangle inequality

||w − wh||H1(Ω) ≤ ||θ − θh||L2(Ω) + ||γ − γh||L2(Ω)

and using (19).

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