Chapter 8 Plates and Shells: Asymptotic Expansions …2 Plates and Shells: Asymptotic Expansions and Hierarchic Models an asymptotic analysis for which we embed our structure in a

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Chapter 8Plates and Shells: Asymptotic Expansions andHierarchic Models

Monique Dauge1, Erwan Faou2 and Zohar Yosibash3

1 IRMAR, Universite de Rennes 1, Campus de Beaulieu, Rennes, France2 INRIA Rennes, Campus de Beaulieu, Rennes, France3 Ben-Gurion University, Beer Sheva, Israel

1 Introduction 1

2 Multiscale Expansions for Plates 4

3 Hierarchical Models for Plates 94 Multiscale Expansions and Limiting Models

for Shells 13

5 Hierarchical Models for Shells 20

6 Finite Element Methods in Thin Domains 21

Acknowledgments 31

Notes 31

References 31

Further Reading 34

1 INTRODUCTION

1.1 Structures

Plates and shells are characterized by (i) their midsurfaceS, (ii) their thickness d . The plate or shell character is thatd is small compared to the dimensions of S. In this respect,we qualify such structures as thin domains. In the case ofplates, S is a domain of the plane, whereas in the case ofshells, S is a surface embedded in the three-dimensionalspace. Of course, plates are shells with zero curvature.

Encyclopedia of Computational Mechanics, Edited by ErwinStein, Rene de Borst and Thomas J.R. Hughes. Volume 1: Funda-mentals. 2004 John Wiley & Sons, Ltd. ISBN: 0-470-84699-2.

Nevertheless, considering plates as a particular class ofshells in not so obvious: They have always been treatedseparately, for the reason that plates are simpler. We think,and hopefully demonstrate in this chapter, that eventually,considering plates as shells sheds some light in the shelltheory.

Other classes of thin domains do exist, such as rods,where two dimensions are small compared to the third one.We will not address them and quote, for example, (Nazarov,1999; Irago and Viano, 1999). Real engineering structuresare often the union (or junction) of plates, rods, shells, andso on. See Ciarlet (1988, 1997) and also Kozlov, Maz’yaand Movchan (1999) and Agratov and Nazarov (2000). Werestrict our analysis to an isolated plate or shell. We assumemoreover that the midsurface S is smooth, orientable, andhas a smooth boundary ∂S. The shell character includesthe fact that the principal curvatures have the same orderof magnitude as the dimensions of S. See Anicic and Leger(1999) for a situation where a region with strong curvature(like 1/d) is considered. The opposite situation is when thecurvatures have the order of d: We are then in the presenceof shallow shells according to the terminology of Ciarletand Paumier (1986).

1.2 Domains and coordinates

In connection with our references, it is easier for us toconsider d as the half-thickness of the structure. We denoteour plate or shell by d . We keep the reference to the half-thickness in the notation because we are going to perform

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2 Plates and Shells: Asymptotic Expansions and Hierarchic Models

an asymptotic analysis for which we embed our structurein a whole family of structures (ε)ε, where the parameterε tends to 0.

We denote the Cartesian coordinates of R3 by x =

(x1, x2, x3), a tangential system of coordinates on S byx = (xα)α=1,2, a normal coordinate to S by x3, with theconvention that the midsurface is parametrized by theequation x3 = 0. In the case of plates (xα) are Cartesiancoordinates in R

2 and the domain d has the tensor productform

d = S × (−d, d)

In the case of shells, x = (xα)α=1,2 denotes a localcoordinate system on S, depending on the choice of alocal chart in an atlas, and x3 is the coordinate alonga smooth unit normal field n to S in R

3. Such a nor-mal coordinate system (also called S-coordinate system)(x, x3) yields a smooth diffeomorphism between d andS × (−d, d). The lateral boundary d of d is char-acterized by x ∈ ∂S and x3 ∈ (−d, d) in coordinates(x, x3).

1.3 Displacement, strain, stress, and elasticenergy

The displacement of the structure (deformation from thestress-free configuration) is denoted by u, its Cartesiancoordinates by (u1, u2, u3), and its surface and transverseparts by u = (uα) and u3 respectively. The transverse partu3 is always an intrinsic function and the surface part udefines a two-dimensional 1-form field on S, depending onx3. The components (uα) of u depend on the choice of thelocal coordinate system x.

We choose to work in the framework of small deforma-tions (see Ciarlet (1997, 2000)) for more general nonlinearmodels e.g. the von Karman model). Thus, we use the straintensor (linearized from the Green–St Venant strain tensor)e = (eij ) given in Cartesian coordinates by

eij (u) = 1

2

(∂ui

∂xj

+ ∂uj

∂xi

)

Unless stated otherwise, we assume the simplest possiblebehavior for the material of our structure, that is, anisotropic material. Thus, the elasticity tensor A = (Aijkl)

takes the form

Aijkl = λδijδkl + µ(δikδj l + δilδjk)

with λ and µ the Lame constants of the material andδij the Kronecker symbol. We use Einstein’s summation

convention, and sum over double indices if they appearas subscripts and superscripts (which is nothing but thecontraction of tensors), for example, σij eij ≡ 3

i,j=1σij eij .

The constitutive equation is given by Hooke’s law σ =Ae(u) linking the stress tensor σ to the strain tensor e(u).Thus

σii = λ(e11 + e22 + e33) + 2µeii , i = 1, 2, 3

σij = 2µeij for i = j(1)

The elastic bilinear form on a domain is given by

a(u, u′) =∫

σ(u) : e(u′) dx =∫

σij (u) eij (u′) dx (2)

and the elastic energy of a displacement u is (1/2)a(u, u).The strain–energy norm of u is denoted by ‖u‖

E()and

defined as (∑

ij

∫|eij (u)|2 dx)1/2.

1.4 Families of problems

We will address two types of problems on our thin domaind : (i) Find the displacement u solution to the equilib-rium equation div σ(u) = f for a given load f, (ii) Findthe (smallest) vibration eigen-modes (, u) of the struc-ture. For simplicity of exposition, we assume in gen-eral that the structure is clamped (this condition is alsocalled ‘condition of place’) along its lateral boundary d

and will comment on other choices for lateral bound-ary conditions. On the remaining part of the boundary∂d \ d (‘top’ and ‘bottom’) traction free condition isassumed.

In order to investigate the influence of the thicknesson the solutions and the discretization methods, we con-sider our (fixed physical) problem in d as part of awhole family of problems, depending on one parameterε ∈ (0, ε0], the thickness. The definition of ε is obvi-ous by the formulae given in Section 1.2 (in fact, ifthe curvatures of S are ‘small’, we may decide that d

fits better in a family of shallow shells, see Section 4.4later). For problem (i), we choose the same right handside f for all values of ε, which precisely means thatwe fix a smooth field f on ε0 and take fε := f|ε foreach ε.

Both problems (i) and (ii) can be set in variational form(principle of virtual work). Our three-dimensional varia-tional space is the subspace V (ε) of the Sobolev spaceH 1(ε)3 characterized by the clamping condition u|ε = 0,and the bilinear form a (2) on = ε, denoted by aε. Thevariational formulations are

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Plates and Shells: Asymptotic Expansions and Hierarchic Models 3

Find uε ∈ V (ε) such that

aε(uε, u′) =∫

ε

fε · u′ dx, ∀u′ ∈ V (ε) (3)

for the problem with external load, and

Find uε ∈ V (ε) , uε = 0, and ε ∈ R such that

aε(uε, u′) = ε

∫ε

uε · u′ dx, ∀u′ ∈ V (ε) (4)

for the eigen-mode problem. In engineering practice, one isinterested in the natural frequencies, ωε = √

ε. Of course,when considering our structure d , we are eventually onlyinterested in ε = d . Taking the whole family ε ∈ (0, ε0]into account allows the investigation of the dependencywith respect to the small parameter ε, in order to knowif valid simplified models are available and how they canbe discretized by finite elements.

1.5 Computational obstacles

Our aim is to study the possible discretizations for areliable and efficient computation of the solutions ud ofproblem (3) or (4) in our thin structure d . An optioncould be to consider d as a three-dimensional body anduse 3-D finite elements. In the standard version of finiteelements (h-version), individual elements should not bestretched or distorted, which implies that all dimensionsshould be bounded by d . Even so, several layers of elementsthrough the thickness may be necessary. Moreover thea priori error estimates may suffer from the behavior ofthe Korn inequality on d (the factor appearing in the Korninequality behaves like d−1 for plates and partially clampedshells; see Ciarlet, Lods and Miara (1996) and Dauge andFaou (2004)).

An ideal alternative would simply be to get rid of thethickness variable and compute the solution of an ‘equiva-lent’ problem on the midsurface S. This is the aim of theshell theory. Many investigations were undertaken around1960–1970, and the main achievement is (still) the Koitermodel, which is a multidegree 3 × 3 elliptic system on S

of half-orders (1, 1, 2) with a singular dependence in d .But, as written in Koiter and Simmonds (1973), ‘Shell the-ory attempts the impossible: to provide a two-dimensionalrepresentation of an intrinsically three-dimensional phe-nomenon’. Nevertheless, obtaining converging error esti-mates between the 3-D solution ud and a reconstructed 3-Ddisplacement Uzd from the deformation pattern zd solutionof the Koiter model seems possible.

However, due to its fourth order part, the Koiter modelcannot be discretized by standard C0 finite elements. The

Naghdi model, involving five unknowns on S, seems moresuitable. Yet, endless difficulties arise in the form of variouslocking effects, due to the singularly perturbed character ofthe problem.

With the twofold aim of improving the precision ofthe models and their approximability by finite elements,the idea of hierarchical models becomes natural: Roughly,it consists of an Ansatz of polynomial behavior in thethickness variable, with bounds on the degrees of the threecomponents of the 3-D displacement. The introduction ofsuch models in variational form is due to Vogelius andBabuska (1981c) and Szabo and Sahrmann (1988). Earlierbeginnings in that direction can be found in Vekua (1955,1965). The hierarchy (increasing the transverse degrees)of models obtained in that way can be discretized by thep-version of finite elements.

1.6 Plan of the chapter

In order to assess the validity of hierarchical models, wewill compare them with asymptotic expansions of solutionsuε when they are available: These expansions exhibit twoor three different scales and boundary layer regions, whichcan or cannot be properly described by hierarchical models.

We first address plates because much more is known forplates than for general shells. In Section 2, we describe thetwo-scale expansion of the solutions of (3) and (4): Thisexpansion contains (i) a regular part each term of which ispolynomial in the thickness variable x3, (ii) a part mainlysupported in a boundary layer around the lateral boundaryε. In Section 3, we introduce the hierarchical modelsas Galerkin projections on semidiscrete subspaces V q(ε)

of V (ε) defined by assuming a polynomial behavior ofdegree q = (q1, q2, q3) in x3. The model of degree (1, 1, 0)

is the Reissner–Mindlin model and needs the introductionof a reduced energy. The (1, 1, 2) model is the lowestdegree model to use the same elastic energy (2) as the 3-Dmodel.

We address shells in Section 4 (asymptotic expansionsand limiting models) and Section 5 (hierarchical models).After a short introduction of the metric and curvature ten-sors on the midsurface, we first describe the three-scaleexpansion of the solutions of (3) on clamped elliptic shells:Two of these scales can be captured by hierarchical models.We then present and comment on the famous classificationof shells as flexural or membrane. We also mention twodistinct notions of shallow shells. We emphasize the uni-versal role played by the Koiter model for the structure d ,independently of any embedding of d in a family (ε)ε.

The last section is devoted to the discretization of the3-D problems and their 2-D hierarchical projections, by p-version finite elements. The 3-D thin elements (one layer of

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elements through the thickness) constitute a bridge between3-D and 2-D discretizations. We address the issue of lockingeffects (shear and membrane locking) and the issue ofcapturing boundary layer terms. Increasing the degree p ofapproximation polynomials and using anisotropic meshes isa way toward solving these problems. We end this chapterby presenting a series of eigen-frequency computations ona few different families of shells and draw some ‘practical’conclusions.

2 MULTISCALE EXPANSIONS FORPLATES

The question of an asymptotic expansion for solutions uε

of problems (3) or (4) posed in a family of plates isdifficult: One may think it is natural to expand uε eitherin polynomial functions in the thickness variable x3, or inan asymptotic series in powers εk with regular coefficientsvk defined on the stretched plate = S × (−1, 1). Infact, for the class of loads considered here or for theeigen-mode problem, both those Ansatze are relevant, butthey are unable to provide a correct description of thebehavior of uε in the vicinity of the lateral boundary ε,where there is a boundary layer of width ∼ ε (except inthe particular situation of a rectangular midsurface withsymmetry lateral boundary conditions (hard simple supportor sliding edge); see Paumier, 1990). And, worse, in theabsence of knowledge of the boundary layer behavior, thedetermination of the terms vk is impossible (except for v0).

The investigation of asymptotics as ε → 0 was first per-formed by the construction of infinite formal expansions;see Friedrichs and Dressler (1961), Gol’denveizer (1962),and Gregory and Wan (1984). The principle of multiscaleasymptotic expansion is applied to thin domains in Maz’ya,Nazarov and Plamenevskii (1991b). A two-term asymp-totics is exhibited in Nazarov and Zorin (1989). The wholeasymptotic expansion is constructed in Dauge and Gruais(1996, 1998a) and Dauge, Gruais and Rossle (1999/00).

The multiscale expansions that we propose differ fromthe matching method in Il’in (1992) where the solutionsof singularly perturbed problems are fully described inrapid variables inside the boundary layer and slow variablesoutside the layer, both expansions being ‘matched’ in anintermediate region. Our approach is closer to that of Vishikand Lyusternik (1962) and Oleinik, Shamaev and Yosifian(1992).

2.1 Coordinates and symmetries

The midsurface S is a smooth domain of the plane R2

(see Fig. 1) and for ε ∈ (0, ε0) ε = S × (−ε, ε) is the

S S

x2

x1

s

r

Figure 1. Cartesian and local coordinates on the midsurface.

x3

+1

−1+ε−ε

ΩΩε

X3 = x3ε

Figure 2. Thin plate and stretched plate.

generic member of the family of plates (see Fig. 2). Theplates are symmetric with respect to the plane . Since theyare assumed to be made of an isotropic material, problems(3) or (4) commute with the symmetry S: u → (u(·,−x3),−u3(·,−x3)). The eigenspaces of S are membrane andbending displacements (also called stretching and flexuraldisplacements), cf. Friedrichs and Dressler (1961):

u membrane iff u(x,+x3) = u(x,−x3)

and u3(x,+x3) = −u3(x,−x3)

u bending iff u(x,+x3) = −u(x,−x3)

and u3(x,+x3) = u3(x,−x3)

(5)

Any general displacement u is the sum um + ub of amembrane and a bending part (according to formulae um =(1/2)(u + Su) and uf = (1/2)(u − Su). They are alsodenoted by uI and uII in the literature).

In addition to the coordinates x in S, let r be the distanceto ∂S in and s an arclength function on ∂S (see Fig. 1).In this way, (r, s) defines a smooth coordinate system ina midplane tubular neighborhood V of ∂S. Let χ = χ(r)be a smooth cut-off function with support in V, equal to 1in a smaller such neighborhood. It is used to substantiateboundary layer terms. The two following stretched (orrapid) variables appear in our expansions:

X3 = x3

εand R = r

ε

The stretched thickness variable X3 belongs to (−1, 1)and is present in all parts of our asymptotics, whereasthe presence of R characterizes boundary layer terms (seeFigure 2).

2.2 Problem with external load

The solutions of the family of problems (3) have a two-scaleasymptotic expansion in regular terms vk and boundary

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layer terms wk , which we state as a theorem (Dauge, Gruaisand Rossle, 1999/00; Dauge and Schwab, 2002). Note thatin contrast with the most part of those references, we workhere with natural displacements (i.e. unscaled), which ismore realistic from the mechanical and computational pointof view, and allows an easier comparison with shells.

Theorem 1. (Dauge, Gruais and Rossle, 1999/00) Forthe solutions of problem (3), ε ∈ (0, ε0], there exist regularterms vk = vk(x, X3), k ≥ −2, and boundary layer termswk = wk(R, s, X3), k ≥ 0, such that

uε ε−2v−2 + ε−1v−1 + ε0(v0 + χw0)

+ ε1(v1 + χw1) + · · · (6)

in the sense of asymptotic expansions: The following esti-mates hold∥∥∥∥∥uε −

K∑k=−2

εk(vk + χwk)

∥∥∥∥∥E(ε)

≤ CK(f) εK+1/2,

K = 0, 1, . . .

where we have set w−2 = w−1 = 0 and the constant CK(f)is independent of ε ∈ (0, ε0].

2.2.1 Kirchhoff displacements and their deformationpatterns

The first terms in the expansion of uε are Kirchhoff dis-placements, that is, displacements of the form (with thesurface gradient ∇ = (∂1, ∂2))

(x, x3) −−−→ v(x, x3) =(ζ(x) − x3∇ζ3(x),

ζ3(x))

(7)

Here, ζ = (ζα) is a surface displacement and ζ3 is afunction on S. We call the three-component field ζ :=(ζ, ζ3), the deformation pattern of the KL displacementv. Note that

v bending iff ζ = (0, ζ3) and v membrane iff ζ = (ζ, 0)

In expansion (6) the first terms are Kirchhoff displacements.The next regular terms vk are also generated by deformationpatterns ζk via higher degree formulae than in (7). We suc-cessively describe the vk, the ζk and, finally, the boundarylayer terms wk .

2.2.2 The four first regular terms

For the regular terms vk , k = −2,−1, 0, 1, there exist bend-ing deformation patterns ζ−2 = (0, ζ−2

3 ), ζ−1 = (0, ζ−13 ),

and full deformation patterns ζ0, ζ1 such that

v−2 = (0, ζ−23 )

v−1 = (−X3∇ζ−23 , ζ−1

3 )

v0 = (ζ0 − X3∇ζ−1

3 , ζ03) + (0, P 2

b (X3)ζ−23 )

v1 = (ζ1 − X3∇ζ03, ζ1

3) + (P 3b (X3)∇ζ−2

3 ,

P 1m(X3) div ζ0

+ P 2b (X3)ζ−1

3 )

(8)

In the above formulae, ∇ = (∂1, ∂2) is the surface gra-dient on S, = ∂2

1 + ∂22 is the surface Laplacian and

div ζ is the surface divergence (i.e. div ζ = ∂1ζ1 + ∂2ζ2).The functions P

b and P m are polynomials of degree ,

whose coefficients depend on the Lame constants accordingto

P 1m(X3) = − λ

λ + 2µX3,

P 2b (X3) =

λ

2λ + 4µ

(X2

3 −1

3

),

P 3b (X3) =

1

6λ + 12µ

((3λ + 4µ) X3

3 − (11λ + 12µ) X3

)(9)

Note that the first blocks in∑

k≥−2 εkvk yield Kirchhoffdisplacements, whereas the second blocks have zero meanvalues through the thickness for each x ∈ S.

2.2.3 All regular terms with the help of formal series

We see from (8) that the formulae describing the successivevk are partly self-similar and, also, that each vk is enrichedby a new term. That is why the whole regular term series∑

k εkvk can be efficiently described with the help of theformal series product.

A formal series is an infinite sequence (a0, a1, . . . , ak,

. . .) of coefficients, which can be denoted in a symbolicway by a[ε] =∑k≥0 εkak, and the product a[ε]b[ε] of thetwo formal series a[ε] and b[ε] is the formal series c[ε] withcoefficients c =∑0≤k≤ akb−k . In other words, the equa-tion c[ε] = a[ε]b[ε] is equivalent to the series of equationc =∑0≤k≤ akb−k , ∀.

With this formalism, we have the following identity,which extends formulae (8):

v[ε] = V[ε]ζ[ε] + Q[ε]f[ε] (10)

(i) ζ[ε] is the formal series of Kirchhoff deformationpatterns

∑k≥−2 εkζk starting with k = −2 .

(ii) V[ε] has operator valued coefficients Vk , k ≥ 0, actingfrom C∞(S)3 into C∞()3:

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V0ζ = (ζ, ζ3)

V1ζ = (−X3∇ζ3, P 1m(X3) div ζ)

V2ζ = (P 2m(X3)∇ div ζ, P 2

b (X3)ζ3)

. . .

V2jζ = (P2jm (X3)∇

j−1 div ζ, P

2jb (X3)

j

ζ3)

V2j+1ζ = (P2j+1b (X3)∇

j

ζ3,

P2j+1m (X3)

j

div ζ)(11)

with P b and P

m polynomials of degree (the firstones are given in (9)).

(iii) f[ε] is the Taylor series of f around the surface x3 = 0:

f[ε] =∑k≥0

εkf k with f k(x, X3) =Xk

3

k!

∂kf

∂xk3

∣∣∣x3=0

(x)

(12)

(iv) Q[ε] has operator valued coefficients Qk acting fromC∞()3 into itself. It starts at k = 2 (we can seenow that the four first equations given by equality(10) are v−2 = V0ζ−2, v−1 = V0ζ−1 + V1ζ−2, v0 =V0ζ0 + V1ζ−1 + V2ζ−2, v1 = V0ζ1 + V1ζ0 + V2ζ−1

+ V3ζ−2, which gives back (8))

Q[ε] =∑k≥2

εkQk (13)

Each Qk is made of compositions of partial derivatives inthe surface variables x with integral operators in the scaledtransverse variable. Each of them acts in a particular waybetween semipolynomial spaces Eq(), q ≥ 0, in the scaleddomain : We define for any integer q, q ≥ 0

Eq() =

v ∈ C∞()3, ∃zn ∈ C∞(S)3, v(x, X3)

=q∑

n=0

Xn3zn(x)

(14)

Note that by (12), f k belongs to Ek().Besides, for any k ≥ 2, Qk acts from Eq() into

Eq+k(). The first term of the series Q[ε]f[ε] is Q2f 0 andwe have:

Q2f 0(x, X3) =(

0,1 − 3X2

3

6λ + 12µf03(x)

)As a consequence of formula (10), combined with thestructure of each term, we find

Lemma 1. (Dauge and Schwab, 2002) With the definition(14) for the semipolynomial space Eq(), for any k ≥ −2the regular term vk belongs to Ek+2().

2.2.4 Deformation patterns

From formula (8) extended by (10) we obtain explicitexpressions for the regular parts vk provided we know thedeformation patterns ζk . The latter solves boundary valueproblems on the midsurface S. Our multiscale expansionapproach gives back the well-known equations of plates(the Kirchhoff–Love model and the plane stress model)completed by a whole series of boundary value problems.

(i) The first bending generator ζ−23 solves the Kirch-

hoff–Love model

Lbζ−23 (x) = f03(x), x ∈ S with ζ−2

3

∣∣∂S

= 0,

∂nζ−23

∣∣∂S

= 0 (15)

where Lb is the fourth-order operator

Lb := 4µ

3

λ + µ

λ + 2µ2

= 1

3(λ + 2µ) 2

(16)

and n the unit interior normal to ∂S. Here λ is the‘averaged’ Lame constant

λ = 2λµ

λ + 2µ(17)

(ii) The second bending generator ζ−13 is the solution of

a similar problem

Lbζ−13 (x) = 0, x ∈ S with ζ−1

3

∣∣∂S

= 0,

∂nζ−13

∣∣∂S

= cbλ,µζ−2

3 (18)

where cbλ,µ is a positive constant depending on the

Lame coefficients.(iii) The membrane part ζ0

of the third deformationpattern solves the plane stress model

Lmζ0(x) = f 0

(x), x ∈ S and ζ0|∂S = 0

(19)

where Lm is the second-order 2 × 2 system(ζ1ζ2

)−−−→

−(

(λ + 2µ)∂11 + µ∂22 (λ + µ)∂12(λ + µ)∂12 µ∂11 + (λ + 2µ)∂22

)×(

ζ1ζ2

)(20)

(iv) Here, again, the whole series of equations over theseries of deformation patterns

∑k≥−2 εkζk can be

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written in a global way using the formal series prod-uct, as reduced equations on the midsurface:

L[ε]ζ[ε] = R[ε]f[ε] in S with d[ε]ζ[ε] = 0 on ∂S

(21)

Here, L[ε] = L0 + ε2L2 + ε4L4 + · · ·, with

L0ζ =(

Lm 00 0

)(ζζ3

)L2ζ =

(L2

m 00 Lb

)(ζζ3

), . . . (22)

where L2mζ has the form c∇ div ζ. The series

of operators R[ε] starts at k = 0 and acts fromC∞()3 into C∞(S)3. Its first coefficient is the mean-value operator

f → R0f with R0f(x) = 1

2

∫ 1

−1f(x, X3) dX3

(23)

Finally, the coefficients of the operator series d[ε] are traceoperators acting on ζ. The first terms are

d0ζ =

ζ · nζ × n

00

, d1ζ =

−cm

λ,µ div ζ000

,

d2ζ =

••ζ3

∂nζ3

, d3ζ =

••0

−cbλ,µζ3

(24)

where cbλ,µ is the constant in (18), cm

λ,µ is another positiveconstant and • indicates the presence of higher orderoperators on ζ.

Note that the first three equations in (21): L0ζ−2 =0, L0ζ−1 = 0, L0ζ0 + L2ζ−2 = R0f 0 on S and d0ζ−2 =0, d0ζ−1 + d1ζ−1 = 0, d0ζ0 + d1ζ−1 + d2ζ−2 = 0 on ∂S,give back (15), (18), and (19) together with the fact thatζ−2 = ζ−1

= 0.

2.2.5 Boundary layer terms

The terms wk have a quite different structure. Their nat-ural variables are (R, s, X3), see Section 2.1 and Fig. 3,

∂S

×

Σ+

+1

−1

R = rε

Figure 3. Boundary layer coordinates in ∂S × +.

and they are easier to describe in boundary fitted compo-nents (wr, ws, w3) corresponding to the local coordinates(r, s, x3). The first boundary layer term, w0 is a bendingdisplacement in the sense of (5) and has a tensor productform: In boundary fitted components it reads

w0s = 0 and (w0

r , w03)(R, s, X3) = ϕ(s) w 0

∗ (R, X3)

with ϕ = ζ−23

∣∣∂S

and w 0∗ is a two component exponentially decreasing profile

on the semi-strip + := (R, X3), R > 0, |X3| < 1: Thereexists η > 0 such that

|eηR w 0∗ (R, X3)| is bounded as R → ∞

The least upper bound of such η is the smallest exponentη0 arising from the Papkovich–Fadle eigenfunctions; seeGregory and Wan (1984). Both components of w 0

∗ arenonzero.

The next boundary layer terms wk are combinations ofproducts of (smooth) traces on ∂S by profiles w k, in(R, X3). These profiles have singularities at the corners(0,±1) of +, according to the general theory of Kon-drat’ev (1967). Thus, in contrast with the ‘regular’ termsvk , which are smooth up to the boundary of , the termswk do have singular parts along the edges ∂S × ±1 ofthe plate. Finally, the edge singularities of the solution uε

of problem (3) are related with the boundary layer termsonly; see Dauge and Gruais (1998a) for further details.

2.3 Properties of the displacement expansionoutside the boundary layer

Let S ′ be a subset of S such that the distance between ∂S′and ∂S is positive. As a consequence of expansion (6) thereholds

uε(x) =K∑

k=−2

εkvk(x, X3) +O(εK+1)

uniformly for x ∈ S ′ × (−ε, ε)

Coming back to physical variables (x, x3), the expan-sion terms vk being polynomials of degree k + 2 in X3(Lemma 1), we find that

uε(x) =K∑

k=−2

εk vK,k(x, x3) +O(εK+1)

uniformly for x ∈ S ′ × (−ε, ε)

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with fields vK,k being polynomials in x3 of degree K − k.This means that the expansion (6) can also be seen as aTaylor expansion at the midsurface, provided we are at afixed positive distance from the lateral boundary.

Let us write the first terms in the expansions of thebending and membrane parts uε

b and uεm of uε:

uεb = ε−2

(−x3∇ζ−2

3 , ζ−23 + λx2

3

2λ + 4µζ−2

3

)

−(

0,λ

6λ + 12µζ−2

3

)+ ε−1

(−x3∇ζ−1

3 ,

ζ−13 + λx2

3

2λ + 4µζ−1

3

)+ · · · (25)

From this formula, we can deduce the following asymp-totics for the strain and stress components

eαβ(uεb) = −ε−2x3∂αβ

(ζ−2

3 + εζ−13

)+O(ε)

e33(uεb) = ε−2 λx3

λ + 2µ(ζ−2

3 + εζ−13

)+O(ε) (26)

σ33(uεb) = O(ε)

Since ε−2x3 = O(ε−1), we see that e33 = O(ε−1). Thus, σ33

is two orders of magnitude less than e33, which means aplane stress limit. To compute the shear strain (or stress),we use one further term in the asymptotics of uε

b and obtainthat it is one order of magnitude less than e33:

eα3(uεb) =

2λ + 2µ

λ + 2µ(ε−2x2

3 − 1)∂αζ−23 +O(ε) (27)

Computations for the membrane part uεm are simpler and

yield similar results

uεm =

(ζ0, − λx3

λ + 2µdiv ζ0

)+ ε

(ζ1, − λx3

λ + 2µdiv ζ1

)+ · · ·

eαβ(uεm) = 1

2(∂αζ

0β + ∂βζ

0α) +

ε

2(∂αζ

1β + ∂βζ

1α) +O(ε2)

e33(uεm) = − λ

λ + 2µdiv(ζ0 + εζ1

)+O(ε2)

(28)

and σ33(uεm) = O(ε2), eα3(u

εm) = O(ε).

In (26)–(28) the O(ε) and O(ε2) are uniform on anyregion S

′ × (−ε, ε) where the boundary layer terms haveno influence. We postpone global energy estimates to thenext section.

2.4 Eigen-mode problem

For each ε > 0, the spectrum of problem (4) is discrete andpositive. Let ε

j , j = 1, 2, . . . be the increasing sequenceof eigenvalues. In Ciarlet and Kesavan (1981) it is provedthat ε−2ε

j converges to the j th eigenvalue KLb,j of the

Dirichlet problem for the Kirchhoff operator Lb, cf. (16). InNazarov and Zorin (1989) and Nazarov (1991c), a two-termasymptotics is constructed for the ε−2ε

j . Nazarov (2000b)proves that |ε−2ε

j − KLb,j | is bounded by an O(

√ε) for a

much more general material matrix A.In Dauge et al. (1999), full asymptotic expansions for

eigenvalues and eigenvectors are proved: For each j thereexist

• bending generators ζ−23 , ζ−1

3 , . . . where ζ−23 is an eigen-

vector of Lb associated with KLb,j

• real numbers 1b,j , 2

b,j , . . .

• eigenvectors uεb,j associated with ε

j for any ε ∈(0, ε0)

so that for any K ≥ 0

εj = ε2KL

b,j + ε31b,j + · · · + εK+2K

b,j +O(εK+3)

uεb,j = ε−2(−x3∇ζ−2

3 , ζ−23 ) + ε−1(−x3∇ζ−1

3 , ζ−13 )

+ · · · + εK(vK+ χwK) +O(εK+1) (29)

where the terms vk and wk are generated by the ζk3, k ≥ 0

in a similar way as in Section 2.2, and O(εK+1) is uniformover ε.

The bending and membrane displacements are the eigen-vectors of the symmetry operator S; see (5). Since S

commutes with the elasticity operator, both have a jointspectrum, which means that there exists a basis of commoneigenvectors. In other words, each elasticity eigenvalue canbe identified as a bending or a membrane eigenvalue. Theexpansion (29) is the expansion of bending eigen-pairs.

The expansion of membrane eigen-pairs can be done ina similar way. Let us denote by ε

m,j the j th membraneeigenvalue on ε and by KL

m,j the j th eigenvalue of theplane stress operator Lm, cf. (20) with Dirichlet boundaryconditions. Then we have a similar statement as above, withthe distinctive feature that the membrane eigenvalues tendto those of the plane stress model:

εm,j = KL

m,j + ε11m,j + · · · + εKK

m,j +O(εK+1) (30)

This fact, compared with (29), explains why the smallesteigenvalues are bending. Note that the eigenvalue formalseries [ε] satisfy reduced equations L[ε]ζ[ε] = [ε]ζ[ε]like (21) with the same L0, L1 = 0 and L2 as in (22). In

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particular, equations(Lm 00 ε2Lb

)(ζζ3

)=

(ζζ3

)(31)

give back the ‘limiting’ eigenvalues KLm and ε2KL

b . Ourlast remark is that the second terms 1

b,j and 1m,j are

positive; see Dauge and Yosibash (2002) for a discussionof that fact.

2.5 Extensions

2.5.1 Traction on the free parts of the boundary

Instead of a volume load, or in addition to it, tractions g±can be imposed on the faces S × ±ε of the plate. Let usassume that g± is independent of ε. Then the displacementuε has a similar expansion as in (6), with the followingmodifications:

• If the bending part of g± is nonzero, then the regularpart starts with ε−3v−3 and the boundary layer partwith ε−1χw−1;

• If the membrane part of g± is nonzero, the membraneregular part starts with ε−1v−1.

2.5.2 Lateral boundary conditions

A similar analysis holds for each of the seven remainingtypes of ‘canonical’ boundary conditions: soft clamping,hard simple support, soft simple support, two types of fric-tion, sliding edge, and free boundary. See Dauge, Gruaisand Rossle (1999/00) for details. It would also be possibleto extend such an analysis to more intimately mixed bound-ary conditions where only moments through the thicknessalong the lateral boundary are imposed for displacement ortraction components; see Schwab (1996).

If, instead of volume load f or tractions g±, we set f ≡ 0,g± ≡ 0, and impose nonzero lateral boundary conditions, uε

will have a similar expansion as in (6) with the remarkablefeature that the degree of the regular part in the thicknessvariable is ≤ 3; see Dauge and Schwab (2002), Rem. 5.4.Moreover, in the clamped situation, the expansion startswith O(1).

2.5.3 Laminated composites

If the material of the plate is homogeneous, but notisotropic, uε will still have a similar expansion; see Daugeand Gruais (1996) and Dauge and Yosibash (2002) fororthotropic plates. If the plate is laminated, that is, formedby the union of several plies made of different homoge-neous materials, then uε still expands in regular parts vk and

boundary layer parts wk , but the vk are no more polynomialsin the thickness variable, only piecewise polynomial in eachply, and continuous; see Actis, Szabo and Schwab (1999).Nazarov (2000a, 2000b) addresses more general materiallaws where the matrix A depends on the variables x andX3 = x3/ε.

3 HIERARCHICAL MODELS FORPLATES

3.1 The concepts of hierarchical models

The idea of hierarchical models is a natural and efficientextension to that of limiting models and dimension reduc-tion. In the finite element framework, it has been firstlyformulated in Szabo and Sahrmann (1988) for isotropicdomains, mathematically investigated in Babuska and Li(1991, 1992a, 1992b), and generalized to laminated com-posites in Babuska, Szabo and Actis (1992) and Actis,Szabo and Schwab (1999). A hierarchy of models consistsof

• a sequence of subspaces V q(ε) of V (ε) with theorders q = (q1, q2, q3) forming a sequence of integertriples, satisfying

V q(ε) ⊂ V q′(ε) if q q′ (32)

• a sequence of related Hooke laws σ = Aqe, cor-responding to a sequence of elastic bilinear formsaε,q(u, u′) = ∫

ε Aqe(u) : e(u′).

Let uε,q be the solution of the problem

Find uε,q ∈ V q(ε) such that

aε,q(uε,q, u′) =∫

ε

fε · u′ dx, ∀u′ ∈ V q(ε) (33)

Note that problem (33) is a Galerkin projection of problem(3) if aε,q = aε.

Any model that belongs to the hierarchical family has tosatisfy three requirements; see Szabo and Babuska (1991),Chap. 14.5:

(a) Approximability. At any fixed thickness ε > 0:

limq→∞‖uε − uε,q‖

E(ε)= 0 (34)

(b) Asymptotic consistency. For any fixed degree q:

limε→0

‖uε − uε,q‖E(ε)

‖uε‖E(ε)

= 0 (35)

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(c) Optimality of the convergence rate. There exists asequence of positive exponents γ(q) with the growthproperty γ(q) < γ(q′) if q ≺ q′, such that ‘in theabsence of boundary layers and edge singularities’:


≤ Cεγ(q)‖uε‖E(ε)

(36)

The substantiation of hierarchical models for plates, ingeneral, requires the choice of three sequences of finitedimensional nested director spaces 0

j ⊂ · · ·Nj ⊂ · · · ⊂

H 1(−1, 1) for j = 1, 2, 3 and the definition of the spaceV q(ε) for q = (q1, q2, q3) as

V q(ε) =

u ∈ V (ε),((x, X3) → uj (x, εX3)

)∈ H 1

0 (S) ⊗ qj

j , j = 1, 2, 3

(37)

We can reformulate (37) with the help of director functions:With dj (N) being the dimension of N

j , let nj = n

j (X3),0 ≤ n ≤ dj (N), be hierarchic bases (the director functions)of N

j . There holds

V q(ε) =

u ∈ V (ε), ∃znj ∈ H 1

0 (S), 0 ≤ n ≤ dj (qj ),

uj (x, x3) =dj (qj )∑n=0

znj (x)n

j

(x3

ε

)(38)

The choice of the best director functions is addressed inVogelius and Babuska (1981c) in the case of second-orderscalar problems with general coefficients (including possi-ble stratifications). For smooth coefficients, the space N

j

coincides with the space PN of polynomial with degree ≤N . The director functions can be chosen as the Legendrepolynomials Ln(X3) or, simply, the monomials Xn

3 (and thenxn

3 can be used equivalently instead (x3/ε)n in (38)).

We describe in the sequel in more detail, the conve-nient hierarchies for plates and discuss the three qualities(34)–(36); see Babuska and Li (1991, 1992a) and Babuska,Szabo and Actis (1992) for early references.

3.2 The limit model (Kirchhoff–Love)

In view of expansion (6), we observe that if the transversecomponent f3 of the load is nonzero on the midsurface,uε is unbounded as ε → 0. If we multiply by ε2, wehave a convergence to (0, ζ−2

3 ), which is not kinematicallyrelevant. At that level, a correct notion of limit uses scalingsof coordinates: If we define the scaled displacement uε byits components on the stretched plate = S × (−1, 1) by

uε := εuε

and uε3 := ε2uε

3 (39)

then uε converges to (−X3∇ζ−23 , ζ−2

3 ) in H 1()3 asε → 0. This result, together with the mathematical deriva-tion of the resultant equation (15), is due to Ciarlet andDestuynder (1979a).

The corresponding subspace of V (ε) is that of bendingKirchhoff displacements or, more generally, of Kirchhoffdisplacements:

V KL(ε) = u ∈ V (ε), ∃ζ ∈ H 10 × H 1

0 × H 20 (S),

u = (ζ − x3∇ζ3, ζ3) (40)

It follows from (40) that e13 = e23 = 0 for which the phys-ical interpretation is that ‘normals to S prior to deforma-tion remain straight lines and normals after deformation’.Hooke’s law has to be modified with the help of what wecall ‘the plane stress trick’. It is based on the assumptionthat the component σ33 of the stress is negligible (notethat the asymptotics (6) of the three-dimensional solutionyields that σ33 = O(ε), whereas eαβ, e33 = O(ε−1) outsidethe boundary layer, cf. (26), which justifies the plane stressassumption). From standard Hooke’s law (1), we extractthe relation σ33 = λ(e11 + e22) + (λ + 2µ)e33, then set σ33

to zero, which yields

e33 = − λ

λ + 2µ(e11 + e22) (41)

Then, we modify Hooke’s law (1) by substituting e33 by itsexpression (41) in σ11 and σ22, to obtain

σii = 2λµ

λ + 2µ(e11 + e22) + 2µeii, i = 1, 2

σij = 2µeij for i = j

(42)

Thus, σii = λ(e11 + e22) + 2µeii , with λ given by (17).Taking into account that e33 = 0 for the elements ofV KL(ε), we obtain a new Hooke’s law given by the sameformulae as (1) when replacing the Lame coefficient λ byλ. This corresponds to a modified material matrix Aijkl

Aijkl = λδijδkl + µ(δikδj l + δilδjk) (43)

and a reduced elastic energy a(u, u) = ∫ε σij (u)eij (u).

Note that for u = (ζ − x3∇ζ3, ζ3)

a(u, u) = 2ε

∫S

Aαβσδeαβ(ζ)eσδ(ζ) dx

+ 2ε3

3

∫S

Aαβσδ∂αβ(ζ3)∂σδ(ζ3) dx (44)

exhibiting a membrane part in O(ε) and a bending part inO(ε3). There hold as a consequence of Theorem 1

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Theorem 2. Let uε,KL be the solution of problem (33) withV q = V KL and aq = a. Then

(i) In general uε,KL = ε−2(−x3∇ζ−2

3 , ζ−23

)+O(1) withζ−2

3 the solution of (15);(ii) If f is membrane, uε,KL = (ζ0

, 0)+O(ε2) with ζ0

thesolution of (19).

Can we deduce the asymptotic consistency for thatmodel? No! Computing the lower-order terms in the expres-sion (35), we find with the help of (25) that, if f03 ≡ 0

‖uε‖E(ε)

O(ε−1/2)

and‖uε − uε,KL‖

E(ε)≥ ‖e33(u

ε)‖L2(ε)

O(ε−1/2)

Another source of difficulty is that, eventually, relation (41)is not satisfied by uε,KL. If f03 ≡ 0 and f 0

≡ 0, we haveexactly the same difficulties with the membrane part.

A way to overcome these difficulties is to considera complementing operator C defined on the elements ofV KL by

Cu = u +(

0,− λ

λ + 2µ

∫ x3

0div u(·, y) dy

)(45)

Then (41) is now satisfied by Cu for any u ∈ V KL. More-over (still assuming f3 ≡ 0), one can show

‖uε − Cuε,KL‖E(ε)

≤ C√

ε‖uε‖E(ε)

(46)

The error factor√

ε is due to the first boundary layer termw0. The presence of w0 is a direct consequence of the factthat Cuε,KL does not satisfy the lateral boundary conditions.

Although the Kirchhoff–Love model is not a memberof the hierarchical family, it is the limit of all models forε → 0.

3.3 The Reissner–Mindlin model

This model is obtained by enriching the space of kinemat-ically admissible displacements, allowing normals to S torotate after deformation. Instead of (40), we set

V RM(ε) = u ∈ V (ε), ∃z ∈ H 10 (S)3, ∃θ ∈ H 1

0 (S)2,

u = (z − x3θ, z3)

With the elasticity tensor A corresponding to 3-D elasticity,the displacements and strain–energy limit of the RM modelas d → 0 would not coincide with the 3-D limit (or theKirchhoff–Love limit).

We have again to use instead the reduced elastic bilinearform a to restore the convergence to the correct limit, byvirtue of the same plane stress trick. The correspondingelasticity tensor is A (43). A further correction can beintroduced in the shear components of A to better representthe fully 3-D shear stresses σ13 and σ23 (and also the strainenergy) for small yet nonzero thickness ε. The materialmatrix entries A1313, A2323 are changed by introducing theso-called shear correction factor κ:

A1313 = κA1313 A2323 = κA2323

By properly chosen κ, either the energy of the RM solution,or the deflection u3 can be optimized with respect to thefully 3-D plate. The smaller the ε, the smaller the influenceof κ on the results. For the isotropic case, two possibleκ’s are (see details in Babuska, d’Harcourt and Schwab(1991a)):

κEnergy = 5

6(1 − ν)or κDeflection = 20

3(8 − 3ν),

with ν = λ

2(λ + µ)(Poisson ratio)

A value of κ = 5/6 is frequently used in engineeringpractice, but for modal analysis, no optimal value of κ isavailable.

Note that, by integrating equations of (33) through thethickness, we find that problem (33) is equivalent to avariational problem for z and θ only. For the elastic energy,we have

a(u, u) = 2ε

∫S

Aαβσδeαβ(z)eσδ(z) dx(membrane energy)

+ ε

∫S

κµ(∂αz3 − θα)(∂αz3 − θα) dx(shear energy)

+ 2ε3

3

∫S

Aαβσδeαβ(θ)eσδ(θ) dx(bending energy)

(47)

Let uε,RM be the solution of problem (33) with V q = V RM

and aq = a. The singular perturbation character appearsclearly. In contrast with the Kirchhoff–Love model, thesolution admits a boundary layer part. Arnold and Falk(1990b, 1996) have described the two-scale asymptotics ofuε,RM. Despite the presence of boundary layer terms, thequestion of knowing if uε,RM is closer to uε than uε,KL hasno clear answer to our knowledge. A careful investigationof the first eigenvalues ε

1, ε,KL1 , and

ε,RM1 of these three

models in the case of lateral Dirichlet conditions showsthe following behavior for ε small enough (Dauge andYosibash, 2002):

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ε,RM1 <

ε,KL1 < ε

1

which tends to prove that RM model is not genericallybetter than KL for (very) thin plates. Nevertheless, anestimate by the same asymptotic bound as in (46) is validfor uε − Cuε,RM.

3.4 Higher order models

The RM model is a (1, 1, 0) model with reduced elasticenergy. For any q = (q, q, q3) we define the space V q

by (compare with (38) for monomial director functions)

V q(ε) =

u ∈ V (ε), ∃zn ∈ H 1

0 (S)2, 0 ≤ n ≤ q,

∃zn3 ∈ H 1

0 (S), 0 ≤ n ≤ q3

u =q∑

n=0

xn3 zn

(x) and u3 =q3∑

n=0

xn3 zn

3(x)

(48)

The subspaces Vq

b and Vq

m of bending and membranedisplacements in V q can also be used, according to thenature of the data. The standard 3-D elastic energy (2) isused with V q and V

qb for any q (1, 1, 2) and with V

qm for

any q (0, 0, 1).

Theorem 3.

(i) If f satisfies f3∣∣S≡ 0, for any q (1, 1, 2) there exists

Cq = Cq(f) > 0 such that for all ε ∈ (0, ε0)


≤ Cq

√ε ‖uε‖

E(ε)(49)

(ii) If f is membrane and f∣∣S≡ 0, for any q (0, 0, 1)

there exists Cq = Cq(f) > 0 such that for all ε ∈ (0, ε0)

(49) holds.

Proof. Since the energy is not altered by the model, uε,q

is a Galerkin projection of uε on V q(ε). Since the strainenergy is uniformly equivalent to the elastic energy on anyε, we have by Cea’s lemma that there exists C > 0


≤ C‖uε − vq‖E(ε)

∀vq ∈ V q(ε)

(i) We choose, compare with (25),

vq = ε−2

(−x3∇ζ−2

3 , ζ−23 + λx2

3

2λ + 4µζ−2

3

)

− ε−2λx23

2λ + 4µϕ(s)

(0, ξ(R)

)

with ϕ = ζ−23

∣∣∂S

and ξ a smooth cut-off functionequal to 1 in a neighborhood of R = 0 and 0 forR ≥ 1. Then vq satisfies the lateral boundary condi-tions and we can check (49) by combining Theorem 1with the use of Cea’s lemma.

(ii) We choose, instead

vq =(ζ0,− λx3

λ + 2µdiv ζ0

)+ λx3

λ + 2µϕ(s)

(0, ξ(R)

)with ϕ = div ζ0

∣∣∂S

It is worthwhile to mention that for the (1, 1, 2) modelthe shear correction factor (when ν → 0, κ(1,1,2) tends to5/6, just like for the two shear correction factors of the RMmodel)

κ(1,1,2) =12 − 2ν

ν2

−1 +√

1 + 20ν2

(12 − 2ν)2

can be used for optimal results in respect with the error inenergy norm and deflection for finite thickness plates; seeBabuska, d’Harcourt and Schwab (1991a). For higher platemodels, no shear correction factor is furthermore needed.

The result in Schwab and Wright (1995) regarding theapproximability of the boundary layers by elements ofV q, yields that the constant Cq in (49) should rapidlydecrease when q increases. Nevertheless the factor

√ε is

still present, for any q, because of the presence of theboundary layer terms. The numerical experiments in Daugeand Yosibash (2000) demonstrate that the higher the degreeof the hierarchical model, the better the boundary layerterms are approximated.

If one wants to have an approximation at a higher orderin ε one should

• either consider a problem without boundary layer, asmentioned in requirement (c) (36), that is, a rectan-gular plate with symmetry boundary conditions: Inthis case, the convergence rate γ(q) in ε is at leastminj qj − 1,

• or combine a hierarchy of models with a three-dimensional discretization of the boundary layer; seeStein and Ohnimus (1969) and Dauge and Schwab(2002).

The (1, 1, 2) is the lowest order model which is asymp-totically consistent for bending. See Paumier and Raoult(1997) and Rossle et al. (1999). It is the first model in thebending model hierarchy

(1, 1, 2), (3, 3, 2), (3, 3, 4), . . .

(2n − 1, 2n − 1, 2n), (2n + 1, 2n + 1, 2n), . . .

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The exponent γ(q) in (36) can be proved to be 2n − 1if q = (2n − 1, 2n − 1, 2n) and 2n if q = (2n + 1, 2n +1, 2n), thanks to the structure of the operator series V[ε]and Q[ε] in (11). If the load f is constant over the wholeplate, then the model of degree (3, 3, 4) captures the wholeregular part of uε, (Dauge and Schwab (2002), Rem. 8.3)and if, moreover, f ≡ 0 (in this case, only a lateral boundarycondition is imposed), the degree (3, 3, 2) is sufficient.

3.5 Laminated plates

If the plate is laminated, the material matrix A = Aε hasa sandwich structure, depending on the thickness variablex3: We assume that Aε(x3) = A(X3), where the coeffi-cients of A are piecewise constant. In Nazarov (2000a)the asymptotic analysis is started, including such a sit-uation. We may presume that a full asymptotic expan-sion like (6) with a similar internal structure, is stillvalid.

In the homogeneous case, the director functions in (38)are simply the monomials of increasing degrees; see (48).In the laminated case, the first director functions are still 1and x3:

01 = 0

2 = 03 = 1; 1

1 = 12 = x3

In the homogeneous case, we have 13 = x3 and 2

j = x23,

j = 1, 2, 3. In Actis, Szabo and Schwab (1999) three morepiecewise linear director functions and three piecewisequadratic director functions are exhibited for the laminatedcase.

How many independent director functions are necessaryto increase the convergence rate γ(q) (36)? In other words,what is the dimension of the spaces

qj

j (cf. (37))? In ourformalism, see (10)–(11), this question is equivalent toknowing the structure of the operators Vj . Comparing withNazarov (2000a), we can expect that

V1ζ =(− X3∇ζ3, P

1,13 (X3)∂1ζ1 + P

1,23 (X3)(∂1ζ2 + ∂2ζ1)

+ P1,33 (X3)∂2ζ2

)V2ζ =

( 3∑k=1

P2,k,1j (X3)∂

21 ζk + P

2,k,2j (X3)∂

212ζk

+ P2,k,3j (X3)∂

22 ζk

)j=1,2,3

(50)

As soon as the above functions Pn,∗j are independent,

they should be present in the bases of the director spacen

j . The dimensions of the spaces generated by the Pn,∗j

have upper bounds depending only on n. But their actualdimensions depend on the number of plies and theirnature.

4 MULTISCALE EXPANSIONS ANDLIMITING MODELS FOR SHELLS

Up to now, the only available results concerning multiscaleexpansions for ‘true’ shells concern the case of clampedelliptic shells investigated in Faou (2001a, 2001b, 2003).For (physical) shallow shells, which are closer to platesthan shells, multiscale expansions can also be proved; seeNazarov (2000a) and Andreoiu and Faou (2001).

In this section, we describe the results for clamped ellipticshells, then present the main features of the classificationof shells as flexural and membrane. As a matter of fact,multiscale expansions are known for the most extremerepresentatives of the two types: (i) plates for flexuralshells, (ii) clamped elliptic shells for membrane shells.Nevertheless, multiscale expansions in the general caseseem out of reach (or, in certain cases, even irrelevant).

4.1 Curvature of a midsurface and otherimportant tensors

We introduce minimal geometric tools, namely, the metricand curvature tensors of the midsurface S, the change ofmetric tensor γαβ, and the change of curvature tensor ραβ.We also address the essential notions of elliptic, hyperbolic,or parabolic point in a surface. We make these notions moreexplicit for axisymmetric surfaces. A general introductionto differential geometry on surfaces can be found in Stoker(1969).

Let us denote by 〈X, Y 〉R3 the standard scalar product

of two vectors X and Y in R3. Using the fact that the

midsurface S is embedded in R3, we naturally define the

metric tensor (aαβ) as the projection on S of the standardscalar product in R

3: Let p be a point of S and X, Y ,two tangent vectors to S in p. In a coordinate systemx = (xα) on S, the components of X and Y are (Xα) and(Y α), respectively. Then the matrix

(aαβ(x)

)is the only

positive definite symmetric 2 × 2 matrix such that for allsuch vectors X and Y⟨

X, Y⟩R3 = aαβ(x)XαY β =:

⟨X, Y

⟩S

The inverse of aαβ is written aαβ and thus satisfies aαβaβσ =δασ, where δα

σ is the Kronecker symbol and where weused the repeated indices convention for the contractionof tensors.

The covariant derivative D is associated with the metricaαβ as follows: It is the unique differential operator suchthat D〈X, Y 〉S = 〈DX, Y 〉S + 〈X, DY 〉S for all vector fieldsX and Y . In a local coordinate system, we have

Dα = ∂α + terms of order 0

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where ∂α is the derivative with respect to the coordinatexα. The terms of order 0 do depend on the choice of thecoordinate system and on the type of the tensor field onwhich D is applied. They involve the Christoffel symbolsof S in the coordinate system (xα).

The principal curvatures at a given point p ∈ S can beseen as follows: We consider the family P of planes P

containing p and orthogonal to the tangent plane to S atp. For P ∈ P, P ∩ S defines a curve in P and we denoteby κ its signed curvature κ. The sign of κ is determinedby the orientation of S. The principal curvatures κ1 andκ2 are the minimum and maximum of κ when P ∈ P. Theprincipal radii of curvature are Ri := |κi |−1. The Gaussiancurvature of S in p is K(p) = κ1κ2.

A point p is said to be elliptic if K(p) > 0, hyperbolicif K(p) < 0, parabolic if K(p) = 0 but κ1 or κ2 isnonzero, and planar if κ1 = κ2 = 0. An elliptic shell isa shell whose midsurface is everywhere elliptic up tothe boundary (similar definitions hold for hyperbolic andparabolic shells. . . and planar shells that are plates).

The curvature tensor is defined as follows: Let : x →(x) be a parameterization of S in a neighborhood of agiven point p ∈ S and n((x)) be the normal to S in(x). The formula

bαβ :=⟨n((x)) ,

∂

∂xα∂xβ

(x)⟩R3

defines, in the coordinate system x = (xα), the compo-nents of a covariant tensor field on S, which is called thecurvature tensor.

The metric tensor yields diffeomorphisms between tensorspaces of different types (covariant and contravariant): Wehave, for example, b

βα = aασbσβ. With these notations, we

can show that in any coordinate system, the eigenvalues ofbα

β at a point p are the principal curvatures at p.In the special case where S is an axisymmetric surface

parametrized by

: (x1, x2) → (x1 cos x2, x1 sin x2, f (x1)) ∈ R3 (51)

where x1 ≥ 0 is the distance to the axis of symmetry,x2 ∈ [0, 2π[ is the angle around the axis, and f : R → R

a smooth function, we compute directly that

(bαβ) =

1√1 + f ′(x1)

2

f ′′(x1)

1 + f ′(x1)2

0

0f ′(x1)

x1

whence

K = f ′(x1)f′′(x1)

x1(1 + f ′(x1)2)2

A deformation pattern is a three-component field ζ =(ζα, ζ3) where ζα is a surface displacement on S, andζ3 a function on S. The change of metric tensor γαβ(ζ)

associated with the deformation pattern ζ has the followingexpression:

γαβ(ζ) = 12 (Dαζβ + Dβζα) − bαβζ3 (52)

The change of curvature tensor associated with ζ writes

ραβ(ζ) = DαDβζ3 − bσαbσβζ3 + bσ

αDβζσ + Dαbσβζσ (53)

4.2 Clamped elliptic shells

The generic member ε of our family of shells is definedas

S × (−ε, ε) (p, x3) −−−→ p + x3 n(p) ∈ ε ⊂ R3

(54)

where n(p) is the normal to S at p. Now three stretchedvariables are required (cf. Section 2.1 for plates):

X3 = x3

ε, R = r

εand T = r√

ε

where (r, s) is a system of normal and tangential coordi-nates to ∂S in S.

4.2.1 Three-dimensional expansion

The solutions of the family of problems (3) have a three-scale asymptotic expansion in powers of ε1/2, with regularterms vk/2, boundary layer terms wk/2 of scale ε likefor plates, and new boundary layer terms ϕk/2 of scaleε1/2.

Theorem 4. (Faou, 2003) For the solutions of problems(3), there exist regular terms vk/2(x, X3), k ≥ 0, boundarylayer terms ϕk/2(T, s, X3), k ≥ 0 and wk/2(R, s, X3), k ≥ 2,such that

uε (v0 + χϕ0) + ε1/2(v1/2 + χϕ1/2) + ε(v1 + χϕ1

+ χw1) + · · · (55)

in the sense of asymptotic expansions: There holds thefollowing estimates∥∥∥∥∥uε −

2K∑k=0

εk/2(vk/2 + χϕk/2 + χwk/2)

∥∥∥∥∥E(ε)

≤ CK(f) εK+1/2, K = 0, 1, . . .

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where we have set w0 = w1/2 = 0 and the constant CK(f)is independent of ε ∈ (0, ε0].

Like for plates, the terms of the expansion are linked witheach other, and are generated by a series of deformationpatterns ζk/2 = ζk/2(x) of the midsurface S. They solvea recursive system of equations, which can be written in acondensed form as an equality between formal series, likefor plates. The distinction from plates is that, now, half-integer powers of ε are involved and we write, for example,ζ[ε1/2] for the formal series

∑k εk/2ζk/2.

4.2.2 Regular terms

The regular terms series v[ε1/2] =∑k εk/2vk/2 is deter-mined by an equation similar to (10):

v[ε1/2] = V[ε1/2]ζ[ε1/2] + Q[ε1/2]f[ε1/2]

(i) The formal series of deformation patterns ζ[ε1/2]starts with k = 0 (instead of degree −2 for plates).

(ii) The first terms of the series V[ε] are

V0ζ = ζ, V1/2 ≡ 0,

V1ζ = (−X3(Dαζ3 + 2bβαζβ), P 1

m(X3)γαα(ζ))

(56)

where P 1m is the polynomial defined in (9), and the

tensors D (covariant derivative), b (curvature), andγ (change of metric) are introduced in Section 4.1:Even if the displacement V1ζ is given through itscomponents in a local coordinate system, it indeeddefines an intrinsic displacement, since Dα, bα

β , andγα

β are well-defined independently of the choice of alocal parameterization of the surface. Note that γα

α(ζ)

in (56) degenerates to div ζ in the case of plateswhere bαβ = 0. More generally, for all integer k ≥ 0,all ‘odd’ terms Vk+1/2 are zero and, if b ≡ 0, alleven terms Vk degenerate to the operators in (11).In particular, their degrees are the same as in (11).

(iii) The formal series Q[ε1/2] appears as a generalizationof (13) and f[ε1/2] is the formal Taylor expansion of faround the midsurface x3 = 0, which means that forall integer k ≥ 0, fk+1/2 ≡ 0 and fk is given by (12).

4.2.3 Membrane deformation patterns

The first term ζ0 solves the membrane equation

ζ0 ∈ H 10 × H 1

0 × L2(S), ∀ ζ′ ∈ H 10 × H 1

0 × L2(S),

aS,m(ζ0, ζ′) = 2∫

S

ζ′ · f 0 (57)

where f 0 = f|S and aS,m is the membrane form

aS,m(ζ, ζ′) = 2∫

S

Aαβσδγαβ(ζ)γσδ(ζ′) dS (58)

with the reduced energy material tensor on the midsurface(with λ still given by (17)):

Aαβσδ = λaαβaσδ + µ(aασaβδ + aαδaβσ)

Problem (57) can be equivalently formulated as L0ζ0 = f 0

with Dirichlet boundary conditions ζ0 = 0 on ∂S and is

corresponding to the membrane equations on plates (com-pare with (19) and (22)). The operator L0 is called mem-brane operator and, thus, the change of metric γαβ(ζ) withrespect to the deformation pattern ζ appears to coincide withthe membrane strain tensor; see Naghdi (1963) and Koi-ter (1970a). If b ≡ 0, the third component of L0ζ vanisheswhile the surface part degenerates to the membrane opera-tor (20). In the general case, the properties of L0 dependson the geometry of S: L0 is elliptic (of multidegree (2, 2, 0)in the sense of Agmon, Douglis and Nirenberg, 1964) in xif and only if S is elliptic in x; see Ciarlet (2000), Genevey(1996), and Sanchez-Hubert and Sanchez-Palencia (1997).

As in (21), the formal series ζ[ε1/2] solves a reducedequation on the midsurface with formal series L[ε1/2],R[ε1/2], f[ε1/2] and d[ε1/2], degenerating to the formal series(21) in the case of plates.

4.2.4 Boundary layer terms

Problem (57) cannot solve for the boundary conditionsζ0

3|∂S = ∂nζ03|∂S = 0 (see the first terms in (24)). The two-

dimensional boundary layer terms ϕk/2 compensate thesenonzero traces: We have for k = 0.

ϕ0 = (0,ϕ03(T, s)) with ϕ0

3(0, s) = −ζ03|∂S

and ∂nϕ03(0, s) = 0

For k = 1, the trace ∂nζ03|∂S is compensated by ϕ

1/23 : The

scale ε1/2 arises from these surface boundary layer terms.More generally, the terms ϕk/2 are polynomials of degree[k/2] in X3 and satisfy

|eηTϕ(T, s, X3)| bounded as T → ∞

for all η < (3µ(λ + µ))1/4 (λ + 2µ)−1/2bss(0, s)1/2 wherebss(0, s) > 0 is the tangential component of the curvaturetensor along ∂S.

The three-dimensional boundary layer terms wk/2 havea structure similar to the case of plates. The first nonzeroterm is w1.

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4.2.5 The Koiter model

Koiter (1960) proposed the solution zε of following surfaceproblem

Find zε ∈ VK(S) such that

εaS,m(zε, z′) + ε3aS,b(zε, z′) = 2ε

∫S

z′ · f 0, ∀ z′ ∈ VK(S)

(59)

to be a good candidate for approximating the three-dimensional displacement by a two-dimensional one. Herethe variational space is

VK(S) := H 10 × H 1

0 × H 20 (S) (60)

and the bilinear form aS,b is the bending form:

aS,b(z, z′) = 2

3

∫S

Aαβσδραβ(z)ρσδ(z′) dS (61)

Note that the operator underlying problem (59) has the formK(ε) = εL0 + ε3B where the membrane operator L0 is thesame as in (57) and the bending operator B is associatedwith aS,b. Thus, the change of curvature tensor ραβ appearsto be identified with the bending strain tensor. Note thatK(ε) is associated with the two-dimensional energy (com-pare with (44))

2ε

∫S

Aαβσδγαβ(z)γσδ(z) dS + 2ε3

3

∫S

Aαβσδραβ(z)ρσδ(z) dS

(62)

For ε small enough, the operator K(ε) is elliptic of multide-gree (2, 2, 4) and is associated with the Dirichlet conditionsz = 0 and ∂nz3 = 0 on ∂S. The solution zε of the Koi-ter model for the clamped case solves equivalently theequations

(L0+ε2B)zε(x) = f 0(x) on S and zε|∂S = 0, ∂nzε3|∂S =0

(63)

This solution has also a multiscale expansion given by thefollowing theorem.

Theorem 5. (Faou, 2003) For the solutions of prob-lem (63), ε ∈ (0, ε0], there exist regular terms zk/2(x) andboundary layer terms ψk/2(T, s), k ≥ 0, such that

zε z0 + χψ0 + ε1/2(z1/2 + χψ1/2) + ε1(z1 + χψ1) + · · ·(64)

in the sense of asymptotic expansions: The following esti-mates hold∥∥∥∥∥zε −

2K∑k=0

εk/2(zk/2 + χψk/2)

∥∥∥∥∥ε,S

≤ CK(f) εK+1/4,

K = 0, 1, . . .

where ‖z‖2

ε,S= ‖γ(z)‖2

L2(S)+ ε2‖ρ(z)‖2

L2(S)and CK(f) is

independent of ε ∈ (0, ε0].

The precise comparison between the terms in the expan-sions (55) and (64) shows that [1] ζ0 = z0, ζ1/2 = z1/2,ϕ0 = ψ0, ϕ

1/2 = ψ

1/2 , while ζ1 and z1, ϕ

1/23 and ψ

1/23 are

generically different, respectively. This allows obtainingoptimal estimates in various norms: Considering the scaleddomain S × (−1, 1), we have

‖uε − zε‖H 1×H 1×L2()

≤ ‖uε − ζ0‖H 1×H 1×L2()

+ ‖zε − ζ0‖H 1×H 1×L2()

≤ Cε1/4

(65)

This estimate implies the convergence result of Ciarletand Lods (1996a) and improves the estimate in Mardare(1998a). To obtain an estimate in the energy norm, weneed to reconstruct a 3-D displacement from zε : First, theKirchhoff-like [2] displacement associated with zε writes,cf. (56)

U1,1,0KL zε = (zε

α − x3(Dαzε3 + 2bσ

αzεσ), zε

3

)(66)

and next, according to Koiter (1970a), we define the recon-structed quadratic displacement [3]

U1,1,2K zε = U1,1,0

KL zε + λ

λ + 2µ

(0,−x3γ

αα(z

ε) + x23

2ρα

α(zε)

)(67)

Then there holds (compare with (46) for plates):

‖uε − U1,1,2K zε‖

E(ε)≤ C

√ε‖uε‖

E(ε)(68)

and similar to plates, the error factor√

ε is optimal andis due to the first boundary layer term w1. Moreover,expansion (64) allows proving that the classical modelsdiscussed in Budiansky and Sanders (1967), Naghdi (1963),Novozhilov (1959), and Koiter (1970a) have all the sameconvergence rate (68).

4.3 Convergence results for general shells

We still embed d in the family (54) with S the mid-surface of d . The fact that all the classical models areequivalent for clamped elliptic shells may not be true inmore general cases, when the shell becomes sensitive (e.g.for a partially clamped elliptic shell with a free portionin its lateral surface) or produces bending effects (case ofparabolic or hyperbolic shells with adequate lateral bound-ary conditions).

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4.3.1 Surface membrane and bending energy

Nevertheless, the Koiter model seems to keep good approx-imation properties with respect to the 3-D model. Thevariational space VK of the Koiter model is, in the totallyclamped case given by the space VK(S) (60) (if the shellε is clamped only on the part γ0 × (−ε, ε) of its bound-ary (with γ0 ⊂ ∂S), the Dirichlet boundary conditions in thespace VK have to be imposed only on γ0). As already men-tioned (62), the Koiter model is associated with the bilinearform εaS,m + ε3aS,b with aS,m and aS,b the membrane andbending forms defined for z, z′ ∈ VK(S) by (58) and (61)respectively.

From the historical point of view, such a decomposi-tion into a membrane (or stretching) energy and a bendingenergy on the midsurface was first derived by Love (1944)in principal curvature coordinate systems, that is, for whichthe curvature tensor (bα

β) is diagonalized. The expression ofthe membrane energy proposed by Love is the same as (61),in contrast with the bending part for which the discussionwas very controversial; see Budiansky and Sanders (1967),Novozhilov (1959), Koiter (1960), and Naghdi (1963) andthe reference therein. Koiter (1960) gave the most naturalexpression using intrinsic tensor representations: The Koi-ter bending energy only depends on the change of curvaturetensor ραβ, in accordance with Bonnet theorem character-izing a surface by its metric and curvature tensors aαβ andbαβ; see Stoker (1969).

For any geometry of the midsurface S, the Koiter modelin its variational form (59) has a unique solution; seeBernadou and Ciarlet (1976).

4.3.2 Classification of shells

According to this principle each shell, in the zero thicknesslimit, concentrates its energy either in the bending surfaceenergy aS,b (flexural shells) or in the membrane surfaceenergy aS,m (membrane shells).

The behavior of the shell depends on the ‘inextensionaldisplacement’ space

VF(S) := ζ ∈ VK(S) | γαβ(ζ) = 0

(69)

The key role played by this space is illustrated by thefollowing fundamental result:

Theorem 6.

(i) (Sanchez-Hubert and Sanchez-Palencia, 1997; Ciarlet,Lods and Miara, 1996) Let uε be the solution ofproblem (3). In the scaled domain S × (−1, 1),the displacement ε2uε(x, X3) converges in H 1()3 asε → 0. Its limit is given by the solution ζ−2 ∈ VF(S)

of the bending problem

∀ ζ′ ∈ VF(S) aS,b(ζ−2, ζ′) = 2

∫S

ζ′ · f 0 (70)

(ii) (Ciarlet and Lods, 1996b) Let zε be the solution ofproblem (59). Then ε2zε converges to ζ−2 in VK(S) asε → 0.

A shell is said flexural (or noninhibited) when VF(S) isnot reduced to 0. Examples are provided by cylindricalshells (or portions of cones) clamped along their generatri-ces and free elsewhere. Of course, plates are flexural shellsaccording to the above definition since in that case, VF(S)

is given by ζ = (0, ζ3) | ζ3 ∈ H 20 (S) and the bending oper-

ator (70) coincides with the operator (16).In the case of clamped elliptic shells, we have VF(S) =

0. For these shells, uε and zε converge in H 1 × H 1 × L2

to the solution ζ0 of the membrane equation (57); seeCiarlet and Lods (1996a) and (65): Such shells are calledmembrane shells. The other shells for which VF(S) reducesto 0 are called generalized membrane shells (or inhibitedshells) and for these also, a delicate functional analysisprovides convergence results to a membrane solution inspaces with special norms depending on the geometry ofthe midsurface; see Ciarlet and Lods (1996a) and Ciarlet(2000), Ch. 5. It is also proved that the Koiter modelconverges in the same sense to the same limits; see Ciarlet(2000), Ch. 7.

Thus, plates and elliptic shells represent extreme situa-tions: Plates are a pure bending structures with an inex-tensional displacement space as large as possible, whileclamped elliptic shells represent a pure membrane situa-tion where VF(S) reduces to 0 and where the membraneoperator is elliptic.

4.4 Shallow shells

We make a distinction between ‘physical’ shallow shells inthe sense of Ciarlet and Paumier (1986) and ‘mathematical’shallow shells in the sense of Pitkaranta, Matache andSchwab (2001). The former involves shells with a curvaturetensor of the same order as the thickness, whereas the latteraddresses a boundary value problem obtained by freezingcoefficients of the Koiter problem at one point of a standardshell.

4.4.1 Physical shallow shells

Let R denote the smallest principal radius of curvature ofthe midsurface S and let D denote the diameter of S. Asproved in Andreoiu and Faou (2001) if there holds

R ≥ 2-D (71)

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then there exists a point p ∈ S such that the orthogonalprojection of S on its tangent plan in p allows therepresentation of S as a C∞ graph in R

3:

ω (x1, x2) →(x1, x2, (x1, x2)

):= x ∈ S ⊂ R

3 (72)

where ω is an immersed (in particular, ω may have self-intersection) domain of the tangent plane in p, and where is a function on this surface. Moreover, we have

|| ≤ CR−1 and ‖∇‖ ≤ CR−1 (73)

with constants C depending only on D.We say that d is a shallow shell if S satisfies a condition

of the type

R−1 ≤ Cd (74)

where C does not depend on d . Thus, if S is a surfacesatisfying (74), for d sufficiently small S satisfies (71)whence representation (72). Moreover, (73) yields that

and ∇ are d . In these conditions, we can choose toembed d into another family of thin domains than (54):We set θ = d−1 and define for any ε ∈ (0, d] the surfaceSε by its parameterization (cf. (72))

ω (x1, x2) →(x1, x2, εθ(x1, x2)

):= x ∈ Sε

It is natural to consider d as an element of the family ε

given as the image of the application

ω × (−ε, ε) (x1, x2, x3) →(x1, x2, ε θ(x1, x2)

)+ x3 nε(x) (75)

where nε(x) denotes the unit normal vector to the mid-surface Sε. We are now in the framework of Ciarlet andPaumier (1986).

A multiscale expansion for the solution of (3) is given inAndreoiu and Faou (2001). The expansion is close to thatof plates, except that the membrane and bending operatorsyielding the deformation patterns are linked by lower-order terms: The associated membrane and bending straincomponents γαβ and ραβ are respectively given by

γαβ := 12 (∂αzβ + ∂βzα) − ε∂αβθ z3 and ραβ := ∂αβz3

(76)

It is worth noticing that the above strains are asymptoticapproximations of the Koiter membrane and bending strainsassociated with the midsurface S = Sε. As a consequence,the Koiter model and the three-dimensional equations con-verge to the same Kirchhoff–Love limit.

4.4.2 Mathematical shallow shells

These models consist in freezing coefficients of standardtwo-dimensional models at a given point p ∈ S in a prin-cipal curvature coordinate system. That procedure yields,with bi := κi(p):

γ11 = ∂1z1 − b1z3, γ22 = ∂2z2 − b2z3,

γ12 = 12 (∂1z2 + ∂2z1) (77)

for the membrane strain tensor, and

κ11 = ∂21 z3 + b1∂1z1, κ22 = ∂2

2 z3 + b2∂2z2,

κ12 = ∂1∂2z3 + b1∂2z1 + b2∂1z2 (78)

as a simplified version of the bending strain tensor. Such alocalization procedure is considered as valid if the diameterD is small compared to R

R D (79)

and for the case of cylindrical shells where the strains havealready the form (77)–(78) in cylindrical coordinates (seeequation (80) below). In contrast with the previous one, thisnotion of shallowness does not refer to the thickness. HereR is not small, but D is. Such objects are definitively shellsand are not plate-like.

These simplified models are valuable so to developnumerical approximation methods, (Havu and Pitkaranta,2002, 2003) and to find possible boundary layer lengthscales, (Pitkaranta, Matache and Schwab, 2001): Theselength scales (width of transition regions from the boundaryinto the interior) at a point p ∈ ∂S are ε1/2 in the nonde-generate case (bss(p) = 0), ε1/3 for hyperbolic degenera-tion (p hyperbolic and bss(p) = 0) and ε1/4 for parabolicdegeneration (p parabolic and bss(p) = 0).

To compare with the standard shell equations, note thatin the case of an axisymmetric shell whose midsurface isrepresented by

: (x1, x2) → (f (x1) cos x2, f (x1) sin x2, x1)

where x1 ∈ R, x2 ∈ [0, 2π[ and f (x1) > 0 is a smoothfunction, we have

γ11(z) = ∂1z1 −f ′(x1)f

′′(x1)

1 + f ′(x1)2

z1 +f ′′(x1)√

1 + f ′(x1)2

z3

γ22(z) = ∂2z2 +f (x1)f

′(x1)

1 + f ′(x1)2

z1 −f (x1)√

1 + f ′(x1)2

z3

γ12(z) = 1

2(∂1z2 + ∂2z1) −

f ′(x1)

f (x1)z2

(80)

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Hence the equation (77) is exact for the case of cylindricalshells, where f (x1) ≡ R > 0, and we can show that thesame holds for (78).

4.5 Versatility of Koiter model

On any midsurface S, the deformation pattern zε solution ofthe Koiter model (59) exists. In general, the mean value ofthe displacement uε through the thickness converges to thesame limit as zε when ε → 0 in a weak sense dependingon the type of the midsurface and the boundary conditions;see Ciarlet (2000). Nevertheless, actual convergence resultshold in energy norm when considering reconstructed dis-placement from the deformation pattern zε.

4.5.1 Convergence of the Koiter reconstructeddisplacement

On any midsurface S, the three-dimensional Koiter recon-structed displacement U1,1,2

K zε is well-defined by (66)–(67).Let us set

e(S, ε, zε, uε) :=‖uε − U1,1,2

K zε‖E(ε)

‖zε‖Eε(S)

(81)

with ‖z‖Eε(S)

, the square root of the Koiter energy (62).In Koiter (1970a, 1970b), an estimate is given: e(S, ε, zε,

uε)2 would be bounded by εR−1 + ε2L−2, with R thesmallest principal radius of curvature of S, and L thesmallest wavelength of zε. It turns out that in the caseof plates, we have L = O(1), R−1 = 0 and, since (46)is optimal, the estimate fails. In contrast, in the case ofclamped elliptic shells, we have L = O(

√ε), R−1 = O(1)

and the estimate gives back (68).Two years after the publications of Koiter (1970a,

1970b), it was already known that the above estimate doesnot hold as ε → 0 for plates. We read in Koiter and Sim-monds (1973) ‘The somewhat depressing conclusion formost shell problems is, similar to the earlier conclusions ofGOL’DENWEIZER, that no better accuracy of the solutions canbe expected than of order εL−1 + εR−1, even if the equa-tions of first-approximation shell theory would permit, inprinciple, an accuracy of order ε2L−2 + εR−1.’

The reason for this is also explained by John (1971)in these terms: ‘Concentrating on the interior we sidestepall kinds of delicate questions, with an attendant gain incertainty and generality. The information about the inte-rior behavior can be obtained much more cheaply (in themathematical sense) than that required for the discussionof boundary value problems, which form a more ‘transcen-dental’ stage’.

Koiter’s tentative estimate comes from formal compu-tations also investigated by John (1971). The analysis byoperator formal series introduced in Faou (2002) is in thesame spirit: For any geometry of the midsurface, there existformal series V[ε], R[ε], Q[ε], and L[ε] reducing the three-dimensional formal series problem to a two-dimensionalproblem of the form (21) with L[ε] = L0 + ε2L2 + · · ·where L0 is the membrane operator associated with the form(58). The bending operator B associated with aS,b can becompared to the operator L2 appearing in the formal seriesL[ε]: We have

∀ ζ, ζ′ ∈ VF(S)⟨L2ζ, ζ′

⟩L2(S)3 =

⟨Bζ, ζ′

⟩L2(S)3 (82)

Using this formal series analysis, the first two authorsare working on the derivation of a sharp expression ofe(S, ε, zε, uε) including boundary layers effects, and opti-mal in the case of plates and clamped elliptic shells; seeDauge and Faou (2004).

In this direction also, Lods and Mardare (2002) prove thefollowing estimate for totally clamped shells

‖uε − (U1,1,2K zε + w)‖

E(ε)≤ Cε1/4‖uε‖

E(ε)(83)

with w an explicit boundary corrector of U1,1,2K zε.

4.5.2 Convergence of Koiter eigenvalues

The operator ε−1K(ε) has a compact inverse, therefore itsspectrum is discrete with only an accumulation point at+∞. We agree to call Koiter eigenvalues the eigenvaluesof the former operator, that is, the solutions µε of

∃ zε ∈ VK(S) \ 0 such that

aS,m(zε, z′) + ε2aS,b(zε, z′) = 2µε

∫S

zε · z′, ∀ z′ ∈ VK(S)

(84)

As already mentioned in Section 2.4, cf. (31), this spec-trum provides the limiting behavior of three-dimensionaleigenvalues for plates. Apparently, very little is known forgeneral shells.

Concerning Koiter eigenvalues, interesting results areprovided by Sanchez-Hubert and Sanchez-Palencia (1997),Ch. X: The µε are attracted by the spectrum of the mem-brane operator S(M) where M is the self-adjoint unboundedoperator associated with the symmetric bilinear form aS,m

defined on the space H 1 × H 1 × L2(S). There holds (westill assume that S is smooth up to its boundary):

Theorem 7. The operator M + µ Id is elliptic of multide-gree (2, 2, 0) for µ > 0 large enough. Moreover its essentialspectrum Ses(M) satisfies:

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(i) If S is elliptic and clamped on its whole boundary,Ses(M) is a closed interval [a, b], with a > 0,

(ii) If S is elliptic and not clamped on its whole boundary,Ses(M) = 0 ∪ [a, b] with a > 0,

(iii) For any other type of shell (i.e. there exists at least onepoint where the Gaussian curvature is ≤ 0) Ses(M) isa closed interval of the form [0, b], with b ≥ 0.

If the shell is of flexural type, the lowest eigenvaluesµε tend to 0 like O(ε2), same as for plates; see (29). If theshell is clamped elliptic, the µε are bounded from below bya positive constant independent of ε. In any other situation,we expect that the lowest µε still tends to 0 as ε → 0.

5 HIERARCHICAL MODELS FORSHELLS

The idea of deriving hierarchical models for shells goesback to Vekua (1955, 1965, 1985) and corresponds toclassical techniques in mechanics: Try to find asymptoticexpansions in x3 by use of Taylor expansion around themidsurface S. An alternative approach consists in choosingthe coefficients zn

j in (38) as moments through the thicknessagainst Legendre polynomials Ln(x3/ε).

Vogelius and Babuska (1981a, 1981b, 1981c) laid thefoundations of hierarchical models in view of their appli-cations to numerical analysis (for scalar problems).

5.1 Hierarchies of semidiscrete subspaces

The concepts mentioned in Section 3 can be adapted to thecase of shells. In contrast with plates for which there existconvenient Cartesian system of coordinates fitting with thetangential and normal directions to the midsurface, morenonequivalent options are left open for shells. They are;for example;

The direction of semidiscretization: The intrinsic choice isof course the normal direction to the midsurface (variablex3), nevertheless for shells represented by a single localchart like in (72), any transverse direction could be cho-sen. In the sequel, we only consider semidiscretizationsin the normal direction.

The presence or absence of privileged components for thedisplacement field in the Ansatz (38). If one privilegedcomponent is chosen, it is of course the normal one u3and the two other ones are (uα) = u. Then the sequenceof orders q is of the form q = (q, q, q3), and thespace V q(ε) has the form (48). Note that this spaceis independent of the choice of local coordinates on S.

If there is no privileged component, q has to be of theform (q, q, q) and the space V q(ε) can be written

V q(ε) =

u = (u1, u2, u3) ∈ V (ε),

∃zn = (zn1, z

n2, z

n3) ∈ H 1

0 (S)3, 0 ≤ n ≤ q,

uj =q∑

n=0

xn3 zn

j (x), j = 1, 2, 3

(85)

Here, for ease of use, we take Cartesian coordinates,but the above definition is independent of any choice ofcoordinates in ε ⊂ R

3. In particular, it coincides withthe space (48) for q = q3.

Then the requirements of approximability (34), asymptoticconsistency (35), and optimality of the convergence rate(36) make sense.

5.2 Approximability

For any fixed thickness ε, the approximability issue isas in the case of plates. By Cea’s lemma, there existsan adimensional constant C > 0 depending only on thePoisson ratio ν, such that


≤ C‖uε − vq‖E(ε)

∀ vq ∈ V q(ε)

and the determination of approximability properties relieson the construction of a best approximation of uε byfunctions in V q(ε).

In Avalishvili and Gordeziani (2003), approximability isproved using the density of the sequence of spaces V q(ε)

in H 1(ε)3. But the problem of finding a rate for theconvergence in (33) is more difficult, since the solutionuε has singularities near the edges and, consequently, doesnot belong to H 2(ε) in general. For scalar problems,Vogelius and Babuska (1981a, 1981b, 1981c) prove bestapproximation results using weighted Sobolev norms [4].Up to now, for elasticity systems, there are no such resultstaking the actual regularity of uε into account.

It is worth noticing that, in order to obtain an equal-ity of the form (36), we must use Korn inequality, sincemost approximation results are based on Sobolev norms.But due to blow up of the Korn constant when ε → 0, itseems hard to obtain sharp estimates in the general case.(Let us recall that it behaves as ε−1 in the case of partiallyclamped shells.)

5.3 Asymptotic consistency

Like for plates, the presence of the nonpolynomial three-dimensional boundary layers wk generically produces a

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limitation in the convergence rate in (35). As previouslymentioned, the only case where a sharp estimate is avail-able, is the case of clamped elliptic shells. Using (68), weindeed obtain the following result (compare with Theo-rem 3):

Theorem 8. If the midsurface S is elliptic, if the shell isclamped along its whole lateral boundary, and if f

∣∣S≡ 0,

then for any q (1, 1, 2) with definition (48), and for anyq (2, 2, 2) with (85), and with the use of the standard 3-Delastic energy (2), there exists Cq = Cq(f) > 0 such that forall ε ∈ (0, ε0)


≤ Cq

√ε ‖uε‖

E(ε)(86)

Note that in this case, the two-dimensional boundarylayers are polynomial in x3, Therefore they can be capturedby the semidiscrete hierarchy of spaces V q.

Using estimate (83) of Lods and Mardare (2002), togetherwith the fact that the corrector term w is polynomial in x3of degree (0, 0, 2), we obtain a proof for the asymptoticconsistency for any (smooth) clamped shell without assum-ing that the midsurface is elliptic:

Theorem 9. If the shell is clamped along its whole lateralboundary, and if f

∣∣S≡ 0, then for q as in Theorem 8 and

the standard energy (2), there exists Cq = Cq(f) > 0 suchthat for all ε ∈ (0, ε0)


≤ Cq ε1/4 ‖uε‖E(ε)

(87)

5.4 Examples of hierarchical models

Various models of degree (1, 1, 0), (1, 1, 2), and (2, 2, 2)

are introduced and investigated in the literature. Note thatthe model (1, 1, 1) is strictly forbidden for shells because itcannot be associated with any correct energy; see Chapelle,Ferent and Bathe (2004).

5.4.1 (1, 1, 0 ) models

One of the counterparts of Reissner–Mindlin model forplates is given by the Naghdi model: see Naghdi (1963,1972). The space of admissible displacements is

V N(ε) = u ∈ V (ε), ∃z ∈ H 10 (S)3, ∃θα ∈ H 1

0 (S)2,

u = (zα − x3(θα + bβαzβ), z3) (88)

As in (47) the energy splits into three parts (with the shearcorrection factor κ):

a(u, u) = 2ε

∫S

Aαβσδγαβ(z)γσδ(z) dS

(membrane energy)

+ εκ

∫S

µaασ(Dαz3 + bδαzδ − θα)

× (Dσz3 + bβσzβ − θσ) dS

(shear energy)

+ 2ε3

3

∫S

Aαβσδραβ(z, θ)ρσδ(z, θ) dS

(bending energy)

(89)

where

ραβ(z, θ) = 12 (Dαθβ + Dβθα)−cαβz3 + 1

2bσαDβzσ + 1

2bσβDαzσ

Note that when the penalization term in the shear energygoes to zero, we get θσ = Dσz3 + b

βσzβ and the displacement

u in (88) coincides with (66). In Lods and Mardare (2002),an estimate of the error between the solution of the Naghdimodel and the solution of the 3-D model is provided in asubenergetic norm.

A more recent (1, 1, 0) model (called general shell ele-ment; see Chapelle and Bathe, 2000) consists of the reducedenergy projection on the space V (1,1,0)(ε). Indeed, it doesnot coincide with the Naghdi model but both models pos-sess similar asymptotic properties and they are preferred toKoiter’s for discretization.

5.4.2 Quadratic kinematics

In accordance with Theorems 8 and 9, it is relevant to usethe standard 3-D elastic energy (2) for such kinematics.Quadratic models based on the (1, 1, 2) model are inves-tigated in Bischoff and Ramm (2000). The enrichment ofthe general shell element by the introduction of quadraticterms – model (2, 2, 2) – is thoroughly studied from bothasymptotic and numerical point views in Chapelle, Fer-ent and Bathe (2004) and Chapelle, Ferent and Le Tallec(2003).

6 FINITE ELEMENT METHODS IN THINDOMAINS

We herein address some of the characteristics of finiteelement methods (FEM), mainly the p-version of the FEM,when applied to the primal weak formulations (3) and (33)for the solution of plate and shell models. We only addressisotropic materials, although our analysis could be extendedto laminated composites.

As illustrative examples, we present the results of somecomputations performed with the p-version FE computerprogram StressCheck. (StressCheck is a trade mark ofEngineering Software Research and Development, Inc.,10845 Olive Blvd., Suite 170, St. Louis, MO 63141, USA.)

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6.1 FEM discretizations

Let us recall that, when conformal, the FEM is a Galerkinprojection into finite dimensional subspaces VN of the varia-tional space associated with the models under consideration.In the p-version of the FEM, subspaces are based on onepartition of the domain into a finite number of subdomainsK ∈ T (the mesh) in which the unknown displacement isdiscretized by mapped polynomial functions of increasingdegree p. The subdomains K are mapped from referenceelement(s) K .

6.1.1 Meshes

All finite element discretizations we consider here are basedon a mesh TS of the midsurface S: We mean that the3-D mesh of ε has in normal coordinates (x, x3) thetensor product form [5] TS ⊗ Iε where Iε represents apartition of the interval (−ε, ε) in layers, for example, thetwo halves (−ε, 0) and (0, ε), or – this case is importantin the sequel – , the trivial partition by only one elementthrough the thickness. We agree to call that latter mesh athin element mesh.

The 3-D elements K are thus images by maps ψK

from reference elements K , which are either pentahedral(triangle × interval) or hexahedral:

ψK : K = T × [0, 1] (x1, x2, x3) → x ∈ K

with T the reference triangle or the reference square. Forthe 2-D FEM, we denote by T the elements in TS : Theyare the image of T by maps ψT

ψT : T (x1, x2) → x ∈ T

If ε is a plate, the midsurface S is plane but itsboundary ∂S is not straight. For some lateral boundaryconditions, for example, the hard, simple supported plate,the approximation of ∂S by a polygonal lines produces, ingeneral, wrong results. This effect is known as the Babuskaparadox (Babuska and Pitkaranta, 1990). If ε is a shell,the geometric approximation of S by ‘plane’ elements isalso an issue: If the mappings are affine, the shell isapproximated by a faceted surface which has quite differentrigidity properties than the smooth surface; see Akian andSanchez-Palencia (1992) and Chapelle and Bathe (2003),Section 6.2.

As a conclusion, good mappings have to be used for thedesign of the elements K (high degree polynomials or otheranalytic functions).

6.1.2 Polynomial spaces for hierarchical models

For hierarchical models (33), the discretization is indeedtwo-dimensional: The degree q of the hierarchy being fixed,the unknowns of (33) are the functions zn

j defined on S andrepresenting the displacement according to (38), where thedirector functions n

j form adequate bases of polynomialsin one variable, for example, Legendre polynomials Ln.

We have already mentioned in Section 5 that the onlyintrinsic option for the choice of components is takingj = (α, 3), which results into the Ansatz (written here withLegendre polynomials)

u =q∑

n=0

zn(x) Ln

(x3

ε

)and

u3 =q3∑

n=0

zn3(x) Ln

(x3

ε

)

Now the discretization consists in requiring that znα|T ψT ,

α = 1, 2, and zn3|T ψT belong to the space Pp(T ) for some

p where Pp(T ) is the space of polynomials in two variables

• of degree ≤ p if T is the reference triangle,• of partial degree ≤ p if T is the reference square

[0, 1] × [0, 1].

It makes sense to fix different degrees pj in relation withj = α, 3, and we set p = (p1, p2, p3). When plugged backinto formula (38), this discretization of the zn

j , j = α, 3,yields a finite dimensional subspace V

qp (ε) of V q(ε). As

already mentioned for the transverse degrees q, cf. (48) andSection 5, we have to assume for coherence that p1 = p2for shells. In the situation of plates, if T is affinely mappedfrom the reference square, the zn

j |T are simply given by

zn1(x) =

p1∑i,k=0

zn1,ikPi(x1)Pk(x2)

zn2(x) =

p2∑i,k=0

zn2,ikPi(x1)Pk(x2)

zn3(x) =

p3∑i,k=0

zn3,ikPi(x1)Pk(x2)

where the znj,ik are real coefficients and Pi denotes a

polynomial of degree i which is obtained from Legendrepolynomials; see for example Szabo and Babuska (1991).

The discretization of hierarchical models (33) can alsobe done through the h-version or the h-p versions of FEM.

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6.1.3 Polynomial spaces for 3-D discretization.Case of thin elements

In 3-D, on the reference element K = T × [0, 1], we canconsider any of the polynomial spaces Pp,q(K) = Pp(T ) ⊗Pq([0, 1]), p, q ∈ N, For the discretization of (3), eachCartesian component ui of the displacement is soughtfor in the space of functions v ∈ H 1(ε) such that forany K in the mesh, v|K ψK belongs to Pp,q(K). Wedenote by Vp,q(

ε) the corresponding space of admissibledisplacements over ε.

In the situation where we have only one layer of elementsover ε in the thickness (thin element mesh) with a (p, q)

discretization, let us set q = (q, q, q) and p = (p, p, p).Then it is easy to see that, in the framework of semidiscretespaces (85), we have the equality between discrete spaces:

Vp,q(ε) = V q

p (ε) (90)

In other words, thin elements are equivalent to the dis-cretization of underlying hierarchical models. Let us insiston the following fact: For a true shell, the correspondencebetween the Cartesian components uj and the tangentialand transverse components (u, u3) is nonaffine. As a con-sequence, equality (90) holds only if the space V

qp (ε)

corresponds to the discretization of a hierarchical modelin Cartesian coordinates.

Conversely, hierarchical models of the type q = (q, q, q)

with the ‘Cartesian’ unknowns znj , n = 0, . . . , q, j = 1, 2, 3

can be discretized directly on S, or inherit a 3-D discretiza-tion; see Chapelle, Ferent and Bathe (2004). Numericalevidence that the p-version with anisotropic Ansatz spacesallows the analysis of three-dimensional shells with highaccuracy was firstly presented in Duster, Broker and Rank(2001).

6.1.4 FEM variational formulations

Let us fix the transverse degree q of the hierarchical model.Its solution uε,q solves problem (33). For each ε > 0 andeach polynomial degree p, (33) is discretized by its finitedimensional subspace V

qp (ε). Let uε,q

p be the solution of

Find uε,qp ∈ V

qp (ε) such that

aε,q(uε,qp , u′) =

∫ε

fε · u′ dx, ∀u′ ∈ V qp (ε) (91)

We can say that (91) is a sort of 3-D discretization of (33).But, indeed, the actual unknowns of (91) are the zn

α, n =0, . . . , q, and zn

3, n = 0, . . . , q3, or the znj for n = 0, . . . , q

and j = 1, 2, 3. Thus, (91) can be alternatively formulatedas a 2-D problem involving spaces Z

qp(S) independent of

ε, and a coercive bilinear form aqS(ε) polynomial in ε.

Examples are provided by the Reissner–Mindlin model,cf. (47), the Koiter model (84), and the Naghdi model, cf.(89). The variational formulation now takes the form

Find Z =: (zn)0≤n≤q ∈ Zqp(S) such that

aqS(ε)(Z, Z′) = F(ε)(f, Z′), ∀Z′ ∈ Zq

p(S) (92)

where F(ε)(f, Z′) is the suitable bilinear form couplingloadings and test functions. Let us denote by Zε,q

p thesolution of (92).

6.2 Locking issues

In the framework of the family of discretizations consid-ered above, the locking effect is said to appear when adeterioration in the resulting approximation of uε,q by uε,q

p ,p → ∞ tends to ∞, occurs as ε → 0. Of course, a similareffect is reported in the h-version of FEM: The deteriora-tion of the h-approximation also occurs when the thicknessε approaches zero.

Precise definition of locking may be found in Babuskaand Suri (1992): It involves the locking parameter (thethickness ε in the case of plates), the sequence of finiteelement spaces V

qp that comprise the extension procedure

(the p-version in our case, but h and h-p versions canalso be considered), and the norm in which error is to bemeasured. Of course, in different error measures differentlocking phenomena are expected.

6.2.1 Introduction to membrane locking

A locking-free approximation scheme is said to be robust.For a bilinear form aS(ε) of the form a0 + ε2a1 like Koi-ter’s, a necessary condition for the robustness of the approx-imation is that the intersections of the discrete spaces withthe kernel of a0 are a sequence of dense subspaces forthe whole kernel of a0; see Sanchez-Hubert and Sanchez-Palencia (1997), Ch. XI. In the case of the Koiter model,this means that the whole inextensional space VF(S) (69)can be approximated by the subspaces of the inextensionalelements belonging to FE spaces. For hyperbolic shells,the only inextensional elements belonging to FE spaces arezero; see Sanchez-Hubert and Sanchez-Palencia (1997) andChapelle and Bathe (2003), Section 7.3, which prevents allapproximation property of VF(S) if it is not reduced to 0.

This fact is an extreme and general manifestation of themembrane locking of shells, also addressed in Pitkaranta(1992) and Gerdes, Matache and Schwab (1998) for cylin-drical shells, which are a prototype of shells having anonzero inextensional space. Plates do not present mem-brane locking since all elements z = (0, z3) are inexten-sional, thus can be approximated easily by finite element

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subspaces. Nevertheless, as soon as the RM model is used,as can be seen from the structure of the energy (47), a shearlocking may appear.

6.2.2 Shear locking of the RM and hierarchicalplate models

Shear locking occurs because the FE approximation usingC0 polynomials for the RM family of plates at the limitwhen ε → 0 has to converge to the KL model in energynorm Suri, 2001, requiring C1 continuity. Let us considerthe three-field RM model on the subspace of V RM(ε),cf. Section 3.3, of displacements with bending parity: u ∈V (ε), u = (−x3θ, z3). According to Suri, Babuska andSchwab (1995) we have the following:

Theorem 10. The p-version of the FEM for the RM platemodel without boundary layers, on a mesh of trianglesand parallelograms, with polynomial degrees of p ≥ 1 forrotations θ and p3 ≥ p for z3 is free of locking in theenergy norm.

For the h-version over a uniform mesh consisting eitherof triangles or rectangles, to avoid locking the tangentialdegree p has to be taken equal to four or larger, withthe transverse degree p3 being chosen equal to p + 1. Asimilar phenomenon was earlier found in connection with‘Poisson Ratio’ locking for the equations of elasticity. (i.e.conforming elements of degree four or higher encounter nolocking); see Scott and Vogelius (1985). In Suri, Babuskaand Schwab (1995), it is proven that locking effects (andresults) for the (1, 1, 2) plate model are similar to theRM model because no additional constraints arise as thethickness ε → 0. Furthermore, it is stated that lockingeffects carry over to all hierarchical plate models.

Here we have discussed locking in energy norm. How-ever, if shear stresses are of interest, then locking is signif-icantly worse because these involve an extra power ε−1.

For illustration purposes, consider a clamped plate withellipsoidal midsurface of radii 10 and 5, Young mod-ulus (we recall that the Young modulus is given byE = µ(3λ + 2µ)/2(λ + µ) and the Poisson ratio by ν =λ/2(λ + µ)) E = 1 and Poisson ratio ν = 0.3; see Figure 4.The plate is loaded by a constant pressure of value (2ε)2.

The discretization is done over a 32 p-element mesh (seeFigure 4(a) and (b) for 2ε = 1 and 0.1) using two layers,each of dimension ε in the vicinity of the boundary. TheFE space is defined with p3 = p ranging from 1 to 8. Weshow in Figure 5 the locking effects for the RM model withκEnergy.

The error plotted in ordinates is the estimated relativediscretization error in energy norm between the numericaland exact solution of the RM plate model for each fixed

X

Y

ZX

Y

Z X

Y

Z

(a) (b) (c)

Figure 4. p-FE mesh for 2ε = 1, 0.1 for RM model and 2ε = 1for 3-D model. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

||e||

Thickness = 1Thickness = 0.1Thickness = 0.01Thickness = 0.001Thickness = 0.0001Thickness = 0.00001

1 2 3 4 5 6 7 8

Polynomial order

1.E+00

1.E–01

1.E–02

1.E–03

1.E–04

Figure 5. Discretization error versus polynomial degree p forRM plates of various thicknesses ε. A color version of this imageis available at http://www.mrw.interscience.wiley.com/ecm

thickness ε (it is not the error between the RM numericalsolution and the exact 3-D plate model). A similar behaviorcan be observed with the model q = (1, 1, 2).

To illustrate both the locking effects for the hierarchicalfamily of plates and the modeling errors between the platemodels and their 3-D counterpart, we have computed fortwo thicknesses of plates (2ε = 1 or 2ε = 0.01), the solutionfor the first four plate models (see Table 1 [6]), and for thefully 3-D plate with the degrees p = p3 = 1, 2, . . . , 8 withthe model represented in Figure 4(c) for 2ε = 1.

The relative errors between energy norms of the hierar-chical models and the 3D plate model versus the polynomialdegree p is shown in Figure 6. As predicted, increas-ing the order of the plate model does not improve thelocking ratio, and as the hierarchical model number isincreased the relative error decreases. We note that when

Table 1. Hierarchical plate-model definitions for bending sym-metry.

Model # 1 (RM) 2 3 4

Degrees q = (q1, q2, q3) (1,1,0) (1,1,2) (3,3,2) (3,3,4)# independent

fields d = (d1, d2, d3) (1,1,1) (1,1,2) (2,2,2) (2,2,3)

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1.E+00

1.E–01

1.E–02

1.E–03

1.E–04

1.E–05

1.E–061 2 3 4 5 6 7 8

||e||

RM, thick = 1Model 2,thick = 1Model 3,thick = 1Model 4,thick = 1RM, thick = 0.01Model 2, thick = 0.01Model 3, thick = 0.01Model 4, thick = 0.01

Polynomial order

Figure 6. Relative error versus polynomial degree for 2ε = 1 and0.01 for the first 4 hierarchical models. A color version of thisimage is available at http://www.mrw.interscience.wiley.com/ecm

2ε = 1 the relative error of the four models converges tothe modeling error, which is still quite big since ε is notsmall, whereas when 2ε = 0.01, the error stays larger that15% for all models when p ≤ 4, and starts convergingfor p ≥ 5.

6.3 Optimal mesh layout for hierarchical modelswith boundary layers

All hierarchical plate models (besides KL model) exhibitboundary layers. These are rapidly varying components,which decay exponentially with respect to the stretcheddistance R = r/ε from the edge, so that at a distanceO(2ε) these are negligible. Finite element solutions shouldbe able to capture these rapid changes. Using the p-version of the finite element method, one may realizeexponential convergence rates if a proper design of meshesand selection of polynomial degrees is applied in thepresence of boundary layers.

In a 1-D problem with boundary layers, it has beenproven in Schwab and Suri (1996) that the p-version overa refined mesh can achieve exponential convergence forthe boundary layers, uniformly in ε. The mesh has to bedesigned so to consist of one O(p(2ε)) boundary layerelement at each boundary point. More precisely, the optimalsize of the element is αp(2ε), where, 0 < α < 4/e (seeFig. 7).

This result carries over to the heat transfer problem on2-D domains as shown in Schwab, Suri and Xenophontos(1998), and to the RM plate model, as demonstrated bynumerical examples. Typical boundary layer meshes areshown in Figure 4 for 2ε = 1 and 0.1: In practice, for easeof computations, two elements in the boundary layer zoneare being used, each having the size in the normal directionof ε, independent of the polynomial degree used. This,

αp(2ε)

Figure 7. A typical design of the mesh near the boundary for thep-version of the FEM.

although not optimal, still captures well the rapid changesin the boundary layer.

In order to realize the influence of the mesh designover the capture of boundary layer effects, we have againsolved numerically the RM plate model for a thickness of2ε = 0.01 (and κDeflection as shear correction factor). Threedifferent mesh layouts have been considered, with twolayers of elements in the vicinity of the edge of dimension0.5, 0.05, and 0.005 (the first two ones are representedin Figure 4). For comparison purposes, we have computedthe 3-D solution over a domain having two layers in thethickness direction and two elements in the boundary layerzone of dimension 0.005. We have extracted the verticaldisplacement u3 and the shear strain e23 along the linestarting at (x1, x2) = (9.95, 0) and ending at the boundary(x1, x2) = (10, 0), that is, in the boundary layer region.Computations use the degrees p = p3 = 8. It turns outthat the vertical displacement u3 is rather insensitive tothe mesh, whereas the shear strain e23 is inadequatelycomputed if the mesh is not properly designed: With themesh containing fine layers of thickness 0.005, the averagerelative error is 10%, but this error reaches 100% withmesh layer thickness 0.05 and 400% for the mesh layerthickness 0.5.

Concerning shells, we have seen in Section 4.2 that theKoiter model for clamped elliptic shells admits boundarylayers of length scale

√ε, and in Section 4.4 that other

length scales may appear for different geometries (ε1/3

and ε1/4). Moreover, for Naghdi model, the short lengthscale ε is also present; see Pitkaranta, Matache and Schwab(2001). Nevertheless, the “long” length scales ε1/3 andε1/4 appear to be less frequent. We may expect a similarsituation for other hierarchical models. As a conclusion themesh design for shell of small thicknesses should (at least)take into account both length scales ε and

√ε. Another

phenomenon should also be considered: Hyperbolic andparabolic shells submitted to a concentrated load or asingular data are expected to propagate singularities along

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their zero curvature lines, with the scale width ε1/3; seePitkaranta, Matache and Schwab (2001).

6.4 Eigen-frequency computations

Eigen-frequency computations are, in our opinion, a verygood indicator of (i) the quality of computations, (ii) thenature of the shell (or plate) response. In particular, thebottom of the spectrum indicates the maximal possiblestress–strain energy to be expected under a load of givenpotential energy. From Theorem 7, we may expect that,except in the case of clamped elliptic shells, the ratiobetween the energy of the response and the energy of theexcitation will behave almost as O(ε−2).

6.4.1 Eigen-frequency of RM versus 3-D for plates

Eigen-frequencies obtained by the p-version finite ele-ment method for clamped RM plates and their counterpart3-D eigen-frequencies have been compared in Dauge andYosibash (2002), where rectangular plates of dimensions1 × 2 × 2ε have been considered. For isotropic materials

with Poisson coefficient ν = 0.3, the relative error for thefirst three eigen-frequencies was found negligible (less than0.12%), for thin plates with slender ratio of less than 1%,and still small (0.2%) for moderately thick plates (slanderratio about 5%).

For some orthotropic materials, much larger relativeerrors between the RM eigen-frequencies and their 3-Dcounterparts have been observed even for relatively thinplates. In one of the orthotropic rectangular plate examplesin Dauge and Yosibash (2002), for which the boundarylayer effect on the eigen-frequencies should be the mostpronounced, a very large relative error of 25% has beenreported for the first eigen-frequency at ε = 0.1. This isa significant deviation, whereas the RM model underes-timates the ‘true’ 3-D by 25%, and is attributed to theboundary layer effect.

6.4.2 3-D eigen-frequency computations for shells

We present computations on three families of shells, seeFigure 8: (a) clamped spherical shells, (b) sensitive spher-ical shells, (c) flexural cylindrical shells, all with material

X

Y

Z

(a)

X

Y Z

(b)

Y X

Z

(c)

Figure 8. Shell models (a), (b) and (c) for ε = 0.04. A color version of this image is available at http://www.mrw.interscience.wiley.com/ecm

StressCheck V6.2.0uD = SOL,Run = 6,DOF = 995

N-Freq. = 2.386e−001Mode Shape = 1 (Uz)Max = 1.193e − 002Min = −4.764e + 000

1.193e − 002−4.656e − 001−9.432e − 001−1.421e + 000−1.898e + 000−2.376e + 000−2.853e + 000−3.331e + 000−3.808e + 000−4.286e + 000−4.764e + 000

X

Y

Z

StressCheck V6.2.0uD = SOL,Run = 6,DOF = 995N-Freq. = 2.458e − 001Mode Shape = 2 (Uz)Max = 2.967e + 000Min = −2.967e + 000

2.967e + 0002.374e + 000 1.780e + 0001.187e + 0005.935e − 0011.192e − 007

−5.935e − 001−1.187e + 000−1.780e + 000−2.374e + 000−2.967e + 000


N-Freq. = 3.618e−001Mode Shape = 4 (Uz)Max = 3.114e + 000Min = −3.114e + 000

3.114e + 0002.491e + 0001.869e + 0001.246e + 0006.228e − 001

−2.801e − 008−6.229e − 001−1.246e + 000 −1.869e + 000−2.491e + 000−3.114e + 000

X

Y

Z

XY

Z

(a) (b) (c)

Figure 9. Model (a). vertical components of eigen-modes 1, 2 and 4 for ε = 0.08. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

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parameters ν = 0.3 and E = 1. These three families illus-trate the three cases (i), (ii) and (iii) in Theorem 7: Theshells (a) are elliptic clamped on their whole boundary, (b)are elliptic, but clamped only on a part of their boundaries,and (c) are parabolic. Note that Theorem 7 states resultsrelating to Koiter eigenvalues and not for 3-D eigenvalues.Nevertheless, a similar behavior can be expected for 3-Deigenvalues.

Family (a). The midsurface S is the portion of the unitsphere described in spherical coordinates by ϕ ∈ [0, 2π)

and θ ∈ (π/4, π/2]. Thus S a spherical cap containing thenorth pole. The family of shells ε has its upper and lowersurfaces characterized by the same angular conditions, andthe radii ρ = 1 + ε and ρ = 1 − ε, respectively. We clampε along its lateral boundary θ = π/4.

We have computed the first five eigen-frequencies of the3-D operator (4) by a FE p-discretization based on twolayers of elements in the transverse direction and 8 × 5elements in the midsurface, including one thin layer of ele-ments in the boundary layer. The vertical (i.e. normal to thetangent plane at the north pole, not transverse to the mid-surface!) component u3 for three modes are representedin Figure 9 for the (half)-thickness ε = 0.08. Mode 3 isrotated from mode 2, and mode 5 from mode 4 (dou-ble eigen-frequencies). The shapes of the eigen-modes forsmaller values of the thickness are similar. Figure 10 pro-vides the three first distinct eigen-frequencies as a functionof the thickness in natural scales. In accordance with

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.1

0.2

0.3

0.4

124

0.35

0.25

0.15

0.05

0

Figure 10. Model (a). Eigen-frequencies versus thickness (2ε).A color version of this image is available at http://www.mrw.interscience.wiley.com/ecm

Theorem 7 (i), the smallest eigen-frequencies all tend tothe same nonzero limit, which should be the (square rootof the) bottom of the membrane spectrum.

Family (b). The midsurface S is the portion of the unitsphere described in spherical coordinates by ϕ ∈ [0, 2π)

and θ ∈ (π/4, 5π/12]. The family of shells ε has its upperand lower surfaces characterized by the same angular con-ditions, and the radii ρ = 1 + ε and ρ = 1 − ε, respectively.


N-Freq. = 5.640e−002Mode Shape = 1(Uz)Max = 5.915e + 000Min = −5.915e + 000

5.915e + 0004.732e + 0003.549e + 0002.366e + 0001.183e + 0000.000e + 000

−1.183e + 000−2.366e + 000−3.549e + 000−4.732e + 000−5.915e + 000

X

YZ


N-Freq. = 6.409e−002Mode Shape = 3(Uz)Max = 5.492e + 000Min = −5.492e + 000

5.492e + 0004.394e + 0003.295e + 0002.197e + 0001.098e + 0001.192e + 007

−1.098e + 000−2.197e + 000−3.295e + 000−4.394e + 000−5.492e + 000

X

YZ



6.190e + 0004.952e + 0003.714e + 0002.476e + 0001.238e + 0001.192e + 007

−1.238e + 000−2.476e + 000−3.714e + 000−4.952e + 000−6.190e + 000

X

YZ

StressCheck V6.2.0uD = SOL,Run = 7,DOF = 3096N-Freq. = 1.072e −001Mode Shape = 7 (Uz)Max = 4.053e + 000Min = −6.377e−019

4.053e + 0003.648e + 0003.242e + 0002.837e + 0002.432e + 0002.027e + 0001.621e + 0001.216e + 0008.106e − 0014.053e − 001

−6.377e − 019

X

YZ

X

YZ

StressCheck V6.2.0uD = SOL,Run = 7,DOF = 3096N-Freq. = 1.441e − 001Mode Shape = 8 (Uz)Max = 2.613e − 004Min = −2.418e −004

2.613e − 0042.110e − 0041.607e − 0041.104e − 0046.005e − 0059.745e − 006

−4.056e − 005−9.086e − 005−1.412e − 004−1.915e − 004−2.418e − 004

X

YZ



6.607e + 0005.286e + 0003.964e + 0002.643e + 0001.321e + 000

−4.578e − 005−1.321e + 000−2.643e + 000−3.964e + 000−5.286e + 000−6.607e + 000

Figure 11. Model (b). Vertical components of modes 1, 3, 5, 7, 8, 9 for ε = 0.04. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

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StressCheck V6.2.1fD = SOL,Run = 7,DOF = 5299


3.509e + 0012.807e + 0012.105e + 0011.403e + 0017.017e + 0009.537e − 007

−7.017e + 000−1.403e + 001−2.105e + 001−2.807e + 001−3.509e + 001

X

Y Z

StressCheck V6.2.1fD = SOL,Run = 7,DOF = 5299N-Freq. = 5.909e − 003Mode Shape = 3 (Uz)Max = 3.860e + 001Min = −3.860e + 001

3.860e + 0013.088e + 0012.316e + 0011.544e + 0017.719e + 000

−4.768e − 006−7.719e + 000−1.544e + 001−2.316e + 001−3.088e + 001−3.860e + 001

Y Z

X


4.150e + 0013.320e + 0012.490e + 0011.660e + 0018.300e + 0002.861e − 006

−8.300e + 000−1.660e + 001−2.490e + 001−3.320e + 001−4.150e + 001

Y Z

X

Figure 12. Model (b). Vertical components of modes 1, 3, 5 for ε = 0.00125. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

We clamp ε along its lateral boundary θ = 5π/12 and letit free along the other lateral boundary θ = π/4. This shellis a sensitive one in the sense of Pitkaranta and Sanchez-Palencia (1997), which means that it is sensitive to thethickness and answers differently according to the valueof ε.

We have computed the first five (or first ten) eigen-frequencies of the 3-D operator (4) by a FE p-discretizationsimilar to that of (a) (two layers in the transverse directionand 8 × 4 elements in the surface direction – for the ‘small’thickness, a globally refined mesh of 16 × 6 elements hasbeen used). In Figure 11, we plot the vertical componentsof modes number 1, 3, 5, 7, 8, and 9 for ε = 0.04 andin Figure 12, modes number 1, 3, 5 for ε = 0.00125. Inboth cases, modes 2, 4, and 6 are similar to modes 1, 3,and 5 respectively and associated with the same (double)eigen-frequencies.

For ε = 0.04, we notice the axisymmetric mode at posi-tion 7 (it is at position 5 when ε = 0.08, and 9 for ε = 0.02).Mode 8 looks odd. Indeed, it is very small (less than 10−4)for normalized eigenvectors in O(1). This means that thismode is mainly supported in its tangential components (wehave checked they have a reasonable size). Mode 8 is in facta torsion mode, which means a dominant stretching effect,whereas the other ones have a more pronounced bendingcharacter.

Figure 13 provides the first distinct eigen-frequenciesclassified by the nature of the eigenvector (namely the num-ber of nodal regions of u3) as a function of the thickness innatural scales. The organization of these eigen-frequenciesalong affine lines converging to positive limits as ε → 0is remarkable. We may expect a convergence as ε → 0 ofthe solution uε of problem (3) provided the loading hasa finite number of angular frequencies in ϕ (the displace-ment will converge to the highest angular frequency ofthe load). Nevertheless, such a phenomenon is specific tothe axisymmetric nature of the shell (b) and could not begeneralized to other sensitive shells. Computations with a

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

00 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1

Torsion

246810

Figure 13. Model (b). Eigen-frequencies versus thickness (2ε). Acolor version of this image is available at http://www.mrw.interscience.wiley.com/ecm

concentrated load (which, of course, has an infinite numberof angular frequencies) display a clearly nonconvergingbehavior (Chapelle and Bathe (2003), Section 4.5.3).

Family (c). The midsurface S is a half-cylinder descri-bed in cylindrical coordinates (r, θ, y) by θ ∈ (0, π), r = 1and y ∈ (−1, 1). The family of shells ε has its upperand lower surfaces characterized by the same angular andaxial condition, and the radii r = 1 + ε and r = 1 − ε,respectively. We clamp ε along its lateral boundariesθ = 0 and θ = π and leave it free everywhere else. Thisis a well-known example of flexural shell, where the spaceof inextensional displacements contains the space, cf. (80)(note that, below, zr = z3)

VF,0 := z = (zr , zθ, zy); zy = 0, zr = zr (θ), zθ = zθ(θ)

with ∂θzθ = zr and zθ = zr = ∂θzr = 0 in θ = 0, π

(93)

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Besides these patterns independent of the axial variable y,there is another subspace VF,1 of inextensional displace-ments, where zy is independent on y and zr , zθ are linearin y:

VF,1 :=

z = (zr , zθ, zy); zy = zy(θ),

zθ = −y∂θzy(θ), zr = −y∂2θ zy(θ)

with zy = zθ = zr = ∂θzr = 0 in θ = 0, π

(94)

and VF = VF,0 ⊕ VF,1. We agree to call ‘constant’ the dis-placements associated with VF,0 and ‘linear’ those associ-ated with VF,1.

We have computed the first ten eigen-frequencies (4) bya FE p-discretization based on two layers of elements in

the transverse direction and a midsurface mesh of 8 × 6curved quadrangles. For the half-thickness ε = 0.0025, weplot the vertical component uz = ur sin θ + uθ cos θ of theeigenmodes u: In Figure 14, the first six constant flexuraleigenmodes and in Figure 15, the first three linear flexuraleigen-modes (their components uy clearly display a nonzeroconstant behavior in y). The shapes of the eigen-modes forlarger values of the thickness are similar. In Figure 16, wehave plotted in logarithmic scale these eigen-frequencies,classified according to the behavior of the flexural eigen-modes (‘constant’ and ‘linear’). The black line has theequation ε → ε/4: Thus we can see that the slopes ofthe eigen-frequency lines are close to 1, as expected bythe theory (at least for Koiter model). In Figure 17, werepresent the first nonflexural modes (with rank 10 forε = 0.01 and rank 8, 9 for ε = 0.04).


3.488e + 0002.790e + 0002.093e + 0001.395e + 0006.976e − 0011.192e − 007

−6.976e − 001−1.395e + 000−2.093e + 000−2.790e + 000−3.488e + 000

Y

Z

X



1.846e + 0006.626e − 001

−5.207e − 001−1.704e + 000−2.887e + 000−4.070e + 000−5.254e + 000−6.437e + 000−7.620e + 000−8.803e + 000−9.987e + 000

Y

Z

X



8.326e + 0006.661e + 0004.996e + 0003.330e + 0001.665e + 0001.192e − 007

−1.665e + 000−3.330e + 000−4.996e + 000−6.661e + 000−8.326e + 000

Y

Z

X



6.918e + 0005.405e + 0003.892e + 0002.379e + 0008.658e − 001

−6.472e − 001−2.160e + 000−3.673e + 000−5.186e + 000−6.699e + 000−8.212e + 000

Y

Z

X


7.517e + 0006.014e + 0004.510e + 0003.007e + 0001.503e + 000

−4.768e − 007−1.503e + 000−3.007e + 000−4.510e + 000−6.014e + 000−7.517e + 000

Y

Z

X


N-Freq. = 1.284e−002Mode Shape = 10 (Uz)

Max = 6.616e + 000Min = −9.213e + 000

6.616e + 0005.033e + 0003.450e + 0001.867e + 0002.845e − 001

−1.298e + 000−2.881e + 000−4.464e + 000−6.047e + 000−7.630e + 000−9.213e + 000

Y

Z

X

Figure 14. Model (c). Vertical components of modes 1, 2, 5, 6, 9 and 10 for ε = 0.0025. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm



1.542e + 0011.234e + 0019.253e + 0006.168e + 0003.084e + 000

−2.146e − 006−3.084e + 000−6.168e + 000−9.253e + 000−1.234e + 001−1.542e + 001

Y

Z

X



1.449e + 0011.159e + 0018.695e + 0005.796e + 0002.898e + 0004.768e − 007

−2.898e + 000−5.796e + 000−8.695e + 000−1.159e + 001−1.449e + 001



1.328e + 0011.063e + 0017.970e + 0005.313e + 0002.657e + 000

−4.768e − 007−2.657e + 000−5.313e + 000−7.970e + 000−1.063e + 001−1.328e + 001

Y

Z

XY

Z

X

Figure 15. Model (c). Vertical components modes 3, 4 and 7 for ε = 0.0025. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

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0.0025 0.01 0.04 0.16

0.001

0.002

0.01

0.02

0.1

Constant flexural modesLinear flexural modes eps/4

Figure 16. Model (c). Eigen-frequencies versus ε in log–logscale. A color version of this image is available at http://www.mrw.interscience.wiley.com/ecm

6.4.3 Thin element eigen-frequency computations

We present in Tables 2–4 the computation of the firsteigen-frequency of the shell ε in families (a) and (b)for a moderate thickness (ε = 0.02) and a small thickness(ε = 0.00125) and for family (c) for a moderate thickness(ε = 0.04) and a small thickness (ε = 0.0025), respectively,for a moderate thickness (ε = 0.04) and a small thickness

(ε = 0.0025). The degree q is the degree in the transversedirection (according to Section 6.1.3 there is one layer ofelements). We notice that, for an accuracy of 0.01% andε = 0.02, the quadratic kinematics is not sufficient, whereasit is for ε = 0.00125. No locking is visible there. In fact,the convergence of the q-models to their own limits is morerapid for ε = 0.02.

6.5 Conclusion

It is worthwhile to point out that the most serious difficul-ties we have encountered in computing all these modelsoccurred for ε = 0.00125 and model (b) – the sensitiveshell: Indeed, in that case, when ε → 0, the first eigen-mode is more and more oscillating, and the difficulties ofapproximation are those of a high-frequency analysis. It isalso visible from Tables 3 and 4 that the computationaleffort is lower for the cylinder than for the sensitive shell,for an even better quality of approximation.

It seems that, considering the high performance ofthe p-version approximation in a smooth midsurface (foreach fixed ε and fixed degree q we have an expo-nential convergence in p), the locking effects can beequilibrated by slightly increasing the degree p as ε

decreases.


N-Freq. = 4.785e−002Mode Shape = 10 (Uz)

Max = 6.988e + 000Min = −1.090e + 001

6.988e + 0005.200e + 0003.411e + 0001.623e + 000

−1.657e − 001−1.954e + 000−3.743e + 000−5.531e + 000−7.320e + 000−9.108e + 000−1.090e + 001

Y

Z

X



2.588e + 0001.871e + 0001.155e + 0004.380e + 001

−2.788e − 001−9.955e − 001−1.712e + 000−2.429e + 000−3.146e + 000−3.862e + 000−4.579e + 000

Y

Z

X



3.426e + 0002.816e + 0002.205e + 0001.595e + 0009.840e − 0013.734e − 001

−2.372e − 001−8.478e + 001−1.458e + 000−2.069e + 000−2.680e + 000

Y

Z

X

Figure 17. Model (c). First nonflexural modes for ε = 0.01 and ε = 0.04. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

Table 2. Thin element computations for the first eigen-frequency of model (a).

p ε = 0.02 and q = 2 ε = 0.02 and q = 3 ε = 0.00125 and q = 2

DOF e-freq. % err. DOF e-freq. % err. DOF e-freq. % err.

1 297 0.2271659 37.967 396 0.2264908 37.557 297 0.2055351 36.4372 729 0.1694894 2.938 828 0.1694269 2.900 729 0.1560694 3.6013 1209 0.1652870 0.386 1308 0.1652544 0.366 1209 0.1537315 2.0494 2145 0.1648290 0.108 2244 0.1648001 0.090 2145 0.1517604 0.7415 3321 0.1646992 0.029 3636 0.1646693 0.011 3321 0.1508741 0.1526 4737 0.1646859 0.021 5268 0.1646555 0.002 4737 0.1506988 0.0367 6393 0.1646849 0.020 7140 0.1646544 0.002 6393 0.1506544 0.0078 8289 0.1646849 0.020 9252 0.1646543 0.002 8289 0.1506447 0.000

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Table 3. Thin element computations for the first eigen-frequency of model (b).



1 864 0.0597700 89.68 1152 0.0595287 88.91 864 0.0462144 932.22 2016 0.0326855 3.73 2304 0.0326036 3.46 2016 0.0129819 189.93 3168 0.0318094 0.95 3456 0.0317325 0.70 3168 0.0064504 44.064 5472 0.0316330 0.39 5760 0.0315684 0.18 5472 0.0047030 5.045 8352 0.0316071 0.30 9216 0.0315319 0.06 8352 0.0045085 0.696 11 808 0.0316011 0.28 13 248 0.0315223 0.03 11 808 0.0044800 0.067 15 840 0.0316000 0.28 17 856 0.0315200 0.03 15 840 0.0044780 0.018 20 448 0.0315998 0.28 23 040 0.0315195 0.03 20 448 0.0044779 0.01

Table 4. Thin element computations for the first eigen-frequency of model (c).



1 567 0.0514951 210.2 756 0.0510683 208.7 567 0.0397025 3666.2 1311 0.0207290 24.9 1500 0.0206911 24.7 1311 0.0079356 653.13 2055 0.0167879 1.2 2244 0.0167596 0.98 2055 0.0011505 9.1884 3531 0.0166354 0.02 3720 0.0166091 0.08 3531 0.0010578 0.3955 5367 0.0166293 0.02 5928 0.0166011 0.03 5367 0.0010548 0.1086 7563 0.0166289 0.02 8496 0.0166004 0.02 7563 0.0010541 0.0457 10 119 0.0166288 0.02 11 424 0.0166003 0.02 10 119 0.0010538 0.0128 13 035 0.0166288 0.02 14 712 0.0166002 0.02 13 035 0.0010537 0.002

Of course, there exist many strategies to overcome lock-ing in different situations: Let us quote here (Bathe andBrezzi, 1985; Brezzi, Bathe and Fortin, 1989; Arnold andBrezzi, 1997) as ‘early references’, on mixed methods,which result in a relaxation of the zero-membrane-energyconstraint. These methods are addressed in other chaptersof the Encyclopedia.

ACKNOWLEDGMENTS

The authors wish to thank Domique Chapelle (INRIA)for stimulating discussions, Martin Costabel and YvonLafranche (fig4tex macro package for drawing figures, seehttp://perso.univ-rennes1.fr/yvon.lafranche/fig4tex/ReferenceGuide.html) (University of Rennes) for their valuabletechnical support.

NOTES

[1] We have a similar situation with plates, where thesolution uε,KL of the Kirchhoff–Love model gives backthe first generating terms on the asymptotics of uε, cf.Theorem 2.

[2] The actual Kirchhoff–Love displacement (satisfyingei3 = 0) is slightly different, containing an extra quad-ratic surface term.

[3] The complementing operator C defined in (45) forplates satisfies CU1,1,0

KL = U1,1,2K .

[4] These norms are those of the domains of the frac-tional powers As of the Sturm–Liouville operatorA: ζ → ∂x((1 − x2)∂xζ) on the interval (−1, 1). Suchan approach is now a standard tool in the p-versionanalysis.

[5] Of course, different mesh designs are possible onthin domains. If one wants to capture boundary layerterms with an exponential rate of convergence, a h-prefinement should be implemented near the edges ofε, Dauge and Schwab (2002).

[6] Here, for ease of presentation, we use the numberingsystem for plate models displayed in Table 1, where wealso provide the number dj of fields in each directionfor bending models, that is, for which the surface com-ponents are odd and the normal component even in x3.

REFERENCES

Actis RL, Szabo BA and Schwab C. Hierarchic models for lam-inated plates and shells. Comput. Methods Appl. Mech. Eng.1999; 172(1–4):79–107.

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Abstract: Concerning thin structures, such as plates and shells, the idea of reducing the equations of elasticity to two-dimensional models defined on the midsurface seems relevant. Such a reduction was first performed thanks to kinematicalhypotheses about the transformation of normal lines to the midsurface. As nowadays, the asymptotic expansion of thedisplacement solution of the three-dimensional linear model is fully known at least for plates and clamped elliptic shells, westart from a description of these expansions in order to introduce the two-dimensional models known as hierarchical models:These models extend the classical models, and presuppose the displacement to be polynomial in the thickness variable,transverse to the midsurface. Because of the singularly perturbed character of the elasticity problem as the thicknessapproaches zero, boundary, or internal layers may appear in the displacements and stresses, and so may numerical lockingeffects. The use of hierarchical models, discretized by higher degree polynomials (p-version of finite elements) may helpovercome these severe difficulties.

Keywords: shells, plates, hierarchical models, asymptotic expansion, modal analysis, eigen-frequencies, finite elements

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