FREE VIBRATIONS OF AXISYMMETRIC SHELLS: PARABOLIC AND ELLIPTIC … · · 2016-02-26FREE...

FREE VIBRATIONS OF AXISYMMETRIC SHELLS :

PARABOLIC AND ELLIPTIC CASES

MARIE CHAUSSADE-BEAUDOUIN, MONIQUE DAUGE, ERWAN FAOU, AND ZOHAR YOSIBASH

ABSTRACT. Approximate eigenpairs (quasimodes) of axisymmetric thin elastic domains with lat-erally clamped boundary conditions (Lame system) are determined by an asymptotic analysis asthe thickness (2) tends to zero. The departing point is the Koiter shell model that we reduce byasymptotic analysis to a scalar model that depends on two parameters: the angular frequency k andthe half-thickness . Optimizing k for each chosen , we find power laws for k in function of that provide the smallest eigenvalues of the scalar reductions. Corresponding eigenpairs generatequasimodes for the 3D Lame system by means of several reconstruction operators, including bound-ary layer terms. Numerical experiments demonstrate that in many cases the constructed eigenpaircorresponds to the first eigenpair of the Lame system.

Geometrical conditions are necessary to this approach: The Gaussian curvature has to be non-negative and the azimuthal curvature has to dominate the meridian curvature in any point of themidsurface. In this case, the first eigenvector admits progressively larger oscillation in the angularvariable as tends to 0. Its angular frequency exhibits a power law relation of the form k =

with = 14 in the parabolic case (cylinders and trimmed cones), and the various s25 ,

37 , and

13 in

the elliptic case. For these cases where the mathematical analysis is applicable, numerical examplesthat illustrate the theoretical results are presented.

1. INTRODUCTION

A shell is a 3D body determined by a surface S R3 and a (small) parameter : Such a body isobtained by thickening S on either side by along a unit normal field N to S (see (2.1) supra fora precise definition). The shell, denoted by is supposed to be made of a linear homogeneousisotropic material. We investigate the first free vibration mode of such a shell when its lateralboundary is clamped and 0. A vibration mode is the eigenpair (,u) where is the square ofthe eigenfrequency and u the eigen-displacement. The mode is denoted as exact if it is an eigenpairof the 3D Lame system in complemented by suitable boundary conditions, see equations (2.2)(2.7) for an explicit formulation. The thin domain limit 0 pertains to shell theory.

Shell theory consists of finding surface models, i.e., systems of equations posed on S, approximat-ing the 3D Lame system when tends to 0. This approach was started for plates (the case whenS is flat) by Kirchhoff, Reissner and Mindlin see for instance [24, 34, 30] respectively. When thestructure is a genuine shell for which the midsurface has nonzero curvature, the problem is evenmore difficult and was first tackled in the seminal works of Koiter, John, Naghdi and Novozhilovin the sixties [25, 26, 27, 23, 32, 31]. A large literature developed afterwards aimed at laying

Date: 27 January 2016.2010 Mathematics Subject Classification. 74K25, 74H45, 74G10, 35Q74, 35C20, 74S05.Key words and phrases. Lame, Koiter, asymptotic analysis, scalar reduction.

1

2 MARIE CHAUSSADE-BEAUDOUIN, MONIQUE DAUGE, ERWAN FAOU, AND ZOHAR YOSIBASH

more rigorous mathematical bases to shell theory see for instance the works of Sanchez-Palencia,Sanchez-Hubert [36, 37, 38, 35], Ciarlet, Lods, Mardare, Miara [11, 13, 12, 28] and the book [9],and more recently Dauge, Faou [20, 21, 15]. Most of these works apply to the static problem, andthe results strongly depend on the geometrical nature of the shell (namely parabolic, elliptic orhyperbolic according to the Gaussian curvature K of S being zero, positive or negative).

Much fewer works were devoted to free vibrations of thin shells. Plates were addressed before-hand, see [10, 14]. To the best of our knowledge, all theoretical works devoted to the asymptoticanalysis of eigenmodes in thin elastic shells were associated with a surface model, as the Koitermodel [25, 26]:

K() = M + 2B, (1.1)

where M is the membrane operator, B the bending operator, and the half-thickness of the shell.These two operators are 3 3 systems posed on S, thus with two tangent variables, but acting on3-component vector fields . At this point, the vector fields have to be represented in surface fittedcomponents, the tangential and normal components and 3. The membrane operator M is oforder 2 on tangential components , but of order 0 on the normal component 3 (for plates, Mdoes not act on 3 at all). The bending operator B has a complementary role: It is order 4 on 3(for plates, B is a multiple of the biharmonic operator 2 acting on the sole normal component).We will briefly recall these operators in section 2.2.

In [35], the essential spectrum of the membrane operator M (the set of s such that M is notFredholm) was characterized in the elliptic, parabolic, and hyperbolic cases. The series of papersby Artioli, Beirao Da Veiga, Hakula and Lovadina [7, 2, 3] investigated the first eigenvalue ofmodels like K(). Effective results hold for axisymmetric shells with clamped lateral boundary:Defining the order of a positive function 7 (), continuous on (0, 0], by the conditions

> 0, lim0+

() + = 0 and lim0+

() = (1.2)

they proved that = 0 in the elliptic case, = 1 for parabolic case, and = 23

in the hyperboliccase.

Concentrating on axisymmetric shells, it is possible to apply a full Fourier decomposition withrespect to the azimuthal angle , allowing all eigenpairs to be classified by their azimuthal fre-quency k(). For such shells Beirao et al. and Artioli et al. [7, 2, 3] investigated by numericalsimulations the azimuthal frequency k() of the first eigenvector of K(): Like in the phenomenonof sensitivity [35], the lowest eigenvalues are associated with eigenvectors with growing angularfrequencies and k() exhibits a negative power law of type , for which [3] identifies the ex-ponents = 1

4for cylinders (see also [6] for some theoretical arguments), = 2

5for a particular

family of elliptic shells, and = 13

for another particular family of hyperbolic shells.

Similarly to the aforementioned publications, we consider here axisymmetric shells whose mid-surface S is parametrized by a smooth positive function f representing the radius as a function ofthe axial variable (see Figure 1):

F : I T S(z, ) 7 (f(z) cos, f(z) sin, z).

(1.3)

Here I is a bounded interval and T is the torus R/2Z. We focus on cases when sensitivity shows

FREE VIBRATIONS OF AXISYMMETRIC SHELLS : PARABOLIC AND ELLIPTIC CASES 3

FIGURE 1. An axisymmetric shell with notations.

up, i.e., when the azimuthal frequency k() of the first eigenvector tends to infinity as the thicknesstends to 0. As will be shown, the rules driving this phenomenon are far to be straightforward, anddepend in a non trivial manner on the geometry of the shell: In the sole elliptic case, we showthat there exist at least three distinct power laws for k(). This is the expression of some bendingeffects and may sound as a paradox since for elliptic shells the membrane is an elliptic system inthe sense of Agmon, Douglis and Nirenberg [1], see [22]. However, there also exist elliptic shellsfor which k() remains constant, see the computations for a spherical cap in [16].

1.1. High frequency analysis. Our departing point is a high frequency analysis of the membraneoperator M on surfaces S with a parametrization of type (1.3). By the Fourier decomposition nat-urally induced by the cylindrical symmetry, we define Mk as the membrane operator acting at thefrequency k N and we perform a scalar reduction of the eigenproblem by a special factorizationin a formal series algebra in powers of the small parameter 1

k. This mathematical tool, developed

for cylindrical shells in the PhD thesis [5] of the first author, reduces the original eigenproblem(which is a 3 3 system) to a scalar eigenproblem posed on the transverse component of thedisplacement. This way, we can construct in a variety of parabolic and elliptic cases a new ex-plicit scalar differential operator Hk whose first eigenvalue 1(Hk) has a computable asymptoticsas k

1(Hk) = h0 + h1k

1 +O(k2), 0 < 1 < 2. (1.4)In (1.4) all coefficients and exponents depend on shells geometry, i.e. on the function f in (1.3).This leads to a quasimode construction for Mk that is valid for all parabolic shells of type (1.3)and all elliptic shells with azimuthal curvature dominating. The operator Hk strongly depends onthe nature of the shell:{

Hk = k4H4 in the parabolic case (i.e., when f = 0),Hk = H0 + k

2H2 in the elliptic case (i.e., when f < 0).(1.5)

with explicit operators H0, H2 and H4, cf. sect. 6.1 for formulas. Let us mention at this point thatfor hyperbolic shells such a suitable scalar reduction Hk cannot be found.

This membrane scalar reduction induces a Koiter-like scalar reduced operator A() for the shellthat we define at the frequency k by

Ak() = Hk + 2k4B0 (1.6)


where the function B0 is positive and explicit (k4 corresponding to the leading order in the Fourierexpansion of the bending operator B). Then we shall show that the first eigenvalue of A() is theinfimum on all angular frequencies of the first eigenvalues of Ak():

1(A()) = infkN

1(Ak()

). (1.7)

In all relevant parabolic cases (i.e., cylinders and cones) and a variety of elliptic cases, we provein this paper:

(i) The infimum in (1.7) is reached for k = k() satisfying a power law of the form1

k() = , (1.8)

with depending only on f and positive. The exponent is calculated so to equilibratek1 (cf. (1.4)) and 2k4 k42/ , which yields:

=2

4 + 1(1.9)

(ii) The smallest eigenvalue of the reduced scalar model A() has an asymptotic expansion ofthe form, as 0

1(A()) = a0 + a11 +O(2), 0 < 1 < 2, (1.10)

where a0 coincides with the coefficient h0 present in (1.4) and 1 is given by the formula(replace k with into the term k1 in (1.4))

1 = 1 =21

4 + 1(1.11)

(iii) The corresponding eigenvector 1() has a leading term that involves another exponentdenoted by which characterizes its concentration scale with respect to . The value = 0 means no concentration at all, and > 0 means that 1() develops layers aroundsome points z0 I. The latter situation occurs when the reduced membrane operator Hkhas a singular perturbation character as k .

Once the asymptotic expansions for the Koiter scalar reduced operator A() is resolved we con-struct quasimodes for the full Koiter model K(). Then, by energy estimates linking 2D and 3Dmodels similar to those of [15], we find an asymptotic upper bound m1() for the first eigenvalueof the 3D Lame system in the shell. This upper bound is given by the first two terms in (1.10):

m1() = a0 + a11 . (1.12)

To make the analysis more complete, we perform numerical simulations showing that, in a numberof cases, the upper bound m1() is actually a good approximation of the true first eigenvalue. Tothis end computations are performed at three different levels:

(1D) We calculate a0, a1 of (1.10) and of (1.8). We either use explicit formulas when available(such as the cylinder and specific elliptic cases), or compute numerically the spectrum ofthe 1D reduced operators Ak() by 1D finite element methods.

1This equality has to be understood up to a O(1) term coming from rounding to the nearest integer.


(2D) Using a Fourier spectral method we compute the eigenvalues of the three-dimensionaloperator on the cylindrical shell by decomposing the operator in Fourier, which yields 2Doperators acting on the meridian domain for all k. For a given thickness , we computethe lowest eigenvalue of these operators for a collection of k which provides the lowesteigenvalue of the three-dimensional operator and the value of k().

(3D) We compute the eigenvalues of the Lame system using 3D finite element methods.

1.2. Specification in the parabolic and elliptic cases. In the Lame system we use the engineer-ing notations of the material parameters: E is the Young modulus and is the Poisson ratio. Theshells to which our analysis apply are uniquely defined by f in (1.3). The inverse parametrization(the axial variable function of the radius) would not provide distinct cases where our analysis isapplicable.

The parabolic cases are those for which f = 0 on I. So f is affine. We classify parabolic casesin two types:

(1) Cylinder f is constant;(2) Cone f is affine and not constant.

The elliptic cases are those for which f < 0 on I. To conduct our analysis, we assume moreoverthat the azimuthal curvature dominates the meridian curvature (admissible cases), which amountsto

1 + f 2 + ff 0. (1.13)Then elliptic cases are discriminated by the behavior of the function H0 that is the first term of thescalar reduction Hk, cf. (1.5),

H0 = Ef 2

(1 + f 2)3. (1.14)

We classify admissible elliptic cases in three types:

(1) Toroidal H0 is constant.(2) Gauss H0 is not constant and reaches its minimum at z0 inside I and not on its boundary

I, with the exception of cases for which H0 or 1 + f 2 + ff are zero at z0.(3) Airy H0 is not constant and reaches its minimum at z0 in the boundary I, with the

exception of cases for which H0 or 1 + f2 + ff are zero at z0.

We summarize in Table 1 our main theoretical results on the exponents 1, , 1, on the azimuthalfrequency k() and the scale of the quasi-eigenvector, and on the quasi-eigenvalue (qev) m1().The exponents of [2] are confirmed. The names of models used for numerical simulations arealso mentioned in this table, whereas in Figure 2 we represent these models in their 3D versionfor = 0.2.

Our paper divides in three parts: The first part is devoted to the setting of the problem and to basicson shells: After the present introduction, we revisit in sect. 2 the linear shell theory in generalwith a brief introduction of the 3D and 2D problems, and in sect. 3 we particularize formulas foraxisymmetric shells. The second part gathers our theoretical study involving the scalar reductionand its first eigenpair: In sect. 4 we set the principles of the high frequency analysis, in sect. 5 and


Type (Model) 1 1 a0 a1 k() m1() [2]

PARABOLICCylinder (A) 4 14 1 0 explicit wrt 1D evs

1/4 0 a1 1

Cone (B) 4 14 1 0 optimization of 1D evs 1/4 0 a1

ELLIPTICToroidal (D) 2 13

23 H0 optimization of 1D evs

1/3 0 a0 + a12/3 0

Gauss (H) 1 2525 H0(z0) explicit

2/5 12 a0 + a1

2/5 0

Airy (L) 2337

27 H0(z0) explicit

3/7 23 a0 + a1

2/7 0

TABLE 1. Summary of exponents 1, , 1, frequency k(), scale , and qev m1()

FIGURE 2. The five models A, B, D, H, L, used for computations (here = 0.2).

6 we address more particularly the parabolic and elliptic cases, respectively. Finally in the lastpart we present numerical experiments (sect. 7) addressing a model for each of the five main typesdescribed above. We conclude in sect. 8. We provide in Appendix A details on the factorizationin formal series leading to the scalar reduction and in Appendix B variational formulations in themeridian domain after Fourier decomposition of the 3D Lame system.

1.3. Overview of main notation. To relieve the complexity of notation, we gather here somedefinitions relating to coordinate systems, operators, and spectrum.

1.3.1. Coordinates. We use three systems of coordinates:

Cartesian coordinates t = (t1, t2, t3) R3 with coordinate vectors Et1 , Et2 , Et3 .

Cylindrical coordinates (r, , ) R+ T R related to Cartesian coordinates by relations(t1, t2, t3) = T (r, , ) with t1 = r cos, t2 = r sin, t3 = . (1.15)

The coordinate vectors associated with the transformation T are Er = rT , E = T , andE = T . We have

Er = Et1 cos+ Et2 sin, E = rEt1 sin+ rEt2 cos, and E = Et3 . (1.16)

Normal coordinates (x1, x2, x3), specified as (z, , x3) in our case. Such coordinates are relatedto the surface S and a chosen unit normal field N to S. The variable x3 is the coordinate along N.


The variables (x1, x2), specified as (z, ) in our case, parametrize the surface. The full transfor-mation F : (z, , x3) 7 (t1, t2, t3) sends the product I T (, ) onto the shell and isexplicitly given by

t1 =(f(z) + x3

1s(z)

)cos, t2 =

(f(z) + x3

1s(z)

)sin, t3 = z x3 f

(z)s(z)

, (1.17)

where s =

1 + f 2. The restriction of F on the surface S (corresponding to x3 = 0) gives backF (1.3). The coordinate vectors associated with the transformation F are zF =: Ez, T thatcoincides with E above, and 3F =: E3. On the surface S, x3 = 0 and E3 coincides with N,whereas Ez and E are tangent to S.

These three systems of coordinates determine the contravariant components of a displacement uin each of these systems by identities

u = ut1Et1 + ut2Et2 + u

t3Et3 = urEr + u

E + uE = u

zEz + uE + u

3E3 . (1.18)

The cylindrical and normal systems of coordinates are suitable for angular Fourier decompositionT 3 7 k Z. The Fourier coefficient of rank k of a function u is denoted by uk

uk =1

2

20

u() eik d.

For functions on , the Fourier coefficients are defined on the meridian domain R2 of .

For a 3D displacement u defined on or a surface displacement defined on S, we have first toexpand them in a suitable system of coordinates (cylindric or normal) and then calculate Fouriercoefficients of their components, for instance

uk = (ur)kEr + (u)kE + (u

)kE with (ur)k(r, ) =1

2

20

ur(r, , ) eik d, etc...

or

k = (z)kEz + ()kE + (

3)kN with (z)k(z) =1

2

20

z(z, ) eik d, etc...

1.3.2. Operators. We manipulate a collection of operators and their Fourier symbols. On theshell we have the Lame system L acting on 3D displacements u. After angular Fourier de-composition, we obtain the family of 33 operators (L)k defined on the meridian domain . Onthe surface S we have the membrane, bending and Koiter operators M, B and K(). They act on3-component surface displacements . On the meridian curve C of S, we have the correspondingfamilies Mk, Bk and Kk(). Finally, on the meridian curve C, we have our scalar reductionsHk and Ak() acting on functions . We go from a higher model to a lower one by reduction,and the converse way by reconstruction. For instance we go from u to by restriction to S. Theconverse way uses the reconstruction operator U (2.14). We go from k to k by selecting thenormal component of k. The converse way uses the reconstruction operators V[k] that we willconstruct.

1.3.3. Spectrum. We denote by (A) and ess(A) the spectrum and the essential spectrum of aselfadjoint operator A, respectively, which means the set of s such that A is not invertibleand not Fredholm, respectively.


2. ESSENTIALS ON SHELL THEORY

Recall that Cartesian coordinates of a point P R3 are denoted by t = (t1, t2, t3). A shell is athree-dimensional object defined by its midsurface S and its thickness parameter in the followingway: We assume that S is smooth and orientable, so that there exists a smooth unit normal fieldP 7 N(P) on S and so that for > 0 small enough the following map is one to one and smooth

: S (, )

(P, x3) 7 t = P + x3 N(P).(2.1)

The boundary of has two parts:

(1) Its lateral boundary 0 := (S (, )

),

(2) The rest of its boundary (natural boundary) 1 := \ 0.

2.1. 3D vibration modes. On the domain , we consider the Lame operator associated with anisotropic and homogeneous material with Young coefficient E and Poisson coefficient . Thismeans that the material tensor is given by

Aijk` =E

(1 + )(1 2)ijk` +

E

2(1 + )(ikj` + i`jk). (2.2)

For clamped boundary conditions the variational space is

V () := {u = (ut1 , ut2 , ut3) H1()3 , u = 0 on 0}. (2.3)For a given displacement field u let eij(u) = 12(iutj + juti) be the strain tensor, where i standsfor the partial derivative with respect to ti. The energy scalar product between two displacementsu and u is given by

a3D(u,u) =

Aijk`eij(u) ek`(u

) d , (2.4)

using the summation convention of repeated indices. The three-dimensional modal problem canbe written in variational form as:

Find (u, ) in V () R with u 6= 0 such that

u V (), a3D(u,u) =

utiuti d

. (2.5)

The strong formulation of (2.5) can be written as

Lu = u, (2.6)

where L is the Lame system

L = E2(1 + )(1 2)

((1 2) + div

)on , (2.7)

associated with Dirichlet BCs on 0 and natural BCs on the rest of the boundary. Its spectrum(L) is discrete and positive. So its eigenvalues in (2.5) can be ordered as an increasing sequence(n)n1

0 < 1 2 n


It would be desired to obtain an asymptotic series as 0 for any chosen n 1 of n, at least inthe axisymmetric case. This has been realized for plates [14]:

n = 2n +O(3) (2.8)

where n is the n-th bending eigenvalue. Presently we cannot achieve this goal in other situations.We thus concentrate on estimates for the first eigenvalue that is obtained by the minimum Rayleighquotient

1 = minuV ()

a3D(u,u)

u2L2().

In this paper we perform two tasks:

[A] Obtain asymptotic upper bounds for 1 via the construction of quasimodes for reducedmodels and reconstruction operators,

[B] Compute numerical approximation of the smallest eigenpair in (2.5) by finite elementmethods.

2.2. 2D shell models. The 2D reduction to the midsurface S , namely the membrane and bendingoperators, are defined via intrinsic geometrical objects attached to S. To introduce them, weneed generic parametrizations F : (x)1,2 t acting from maps neighborhoods V into themidsurface S. Associated tangent coordinate vector fields are

E = F , = 1, 2, with =

x.

Completed by the unit normal field N they form a basis {E1,E2,N} in each point of S. The metrictensor (a) and the curvature tensor (b) are given by

a = E,E and b = F ,N .

Denoting by (a) the inverse of (a), the curvature (symmetric) matrix is defined by

(b) with b = a

b.

The eigenvalues 1 and 2 of the matrix (b) are called the principal curvatures of S and theirproduct is the Gaussian curvature K. Here comes the classification of shells: If K 0, the shellis parabolic, if K > 0, the shell is elliptic, if K < 0, the shell is hyperbolic. Finally let R denotethe minimal radius of curvature of S

R = infPS

{min{|1(P)|1, |2(P)|1}

}. (2.9)

The basis {E,N} determines contravariant components (, 3) of a vector field on S:

= tiEti = E +

3N .

The covariant components are (, 3) with = a and 3 = 3. The 2D rigidity tensor on Sis given by

M =E

1 2aa +

E

2(1 + )(aa + aa).

Note that, even if S is flat (a = ), M is different than the 3D rigidity tensor A.


2.2.1. Membrane operator. The variational space associated with the membrane operator is

VM(S) = H10 (S)H10 (S) L2(S). (2.10)For an element = (, 3) in VM(S), the change of metric tensor = () is given by

() =12(D + D) b3,

where D is the covariant derivative on S, see [18, 19, 39]. The membrane energy scalar productis defined as

aM(, ) =

SM() (

) dS .

Here the volume form dS is| det(a)| dx1dx2. The variational formulation of the modal

problem associated with the membrane operator M is given by

Find (,) with VM(S) \ {0} and R such that for all VM(S),

aM(, ) =

S( +

33 ) dS. (2.11)

2.2.2. Bending operator and Koiter model. The variational space associated with the bendingoperator is

VB(S) = H10 (S)H10 (S)H20 (S). (2.12)For an element = (, 3) in VB(S), the change of curvature tensor = () is given by

() = DD3 + D(b) + b

D bb3.

The bending operator B acts on the variational space VB(S) and its energy scalar product is

aB(, ) =

1

3

SM()(

) dS .

For any positive , the Koiter operator K() is defined as M + 2B. It can be shown, see [8],that K() is elliptic with multi-order on VB(S) in the sense of Agmon-Douglis-Nirenberg [1]. Thecorresponding physical energy scalar product is

a2D(, ) = 2 aM(,

) + 23aB(, ). (2.13)

2.3. Reconstruction operators from 2D to 3D. The parametrizations of the midsurface inducelocal system of normal coordinates (x, x3) inside the shell and, correspondingly, the covariantcomponents u and u3 of a displacement u. The rationale of the shell theory is to deduce by anexplicit procedure a solution u of the 3D Lame system posed on the shell from a 2D solution of the Koiter model posed on the midsurface. This is done via a reconstruction operator U, cf[26, 27] and [15]. With any 2D displacement (x) defined on the midsurface S, U associates a3D displacement u depending on the three coordinates (x, x3) in . The operator U is definedby

U = T W (2.14)where W is the shifted reconstruction operator

W =

x3(D3 + b),

3 1 x3 () +

22 x

23

(),

(2.15)


and T : 7 T is the shifter defined as (T) = x3b and (T)3 = 3, see [31]. The2D elastic energy of is a good approximation of the 3D elastic energy of U, cf. [15, TheoremA.1]: For any (H2 H2 H3) VB(S), there holds, with non-dimensional constant Aa2D(, ) a3D(U,U) Aa2D(, )( R + 2L2), (2.16)where R is the minimal radius of curvature (2.9) of S, and L is the wave length for defined asthe largest constant such that the following inverse estimates hold

L | |H1(S) L2(S) and L ||H1(S) L2(S) . (2.17)

Note that for VB(S), the first two components of U satisfy the Dirichlet condition on 0,whereas the third one does not need to satisfy it. In order to remedy that, we add a correctorterm ucor to W to compensate for the nonzero trace g =

1 x3 () +

22 x

23

()

S .

This corrector term is constructed and its energy estimated in [15, sect. 7]. It has a simple tensorproduct form and exhibits the typical 3D boundary layer scale d/ with d = dist(P, 0):

ucor =(

0, 0, g (d

))>with C0 (R), (0) = 1.

True boundary layer terms live at the same scale, decay exponentially, but have a non-tensorform in variables (d, x3), see [17, 14] for plates and [21] for elliptic shells. Nevertheless thisexpression for ucor suffices to obtain good estimates: There holds

a3D(ucor,ucor) Aa2D(, )

(`

+3

`3

),

for ` the lateral wave length of defined as the largest constant such that

`| |2L2(S) + `

3| |2H1(S)

2L2(S) and `||

2L2(S) + `

3||2H1(S)

2L2(S) . (2.18)

Example 2.1. Let G in H2(R+) be such that G 0 for t 1. Let k N and 0 < < 0 for 0small enough. The function g(z, ) defined on S as

g(z, ) = eikG(d

)satisfies the estimates L |g|H1(S) gL2(S) and `|g|2L2(S) + `3|g|2H1(S) g2L2(S) for L and `larger than c(G) min{, k1} where the positive constant c(G) is independent of and k.

In the present work, we are interested in comparing 2D and 3D Rayleigh quotients so we introducethe following notations

Q3D(u) =a3D(u,u)

u2L2(), and Q2D() =

a2D(, )

22L2(S).

By similar inequalities as in [15] we can prove the following relative estimate

Theorem 2.2. (i) For all (H2 H2 H3) VB(S) and with U defined in (2.14) we setU = U ucor.

ThenU belongs to the 3D variational space V (). With L and ` the wave lengths (2.17) and

(2.18), let us assume L and `. We also assume Q2D() EM for a chosen constant


M 1 independent of . Then we have the relative estimates between Rayleigh quotients for small enough Q2D()Q3D( U) AQ2D()( R + 2L2 + (`)1/2 + M ) , (2.19)with a constant A independent of and .

(ii) If belongs to (H20 H20 H30 )(S), the boundary corrector ucor is zero and the aboveestimates do not involve the term

/` any more.

This theorem allows to find upper bounds for the first 3D eigenvalue 1 if we know convenientenergy minimizers for the Koiter model K() and if we have the relevant information abouttheir wave lengths.

3. AXISYMMETRIC SHELLS

An axisymmetric shell is invariant by rotation around an axis that we may choose as t3. Recall that(r, , ) R+ T R denote associated cylindrical coordinates satisfying relations (1.15) andcoordinate vectors are Er, E, and E given by (1.16). Accordingly, the (contravariant) cylindricalcomponents of a displacement u = utiEti are (u

r, u, u ) so that u = urEr + uE + uE . Inparticular the radial component of u is given by

ur = ut1 cos+ ut2 sin. (3.1)

The components u and u are called azimuthal and axial, respectively.

An axisymmetric domain R3 is associated with a meridian domain R+ R so that = {x R3, (r, ) and T}. (3.2)

3.1. Axisymmetric parametrization. For a shell that is axisymmetric, let be its meridiandomain. The midsurface S of is axisymmetric too. Let C be its meridian domain. We have arelation similar to (2.1)

: C (, ) 3((r, ), x3

)7 (r, ) + x3N(r, ) . (3.3)

The meridian midsurface C is a curve in the halfplane R+ R.Assumption 3.1. Let I denote any bounded interval and let z be the variable in I.(i) The curve C can be parametrized by one map defined on I by a smooth function f :

I Cz 7 (r, ) = (f(z), z)

with f : z 7 r = f(z). (3.4)

(ii) The shells are disjoint from the rotation axis, i.e., there exists Rmin > 0 such that f Rmin.Remark 3.2. We impose condition (ii) to avoid technical difficulties due to the singularity at theorigin. We have observed that, if we keep this condition, the inverse parametrization z = g(r)does not bring new examples in the framework that we investigate in this paper. For instanceannular plates pertain to this inverse parametrization, but they fall in [14] that provides a completeeigenvalue asymptotics.


The parametrization (3.4) of the meridian curve C provides a parametrization of the meridiandomain by I (, ): Let us introduce the arc-length

s(z) =

1 + f (z)2, z I . (3.5)

The unit normal vector N to C at the point (r, ) = (f(z), z) is given by ( 1s(z)

,f(z)s(z)

) and theparametrization by

I (, ) 3 (z, x3) 7(f(z) + x3

1s(z)

, z x3 f(z)s(z)

) .

The parametrization (3.4) also induces the parametrization F (1.3) of the midsurface S by thevariables (z, ) I T. The unit normal vector N to S at the point F (z, ) is given by

N = s(z)1(Er f (z)E )

while tangent coordinate vectors are Ez = zF and E = F , i.e.

Ez = f(z)Er + E

while E coincides the coordinate vector of same name corresponding to cylindrical coordinates(1.16). The metric tensor (a) is given by E,E with , {z, }, i.e.(

azz az

az a

)(z) =

(s(z)2 0

0 f(z)2

). (3.6)

The curvature tensor and Gaussian curvature K are respectively given by(bzz b

z

bz b

)(z) =

(f (z)s(z)3 0

0 f(z)1s(z)1

)and K(z) = f

(z)

f(z)s(z)4. (3.7)

So the curvature tensor is in diagonal form, and K is simply the product of its diagonal elements.

Definition 3.3. We call bzz the meridian curvature and b the azimuthal curvature.

Since we have assumed that f R0 > 0, all terms are bounded and we find that

(1) If f 0, i.e. f is affine, the shell is (nondegenerate) parabolic. If f is constant, the shellis a cylinder, if not it is a truncated cone (without conical point!).

(2) If f < 0, the shell is elliptic.(3) If f > 0, the shell is hyperbolic.

3.2. 2D axisymmetric models in normal coordinates. Relations (1.3) and (2.1) define normalcoordinates (z, , x3) in the thin shell . For example when the midsurface S is a cylinder (fconstant), the normal coordinates are a permutation of standard coordinates: (z, , x3) = (, , r).The associate (contravariant) decomposition of surface displacement fields is written as =zEz + E + 3N, where 3 is the component of the displacement in the normal direction Nto the midsurface, z and the meridian and azimuthal components respectively, defined so thatthere holds

t1Et1 + t2Et2 +

t3Et3 = zEz +

E + 3N .


Note that the azimuthal component is the same as defined by cylindrical coordinates. The covari-ant components are

z = s2z, = f

2, and 3 = 3.The change of metric tensor () has the expression in normal coordinates

zz() = zz f f

s2z

f

s3

z() =1

2(z + z)

f

f

() = +ff

s2z +

f

s3 ,

(3.8)

while the change of curvature tensor () is written as

zz() = 2z3

f 2

s43 +

2f

s3zz +

f s2 5f f 2

s5z

() = 23

1

s23

2

fs

2f

s3z

z() = z3 +f

s3z

1

fsz +

2f

f 2s .

(3.9)

4. PRINCIPLES OF CONSTRUCTION: HIGH FREQUENCY ANALYSIS

The construction is based on the following postulate:

Postulate 4.1. The eigenmodes associated with the smallest vibrations are strongly oscillating inthe angular variable and this oscillation is dominating.

This means that if this postulate happens to be true for certain families of shells, our constructionwill provide rigorous quasimodes and, moreover, these quasimodes are candidates to be associatedwith lowest energy eigenpairs.

We may notice that Postulate 4.1 is wrong for planar shells. But it appears to be true for nonde-generate parabolic shells and some subclasses of elliptic shells.

4.1. Angular Fourier decomposition. We can perform a discrete Fourier decomposition in theshell T and in its midsurface S C T = I T. For a function u(z, ) in L2(I T)we denote its Fourier coefficients by

uk(z) =1

2

20

u(z, ) eik d, k Z, z I.

This Fourier decomposition diagonalizes any operator L with coefficient independent of withrespect to the angular modes eik, k Z, due to the relation:

(Lu)k = Lkuk.

Here the operators of interest are the Lame system L written in normal components, togetherwith the membrane and bending operators M and B defined on the spaces VM(S) and VB(S),


composing the Koiter operator K(). We denote by (L)k, Mk, Bk and Kk(), k Z, theangular Fourier decomposition of L, M, B and K(), respectively.

The (non decreasing) collections of the eigenvalues ,kj of (L)k for all k Z gives back all

eigenvalues of L. Note that since L is real valued, the eigenvalues for k and k are identical.Thus 1 = infkN

,k1 and we denote by k() the smallest natural integer k such that (if exists)

1 = ,k()1 .

Postulate 4.1 means that k() as 0.

4.2. High frequency analysis of the membrane operator. The eigenmode membrane equation(2.11) at azimuthal frequency k takes the form

Mkk = kAk (4.1)

where A is the mass matrix

A =

azz 0 0

0 a 0

0 0 1

=s2 0 0

0 f2 0

0 0 1

. (4.2)We construct quasimodes for Mk as k , i.e. pairs (k, k) with k in the domain of theoperator Mk and satisfying the estimates

(Mk k) kL2(S) (k)kL2(S) with (k)/

k 0 as k .

Now we consider the membrane operator as a formal series with respect to k

Mk = k2M0 + kM1 + M2 M[k], with M[k] = k2nN

knMn , (4.3)

and try to solve (4.1) in the formal series algebra:

M[k][k] = [k]A[k]. (4.4)

Here the multiplication of formal series is the Cauchy product: For two formal series a[k] =n knan and b[k] =

n knbn, the coefficients of the series a[k] b[k] =

n kncn are given by

cn =

`+m=n a`bm.

The director M0 of the series M[k] is given in parametrization r = f(z) by

M0 =E

1 2

1

2f2s20 0

0 1f4

0

0 0 0

. (4.5)Its kernel is given by all triples of the form (0, 0, 3)>. This is the reason why we look fora reduction of the eigenvalue problem for M to a scalar eigenvalue problem set on the normalcomponent 3. The key is a factorization process in the formal series algebra proved in [5, Chap.3],

M[k]V[k] [k]AV[k] = V0 (H[k] [k]) . (4.6)


Here V(k) is a (formal series of) reconstruction operators whose first term V0 is the embeddingV0 = (0, 0, )> in the kernel of M0, and H[k] is the scalar reduction.

Theorem 4.2. Let be a formal series with real coefficients :

[k] =n0

knn.

For n 1, there exist operators Vn,z,Vn, : C(I) C(I) of order n 1, polynomial in j ,for j n 3, and for n 0 scalar operators Hn : C(I) C(I) of order n, polynomial inj , for j n 2 such that if we set :

V[k] =n0

knVn with Vn = (Vn,z,Vn,, 0)>, and H[k] =n0

knHn

we have (4.6) in the sense of formal series.

See Appendix A for more details on this theorem.

With the scalar reduction H[k] is associated the formal series problem

H[k][k] = [k][k] (4.7)

where [k] =

n0 knn is a scalar formal series. The previous theorem shows that any solution

to (4.7) provides a solution [k] = V[k][k] to (4.4).

The cornerstone of our quasimodes construction for Mk as k is to construct a solution [k]of the problem (4.7). This relies on the possibility to extract an elliptic operator Hk with compactresolvent from the first terms of the series H[k] as we describe in several geometrical situationslater on.

Remark 4.3. The essential spectrum ess(Mk) of the membrane operator Mk at frequency kcan be determined explicitly thanks to [4, Th.4.5]. It depends only on its principal part, whichcoincides with the (multi-degree) principal part of M2, and is given by the range of Ef(z)2s(z)2 forz I, see [5, sect. 2.7] for details. With formula (3.7), we note the relation with the azimuthalcurvature

ess(Mk) =

{E b(z)

2 , z I}. (4.8)

As a consequence of Assumption 3.1, the minimum of ess(Mk) is positive.

4.3. High frequency analysis of the Koiter operator. Similar to the membrane operator M[k],the bending operator expands as

Bk = k4B0 +4

n=1

k4nBn B[k],

with first term

B0 =

0 0 00 0 00 0 B0

with B0 = E1 2

1

3f 4. (4.9)


We notice that we have the commutation relation

B0V[k] = V0B0 .

Therefore the identity (4.6) implies for all the identity(M[k] + 2k4B0

)V[k] [k]AV[k] = V0

(H[k] + 2k4B0 [k]

). (4.10)

Thus the same factorization as for the membrane operator will generate the quasimode construc-tions for the Koiter operator as soon as the higher order terms of Bk correspond to perturbationterms. This is related to Postulate 4.1. The identity (4.10) motivates the formula (1.6) defining thereduced Koiter operator Ak() = Hk + 2k4B0. In the following two sections we provide Ak()and its lowest eigenvalues in several well defined cases.

5. NONDEGENERATE PARABOLIC CASE.

We assume in addition to Assumption 3.1

f(z) = Tz +R0, z I, with R0 > 0, T R . (5.1)

If T = 0, the corresponding surface S is a cylinder of radius R0 and the minimal radius ofcurvature R (2.9) equals to R0. So we write f = R in the cylinder case. If T 6= 0, the surface S isa truncated cone. The arc length (3.5) is s =

1 + T 2.

5.1. Membrane scalar reduction in the parabolic case. The first terms Hn of the scalar formalseries reduction of the membrane operator have been explicitly calculated in [5] in the cylindricalcase T = 0 and have the following expression in the general parabolic case:

H0 = H1 = H2 = H3 = 0 and H4(z, z) = E(f 2s64z +

6f f

s63z +

6f 2

s62z

). (5.2)

It is relevant to notice that H4 is selfadjoint on H20 (I) with respect to the natural measure dI =f(z)s(z) dz , since there holds

H4, I =

E

(1 + T 2)3

If(z)2 2z

2z dI . (5.3)

This also proves that H4 is positive. The Dirichlet boundary conditions = z = 0 on I arethe right conditions to implement the membrane boundary condition = 0 on I through thereconstruction operators Vn, see (A.6) (A.7). The eigenvalue formal series [k] starts with 4that is the first eigenvalue of H4:

0 = 1 = 2 = 3 = 0 and 4 > 0. (5.4)

The pair (k, k)

k =

0n+m6

knmVnm and k = k44 (5.5)


with (4, 0) an eigenpair of H4, and m (m = 1, . . . , 6) constructed by induction so that themembrane boundary conditions k = 0 are satisfied, is a quasimode for M

k. For instance, in thecylindrical case f = R, the triple k takes the form

k =

000

+ ik

0R00

+ 1k2

R002

+ ik3

0R30 +R23

1k4

(+2)R30 +R2R34

+ . . .(5.6)

and the boundary conditions are, for z I0(z) = 0,

0(z) = 0, 2(z) = R

20(z), 2(z) = ( + 2)R

20 (z), 3(z) = 0, . . . (5.7)

Recall that the minimum of the essential spectrum of Mk is positive by Remark 4.3. For |k| largeenough, Mk has therefore at least an eigenvalue = 4k4 under its essential spectrum and

dist(k44, (M

k)). k5, k . (5.8)

5.2. Koiter scalar reduction in the parabolic case. The leading term of the series H(k) is Hk =k4H4, as mentioned in the introduction, see (1.5). So, the leading term of the scalar reduction ofthe Koiter operator is, cf. (4.10)

Ak() = k4H4 + 2k4B0 = k

4H4 +2

3

E

1 2k4

f 4. (5.9)

5.2.1. Optimizing k. The operator Ak() is self-adjoint on H20 (I) and positive. Let A1 (, k) beits smallest eigenvalue. For any chosen we look for kmin = k() realizing the minimum A1 ()of A1 (, k) if it exists:

A1 () = A1 (, k()) = min

kR+A1 (, k).

To homogenize the terms k4 and 2k4 let us define () by setting

() = 1/4k(), (5.10)

so that we look equivalently for (). There holds

k()4H4 + 2k()4B0 =

( 1()4

H4 + ()4B0).

Therefore () does not depend on . Let 1() be the first eigenvalue of the operator1

4H4 +

4B0 . (5.11)

The function 7 1() is continuous and, since H4 and B0 are positive, it tends to infinity as tends to 0 or to +. Therefore we can define min as the (smallest) positive constant such that1() is minimum

1(min) = minR+

1() =: a1 . (5.12)

Thus k() satisfies a power law that yields a formula for the minimal first eigenvalue A1 ():

k() = 1/4min and A1 () = a1. (5.13)


5.2.2. Case of cylinders. In the cylindrical case T = 0, things are easier because f is constant.So everything can be written as a function of the first Dirichlet eigenvalue bilap1 of the bilaplacianoperator 2 on H20 (I) as we explain now. We have

H4 = ER2 2 and B0 =

E

1 21

3R4.

So the eigenvalue of 14H4 +

4B0 is

1() =1

4ER2bilap1 +

4 E

1 21

3R4.

It is minimum for min such that

4min = R3

3(1 2)bilap1

and we find that the minimum eigenvalue (5.12) is

1(min) =2E

R

bilap1

3(1 2)=: a1 . (5.14)

Thusk() = 1/4R3/4

(3(1 2)bilap1

)1/8. (5.15)

Remark 5.1. Denote by bilap the first eigenvalue of 2 on the unit interval (0, 1). We have therelation bilap1 =

bilap L4 with the length L of the interval I.

5.2.3. 2D reconstruction. We convert the law (5.13) giving k as a function of into a law giving as a function of k

= k44min (5.16)and insert it into the identity (4.10). We obtain(

M[k] + 8mink4B0

)V[k] [k]AV[k] = V0

(H[k] + 8mink

4B0 [k]). (5.17)

So the series [k] starts with the first eigenvalue 4 = 4mina1 of H4 + 8minB0. Let 0 be an

associated eigenvector. Like before, but now with this new 0, the pair (k, k) defined by (5.5)is a quasimode for Mk + 8mink

4B0 = Mk + 2k4B0 with membrane boundary conditions. To

restore the dependency on (via law (5.16)), we write

k = k() and k = k(), with k = k().

Since with law (5.16) the terms 2(Bk k4B0) are of order k5 or higher, the same pair is aquasimode for the full Koiter operator Kk(), but still with membrane boundary conditions.

5.2.4. Quasimodes for the Koiter model. Bending boundary layers. The full bending boundaryconditions 3 = 0 and 3 = 0 on I cannot be implemented in general for the quasimodes(k(), k()). The singularly perturbed nature of the Koiter operator causes the loss of theseboundary conditions between the bending and membrane operator. Solutions of the Koiter model,just as eigenvectors, incorporate boundary layer terms. In all cases investigated in this paper, theseterms exist at the scale d/

with d = dist(z, I). Such a scaling appears in [33] in a variety of

nondegenerate cases (the boundary of S is noncharacteristic for the curvature). It is rigorouslyanalyzed in [21] in the case of static clamped elliptic shells.


More precisely, the scaled variable is (for I = (z, z+))

Z =d

with d = z+ z or z z, (5.18)

according as we consider the localization at the end z0 = z+ or z0 = z of the interval I. In viewof law (5.13), we can write the operator Kk() as a series in powers of 1/4. In the rapid variable Z,there holds zG(Z) = 1/2G for any profile G(Z), which provides a new formal series K[1/4].Its leading term K0 is compatible with the full bending boundary conditions at Z = 0. It has thefollowing form in the cylindrical case f = R

K0 =E

1 2

2Z 0

RZ

0 12R2

2Z 0

RZ 0

1R2

+ 134Z

.It allows to construct a series of exponentially decreasing vector functions G[1/4] satisfying a for-mal series relation of the type K[1/4]G[1/4] = [1/4]G[1/4], that compensate for the missingtraces of k(), see [5, Section 5.6]. Our true quasimode has now the form (k(),

k()) with

k()(z) = k()(z) + (d)

6n=2

n/4Gn(Z) with the same k() = A1 () = a1 . (5.19)

Here is a smooth cut-off that localizes near the boundary I. The outcome is the spectralestimate

dist(a1 , (K

k())). 5/4 with k() = 1/4min, as 0. (5.20)

5.3. 3D reconstruction and Rayleigh quotients. We construct a three-component vector fieldon the surface S by setting in normal coordinates

(z, ) = eikk(z) with k given in (5.19) and k = k() = 1/4min.

By construction, belongs to the variational space VB(S), and by the elliptic regularity of theKoiter problem, it also belongs to (H2 H2 H3)(S). So we may apply the reconstructionoperator introduced in Theorem 2.2: Set

u =U.

To take advantage of the comparison (2.19) between the Rayleigh quotients of and u, we haveto exhibit the behavior of the wave lengths L = L (2.17) and ` = ` (2.18) of as 0.Following the construction of the fields , we see that they all originate from an eigenfunction 0that does not depend on . The nontrivial behavior of L and ` arises from, cf. Example 2.1:

The Koiter boundary layer terms Gn(Z) = Gn(d/1/2) that contribute a term in 1/2, The azimuthal oscillation eik that contributes a term in k1 ' 1/4.

As a result we find in the nondegenerate parabolic case

L, ` & 1/2.


So the assumptions of Theorem 2.2 are uniformly satisfied for the family () and the estimate(2.19) reads now Q2D()Q3D(u) . 1/4Q2D() . 5/4 .Stricto sensu, we have at hand a family of 3D displacements u with azimuthal frequency k() 1/4min such that Q3D(u) a1 . 5/4.Moreover the trace of the transverse component u3 on the midsurface (x3 = 0) is close to thegenerating eigenvector 0. We have proved the results summarized in the first two lines of Table1. Our numerical experiments (Model A, sect. 7.1, and Model B, sect. 7.2) suggest that, in fact,(a1,u) is an approximation of the first 3D eigenpair.

6. ELLIPTIC CASE (SMALL MERIDIAN CURVATURE)

The elliptic case in parametrization r = f(z), z I, corresponds to the situation f < 0 on I.

6.1. Membrane reduction in the general case. When the parametrizing function f is not affine,i.e., when f 6 0, the scalar reduction of the membrane operator has non-vanishing first terms asfollows:

H0(z, z) = Ef 2

s6, H1(z, z) = 0, H2(z, z) = H

(2)2 (z)

2z + H

(1)2 (z) z + H

(0)2 (z) (6.1)

with

H(2)2 (z) = 2E

(ff s6

+f 2f 2

s8

)H

(1)2 (z) = 2E

(2f f s6

+ff

s6 2ff

f 2

s8+

2f 2f f

s8 7f

2f f 3

s10

)H

(0)2 (z) = E

( 10f

2f 2

s8+

4f f

s6+

2f 2f

fs6 ( 2)ff

2f 3

s10 5ff

f f

s8

+ff (4)

s6+

2f 2f f (4)

s8+

36f 2f 2f 4

s12+

( 2)ff 3

s8 6f

2f 4

s10

20f2f f 2f

s10

) 0

(1s f

f

s3

)2.

(6.2)

The rank-3 operator in the formal series H[k] is given by

H3(z, z) =( 1s2

+2ff

s4

2f 2f 2

s6

)1, (6.3)

and the rank-4 operator can be written as

H4(z, z) =4j=0

H(j)4 (z)

jz , with H

(4)4 (z) = E

(4f 3f s8

+3f 4f 2

s10+f 2

s6

)(6.4)

where the other terms H(j)4 (z) are smooth functions of z.


So, H0(z, z) = H0(z) is the multiplication by a function (which can be seen as a potential) andwe check that H2 is a selfadjoint operator of order 2 on H10 (I) with respect to the natural measuredI = f(z)s(z) dz :

H2, I =

I

( H(2)2 (z) z z + H

(0)2 (z)

)

dI . (6.5)

We recall from (3.7) that the principal curvatures are bzz =f

s3and b = 1fs . Note that both are

negative in the elliptic case.

Remark 6.1. (i) The function H0/E coincides with the square of the meridian curvature

H0 = E (bzz)

2.

(ii) There holds the following relation between H(2)2 and the principal curvatures

H(2)2 = 2Ef 2

s2bzz(b

bzz). (6.6)

(iii) Similarly

H(4)4 = E

f 4

s4(b 3bzz)(b bzz). (6.7)

6.2. High frequency analysis of the membrane operator in the elliptic case. As mentionedabove, we have to select one or several terms starting the series H[k] that will play the role ofan engine to work out a recurrence and allow to solve the formal series problem (4.7). In theparabolic case, this engine is h4H4. In the elliptic case, H0 is the multiplication by the positivefunction E(bzz)

2. Its spectrum is essential and its bottom determines 0

0 = EminzI

(bzz)2. (6.8)

We have to complete H0 by further terms so that to obtain an operator with discrete spectrum closeto the minimum energy 0. This will be the case for the operator

Hk = H0 + k2H2 (6.9)

if, cf. condition (1.13),

H(2)2 0 on I, i.e. |b| |bzz| on I, (6.10)(use (6.6)), with strict inequalities for the values of z where H0 attains its minimum 0. It isinteresting to note that the latter condition implies that, cf. (6.8) and (4.8),

minzI

(bzz)2 < min

zI(b)

2, i.e. 0 < miness(Mk),

which means that the expected limit at high frequency will be attained by eigenvalues below theessential spectrum.

Remark 6.2. We note that in the hyperbolic case, f > 0, so bzz > 0. Hence the coefficient H(2)2

is always negative and our analysis never applies in the hyperbolic case. Besides, in this case,0 is not the membrane high frequency limit, that is indeed 0 (recall that the exponent in (1.2) is = 2

3in hyperbolic case).


From now on, we assume that (6.10) holds and we discuss the lowest eigenpairs of the operatorsHk defined in (6.9) and

Ak() = H0 + k2H2 +

2k4B0, where B0 =1

3

E

1 21

f 4(6.11)

in relation with properties of the potential H0. For simplicity we denote

g(z) := H(2)2 (z), (6.12)and consider successively the cases when H0 has a non-degenerate minimum inside or on theboundary of the interval I, or when it is constant.

6.3. Internal minimum of the potential (Gaussian case). Besides (6.10), we assume that H0has a (unique) nondegenerate minimum in z0 I. Thus

0 = H0(z0) and 2zH0(z0) > 0.

We assume moreover g(z0) > 0.

6.3.1. High frequency analysis for the membrane operator. Then the lowest eigenpairs of themembrane reduction Ak = H0 + k2H2 as k are driven by the harmonic oscillator

g(z0) 2Z +Z2

22zH0(z0) . (6.13)

Here, the new homogenized variable Z spans R and is linked to the physical variable z by therelation

Z =k (z z0) . (6.14)

This change of variable can be applied to the formal series reduction (4.6) as follows: Let L[k] =k0 k

nLn(z, z) be a formal series such that Ln is an operator of order n. By Taylor expan-sion around z0, we can expand for all n the operator Ln(z, z) =

jn k

j/2Ln,j(Z, Z). Byreordering the powers of kj/2, we thus see that we can write

L[k] L[k1/2] =n0

kn/2Ln(Z, Z),

where the operators Ln have polynomial coefficients in Z. Applying this change of variable to theformal series reduction given in Theorem 4.2, we obtain the new identity

M[k1/2]V [k1/2] [k1/2]AV [k1/2] = V0 (H[k1/2] [k1/2]) , (6.15)where M[k1/2], V [k1/2] and H[k1/2] are the formal series induced by the formal series M[k],V[k] and H[k] respectively. We also agree that [k1/2] is related with the old series old[k] =

n0 knn,old by the identities n = 0 if n is odd, and n = n/2,old if n is even. Moreover, we

calculate that

H0 = H0(z0), H1 = 0, and H2 = g(z0) 2Z +Z2

22zH0(z0).

Like for (4.7), the previous reduction leads to consider the formal series problem

H[k1/2][k1/2] = [k1/2][k1/2] . (6.16)


The first equation induced by this identity is H00 = 00, hence we have found again 0 =H0 = H0(z0). Since for any we have nowH0 = 0, the next equations yield

H10 = 10 and H20 = 20.Therefore 1 = 0 (which is coherent with what was agreed in identity (6.15)) and 0 is an eigen-vector of the harmonic oscillator (6.13). The eigenvalues of this latter operator are

(2` 1) c , ` = 1, 2, . . . with c = 12

g(z0) 2zH0(z0) (6.17)

and the corresponding eigenvectors are Gaussian functions. Taking 0(Z) as the first eigenmode(` = 1) we can construct the first terms 1, 2, . . . of the formal series problem (6.16). As the coef-ficients of the operators Hj depend polynomially on Z, these terms are exponentially decreasingwith respect to Z. We can then define the pair (k, k) by the formula

k = (z)(V00 +

1n+m6

k(n+m)/2(Vnm)(k(z z0))

)k = H0(z0) + k

1c ,(6.18)

where C0 (I) is identically equal to 1 in a neighborhood of z0. This pair is a quasimode forthe full membrane operator Mk as k , and we obtain that

dist(H0(z0) + k

1c , (Mk)). k3/2, k . (6.19)

Note that in this case, the boundary conditions are automatically fulfilled as the quasimode con-structed is localized near z0.

6.3.2. High frequency analysis for Koiter and Lame operators. Now we consider the operatorAk() defined in (6.11). We look for its smallest eigenvalue A1 (, k) and for each > 0 smallenough, look for k() such that A1 (, k()) is minimum. Setting :=

2k4, we see that theoperator Ak() has the form

W + k2H2 with W = H0 + B0.

If is small enough, the function W has the same property as H0, i.e., it has a (unique) nonde-generate minimum. Let z0() be the point where this minimum is attained. By implicit functiontheorem, the correspondence z0() is smooth for small enough and there holds

A1 (, k) = H0(z0()) + B0(z0()) +k1

2

g(z0()) 2z (H0 + B0)(z0()) +O(k3/2)

ButH0(z0()) = H0(z0) +O(2), 2zH0(z0()) = 2zH0(z0) +O(),B0(z0()) = B0(z0) +O(), g(z0()) = g(z0) +O().

Hence

A1 (, k) = H0(z0) + B0(z0) +k1

2

g(z0) 2zH0(z0) +O(2) +O(k1) +O(k3/2).

Let us set

b = B0(z0) and c =12

g(z0) 2zH0(z0) . (6.20)


So, replacing by its value 2k4, we look for k = k() such that 2k4b + k1c is minimum andsuch that = 2k4 is small2. We homogenize the powers of k by letting () = k() 2/5, andsetting A1 () =

A1 (, k()) we find

k() = 2/5 and A1 () = H0(z0) + a1 2/5 +O(3/5), (6.21)

with the explicit constants and a1:

=( c

4b

)1/5and a1 = (4bc4)1/5(1 +

1

4) . (6.22)

Along the same lines as in the parabolic case, we convert the power law for k (6.21) into thepower law = (/k)5/2. We can then consider a formal series reduction as in (5.17) and combineit with the change of variable Z =

k(z z0). The same analysis as before yields quasimodes

(k(), k()). Here k() = A1 () and k() has a form similar to (6.18), whereas k = k().

Note that these quasimodes remain localized around z0 and hence bending boundary layers do notshow up as they did in the parabolic case. We thus obtain

dist(m1() , (K())

). 3/5 with m1() = H0(z0) + a1 2/5. (6.23)

Remark 6.3. If H0 attains its minimum a0 in a finite number of points z(i)0 , we can construct

quasimodes attached to each of these points of the same form as above, and with disjoint supports.The associated quantities obey to the same formulas as in (6.21)-(6.23)

k(i)() = (i) 2/5 and m(i)1 () = a0 + a(i)1

2/5,

with (i) and a(i)1 defined by (6.22) with the values of quantities b and c at point z(i)0 . Then

m1() = minim(i)1 () and k() = k

(i0)() for i0 such that the previous minimum is attained.

6.3.3. 3D reconstruction and Rayleigh quotients. As in the parabolic case, we construct a three-component vector field on the surface S by setting in normal coordinates

(z, ) = eikk(z) with k given in (6.18) and k = 2/5min,

and by setting

u =U.

Since all traces of any order of vanish on I, there is no boundary corrector and we are in case(ii) of Theorem 2.2. So we only have to estimate the behavior of the wave length L = L (2.17)of as 0. We note the influence of:

The profiles Gn(k(z z0)) with k ' 2/5, that contribute a term in 1/

k ' 1/5,

The azimuthal oscillation eik that contributes a term in k1 ' 2/5.

As a result we find in the nondegenerate parabolic case

L & 2/5.

So the assumptions of Theorem 2.2 are uniformly satisfied for the family () and the estimate(2.19) reads now Q2D()Q3D(u) . Q2D() . .

2We check that = O(2/5)


Thus, we have exhibited a family of 3D displacements u with azimuthal frequency k() 2/5such that Q3D(u)m1() . 3/5 with m1() = H0(z0) + a1 2/5.So we have proved the results summarized in the third line of Table 1. The numerical experiments(Model H, sect. 7.4) suggest that (m1(),u) is indeed an approximation of the first 3D eigenpair.

6.4. Minimum of the potential on the boundary (Airy case). We assume that H0 attains itsminimum at a point z0 I with zH0(z0) 6= 0. Let us agree that z0 is the left end of I, i.e.,z0 = z, so that we have

0 = H0(z0) and zH0(z0) > 0.

We still assume g(z0) > 0. The analysis is somewhat similar to the previous case, though a littlemore tricky.

6.4.1. High frequency analysis for the membrane operator. Membrane boundary layers. We meetthe Airy-like operator

g(z0)2Z + Z zH0(z0) (6.24)on H10 (R+) instead the harmonic oscillator (6.13). The homogenized variable Z is given by

Z = (z z0)k2/3. (6.25)

We can perform an analysis very similar to the previous case by doing a change of variable in theformal series reduction. This yields formal series problem in powers of k1/3 whose first termsare given byH0 + k2/3H2 whereH0 = H0(z0) andH2 is the operator (6.24). The eigenvalues ofthe model operator (6.24) are given by

z(`)Airy

(g(z0)

)1/3 (zH0(z0)

)2/3, ` = 1, 2, . . . (6.26)

where z(`)Airy is the `-th zero of the reverse Airy function Ai. We find that the first eigenvalue of themembrane reduction Hk satisfies

H1 (k) = H0(z0) + k2/3c +O(k1) with c = z(1)Airy

(g(z0)

)1/3 (zH0(z0)

)2/3. (6.27)

Using the reconstruction operators V [k1/3] in the scaled variable allows to construct displacementk from an eigenfunction profile 0(Z) of the Airy operator. However, the first terms of thereconstruction take the form:k

4/3kzk1kk3

where

kz

kk3

=f

2(b ( + 2)bzz)Z0if 2(b + bzz)0

0

+O(k1/3) .While we can impose 0(0) = 0 to ensure that k = 0 at first order, we see that we have in generalkz 6= 0. To construct a quasimode, we have to add new boundary layer terms to k.

To determine such boundary layers near z0 = z, we introduce the scaled variable Z = kd withd = zz. Like already seen for the Koiter operator (sect. 5.2.4), the formal series operator M[k]


is changed to a new formal series M [k] whose first term is given by

M 0 =

1s42Z +

12

(b)2 1+

2i(b)

2Z1s2

(bzz + b)Z

1+2i(b)

2Z 12 (b)

22Z +1f4

i 1f2

(b + bzz)

1s2

(bzz + b)Z i 1f2 (b

+ b

zz) (b

+ b

zz)

2

where the quantities are evaluated in z0 = z. We can prove that this operator yields boundarylayer profiles G(Z) exponentially decreasing with respect to Z = kd, and satisfying Gz(0) = azfor any given number az, which allows to compensate for the trace of the first term of kz .

We obtain a compound quasimode combining terms at scale k2/3d and terms at scale kd, anddeduce in the end

dist(H0(z0) + k

2/3c , (Mk)). k1, k . (6.28)

6.4.2. High frequency analysis for Koiter operator. The Koiter scalar reduction operator Ak(),see (6.11) is still an Airy-like operator because the minimum of H0 + 2k4 is still z0 for 2k4 smallenough. We look for k = k() such that 2k4b + k2/3c is minimum. We homogenize 2k4 withk2/3. We find

k() = 3/7 and A1 () = H0(z0) + a1 2/7 +O(3/7), (6.29)

with the explicit constants and a1, with b = B0(z0) and c defined in (6.27):

=( c

6b

)3/14and a1 = (6bc6)1/7(1 +

1

6) . (6.30)

Note that in this case, two types of boundary layer terms are present: the one constructed above(membrane boundary layer) and the bending boundary layers terms associated with the Koiteroperator, see sect. 5.2.4. We obtain

dist(m1() , (K())

). 3/7 with m1() = H0(z0) + a1 2/7.

At this point, the reconstruction operatorU is not precise enough to allow us to conclude as in the

parabolic and Gaussian cases. Nevertheless numerical experiments (Model L, sect. 7.5) tend toconfirm that the first eigenvalue of the Lame operator L behaves like m1().

6.5. Constant potential (toroidal case). Let us assume that H0 is constant. We recall that H0 =E(bzz)

2. But bzz coincides with the curvature of the arc C of equation r = f(z) in the meridianplane. So, bzz is constant if and only if C is a circular arc. Let R be its radius and (r, z) R2 beits center. Notice that the center of the circular arc may be at negative r. Then, in the elliptic casef < 0,

f(z) = r +R2 (z z)2, (6.31)

and the principal curvatures are given by

bzz = 1

Rand b(z) =

1

R

(1 r

f(z)

). (6.32)

So in this case, we have

H0 =E

R2= 0 and g = H(2)2 = 2E

f

s2rR2

. (6.33)


Now the Koiter scalar reduction operator Ak() is H0 + k2H2 + 2k4B0 where H0 is a constantfunction acting as a simple shift on the spectrum. In this case, no concentration occurs, and wehave simply to come back to the approach used for the parabolic case mutatis mutandis, with k2

instead of k4.

6.5.1. Membrane scalar reduction. We assume the sharp version of condition (6.10) (strict in-equalities) that ensures that H2 has a compact resolvent and is semibounded from below. Thanksto (6.33), we find that such condition is equivalent to

r < 0 . (6.34)

Let 2 be the first eigenvalue of H2. Then the first eigenvalue of the membrane scalar reductionoperator Hk = H0 + k2H2 is 0 + k22 and we can deduce that

dist(0 + k

22 , (Mk)). k5/2, k . (6.35)

6.5.2. Koiter scalar reduction. The operator Ak() = H0 + k2H2 + 2k4B0 is self-adjoint onH10 (I). Let A1 (, k) be its first eigenvalue. For any chosen we look for k() realizing theminimum of A1 (, k). We set

() = k() 1/3 (6.36)so that our operator becomes

H0 + 2/3( 1()2

H2 + ()4 B0

).

Therefore does not depend on . Let 1() be the first eigenvalue of the operator1

2H2 +

4 B0 . (6.37)

The function 7 1() is continuous. At this point we need the following extra assumption:2 > 0, i.e. H2 > 0. (6.38)

Then the same argument as in the parabolic case allows to define min as the (smallest) positiveconstant such that 1() is minimum

1(min) = minR+

1() =: a1 . (6.39)

As an illustration of the non-trivial behavior of the quantities 2, min and a1, we plot them versusr in Figure 3 (we choose R = 2 and z = 0).

Thus k() satisfies a power law that yields a formula for the minimal first eigenvalue A1 ():

k() = 1/3min and A1 () = m1() = H0 + 2/3a1 , (6.40)

and after adding membrane and bending boundary layer terms as in the Airy case we arrive to

dist(m1() , (K())

). with m1() = H0 + 2/3a1.

We note that, in contrast with the two previous cases when H0 is not constant, the lower order termH

(0)2 of the operator H2 is involved in the asymptotics. Finally, like in the Airy case, we would

need a more complete reconstruction operator combined with plates boundary layer terms to


2 1.5 1 0.5 00.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

r

First ev of H2

0

a1

min

FIGURE 3. Quantities 2, min and a1 vs r (with R = 2 and z = 0).

conclude the construction of 3D minimizers or 3D quasimodes. Nevertheless, numerical experi-ments (Model D, sect. 7.3) prove that, at least for some values of r, the lowest 3D eigenpairs fitthe asymptotics m1() = H0 + 2/3a1 and k() = 1/3min.

7. MODELS

We present in this section five models: cylinders (Model A), cones (Model B), toroidal barrels(Model D), Gaussian barrels (Model H), and Airy barrels (Model L), representing each of thefive types that we could investigate from a theoretical point of view. We choose for all models

E = 1 and = 0.3

and perform 1D, 2D and 3D computations for each model. The 2D and 3D computations areperformed with finite element codes (MELINA [29] for 2D and STRESSCHECK3 for 3D) and for afinite set of values of ranging from 0.2 to 104 (in general this set contains the values 0.2, 0.1,and 5 10j , 2 10j , 10j for j = 2, 3, 4).

The 1D calculations consist in computing the coefficients a0, a1 of (1.10) and of (1.8). ForGaussian and Airy barrels we use our explicit formulas (6.22) and (6.30). For cylinders, conesand toroidal barrels, we compute with MATLAB the spectrum of the one dimensional reducedoperators (5.11) and (6.37) and optimize with respect to the parameter .

The 2D calculations solve the Lame system at azimuthal frequency k on meridian domains ,see the corresponding variational formulations in Appendix B. For each thickness parameter ,any integer value of k from 0 to a certain cut-off frequency kmax = kmax() is used. The cut-offfrequency kmax is determined so that we can observe a minimum for the first eigenvalue dependingon k. This provides the numerical value for k(). The domain is meshed by curvilinearquadrilaterals of geometric degree 3. The meshes contain 2 elements in the thickness direction,

3Stress Check 9.0 is a trade mark of Engineering Software Research and Development, Inc., St. Louis, MO 63141,U.S.A.


Model k() m1() a0 a1 Remainder

PARABOLICA / Cylinder 1/4 2.9323126 a1 0 3.385232 15 3/2 (asymptotics)B / Cone 1/4 2.1247294 a1 0 3.446424 15 3/2 (asymptotics)ELLIPTICD / Toroidal 1/3 0.8593528 a0 + a12/3 0.2500000 0.715000 0.2 (upper bound)H / Gauss 2/5 0.7590108 a0 + a12/5 0.0625000 0.607854 0.05 3/5 (asymptotics)L / Airy 3/7 0.8514053 a0 + a12/7 0.1780449 1.554722 2.9 4/7 (asymptotics)

TABLE 2. Numerical values for asymptotic quantities k() and m1(). Observedasymptotics for the remainder 1 m1() (with 1 obtained by 2D computations).

and 8, 12 or 16 in the meridian direction. The polynomial interpolation degree of the FEM is 6 ineach direction.

The 3D calculations solve the Lame system L on the 3D shells . The azimuthal frequency k()is observed by counting the oscillations of the radial component (3.1) of the first eigenmode.

We represent in figures 4, 7 and 10, the meridian domains in the (r, z) plane for models A,B and D, respectively. The curve C is dotted. Figures 15 and 18 provide for = 0.2 formodels H and L. Figures 5, 8, 11, 13, and 16 show the lowest computed eigenvalue 1 and theassociated azimuthal frequency k() versus in loglog scale (in base 10). For elliptic models D,H and L, the difference 1 0 is plotted. The 1D asymptotics is the line 7 m1() (1.12) (or 7 m1()0 in elliptic models). Figures 6, 9, 12, 14, and 17 show the radial component of thefirst 3D eigenvector for three values of and the five models, respectively. Figures 15 and 18 aresurface plots of the first 2D eigenvectors in Gaussian and Airy barrels, respectively, exhibiting theconcentration of the modes as decreases. For visibility, they are scaled with respect to the widthin order to be represented on the meridian domain with thickness parameter = 0.2.

We summarize in Table 2 the numerical values of the asymptotic quantities m1() and k(), aswell as the observed asymptotics for the remainder 1 m1() for each of the five models.

The formulas providing parameters , a0 and a1 are given in the following equations: (5.12)(5.15) for models A and B, (6.31), (6.39) for model D. Concerning models H and L, the functionH0/E =

f 2

s6is equal to (1

4+ 3

4z2)2/(1 + (1

4z + 1

4z3)2)3 and reaches its minimum in the interior

point z0 = 0 for model H, and in the boundary point z0 = 0.5 for model L. Formulas for and a1are given in (6.22) with (6.20), and (6.30) with (6.27) for models H and L, respectively.


7.1. Model A: Cylindrical shells. The midsurface parametrization, cf (5.1), is given byf(z) = R, z (1, 1), R = 2.

axis = 0.2

axis = 0.1

axis = 0.05

FIGURE 4. Model A: Meridian domains for several values of .

4 3 2 1 03.5

3

2.5

2

1.5

1

0.5

0

log10(epsilon)

log10(e

igenvalu

e)

Model A: Cylinder

2D computations

3D computations

Asymptotics

4 3 2 1 00

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

log10(epsilon)

log10(k

)

Model A: Cylinder

2D computations

3D computations

Asymptotics

FIGURE 5. Model A: First eigenvalue 1 and associated azimuthal frequency k().

FIGURE 6. Model A: First eigenmode (radial component) for = 102, 103, 104.


7.2. Model B: Conical shells. The midsurface parametrization, cf (5.1), is given byf(z) = Tz +R, z (1, 1), T = 0.5, R = 1.5.

axis = 0.2

axis = 0.1

axis = 0.05

FIGURE 7. Model B: Meridian domains for several values of .

4 3.5 3 2.5 2 1.5 1 0.53.5

3

2.5

2

1.5

1

0.5

0

log10(epsilon)

log10(e

igenvalu

e)

Model B

2D computations

3D computations

Asymptotics

4 3.5 3 2.5 2 1.5 1 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

log10(epsilon)

log10(k

)

Model B

2D computations

3D computations

Asymptotics

FIGURE 8. Model B: First eigenvalue 1 and associated azimuthal frequency k().

FIGURE 9. Model B: First eigenmode (radial component) for = 102, 103, 104.


7.3. Model D: Toroidal barrels. The midsurface parametrization is, cf (6.31)

f(z) = r +R2 z2, z (1, 1), R = 2 and r = 1

axis = 0.2

axis = 0.1

axis = 0.05

FIGURE 10. Model D: Meridian domains for several values of .

4 3.5 3 2.5 2 1.5 1 0.53

2.5

2

1.5

1

0.5

log10(epsilon)

log10(e

igenvalu

e

H0)

Model D

2D computations

3D computations

Asymptotics

4 3.5 3 2.5 2 1.5 1 0.50

0.2

0.4

0.6

0.8

1

1.2

log10(epsilon)

log10(k

)

Lam on Model D

2D computations

3D computations

Asymptotics

FIGURE 11. Model D: Difference 1 0 and associated azimuthal frequency k().

FIGURE 12. Model D: First eigenmode (radial component) for = 102, 103, 104.


7.4. Model H: Gaussian barrel. The midsurface parametrization is

f(z) = 1 z28 z4

16, z (1, 1).

4 3.5 3 2.5 2 1.5 1 0.51.8

1.6

1.4

1.2

1

0.8

0.6

0.4

log10(epsilon)

log10(e

igenvalu

e

0)

Model H

2D computations

3D computations

Asymptotics

4 3.5 3 2.5 2 1.5 1 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

log10(epsilon)

log10(k

)

Model H

2D computations

3D computations

Asymptotics

FIGURE 13. Model H: Difference 1 0 and associated azimuthal frequency k().

FIGURE 14. Model H: First eigenmode (radial component) for = 102, 103, 104.

FIGURE 15. Model H: 2D first eigenmode (radial component) for = 103 andk = 12, = 3 104 and k = 19, = 104 and k = 30. Represented on 0.2.


7.5. Model L: Airy barrel. The midsurface parametrization is

f(z) = 1 z28 z4

16, z (0.5, 1.5).

5 4 3 2 1 01.4

1.2

1

0.8

0.6

0.4

0.2

0

0.2

0.4

log10(epsilon)

log10(e

igenvalu

e

0)

Model L

2D computations

3D computations

Asymptotics

5 4 3 2 1 00

0.5

1

1.5

2

2.5

log10(epsilon)

log10(k

)

Lam on Model L

2D computations

3D computations

Asymptotics

FIGURE 16. Model L: Difference 1 0 and associated azimuthal frequency k().

FIGURE 17. Model L: 3D first eigenmode (radial component) for = 102, 103, 104.

FIGURE 18. Model L: 2D first eigenmode (radial component) for = 103 andk = 15, = 3 104 and k = 26, = 104 and k = 43. Represented on 0.2.


8. CONCLUSION

For five categories of clamped axisymmetric shells, we have exhibited a scalar 1D operator thatdetermines the asymptotic expansion of the azimuthal frequency k() of the first vibration mode,and a two-term asymptotic expansion for m1() = a0 + a11 for the first eigenvalue of the full3D Lame system in the shell. These five categories are the cylinders and the trimmed cones (par-abolic shells), as well as what we denote toroidal, Gaussian and Airy barrels (elliptic shells). Themost striking outcome of our analysis is the extremely good agreement of the three computationmethods (3D, 2D and 1D) in all the five cases described above, strengthening the relevance of ourconstructions. The presented methods demonstrate that the smallest eigenpairs for the Lame sys-tem can be estimated for specific shells by the reduced 1D model. Furthermore, the spreading orthe concentration of the first eigenmode can be predicted accurately: The cases for which concen-tration occurs are the Gauss and Airy barrels and for those shells, the first eigenmode concentratesaround a ring whose location (f(z0), z0) is analytically known.

Another interesting observation is the comparison with the computations in [3]. The elliptic casethat is considered there is f(z) = 1 1

2z2 on the interval I = (a, a) with a = 0.892668. For

this example we find

g(z) = 32z2(1 1

2z2) and H0(z) = (1 + z2)3.

So we see that we are in our admissible Airy case, predicting a behavior in 3/7 for the azimuthalfrequency k() of the first eigenvector. Noting that 3

7' 0.43 and 2

5= 0.4, we believe that this

explains what have observed the authors [3, p.55]: We also notice that in the elliptic case, Kt[k()] is probably growing slightly faster than exactly t2/5 [2/5].

APPENDIX A. HIGH FREQUENCY REDUCTION OF THE MEMBRANE OPERATOR

Recall that the membrane operator is written as M[k] = k2M0 + kM1 +M2. Let us give elementsof the proof of Theorem 4.2. We write the reduction formula (4.6) in the form

M[k]V[k] = V0 (H[k] [k]) + [k]AV[k] .

This formula holds in the sense of the formal series algebra: This means that it is equivalent to thecollection of equations: For all n 0,

M0Vn + M1Vn1 + M2Vn2 = V0 (Hn2 n2) +

p+q=n2

pAVq. (A.1)

with the convention that Vn, Hn, and n are 0 for n < 0, where Vn and Hn are the unknowncoefficients of the formal series V[k] =

n0 h

nVn and H[k] =

n0 hnHn. Here the series

[k] =

n0 hnn is given. The mass matrix A is given by (4.2), and the operators M` by

M0 =E

1 2

1

2f2s20 0

0 1f4

0

0 0 0

=:M

zz0 0 0

0 M0 0

0 0 0

. (A.2)


M1 = iE

1 2

0 1+

2f2s2z +

2f

f3s20

1+2f2s2

z

+(

(3)f 2f3s2

+ (1+)ff

2f2s4

) 0 1f3s

+ f

f2s3

0 1f3s f

f2s30

(A.3)and

M2 =E

1 2

1s42z +

(3f f

s6 f

fs4

)z

+f2+f f

s6 4f 2f 2

s8

f fs4

+ (1+)f2f

fs6+ f

2

f2s4

0

(f

s5

fs3

)z

+ f

f2s3+ f

s5

3f f 2s7

+ ff

fs5

0

12f2s2

2z

+(

(1)f f 2f2s4

+ (1)f

2f3s2

)z

+ (1)f

f3s2 (1)f

2f

f3s4

0

( f

s5+

fs3

)z

+ff 2

s7+ f

f2s3 2f f

fs5

0 f2

s6+ 1

f2s2 2f

fs4

.

(A.4)Let us emphasize that the operators M1 and M2 have the following structure:

M1 =

0 Mz1 0

Mz1 0 M31

0 M31 0

and M2 =M

zz2 0 M

z32

0 M2 0

M3z2 0 M332

. (A.5)Let us now examine the collection of equation (A.1). For n = 0, this equation reduces to

M0V0 = 0,

which is satisfied with the choice V0 = (0, 0, Id)>. For n = 1, using the structure (A.5) of theoperator M1, the equation is M0V1 = M1V0 that can be written as the systemM

zz0 V1,z

M0 V1,

0

= 0M31

0

.Hence, solving this equation we find V1,z = 0 and V1, = (M0 )1M

31 , i.e.,

V1,z = 0 V1, =if

s if

f 2

s3. (A.6)

The equation for n = 2 is written as

M0V2 = M1V1 M2V0 + V0 (H0 0) + 0AV0 .Using the structure (A.5) and the expressions of V0 and V1, it is equivalent to the system

Mzz0 V2,z = Mz1 V1, Mz32

M0 V2, = 0

0 = M31 V1, M332 + H0 .


The last equation of the previous system joint with (A.6) gives the expression of the operator H0H0 = M

332 M

31 (M

0 )1M31 ,

and using the first two equations, we can solve for V2 by setting

V2,z = (Mzz0 )1(Mz1 (M

0 )1M31 Mz32 ) and V2, = (M

0 )1a0.

Thus we find that H0 = E f2

s6and that the components of V2 are

V2, = 0,

V2,z =( fs ( + 2)f

2f

s3

)z

+f

s+

3( + 2)f 2f f 2

s5 ( + 2)f

2f

s3 (2 + 1)ff

f

s3. (A.7)

Now let us assume that the operators Ln and Vn+1 are constructed for n 1. Then writing theequation (A.1) for n+ 2, we obtain the relation (using the fact that V0n + nAV0 = 0)

M0Vn+2 V0 Hn = M1Vn+1 M2Vn +n1p=0

pAVnp .

This equation is equivalent to the system (using the fact that Vn,3 = 0 for n 1)M0 Vn+2,z = M

z1 Vn+1, Mzz2 Vn,z +

n1p=0 pa

zzVnp,z

M0 Vn+2, = Mz1 Vn+1,z M

2 Vn, +

n1p=0 pa

Vnp,

Hn = M31 Vn+1, + M

3z2 Vn,z ,

which gives the existence of the operators Vn+2,z, Vn+2, and Hn. This shows the existence of theoperators Vn = (Vn,z,Vn,, 0)>. Moreover, we can check that Vn is an operator of order n 1and is polynomial in j , for j n 3. The scalar operators Hn are of order n, polynomial in j ,for j n 2.

Expressions of the operators Hn for n = 0, . . . , 4 are given by the formulas (6.1)(6.4) and thecomponents of the operator V3 are given by

V3,z = 0,

V3, = i( f

3

s3 (1 + 2)f

4f

s5

)2z

+ i( (4 + 6)f

3f f

s5 (4 + 2)f

4f

s5+

7(2 + 1)f 4f f 2

s7 f

2f

s3

)z

+ i((2 + 19 + 19)f 3f 2f 2

s7 (6 + 6)f

3f f

s5 (5 + 3)f

2f 2f

s5

(36 + 18)f4f 2f 3

s9+

(20 + 10)f 4f f f

s7+f f 2

s3+f 2f

s3

+(6 + 3)f 4f 3

s7 (2 + 1)f

4f (4)

s5 (

2 + + 1)f 3f 2

s5

)+ i

1 2

E0

(f 3s f

4f

s3

). (A.8)


APPENDIX B. VARIATIONAL FORMULATIONS ON THE MERIDIAN DOMAIN

The 3D Lame operator is independent of the azimuthal coordinate if expressed in cylindricalcomponents of displacements. The contravariant cylindrical components used in this paper are

ur = ut1 cos+ ut2 sin, (radial)u = ut1 1

rsin+ ut2 1

rcos, (azimuthal)

u = ut3 (axial).

For a displacement u defined on , denote by uk = (urk, uk , u

k) the Fourier coefficients of these

components:

uak(r, ) =1

2

20

ua(r, , ) eik d, a {r, , }, k Z, (r, ) .

The energy bilinear form a3D is decomposed in Fourier coefficients as follows

a3D(u, v) =kZ

ak(uk, vk) .

As soon as k 6= 0, the energy bilinear form ak at azimuthal frequency k as non real coefficients.Nevertheless, by a simple change of components, the coefficients are back to real:

ak((ur, iu, u ), (vr, iv, v )

)=

E

1 2

{(1 )2

1 2

(rru

rrvr + ru

v +

k2

r3uv +

1

rurvr

)+(1 )1 2

r(ru

rv + u

rvr)

+(1 )1 2

(ru

rvr + urrvr + urv

+ uvr)

+1

2

k2

r

(urvr + uv

)+

1 2

1

r

(ru

rv 2

rru

v 2rurv

+4

r2uv + u

v)

+1

2r(ru

vr + u

rrv + ru

rv + u

rvr)

+ k

[(1 )2

1 21

r2

(uvr + urv

)+

1 r2

(uvr + urv

)+(1 )1 2

1

r

(urv

r + rurv + uv

+ uv)

1 2

1

r

(uv

+ uv + urrv

+ ruvr)]}

drd

The associate eigenproblem is: Find and a nonzero (ur, u, u ) V () such that for all(vr, v, v ) V ()

ak((ur, iu, u ), (vr,iv, v )

)=

[urvr +1

r2uv + uv ] rdrd.


Here the variational space V () corresponds to V ()

V () := {u = (ur, u, u ) H1()3 , u = 0 on 0}.Note that the eigenvalues of ak are the same as those of a

k because of the identity

ak((ur, iu, u ), (vr,iv, v )

)= ak

((ur,iu, u ), (vr, iv, v )

).

REFERENCES

[1] S. AGMON, A. DOUGLIS, AND L. NIRENBERG, Estimates near the boundary for solutions of elliptic partialdifferential equations satisfying general boundary conditions II, Comm. Pure Appl. Math., 17 (1964), pp. 3592.

[2] E. ARTIOLI, L. BEIRAO DA VEIGA, H. HAKULA, AND C. LOVADINA, Free vibrations for some Koiter shellsof revolution., Appl. Math. Lett., 12 (2008 (21)), pp. 12451248.

[3] , On the asymptotic behaviour of shells of revolution in free vibration., Computational Mechanics, 44(2009), pp. 4560.

[4] F. ATKINSON, H. LANGER, R. MENNICKEN, AND A. A. SHKALIKOV, The essential spectrum of some matrixoperators., Math. Nach., 167 (1994), pp. 520.

[5] M. BEAUDOUIN, Modal analysis for thin axisymmetric shells, theses, Universite Rennes 1, Nov. 2010,https://tel.archives-ouvertes.fr/tel-00541467.

[6] L. BEIRAO DA VEIGA, H. HAKULA, AND J. PITKARANTA, Asymptotic and numerical analysis of the eigen-value problem for a clamped cylindrical shell., Math. Models Methods Appl. Sci., 18 (11) (2008), pp. 19832002.

[7] L. BEIRAO DA VEIGA AND C. LOVADINA, An interpolation theory approach to shell eigenvalue problems.,Math. Models Methods Appl. Sci., 18 (12) (2008), pp. 20032018.

[8] M. BERNADOU AND P. G. CIARLET, Sur lellipticite du modele lineaire de coques de W.T.Koiter., in Comput-ing Methods in Applied Sciences and Engineering, R.Glowinski and J.L.Lions, eds., Lecture Notes in Econom-ics and Mathematical Systems, Vol.134, Springer-Verlag, Heidelberg, 1976, pp. 89136.

[9] P. G. CIARLET, Mathematical elasticity. Vol. III, North-Holland Publishing Co., Amsterdam, 2000. Theory ofshells.

[10] P. G. CIARLET AND S. KESAVAN, Two-dimensional approximation of three-dimensional eigenvalue problemsin plate theory, Comp. Methods Appl. Mech. Engrg., 26 (1981), pp. 149172.

[11] P. G. CIARLET AND V. LODS, Asymptotic analysis of linearly elastic shells. I. Justification of membrane shellequations, Arch. Rational Mech. Anal., 136 (1996), pp. 119161.

[12] , Asymptotic analysis of linearly elastic shells. III. Justification of Koiters shell equations, Arch. RationalMech. Anal., 136 (1996), pp. 191200.

[13] P. G. CIARLET, V. LODS, AND B. MIARA, Asymptotic analysis of linearly elastic shells. II. Justification offlexural shell equations, Arch. Rational Mech. Anal., 136 (1996), pp. 163190.

[14] M. DAUGE, I. DJURDJEVIC, E. FAOU, AND A. ROSSLE, Eigenmode asymptotics in thin elastic plates, J.M

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