Comput Mech (2008) 42:287–304DOI 10.1007/s00466-007-0204-8

ORIGINAL PAPER

Computing singular perturbations for linear elliptic shells

F. Béchet · E. Sanchez-Palencia · O. Millet

Received: 9 January 2007 / Accepted: 10 June 2007 / Published online: 17 July 2007© Springer-Verlag 2007

Abstract This paper is devoted to the theoretical andnumerical study of singular perturbation problems for ellip-tic inhibited shells. We present a reduction of the classicalmembrane equations to a partial differential equation withrespect to the bending displacement, which is well adaptedto the study of singularities of the limit problem. For a dis-continuous loading or when the boundary of the loadingdomain presents corners, we put in a prominent position theexistence of two kinds of singularities. One of them is notclassical; it reduces to a logarithmic point singularity at thecorner of the loading domain. To finish numerical simula-tions are performed with a finite element software coupledwith an anisotropic adaptive mesh generator. They enable tovisualize precisely the singularities predicted by the theorywith only a very small number of elements.

Keywords Shell theory · Singular perturbation ·Anisotropic mesh generator · Logarithmic point singularity

1 Introduction

Thin shells are present in numerous parts in mechanicalstructure area. Since the pioneering models of Novozhilov–

F. BéchetLaboratoire de Mécanique de Lille,Université Lille 1, Villeneuve, France

E. Sanchez-PalenciaLaboratoire de Modélisation en Mécanique,Université de Jussieu, Jussieu, France

O. Millet (B)Laboratoire d’Etude des Phénomènes de Transfert Appliqués auxBâtiments, Université de La Rochelle, La Rochelle, Francee-mail: olivier.millet@univ-lr.fr

Donnell [26] and Koiter [18,19], numerous works have beendevoted to establish linear and non linear elastic shell modelusing direct or surfacic approaches [5,6,17,34], and evenelasto-plastic or plastic shell models [1,32,33]. Morerecently, the asymptotic methods have been used, to try tojustify rigorously from the three-dimensional equations theshell models obtained by direct approaches relying on apriori assumption, and to construct new models [15,16]. Thisway, numerous shell models have been justified in linear andnon linear elasticity [11,14–16,23–25,27,29].

In linear elasticity, the Koiter shell model cannot be jus-tified by asymptotic methods, whereas it is one of the mostused by engineers for numerical computations. It containsmembrane and bending effects coupled at different orders ofmagnitude: the bending energy is proportional to ε3, whereasthe membrane energy is only proportional to ε, ε being thethickness of the shell (see Eq. 1). Thus the variational formu-lation of Koiter shell model is a singular perturbation problem[22] when the thickness tends towards zero. Some theoreti-cal results of convergence have been established in particularcases: when the shell is inhibited or non inhibited1 accordingto the terminology of [27–29]. For well inhibited shells,2 itcan be proved that the solution of the Koiter shell model tends(in appropriate spaces) towards the solution of the membranemodel (see [27,30]). However, when ε approaches zero but isnot exactly zero, boundary layers and singular perturbationsappear leading to numerical troubles [9,30]. These singularperturbations depend directly on the nature of the middlesurface of the shell (elliptic, parabolic or hyperbolic) and

1 We recall that a shell is said to be inhibited when there does not existany pure bending displacement which let invariant the metric of themiddle surface, equivalently when the space G defined in (8) reducedto {0}. The shell is said to be non inhibited in the contrary case.2 It is the case of an elliptic shell clamped on all its lateral surface.

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288 Comput Mech (2008) 42:287–304

on the boundary conditions.The singularities resulting arevisible on the numerical simulation performed on Fig. 6 foran elliptic shell.

In this paper, we will focus on the theoretical study andon the numerical computation of singular perturbations forelliptic boundary value problems. It is the case when themiddle surface has both principal curvatures of the samesign. For well-inhibited shells (elliptic and clamped on all itsboundary), the limit membrane model enters in the frame-work of well-posed elliptic problems. Oppositely, as soon asthe middle surface of the shell is free on a part of its lateralboundary, the limit problem is out of the general theory ofelliptic boundary value problems. We have ill-posed or sen-sitive problems. More complex phenomena appear, leadingto large oscillations of the bending displacement on the freeboundary [10]. However this is out of the scope of this paper.

In the first section of this paper, we propose a reductionof the membrane system to a partial differential equation inu3 (Eq. 21) which is well adapted to the study of singular-ities. In the following sections, starting from this reducedformulation, we study the structure of the singularities of thesolution u3 of the elliptic3 membrane problem. In particular,we put in a prominent position the existence of two kinds ofsingularities, when the loading is not continuous, or whenthe boundary of the loading domain is not regular enoughand has corners. In the first case, it can be easily proved thatu3 has the same singularity as the loading f3 and we get aclassical internal layer. On the other hand, when the loadingdomain has corners, other not classic logarithmic point sin-gularities appear at the corners but only when the principalcurvature are different. Indeed, when the principal curvatureare identical, these logarithmic singularity vanish. After thetheoretical study of these singularities, we perform numericalsimulations which enable to apprehend and to visualize themprecisely. These numerical simulations are performed withthe software MODULEF coupled with the anisotropic adap-tive mesh generator BAMG, which enables to refine the meshin the directions perpendicular to the singularities and to theboundary or internal layers. This way, we can approach veryprecisely these two kinds of singularities with only a quitesmall number of elements.

2 The membrane system inside �

2.1 Recall on Koiter shell theory

Let us consider the 2D-domain� and (y1, y2) a current pointof �, where y1 and y2 are called the local coordinates. Themapping �(y1, y2) (see Fig. 1) then define the middle sur-face S of the shell. We consider an elastic isotropic shell with

3 When both principal curvatures are of the same sign.

,

Fig. 1 Domain � and associated mapping

a constant relative thickness ε, subjected to a loading f . Thevariational formulation of the Koiter shell model classicallywrites [2,30]

Find u ε V, such as, ∀v ε V :ε

∫

S

Aαβγ δγγ δ(u)γαβ(v)d S

+ ε3

12

∫

S

Aαβγ δργ δ(u)ραβ(v)d S =∫

S

f ivi d S (1)

with V = {v = (v1, v2, v3) ∈ H1(�)× H1(�)× H2(�) ;

v statis f ying the kinematic boundary conditions}Aαβγ δ denote the elasticity coefficients which depend on

the first fundamental form aαβ and also on the shell shape.These coefficients satisfy the property of positivity. They aredefined by

Aαβγ δ = E

2(1 + ν)

[aαγ aβδ + aαδaβγ + 2ν

1 − νaαβaγ δ

]

(2)

The components γαβ and ραβ of the membrane strain tensorand of the tensor of curvature variation are given by

γαβ(u) = 1

2(Dαuβ + Dβuα)− bαβu3 (3)

and

ραβ(u) = ∂α∂βu3 − �γαβ∂γ u3 − bγαbγβu3

+Dα(

bγβuγ)

+ bγα Dβuγ (4)

where

Dαuβ = ∂αuβ − �λαβuλ (5)

denotes the covariant derivative of uβ , bαβ being the coef-ficients of the second fundamental form accounting for cur-vatures. Finally, �λαβ denote the Christoffel symbols of themiddle surface.

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Comput Mech (2008) 42:287–304 289

The Koiter shell model writes classically on the reducedform

am(u, v)+ ε2a f (u, v) = b(v) (6)

where

am(u, v) =∫

S

Aαβγ δγγ δ(u)γαβ(v)d S

represents the membrane energy bilinear form and

a f (u, v) = 1

12

∫

S

Aαβγ δργ δ(u)ραβ(v)d S

the bending energy bilinear form and where we have set

b(v) = ∫S f ivi d S with

f = ε f(7)

In this paper, we will focus on the case of inhibited shells(geometrically rigid) where the space

G = {v ε V ; γαβ(v) = 0

}(8)

reduces to {0}. The space G contains the displacements whichlet invariant the metric of the middle surface. For inhibitedshells, when ε tends towards zero, the limit problem (6) isa singular perturbation problem called “the membrane prob-lem” which writes

am(u, v) = b(v) (9)

The nature of this problem depends on the shape of theshell. It is elliptic when the shell is itself elliptic. As we willonly study well-inhibited shells, this limit problem is “well-posed”.

2.2 Reduction of the membrane system to a differentialequation in u3

In this section, we consider a simplified membrane prob-lem with constant coefficients for the shell geometry. Thefollowing results will hold true in the general case but onlyfor the leading order terms. According to the notations of

the previous section, the variational formulation of the mem-brane problem writes

Find u ε V, such as, ∀ v ε V :∫

S

Aαβγ δγγ δ(u)γαβ(v)d S

=∫

S

f ivi d S (10)

After some integrations by parts (v1, v2, v3 being arbitraryin V), we obtain a differential system of three equations con-taining the displacements ui and the surfacic forces f i

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−A1βγ 1∂β∂γ u1 − A1βγ 2∂β∂γ u2 + A1βγ δbγ δ∂βu3 = f 1

−A2βγ 1∂β∂γ u1 − A2βγ 2∂β∂γ u2 + A2βγ δbγ δ∂βu3 = f 2

−A1βγ δbγ δ∂βu1− A2βγ δbγ δ∂βu2+ Aαβγ δbαβbγ δu3 = f 3

(11)

We can write this system on the following form:

A�u = �f (12)

with A = (ai j

)and where the orders of differentiation of(

ai j)

are the following:⎛⎝ 2 2 1

2 2 11 1 0

⎞⎠ (13)

Now let us explicit the matrix A = (ai j

)of Eq. (12) in the

case of the membrane problem. Using the definition of Aαβγ δ

and considering the simplified case of an orthonormal coor-dinate system, we get

A = −

⎛⎜⎜⎜⎜⎜⎜⎜⎝

k∂21 + 2n∂2

2 (m + 2n)∂1∂2 −(kb11 + mb22)∂1 − 4nb12∂2

(k + 2n)∂1∂2 k∂22 + 2n∂2

1 −(mb11 + kb22)∂2 − 4nb12∂1

(kb11 + mb22)∂1 (mb11 + kb22)∂2 −k(b211 + b2

22)

+4nb12∂2 +4nb12∂1 −2mb11b22 − 8nb212

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(14)

with k = E

1 − ν2 , m = Eν

1 − ν2 and n = E

4(1 + ν).

Thanks to an analogous to the Cramer’s rule for operators(which can only be applied to systems with constant coef-ficients) applied to the membrane system (11), we get thefollowing equation for u3:

Det (A)u3 = A13 f 1 + A23 f 2 + A33 f 3 (15)

where the Ai j are the cofactors of ai j .We consider in what follows the particular case of a prob-

lem with only a normal load (which gives the most singularresults):

f 1 = f 2 = 0 and f 3 �= 0 (16)

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290 Comput Mech (2008) 42:287–304

so that we have

Det (A)u3 = A33 f 3 (17)

After some calculations which are technical but not difficult,so that they are not detailed here, we obtain

Det (A) = E3

2(1 − ν2)(1 + ν)

([b22∂

21 + b11∂

22

](2)

+4[b2

12∂21∂

22 − b22b12∂

31∂2 − b11b12∂1∂

32

])(18)

and

A33 = E2

2(1 − ν2)(1 + ν)�2 (19)

Thus Eq. (17) takes finally the form

E

([b22∂

21 + b11∂

22

](2)

+4[b2

12∂21∂

22 − b22b12∂

31∂2 − b11b12∂1∂

32

])u3 =�2 f 3

(20)

To simplify this equation, we consider the axes X1,X2 in thedirection of the principal curvatures (assumed to be constant),so that, in the new coordinate system we have b12 = 0. Sowe get the simple equation

E[b22∂

21 + b11∂

22

](2)u3 = �2 f 3 (21)

Some explanations on the physical interpretation of equa-tion (21) [or equivalently (20)] should be worthwhile. Thecoefficients of the operator in the left-hand side are the cur-vatures. In other words, the main qualitative structure of thisoperator follows from the geometry of the shell. Oppositely,the operator �2 in the right-hand side is associated with thecofactor A33 in the matrix-operator of (11).

From (19), A33 is elliptic and isotropic, in other words,independent of the direction of the axes.

To finish, let us consider the case of an elliptic middle sur-face where both principal curvatures are of the same sign. Weassume here without loss of generality that they are strictlypositive. Then, using an anisotropic change in the X1, X2

coordinates, the operator in the left hand side of (21) becomes�2. The operator of the right hand side becomes another 4thoperator, whose specific form is not interesting. Indeed, letus perform the change of variables

(X1, X2) ⇒ (Y1,Y2) with

⎧⎪⎨⎪⎩

Y1 = E−1/4 b−1/222 X1

Y2 = E−1/4 b−1/211 X2

(22)

which gives the new equation

�2u3 = C4 f 3 (23)

C4 being a 4th order operator. This operator can be writtenC4 = (D2)

2 with

D2 = E−1/2(

1

b22

∂2

∂Y12 + 1

b11

∂2

∂Y22

)(24)

3 Structure of singularities of an elliptic equationsolution along a curve

In this section, we shall only consider an elliptic system withconstant coefficients (i.e. without the lower order terms).Obvious modifications, not changing the leading order termsallow us to consider general equations and especially themembrane shell equations (see Sect. 4 for more details). Wewrite this equation on the general form

Au = F (25)

A being a differential operator, u being the unknown, and F ,the right hand side. Let us consider some singularities of theright hand side inside the 2D-domain, along a front (i.e. acurve depending of two variables X1 and X2). In the ellip-tic case, there is no characteristic curve. The problem is thesame for each curve inside the domain.

We know that the general case of a system of equationwith constant coefficients reduces, with an analogous of theCramer’s rule for operators, to equations (see Sect. 2.2). Inthis case, the right hand side takes the form of a certain differ-ential operator acting upon the right hand side of the system.We shall consider a straight front, specifically X2 = 0. Thecase of a curved front is analogous in adapted curvilinearcoordinates.4 To fix ideas, we consider a 4th order operatorof the form

A = a0 ∂42 + a1 ∂1∂2

3 + a2 ∂12∂2

2 + a3 ∂13∂2 + a4 ∂1

4 (26)

with a0 �= 0. Indeed, an operator is elliptic if we only con-sider the terms of higher order and replace the derivatives asfollows:

∂1 = iξ1 and ∂2 = iξ2 (27)

If the resulting polynomial has a solution in R2 − {0, 0},

the operator is not elliptic. That is the case here, if a0 = 0,the polynomial has the solution (0, 1) and the operator isnot elliptic. In what follows, we consider a basic singularityS0(X2) and the corresponding chain

. . . S−2(X2), S−1(X2), S0(X2), S1(X2), S2(X2), S3(X2), . . .

(28)

4 Note that in curvilinear coordinates, the coefficients are no longerconstant but according to the previous remark, in that case the corre-sponding results will only hold true at the leading order.

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Comput Mech (2008) 42:287–304 291

with Sk+1 = d

d X2Sk . As usual, this chain of singularities is

understood in the sense of functions (or distributions) definedup to an additive function (or distribution) which is smoothin the neighborhood of X2. An example is

. . . H(X2) X2, H(X2), δ(X2), δ′(X2), . . . (29)

H(.) being the Heavyside jump function and δ(.) the Diracfunction, but there are many others. Let us now replace F in(25) by

F(X1, X2) = S0(X2)�(X1)+ · · · (30)

where “+ · · · ” denotes a smooth function [or even a functioncontaining lower order singularities in the sense of (28)]. Thecoefficient�(X1) is a given smooth function (comments fornon-smooth function will be given later).

We then try the Ansatz

u(X1, X2)=U−4(X1)S−4(X2)+ U−3(X1)S−5(X2)+ · · ·(31)

i.e. a formal sequence of singularities taken in (28) withhigher order than the one of (30), multiplied by unknown“coefficients” U−4(X1) or U−3(X1). Replacing (30) and (31)in (25), we have according to (26)

a0

[U−4(X1)S0(X2)+ U−3(X1)S−1(X2)+ · · ·

]

+a1

[U (1)

−4 (X1)S−1(X2)+ · · ·]

+ · · ·+a4

[U (4)

−4 (X1)S−4(X2)+ · · ·]

= �(X1)S0(X2)+ · · ·(32)

where U (k)i = ∂kUi

∂X1k

So that equating the coefficients of the singularities (westart with the most singular S0, then S−1 and so on).⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

for S0 : a0 U−4(X1) = �(X1)

for S−1 : a0 U−3(X1)+ a1 U (1)−4 (X1) = 0

. . .

(33)

⇔

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

U−4(X1) = 1

a0�(X1)

U−3(X1) = −a1

a0U (1)

−4 (X1)

. . .

(34)

so that, as a0 �= 0, all the unknown coefficientsU−4, U−3, . . .

are uniquely determined.We may justify this formal expansion by truncating it and

considering the smoothness of the complementing term. For

instance, let us consider only the leading term in (31). Wewrite

u(X1, X2) = U−4(X1)S−4(X2)+ v (35)

where U−4(X1) has the expression of (34) and v is a newunknown. Proceeding as before in (32), we have

Av = −a1 U (1)−4 (X1) S−1(X2)− · · ·

−a4 U (4)−4 (X1) S−4(X2)+ · · · (36)

where U−4(X1) denotes a smooth function given by (34) and“+ · · · ” denotes the smooth terms of (30).

We obviously assume that the considered singularities ofF are the only ones in a neighbourhood of (X1, X2) = (0, 0).We may then use local regularity theory for elliptic equations(or microlocally elliptic, as X2 = 0 is not a characteristic) toprove that the local regularity of v is higher than that of u asthe right-hand side of (36) is smoother than that of (25) (see30). Nevertheless, in the case when �(X1) is not of classC∞ in a neighbourhood of X1 = 0, we see that the coeffi-cients of the singularities at the right-hand side of (36) aremore singular than in (25) (see 30). It means that the previ-ous technique is only effective (always in a neighbourhood ofX1 = 0) provided that the smoothness of�(X1) is sufficientto ensure that the regularity theory is effectively applicable in(36). Otherwise, the local structure of u relies on singularitytheory [12,20].

In the case of shells, as we know, the limit (membrane)problem for u3 takes the form

A u3 = B f (37)

where A and B are two elliptic operators of 4th order (see20). We then use the previous framework with

F = B f (38)

taking for instance

f = �(X1) S0(X2) (39)

This time, due to B, which is also a 4th order operator, weuse the Ansatz

u3 = U0(X1) S0(X2)+ U1(X1) S−1(X2)+ · · · (40)

and we have an analogous to (32). However, in the right-handside there is an analogous operator B (of coefficients bi )acting upon f given by (38). Finally, we have

a0 U0(X1) = b0 �(X1) (41)

U0(X1) = b0

a0�(X1) (42)

Therefore, for that case, u3 has the same order of singularityas f3. The comments on the smoothness of�(X1) hold true.

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292 Comput Mech (2008) 42:287–304

4 Study of point singularities of the elliptic membraneequation

We propose now to apply the development of the Sect. 2.2to the study of the point singularities of the solution of themembrane shell equation. To do this, we start from the welladapted elliptic formulation (23). For such equations, it isknown (see for instance [12,20,21]) that for specific charac-teristic exponents λk , singularity of the kind:5

rλk ϕk(θ) (43)

or

rλk [ϕk(θ)+ log(r) ψk(θ)] (44)

or even higher powers of log(r)may appear. The specific val-ues of λk , which give the complete description of the singu-larities, depend on the coefficients of the equation. In general,these values (which are the eigenvalues of certain operator)may only be obtained by numerical computation (see [21]).In certain specific examples (involving the Laplacian or otheranalogous operators), a Fourier expansion with respect to θenables us to have a complete description of the structure ofthe solution.

Thanks to the change (22), the new Eq. (23) in the Y vari-ables allows an explicit treatment using the correspondingpolar coordinates and Fourier expansions in the angular var-iable θ . We shall do this study in the following subsections,starting with the simple case of the Laplacian (Sect. 4.1). Wethen consider the complete problem (23) with�2 (Sect. 4.2)and we exhibit the presence of logarithmic singularities whichdisappear in the particular case b11 = b22, i.e. when the localgeometry is itself isotropic (Sect. 4.3).

4.1 Model problem with the Laplacian

Let us consider a current point P of the domain� and let ususe a polar coordinate system with P as origin (see Fig. 2).We will study the problem

�u = E2 f3(θ) in a vicinity of r = 0 (45)

where E2 is a second order operator like � (but not propor-tional to �) and where f3(θ) is independent of r . We justneed to study the singularity of one solution of (45) becauseall the solutions have the same singularities. Indeed, let ustake 2 solutions u and u, we have

�u = E2 f3(θ) (46)

�u = E2 f3(θ) (47)

5 For a polar coordinate system (r , θ).

Fig. 2 Polar coordinates

which gives by linearity �(u − u) = 0. So that, as u − u isa harmonic function (a smooth function), u and u will havethe same singularities.

Let us first study the problem

�v = f3(θ) (48)

without the operator E2. In polar coordinate system, the prob-lem is

1

r2

[(r∂

∂ r

)(2)+ ∂2

∂θ2

]v = f3(θ)r

0 (49)

v and f3 being 2π -periodic in θ , we can perform a Fourierexpansion

v = v0(r)+∞∑

n=1

(vn(r) cos(nθ)+ wn(r) sin(nθ)) (50)

and

f3(θ) = a0 +∞∑

n=1

(an cos(nθ)+ bn sin(nθ)) (51)

The problem (48) becomes

(r

d

d r

)(2)v0(r) = a0r2 (52)

[(r

d

d r

)(2)− n2

]vn(r) = an r2 (53)

[(r

d

d r

)(2)− n2

]wn(r) = bn r2 (54)

After a new change of variable log(r) = t , we obtain

d2

dt2 v0(t) = a0 e2t (55)(

d2

dt2 − n2)vn(t) = ane2t (56)

(d2

dt2 − n2)wn(t) = bn e2t (57)

which can be solved easily. The solution of the non homo-geneous equation will be of the form vn = An e2t except forn = 2. For n = 2, the solutions will be A2 t e2t and B2 t e2t

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Comput Mech (2008) 42:287–304 293

with A2 = a24 and B2 = b2

4 . So we can find v

v = a2

4t e2t cos(2θ)+ b2

4t e2t sin(2θ)

+ terms in ent wi th n �= 2 (58)

Coming back to the variable r , we get:

v = a2

4log(r).r2 cos(2θ)+ b2

4log(r).r2 sin(2θ)

+ terms in rn wi th n �= 2 (59)

Let us now go back to the problem (45) by applying the oper-ator E2 to the both sides of the previous problem. We get:E2�v = E2 f3(θ). As both operators have constant coeffi-cients, they commute. Then we have

�E2 v = E2 f3(θ) (60)

Thus u = E2 v is a solution of (45). As E2 is a second orderoperator, the most singular terms will be those derived twicewith respect to Y1 and/or Y2. In polar coordinates, the powercompared to r loses two orders either because of two deri-vation(s) and/or division(s) by r (see the expressions of ∂

∂Y1

and ∂∂Y2

in polar coordinates). So, finally, we have

u = G(θ) log(r)+ more regular terms (61)

where G(θ) is a function depending only on θ .

4.2 The membrane problem �2u3 = C4 f3(θ)

Let us go back to the initial problem (23) which can be studiedthanks to the previous one. Let us first consider the problemwithout the operator C4. It is equivalent to the problem

�2 u = f3(θ) ⇔ ��u = f3(θ) (62)

⇔{�v = f3(θ)

v = �u(63)

The first equation of (63) has been studied in the previoussection. Its general solution is given by (59). Then replacingv in the second equation of (63), we obtain

�u = a2

4log(r).r2 cos(2θ)+ b2

4log(r).r2 sin(2θ)

+ terms of the f orm rn wi th n �= 2 (64)

Let us keep only the terms which are not regular. We willthen study the problem

�u = a2

4log(r).r2 cos(2θ)+ b2

4log(r).r2 sin(2θ) (65)

After a Fourier expansion of u

u = u0(r)+∞∑1

(un(r) cos(nθ)+ u∗

n(r) sin(nθ))

(66)

we get the system

1

r2

[(r∂

∂ r

)(2)− 4

]u2(r) = a2 r2 log(r) (67)

1

r2

[(r∂

∂ r

)(2)− 4

]u∗

2(r) = b2 r2log(r) (68)

⇔[(r∂

∂ r

)(2)− 4

]u2(r) = a2 r4 log(r) (69)

[(r∂

∂ r

)(2)− 4

]u∗

2(r) = b2 r4log(r) (70)

With the same change of variable log(r) = t as previ-ously, we get[(

∂2

∂ t2

)− 4

]u2(t) = a2 t e4t (71)

[(∂2

∂ t2

)− 4

]u∗

2(t) = b2 t e4t (72)

As e4t is not a solution of the homogeneous equation, u2 (andsimilarly u∗

2) has the form

u2 = A t e4t + B e4t ⇒[

d2

dt2 − 4

]u2

= A(8 + 16t)e4t + 16Be4t (73)

So u2 is solution of (71) only if

⎧⎨⎩

8A + 16B = 0

16A = a2

⇔

⎧⎪⎪⎨⎪⎪⎩

A = a2

16

B = − a2

32

(74)

Now we can determine the most singular terms of u whichwrite

u = a2

16r4 log(r) cos(θ)+ b2

16r4 log(r) sin(θ)

+ more regular terms (75)

Finally, to solve problem (23), let us apply the operator C4 toEq. (62) as we did in the previous section. If we set u3 = C4 u,with C4 a 4th order operator, we obtain

�2u3 = C4 f3(θ) (76)

and as a consequence

u3 = H(θ) log(r)+ more regular terms (77)

where H(θ) is a function depending only on θ .As a conclusion, we get a logarithmic point singularity

for the elliptic membrane problem considered (which is ofthe 4th order). However, this singularity only exists if at least

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294 Comput Mech (2008) 42:287–304

one of the coefficients a2 or b2 (of Fourier expansion of theloading f3(θ)) is not equal to zero. In other words if

a2 = 1

π

2π∫

0

f3(θ) cos(2θ)dθ �= 0 (78)

and/or

b2 = 1

π

2π∫

0

f3(θ) sin(2θ)dθ �= 0 (79)

4.3 Case when b11 = b22

When the two curvatures b11 and b22 (in the principal direc-tions) are equal, things are easier to solve.6 Indeed, if wehave b11 = b22 = b, (21) becomes

2E b[∂2

1 + ∂22

](2)u3 = �2 f3 (80)

⇐⇒ 2E b �2u3 = �2 f3 (81)

⇐⇒ �2 [2E b u3 − f3] = 0 (82)

So 2E b uε3 − f3 is a harmonic function (a smooth function).So as a consequence, u3 and f3 have the same singularities.This means that even if, as seen previously, a2 or b2 is notequal to zero, there cannot exist any logarithmic singularityin a point where the principal curvatures are equal.

5 Anisotropic adaptive numerical procedure

The computations have been performed with two programslinked together [8,9]: MODULEF, a finite element programand Bidimensionnal Anisotropic Mesh Generator (BAMG),an automatic mesh generator, both developed by INRIA.7 Weuse the DKTC element of MODULEF (Discrete KirschhoffTriangle for Shell) [2] for the shell. This triangular elementis based on the Koiter shell theory including membrane andflexion energies in the variational formulation. The speci-ficity of MODULEF is that we mesh the 2D-domain of themapping, but not the middle surface of the shell imbedded inR

3. This avoids the errors due to a geometric approximationof the surface with planar facets.

The anisotropic mesh generator BAMG enables to con-struct an adapted mesh from a simple initial mesh and thesolution of the problem with this first mesh [4,3]. The refin-ing of the mesh is then done in a anisotropic way according tothe specific solved problem. This avoids an isotropic refining

6 We recall that Eq. (21) is established in the particular case of anorthonormal coordinate system to simplify the calculations. In thegeneral case, the condition would be b1

1 = b22.

7 Institut National de Recherche en Informatique et Automatique,France.

which induces a high number of elements. It is particularlyinteresting for shell problems. Indeed, in the problems pre-sented in the previous sections, we especially need to refinethe mesh in the direction normal to the internal layer or tothe discontinuities. So that we can increase the number ofelements only in this direction, while it stays constant in thedirection of the singularity.

BAMG uses the Delaunay method to create meshes. Themain point is to create elements whose edges have the rightlength. In the isotropic case, the edge AB of elements (param-etrized with t) has to satisfy

lh(AB) =∫

t

√−→AB.

−→AB

h2(t)dt = 1 (83)

where the dot denotes the scalar product of R3 and h the size

of the mesh. In a more general way, this condition can bewritten

lM(AB) =∫

t

√−→AB.M(t)

−→ABdt = 1 (84)

with

M(t) =⎛⎜⎝

1

h2(t)0

01

h2(t)

⎞⎟⎠ (85)

in the isotropic case. In the anisotropic case, the conditionlM(AB) = 1 becomes

M(t) = T R(t)

⎛⎜⎜⎝

1

h21(t)

0

01

h22(t)

⎞⎟⎟⎠R(t) (86)

where R(t) denotes the rotation matrix that indicates thedirections of the anisotropy, T the transposition operator,and hi (t) the wished length in the direction i .

The adaption of the mesh is based on an estimate of theerror between the discrete solution and the real one. If weuse a linear interpolation between the results at each node,the error can be estimated for a 2D problem, thanks to theTaylor development (see [13], p. 283), by

|η − ηh |∞ ≤ c0 h2 ‖H(η)‖∞ (87)

H being the Hessian of η, c0 a constant and h the size of themesh. | |∞ denotes the L∞ norm in R

2 and ‖ ‖∞ the matrixL∞ norm.

Then BAMG generates a new mesh using the new metricM(t) calculated as follows:

M(t) = c0

ε|H| (88)

where |H| is the diagonal matrix with the absolute values ofthe eigen values of the Hessian H.

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Comput Mech (2008) 42:287–304 295

Finally, for the adaptation procedure, we have to choosewhich unknown will play the role of the scalar variable η. Forthe shell problems considered here, and computed numeri-cally with MODULEF, we chose η = u3 which is the mostsingular displacement for a given f3. Thus, the results givenby MODULEF for u3 at each iteration are used by BAMG tocalculate the Hessian of u3. From it, an adapted mesh is gen-erated using (88). This way, the mesh will be refined in theareas where the second order derivatives of u3 are the mostimportant. They correspond to the areas where the bendingis important. In general, several iterations are needed to geta better mesh.

6 Numerical results on the well inhibited problem

Let us consider the domain�={(y1, y2) ε [−1, 1]×[−1, 1]}

and the elliptic surface S defined by the local mapping (�,ψ)where

ψ(y1, y2) = (y1, y2, (y1)2 + (y2)2) (89)

The considered shell is defined by S and the thickness εwhich is constant in �. The shell is clamped on its wholeboundary so that it is well-inhibited according to the termi-nology of [29,31]. We then apply a loading on thedomain F defined by F = {

(y1, y2) ε [− 12 ; 1

2 ] × [− 12 ; 1

2 ]}(see Fig. 3). According to (7), we consider a normal loadingf3 = −10 ε MPa proportional to the thickness ε, so thatf3 is constant.We also take the following material constantswhich correspond to a standard steel: E = 210 000 MPa andν = 0, 3.We will perform some computations for different values ofε. As the problem is symmetric compared to the lines y1 = 0and y2 = 0, we only take a quarter of the domain and weapply the following symmetries on the displacements

u1 = 0 and u3,1 = 0 on the line y1 = 0 (90)

u2 = 0 and u3,2 = 0 on the line y2 = 0 (91)

For every ε, we start the computation with an uniformmesh made of 236 elements (Fig. 5). Then, the mesh isrefined automatically (see next section). Figure 6 presentsthe normal displacement u3 on a quarter of the domain afterseven iterations. We can see the two phenomena that happen

– a singularity along the loading domain boundary– a peak or a logarithmic singularity at the corner of the

loading domain F .

The following sections are devoted to the numerical studyof these two singularities. All the numerical results for thedisplacements are given in meters.

Fig. 3 Domain �

-1-0.5

00.5 1y1 -1

-0.5

0

0.5

1

y2

0

0.5

1

1.5

2

(y1)2+(y2)2

Fig. 4 Surface S

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fig. 5 Initial mesh (236 elts)

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296 Comput Mech (2008) 42:287–304

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

-0.00016-0.00014-0.00012-0.0001-8e-005-6e-005-4e-005-2e-005

0 2e-005

Fig. 6 Displacement u3 in the whole domain for ε = 10−4

6.1 The internal layer due to the loading

Using the theoretical developments of Sect. 3, we can easilydeduce the singularity of the solution u3 from the loadingf3. Indeed, the discontinuity of the loading f3 through thedomain F described above can be written

f3 = −10[

H(y1 + 0, 5)− H(y1 − 0, 5)]

×[

H(y2 + 0, 5)− H(y2 − 0, 5)]

(92)

where H denotes the Heaviside function satisfying

H(y1 − a) ={

0 if y1 < a1 if y1 > a

Thus the singularity in the direction y1 at y1 = 0, 5 is equalto H(y1 − 0, 5) for −0, 5 < y2 < 0, 5 and symmetricallyin the direction y2. Now according to (42) and to the mem-brane shell model used, it is easy to see that u3 and f3 havethe same Heaviside-like singularity, except near the cornerswhere we have a logarithmic point singularity. So that thenumerical simulations performed should put in a prominentposition a jump (a Heavyside-like singularity) at y1 = 0, 5on the line y2 = 0 (also on the lines y2 = c with c < 0, 5but not too close to 0, 5 (we will see why in the next section).It is the goal of what follows.

6.2 Mesh adaptation and resolution

We consider the case of a small thickness ε = 10−4. First,let us observe the evolution of the mesh on Figs. 7 and 8.We can see that the mesh is mainly refined along the limit ofthe loading domain and in particular near the corner

[ 12 ; 1

2

].

We saw in the previous sections, that the discontinuity of theloading induces an internal layer along the border of F anda logarithmic singularity at the corner. With the anisotropicmesh generator used (BAMG), the areas where the mesh is

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fig. 7 Mesh at the 3rd iteration (3076 elts)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fig. 8 Mesh at the 7th iteration (5351 elts)

more refined correspond to those where the displacement u3

is singular.The mesh aspect is different according to the part of the

mesh. In the layer, the elements are anisotropic: the lengthof the edge in the direction of the layer is more importantthan the length of the edge in the perpendicular direction(Fig. 9). This is due to the singularity which only exists inthe direction perpendicular to the loading boundary. On theother hand, the mesh around the point singularity is isotro-pic as the logarithmic singularity exists all around the point(0.5, 0.5) (Fig. 10).

Let us now focus on u3 which is the most singular displace-ment of this problem. Figures 11 and 12 show the efficiencyof the remeshing procedure on the results for the compu-tation of u3. Whereas the first mesh gives very bad resultsaround the internal layer (the shape of the singularity is not

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Comput Mech (2008) 42:287–304 297

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.46 0.48 0.5 0.52 0.54

Fig. 9 Mesh in the layer

0.4

0.45

0.5

0.55

0.6

0.4 0.45 0.5 0.55 0.6

Fig. 10 Mesh around the point singularity

well described), the results of the 3rd iteration are close to thetheory. Indeed, we can really notice the jump of u3 near theline y1 = 0.5, which corresponds to a Heavyside-like sin-gularity as predicted by the theory. The next iterations giveagain a better approximation of the singularity shape.

Another important fact to quote is that the number of nodes(or of elements) increases inside the internal layer whereasit is constant outside. In contrary, between the lines y1 = 0and y1 = 0.25, and around y1 = 0.75, the results are quitethe same for each iteration as for the initial mesh. This is dueto the absence of internal layer in these zones where u3 issmooth.

Finally, let us compare the results of the different iterationsto the results obtained with an uniform mesh containing the

same number of elements as the seventh iteration (5582 ele-ments). It is clear on Figs. 11 and 12 that that an uniform meshgives very bad results in the layer, even with a great numberof elements. The approximation of u3 obtained at the thirditeration with about twice less elements (3076 exactly), givesbetter results than the uniform mesh with 5582 elements. Thisenforces the interest of an anisotropic remeshing inside theinternal layers or near the singularities.

6.3 Study of the internal layer thickness

Let us now determine the internal layer thickness by numer-ical simulations performed on the same example as previ-ously. On Fig. 13 are represented the variations of u3 fordifferent values of ε. We recall that we apply a loading pro-portional to ε in the loading domain F .Thus the displacementu3 stays quite the same out of the internal layer (in the smoothregions) and it is easy to compare the results on the samefigure.

When ε tends towards zero which corresponds to the mem-brane model, the theoretical analysis developed in theprevious section proves that u3 tends towards a Heavysidesingularity. For thicker shells (ε = 10−2 for instance), u3 isa smooth function and it is difficult to measure the internallayer thickness. A first possibility would consist in measur-ing the distance between the two extrema near y1 = 0.5. Butthis is not an easy task because the extremum is difficult todetermine precisely for high ε (especially for y1 < 0.5). Sothat it is preferable to measure this distance η (see Fig. 14)for 10−6 ≤ ε ≤ 10−3 and we search a relation of the formη = εα . To do this, in Fig. 15 we plot log(η) with respectto log(ε) for different values of ε.The result is once againvery close to the theory:8 we find η = O(ε0,5163) with aR-squared very close to 1.

6.4 Influence of the loading domain

To study the influence of the loading, we consider the sameproblem as previously, but we change the size of the load-ing domain F . Let us denote L the length of the edge of theloading square domain. We consider decreasing values of Lso that the loading tends towards a Dirac distribution.

We still perform the numerical computations on a quar-ter of the domain. We can see on Figs. 16 to 19 that, as Ldecreases, the discontinuity of u3 follows the boundary ofthe loading domain. Moreover it is clear on Fig. 20 that thedisplacement u3 tends to a Dirac at the origin.

On the other hand, let us notice that the logarithmic sin-gularity tends to disappear when L deceases and approacheszero. This can be explained as follows: when the loading f3

8 We recall that for such problems, the thickness of the internal layers

or of the boundary layers are of the order of ε12 (see [30]).

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298 Comput Mech (2008) 42:287–304

Fig. 11 Convergence of u3during the remeshing at y2 = 0and for y1 ∈ [0, 1]

Fig. 12 Convergence of u3during the remeshing at y2 = 0and for y1 ∈ [0.4, 0.5]

Fig. 13 Displacement u3 fordifferent ε from 10−2 to 10−5

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Comput Mech (2008) 42:287–304 299

Fig. 14 Measure of η

Fig. 15 log(η) versus log(ε)measured on u3

0

0.2

0.4

0.6

0.8

1 0 0.2

0.4 0.6

0.8 1

-3e-005

-2.5e-005

-2e-005

-1.5e-005

-1e-005

-5e-006

0

5e-006

Fig. 16 Displacement u3 for L = 0.5 and ε = 10−4

tends to a Dirac distribution at the origin, the corner of theloading domain tend towards the origin (0,0). As the twoprincipal curvatures are equal at that point, there does notexist any logarithmic singularity (see Sect. 4.3).

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1

-1.6e-005-1.4e-005-1.2e-005

-1e-005-8e-006-6e-006-4e-006-2e-006

0 2e-006

Fig. 17 Displacement u3 for L = 0.2 and ε = 10−4

6.5 The logarithmic singularity at the corner

As we saw in Sect. 4, there may exist logarithmic singulari-ties in some particular points for an elliptic shell problem. At

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300 Comput Mech (2008) 42:287–304

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8 1

-1.4e-005-1.2e-005

-1e-005-8e-006-6e-006-4e-006-2e-006

0 2e-006

Fig. 18 Displacement u3 for L = 0.1 and ε = 10−4

the point P(0.5, 0.5), condition (79) is satisfied for a2 = 0but b2 �= 0. Indeed, around P , using the polar coordinates(see Fig. 2), the loading writes

f3(θ) =⎧⎨⎩

0 f or − π

2≤ θ ≤ π

1 f or π ≤ θ ≤ 3π

2

(93)

and it follows that:

a2 = 1

π

2π∫

0

f3(θ) cos(2θ)dθ

= 1

π

3π/2∫

π

cos(2θ)dθ = 0 (94)

b2 = 1

π

2π∫

0

f3(θ) sin(2θ)dθ

= 1

π

3π/2∫

π

sin(2θ)dθ = 2

π(95)

Moreover, at P the two principal curvatures are different.That ensures the existence of a logarithmic singularity.

Let us now compare the numerical simulations performedto the theory which predicts the existence of a logarithmicsingularity at the corner P . Figure 21 represents the normaldisplacement u3 along the line y2 = y1. We note that themaximum of u3 is at about y1 = 0, 48. As we are not exactlyat the limit problem (ε = 10−4 �= 0), u3 is not exactly a log-arithmic function near by P. There is a smooth link betweenthe two logarithmic-like functions which exist at each sideof P.

To specify the form of u3 at the corner, let us zoom on azone near the peak for y1 > 0.5. On Fig. 22, we remove thepoints near the maximum (that make the smooth link betweenthe 2 sides of the singularity). We take the point where u3

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8 1

-8e-006-7e-006-6e-006-5e-006-4e-006-3e-006-2e-006-1e-006

0 1e-006

Fig. 19 Displacement u3 for L = 0.02 and ε = 10−4

reaches its maximum as origin and we call r the distancebetween the origin and the considered point (we take a cylin-drical coordinate system). The graph of u3 for 0 < r < 0.1(for the side y1 > 0.5) shows that u3 is very close to a loga-rithmic function. Therefore the numerical results performedwith the anisotropic remeshing enable to approach preciselythe logarithmic corner singularity predicted by the theory.

Evolution of the logarithmic singularityFigure 23 shows the evolution of the logarithmic singularitywith respect to ε.

It is clear again that when ε tends to zero, the bending dis-placement tends towards the solution of the membrane modelwhich possesses a logarithmic singularity near the corner fory1 = y2 = 0.5. For a thick shell ε = 10−2, the solutionu3 is much more smooth and does not present any singular-ity. This one appears more clearly for small values of ε (forε < 10−4). The maximum of u3 becomes larger: 3.5 10−5

for ε = 10−2 and 2.1 10−4 for ε = 10−5. It is easy to verifythat it proportional to log(ε).

A second thing to remark is that the smooth link betweenthe two sides of the singularity is more and more narrow.These two things are in relation. Indeed u3 is like a loga-rithmic function (u3 ≈ c log(r)) but not in a zone around itsmaximum. The thickness of this zone decreases as ε

decreases. If the thickness of this zone is of the order O(εα)with a given α, the logarithmic part starts from r = a εα . Asthe maximum of u3 is near the maximum of the logarithmicpart of u3 reached for r = a εα , we have

u3max ≈ c log(a εα) ∼0

c α log(ε) (96)

Finally, let us quote the position of the maximum progres-sively tends towards the point P = (0.5, 0.5) as ε tends tozero.

6.6 A case where the logarithmic singularity vanishes

As seen in Sect. 4.3, there is no logarithmic singularity atthe corner when the principal curvatures are equal, i.e. when

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Comput Mech (2008) 42:287–304 301

Fig. 20 Displacement u3 fordifferent values of L

Fig. 21 Displacement u3 onthe line y2 = y1 for ε = 10−4

Fig. 22 Comparison betweenu3 and a logarithmic function

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302 Comput Mech (2008) 42:287–304

Fig. 23 Evolution of thelogarithmic singularity forvarious ε from 10−2 to 10−5

Fig. 24 Mesh of the last iteration

Fig. 25 u3 on the whole domain

b11 = b2

2 (see Sect. 4.3). To verify this point numerically, letus consider the same elliptic problem as previously, but fora half-sphere with a radius equal to 100 mm. In this case,the principal curvatures are equal on the whole domain. Theloading is still applied on the square domain F with cor-ners. To simplify the numerical computations according tothe symmetry, we consider again only a quarter of the half-sphere.

The mesh and the normal displacement u3 are representedrespectively on Figs. 24 and 25 at the last iteration (with 7559elements).

It is clear that there is no logarithmic singularity at thecorner. We just have a jump in the direction of the loading atthe boundary of the loading domain F , which corresponds aspreviously to a Heavyside-like singularity. Therefore, whenthe principal curvatures are equal on the whole domain, therecannot exist any logarithmic singularity anywhere. But if thecondition (78) or (79) is satisfied at a point, and if the twoprincipal curvatures are punctually equal at this point, thereis no logarithmic singularity at this point.

6.7 A case without logarithmic singularity

Let us consider another elliptic shell problem with a differ-ent mapping and loading domain, respectively� and F (seeFig. 26). The loading domain has no corner.

The middle surface of the considered shell, whose thick-ness is ε = 10−3, is plotted in Fig. 27.

In this case, the coefficients a2 and b2 of the Fourierexpansion of the loading are zero at each point of the bound-ary of the loading domain. Thus, there should be no loga-rithmic singularity according to Sect. 4.3. Once again, thenumerical simulation are in good agreement with the theory(see Fig. 28). We only observe a jump corresponding to aHeavyside-like singularity due to the discontinuity of theloading.

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Comput Mech (2008) 42:287–304 303

Fig. 26 Domain �

-1-0.5

0 0.5

1

-1-0.5

0 0.5

1

0

0.5

1

1.5

2

Fig. 27 Middle surface in R3

0

0.2

0.4

0.6

0.8

1 0 0.2

0.4 0.6

0.8

1

-3e-005-2.5e-005

-2e-005-1.5e-005

-1e-005-5e-006

0 5e-006

Fig. 28 u3 on the whole domain for ε = 10−3

7 Conclusion

In this paper, we have presented a theoretical and numer-ical study of singular perturbations for elliptic membraneshell problems. The reduction of the membrane equationsto a partial differential equation with respect to the normaldisplacement, revealed to be well adapted to this study. Thetheoretical analysis put in a prominent position the existenceof two kinds of singularities for the inhibited shell prob-lems. The first one is directly linked to the discontinuitiesof the loading. The second one reduces to logarithmic pointsingularities at the corner of the loading domain when theprincipal curvature are different at this point. In the secondpart of this paper, we have performed numerical simulationswith the finite element software MODULEF coupled with theanisotropic adaptive mesh generator BAMG. The numericalsimulations obtained enabled to visualize very precisely thesingularities predicted by the theory with only a small num-ber of elements.

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