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Preprint submitted on 11 Jun 2009

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A Finite Difference Element Method for thin elasticShells

Daniel Choï

To cite this version:

Daniel Choï. A Finite Difference Element Method for thin elastic Shells. 2009. hal-00383366v2

A Finite Difference Element Method for thin elastic Shells

Daniel Choıa

aLaboratoire de Mathematiques Nicolas Oresme ,Universite de Caen14032 Caen, France

Abstract

We present, in this paper, a four nodes quadrangular shell element (FDEM4) based on a Finite Difference Element Method procedurethat we introduce. Its stability and robustness with respect to shear locking and membrane locking problems is discussed. Numericaltests including inhibited and non-inhibited cases of thin linear shells are presented and compared with widely used DKT and MITC4elements.

Key words: Finite Element Method, Membrane Locking, Shear Locking, Elastic shell, Galekin Method, Finite Difference

1. Introduction

The Finite Element Method based on polynomial interpo-lation is widely and successfully used in engineering applica-tions. Among major problems encountered in Finite ElementMethods is the numerical locking. Classical examples are in-compressible continuum, shear locking in beam and plates el-ement and shear and membrane locking in thin shells compu-tation [3, 11, 16, 24, 29]. These locking phenomenon has incommon the inability of the (polynomial) shape functions toperform some constraints of the problem (such as incompress-ibility condition displacements or bendings in case of shell).

The finite element computations of thin elastic shells re-mains an active research topic and of utmost interest for en-gineering analysis. The main difficulty in shells computationsis the extreme variety of asymptotic behaviors depending onthe geometry, boundary conditions and external loading : a per-fect shell element must be locking free for bending dominatedbehaviors and in the same time be effective in membrane dom-inated and mixed behaviors [11].

Various finite element methods claims successful with re-gard to the locking problems [2, 7, 4, 3, 12, 22], such as Par-tial Reduced Selective Integration method, Mixed Interpola-tion method, Discrete Strain Gap method and more recentlySmoothed finite element method, [22]. In [14], the shape func-tions has been identified as closely related to numerical lockingfor conformal finite elements. For that reason, we wanted toexplore the possibility to implement a finite element procedurewithout shape functions as an alternative. Based on this idea,we define, in this paper, a Finite Difference Element Methodconcept and we present a very simple four nodes quadrangularFinite Difference Element procedure (FDEM4) for thin elasticshell problems, based on Naghdi’s model.

The paper is organized as follows, first, we present the Fi-nite Difference Element Method concept and we illustrate witha simple scalar Poisson problem.

The Nagdhi’s model of thin elastic shells [6] is exposed in

section 3. The FDEM4 shell element is based on this model, itis presented in section 4. The section 5 is devoted to numericalresults. The robustness with respect to both shear locking andmembrane locking, with regular and distorted meshes is tested.FDEM4’s performance is compared with MITC4 and triangu-lar DKT. In accordance with the asymptotic analysis of thinelastic shells, [27, 10, 11], inhibited and non-inhibited casesof shells are treated. In non-inhibited cases, the computed so-lutions are compared with the solution of asymptotic bendingproblem [15].

We then conclude and discuss the perspectives of the Fi-nite Difference Element method for thin shell computations andmore.

Notations and conventions

In this paper, we employ the Einstein convention of sum-mation on repeated upper or lower indices. The Greek (resp.Latin) indices range over 1, 2 (resp. 1, 2, 3). The partialderivatives with respect to variables xα are denoted in lowerindices preceded by a comma. The ∇ symbol represents thegradient of one function. We use overarrow to indicate spacevectors. The variables x1, x2 (resp. x1, x2, x3) live in boundeddomain Ω ⊂ R2 (resp. R3). The Hilbert functional space Vwill denote a Sobolev space and therefore injects continuouslyinto H = L2(Ω). The Harpoon symbol − will indicate a con-vergence in the weak sense. The O and o symbols representnegligible quantities in the sense that

limh→0

O(h) = 0 and limh→0

o(h)h

= 0.

2. Towards a Galerkin Method without shape functions

How to avoid shape functions and in the same time applya Galerkin method ? To this paradoxical question, one answerwe found is the finite difference element method : we simplypropose to replace the differential operators by finite difference

Preprint submitted to Computer & Structures May 27, 2009

approximations in some Galerkin projection procedure. Moreprecisely, we replace the differential expressions of undefinedshape functions, involved in the variational form problem bysome finite difference approximations at the nodes of integra-tion schemes. Although such idea is not new, the method wepropose is very similar to the finite difference energy method[3, 9, 23], we give here a different point of view where the finitedifference element method is interpreted as a Galerkin method.

A scalar problem example

For sake of simplicity, we shall present the finite differenceelement method on a scalar problem (corresponding mathemat-ically to a heat or a membrane problem) over Ω ⊂ R2:

Find u ∈ V such that∫Ω

∇u.∇u∗ =

∫Ω

f u∗ ∀u∗ ∈ V,(1)

where V is the Sobolev space H10(Ω). The variational problem

(1) is the standard Poisson problem with Dirichlet boundaryconditions.

Shape functions

Actually, we still need shape functions, at least from a the-oretical point of view. The only difference with standard finiteelement method is that, using finite difference approximations,the explicit form or expression of shape functions need not beexplicit.

Let Vh be a subspace of V , generated by a set of shape func-tions φi piecewise defined on Ωh, continuous in Ω, where h de-notes the mean size of the elements Ωi. We can impose somenatural conditions : φi(N j) = δi, j,

φi = 0, on Ωk if Ni < Ωk.(2)

We have then the decomposition :

uh = uiφi,

where the coefficients ui will represent the value of u at nodesNi. We don’t impose a priori shape functions to realize a par-tition of unity, but an approximate partition of unity. At thispoint, we need not to be definitive.

Classically, as h → 0, the subspaces Vh define a set of dis-crete conformal approximations of V : for any u ∈ V , it is easyto construct a sequence un ∈ Vh such that un converge almosteverywhere towards u in Ω. Let’s consider now uh, the Galerkinprojection of the solution u of (1) in Vh :

Find uh ∈ Vh such that∫Ω

∇uh.∇u∗ =

∫Ω

f u∗ ∀u∗ ∈ Vh(3)

It classically reduces to a linear system with the ’stiffness’ ma-trix K :

Ku = L,

where

u = [u1, . . . un]>,L = [l1, . . . ln]>,

and with

Ki j =

∫Ω

∇φi.∇φ j (4)

l j =

∫Ω

fφ j. (5)

In classical finite element method, the computation of (4)need the explicit expression of the shape functions on the nodesof integration scheme. Here, the Finite Difference ElementMethod consists simply in integrating over nodes where we re-place differential expressions by some consistent finite differ-ence approximations :

Finite difference approximations over a 4 nodes quadrangle

If we consider a quadrangle defined by N1, N2, N3, N4, ofcoordinates (xi, yi), and with N0 as the intersection of the twodiagonals, see Figure 1.

1

N0

N4

N

N

N

2

3

Figure 1: A four nodes quadrangle

By definition, we have, for j = 1,2,3,4 :

u(N j) = u(N0) + (∇u(N0),−−−−→N0N j) + O(

−−−−→N0N j).

This gives :

u(N3) − u(N1) = (∇u(N0),−−−−→N1N3) + O(

−−−−→N0N3) − O(

−−−−→N0N1),

u(N4) − u(N2) = (∇u(N0),−−−−→N2N4) + O(

−−−−→N0N4) − O(

−−−−→N0N2).

(6)

We can then solve the linear system (6) to obtain the consistentfinite difference central approximation :

∇u(N0) = B

u1u2u3u4

+ maxj

(O(−−−−→N0N j)), (7)

with

B =1∆

[y2 − y4 y3 − y1 y4 − y2 y1 − y3x4 − x2 x1 − x3 x2 − x4 x3 − x1

]and

∆ = (x3 − x1)(y4 − y2) − (x4 − x2)(y3 − y1).

2

Similarly, it is also possible to write the finite difference oneach nodes Nk instead of NO, for instance :

u(N2) = u(N1) + (∇u(N1),−−−−→N1N2) + O(

−−−−→N1N2),

u(N4) = u(N1) + (∇u(N1),−−−−→N1N4) + O(

−−−−→N1N4).

Then

∇u(N1) = B∗

u1u2u3u4

+ maxi, j

(O(−−−→NiN j)) (8)

with

B∗ =1∆∗

[y2 − y4 y4 − y1 0 y1 − y2x4 − x2 x1 − x4 0 x2 − x1

]and

∆∗ = (x2 − x1)(y4 − y1) − (x4 − x1)(y2 − y1).

With these two finite difference approximation schemes, 2 dif-ferent integration scheme to compute the ’stiffness’ matrix Kfollow : one node integration with (7), four nodes integrationwith (8). In this paper, we shall exclusively focus on the centralfinite difference approximation (7).

With the central finite difference approximation (7), on eachquadrangular element Ek, the elementary ’stiffness’ matrix sim-ply reads :

Kel = B′ ∗ B|Ek |.

where |Ek | is the measure of the element.

Numerical results on the scalar model problem (1)We apply the finite difference method to the Poisson prob-

lem (3) with Ω = [− 12 ,

12 ]2. Due to the symmetries of the prob-

lem we compute on a quarter of the domain only, that is [0, 12 ]2.

With a set of regular and distorted k × k meshes where the pa-rameter k denotes the number of subdivision of the domain, seeFigure 2.

Regular mesh Distorted mesh

Figure 2: Regular and distorted 4x4 meshes

We consider the finite difference element procedure withone degree of freedom (DOF) per node, for which we solelyuse the (7) central finite difference approximation coupled withone node integration scheme; we call this element FDEM4.

In Figure 3, we display the relative error of the solution atthe center of Ω, where it takes its maximal value, comparedto the exact solution (obtained through Fourier series) and thecomputed solution by a standard four node quadrangle polyno-mial element, QUA4.

We can see that FDEM4 element converges with an appar-ent nearly quadratic rate, even though it is clearly outperformedby QUA4. The convergence rate for FDEM4 does not showmuch sensitivity to mesh distorsion, whereas it deteriorates forQUA4 in the case of distorted meshes. This is not surpris-ing since the finite difference scheme (7) is independent of theshape of the quadrangular and the P2 polynomial interpolationof QUA4 is incomplete. These numerical results already indi-cate the behavior of FDEM4 element : it will have average topoor performance on very coarse meshes, but the convergencerate should be nearly quadratic and should not be too sensitiveon mesh distorsion.

2 5 10 200.001

0.01

0.1

1

Relative error on regular meshes

k

rela

tive e

rror

Linear rate

QUA4

FDEM4

Linear rate

QUA4

FDEM4

2 5 10 200.001

0.01

0.1

1

Relative error on distorted meshes

k

rela

tive e

rror

Linear rate

QUA4

FDEM4

Linear rate

QUA4

FDEM4

Figure 3: Relative error uh−uu at (0, 0) for regular and distorted k × k meshes.

Galerkin or not ?

Our interpretation of the Finite Difference Element Method,is that the rough integration scheme can nevertheless be exactfor some class of shape functions. From this point of view, theFinite Difference Element Method is even a conformal finiteelement method. A description of such a class is still underinvestigation, but we can reasonably conjecture such class isnot void in the space of admissible solutions, in this case.

3. Thin elastic shells : the Naghdi’s model

We refer to [6, 11] for a full description and presentationof the theory on thin shells in the Naghdi’s model. We onlyrecall the mechanical problem in static linear elasticity for ahomogeneous and isotropic shell with Young’s modulus E andcoefficient of Poisson ν.

Let’s first define a shell with its geometry, see Figure 4. LetS be the middle surface defined with the map (Ω,−→r ) : Ω ∈ R2

and−→r = x(x1, x2)−→e 1 + y(x1, x2)−→e 2 + z(x1, x2)−→e 3. (9)

The surface map defines the tangent vectors

−→a α = −→r ,α,

and the unit normal vector

−→a 3 =−→a 1 ∧

−→a 2

‖−→a 1 ∧

−→a 2‖.

3

the three vectors −→a i constitute the covariant basis of S . Wekeep in mind that the covariant basis is not a priori orthonormalnor orthogonal, we thus define the dual contravariant basis −→a

i

in order to make scalar products.Let ε be the constant thickness the shell, then any of its point

can be given by the position vector, ∀(x1, x2) ∈ Ω, x3 ∈ [− ε2 ,ε2 ],

−→p (x1, x2, x3) = −→r (x1, x2) + x3−→a 3.

Ω

a

a

a3

2

1

Figure 4: A Shell, the covariant basis of its midsurfcace

The Naghdi’s model involves the displacement −→u and therotation

−→θ to define the deformation of the shell under external

loadings. Although the theory involves covariant components,we shall represent the displacement with its Cartesian compo-nent :

−→u = ui−→ei .

The rotation is tangent to S and thus can conveniently be repre-sented by its covariant components:

−→θ = θα

−→aα.

Under external loadings, represented by a linear form Lε, wedefine the mechanical problem of thin elastic shells in its vari-ational form as follows, where V denotes a Sobolev space ofkinematically admissible displacements and rotations : Find U = (−→u ,

−→θ ) ∈ V such that

aε(U,U∗) = Lε(U∗) ∀U∗ ∈ V.(10)

The bilinear form of deformation energy, aε, for any U = (−→u ,−→θ )

and U∗ = (−→u∗,−→θ∗

), is :

aε(U,U∗) = ε3ab(U,U∗) + εas(U,U∗) + εam(U,U∗) (11)

where the energy form is respectively decomposed in bending,shear and membrane parts :

ab(U,U∗) =

∫Ω

112

Aαβλµραβρλµ (12)

as(U,U∗) =

∫Ω

CαβΛαΛβ (13)

am(U,U∗) =

∫Ω

Aαβλµγαβγλµ (14)

where γαβ and ραβ respectively represent the variation tensorsof the first and second fundamental form of the surface S , mea-suring the variation of length and variation of curvature. The

Λα measure the shear :

γαβ =12

(−→u ,α.−→a β + −→u ,β.

−→a α

)(15)

ραβ =12

(−→θ ,α.−→a β +

−→θ ,β.−→a α −

−→a 3,α.−→u ,β −

−→a 3,β.−→u ,α

)(16)

Λα = θα + −→u ,α.−→a 3. (17)

Since the shell is homogeneous and isotropic, the constitutivelaw is given by :

Aαβλµ =E

2(1 + ν)

[aαλaβµ + aαµaβλ +

2ν1 − ν

aαβaλµ]

(18)

Cαβ =E

4(1 + ν)aαβ. (19)

with the contravariant components of the first fundamental form

aαβ = −→aα.−→a

β.

Asymptotic behavior and Numerical difficultiesThe natural trend for a very thin elastic shell is to perform

bendings as, from (10), shear and membrane deformation en-ergies are penalized when the thickness ε is very small, seealso [27, 11, 26]. But, unlike the bendings in beams prob-lems, bending deformation on a surface is not always admis-sible : it depends on the boundary conditions and the geometry[18]. Thin shells are thus classified as [with bendings] inhib-ited or not-inhibited, following the terminology introduced bySanchez-Palencia[27].

This classification is also referred as membrane dominatedand bending dominated [24, 17]. These two very distinct asymp-totic behaviors lead to strong difficulties in the numerical ap-proximation of very thin elastic shells by finite element proce-dures : boundary layers, propagation and reflexion of singular-ities, sensitivity within the inhibited case and numerical (mem-brane) locking in the non-inhibited or bending dominated case,see [24, 25, 16, 11, 20, 26].

Classically, we have the following asymptotic behavior validfor thin shells linear models such as Koiter’s or Naghdi’s [26,17, 11]:

Theorem 1. If the (scaled) load Lε(−→v ) depends on ε as Lε(−→v ) =

ε3L(−→v ), then the solutions −→uε

of (10) converge (weakly in−→V )

towards the solution −→u0

of the asymptotic (limit) bending prob-lem

find −→u0∈−→G such that∫

S

Aαβλµ

12ραβ(−→u

0)ρλµ(−→v ),= L(−→v

∗), ∀

−→v∗∈−→G ,

(20)

where−→G is the set of kinematically admissible infinitesimal bend-

ings (also called inextensional displacement) :

−→G =

−→u ∈ V such that γαβ(−→u ) = 0

.

and where we replaced θα in ραβ with −−→u ,α.−→a 3.

4

We recall that infinitesimal bendings are solutions of a par-tial differential equation, the nature of which depends on thenature of the surface. In particular, if a surface is defined by aposition vector :

r(x1, x2) = x1−→e 1 + x3

−→e 2 + ϕ(x1, x2)−→e 3,

the vertical component u3 of any infinitesimal bendings on thesurface satisfies the second order PDE :

ϕ,22u3,11 + 2ϕ,12u3,12 + ϕ,11u3,22 = 0, (21)

the characteristics of which corresponds to the asymptotic linesof the surface [18]. Namely, the nature of equation (21) is re-spectively hyperbolic, parabolic, elliptic when the surface is hy-perbolic, developpable, elliptic.

This essential property underlines the fundamental impor-tance of the asymptotic lines and therefore the topology of ameshing when dealing with the membrane locking, see [1, 27,11].

In the inhibited shell cases, that is, when the space of in-finitesimal bendings

−→G reduces to 0 or rigid displacements,

which is the most common situation in practice, the theorem 1simply indicates that the solutions −→u

εconverge to 0, as ε 0.

To fully analyze the membrane dominated asymptotic be-havior, we need to change the scale of the loading from ε3L toεαL with 1 < α < 3, see [11, 20] depending on the case. Weshall not recall here all the subtleties of membrane dominatedcase, we simply recall that singularities and boundary layersappear, which make accurate computations difficult :

Theorem 2. With the loading scaled as Lε = εL, if the shellis inhibited, then as ε decreases towards 0, the solution −→u

εof

problem (1) converge weakly to−→u the solution of the membrane

limit problem :find−→u ∈ V

msuch that∫

SAαβλµγαβ(

−→u )γλµ(−→v

∗),= L(−→v

∗), ∀

−→v∗∈ V

m,

(22)

for any L ∈ Vm′

,

where Vm

is the completion of V for the norm associated witham. In some cases V

mmay be so big that it is not contained in

the space of distributions. The shell is then said ill-inhibited,see [26]. The corollary is that the dual space V

m′can be very

small : a uniform loading may even not be admissible in mostcases with a free edges. This explains the singularities andboundary layer appearing in the inhibited or membrane dom-inated shells or even the instabilities such as the chancellingshells [21]. The asymptotic behavior of inhibited (membranedominated) shells thus may vary depending on the loading [26,11].

4. The FDEM4 shell element

The FDEM4 shell element we propose here is based on theNaghdi’s Shell Model presented in section 3. The element is

simply defined as replacing all first order differentials, in equa-tions (15-17), involved in the variational problem of Naghdi’sshell (10) by the central finite difference scheme (7). The inte-gration is then obtained through a single node integration at theintersection of the diagonals of each quadrangular element.

It is indeed a very simple element, very cheap from a com-putational point of view, since the elementary stiffness matrix isonly evaluated once per element. We believe no shell elementcan possibly be more simple.

Each node have 5 degrees of freedom (DOF) : the 3 Carte-sian components, ui , of the displacement and 2 covariant com-ponents, θα, of the rotation. We remark from (16), that it is notnecessary to compute the derivatives of the covariant compo-nent of the rotation, we simply need to compute the derivativesof the rotation by their expressions in the Cartesian basis, inorder to apply the finite difference scheme (7). We make thecomputation directly on each element without dealing with areference element.

Intregration node

Nodes of DOF

Figure 5: The FDEM4 shell element

In an algebraic way, such ’reduced’ integration scheme re-duces the number of constraints coming from shear and bend-ings condition : two conditions from the shear constraints andanother three from the bending condition, when in bending dom-inated situations. Thus, a simple count of the number of avail-able DOF, suggest that the FDEM4 to be locking free, or at leastless sensitive to the locking effect.

Aside this (necessary) algebraic account, the finite differ-ence element method procedure allows another interpretation :we conjecture there is a class of shape functions such that thefinite difference element schemes coincide with the integral in-volved in (3), we expect then this class of functions to be largeenough to ’contain’ infinitesimal bendings. The existence anddescription of such a class certainly depends on the geometryand the mesh topology.

This work is preliminary, we chose to use exact geometryfor the definition of the surface, that is, we use the exact expres-sions of the covariant basis and theirs derivatives. The develop-ment of FDEM shell procedure using only the coordinates ofthe nodes of the surface is in progress.

5. Numerical results

In this section, we follow [10, 24, 14, 15] to test the perfor-mance of the quadrangular finite difference element, FDEM4,on several cases of inhibited and non-inhibited shells and com-pare with widely used shell elements, namely the 3-nodes trian-gular plane facet Discrete Kirchhoff Triangle (DKT) shell ele-

5

ment within the CasT3M1 software and the 4-nodes quadrangu-lar MIxed Tensorial Components (MITC4) element within thesoftware Scilab2 and the package OpenFEM, [5, 3].

We used the same meshes for FDEM4 and MITC4, but forthe triangular DKT element, we divided each quadrangle of themeshes in 4 triangular parts, thus our computations with DKTinvolve significantly more nodes and DOF than their quadran-gular counterparts.

5.1. A plate as a plane shellWe start the numerical tests with the case of a plate, for

which the Nagdhi’s model reduces to Reissner-Mindlin platemodel. The test is interesting from the numerical locking pointof view because it first evaluates the robustness with respect toshear locking.

We consider the classical example of a circular plate clampedall along its boundary.It is submitted to an uniform verticalloading. To test the shear locking we have taken a very smallthickness, ε = 10−4 (tests with smaller thicknesses give nearlyidentical results). Taking advantage of the axial symmetries ofthe problem, the computational domain is reduced to a quarterof the domain, see figure 6.

In Figure 7, we plot the relative error on the central de-flection, compared to exact analytical solution of the problemwithin Kirchhoff-Love plate model. We display the errors forvarious regular k × k mesh, k being the number of element sub-division of one side of the quarter of the domain. The size ofthe mesh being h = 1

k .The FDEM4 plate element is convergent with quadratic rate

but is outperformed by both DKT and MITC elements. This issimilar to the scalar model case. (1). Nevertheless, the shearlocking is not present in FDEM4 for this test.

Figure 6: A quadrangular k × k mesh of the quarter of the circular plate withrefinement parameter k = 6.

5.2. An inhibited cylindrical shell, the Scordelis-Lo roofWe start the numerical test on shells with a classic: the

Scordelis-Lo roof. It is a cylindrical shell supported by rigiddiaphragms at its end edges, the two other boundaries along thegeneratrices are free. The shell is submitted to a uniform ver-tical loading. Due to the plane symmetries of the problem, the

1Cast3M is available at http://www-cast3m.cea.fr and is developedby the Commissariat a l’Energie Atomique.

2Scilab is available at http://www.scilab.org and is developed by theINRIA.

0.01 0.1 10.0001

0.001

0.01

0.1

1

h

rela

tive

err

or

Quadratic rate

MITC4

FDEM4

DKT

Quadratic rate

MITC4

FDEM4

DKT

Figure 7: Circular clamped plate. The relative error of central displacementuexact−u

uexactcompared to the exact solution.

computed domain is reduced to one forth of the cylinder, seeFigure 8.The original benchmark proposed takes a fixed rela-tive thickness ε = 3

300 = 10−2, the Young’s modulus E = 3.106,the Poisson coefficient ν = 0 and the vertical load L = 0.625,the coherent units being inches and pounds.

40o

300

300

u =u =0x y

free edgeD

C

B

A

Figure 8: Scordelis-Lo roof

We plot the value of the vertical component at midside Bnormalized by a reference value and for k×k regular meshes, seeFigure 9. The reference values are the exact analytical solutionbased on Flugge’s shell theory, see [8], and are given in Table1. We note that the FDEM4 seems to converge but is slightlyoverestimating the deflection, unlike to DKT and MITC4.

We push forward this example by decreasing the relativethicknesses from ε = 10−2 to 10−5. We refer to [6, 11] for acomplete asymptotic analysis of the test, but we emphasize onthe fact it is a case of inhibited shell, more precisely ill-inhibitedshell. The vertical loading does not belong to the space V

m′and

thus is not admissible : boundary layers appear along the freeedges if the thickness is chosen very small. The width of thisboundary layer was shown, in [19], to be proportional to ε

14 .

We then take account of this information, choosing 32ε

14 as the

width of the boundary layer to define adapted meshes equallyrefined inside and outside the boundary layer, see Figure 10.This can be seen as a test on whether the FDEM4 is suitable for

6

6 8 10 12 14 16 18 200.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

relative thickness = 0.01

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

on

mid

sid

e

Reference solution

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

Figure 9: Vertical deflection on midside of Scordelis-Lo roof for regular k × kmeshes, normalized by a reference solution.

adapted meshes with stretched elements. We scaled the loadingby a ε2

9 factor. In Figure 11 we plot the deflection on midsideof the free edge normalized by a reference value, see Table 1.The FDEM4 performs here very well with a satisfactory con-vergence rate, and clearly better than both DKT and MITC4when the relative thickness ε = 10−4 or ε = 10−5.

ε 10−2 10−3 10−4 10−5

u3 3.6285 3.9528 4.0882 4.0805

Table 1: Reference value of vertical deflection on midside of Scordelis-Lo rooffor relative thickness from 10−2 to 10−5, based on Flugge’s shell theory.

Relative thickness = 0.0001 Relative thickness = 0.00001

Figure 10: Adapted (k + k) × k meshes with k = 4.

5.3. A partially clamped non inhibited cylindrical shellWe continue with a cylindrical shell, but choose the bound-

ary conditions so that the infinitesimal bendings are renderedadmissible. We roughly take a simplified Scordelis-Lo cylinderpartially clamped along a generatrice (straight edge) while theother edges are free. We impose a localized loading on a freecorner as shown in Figure 12. The geometry is defined by :

x = x1, y = x2, z = ϕ(x), ∀(x1, x2) ∈ Ω, (23)

with ϕ(x1) =

√1 − x2

1 and Ω = [0,√

22 ] × [−0.5, 0.5]. The shell

is subjected to a vertical force−→Fε = ε3−→e 3 localized at the point

P = (√

22 , 0.5,

√2

2 ); note the ε3 scaling corresponding to a non-inhibited shell.

2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

1.2

k

norm

alized v

ert

ical dis

pla

cem

ent on m

idsid

e

Reference solution

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

relative thickness = 0.0001

2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

1.2

relative thickness = 0.00001

k

norm

alized v

ert

ical dis

pla

cem

ent on m

idsid

e

Reference solution

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

Figure 11: Normalized deflection on midside on Scordelis-Lo roof with adapted(k + k) × k meshes.

A

P

B

Clamped boundary

Vertical localized load

free edges

Figure 12: A Cylinder clamped along AB and subjected to a vertical load local-ized on P.

This test is not classical, we preferred it over the full cylin-der with free ends and subjected to a periodic pressure, see forinstance [10]. The main reason is the more complex shape ofthe solution in our test; the test becomes henceforth ’tougher’than the full cylinder one. To our best knowledge, no ana-lytic solution for this problem is available but it is possible tosolve the associated asymptotic bending problem (20) [15]. Theasymptotic solution can then be used as a reference as the thick-ness we is decreased .

Equation (21) which characterizes the vertical componentof an infinitesimal bending reduces then to

ϕ22u3,11 = 0.

The vertical component of an infinitesimal bending on the cylin-der are then the polynomials in x2 with arbitrary functions g andh of the variable x1 :

u3 = g(x1) + x2h(x1).

The shell being now clamped along the straight line x1 = 0,all vertical displacements u3 = g(x1) + x2h(x1) satisfying theboundary conditions g(0) = g′(0) = h(0) = h′(0) = 0, de-fine admissible infinitesimal bendings. The shell is thus non-inhibited.

The bending limit problem (20) reduces then to a one di-mensional differential problem :

Proposition 3. The vertical component u03 of the asymptotic

bending problem (20) can be computed by solving a variational

7

problem of two functions of one variable :Find u0

3 = g(x1) + x2h(x1) ∈ V∫S

112

[A1111ρ11ρ

∗11 + 4A1212ρ12ρ

∗12

]= L(u∗3),

∀u∗3 = g∗(x1) + x2h∗(x1) ∈ V.

(24)

We don’t have the exact analytical solution of the reducedasymptotic bending problem (24), but it can be solved usingstandard one dimensional finite element procedure. To fix theidea, the value of the vertical displacement at P is set to u0

3 =

2.1168107, for E = 1, ν = 1/3 and unitary vertical load.We note that in this case, the vertical displacement on P of

the limit bending solution equals to the limit flexion energy orthe asymptotic energy :

u03(P) = ab(−→u

0,−→u

0).

From minimizing energy results, we remind that if uε is thesolution of (10) with the scaled loading Lε then, we have nec-essarily :

u03(P) ≤ uε3(P) ∀ε > 0.

The value of the vertical component at P is then a good criteriaas to evaluate a finite element computation in this case. con-vergence for regular and distorted meshes are shown in Figure13, where the distorsion is defined as a geometric sequence ofcommon ratio 1.1. We must keep in mind that with such dis-torted meshes, the asymptotic lines are no more represented bysome sides of the elements. To test the membrane locking, werun computations for thicknesses from 10−2 to 10−4.

Figure 13: Regular and distorted k × k meshes, with k = 4 .

In the Figure 14, we plot the vertical displacement normal-ized by the solution of the asymptotic (bending) limit problem(24) using k × k regular meshes for various values of k, thenumber of subdivisions of the interval [0,

√2

2 ] For the thicknessε = 10−2 we see that DKT, MITC4 and FDEM4 are coher-ent and converge to the same solution. As the thickness de-creases, we clearly see that DKT underestimates the real solu-tion whereas the MITC4 and FDEM4 are both coherent withthe asymptotic analysis and seem to converge. In this case,the membrane locking is absent for both MITC4 and FDEM4whereas present with DKT.

In the Figure 15, we plot the vertical displacement normal-ized by a reference solution using quasi uniform distorted k × kmeshes, see Figure 13 , for various values of mesh refinementparameter k. As a reference solution, we used the value ob-tained with MITC4 element with a 28x28 regular mesh. We

4 6 8 10 120.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

thickness = 0.01

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

4 6 8 10 120.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

thickness = 0.001

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

4 6 8 10 120.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

thickness = 0.0005

4 6 8 10 120.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

thickness = 0.0001

Figure 14: Non-inhibited partially clamped cylinder submitted to vertical local-

ized load, with regular meshes,uε3(P)

u03(P)

u for thicknesses from 10−2 to 10−4.

note in this case that the reference solution is not very differentfrom the asymptotic solution.

The locking is still present for DKT. On the contrary MITC4shows signs of significant deterioration of the approximationfor very small thicknesses (ε = 5.10−4), while FDEM4 seemsunaffected.

5.4. A partially clamped non-inhibited hyperbolic paraboloidWe study here the case of a partially clamped hyperbolic

paraboloid defined for any (x1, x2) ∈ Ω =

[−√

22 ,

√2

2

]2

x = x1, y = x2, z =12

(x21 − x2

2). (25)

Note that the last surface is equivalently defined by

x = x1, y = x2, z = x1x2, ∀(x1, x2) ∈ Ωl (26)

where Ωl is the quadrangle defined by the four points A : (−1, 0),B : (0,−1), C : (1, 0), D : (0, 1), see figure 16

−1 B

0

D1

−1 1

CA

A

B

C

D

Figure 16: A hyperbolic paraboloid, partially clamped, along AB, and submit-ted to uniform vertical load.

8

4 6 8 10 120.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

thickness = 0.01

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

Reference solution

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

4 6 8 10 120.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

Reference solution

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

thickness = 0.001

4 6 8 10 120.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

Reference solution

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

thickness = 0.0005

4 6 8 10 120.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

thickness = 0.0001

Figure 15: Non-inhibited partially clamped cylinder submitted to vertical local-

ized load, with distorted meshes,uε3(P)

u03(P)

u for thicknesses from 10−2 to 10−4

Here again we compare FDEM4 with MITC4 and DKTwith respect to the solution of the bending limit problem. In thecase of hyperbolic paraboloid defined by the map (26), equation(21) characterizing the vertical component of an infinitesimalbending reduces to

ϕ,12u3,12 = 0.

Then any arbitrary functions f and g of one variable define thevertical components of any inextensional displacements (or in-finitesimal bendings) on the paraboloid :

u3(x1, x2) = g(x1) + h(x2) ∀(x1, x2) ∈ Ωl

We recall that when the shell is clamped along the AB boundary,it is not inhibited although bendings necessarily vanishes in thesubdomain (x1, x2) ∈ Ω/ x1 ≤ 0 and x2 ≤ 0. It suffices thenthat g and h and their derivatives vanishes on [−1, 0] and bearbitrary on ]0, 1].

On such shell problem, the limit bending limit problem canbe reduced to a more simple formulation, see [14] :

Find u3 = g(x1) + h(x2)/g, h ∈ V2

112

∫Ωl

[g′′

h′′

]> [A1111 A1122

A2211 A1222

] [g∗′′

h∗′′

]a

32

=

∫Ωl

(g∗(x1) + h∗(x2))a12 ,

∀g∗, h∗ ∈ V2

(27)

whereV = f ∈ H2([0, 1])/ f (0) = f ′(0) = 0.

In Figure, 18 and 19, we normalized the vertical displace-ment on corner node D by the solution of the bending limitproblem. We used regular and distorted meshes, see Figure 17,for various values of mesh refinement parameter k. The meshdistorsion is again defined as geometric sequence of commonratio 1.1.

As we can see no membrane locking arises on regular meshesand the results are consistent with the asymptotic behavior, weobserve that the computed solutions converge to the asymptoticbending limit solution as the thickness decreases towards zero.The convergence rate seems good but FDEM4 is underperform-ing compared to both DTK and MITC4.

Figure 17: regular and distorted k × k meshes, with k = 6.

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

6 8 10 12 14 16 180.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

thickness = 0.01

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

6 8 10 12 14 16 180.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

thickness = 0.001

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

6 8 10 12 14 16 180.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

thickness = 0.0001

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

6 8 10 12 14 16 180.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

thickness = 0.00001

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

Figure 18: Non-inhibited partially clamped hyperbolic paraboloid. Normalizedvertical displacement on k × k regular meshes

In Figure 19, we plot the computed normalized vertical com-ponent on corner node D for distorted k × k meshes. The ver-tical displacement is normalized with by a reference solutionobtained with the MITC4 solution and a very thin regular mesh(28x28), see table 2, except for the smallest thickness ε = 10−5

for which we normalized by the asymptotic solution.It appears that the computations from both DKT and MITC4

presents severe signs of deterioration for thicknesses below 10−3.

9

FDEM4 also presents slight signs of computational deteriora-tion but in a very different magnitude and overall correct exceptfor the smallest thickness for which the error in vertical deflec-tion is nearly 20%.

ε 10−2 10−3 10−4 Asymp. solutionu3 0.86 0.60580 0.52349 0.5006

Table 2: Reference value of vertical deflection at D compared to the asymptoticsolution.

6 8 10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

thickness = 0.01

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

Reference solution

6 8 10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

thickness = 0.001

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

Reference solution

6 8 10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

thickness = 0.0001

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

MITC4

FDEM4

DKT

Reference solution

MITC4

FDEM4

DKT

Reference solution

6 8 10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

k

no

rma

lize

d v

ert

ica

l d

isp

lace

me

nt

MITC4

FDEM4

DKT

Asymptotic solution

MITC4

FDEM4

DKT

Asymptotic solution

thickness = 0.00001

Figure 19: Non-inhibited partially clamped hyperbolic paraboloid. Normalizedvertical displacement on k × k distorted meshes

5.5. A clamped inhibited hyperbolic paraboloid

We consider again the geometry of section 5.4 and uniformvertical loading but the shell is now clamped all along its bound-ary, the shell is with bendings inhibited.

Instead of focusing on the deflection on one particular point,we had rather study the convergence in energy norms. In Fig-ure, we plot the relative errors from computations on regulark × k meshes with respect to the values obtained with a 60 × 60mesh.

In Figure 20, the convergence rate seems almost quadraticagain. For very small thicknesses (ε < 10−3), we detect insta-bilities, illustrated by the poor convergence of the bending andshear energy, that disappear with finer meshes. Although, wedidn’t detect spurious modes, we think the existence of pseudo-bendings [28, 13], explains the instabilities. Note that thoseinstabilities are also present for DKT and MITC4.

1 5 10 20 100−4

10

−3

10

−2

10

−1

10

0

10

1

10

thickness = 0.01

k

rela

tive

err

or

Quadratic rate

Membrane energy

Bending energy

Shear Energy

Quadratic rate

Membrane energy

Bending energy

Shear Energy

1 5 10 20 100−4

10

−3

10

−2

10

−1

10

0

10

1

10

thickness = 0.001

k

rela

tive

err

or

Quadratic rate

Membrane energy

Bending energy

Shear Energy

Quadratic rate

Membrane energy

Bending energy

Shear Energy

Figure 20: Non-inhibited clamped hyperbolic paraboloid. Relative errors ofmembrane, flexion, total energy on k × k regular meshes

Figure 21: Spherical cap with 6 × 6 quadrangular mesh.

5.6. A clamped well-inhibited spherical capWe consider her the spherical cap defined with the position

vector, ∀(x1, x2) ∈ Ω

r(x1, x2) =

√4 − x2

1 − x22

where Ω = (x1, x2) ∈ R2/x21 + x2

2 ≤ 1, see Figure 21. Thegeometric nature is elliptic, there are no asymptotic lines in thatcase. The shell is clamped all along it boundary, the shell isthen inhibited, more precisely well-inhibited and the completedspace V

mis included in H1

0×H10×L

2(Ω) (for tangent and normalcomponents of the displacement), see [26].

We impose a uniform vertical load. As the problem is in-variant by axial rotation, we use the computed solution of anaxisymmetric shell element as reference. We plot, in Figure22, the relative error of the vertical displacement at the topthe spherical cap. In this case, DKT, MITC4 and FDEM4 allpresent instabilities, although less important for MITC4. Whilewe observe convergence, the instabilities grows as the thicknessdecreases for a fixed mesh. For FDEM4, it is due to the exis-tence, in this elliptic case, of a spurious mode for the asymp-totic membrane problem (22) that is stabilized by the bendingenergy.

6. Conclusions and perspectives

We have proposed an original four node quadrangular shellfinite element, FDEM4, using a finite difference element methodprocedure based on the quest of a Galerkin method without ex-plicit shape functions. The FDEM4 shell element derives di-rectly from the Naghdi’s shell model without any corrections.

10

5 10 20 100−4

10

−3

10

−2

10

−1

10

0

10

1

10

relative thickness = 0.01

k

rela

tive e

rror

of vert

ical dis

pla

cem

ent

Quadratic rate

MITC4

FDEM4

DKT

Quadratic rate

MITC4

FDEM4

DKT

5 10 20 100−4

10

−3

10

−2

10

−1

10

0

10

1

10

relative thickness = 0.001

k

rela

tive e

rror

of vert

ical dis

pla

cem

ent

Quadratic rate

MITC4

FDEM4

DKT

Quadratic rate

MITC4

FDEM4

DKT

Figure 22: Well-inhibited clamped spherical cap with uniform loading. Nor-malized vertical displacement at (0,0).

The element is simple and cost effective due one node integra-tion scheme in the construction of the stiffness matrix.

The numerical tests on plates, inhibited and non-inhibitedshells, with regular and distorted meshes are very encouraging :the FDEM4 has constantly showed convergence behavior withalmost quadratic rate in every cases we have tested. In mostcases, FDEM4 also compares favorably to MITC4 or DKT.

In non-inhibited cases, the computed solutions are com-pared with the solution of the asymptotic bending problem. Asthe thickness is decreased, we observe a coherent asymptoticbehavior of FDEM4 with the asymptotic theory of thin shells.

In the case of a spherical cap (elliptic shell), the spuriousmode for the membrane energy, even though stabilized by thebending energy, prevents FDEM4 to be an ideal quadrangularshell element. It also means that the Galerkin interpretationof the FDEM4 element presented here, fails at least for thisgeometry.

Overall, the constant convergence behavior, the good ro-bustness to distorsion and membrane locking makes the FDEM4an interesting and already mostly reliable shell element. It of-fers a perspective for more robust and perhaps more performantshell elements based on this concept. We may for exampleadapt the finite difference scheme with respect to the geometricnature of the shell.

Much work still need to be done apart from improving ro-bustness : a theoretical study of convergence and the interpreta-tion of finite difference element method as part of the Galerkinmethod’s family. We need also to develop facet based FDEMelement instead of an exact definition of the geometry. Of course,one of the most important, of utmost engineering interest, is thedevelopment of triangular shell elements, as quadrangular dis-cretisation cannot always be easily available.

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[2] D. N. Arnold and F. Brezzi. Locking free finite element methods for shellproblem. Mathematics of Computation, 66(217):1–14, 1997.

[3] K.-J. Bathe. Finite Element Procedures. Prentice Hall, 1996.[4] K.-J. Bathe, A. Iosilevich, and D. Chapelle. An evaluation of the mitc

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11