On the effects of using curved elements in the approximation of the Reissner–Mindlin plate by the p version of the finite element method

n

SA

methodmptoticgs willvarious

ceivedchniquesoblem,ofessns that

ineering

Applied Numerical Mathematics 46 (2003) 231–246www.elsevier.com/locate/apnum

On the effects of using curved elements in the approximatioof the Reissner–Mindlin plate by thep version

of the finite element method

Jason Kurtza,1, Christos Xenophontosb,∗

a Division of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13699-5815, USAb Department of Mathematical Sciences, Loyola College, 4501 N. Charles Street, Baltimore, MD 21210-2699, U

Abstract

We consider the approximation of the Reissner–Mindlin plate model by the standard Galerkinp version finiteelement method. Under the assumption of sufficient smoothness on the solution, we illustrate that theis asymptotically free of locking even when certain curvilinear elements are used. The amount of preasylocking is also identified and is shown to depend on the element mappings. We identify which mappinresult in asymptotically locking free methods and through numerical computations we verify the results formappings used in practice. 2003 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords:Finite element method; Curved elements;p version; Reissner–Mindlin plate; Shear locking

1. Introduction

The numerical approximation of the solution to the Reissner–Mindlin (R–M) plate model has remuch attention in recent years (see, e.g., [5,6,14,18] and the references therein). Several tehave been proposed to alleviate the two major computational difficulties associated with this prnamely the presence oflocking andboundary layereffects. The former occurs due to the inabilitythe approximating spaces to satisfy certain constraints imposed on the solution as the thicknt ofthe plate tends to zero. The latter is due to the fact that the system of partial differential equatio

* Corresponding author.E-mail addresses:[email protected] (J. Kurtz), [email protected] (C. Xenophontos).

1 Current address: Texas Institute for Computational and Applied Mathematics, Applied Computational and EngScience Building, Austin, TX 78712-1085, USA.

0168-9274/03/$30.00 2003 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/S0168-9274(03)00032-1

232 J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246

describes the R–M plate model is singularly perturbed. The interplay of both phenomena is a rathercomplicated affair and the question of how to alleviate themboth is not easily answered. It has been

withate witht “theed

n moredetails).e of thecurs inthatrnadoudry of the

eablewhen

smoothmain

f lockingh was

nd itselement.e results

the errorhan onev space

t

y per

shown in [15] that thep version of the Finite Element Method (FEM), on quasi-uniform meshesstraight sided elements, is asymptotically free of locking. These results were established for a plperiodic boundary conditions, hence eliminating the presence of boundary layers. The statemenpversion is asymptotically free of locking” means that asp→ ∞ the rate of convergence is not affectby t → 0. However, forp in the pre-asymptotic (often practical) range, the errordoesincrease, which inthe terminology of [7] means that the locking ratio is greater than 1. This error increase can be evesevere if curved elements are used and/or the error measure changes (see [7,8,12,15] for more

In this article we address the question of how curved elements affect the rate of convergencp version in the presence of locking. Our goal is to characterize the amount of locking that octhe pre-asymptotic range ofp and to verify this through numerical computations. Previous workdealt with curved elements for the approximation of plates (and shells) includes the work of Be[3,4] in which the traditionalh version of the FEM was used. The approximation in [3,4], possesseC1

smoothness and triangular elements with one curved side were used to approximate the boundadomain. In the present study we will focus on the standard Galerkinp version FEM. As a result, wwill not consider amixed/reduced constraintmethod approach even though this is, of course, a vialternative. (For such a study we refer to [10].) In order to avoid further complications that ariseboundary layers are present (see [1,2]), we will restrict ourselves to the case of a sufficientlysolution (one such example being the R–M plate with periodic boundary conditions [15]). Ourresult will be Theorem 1 which will state that the amount of locking depends on themappingused tomap the reference element to the curved elements in the mesh. We will comment on the amount ofor various mappings used in practice. Our approach is closely related to that found in [11] whicused for nearly incompressible elasticity (see also [9]).

The outline of the article is as follows. In Section 2 we describe the R–M plate model adiscretization, and in Section 3 we analyze the method in the case of the mesh consisting of oneThe several curvilinear element mesh is also discussed in that section. In Section 4 we present thof several computations that enable us to see how this phenomenon manifests itself as a “shift incurves”. We consider different mappings used in practice as well as meshes consisting of more telement. Our conclusions will be given in Section 5. Throughout the paper we use the usual SobolenotationHk(Ω), whereΩ ⊂ R

2 is an open bounded simply connected domain. Also,H 0(Ω)= L2(Ω).The norm and seminorm onHk(Ω) will be denoted by‖ · ‖k,Ω and| · |k,Ω , respectively. (The subscripindicating the domain will be dropped when no confusion occurs.) Finally, the letterC will denote ageneric positive constant independent of the thicknesst and any discretization parameters.

2. The model problem and its discretization

Consider the bending of a homogeneous isotropic plate of thicknesst > 0, occupying the regionR=Ω×(−t/2, t/2), whereΩ ⊂ R

2 represents the mid-plane of the plate, under normal load densitunit area given bygt3, with g independent oft . The equations of equilibrium for the rotation

−→φ = (φ1,φ2),

and transverse displacementw are

−D2

(1− ν)−→

φ + (1+ ν)∇∇ · −→φ

− λt−2(∇w− −→φ

) = 0, (1)

J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246 233

−λt−2∇ · (∇w− −→φ

) = g, (2)

which:tohe

mple,

sional

whereν is the Poisson ratio,κ is the shear correction factor,E is the Young’s modulus and

D = E

12(1− ν2), λ= κE

2(1+ ν) .The boundary conditions that often accompany the system (1)–(2) are summarized in Table 1, in−→n denotes the unit vector normal to the boundary, directed outward,−→s denotes the unit vector tangentthe boundary, directed counterclockwise and, withE(−→φ) the matrix denoting the symmetric part of tgradient of

−→φ ,

CE(−→φ

) =D[(1− ν)E(−→

φ) + ν trace

(E(−→φ

))I],

whereI denotes the 2× 2 identity matrix.LetV ⊆ [H 1(Ω)]2 ×H 1(Ω) be a given space that depends on the boundary conditions. (For exa

in the case of a clamped plateV = [H 10 (Ω)]2 ×H 1

0 (Ω), whereH 10 (Ω)= v ∈H 1(Ω): v = 0 on∂Ω.)

Then, the variational formulation of (1)–(2) reads: Findut := (−→φt,wt) ∈ V ⊆ [H 1(Ω)]2 ×H 1(Ω) such

that for allv = (−→θ , ζ ) ∈ VAt(ut , v) := a

(−→φt ,

−→θ) + λt−2

⟨∇wt − −→φt ,∇ζ − −→

θ⟩ = 〈g, ζ 〉, (3)

where〈·, ·〉 denotes the usualL2 inner product anda(·, ·) is given by

a(−→φt ,

−→θ) = D

2

∫Ω

(1− ν)∇−→

φt · ∇−→θ + (1+ ν)(∇ · −→

φt)(∇ · −→

θ)

dx1 dx2. (4)

The standard Galerkin finite element approximation of (3) proceeds by choosing a finite-dimensubspaceV N ⊂ V , of dimensionN , and seekinguNt = (−−→φNt ,wNt ) ∈ V N satisfying

a(−−→φNt ,

−→θ) + λt−2

⟨∇wNt − −−→φNt ,∇ζ − −→

θ⟩ = 〈g, ζ 〉 (5)

Table 1Boundary conditions for various boundary value problems for the Reissner–Mindlinplate model

Essential Natural

Hard Clamped−→φ · −→n = 0−→φ · −→s = 0w = 0

Soft Clamped−→φ · −→n = 0w = 0 −→s · CE(−→φ )−→n = 0

Hard Simply Supported−→φ · −→s = 0 −→n · CE(−→φ )−→n = 0w = 0

Soft Simply Supported −→n · CE(−→φ )−→n = 0

w = 0 −→s · CE(−→φ )−→n = 0

Free −→n · CE(−→φ )−→n = 0−→s · CE(−→φ )−→n = 0∂w∂n

− −→φ · −→n = 0

234 J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246

for all v = (−→θ , ζ ) ∈ V N . We then have∥ ∥

halled

meher

e will,

ingd

f

∥ut − uNt ∥E,t

= infv∈VN

‖ut − v‖E,t , (6)

where

‖ut‖E,t :=(At(ut , ut )

)1/2, (7)

denotes theenergy norm. It may be shown that there existα1, α2> 0 such that

α1‖u‖1,1,Ω ‖u‖E,t α2t−1‖u‖1,1,Ω, (8)

where

‖u‖2s,%,Ω := ∥∥−→

φ∥∥2s,Ω

+ ‖w‖2%,Ω := ‖φ1‖2

s,Ω + ‖φ2‖2s,Ω + ‖w‖2

%,Ω. (9)

It is well known that ast → 0, the solutionut = (−→φt ,wt) tends tou0 = (

−→φ0,w0) which satisfies

Kirchhoff’s constraint−→φ0 − ∇w0 = 0. (10)

The same constraint must be satisfied byuNt = (−−→φNt ,w

Nt ) and if the spaceV N does not have enoug

functions satisfying (10) the approximation will be quite poor, a well-known phenomenon c(shear)locking. The finite-dimensional subspacesV N usually consist of piecewise polynomials on sosubdivision (mesh) of the domainΩ . The dimensionN can be increased to improve accuracy, eitby refining the mesh (h version), increasing the polynomial degree (p version) or both (hp version).In this article we will only consider thep version, which, along with thehp version, is known to beasymptotically free of locking when straight sided elements are used [15]. To emphasize this wfrom this point on, writeV p instead ofV N . We will be considering meshesMp = ΩkMk=1 consistingof, possibly curvilinear, trianglesTk or quadrilateralsSk , such that there exists an invertible mappFk : Ω →Ωk that maps the reference elementΩ = T or S toΩk = Tk or Sk . The reference triangle anquadrilateral are given byT = (ξ1, ξ2): 0< ξ1, ξ2, ξ1 + ξ2 < 1, and S = (ξ1, ξ2): −1< ξ1, ξ2 < 1,respectively (see Fig. 1). We will use the usual reference spaces, i.e.,Pp(T ) will denote the space opolynomials defined onT of total degreep andQp(S) will denote the space of polynomials onS ofseparate degreep. We will also writeQp,q(S) = spanξ i1ξ j2 , 0 i p, 0 j q and forI ⊂ R an

Fig. 1. The reference triangleT and squareS.

J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246 235

interval,Pp(I ) will denote the space of polynomials onI of degree p. With V p any of the referencespaces above, we define

hile thepingunction

s

is

V p = u: u|Ωk = u F−1, u ∈ V p ∀Ωk ∈Mp

. (11)

In what follows we will write−→x = Fk−→ξ as well as

u(x1, x2)= u(x1(ξ1, ξ2), x2(ξ1, ξ2)

) =: u(ξ1, ξ2) (12)

for any functionu defined onΩk . Then, withJ denoting the Jacobian ofFk , we have[∂u/∂x1

∂u/∂x2

]= J−T

[∂u/∂ξ1

∂u/∂ξ2

], (13)

where

J−T =[∂ξ1/∂x1 ∂ξ2/∂x1

∂ξ1/∂x2 ∂ξ2/∂x2

]. (14)

3. Locking and curved elements

In this section we consider the case when a single element is used for the discretization, wpolynomial degreep is increased. Our goal will be to identify what condition(s) the element mapmust satisfy in order for the method to be locking free. To this end, suppose that we are given a fu = (

−→φ,w) with

−→φ = ∇w, defined on an elementΩk . The reason for choosing

−→φ this way is due to

Theorem 3.3 of [15], which states that it suffices to consider the limiting case ast → 0. In our case, thismeans that (10) holds which explains our choice of

−→φ . Then, using (12) and (14), we haveu= (φ1, φ2, w)

on Ωk , with[φ1

φ2

]=

[φ1

φ2

]= J−T

[∂w/∂ξ1

∂w/∂ξ2

], (15)

and x = Fkξ . Suppose now thatup = (φp

1 , φp

2 ,wp) is an approximation tou onΩk , that also satisfie

φp = ∇wp. Note that onΩk we have up = (φp

1 , φp

2 , wp). Assuming φpi ∈ V p, i = 1,2, we will

investigate under what conditions onFk , wp will be a polynomial.Consider, for simplicity, the caseΩk = S with one side ofS being curved, as depicted in Fig. 2. In th

case,Fk : S → S is given by

x1 = ξ1, x2 = −1+ (1+ f (ξ1)

)(1+ ξ2

2

),

whence,

J T =[

1 f ′(ξ1)(1+ ξ2)/20 (1+ f (ξ1))/2

].

Then by multiplying both sides of (15) byJ T we obtain,[∂wp/∂ξ1

∂wp/∂ξ2

]= J T

[φp

1

φp

2

]=

[φp

1 + φp2 f ′(ξ1)(1+ ξ2)/2φp

2 (1+ f (ξ1))/2

]. (16)

236 J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246

the

,

n,

Fig. 2. Quadrilateral with one curved side.

Clearly, f must be a polynomial in order forwp to be a polynomial. In particular, ifφpi ∈ V p =Qp(S), i = 1,2, andf ∈ Pq , then wp ∈ Qp+q,p+1(S). These observations can be generalized tocase of possibly all sides ofΩk being curved, and formulated as the following theorem.

Theorem 1. LetFk : Ω →Ωk ∈ Mp be an invertible polynomial map,Fk ∈ [Vq]2, with Vq = Qq(Ω) ifΩ = S, andVq = Pq(Ω) if Ω = T . Suppose thatut = (−→φt,wt) ∈ [Hr−1(Ωk)]2 ×Hr(Ωk), r 2 satisfies(3) andupt = (−→

φpt ,w

pt ) ∈ [V p(Ωk)]2 × V p+q(Ωk) satisfies(5), whereV p(Ωk) is defined by(11) with

V p =Qp

(Ω

)if Ω = S,

Pp(Ω

)if Ω = T .

Then, there exists a constantC ∈ R independent oft, p andα such that∥∥ut − upt ∥∥1,1,Ωk C(p− α)−r+2‖ut‖r−1,r,Ωk , (17)

whereα = 3q − 2 if Ω = S, andα = 3q − 4 if Ω = T .

Proof. It is sufficient to only consider the limiting caseu0 = (−→φ0,w0) = (∇w0,w0) =: (∇w,w) (see

[15, Theorem 3.3]). Let−→x = Fk−→ξ , with Jk and |Jk| denoting the Jacobian ofFk and its determinant

respectively. We then have

|Jk| = ∂x1

∂ξ1

∂x2

∂ξ2− ∂x1

∂ξ2

∂x2

∂ξ1∈

Q2q−1

(Ω

)if Ω = S,

P2q−2(Ω

)if Ω = T .

SinceFk : Ω →Ωk is invertible, we have|Jk|> 0 onΩ and forw ∈Hr(Ωk)

C1‖w‖r,Ωk ∥∥w∥∥

r,Ω C2‖w‖r,Ωk , (18)

where w = w Fk . Define z = w/|Jk|2. Then, by [13, Lemma 3.1], for any% there existsz% ∈ V %,V % = Q%(Ω) if Ω = S andV % = P%(Ω) if Ω = T , such that∥∥z− z%

∥∥s,Ω

C%−(r−s)∥∥z∥∥r,Ω

(19)

for 0 s r (see [13] for more precise estimates). Letwp = z%|Jk|2 with % to be selected shortly. The∥∥w−wp∥∥2,ΩkC

∥∥w− wp∥∥2,Ω C%−(r−2)∥∥w∥∥

r,Ω C%−(r−2)‖w‖r,Ωk . (20)

J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246 237

Now, [p

] [ ]

, in the

∇wp = ∂w /∂x1

∂wp/∂x2=: ∂w1

∂w2.

We will look at ∂w1 (the other term is almost identical). We have

∂w1 =(∂z%

∂ξ1

∂ξ1

∂x1+ ∂z%

∂ξ2

∂ξ2

∂x1

)|Jk|2 + 2z%|Jk|

(∂|Jk|∂ξ1

∂ξ1

∂x1+ ∂|Jk|∂ξ2

∂ξ2

∂x1

).

Note that[∂ξ1/∂x1 ∂ξ1/∂x2

∂ξ2/∂x1 ∂ξ2/∂x2

]=

[∂x1/∂ξ1 ∂x1/∂ξ2

∂x2/∂ξ1 ∂x2/∂ξ2

]−1

= 1

|Jk|[∂x2/∂ξ2 −∂x1/∂ξ2

−∂x2/∂ξ1 ∂x1/∂ξ1

].

Hence,

∂w1 =(∂z%

∂ξ1

∂x2

∂ξ2− ∂z%

∂ξ2

∂x2

∂ξ1

)|Jk| + 2z%

(∂|Jk|∂ξ1

∂x2

∂ξ2− ∂|Jk|∂ξ2

∂x2

∂ξ1

).

Evidently,

∂w1 ∈Q%+3q−2

(Ω

)if Ω = S,

P%+3q−4(Ω

)if Ω = T ,

hence∇wpt ∈ [Q%+3q−2(S)]2 or ∇wpt ∈ [P%+3q−4(T )]2.

Assume thatΩ = S and letα = 3q−2 andp = %+3q−2, which implies%= p−α. The caseΩ = Tis similar. By (20) we have∥∥w−wp∥∥2,Ωk

C(p− α)−(r−2)‖w‖r,Ωk .Defineup = (∇wp,wp) ∈ [V p(Ωk)]2 × V p+q(Ωk). Then, sinceu= (∇w,w)∥∥u− up∥∥2

1,1,Ωk= ∥∥∇w− ∇wp∥∥2

1,Ωk+ ∥∥w−wp∥∥2

1,Ωk C

∥∥w−wp∥∥22,Ωk

,

from which the result follows. The above theorem predicts the shift seen in the pre-asymptotic range ofp for the caseFk ∈ [Qq(Ω)]2

or [Pq(Ω)]2. This is, in some sense, the “worst case scenario” since a more realistic assumptionpractical sense, would be

Fk ∈[

Q1,q(Ω

)] × [Qq,1

(Ω

)]if Ω = S,[

Qq+1(Ω

)]2 ⊆ [P2q+2

(Ω

)]2if Ω = T . (21)

In this case the argument above can be repeated to yield the following corollary.

Corollary 2. Let the assumptions of Theorem1 hold, with Fk satisfying (21). Then(17) holds withα = 6q − 3 for Ω = T and

α =

0 if q = 0,2q − 1 else,

(22)

for Ω = S.

238 J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246

We see that pre-asymptotic locking cannot be avoided when polynomial mappings are used. There arecertainly other mappings used in practice, e.g., circular arcs, for which the above theorem and corollary

r non-

eneralhe one

d by

timates.

mercialperiodiclar, weconsider

erformlements

cal

lateg

htagegtion thatferenceis veryrm (9),

ted by

ft

do not apply. The numerical results in Section 4, however, will suggest that similar results hold fopolynomial (smooth) mappings as well.

We will only consider the simple case of a single element to avoid the complexities of more gmeshes. Experimentally, the case of several curvilinear elements is quantitatively similar to telement case, as will be demonstrated through our numerical computations.

Remark 1. The numerical experiments in the next section will suggest that the shift predicteTheorem 1 (and Corollary 2) is not visible for values of the thicknesst within the “practical range”.This is due to the fact that in the proof of Theorem 1 we considered the limiting solution (ast → 0),which although a mathematically valid step (see, e.g., [15, Theorem 3.3]), results in non-sharp esNevertheless, the bound (17) shows that the method is asymptotically free of locking, asp→ ∞.

4. Numerical results

In this section we present the results of numerical computations performed, using the comsoftware package StressCheck (E.S.R.D., St. Louis, MO). First, we consider a square plate withboundary conditions in order to see how Theorem 1 and Corollary 2 apply in practice. In particuwish to see what happens when several curvilinear elements are used. The second example weis a circular, soft simply supported plate. Even though the analysis in this case is very difficult to pdue to the presence of both boundary layer and locking phenomena, we wish to see how curved econtribute to the deterioration of the approximation ast → 0. To some extend, this second numericomputation may be viewed as an “extension” of the previous results.

4.1. A periodic plate

We consider a square R–M plate withΩ = [−1,1]2 and periodic boundary conditions−→φ(x1,1)= −→

φ(x1,−1),−→φ(x2,1)= −→

φ(x2,−1),

w(x1,1)=w(x1,−1), w(x2,1)=w(x2,−1),

loaded byg = cos(πx1/2)cos(πx2/2). The computations are performed on a quarter of the pwith symmetry boundary conditions along the linesx = 0, y = 0 and anti-symmetry conditions alonx = 1, y = 1. Both quadrilateral and triangular meshes are investigated as seen Fig. 3, in whicf (x)

is taken to be a polynomial of degree 0, . . . ,4, as well as a circular arc. Figs. 4–7 show the percenrelative error in the energy norm versus the polynomial degreep of the basis functions, in a semi-loscale. The shifts seen in these figures are summarized in Tables 2 and 3. We would like to menwhile there is no known exact solution for this problem, the results we report are based on a resolution obtained using an extrapolation technique (cf. [17]) and, due to the fact that the solutionsmooth, are reliable. Also, the results of Theorem 1 and Corollary 2 are given in terms of the nobut due to (8) we may report our numerical results in terms of the energy norm.

As was noted in Remark 1, when we compare the observed shifts with the ones predicCorollary 2, we see that in practiceα is not as large as (22) states. For example, whent = 0.0005and “quartic quadrilateral elements” are used, Corollary 2 predictsα = 7, while we only observe a shi

J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246 239

elements

he onesmethod

ee thatppears to

erical

Fig. 3. The mesh and boundary conditions on a quarter of the periodic plate; quadrilateral elements (left), triangular(right).

Table 2Shift α for quadrilateral elements

t \ f (x) Constant Linear Quadratic Cubic Quartic Circular

0.5 0 0.13 1.06 2.08 3.06 0.950.05 0 0.26 0.6 1.69 3.41 1.290.005 0 0.28 1.37 2.35 3.74 2.050.0005 0 0.47 1.44 2.6 4.27 2.03

Table 3Shift α for triangular elements

t \ f (x) Linear Quadratic Cubic Quartic Circular

0.5 0 0.71 1.39 3 0.110.05 0 1.15 2.15 3.73 0.320.005 0 1.18 2.36 4.02 0.150.0005 0 1.98 3.11 4.71 1.08

of about 4.27. The same observations hold for triangular elements as well. These shifts (both tobserved in practice and the theoretical ones) are not very severe and the performance of theis not drastically affected by the use of curved (polynomially mapped) elements. Moreover, we seven when circular curved elements are used, the observed shifts are not large and the method aremain locking free asp→ ∞, as was shown in [15] for straight sided elements.

4.2. A circular plate

As a second example, we consider the case of a circular plate (Ω being the unit disk,Ω = (r, θ): 0 r 1, 0 θ 2π), with soft simple support boundary conditions, loaded byg = cos(θ). In this casethe solution will contain boundary layers (see [1]), hence the difficulties involved with the num

240 J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246

Fig. 4. Energy norm convergence for the periodic plate of thicknesst = 0.5,0.05, using quadrilateral elements.

J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246 241

Fig. 5. Energy norm convergence for the periodic plate of thicknesst = 0.005,0.0005, using quadrilateral elements.

242 J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246

Fig. 6. Energy norm convergence for the periodic plate of thicknesst = 0.5,0.05, using triangular elements.

J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246 243

Fig. 7. Energy norm convergence for the periodic plate of thicknesst = 0.005,0.0005, using triangular elements.

244 J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246

ain) aree meshayer of

le fromffect the

rgy normm intremely.

by thetoticallygs areon the

rvilinearedictedserved

ing andoverallce of the

e shearurned

Fig. 8. Mesh and boundary conditions for (a quarter of) the circular plate.

approximation increase. The mesh-design and boundary conditions (on a quarter of the domshown in Fig. 8. The recommendations of [16] were taken into consideration for the design of thin order for the boundary layers to be uniformly approximated, hence the presence of the thin lelements of widthpt along the boundary.

There are two reasons for considering this example. First, an exact solution is readily availab[2] and the reported results are hence reliable. Second, we wish to see how curved elements aconvergence in this more complicated case. Fig. 9 shows the percentage relative error in the eneversus (a) the polynomial degreep in a semi-log plot and (b) versus the number of degrees of freedoa log–log plot. We see that even though there is a shift in the error curves, the method performs exwell and the presence of curved elements does not affect the overall behavior in a significant way

5. Conclusions

We have studied the effect of curved elements in the approximation of the R–M plate modelp version of the finite element method. We have shown that on one element, the method is asympfree of locking, when certain curved elements are used. In particular, if the element mappinpolynomials, then the amount of pre-asymptotic locking was calculated and shown to dependdegree of the mappings. Through numerical computations we examined the case of several cuelements and found it to be quantitatively similar to the one element case. In fact, in theory, the prshifts correspond to the “limiting solution” and the bounds obtained are not sharp; in practice the obshifts are not very pronounced. We have also performed computations for a plate with both lockboundary layer phenomena present. We found that curved elements do not significantly affect theconvergence of the method, provided the proper mesh-design is chosen to account for the presenboundary layer.

We should mention that when another error measure is used, e.g., the point-wise error in thforce −→q = κµt−2(∇w − −→

φ), then the situation changes drastically [19]. In fact, researchers have t

J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246 245

Fig. 9. Energy norm convergence for the circular plate.

246 J. Kurtz, C. Xenophontos / Applied Numerical Mathematics 46 (2003) 231–246

their attention to mixed/reduced constrained methods in order to avoid this additional difficulty [20]. Thequestion of how curved elements affect the convergence of the mixed/reduced constraintp andhp finite

Math.

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ity

urved, 2002.Ph.D.

Mech.

in plate

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ongress

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element methods is the focus of our current research efforts.

References

[1] D.N. Arnold, R.S. Falk, Asymptotic analysis of the boundary layer for the Reissner–Mindlin plate model, SIAM J.Anal. 27 (1996) 486–514.

[2] D.N. Arnold, R.S. Falk, Edge Effects in the Reissner–Mindlin Plate Theory, in: A.K. Noor, T. Belytschko, J.C. Simein: Analytic and Computational Models for Shells, ASME, New York, 1989.

[3] M. Bernadou,C1-curved finite elements with numerical integration for thin plate and thin shell problems, PConstruction and interpolation properties of curvedC1 finite elements, Comput. Methods Appl. Mech. Engrg. 102 (19255–289.

[4] M. Bernadou,C1-curved finite elements with numerical integration for thin plate and thin shell problems, PApproximation of thin plate and thin shell problems, Comput. Methods Appl. Mech. Engrg. 102 (1993) 389–421.

[5] I. Babuška, Li, The problem of plate modeling. Theoretical and computational results, Comput. Methods Appl.Engrg. 100 (1992) 249–273.

[6] I. Babuška, Li, Hierarchic modeling of plates, Comput. and Structures 40 (1991) 419–430.[7] I. Babuška, M. Suri, On locking and robustness in the finite element method, SIAM J. Numer. Anal. 29 (1992) 1261[8] I. Babuška, M. Suri, Locking effects in the finite element approximation of elasticity problems, Numer. Math. 62

439–463.[9] L.K. Chilton, M. Suri, On the construction of stable curvilinearp version elements for mixed formulations of elastic

and Stokes’ flow, Numer. Math. 86 (2000) 29–48.[10] J. Kurtz, A p Version Finite Element Method for Shear Force Computation in Reissner–Mindlin Plates with C

Boundaries, M.S. Thesis, Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY[11] L.K. Chilton, Locking Free Mixed hp Finite Element Methods for Linear and Geometrically Nonlinear Elasticity,

Dissertation, University of Maryland, Baltimore County, 1997.[12] M. Suri, Analytical and computational assessment of locking in the hp finite element method, Comput. Methods

Engrg. 133 (1996) 347–371.[13] M. Suri, Thep-version of the finite element method for elliptic problems of order 2%, RAIRO Modél. Math. Anal.

Numér. 24 (1990) 265–304.[14] C. Schwab, M. Suri, Locking and boundary layer effects in the finite element approximation of the Reissner–Mindl

model, Proc. Sympos. Appl. Math. 48 (1994) 367–371.[15] M. Suri, I. Babuška, C. Schwab, Locking effects in the finite element approximation of plate models, Math. Co

(1995) 461–482.[16] C. Schwab, M. Suri, C. Xenophontos, Thehp finite element method for problems in mechanics with boundary lay

Comput. Methods Appl. Mech. Engrg. 57 (1998) 311–334.[17] B. Szabó, I. Babuška, Finite Element Analysis, Wiley, New York, 1991.[18] C. Xenophontos, Finite element computations for the Reissner–Mindlin plate model, Comm. Numer. Methods E

(1999) 1119–1131.[19] C. Xenophontos, J. Kurtz, S.R. Fulton, On the uniform approximation of the Reissner–Mindlin plate model byp/hp finite

element methods, in: H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner (Eds.), Proceedings of the Fifth World Con Computational Mechanics (WCCM V), July 7–12, 2002, Vienna, Austria.

[20] M. Ainsworth, K. Pinchedez, Thehp-MITC finite element method for the Reissner–Mindlin plate problem, J. ComAppl. Math. 148 (2) (2003) 429–462.