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A locking-free model for Reissner-Mindlin plates:
Analysis and isogeometric implementation via
NURBS and triangular NURPS
L. Beirão da Veiga ∗ T.J.R. Hughes † J. Kiendl ‡
C. Lovadina § J. Niiranen ¶ A. Reali ‖ H. Speleers ∗∗††
January 13, 2015
Abstract
We study a reformulated version of Reissner-Mindlin plate theory
inwhich rotation variables are eliminated in favor of transverse
shear strains.Upon discretization, this theory has the advantage
that the “shear lock-ing” phenomenon is completely precluded,
independent of the basis func-tions used for displacement and shear
strains. Any combination works, butdue to the appearance of second
derivatives in the strain energy expres-sion, smooth basis
functions are required. These are provided by Isogeo-metric
Analysis, in particular, NURBS of various degrees and
quadratictriangular NURPS. We present a mathematical analysis of
the formula-tion proving convergence and error estimates for all
physically interestingquantities, and provide numerical results
that corroborate the theory.
∗Department of Mathematics, University of Milan, Via Saldini 50,
20133 Milan, Italy.Email: [email protected]†Institute for
Computational Engineering and Sciences, University of Texas
at Austin, 201East 24th Street, Stop C0200 Austin, TX
78712-1229, USA.Email: [email protected]‡Department of Civil
Engineering and Architecture, University of Pavia, Via Ferrata
3,
27100, Pavia, Italy. Email: [email protected]§Department of
Mathematics, University of Pavia, Via Ferrata 1, 27100 Pavia,
Italy.
Email: [email protected]¶Department of Civil and
Structural Engineering, Aalto University, PO Box 12100, 00076
AALTO, Finland. Email: [email protected]‖Department of
Civil Engineering and Architecture, University of Pavia, Via
Ferrata 3,
27100, Pavia, Italy. Email:
[email protected]∗∗Department of Computer Science,
University of Leuven, Celestijnenlaan 200A, 3001 Hev-
erlee (Leuven), Belgium. Email:
[email protected]††Department of Mathematics,
University of Rome ‘Tor Vergata’, Via della Ricerca Scien-
tifica, 00133 Rome, Italy. Email: [email protected]
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1 Introduction
Finite element thin plate bending analysis, based on the
Poisson-Kirchhoff the-ory, began with the MS thesis of Papenfuss at
the University of Washington in1959 [24]. This four-node
rectangular element employed C1-continuous inter-polation
functions, but was deficient in the sense that its basis functions
werenot complete through quadratic polynomials. Throughout the
1960s the devel-opment of quadrilateral and triangular thin plate
elements was a focus of finiteelement research. The problem was
surprisingly difficult. Eventually there weretechnical successes
but the elements were complicated, hard to implement, anddifficult
to use in practice. For an early history, we refer to Felippa [16].
Tocircumvent the difficulties, in the 1970s attention was
redirected to the “thick”plate bending theory of Reissner-Mindlin.
In this case only C0 basis functionswere required for the
displacement and rotations, so the continuity and com-pleteness
requirements were easily satisfied with standard isoparametric
basisfunctions, but new difficulties arose associated with “shear
locking” in the thinplate limit in which the transverse shear
strains must vanish. Nevertheless,through a number of clever ideas,
some tricks and some fundamental, successfulelements for many
applications became available. The simplicity and efficiencyof
these elements led to immediate incorporation in industrial and
commer-cial structural analysis software programs, and that has
been the situation eversince. Research in the subject then became
somewhat stagnant for a number ofyears, but recently things
changed.
Isogeometric Analysis was proposed by Hughes, Cottrell &
Bazilevs in 2005[18]. The motivation for its development was to
simplify and render more effi-cient the design-through-analysis
process. It is often said that the developmentof suitable Finite
Element Analysis (FEA) models from Computer Aided Design(CAD) files
occupies over 80% of overall analysis time [12]. Some
design/analysisengineers claim that this is an underestimate and
the situation is actually worseperhaps 90%, or even more. Whatever
the precise percentage, it is clear that theinterface between
design and analysis is broken, and it is the stated aim of
Isoge-ometric Analysis to repair it. The way this has been
approached in IsogeometricAnalysis is to reconstitute analysis
within the functions utilized in engineeringCAD, such as, for
example, NURBS and T-splines, making it possible, at thevery least,
to perform analysis within the representations provided by
design,eliminating redundant data structures and unnecessary
geometric approxima-tions. This has been the primary focus of
Isogeometric Analysis, but in theprocess of its development new
analysis opportunities have also presented them-selves. One
emanates from a basic property of the functions utilized in CADthey
are smooth, usually at least C1-continuous, more often
C2-continuous, andthey do not require derivative
degrees-of-freedom, one of the paradigmatic de-ficiencies of the
early thin plate bending elements. CAD functions also haveother
advantageous properties, but we will not go into these here. It
turns outthat smoothness alone has created new opportunities in
plate bending elementresearch.
The first and most obvious to be exploited by Isogeometric
Analysis is re-
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newed interest in the long abandoned Poisson-Kirchhoff theory,
as anticipated in[18]. With convenient, smooth basis functions
provided by Isogeometric Analy-sis, the role of Poisson-Kirchhoff
theory is being reconsidered. A prime advan-tage is that no
rotational degrees-of-freedom are required. This immediatelyreduces
the size of equation systems and the computational effort
necessaryto solve problems. In finite deformation analysis there is
a further advantage.This stems from the fact that finite rotations
involve a product-group structure,SO(3), a significant
complication. It is completely obviated by rotationless thinbending
elements.
Another innovation that occurred, due to Long, Bornemann &
Cirak [21]and Echter, Oesterle & Bischoff [15] in the context
of shells, was again moti-vated by the existence of smooth basis
functions. It was observed that if onecould deal with second
derivatives appearing in strain energy expressions, thena change of
variables from rotations in Reissner-Mindlin plate theory to
trans-verse shear strains would eliminate transverse shear locking,
independent of thebasis functions employed. Why wasn’t this amazing
formulation utilized previ-ously for the development of plate and
shell elements? The answer seems to bethere simply were not
convenient smooth basis functions available within thefinite
element paradigm. Shear locking has been the fundamental obstacle
tothe design of effective plate elements based on Reissner-Mindlin
theory. It isremarkable that the discovery of a complete and
general solution has occurrednearly 50 years after the widespread
adoption of the theory as a framework forthe derivation of plate
and shell elements.
Given the above, the question arises as to what other
opportunities might beprovided by clever changes of variables? One
answer has been presented in thework of Kiendl et al. [19] who have
shown that smooth basis functions with onlytranslational
displacement degrees-of-freedom can also be employed
successfullyfor “thick” bending elements. The price to pay in the
formulation of [19] is thatsquares of third derivatives appear in
the strain energy expression, but theseare no problem for C2
continuous Isogeometric Analysis basis functions.
All these new ideas have created a renaissance in the
development of methodsto solve problems involving thin and thick
bending elements. It is clear thatthese and other related concepts
will generate considerable interest in the comingyears.
In this paper we undertake the mathematical analysis of a class
of methodsfor Reissner-Mindlin plate theory based on the change of
variables introducedin [15, 21]. The dependent variables are then
the transverse displacement ofthe plate and the transverse shear
strain vector. The change of variables re-sults in squares of
second derivatives of the displacement in the strain energy,which
yield to spline discretizations of C1, or higher, continuity. It is
apparentthat the shear strain vector is not as physically appealing
or implementation-ally convenient as the rotation vector. However,
by utilizing weakly enforcedrotation boundary conditions, by way of
Nitsche’s method [23], these issues arecircumvented. Nitsche’s
method also provides other analytical benefits in that italleviates
“boundary locking”, a potential problem encountered in the
presenttheory for clamped boundary conditions. In addition to Cp−1
NURBS dis-
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cretizations, we consider quadratic C1 triangular NURPS, that
is, non-uniformrational Powell-Sabin splines. We note in passing
that Isogeometric Analysishas previously been applied to the
standard displacement-rotation forms of theReissner-Mindlin plate
and shell theories [6, 8].
An outline of the remainder of the paper follows: In Section 2
we presentthe Reissner-Mindlin plate model, first in terms of
displacement and rotationvariables and then in an equivalent form
in terms of displacement and transverseshear strain variables. We
introduce the discrete form of the theory and summa-rize stability
and convergence results that apply to displacements, shear
strains,rotations, bending moments, and transverse shear force
resultants. In Section 3we describe and perform numerical tests
with NURBS spaces. In Section 4 wedo likewise with quadratic NURPS,
and in Section 5 we present concluding re-marks. The technical
details of proofs are postponed until Appendix A so asnot to
interrupt the flow of the main ideas and results in the body of the
paper.
2 The model and discretization
We start by considering the classical Reissner-Mindlin model for
plates. Let Ωbe a bounded and piecewise regular domain in R2
representing the midsurfaceof the plate. We subdivide the boundary
Γ of Ω in three disjoint parts (suchthat each is either void or the
union of a finite sum of connected components ofpositive
length),
Γ = Γc ∪ Γs ∪ Γf .
The plate is assumed to be clamped on Γc, simply supported on Γs
and freeon Γf , with Γc,Γs defined to preclude rigid body motions.
We consider forsimplicity only homogeneous boundary conditions. The
variational space ofadmissible solutions is given by
X̃ ={
(w,θ) ∈ H1(Ω)× [H1(Ω)]2 : w = 0 on Γc ∪ Γs, θ = 0 on Γc}.
Following the Reissner-Mindlin model, see for instance [4] and
[17], the platebending problem reads as{
Find (w,θ) ∈ X̃, such that
a(θ,η) + µkt−2(θ −∇w,η −∇v) = (f, v), ∀(v,η) ∈ X̃,(1)
where µ is the shear modulus and k is the so-called shear
correction factor.In the above model, t represents the plate
thickness, w the deflection, θ therotation of the normal fibers and
f the applied scaled normal load. Moreover,(·, ·) stands for the
standard scalar product in L2(Ω) and the bilinear form a(·, ·)is
defined by
a(θ,η) = (Cε(θ), ε(η)), (2)
with C the positive definite tensor of bending moduli and ε(·)
the symmetricgradient operator.
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Provided the boundary conditions satisfy the conditions above,
the bilinearform appearing in problem (1) is coercive, in the
following sense (see Proposi-tion A.1 in Appendix A). There exists
a positive constant α depending only onthe material constants and
the domain Ω such that
a(η,η) + µkt−2(η −∇v,η −∇v)
≥ α(||η||2H1(Ω) + t
−2||η −∇v||2L2(Ω) + ||v||2H1(Ω)
), ∀(v,η) ∈ X̃.
(3)
The above result states the coercivity of the bilinear form on
the product spaceX̃. It is well known that the discretization of
the Reissner-Mindlin model posesdifficulties, due to the
possibility of the locking phenomenon when the thicknesst
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where we recall that we now have µk = 1 for notational
simplicity.Problem (6) is clearly equivalent to (1) up to the above
change of variables.
In particular, the coercivity property (3) translates
immediately into
a(∇v + τ ,∇v + τ ) + t−2(τ , τ ) ≥ α|||τ , v|||2, ∀(v, τ ) ∈ X,
(7)
for the same positive constant α.
2.2 Discretization of the model
We introduce a pair of finite dimensional spaces for deflections
and shear strains:
Wh ⊂ H2(Ω), Ξh ⊂ [H1(Ω)]2.
We assume that the two spaces above are generated either by some
finite elementor isogeometric technology, and are therefore
associated with a (physical) meshΩh. In the following, we will
indicate by K ∈ Ωh a typical element of themesh, and denote by hK
its diameter. We denote by h the maximum mesh size.Moreover, we
will indicate by Eh the set of all (possibly curved) edges of
themesh, by e ∈ Eh the generic edge, and by he its length. As usual
we assumethat the boundary parts Γc,Γs,Γf are unions of mesh edges.
Furthermore, wewill assume that the set of elements K in the family
{Ωh}h is uniformly shaperegular in the classical sense.
We then consider the discrete space with (partial) boundary
conditions
Xh ={
(vh, τh) ∈Wh × Ξh : vh = 0 on Γc ∪ Γs}
;
see also Remark 2.3 below. The rotation boundary condition on Γc
will beenforced with a penalized formulation in the spirit of
Nitsche’s method [23],through the introduction of the following
modified bilinear form. Let
ah(∇wh + γh,∇vh + τh) = a(∇wh + γh,∇vh + τh)
−∫
Γc
(Cε(∇wh + γh)ne
)· (∇vh + τh)
−∫
Γc
(Cε(∇vh + τh)ne
)· (∇wh + γh)
+ β tr(C)∑
e∈Eh∩Γc
c(e)−1∫e
(∇wh + γh)(∇vh + τh),
(8)
for all (wh,γh), (vh, τh) in Xh and for β > 0 a stabilization
parameter. Above,c(e) is a characteristic quantity depending on the
side e.
Remark 2.2. For a shape regular family of meshes, one can choose
c(e) = heor c(e) = (AreaKe)
1/2, where Ke is the element containing e. This latter choicehas
been employed in the numerical tests of Section 3, while the former
has beenused in Section 4 . However, when a side e belongs to an
element with a largeaspect ratio, a different choice could be
preferable. For example, significantly
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thin elements may be used in the presence of boundary layers,
and one maychoose c(e) = h⊥e , where h
⊥e is the element size in the direction perpendicular
to the boundary.
We are now able to present the proposed discretization of the
model (6):{Find (wh,γh) ∈ Xh, such thatah(∇wh + γh,∇vh + τh) +
t−2(γh, τh) = (f, vh), ∀(vh, τh) ∈ Xh.
(9)
Remark 2.3. It is not wise to use a direct discretization of the
space X byenforcing all the boundary conditions on Wh×Ξh. Indeed,
the clamped conditionon rotations
∇wh + γh = 0 on Γc, ∀wh ∈Wh, γh ∈ Ξh,is very difficult to
implement, and can be a source of boundary locking unlessthe two
spaces Wh,Ξh are chosen in a very careful way. To illustrate why a
poorapproximation might occur on Γc, let us consider the following
example.
Let Ω = (0, 1) × (0, 1) and Γc = [0, 1] × {1}. Select Wh =
Sp,pp−1,p−1(Ω) andΞh = S
p−1,p−1p−2,p−2(Ω)× S
p−1,p−1p−2,p−2(Ω), where S
p,qr,s (Ω) denotes the space of B-splines
of degree p and regularity Cr with respect to the x direction,
and of degree q andregularity Cs with respect to the y
direction.
Imposing ∇wh + γh = 0 on Γc implies in particular∂wh∂y
(x, 1) = −γ2,h(x, 1) ∀x ∈ [0, 1],
where γ2,h is the second component of the vector field γh.
Since∂wh∂y (x, 1) ∈
Spp−1(0, 1) and γ2,h(x, 1) ∈ Sp−1p−2(0, 1), it follows that
γ2,h(x, 1) ∈ Sp−1p−2(0, 1) ∩ Spp−1(0, 1) = S
p−1p−1(0, 1).
Hence γ2,h(x, 1) is necessarily a global polynomial of degree at
most p − 1 onΓc, regardless of the mesh. This means that on Γc γ2,h
cannot converge to γ2,second component of γ, as the mesh size tends
to zero, in general.
Instead, as we will prove in the next section, the method
proposed above isfree of locking for any choice of the discrete
spaces Wh,Ξh.
2.3 Stability and convergence results
In the present section we show the stability and convergence
properties of theproposed method. All the proofs can be found in
Appendix A.
In what follows, we set c(e) = he in (8), which is a suitable
choice for shaperegular meshes, see Remark 2.2. We start by
introducing the following discretenorm
|||vh, τh|||2h = |||vh, τh|||2 +∑
e∈Eh∩Γc
h−1e ||∇vh + τh||2L2(e), (10)
for all (vh, τh) ∈ Xh.In the theoretical analysis of the method
we will make use of the following
assumptions on the solution regularity and space approximation
properties.
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A1) We assume that the solution w ∈ H2+s(Ω) and γ ∈ H1+s(Ω) for
somes > 1/2.
A2) We assume that the following standard inverse estimates
hold
|vh|H1(K) ≤ Ch−1K ||vh||L2(K), |τh|H1(K) ≤ Ch−1K
||τh||L2(K),
for all vh ∈Wh and τh ∈ Ξh with C independent of h.
A3) We assume the following approximation properties for Xh. Let
s = 2, 3.For all (v, τ ) ∈ (Hs(Ω) × [H1(Ω)]2) ∩X there exists (vh,
τh) ∈ Xh suchthat
||τ − τh||Hj(Ω) ≤ Ch1−j ||τ ||H1(Ω), j = 0, 1,||v − vh||Hj(Ω) ≤
Chs−j ||v||Hs(Ω), j = 0, . . . , s,
with C independent of h.
Moreover, we will make use of the following natural assumption,
in order toavoid rigid body motions:
A4) We assume that Γc ∪ Γs has positive length and that eitheri)
Γc has positive length, orii) Γs is not contained in a straight
line.
We now introduce a coercivity lemma stating in particular the
invertibilityof the linear system associated with (9).
Lemma 2.1. Let hypotheses A2 and A4 hold. There exist two
positive constantsβ0, α
′ such that, for all β ≥ β0, we have
ah(∇vh+τh,∇vh+τh)+t−2(τh, τh) ≥ α′|||τh, vh|||2h, ∀(vh, τh) ∈
Xh. (11)
The constant α′ only depends on the material parameters and the
domain Ω,while the constant β0 depends only on the shape regularity
constant of Ωh.
Let A1 hold. By an integration by parts, it is immediately
verified that thescheme (9) is consistent, in the sense that
ah(∇w + γ,∇vh + τh) + t−2(γ, τh) = (f, vh), ∀(vh, τh) ∈ Xh,
(12)
where (w,γ) is the solution of problem (6) and the left-hand
side makes sense dueto the regularity assumption A1. Note that
condition A1 could be significantlyrelaxed by interpreting the
integrals on Γc appearing in (8) in the sense ofdualities.
By combining the coercivity in Lemma 2.1 with the consistency
property(12), the following convergence result follows.
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Proposition 2.1. Let A1 and A2 hold. Let (w,γ) be the solution
of problem(6) and (wh,γh) ∈ Xh be the solution of problem (9).
Then, if β ≥ β0, we have
|||w − wh,γ − γh|||2h
≤ C inf(vh,τh)∈Xh
2∑j=0
( ∑K∈Ωh
h2(j−1)K |w − vh|
2Hj+1(K)
+∑K∈Ωh
h2(j−1)K |γ − τh|
2Hj(K) + t
−2||γ − τh||2L2(Ω)),
(13)
with C depending only on the material parameters and the domain
Ω.
We now state a convergence result concerning bending moments and
shearforces, the quantities of interest in engineering
applications. To this aim, wefirst define the scaled bending
moments and the scaled shear forces as follows:
M = −Cε(∇w + γ), Q = t−2µκγ = t−2γ, (14)
where (w,γ) is the solution to problem (6). The above quantities
are knownto converge to non-vanishing limits, as t → 0 (see, e.g.,
[3] and [9]). Onceproblem (9) has been solved, we can define the
scaled discrete bending momentsand the scaled discrete shear
moments as
Mh = −Cε(∇wh + γh), Qh = t−2γh. (15)
We have the following estimates.
Proposition 2.2. Let A1 and A2 hold. Let (w,γ) be the solution
of problem(6) and (wh,γh) ∈ Xh be the solution of problem (9).
Then, if β ≥ β0, we have
||M −Mh||L2(Ω) + t ||Q−Qh||L2(Ω) ≤ C|||w − wh,γ − γh|||h.
(16)
Moreover, let the mesh family {Ωh}h be quasi-uniform. Then, we
have
h ||Q−Qh||L2(Ω) ≤ C(|||w − wh,γ − γh|||h + h inf
sh∈Ξh||Q− sh||L2(Ω)
), (17)
||Q−Qh||H−1(Ω) ≤ C(|||w − wh,γ − γh|||h + h inf
sh∈Ξh||Q− sh||L2(Ω)
). (18)
In addition, we can formulate the following improved result
regarding theerror for the rotations in the L2-norm and the error
for the deflections in theH1-norm. An analogous result (possibly
with a smaller improvement in terms ofh, t) also holds if the
additional hypotheses in the proposition are not satisfied;we do
not detail here this more general case.
Proposition 2.3. Let the same assumptions and notation of
Proposition 2.1hold. Moreover, let assumption A3 hold, Γc = Γ and
let the domain Ω be
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either regular, or piecewise regular and convex. Then, the
following improvedapproximation result holds
||θ − θh||L2(Ω) ≤ C(h+ t) |||w − wh,γ − γh|||h, (19)||w −
wh||H1(Ω) ≤ C(h+ t) |||w − wh,γ − γh|||h + ||γ − γh||L2(Ω),
(20)
where θh = ∇wh +γh and the constant C depends only on the
material param-eters and the domain Ω.
Note that the last term appearing in (13) is not a source of
locking sincet−1γ = tQ is a quantity that is known to be uniformly
bounded in the correctSobolev norms (see, e.g., [3] and [9]). The
constants appearing in Propositions2.1 and 2.3 are independent of
t, and so the results shown state that the proposedmethod is
locking-free regardless of the discrete spaces adopted. This is a
veryinteresting property that is missing in the standard methods
for the Reissner-Mindlin problem. The accuracy of the discrete
solution (9) will only dependon the approximation properties of the
adopted discrete spaces, and will not behindered by small values of
the plate thickness. We also remark that the normsfor the scaled
shear forces appearing in the left-hand side of (17) and (18),
areindeed the usual norms for which a convergence result can be
established (see,e.g., [9] and [10]).
In the following two sections we will present a pair of
particular choices forWh,Ξh within the framework of
• standard tensor-product NURBS-based isogeometric analysis
(Section 3);
• triangular NURPS-based isogeometric analysis (Section 4).
For such choices we can apply Propositions 2.1, 2.3 in order to
obtain the ex-pected convergence rates in terms of h. For example,
in the case of standardtensor-product NURBS, combining Proposition
2.1 with the approximation es-timates in [5, 7] we obtain the
following convergence result.
Corollary 2.1. Let the same assumptions and notation of
Proposition 2.1 hold.Let standard isoparametric tensor-product
NURBS of polynomial degree p for allvariables be used for the space
Xh. Then, provided that the solution of problem(6) is sufficiently
regular for the right-hand side to make sense, the followingerror
estimates holds for all 2 ≤ s ≤ p:
|||w − wh,γ − γh|||h ≤ Chs−1(||w||Hs+1(Ω) + ||θ||Hs(Ω) +
t||γ||Hs−1(Ω)
), (21)
||M −Mh||L2(Ω)+t ||Q−Qh||L2(Ω)≤ Chs−1
(||w||Hs+1(Ω) + ||θ||Hs(Ω) + t||γ||Hs−1(Ω)
).
(22)
Moreover, let the mesh family {Ωh}h be quasi-uniform. Then,
h ||Q−Qh||L2(Ω) + ||Q−Qh||H−1(Ω)≤ Chs−1
(||w||Hs+1(Ω) + ||θ||Hs(Ω) + t||γ||Hs−1(Ω) + ||γ||Hs−2(Ω)
).
(23)
The constant C depends only on p, the material parameters and
the domain Ω.
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The above corollary can also be combined with Proposition 2.3 to
obtainimproved error estimates for the L2-norm of the rotations and
the H1-norm ofthe deflections. Also recalling definition (5), one
can easily obtain a convergencerate of (h+ t)hs−1 for such
quantities. Finally, note that, like for all high ordermethods, the
regularity requirement in Corollary 2.1 may be too demanding dueto
the presence of layers in the solution. This situation is typically
dealt withby making use of refined meshes, an example of which is
shown in the numericaltests.
3 Isogeometric discretization with NURBS
In this section, NURBS-based isogeometric analysis is used to
perform numericalvalidations of the presented theory. We begin with
a brief summary of B-splinesand NURBS (Non-Uniform Rational
B-Splines).
3.1 B-splines and NURBS
B-splines are piecewise polynomials defined by the polynomial
degree p and aknot vector [ξ1, ξ2, . . . , ξn+p+1], where n is the
number of basis functions. Theknot vector is a set of parametric
coordinates ξi, called knots, which divide theparametric space into
intervals called knot spans. A knot can also be repeated,in this
case it is called a multiple knot. At a single knot the B-splines
are Cp−1-continuous, and at a multiple knot of multiplicity k the
continuity is reduced toCp−k.
The B-spline basis functions of degree p are defined by the
following recursionformula. For p = 0,
Ni,0(x) =
{1, ξi ≤ x < ξi+1,0, otherwise.
For p ≥ 1,
Ni,p(x) =x− ξiξi+p − ξi
Ni,p−1(x) +ξi+p+1 − xξi+p+1 − ξi+1
Ni+1,p−1(x).
A bivariate NURBS function Rp,qi,j is defined as the weighted
tensor-product ofthe B-spline functions Ni,p and Mj,q with
polynomial degrees p and q,
Rp,qi,j (x, y) =Ni,p(x)Mj,q(y)ωi,j
n∑l=1
m∑r=1
Nl,p(x)Mr,q(y)ωl,r
,
where ωi,j are called control weights. Following the
isogeometric concept, NURBSare employed to both represent the
geometry and to approximate the solution,i.e. the isoparametric
concept is invoked. Accordingly, the unknown variables
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w and γ are approximated by
wh(x, y) =
nw∑i=1
mw∑j=1
Rpw,qwi,j (x, y)ŵi,j , γh(x, y) =
nγ∑i=1
mγ∑j=1
Rpγ ,qγi,j (x, y)γ̂i,j ,
with nw, mw the numbers of basis functions in the two parametric
directions forwh, and nγ , mγ the numbers for γh. The test
functions v and τ are discretizedaccordingly.
Since the rotations θ are not discretized in this approach,
rotational bound-ary conditions cannot be imposed in a standard way
by applying them directlyon the respective degrees of freedom at
the boundary. Instead, such boundaryconditions are enforced by the
modified bilinear form introduced in equation(8). Displacement
boundary conditions are enforced in a standard way throughthe
displacement degrees of freedom on the boundary.
3.2 Numerical tests
In this section, the proposed method is tested on different
numerical examples.We always employ the same polynomial order and
the highest regularity for allthe unknowns, but different choices
can be made. In order to demonstrate thelocking-free behavior of
this method we consider both thick and thin plates.Furthermore, we
investigate an example which exhibits boundary layers. As anerror
measure, an approximated L2-norm error for a variable u is computed
as
||uex − uh||L2||uex||L2
=
√∑(xi,yi)∈G(uex − uh)
2∑(xi,yi)∈G u
2ex
,
where G is a 101 × 101 uniform grid in the parameter domain [0,
1]2 mappedonto the physical domain.
3.2.1 Square plate with clamped boundary conditions
The first example consists of a unit square plate [0, 1]2 with
an analytical solutionas described in [11]. The plate is clamped on
all four sides, and subject to aload given by
f(x, y) =E
12(1− ν2)[12y(y − 1)(5x2 − 5x+ 1)(2y2(y − 1)2
+ x(x− 1)(5y2 − 5y + 1))+ 12x(x− 1)(5y2 − 5y + 1)(2x2(x− 1)2
+ y(y − 1)(5x2 − 5x+ 1))].
12
-
The analytical solution for the displacement w is given by
w(x, y) =1
3x3(x− 1)3y3(y − 1)3
− 2t2
5(1− ν)[y3(y − 1)3x(x− 1)(5x2 − 5x+ 1)
+ x3(x− 1)3y(y − 1)(5y2 − 5y + 1)].
We perform an h-refinement study using equal polynomial degrees
p = 2, 3, 4, 5for wh and γh, for the case of a thick plate with t =
10
−1 and a thin plate witht = 10−3. The material parameters are
taken to be E = 106 and ν = 0.3.Figure 1(a) shows the convergence
plots for the thick plate, whereas Figure 1(b)those for the thin
plate. Dashed lines indicate the reference order of convergence.As
can be seen, the convergence rates for all polynomial orders are at
least oforder p.
In addition, we study the convergence for bending moments and
shear forcessince these are of prime interest in the engineering
design of plates. Bendingmoments m and shear forces q are obtained
as
m = −t3Cε(∇w + γ), q = kµtγ.
The exact solution for bending moments and shear forces is
mxx = −Kb2(y3(y − 1)3(x− x2)(5x2 − 5x+ 1)
+ ν(x3(x− 1)3(y − y2)(5y2 − 5y + 1))),
myy = −Kb2(ν(y3(y − 1)3(x− x2)(5x2 − 5x+ 1))
+ x3(x− 1)3(y − y2)(5y2 − 5y + 1)),
mxy = myx = −Kb(1− ν)3y2(y − 1)2(2y − 1)x2(x− 1)2(2x− 1),
qx = −Kb2(y3(y − 1)3(20x3 − 30x2 + 12x− 1)
+ 3y(y − 1)(5y2 − 5y + 1)x2(x− 1)2(2x− 1)),
qy = −Kb2(x3(x− 1)3(20y3 − 30y2 + 12y − 1)
+ 3x(x− 1)(5x2 − 5x+ 1)y2(y − 1)2(2y − 1)),
where Kb =Et3
12(1− ν2)is the plate bending stiffness. For the error measure,
the
Euclidean norms of m and q are used, m =
√2∑i=1
2∑j=1
m2ij and q =
√2∑i=1
q2i . The
13
-
1.4 1.6 1.8 2−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 /
||w
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#dof)−1
c2*(#dof)−3/2
c3*(#dof)−2
c4*(#dof)−5/2
(a)
1.4 1.6 1.8 2−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 /
||w
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#dof)−1
c2*(#dof)−3/2
c3*(#dof)−2
c4*(#dof)−5/2
(b)
Figure 1: Square plate with clamped boundary conditions. L2-norm
approxi-mation error of displacements with tensor-product B-splines
for (a) t = 10−1
and (b) t = 10−3.
1.4 1.6 1.8 2−7
−6
−5
−4
−3
−2
−1
0
log10
(#dof1/2
)
log
10(
||m
ex −
mh||
L2 / ||m
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#cp)−1/2
c2*(#cp)−1
c3*(#cp)−3/2
c4*(#cp)−2
(a)
1.4 1.6 1.8 2−7
−6
−5
−4
−3
−2
−1
0
log10
(#dof1/2
)
log
10(
||m
ex −
mh||
L2 / ||m
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#cp)−1/2
c2*(#cp)−1
c3*(#cp)−3/2
c4*(#cp)−2
(b)
Figure 2: Square plate with clamped boundary conditions. L2-norm
approxi-mation error of bending moments with tensor-product
B-splines for (a) t = 10−1
and (b) t = 10−3.
convergence plots for bending moments are presented in Figure 2,
and those forshear forces in Figure 3. Recalling that m = t3M
(resp. mh = t
3Mh) andq = t3Q (resp. qh = t
3Qh), we notice that the convergence rates for the
relativeerrors displayed in Figures 2 and 3 are in accordance with
the theoretical resultsof Corollary 2.1 (see estimates (22) and
(23)). In particular, we remark thatFigure 3(b) displays an O(1)
convergence rate for the L2-norm of the shear forceerrors, when p =
2, in agreement with estimate (23), for s = 2. In other words,there
is no convergence.
14
-
1.4 1.6 1.8 2−8
−7
−6
−5
−4
−3
−2
−1
log10
(#dof1/2
)
log
10(
||q
ex −
qh||
L2 / ||q
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#cp)−1/2
c2*(#cp)−1
c3*(#cp)−3/2
c4*(#cp)−2
(a)
1.4 1.6 1.8 2−7
−6
−5
−4
−3
−2
−1
0
log10
(#dof1/2
)
log
10(
||q
ex −
qh||
L2 / ||q
ex||
L2 )
p=2
p=3
p=4
p=5
c2*(#cp)−1/2
c3*(#cp)−1
c4*(#cp)−2
(b)
Figure 3: Square plate with clamped boundary conditions. L2-norm
approxi-mation error of shear forces with tensor-product B-splines
for (a) t = 10−1 and(b) t = 10−3.
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
x
y
Figure 4: Quarter of annulus plate. Geometry setup.
3.2.2 Quarter of an annulus with clamped and simply
supportedboundary conditions
The second test consists of a quarter of an annulus with an
inner diameter of 1.0and outer diameter of 2.5, as shown in Figure
4. The plate thickness is t = 0.01and the material parameters are E
= 106 and ν = 0.3. The plate is loaded witha uniform load f(x, y) =
1 and two boundary conditions are considered: (a)all edges are
clamped, (b) all edges are simply supported. For both cases,
thisexample exhibits boundary layers. Therefore, we adopt a
refinement strategyin order to better capture the boundary layers.
Given that the knot vectorsrange from 0 to 1, we introduce as a
first step additional knots at 0.1 and 0.9 in
15
-
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
x
y
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
x
yFigure 5: Quarter of annulus with boundary refinement. (a)
Initial model, (b)boundary refined mesh.
1.2 1.4 1.6 1.8 2−8
−7
−6
−5
−4
−3
−2
−1
0
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 / ||w
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#dof)−1
c2*(#dof)−3/2
c3*(#dof)−2
c4*(#dof)−5/2
(a)
1.4 1.6 1.8 2−9
−8
−7
−6
−5
−4
−3
−2
−1
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 / ||w
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#cp)−1
c2*(#cp)−3/2
c3*(#cp)−2
c4*(#cp)−5/2
(b)
Figure 6: Quarter of annulus with clamped boundary conditions.
L2-norm ap-proximation error of displacements with tensor-product
NURBS and (a) bound-ary refinement, (b) uniform refinement.
both directions, see Figure 5. Then, we perform uniform
refinement of the givenknot spans. In the following, we perform
convergence studies with and withoutthe boundary refinement
strategy. The refinement is performed such that thetotal number of
degrees of freedom is comparable in both cases. Since
analyticalsolutions are not available for these problems, we use as
reference the solutionsobtained on a very fine mesh (an “overkill”
solution) with 100 × 100 quinticelements and compute the L2-norm
approximation errors for the displacement.
Figure 6 shows the results for the clamped case, (a) with
boundary refine-ment and (b) with uniform refinement, while in
Figure 7 the results for the
16
-
1.2 1.4 1.6 1.8 2−6
−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 /
||w
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#dof)−1
c2*(#dof)−3/2
c3*(#dof)−2
c4*(#dof)−5/2
(a)
1.4 1.6 1.8 2−4
−3.8
−3.6
−3.4
−3.2
−3
−2.8
−2.6
−2.4
−2.2
−2
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 /
||w
ex||
L2 )
p=2
p=3
p=4
p=5
c1*(#dof)−1
(b)
Figure 7: Quarter of annulus with simply supported boundary
conditions. L2-norm approximation error of displacements with
tensor-product NURBS and(a) boundary refinement, (b) uniform
refinement.
simply supported case are plotted, (a) with boundary refinement
and (b) withuniform refinement. As expected, due to the presence of
boundary layers, inboth the clamped and the simply supported case,
the tests confirm that a suit-able boundary refined mesh is needed
to achieve optimal orders of convergence.With a uniform refinement,
suboptimal results are instead obtained; this is morepronounced in
the simply supported case.
4 Isogeometric discretization with NURPS
In this section, we use a triangular NURPS-based isogeometric
discretization toperform numerical validations. We first briefly
summarize the construction andsome properties of quadratic
B-splines over a Powell-Sabin (PS) refinement of atriangulation and
their rational generalization, the so-called NURPS B-splines.Then,
the discretized model is tested on the same examples as described
in theprevious section.
4.1 Quadratic PS and NURPS B-splines
Let T be a triangulation of a polygonal (parametric) domain Ω̂
in R2, and letV i = (V
xi , V
yi ), i = 1, . . . , NV , be the vertices of T . A PS refinement
T ∗ of T
is the refined triangulation obtained by subdividing each
triangle of T into sixsubtriangles as follows (see also Figure
8).
• Select a split point Ci inside each triangle τi of T and
connect it to thethree vertices of τi with straight lines.
• For each pair of triangles τi and τj with a common edge,
connect the twopoints Ci and Cj . If τi is a boundary triangle,
then also connect Ci to
17
-
Figure 8: A triangulation T and a Powell-Sabin refinement T ∗ of
T .
an arbitrary point on each of the boundary edges.
These split points must be chosen so that any constructed line
segment [Ci,Cj ]intersects the common edge of τi and τj . Such a
choice is always possible: forinstance, one can take Ci as the
incenter of τi, i.e. the center of the circleinscribed in τi.
Usually, in practice, the barycenter of τi is also a valid
choice,but not always.
The space of C1 piecewise quadratic polynomials on T ∗ is called
the Powell-Sabin spline space [26] and is denoted by S12(T ∗). It
is well known that thedimension of S12(T ∗) is equal to 3NV .
Moreover, any element of S12(T ∗) isuniquely specified by its value
and its gradient at the vertices of T , and can belocally
constructed on each triangle of T once these values and gradients
aregiven.
Dierckx [13] has developed a B-spline like basis {Bi,j , j = 1,
2, 3, i =1, . . . , NV } of the space S12(T ∗) such that
Bi,j(x, y) ≥ 0,NV∑i=1
3∑j=1
Bi,j(x, y) = 1, (x, y) ∈ Ω̂. (24)
The functions Bi,j will be referred to as Powell-Sabin (PS)
B-splines. The PSB-splines Bi,j , j = 1, 2, 3, are constructed to
have their support locally in the
molecule Ω̂i of vertex V i, which is the union of all triangles
of T containing V i.It suffices to specify their values and
gradients at any vertex of T . Due to thestructure of the support
Ω̂i, we have
Bi,j(V k) = 0,∂
∂xBi,j(V k) = 0,
∂
∂yBi,j(V k) = 0,
for any vertex V k 6= V i. Moreover, we set
Bi,j(V i) = αi,j ,∂
∂xBi,j(V i) = βi,j ,
∂
∂yBi,j(V i) = γi,j .
18
-
Figure 9: Location of the PS points (black bullets), and a
possible PS triangleassociated with the central vertex
(shaded).
The triplets (αi,j , βi,j , γi,j) can be specified in a
geometric way in order to satisfy(24). To this aim, for each vertex
V i, i = 1, . . . , NV , we define three points
{Qi,j = (Qxi,j , Qyi,j), j = 1, 2, 3},
such that αi,1 αi,2 αi,3βi,1 βi,2 βi,3γi,1 γi,2 γi,3
Qxi,1 Qyi,1 1Qxi,2 Qyi,2 1Qxi,3 Q
yi,3 1
=V xi V yi 11 0 0
0 1 0
.The triangle with vertices {Qi,j , j = 1, 2, 3} will be
referred to as the PS triangleassociated with the vertex V i and
will be denoted by Ti. Finally, for each vertexV i we define its PS
points as the vertex itself and the midpoints of all the edgesof
the PS refinement T ∗ containing V i, see Figure 9. It has been
proved in[13] that the functions Bi,j , j = 1, 2, 3, are
non-negative if and only if the PStriangle Ti contains all the PS
points associated with the vertex V i. From astability point of
view, it is preferable to choose PS triangles with a small
area.
Being equipped with a B-spline like basis, PS splines admit a
straightforwardrational extension. A NURPS (Non-Uniform Rational
PS) basis function isdefined as
Ri,j(x, y) =Bi,j(x, y)ωi,j
NV∑l=1
3∑r=1
Bl,r(x, y)ωl,r
,
where ωi,j are positive control weights. In our plate context,
similar to thediscretization with NURBS, the unknown variables w
and γ are approximatedby
wh =
NV w∑i=1
3∑j=1
Ri,j(x, y)ŵi,j , γh =
NV γ∑i=1
3∑j=1
Ri,j(x, y)γ̂i,j ,
19
-
where NV w is the number of vertices for wh and NV γ is the
number of verticesfor γh. The test functions v and τ are
discretized accordingly. Displacementboundary conditions are
enforced in a standard way through the displacementdegrees of
freedom on the boundary while rotation boundary conditions
areenforced by the modified bilinear form introduced in equation
(8).
PS and NURPS B-splines have already been successfully employed
to solvepartial differential problems [30], in particular in the
isogeometric environment[32, 31]. Certain spline spaces of higher
degree and smoothness (regularity) havealso been defined on
triangulations endowed with a PS refinement, and they canbe
represented in a similar way as in the quadratic case. We refer to
[27] for C2
quintics and to [28] for a family of splines with arbitrary
smoothness. Moreover,the quadratic case has been extended to the
multivariate setting in [29].
Unfortunately, they are lacking the same flexibility of any
combination ofpolynomial degree and smoothness in contrast with the
tensor-product B-splinecase. On the one hand, it is known how to
construct stable spline spaces ontriangulations with a sufficiently
high polynomial degree with respect to theglobal smoothness (see,
e.g., [20]). In particular, one can quite easily do
degree-elevation for the above mentioned existing spaces (i.e.,
raising the polynomialdegree and keeping the original smoothness).
On the other hand, it is extremelychallenging to construct spline
spaces on triangulations with a very high smooth-ness relatively to
the degree (like the highest continuity Cp−1 for a degree p ≥
2).Another interesting point of further investigation is the
construction of splinespaces with mixed smoothness.
4.2 Numerical tests
In this section, we solve the same examples as illustrated in
Section 3.2, usingquadratic PS or NURPS B-splines. In particular,
the same error measure asdescribed before is adopted. Despite the
fact that PS/NURPS splines can bedefined on arbitrary
triangulations, we will only consider regular meshes in
ourexamples, in order to be able to make a fair comparison with
tensor-productsplines. Of course, in real applications one should
exploit this feature anduse triangulations generated by an adaptive
refinement strategy. For resultswith adaptive PS/NURPS
approximations in isogeometric analysis, we refer to[32, 31].
4.2.1 Square plate with clamped boundary conditions
We perform the same test described in Section 3.2.1 using
quadratic PS B-splines both for deflections and rotations defined
on uniform triangulations.The coarsest triangulation is depicted in
Figure 10 (left), and the approximationerror for the displacement
is shown in Figure 11. The dashed line indicates thereference order
of convergence. As can be seen, the convergence rate is of order
2.Figure 12 represents the approximation error for bending moments
and shearforces. We remark that shear forces seem to converge like
O(h) in the L2-normfor this case. However, other numerical tests
(not reported here) exhibit an
20
-
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
Figure 10: Uniform triangulation and its mapping to a quarter of
an annulus.
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−4
−3.5
−3
−2.5
−2
−1.5
−1
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 / ||w
ex||
L2 )
t=10−1
t=10−3
c1*(#dof)−1
Figure 11: Square plate with clamped boundary conditions.
L2-norm approx-imation error of displacements for t = 10−1 and t =
10−3 with triangular PSsplines.
O(1) convergence rate, in agreement with the theoretical
estimate (23), withs = 2.
4.2.2 Quarter of an annulus with clamped and simply
supportedboundary conditions
We perform the same test described in Section 3.2.2 using NURPS
B-splines.As before we consider a clamped and a simply supported
case and in both caseswe perform boundary refinement and uniform
refinement. The coarsest uniformmesh and its image are shown in
Figure 10. The images of some of the boundaryrefined meshes are
shown in Figure 13. Since there are no analytical
solutionsavailable, we have taken as reference solutions the NURPS
approximations ona fine mesh (an overkill solution): we have used a
triangulation consisting of
21
-
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−2.5
−2
−1.5
−1
−0.5
0
0.5
log10
(#dof1/2
)
log
10(
||m
ex −
mh||
L2 / ||m
ex||
L2 )
m, t=10−1
m, t=10−3
c*(#dof)−1/2
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−2.5
−2
−1.5
−1
−0.5
0
0.5
log10
(#dof1/2
)
log
10(
||q
ex −
qh||
L2 / ||q
ex||
L2 )
q, t=10−1
q, t=10−3
c*(#dof)−1/2
Figure 12: Square plate with clamped boundary conditions.
L2-norm approx-imation error of bending moments (left) and shear
forces (right) for t = 10−1
and t = 10−3 with triangular PS splines.
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
Figure 13: Some boundary refined meshes.
20000 triangles according to the two refinement schemes. Figure
14 shows theapproximation error for the displacement. The dashed
line indicates the refer-ence order. As can be seen, boundary
refinement yields improved results forboth cases. In particular,
the following remark holds for the investigated rangeof degrees of
freedom. For the simply supported case, the boundary
refinementscheme achieves the correct convergence rate, whereas
uniform refinement pro-duces a sub-optimal convergence rate. For
the clamped case, both the boundaryrefinement scheme and the
uniform refinement scheme give optimal convergencerates, but the
former procedure exhibits a numerical better constant in the
errorplots.
22
-
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−4
−3.5
−3
−2.5
−2
−1.5
−1
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 / ||w
ex||
L2 )
uniform
boundary
c1*(#dof)−1
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2−4
−3.5
−3
−2.5
−2
−1.5
−1
log10
(#dof1/2
)
log
10(
||w
ex −
wh||
L2 / ||w
ex||
L2 )
uniform
boundary
c1*(#dof)−1
Figure 14: Quarter of annulus with clamped (left) and simply
supported (right)boundary conditions. Uniform and boundary refined
meshes are considered. L2-norm approximation error of displacements
computed with triangular NURPS.
5 Conclusions
In this paper we mathematically and numerically investigated the
reformulatedvariational formulation of Reissner-Mindlin plate
theory in which the rotationvariables are eliminated in favor of
the transverse shear strains. Boundary con-ditions on the rotations
were enforced weakly by way of Nitsche’s method tomake the
implementation easier and to overcome possible boundary
lockingphenomena (see Remark 2.3). A distinct advantage of this
theory is that shearlocking is precluded for any combination of
trial functions for displacement andtransverse shear stains.
However, second derivatives of the displacement ap-pear in the
strain energy expression and these require basis functions of at
leastC1 continuity. To deal with the smoothness requirements we
employed Isogeo-metric Analysis, specifically various degree NURBS
of maximal continuity, andquadratic triangular NURPS. The numerical
results corroborated the theoreti-cal error estimates for
displacement, bending moments and transvers shear
forceresultants.
Acknowledgements
L. Beirão da Veiga, C. Lovadina and A. Reali were supported by
the EuropeanCommission through the FP7 Factory of the Future
project TERRIFIC (FoF-ICT-2011.7.4, Reference: 284981).T. J. R.
Hughes was supported by grants from the Office of Naval
Research(N00014- 08-1-0992), the National Science Foundation
(CMMI-01101007) andSINTEF (UTA10- 000374), with the University of
Texas at Austin.J. Kiendl and A. Reali were supported by the
European Research Councilthrough the FP7 Ideas Starting Grant n.
259229 ISOBIO.Jarkko Niiranen was supported by Academy of Finland
(decision number 270007).H. Speleers was supported by the Research
Foundation Flanders and by the
23
-
MIUR ‘Futuro in Ricerca’ Programme through the project
DREAMS.
A Proofs of the theoretical results
In the present section we prove all the theoretical results
previously presentedin the body of the paper. In the following we
will assume the obvious conditionthat 0 < t < diam(Ω), where
diam(Ω) denotes the diameter of Ω. We will needthe following
results.
• First Korn’s inequality (see [14]). There exists a positive
constant C suchthat
||ε(η)||2L2(Ω) + ||η||2L2(Ω) ≥ C||η||
2H1(Ω), ∀η ∈ H1(Ω)2. (25)
• Second Korn’s inequality (see [14]). Suppose that |Γc| > 0.
Then, thereexists a positive constant C such that
||ε(η)||2L2(Ω) ≥ C||η||2H1(Ω), ∀η ∈ H1(Ω)2, such that η|Γc = 0.
(26)
• Agmon’s inequality (see [1, 2]). Let e be an edge of an
element Ke. Then∃Ca(Ke) > 0 only depending on the shape of Ke
such that
||ϕ||2L2(e) ≤ Ca(Ke)(h−1e ||ϕ||2L2(Ke) + he|ϕ|
2H1(Ke)
), ϕ ∈ H1(Ke). (27)
Clearly, (27) also holds for vector-valued and tensor-valued
functions.
A.1 Coercivity of the continuous problem
Proposition A.1. Let assumption A4 hold. Then there exists a
positive con-stant α depending only on the material constants and
the domain Ω such that
a(η,η) + µkt−2(η −∇v,η −∇v)
≥ α(||η||2H1(Ω) + t
−2||η −∇v||2L2(Ω) + ||v||2H1(Ω)
), ∀(v,η) ∈ X̃.
(28)
Proof. It is easy to see that the hypotheses on Γc and Γs are
sufficient to preventrigid body motions. We proceed by considering
the two different cases.
i) Let Γc have positive length. Then, from the
positive-definiteness of C andthe second Korn’s inequality, we
get
a(η,η) = (Cε(η), ε(η)) ≥ C1||ε(η)||2L2(Ω) ≥ C2||η||2H1(Ω).
Therefore, estimate (28) follows from a little algebra and the
Poincaré inequalityfor v.
ii) Let Γc have zero length. Then Γs is not contained in a
straight line, and,since Γc ∪ Γs has positive length, it follows
that Γs has positive length. It isenough to prove that one has
a(η,η) + ||η −∇v||2L2(Ω) ≥ C(||η||2H1(Ω) + ||v||
2H1(Ω)
), ∀(v,η) ∈ X̃. (29)
24
-
By contradiction, suppose that estimate (29) does not hold.
Then, there exists
a sequence {(vk,ηk)} ∈ X̃ such that{a(ηk,ηk) + ||ηk −∇vk||2L2(Ω)
→ 0, for k → +∞;
||ηk||2H1(Ω) + ||vk||2H1(Ω) = 1.
(30)
Up to extracting a subsequence, the second equation of (30)
shows that
ηk ⇀ η0 weakly in H1(Ω)2; vk ⇀ v0 weakly in H
1(Ω). (31)
By Rellich’s Theorem we infer that
ηk → η0 in L2(Ω)2; vk → v0 in L2(Ω). (32)
Therefore, recalling that C is positive-definite, from a(ηk,ηk)
→ 0 (cf. (30)),(32) and (25), we get that {ηk} is a Cauchy sequence
in H1(Ω)2. Thus, we have
ηk → η0 in H1(Ω)2 and ε(η0) = 0. (33)
Moreover, since {ηk} is a Cauchy sequence in L2(Ω)2, from
||ηk−∇vk||2L2(Ω) → 0(cf. (30)), we have that also {∇vk} is a Cauchy
sequence in L2(Ω)2. Therefore,from (31) and (32) we obtain that
vk → v0 in H1(Ω) and ∇v0 = η0.
Hence, from (33) we get ε(∇v0) = 0, which implies that v0 is an
affine function.Since v0 = 0 on Γs and Γs is not contained in a
straight line, it follows thatv0 = 0 in Ω. Therefore, η0 = 0 and we
have proved that (ηk, vk) → (0, 0) inH1(Ω)2 ×H1(Ω), which
contradicts the second equation of (30).
A.2 Stability and convergence analysis
In the present section we give the proofs of the results in
Section 2.3. We needthe following Korn’s type inequality.
Lemma A.1. Suppose that Γc has positive length. Then, there
exists a positiveconstant C such that
||ε(v)||2L2(Ω) + ||v||2L2(Γc)
≥ C||v||2H1(Ω), ∀v ∈ H1(Ω)2. (34)
Proof. By contradiction. If (34) does not hold, then there
exists a sequence{vk} in H1(Ω)2 such that{
||ε(vk)||2L2(Ω) + ||vk||2L2(Γc)
→ 0, for k → +∞;
||vk||2H1(Ω) = 1.(35)
Up to extracting a subsequence, the second equation of (35) and
Rellich’s the-orem show that there exists v0 ∈ H1(Ω)2 such that
25
-
{vk ⇀ v0 weakly in H
1(Ω)2;
vk → v0 strongly in L2(Ω)2.(36)
From (35) and (36) we get that {(vk, ε(vk))} is a Cauchy
sequence in L2(Ω)2×L2(Ω)4s. Using the first Korn inequality (25) we
deduce that {vk} is a Cauchysequence also in H1(Ω)2, and thus vk →
v0, strongly in H1(Ω)2. Therefore,from the first equation of (35)
we have
ε(v0) = 0 in Ω; v0|Γc = 0. (37)
Equation (37) easily implies v0 = 0. Therefore, vk → 0, which is
in contradic-tion with ||vk||H1(Ω) = 1 (cf. (35)).
Proof of Lemma 2.1. We distinguish two cases.i) Γc has zero
length. In this case we have
ah(∇vh + τh,∇vh + τh) = a(∇vh + τh,∇vh + τh),|||vh, τh|||h =
|||vh, τh|||,
for every (vh, τh) ∈ Xh; see (8) and (10). Therefore, estimate
(11) immediatelyfollows from estimate (7), since Γc with vanishing
length implies Xh ⊂ X.
ii) Γc has positive length. First, for every (vh, τh) ∈ Xh we
will show that(cf. (10)):
ah(∇vh + τh,∇vh + τh)
≥ C
(||∇vh + τh||2H1(Ω) +
∑e∈Eh∩Γc
h−1e ||∇vh + τh||2L2(e)
).
(38)
For notational simplicity, we set θh := ∇vh + τh. Then,
recalling (8), we have
ah(θh,θh) = a(θh,θh)− 2∫
Γc
(Cε(θh)ne
)· θh + β tr(C)
∑e∈Eh∩Γc
h−1e
∫e
|θh|2
= a(θh,θh) +β
2tr(C)
∑e∈Eh∩Γc
h−1e
∫e
|θh|2
− 2∫
Γc
(Cε(θh)ne
)· θh +
β
2tr(C)
∑e∈Eh∩Γc
h−1e
∫e
|θh|2.
Applying (34) with v = θh, we obtain
ah(θh,θh) ≥ CK ||θh||2H1(Ω)
− 2∫
Γc
(Cε(θh)ne
)· θh +
β
2tr(C)
∑e∈Eh∩Γc
h−1e
∫e
|θh|2,(39)
26
-
for a suitable positive constant CK . For each edge e ∈ Γc ∩ Eh,
let the symbolKe denote an element of Ωh such that e ∈ ∂K. We now
have, by simple algebraand using (27):
− 2∫
Γc
(Cε(θh)ne
)· θh =
∑e∈Eh∩Γc
(−2∫e
(Cε(θh)ne
)· θh
)≥
∑e∈Eh∩Γc
(−2||Cε(θh)ne||L2(e)||θh||L2(e)
)≥
∑e∈Eh∩Γc
(−2CC||ε(θh)||L2(e)||θh||L2(e)
)≥ −
∑e∈Eh∩Γc
CC
(γhe||ε(θh)||2L2(e) +
1
γehe||θh||2L2(e)
)≥ −
∑e∈Eh∩Γc
(CCCa(Ke)γe
(||ε(θh)||2L2(Ke) + h
2e|ε(θh)|2H1(Ke)
)+
CCγ2he
||θh||2L2(e)),
(40)for positive constants {γe}e∈Γc∩Eh to be chosen. By using
the inverse inequality
|ε(θh)|2H1(Ke) ≤ Cinv(Ke)h−2Ke||ε(θh)||2L2(Ke),
and setting
C(Ke) = CCCa(Ke)
(1 + Cinv(Ke)
h2eh2Ke
),
from (40) it follows that
−2∫
Γc
(Cε(θh)ne
)· θh ≥ −
∑e∈Eh∩Γc
(C(Ke)γe||ε(θh)||2L2(Ke) +
CCγehe
||θh||2L2(e)).
(41)Therefore, from (39) and (41) we get
ah(θh,θh) ≥ CK ||θh||2H1(Ω) −∑
e∈Eh∩Γc
C(Ke)γe||ε(θh)||2L2(Ke)
+∑
e∈Eh∩Γc
(β
2tr(C)− CC
γe
)h−1e
∫e
|θh|2 ≥(CK −
∑e∈Eh∩Γc
C(Ke)γe
)||θh||2H1(Ω) +
(β
2tr(C)− CC
γe
) ∑e∈Eh∩Γc
h−1e
∫e
|θh|2.
(42)Choosing
γe =CK2C(Ke)
−1 and β0 =γCK + 2CCγtr(C)
with γ = mine∈Eh∩Γc
γe,
from (42) we deduce that, for every β ≥ β0, we have
ah(θh,θh) ≥CK2
(||θh||2H1(Ω) +
∑e∈Eh∩Γc
h−1e
∫e
|θh|2).
27
-
Recalling that θh := ∇vh + τh, we get that (38) holds.
Therefore, (11) followsfrom (38), (5), (10) and the Poincaré
inequality applied to vh (recall that vh|Γc =0 and |Γc| > 0).
Finally note that, due to the uniform shape regularity of
theelements K in {Ωh}h, it is easy to check that the constant β0 is
uniformlybounded from above independently of the mesh size h.
Proof of Proposition 2.1. In the following, C will denote a
generic positiveconstant independent of h. Given any pair (vh, τh)
in Xh, we denote by wE =wh − vh, γE = γh − τh and by wA = w − vh,
γA = γ − τh. By applying firstthe coercivity Lemma 2.1 and then
using the linearity of the bilinear forms andthe consistency
condition (12), we get
α′|||γE , wE |||2h ≤ ah(∇wE + γE ,∇wE + γE) + t−2(γE ,γE)=
ah(∇wA + γA,∇wE + γE) + t−2(γA,γE).
(43)
By definitions (8) and (2) and standard algebra we get from
(43)
|||γE , wE |||2h ≤ C T1/2A · T
1/2E + t
−2||γA||L2(Ω)||γE ||L2(Ω), (44)
where the scalar terms are given by
TA = ||∇wA + γA||2H1(Ω) +∑
e∈Eh∩Γc
he||Cε(∇wA + γA)ne||2L2(e)
+∑
e∈Eh∩Γc
h−1e ||∇wA + γA||2L2(e),
and
TE = ||∇wE + γE ||2H1(Ω) +∑
e∈Eh∩Γc
he||Cε(∇wE + γE)ne||2L2(e)
+∑
e∈Eh∩Γc
h−1e ||∇wE + γE ||2L2(e).
Term TA can be bounded by using (27), as already done in (40).
Without againshowing the details, and following the same notation
for Ke introduced in (40),we get
TA ≤ C(||∇wA + γA||2H1(Ω) +
∑e∈Eh∩Γc
(h2e|wA|2H3(Ke) + |wA|
2H2(Ke)
+ h2e|γA|2H2(Ke) + |γA|2H1(Ke)
+ h−2e |wA|2H1(Ke) + h−2e ||γA||2L2(Ke)
)).
From the above bound, a triangle inequality, and the definition
of wA, γA, weget
TA ≤ C2∑j=0
( ∑K∈Ωh
h2(j−1)K |w− vh|
2Hj+1(K) +
∑K∈Ωh
h2(j−1)K |γ − τh|
2Hj(K)
). (45)
28
-
We now bound TE . Again, using the Agmon inequality (27) and
inverse esti-mates as done in (40), we get for all e ∈ Eh ∩ Γc:
he||Cε(∇wE + γE)ne||2L2(e) ≤ C(h2e|∇wE + γE |2H2(Ke)+
|∇wE + γE |2H1(Ke))≤ C|∇wE + γE |2H1(Ke).
Combining the above bound with the definition of TE and (10)
yields
TE + t−2||γE ||2L2(Ω) ≤ C|||wE ,γE |||
2h. (46)
Now, recalling (44) and using (46) we easily get
|||γE , wE |||h ≤ C(T
1/2A + t
−1||γA||L2(Ω)).
Finally, recalling that the above inequality holds for all (vh,
τh) ∈ Xh, thebound in (45) concludes the proof.
Proof of Proposition 2.2. We only sketch the proof. Estimate
(16) immedi-ately follows from (5), by recalling (14) and (15). We
prove (17) only for thecase of a simply supported plate. In this
case, ah(·, ·) = a(·, ·), because Γc is theempty set. However, we
notice that different boundary conditions can be dealtwith using a
similar technique. Let sh and τh be given in Ξh. Using (6) and(9),
we get
(Qh − sh, τh) = (Qh −Q, τh) + (Q− sh, τh)= a
(∇(w − wh) + (γ − γh), τh
)+ (Q− sh, τh).
Choosing ψh = h2(Qh − sh), and using the inverse inequality
|Qh − sh|H1(Ω) ≤ Ch−1||Qh − sh||L2(Ω),
we have
h2||Qh − sh||2L2(Ω) = h2a(∇(w − wh) + (γ − γh), (Qh − sh)
)+ h2(Q− sh,Qh − sh) ≤ C|||w − wh,γ − γh|||hh ||Qh −
sh||L2(Ω)
+ h ||Q− sh||L2(Ω) h ||Qh − sh||L2(Ω).
Hence, we obtain
h ||Qh − sh||L2(Ω) ≤ C|||w − wh,γ − γh|||h + h ||Q−
sh||L2(Ω).
Therefore, the triangle inequality gives
h ||Q−Qh||L2(Ω) ≤ C|||w − wh,γ − γh|||h + 2h ||Q− sh||L2(Ω),
from which we infer
h ||Q−Qh||L2(Ω) ≤ C|||w − wh,γ − γh|||h + 2h infsh∈Ξh
||Q− sh||L2(Ω),
29
-
i.e. (17).To prove (18), we use the so-called
Pitkäranta-Verfürth trick (see [25] and [33]).
We start with noticing that
||Q−Qh||H−1(Ω) = supτ∈H10 (Ω)2
(Q−Qh, τ )|τ |H1(Ω)
. (47)
Then, fix τ ∈ H10 (Ω)2 and take τ I as its best approximation in
Ξh with respectto the H1-norm. Then, we have
(Q−Qh, τ ) = (Q−Qh, τ − τ I) + (Q−Qh, τ I)≤ Ch
||Q−Qh||L2(Ω)|ψ|H1(Ω) + (Q−Qh, τ I).
(48)
Using (6) and (9), we get
(Q−Qh, τ I) = a(∇(wh − w) + (γh − γ), τ I
)≤ C|||w − wh,γ − γh|||h|τ I |H1(Ω)≤ C|||w − wh,γ − γh|||h|τ
|H1(Ω).
(49)
Therefore, from (47), (48) and (49) we deduce
||Q−Qh||H−1(Ω) ≤ C(h ||Q−Qh||L2(Ω) + |||w − wh,γ − γh|||h
). (50)
Estimate (18) now follows from (50) and (17).
Proof of Proposition 2.3. We begin with introducing the
following auxiliaryproblem:{
Find (w̃, γ̃) ∈ X, such thata(∇w̃ + γ̃,∇v + τ ) + t−2(γ̃, τ ) =
(θ − θh,∇v + τ ), ∀(v, τ ) ∈ X.
(51)
Note that problem (51) is equivalent, up to the usual
transformation (4), to aclassical Reissner-Mindlin problem with the
load acting on the rotations. There-fore, also recalling the
hypotheses above and the definition of γ, the followingregularity
result holds [22]
||w̃1||H3(Ω) + t−1||w̃2||H2(Ω) + t−1||γ̃||H1(Ω) ≤ C||θ −
θh||L2(Ω), (52)
with w̃ = w̃1 + w̃2 denoting a splitting of the deflections.The
consistency of the discrete bilinear form, analogously as in (12),
implies
thatah(∇w̃ + γ̃,∇vh + τh) + t−2(γ̃, τh) = (θ − θh,∇vh + τh),
(53)
for all (vh, τh) ∈ Xh. Note that, whenever both entries are in
X, the bilinearform ah(·, ·) is equal to the original form a(·, ·)
since all the additional termsvanish. Therefore, from (51) we also
have
ah(∇w̃ + γ̃,∇v + τ ) + t−2(γ̃, τ ) = (θ − θh,∇v + τ ), ∀(v, τ )
∈ X. (54)
30
-
Combining (53) and (54) we can take (w − wh,γ − γh) as a test
function, andobtain
||θ − θh||2L2(Ω) = ah(∇w̃ + γ̃,∇(w − wh) + γ − γh) + t−2(γ̃,γ −
γh) (55)
where we recall the usual notation θ = ∇w + γ and θh = ∇wh + γh.
Theremaining steps are standard in Aubin-Nitsche duality arguments
and thereforeonly described briefly. By the symmetry of the
bilinear form and using (9), (12),from (55) we get
||θ − θh||2L2(Ω) = ah(∇(w̃ − w̃I) + γ̃ − γ̃I ,∇(w − wh) + γ −
γh)+ t−2(γ̃ − γ̃I ,γ − γh),
(56)
where (w̃I , γ̃I) ∈ Xh is an approximant of (w̃, γ̃).Due to the
||| · |||h-norm continuity of the bilinear form in (56), all we
need
to show in order to prove (19) is that
|||w̃ − w̃I ,γ − γI |||h ≤ C (h+ t) ||θ − θh||L2(Ω), (57)
for some uniform constant C. The bound in (57) can be shown by
making use ofthe regularity estimate (52) and the approximation
properties A3 of the spaceXh; we do not show the details.
Finally, estimate (20) follows easily by the triangle
inequality
||w − wh||H1(Ω) ≤ ||θ − θh||L2(Ω) + ||γ − γh||L2(Ω)
and using (19).
References
[1] S. Agmon, Lectures on elliptic boundary value problems, Van
NostrandMathematical Studies, Princeton, NJ, 1965.
[2] D.N. Arnold, An interior penalty finite element method with
discontinuouselements, SIAM J. Numer. Anal., 19, 742–760, 1982.
[3] D.N. Arnold and R.S. Falk, A uniformly accurate finite
element method forthe Reissner-Mindlin plate, SIAM J. Numer. Anal.,
26, 1276–1290, 1989.
[4] K.J. Bathe, Finite Element Procedures in Engineering
Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982.
[5] Y. Bazilevs, L. Beirão da Veiga, J.A. Cottrell, T.J.R.
Hughes and G. San-galli, Isogeometric analysis: approximation,
stability and error estimatesfor h-refined meshes, Math. Mod. Meth.
Appl. Sci., 16, 1–60, 2006.
[6] L. Beirão da Veiga, A. Buffa, C. Lovadina, M. Martinelli
and G. Sangalli.An isogeometric method for the Reissner-Mindlin
plate bending problem,Comput. Methods Appl. Mech. Engrg., 209,
2012, 45-53.
31
-
[7] L. Beirão da Veiga, D. Cho and G. Sangalli. Anisotropic
NURBS approxi-mation in isogeometric analysis, Comput. Meth. Appl.
Mech. Engrg., 209–212, 1–11, 2012.
[8] D.J. Benson, Y. Bazilevs, M.-C. Hsu, and T. J. R. Hughes.
Isogeometricshell analysis: The Reissner-Mindlin shell, Comput.
Meth. Appl. Mech.Engrg., 199, 276–289, 2010.
[9] F. Brezzi, M. Fortin and R. Stenberg, Error analysis of
mixed-interpolatedelements for Reissner-Mindlin plates, Math.
Models Meth. Appl. Sci., 1,125–151, 1991.
[10] D. Chapelle and R. Stenberg, An optimal low-order
locking-free finite ele-ment method for Reissner-Mindlin plates,
Math. Models Meth. Appl. Sci.,8, 407–430, 1998.
[11] C. Chinosi and C. Lovadina, Numerical analysis of some
mixed finite el-ement methods for Reissner-Mindlin plates, Comput.
Mech., 16, 36–44,1995.
[12] J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs, Isogeometric
Analysis: To-ward Integration of CAD and FEA, John Wiley &
Sons, 2009.
[13] P. Dierckx, On calculating normalized Powell-Sabin
B-splines, Comput.Aided Geom. Design, 15, 61–78, 1997.
[14] G. Duvaut and J.L. Lions, Inequalities in Mechanics and
Physics, Springer,Berlin Heidelberg, 1976.
[15] R. Echter, B. Oesterle and M. Bischoff, A hierarchic family
of isogeometricshell finite elements, Comput. Methods Appl. Mech.
Engrg., 254, 170–180,2013.
[16] C.S. Felippa, Advanced Finite Element Methods (ASEN6367) -
Spring 2013, Department of Aerospace Engineer-ing Sciences,
University of Colorado at Boulder, Chapter
22:http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/Home.html
[17] T.J.R. Hughes. The finite element method: Linear static and
dynamic finiteelement analysis. Dover Publications, 2000.
[18] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, Isogeometric
analysis: CAD,finite elements, NURBS, exact geometry and mesh
refinement, Comput.Methods Appl. Mech. Engrg., 194, 4135–4195,
2005.
[19] J. Kiendl, F. Auricchio, T.J.R. Hughes, and A. Reali,
Single-variable formu-lations and isogeometric discretizations for
shear deformable beams, Com-put. Methods Appl. Mech. Engrg., 284,
988–1004, 2015.
[20] M.J. Lai and L.L. Schumaker, Spline Functions on
Triangulations, Cam-bridge University Press, 2007.
32
-
[21] Q. Long, P.B. Bornemann and F. Cirak, Shear-flexible
subdivision shells,Int. J. Numer. Meth. Engrg., 90, 1549–1577,
2012.
[22] M. Lyly, J. Niiranen and R. Stenberg. A refined error
analysis of MITCplate elements, Math. Models Methods Appl. Sci. 16,
967-977, 2006.
[23] J.A. Nitsche. Über ein Variationsprinzip zur Lösung
Dirichlet-Problemenbei Verwendung von Teilräumen, die keinen
Randbedingungen unteworfensind, Abh. Math. Sem. Univ. Hamburg 36,
9–15, 1971.
[24] S.W. Papenfuss, Lateral plate deflection by stiffness
matrix methods withapplication to a marquee, M.S. Thesis,
Department of Civil Engineering,University of Washington, Seattle
(December 1959).
[25] J. Pitkäranta, Boundary subspaces for the finite element
method with La-grange multipliers, Numer. Math., 33, 273–289,
1979.
[26] M.J.D. Powell and M.A. Sabin, Piecewise quadratic
approximations on tri-angles, ACM Trans. Math. Software, 3,
316–325, 1977.
[27] H. Speleers, A normalized basis for quintic Powell-Sabin
splines, Comput.Aided Geom. Design, 27, 438–457, 2010.
[28] H. Speleers, Construction of normalized B-splines for a
family of smoothspline spaces over Powell-Sabin triangulations,
Constr. Approx., 37, 41–72,2013.
[29] H. Speleers, Multivariate normalized Powell-Sabin B-splines
and quasi-interpolants, Comput. Aided Geom. Design, 30, 2–19,
2013.
[30] H. Speleers, P. Dierckx and S. Vandewalle, Numerical
solution of partialdifferential equations with Powell-Sabin
splines, J. Comput. Appl. Math.,189, 643–659, 2006.
[31] H. Speleers, C. Manni and F. Pelosi, From NURBS to NURPS
geometries,Comput. Methods Appl. Mech. Engrg., 255, 238–254,
2013.
[32] H. Speleers, C. Manni, F. Pelosi and M.L. Sampoli,
Isogeometric analysiswith Powell-Sabin splines for
advection-diffusion-reaction problems, Com-put. Methods Appl. Mech.
Engrg., 221–222, 132–148, 2012.
[33] R. Verfürth, Error estimates for a mixed finite element
approximation ofthe Stokes equations, RAIRO Anal. Numer., 18,
175–182, 1984.
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