Port-Hamiltonian formulation and symplectic discretization of plate models. Part II : Kirchhoff model for thin plates



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an author's

https://oatao.univ-toulouse.fr/23901

https://doi.org/10.1016/j.apm.2019.04.036

Brugnoli, Andrea and Alazard, Daniel and Pommier-Budinger, Valérie and Matignon, Denis Port-Hamiltonian

formulation and symplectic discretization of plate models. Part II : Kirchhoff model for thin plates. (2019) Applied

Mathematical Modelling, 75. 961-981. ISSN 0307-904X

Port-Hamiltonian

formulation

and

symplectic

discretization

of

plate

models

Part

II:

Kirchhoff model

for

thin

plates

Andrea

Brugnoli

∗

,

Daniel

Alazard

,

Valérie

Pommier-Budinger

,

Denis

Matignon

ISAE-SUPAERO, Université de Toulouse, 10 Avenue Edouard Belin, BP-54032, Cedex 4, Toulouse 31055, France

a

r

t

i

c

l

e

i

n

f

o

Keywords:

Port-Hamiltonian systems Kirchhoff plate

Partitioned Finite Element Method Geometric spatial discretization Boundary control

a

b

s

t

r

a

c

t

Themechanicalmodelofathinplatewithboundarycontrolandobservationispresented asaport-Hamiltoniansystem(PHs1_{),bothinvectorialandtensorialforms:the} Kirchhoff-Lovemodel ofaplateisdescribedbyusingaStokes-Diracstructure andthisrepresents anovelty withrespecttotheexisting literature.Thisformulation iscarriedoutbothin vectorialand tensorialforms.Thankstotensorialcalculus,thismodelisfoundtomimic theinterconnectionstructure ofitsone-dimensionalcounterpart, i.e.theEuler-Bernoulli beam.

ThePartitionedFiniteElementMethod(PFEM2_{)isthenextendedtoobtainasuitable,i.e.} structure-preserving,weak form.The discretization procedure, performedon the vecto-rialformulation,leadstoafinite-dimensionalport-Hamiltoniansystem.ThispartIIofthe companionpaperextendspartI,dedicatedtotheMindlinmodelforthickplates.Thethin platemodelcomesalongwithadditional difficulties,becauseofthehigher orderofthe differentialoperatorunderconsideration.

Introduction

AspresentedinpartIofthiscompanionpaper,theport-Hamiltonian(PH)formalism[1–3]allowsthestructured model-inganddiscretizationofmulti-physicsapplicationsinvolvinginterconnectedfinite-andinfinite-dimensionalsystems[4,5]. Preservingthe port-Hamiltonianstructure inthe discretizationprocess is akeypoint to take benefitof thispowerful for-malism.Thisissuewasfirst addressedin [6],witha mixedfinite elementspatialdiscretization method,andin[7],with pseudo-spectralmethodsrelying onhigher-orderglobalpolynomialapproximations. Allthosemethodsare difficultto im-plement,especially forthosesystemthe spatialdimensionof whichisbiggerthan one. Veryrecentlyweak formulations whichleadtoGalerkinnumericalapproximationsbegantobeexplored:in[8],astructurepreservingfiniteelementmethod wasintroducedforthewaveequationinatwo-dimensionaldomain;thismethodexhibitsgoodresults,bothinthespectral analysisandsimulationpart,thoughrequiringofaprimal anda dualmeshonthegeometryoftheproblem. Another ap-proachisthePartitionedFiniteElementMethod(PFEM)proposedin[9],alreadylargelyexploredinpartIofthiscompanion paper.Theadvantagesofthislattermethodology areits simplicityofimplementationandits potential tocarry overto a

∗ _{Corresponding author.}

E-mail addresses: Andrea.Brugnoli@isae.fr (A. Brugnoli), Daniel.Alazard@isae.fr (D. Alazard), Valerie.Budinger@isae.fr (V. Pommier-Budinger),

Denis.Matignon@isae.fr (D. Matignon).

1 PHs stands for port-Hamiltonian systems. 2 PFEM stands for partitioned finite element method.

widesetofexamples,nomatterthespatialdimensionoftheproblem.ThepossibleuseofopensourcesoftwarelikeFEniCS

[10]orFiredrake[11]isalsoanappealingfeatureofthislattermethod.

InpartIIofthiscompanionpaper,themodelinganddiscretizationofthinplatesdescribedby theKirchhoff-Loveplate modeliscarriedoutwithinthePHframework,allowingforboundarycontrolandobservation.Theexistingliterature deal-ingwiththesymplecticHamiltonianformulationoftheKirchhoff plate[12,13]focusedmainlyonanalyticalsolutionforthe free vibrationproblem. Thisapproach ispowerfulwhenevereasy solutionare sought forbutdoesnot extendtosystems

interconnectedincomplexmanners.Furthermore,platemodelswereinvestigatedwithingtheport-Hamiltonianframework

using jet theory [14,15],butthe numerical implementationof such models remains cumbersome.The main contribution

ofthispaperconcernstherepresentationoftheKirchhoff plateusingtheconceptofStokes-Diracstructure,sototake ad-vantagefromthemodularityofthisgeometricstructure.Thisformalism ispresentedbothinvectorialandtensorialforms. Moreover, thetensorial formalism [16, Chapter16] highlights that this model mimicsthe interconnection structureof its one-dimensional counterpart,i.e.theEuler-Bernoulli beam.Comparedto partIdedicatedto thickplateMindlinmodelin whichfirst-orderdifferential operatorsare exploredin dimensiontwo,andcompared to[17]in whichsecond-or higher-order differential operators were explored indimension one only, the contributionof thispaper is the PHformalism of systemsofdimensiontwodescribedwithsecond-orderdifferentialoperators,suchastheKirchhoff-Lovemodel.Themodel, oncewritteninatensorialform,highlightsnewinsightsonsecond-orderdifferentialoperators:especiallythedouble diver-genceandtheHessianareprovedtobeadjointoperatorsoneofanother,whichrepresentsanotherimportantcontribution ofthispaper.Finally,theextensionofthePFEMmethodtothestructure-preservingdiscretizationoftheKirchhoff modelis alsoanoveltyofthepaper.Itallows simpleimplementationofnumericalschemescomparedtothejet theoryformalism, whilepreservingthestructureofPHSatthediscretelevel.Thelastsectionisdedicatedtonumericalstudiesofthismodel usingFiredrake[11].

1. Second-orderdistributedPHsystems:Euler-Bernoullibeam

TheEuler-Bernoullibeamistheone-dimensionalequivalentoftheKirchhoff-Loveplate.ThismodelconsistsofonePDE, describingtheverticaldisplacementalongthebeamlength:

ρ

(

x

)

∂

2_w

∂

t2

(

x,t

)

+

∂

2

∂

x2



EI

(

x

)

∂

2_w

∂

x2



=0, x∈

(

0,L

)

,t≥ 0, (1)

wherew(x,t)isthetransversedisplacementofthebeam. Thecoefficients

ρ

(x),E(x)andI(x)arethemassperunit length,

Young’s modulus of elasticity and the moment of inertia of a cross section. The energy variables are then chosen as

follows:

α

w=

ρ

(

x

)

∂

w

∂

t

(

x,t

)

, LinearMomentum,

α

κ =

∂

2_w

∂

x2

(

x,t

)

, Curvature. (2)

Those variablesarecollected inthevector

α

₌

(

α

w,

α

κ

)

T,sothatthe Hamiltoniancan bewrittenasaquadratic

func-tionalintheenergyvariables:

H=1 2 L 0

α

T_Q

_α

_{dx, where Q=}



₁ ρ(x) 0 0 EI

(

x

)



. (3)

Theco-energyvariablesarefoundbycomputingthevariationalderivativeoftheHamiltonian:

ew:=

∂

H

∂

α

w =

∂

w

∂

t

(

x,t

)

, Verticalvelocity, e_κ :=

∂

H

∂

α

κ =EI

(

x

)

∂

2_w

∂

x2

(

x,t

)

, Flexuralmomentum. (4)

Those variablesareagaincollected invectore=

(

ew,eκ

)

T, sothatthe underlyinginterconnectionstructure isthenfound

tobe:

∂

α

∂

t =Je, where J=



0 −∂2 ∂x2 ∂2 ∂x2 0



. (5)

Foraninfinite-dimensionalsystem,boundaryvariableshavetobedefinedaswell.Thosecanbefoundbyevaluatingthe energyrateflow acrosstheboundary. Onepossible choiceamongothers(see [18]foramore exhaustiveexplanation)for

Fig. 1. Kinematic assumption for the Kirchhoff plate.

thismodelisthefollowing:

f_∂=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

ew

(

0

)

∂

ew

∂

x

(

0

)

∂

eκ

∂

x

(

L

)

e_κ

(

L

)

⎞

⎟

⎟

⎟

⎟

⎟

⎠

, e_∂=

⎛

⎜

⎜

⎜

⎝

∂

e_κ

∂

x

(

0

)

−eκ

(

0

)

−ew

(

L

)

∂

ew

∂

x

(

L

)

⎞

⎟

⎟

⎟

⎠

. (6)

Thepowerflowistheneasilyevaluatedas:

d dtH

(

t

)

=  L 0

∂

α

∂

t · edx=



e∂,f∂



IR4. (7)

Theflowvariablescannowbedefinedasf₌₋∂_∂α_t,sothattheflowspaceisgivenbythetuples

(

f_,f_∂

)

_∈F.Equivalently theeffortspaceisgivenby

(

e,e_∂

)

∈E.ThebondspaceisthereforetheCartesianproductofthesetwospaces:

B:=

{

(

f,f_∂,e,e_∂

)

∈F× E

}

. (8)

ThedualitypairingbetweenelementsofB isthendefinedasfollows: 

((

fa,fa_∂

)

,

(

ea_,ea ∂

))

,

((

fb,fb∂

)

,

(

eb,eb∂

))

:= L 0

(

fa

)

T_eb₊

₍

_fb

₎

T_ea



_dx+

₍

_fa ∂

)

Teb∂+

(

fb∂

)

Tfa∂. (9)

TheStokes-DiracstructurefortheEuler-Bernoullibeamistherefore:

Theorem1 (From [17], Stokes-Diracstructure for the Bernoulli beam). Considerthe space of powervariables B definedin

(8)andthebilinearform(+pairingoperator)  ,  givenby(9).DefinethefollowinglinearsubspaceD⊂ F× E:

D=

{

(

f,f_∂,e,e_∂

)

∈F× E

|

f =−Je

}

, (10)

wheref_∂ ande_∂ weredefinedin(6).Then,itholdsD=D⊥_,whereD⊥_{isunderstoodin thesenseoforthogonalitywithrespect} tothebilinearproduct _{ ,  ,}i.e_{D is}aStokes-Diracstructure.

Remark1. Forwhatconcernstheuseofthismodelforcontrolandsimulationpurposes,thereadercanreferto[19]fora stabilityandstabilizationproofoftheEuler-Bernoullibeamorto[20]foranillustrationofarotatingspacecraftwithflexible

appendagesmodelasPHBernoullibeams.

2. Kirchhoff-Lovetheoryforthinplates

Inthissectiontheclassicalvariationalapproach(Hamilton’sprinciple)toderivetheequationofmotionsisfirstdetailed. Thephysicalquantitiesinvolvedandthedifferentenergies,ofutmostimportanceforthePHformalism,arereminded.

2.1. Modelandassociatedvariationalformulation

TheKirchhoff-Love plateformulationrestsonthe hypothesis ofsmallthicknesscomparedto theinplane dimensions.

The notationsandsymbolsare borrowedform Cook etal.[21,22].The displacement field andthe strainsare defined by

assumingthatfibersorthogonaltothemiddleplaneremainorthogonal(seeFig.1).Thisleadstothefollowingrelationsfor thedisplacementfield

u

(

x,y,z

)

=−z

∂

_∂

w

x,

v

(

x,y,z

)

=−z

∂

w

∂

y, w

(

x,y,z

)

=w

(

x,y

)

(11)

andforthestrains



=



_

xx



yy

γ

xy



=

⎛

⎝

∂ ∂x 0 0 _∂∂_y ∂ ∂y ∂∂x

⎞

⎠



−z∂∂wx −z∂w ∂y



=−z

⎛

⎝

∂2_w ∂x2 ∂2_w ∂y2 2∂2w ∂x∂y

⎞

⎠

. (12)

Thecurvaturevectorisdefinedas:

κ

=



_κ

xx

κ

yy

κ

xy



=

⎛

⎝

∂2_w ∂x2 ∂2_w ∂y2 2∂2w ∂x∂y

⎞

⎠

. (13)

Hooke’sconstitutivelawforisotropicmaterialisconsideredfortheconstitutiverelation:

σ

=E



, E:= E 1−

ν

2



₁

_ν

₀

ν

1 0 0 0 1−ν 2



. (14)

where

ν

isPoisson’sratioandEYoung’smodulus.Thesephysicalparametersmaybeinhomogeneous,i.e.

ν

=

ν

(

x,y,z

)

,E=

E

(

x_,y_,z

)

.Thegeneralizedmomentaarefoundbyintegratingthestressesalongthefiber:

M=



_M xx Myy Mxy



=



 h 2 −h 2 Ez2_dz



κ

,

wherehistheplatethickness.Therelationbetweenmomentaandcurvaturesisexpressedbythebendingrigiditymatrix D: M=D

κ

D:=  h 2 −h 2 Ez2dz. (15)

NowtheclassicalKirchhoff-Lovemodelforthinplatescanberecalled[23]:

μ∂

_∂

2_tw2 +

∂

2_M xx

∂

x2 +2

∂

2_M xy

∂

x

∂

y +

∂

2_M yy

∂

y2 =0, (16)

where

μ

=

ρ

histhesurfacedensityand

ρ

themassdensity.IftheE and

ν

coefficientsareconstant,thentherulingPDE becomes:

μ∂

_∂

2_tw2 +D

2_{w=p, (17)} where

2₌ ∂4 ∂x4 +2 ∂ 4 ∂x4_∂y4+ ∂ 4

∂y4 isthebiLaplacianandD=

Eh3

12(1−ν2₎ isthebendingrigiditymodulus.Thekineticand poten-tialenergydensitiesperunitarea_{K andU,}arerespectivelygivenby:

K= 1 2

μ



∂

w

∂

t



2 , U=1 2M·

κ

.

Thetotalenergydensityissplitintokineticandpotentialenergy

H=K+U, (18)

andthecorrespondingtotalenergiesgivenbythefollowingrelations:

H=  Hd

,

K=  Kd

,

U=  U d

.

(19)

3. PHformulationoftheKirchhoff plate

In this section the port-Hamiltonian formulation of the Kirchhoff plate is presented first in vectorial form in

3.1. PHvectorialformulationoftheKirchhoff plate

Toobtainaport-Hamiltoniansystem(PHs)theenergyvariablesaswellastheunderlyingStokes-Diracstructure, associ-atedwiththeskew-adjointoperatorJ,havetobeproperlydefined.ConsidertheHamiltonianenergy:

H=  1 2



μ



∂

w

∂

t



2 +M·

κ



d

=  1 2



μ



∂

_∂

w_t



2 +

κ

T_D

_κ



d

.

(20)

Theenergyvariablesarethenselectedtobethelinearmomentum

μ

∂w

∂t andthecurvatures

κ

,inananalogousfashionwith

respecttothe one-dimensionalcounterpartofthismodel,the Euler-Bernoullibeam. Theenergyvariables arecollected in vector

α

:=

(

μ

wt,

κ

xx,

κ

yy,

κ

xy

)

T, (21)

wherewt= ∂_∂w_t.TheHamiltoniandensityisgivenbythefollowingexpression:

H=1 2

α

T



₁ μ 0 0 D



α

, H=  Hd

.

(22)

Soitsvariationalderivativeprovidesasco-energyvariables:

e:=

∂

H

∂

α

=

(

wt, Mxx, Myy, Mxy

)

T, (23)

Theport-Hamiltoniansystemandskew-symmetricoperatorrelatingenergyandco-energyvariablesarefoundtobe:

∂

α

∂

t =Je and J:=

⎡

⎢

⎢

⎣

0 −∂2 ∂x2 −∂ 2 ∂y2 −



_∂2 ∂x∂y+ ∂ 2 ∂y∂x



∂2 ∂x2 0 0 0 ∂2 ∂y2 0 0 0 ∂2 ∂x∂y+ ∂ 2 ∂y∂x 0 0 0

⎤

⎥

⎥

⎦

. (24)

Thefirstlineoftheskew-symmetricoperatorin(24)isfoundbyconsideringEq.(16).TheremaininglinesexpressClairaut’s theoremfortheverticaldisplacement.Thistheoremstatesthat,forsmoothfunctions,higherorderpartialderivative com-mute.

Remark 2. From theSchwarz theorem forC2 _{functionsthe mixedderivative could be be expressed as2} ∂2

∂x∂y,instead of

∂2

∂y∂x+ ∂

2

∂x∂y.However, inthiswaythesymmetryintrinsicallypresentin

κ

xy= ∂

2_w

∂y∂x+ ∂

2_w

∂x∂y wouldbelost.Themixed

deriva-tiveisheresplittoreestablishthesymmetricnatureofcurvaturesandmomenta(thatareoftensorialnatureasexplained laterinSection3.2).

TheboundaryvariablesareobtainedbyevaluatingthetimederivativeoftheHamiltonian: ˙ H= 

∂

H

∂

α

·

∂

∂

α

t d

= 



wt



−

∂

2Mxx

∂

x2 −

∂

2_M xx

∂

y2 − 2

∂

2_M xy

∂

x

∂

y



+Mxx

∂

2_w t

∂

x2 +Myy

∂

2_w t

∂

y2 +2Mxy

∂

2_w t

∂

x

∂

y



d

= 



e1



−

∂

2e2

∂

x2 −

∂

2_e 3

∂

y2 − 2

∂

2_e 4

∂

x

∂

y



+e2

∂

2_e 1

∂

x2 +e3

∂

2_e 1

∂

y2 +2e4

∂

2_e 1

∂

x

∂

y



d

InFig.2thenotationsforthedifferentreferenceframesareintroduced.ByapplyingGreentheorem,consideringthesplit mixedderivative(2_∂∂_x∂2_y=_∂∂x∂2y+ ∂ 2 ∂y∂x): ˙ H=  ∂



nx



e2

∂

e1

∂

x +e4

∂

e1

∂

y − e1

∂

e2

∂

x − e1

∂

e4

∂

y



+ny



e3

∂

e1

∂

y +e4

∂

e1

∂

x − e1

∂

e3

∂

y − e1

∂

e4

∂

x



ds. (25)

wherenx,nyarethecomponentsalongthex− andthey−axisofthenormaltotheboundary.Thevariableofintegrations

Fig. 2. Reference frames and notations.

Fig. 3. Cauchy law for momenta and forces at the boundary.

Ifthephysicalvariablesareintroduced,then ˙ H=  ∂



nx



Mxx

∂

wt

∂

x +Mxy

∂

wt

∂

y − wt

∂

Mxx

∂

x − wt

∂

Mxy

∂

y



+ny



Myy

∂

wt

∂

y +Mxy

∂

wt

∂

x − wt

∂

Myy

∂

y − wt

∂

Mxy

∂

x



ds. (26)

Nowthefollowingquantities,representedinFig.3,aredefined: ShearForce qn:=nxqx+nyqy, Flexuralmomentum Mnn:=nT



Mxxnx+Mxyny Mxynx+Myyny



, Torsionalmomentum Mns:=sT



Mxxnx+Mxyny Mxynx+Myyny



, n=



nx ny



, s=



−ny nx



, (27)

whereqx=−∂M_∂xxx−∂M_∂yxy and qy=−∂_∂M_yyy−∂M_∂xxy. Thegradient ofthe verticalvelocity can beprojected upon thenormal

andtangentialdirectionstotheboundary:

∇

wt=

(

∇

wt· n

)

n+

(

∇

wt· s

)

s=

∂

wt

∂

n n+

∂

wt

∂

s s. (28)

SothetimederivativeoftheHamiltoniancanbefinallywrittenas: ˙ H=  ∂



wtqn+

∂

wt

∂

s Mns+

∂

wt

∂

n Mnn



ds. (29)

Variableswtand∂_∂w_st arenotindependentastheyaredifferentiallyrelatedwithrespecttoderivationalongs,thecurvilinear

abscissaoftheboundarydomain(seeforinstance[23]).Anotherintegrationbypartisneededtohighlightappropriate in-dependentpowerconjugatedvariables.Letussupposethattheboundaryisaclosedandregularcurve.Thentheintegration bypartsalongaclosedboundaryleadsto:

 ∂

∂

wt

∂

s Mnsds=−  ∂

∂

Mns

∂

s wt ds. (30)

Theenergybalancecanbefinallywrittenas: ˙ H=  ∂



wtqn+

∂

wt

∂

n Mnn



ds, (31)

whereqn:=qn−∂M_∂sns istheeffectiveshearforce.Eq.(31)isofutmostimportance,sinceitcontainstheboundaryvariables

3.1.1. UnderlyingStokes-Diracstructure

LetF denote the flow spaceand let E denote theeffort space.Forsimplicity we take F≡ E=C∞

₍

_,R4

₎

_{, thespace}

of smooth vector-valued functionsin R4_{. Eq.(31) allows identifying the boundary terms of the underlying Stokes-Dirac}

structures.Thespaceofboundaryvariablesisavectoroffourcomponentsgivenby:

Z=

{

z

|

z=B_∂

(

e

)

,

∀

e∈E

}

, z=



 qn,wt,Mnn,

∂

wt

∂

n



T .

InthecasewherethedifferentialJoperatoroforderone,theB_∂ operatorisalinearoperatoroverthetraceoftheeffort variables..Here,sincethedifferentialJisofordertwo,B_∂ containsthenormalandtangentialderivativesattheboundary andsomoreregularityisrequiredfortheboundaryvariables.

Remark3. Thisfactwasalreadystatedfor1-Dsystemsin[17];hereitistheextensionto2-Dsystemwithasecond-order differentialoperatorJ.

Thisoperatorreads:

B_∂

(

e

)

=

⎡

⎢

⎣

0 0 0 0 1 0 0 0 0 n2 x n2y 2nxny 0 0 0 0

⎤

⎥

⎦

e−

⎡

⎢

⎣

0 nx 0 ny 0 0 0 0 0 0 0 0 0 0 0 0

⎤

⎥

⎦

∂

_∂

e_x−

⎡

⎢

⎣

0 0 ny nx 0 0 0 0 0 0 0 0 0 0 0 0

⎤

⎥

⎦

_∂

∂

e_y +

∂

∂

n

⎛

⎜

⎝

⎡

⎢

⎣

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

⎤

⎥

⎦

e

⎞

⎟

⎠

−

∂

∂

s

⎛

⎜

⎝

⎡

⎢

⎣

0 −nxny nxny n2x− n2y 0 0 0 0 0 0 0 0 0 0 0 0

⎤

⎥

⎦

e

⎞

⎟

⎠

. (32) Theorem2(Stokes-DiracstructurefrtheKirchhoff Plate). Theset

D:=



(

f,e,z

)

∈F× E× Z

|

f =−

∂

_∂

α

t =−Je, z=B∂

(

e

)



(33)

isaStokes-Diracstructurewithrespecttothepairing



(

f1,e1,z1

)

,

(

f2,e2,z2

)

 =  e T 1f2+eT2f1

!

d

+  ∂BJ

(

z1,z2

)

ds, (34)

whereBJisasymmetricoperator,arisingfromadoubleapplicationoftheGreentheorem.Itreads

BJ

(

z1,z2

)

=qn,2 wt,1+Mnn,2

∂

w_t,1

∂

n +q_n,1 wt,2+Mnn,1

∂

wt,2

∂

n =zT 1BJz2 , BJ=

⎡

⎢

⎣

0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0

⎤

⎥

⎦

. (35)

Proof.Aregularboundarywillbeassumedfortheproof. StepI.ThefirstimplicationisD⊆ DT_{.Thisistrueif}

_∀

_ω

α=

(

f_α,e_α,z_α

)

and

ω

_β=

(

f_β,e_β,z_β

)

∈Dthen

ω

α,

ω

β=0. Theintegraloverthedomainreads:

 e T αfβ+eTβfα

!

d

= 



eα₁



∂

2_eβ 2

∂

x2 +

∂

2_eβ 3

∂

y2 +

∂

2_eβ 4

∂

x

∂

y



− eα2

∂

2_eβ 1

∂

x2 − eα3

∂

2_eβ 1

∂

y2 − 2eα₄

∂

2_eβ 1

∂

x

∂

y+e β 1



∂

2_eα 2

∂

x2 +

∂

2_eα 3

∂

y2 +

∂

2_eα 4

∂

x

∂

y



− eβ₂

∂

2eα1

∂

x2 − e β 3

∂

2_eα 1

∂

y2 − 2e β 4

∂

2_eα 1

∂

x

∂

y



d

.

OncetheGreentheoremhasbeenapplied,therelevantquantitiesexpressedbyEq.(27)popupasfollows:  e T αfβ+eTβfα

!

d

=  ∂



eα₁



∂

eβ₂

∂

x nx+

∂

eβ₃

∂

yny+

∂

eβ₄

∂

ynx+

∂

eβ₄

∂

x ny



+eβ₁



∂

eα₂

∂

x nx+

∂

eα₃

∂

yny+

∂

eα₄

∂

ynx+

∂

eα₄

∂

x ny



− eα2

∂

eβ₁

∂

x nx− e β 2

∂

eα₁

∂

xnx −eα3

∂

eβ₁

∂

yny− e β 3

∂

eα₁

∂

yny− eα4



∂

eβ₁

∂

ynx+

∂

eβ₁

∂

x ny



− eβ4



∂

eα₁

∂

ynx+

∂

eα₁

∂

xny



ds. (36)

−  ∂



wα_tqβn+wβtqαn+

∂

wα_t

∂

n M β nn+

∂

w_tβ

∂

n Mnnα



ds=−  ∂BJ

(

zα,zβ

)

ds. (37)

Thisconcludesthefirstpartoftheproof.

StepIIForthesecond implication,i.e._D⊥_{⊆ D.}Letustake

ω

_α∈D⊥_,

∀

ω

_β∈D. Thenthe bilinearform, oncetheGreen theoremhasbeenapplied,providesthefollowing





eβ₁



f₁α−

∂

2eα2

∂

x2 −

∂

2_eα 3

∂

y2 − 2

∂

2_eα 4

∂

x

∂

y



+eβ₂



f₂α+

∂

2eα1

∂

x2



+eβ₃



f₃α+

∂

2eα1

∂

y2



+ eβ₄



f₄α+2

∂

2_eα 1

∂

x

∂

y



d

+  ∂



wβ_t



∂

eα₂

∂

x nx+

∂

eα₃

∂

yny+

∂

eα₄

∂

ynx+

∂

eα₄

∂

yny



−qβneα1−

∂

wβ_t

∂

x eα2nx−

∂

wβ_t

∂

y eα3ny− eα4



∂

wβ_t

∂

y nx+

∂

wβ_t

∂

x ny



−

∂

eα1

∂

x M β xxnx−

∂

eα₁

∂

yM β yyny −Mβxy



∂

eα₁

∂

ynx+

∂

eα₁

∂

x ny



+qβnz1α+wtβzα2+Mβnnzα3+

∂

wβ_t

∂

n z β 4



ds=0. (38) Sincetherelationhastobevalidforeach

ω

_β∈D thefluxandeffortvariablesareinD.Fortheboundarytermsthesame procedureasbeforehastobeappliedbyconsideringthedefinitionofthemomentaovertheboundary(seeEq.(27)).Then itcanbestatedthat

ω

_{α∈D. }

3.1.2. Includingdissipationandexternalforcesinthemodel

Distributedforcesorcontrol anddissipativerelationscanbeeasilyincludedinanaugmentedStokes-Diracstructureby simplydefiningtheappropriateconjugatedvariables.

Ifdistributedforceshavetobeconsidered,thentheset

Dd:=

(

f,fd,e,ed,z

)

∈F× Fd× E× Ed× Z

|

f =−

∂

_∂

α

t =−Je− Gdfd, ed=G ∗ de, z=B∂

(

e

)



(39) isaStokes-Diracstructurewithrespecttotheparing



(

f1,fd,1e1,ed,1,z1

)

,

(

f2,fd,2e2,ed,2,z2

)

 =  e T 1f2+eT2f1+ed,T1fd,2+eTd,2fd,1

!

d

+  ∂BJ

(

z1,z2

)

ds. (40)

Ifgravityhastobeincluded,thenGd=[1,0,0,0]T,fd=−

μ

g.

AnalogouslydissipationcanbeincludedinanaugmentedDiracstructure.Asanexample,therulingPDE,oncea dissipa-tivetermoffluiddampingtypeisconsidered,reads:

μ∂

_∂

2_tw2 +r

∂

w

∂

t +

∂

2_M xx

∂

x2 +2

∂

2_M xy

∂

x

∂

y +

∂

2_M yy

∂

y2 =0, (41)

wherer>0isthedampingcoefficient.Ifthisequationisrewrittenusingtheport-Hamiltonianformalismthenweget:

∂

α

∂

t =

(

J− R

)

e, R:=

⎡

⎢

⎣

r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎤

⎥

⎦

. (42)

TheRmatrix,whichisasymmetric,semi-positivedefiniteoperator,canbedecomposedas

R=GRSG∗R, (43)

whereS=r isacoerciveoperator(inthiscasesimplyapositivescalar),GR=

(

1000

)

T andG∗Rdenotestheadjointoperator

toGR.Theaugmentedstructure

Dr:=

(

f,f_r

)

∈F,

(

e,er

)

∈E, z∈Z

|

f=−

∂

α

∂

t =−Je− GRfr, fr=−Ser, er=G ∗ Re, z=B∂

(

e

)



(44) isaStokes-Diracstructurewithrespecttotheparing



(

f1,fr,1,e1,er,1,z1

)

,

(

f2,fr,2,e2,er,2,z2

)

 =  e T 1f2+eT2f1+eTr,1fr,2+eTr,2fr,1

!

d

+  ∂BJ

(

z1,z2

)

ds. (45)

Remark4. Moreinvolveddissipationmodelscanbefoundin[24].Morespecifically,forKirchhoff plate,somespecific damp-ingmodelscanbefoundin[25].

3.2.PHtensorialformulationoftheKirchhoff plate

InSection3.1theStokes-DiracstructureoftheKirchhoff platewasfoundbyusingavectorialnotationforthecurvatures andmomenta.Infactthesevariablesareoftensorialnatureandinthefollowingthetensorialformulationtakestheplace ofthevectorialone.Firstletusrewritethemomentaandcurvaturesassymmetricmatrices(correspondingtothechoiceof aCartesianframefortherepresentationoftensors):

K=



κ

xx

κ

xy

κ

xy

κ

yy



, M=



Mxx Mxy Mxy Myy



, (46)

wherenow,withaslightabuseofnotation,

κ

xy differsby 1/2fromthedefinitiongiveninEq.(13),i.e.

κ

xy= ∂

2_w

∂x∂y.Allthe

otherquantitiesstaythesamewithrespecttowhatstatedinSection2.1.TheHamiltonianenergyiswrittenas:

H= 



1 2

μ



∂

w

∂

t



2 +1 2M:K



d

,

(47)

wherethetensorcontractioninCartesiancoordinatesisexpressedas M:K=

2

"

i, j=1

Mi j

κ

i j=Tr

(

MTK

)

.

Forwhatconcernsthechoiceoftheenergyvariables,ascalarandatensorvariableareconsidered:

α

w=

μ∂

w

∂

t , Aκ =K. (48)

Theco-energyvariablesarefoundbycomputingthevariationalderivativeoftheHamiltonian:

ew:=

∂

H

∂

α

w =

∂

w

∂

t :=wt, Eκ :=

∂

H

∂

Aκ =M. (49)

Remark5. Forthevariationalderivativewithrespecttoatensor,seePropostion1in[26]. Theport-Hamiltoniansystem(24)isnowrewrittenas:

⎧

⎨

⎩

∂

α

w

∂

t =−div

(

Div

(

Eκ

))

,

∂

Aκ

∂

t =Grad

(

grad

(

ew

))

, (50)

wheredivandDivdenotethedivergenceofavectorandofatensorrespectively.TheoperatorGraddenotesthesymmetric gradient:

Grad

(

a

)

= 1 2



∇

a+

(

∇

a

)

T



. (51)

TheoperatorGrad_◦gradcorrespondstotheHessianoperator.InCartesiancoordinatesitreads: Grad◦ grad=



_∂2 ∂x2 ∂ 2 ∂x∂y ∂2 ∂y∂x ∂ 2 ∂y2



. (52)

Theorem3. TheoperatorGrad◦grad,correspondingtotheHessianoperator,istheadjointofthedoubledivergencediv◦Div.

Proof.LetusconsidertheHilbertspaceofthesquareintegrablesymmetricsquaretensorsofsizen× noveranopen con-nectedset

.ThisspacewillbedenotedbyH1=L2

(

,Rnsym×n

)

.Thisspaceisendowedwiththeintegralofthetensor

con-tractionasscalarproduct:



E,F



H1=  E:Fd

=  Tr

(

E T_F

₎

_d

_,

_∀

_E,F∈[L2 sym

()

]n×n.

ConsidertheHilbertspace_H2=L2

()

ofscalarsquareintegrablefunctions,endowedwiththeinnerproduct:



e,f



_H2 =



Letusconsiderthedoubledivergenceoperatordefinedas:

A: H1→H2,

E→div

(

Div

(

E

))

=e, withe=div

(

Div

(

E

))

=

n " i=1 n " j=1

∂

2_E i j

∂

xi

∂

xj. WeshallidentifyA∗ A∗: H2→H1, f→A∗f=F, suchthat



AE,f



_H2=



E,A∗f



_H1,

∀

E∈Domain

(

A

)

⊂H1

∀

f∈Domain

(

A∗

)

⊂H2.

Thefunction havetobelong totheoperatordomain,soforinstance f∈C2

0

()

∈Domain

(

A∗

)

thespaceoftwice

differ-entiablescalar functionswithcompactsupport onan open simply connectedset

andadditionallyE canbe chosen in

thesetC2

0

(

,R2sym×2

)

∈Domain

(

A

)

,thespaceoftwicedifferentiable 2× 2 symmetrictensorswithcompactsupporton

.A

classicalresultisthefactthattheadjointofthevectordivergenceisdiv∗=−gradasstatedin[27].Thismaybegeneralized totheadjointofthetensordivergenceDiv∗=−Grad(seeTheorem4of[26]).ConsideringthatAisthecompositionoftwo differentoperatorsA=div◦ Divandthattheadjointofacomposedoperatoristheadjointofeachoperatorinreverseorder, i.e.

(

B_{◦ C}

)

∗₌C∗_{◦ B}∗_,thenitcanbestated

A∗=

(

div◦ Div

)

∗_=Div∗_{◦ div}∗_{=Grad◦ grad.}

Sinceonlyformaladjointsarebeinglookedfor,thisconcludestheproof. 

Ifthevariablesinsystem(50)aregatheredtogethertheformallyskew-symmetricoperatorJcanbehighlighted:

∂

∂

t



α

w Aκ



=



0 −div◦ Div Grad◦ grad 0



&

'(

)

J



ew Eκ



. (54)

whereallzerosareintendedasnullifyingoperatorfromthespaceofinputvariablestothespaceofoutputvariables.

Remark6. TheinterconnectionstructureJnowresemblesthatoftheBernoullibeam.Thedoubledivergenceandthedouble gradientcoincide,indimensionone,withthesecondderivative.

AgaintheboundaryportvariablescanbefoundbyevaluatingthetimederivativeoftheHamiltonian: ˙ H= 



∂

α

w

∂

t ew+

∂

Aκ

∂

t :Eκ



d

= 

{

−div

(

Div

(

Eκ

))

ew+Grad

(

grad

(

ew

))

:Eκ

}

d

, Integrationbyparts

=  ∂

⎧

⎨

⎩

−n

&

· Div

'(

(

Eκ

)

)

qn ew+[ngrad

(

ew

)

]:Eκ

⎫

⎬

⎭

ds, See(27)and(28) =  ∂

⎧

⎨

⎩

qnew+

∂

ew

∂

n

(

&

n

'(

n

)

:Eκ

)

Mnn +

∂

ew

∂

s

(

&

n

'(

s

)

:Eκ

)

Mns

⎫

⎬

⎭

ds, Dyadicproperties =  ∂



qnwt+

∂

wt

∂

nMnn+

∂

wt

∂

s Mns



ds. (55)

Remark7. Thedefinitions

qn=−n· Div

(

Eκ

)

, Mnn=

(

nn

)

:Eκ, Mns=

(

ns

)

:Eκ

areexactlythesameasthosegivenin(27).Thetensorialformalismallowsamorecompactwriting.

The kinematically independent variables must be highlighted. The tangential derivative has to be moved on the

tor-sional momentum. In order to do that, the boundary needs to be split in a collection of regular subsets



i, such that

 ∂

∂

wt

∂

s Mns ds= " i⊂∂  i

∂

wt

∂

s Mns ds = " i⊂∂ [Mnswt]_∂i−  ∂

∂

Mns

∂

s wt ds. (56)

Ifaregularboundaryisconsideredthefinalenergybalanceisexactlythesameastheobtainedwiththevectorial nota-tion,namely: ˙ H=  ∂



wtqn+

∂

wt

∂

n Mnn



ds, whereqn:=qn−

∂

Mns

∂

s . (57)

Thetensorialformulationallows highlightingtheintrinsicdifferentialoperators.Furthermorethesymmetricnature ofthe variablesis explicitlyexpressedby theusage ofsymmetrictensors.Now that theenergybalance hasbeenestablished in termsoftheboundaryvariablestheStokes-DiracstructurefortheKirchhoff plateintensorialformcanbedefined.Consider

nowthebondspace:

B:=

{

(

f,e,z

)

∈F× E× Z

}

, (58)

whereF=L2

₍₎

_:=L2

₍₎

_{× L}2

₍

_,R2×2

sym

)

andE=H2

()

=H2

()

× HdivDiv

(

,Rsym2×2

)

.ThespaceHdivDiv

(

,R2sym×2

)

issuch

that

HdivDiv

_(,

_R2×2 sym

)

=

.

A∈L2

_(,

_R2×2

sym

)

|

div

(

Div

(

A

))

∈L2

()

/

.

Considerthespaceofboundaryportvariables:

Z:=



z

|

z=



f_∂ e_∂



, with f_∂=



wt ∂wt ∂n



, e_∂=



 qn Mnn



. (59)

ThedualitypairingbetweenelementsofB isthendefinedasfollows:





(

f₁,e1,z1

)

,

(

f2,e2,z2

)





:=



e1,f2



L2₍₎+



e2,f1



L2₍₎+



∂BJ

(

z1,z2

)

ds, (60)

wherethepairing



·,·



L2₍₎istheL2innerproductonspaceL2

()

andBJ

(

z1,z2

)

:=

(

f_∂,1

)

Te∂,2+

(

f∂,2

)

Te∂,1.

Theorem4(Stokes-DiracStructurefortheKirchhoff plateintensorialform). ConsiderthespaceofpowervariablesB defined in(58)andthematrixdifferentialoperatorJin(54).ByTheorem2in[26]thelinearsubspaceD⊂ B

D=



(

f,e,z

)

∈B

|

f =−

∂

α

∂

t =−Je, z=



f_∂ e_∂



, (61)

isaStokes-Diracstructurewithrespecttothepairing



· , ·



givenby(60).

4. DiscretizationoftheKirchhoff plateusingaPartitionedFiniteElementMethod

Followingtheprocedureillustratedin[9]theKirchhoff platewrittenasaport-Hamiltoniansystemcanbediscretizedby usingaPartitionedFiniteElementMethod(PFEM).ThismethodisanextensionoftheMixedFiniteElementMethodtothe caseofpHsystemsandrequirestheintegrationbypartstobeperformed,sothatthesymplectic structureispreserved. It consistsofthreedifferentsteps:

1.thesystemisfirstputintoweakform;

2.oncetheboundarycontrolofinterestisselected,thecorrespondingsubsystemisintegratedbyparts; 3.theproblemisdiscretizedbyusingaMixedFiniteElementmethod.

Theweakformisillustratedusingthetensorialformulation.Twodifferentkindofboundarycontrolswillbeshown: 1.boundarycontrolthroughforcesandmomenta,inthiscasethefirstlineof(54)isintegratedbyparts(inSection4.1.1); 2.boundarycontrolthroughkinematicvariables,inthiscasethesecondlineof(54)isintegratedbyparts(inSection4.1.2).

4.1. Weakform

Thesameproceduredetailedabovecanbeusedonsystem(54).Inthiscasethetestfunctionsareofscalarortensorial nature.KeepingthesamenotationthaninSection3.2thescalartestfunctionisdenotedbyvw,thetensorialonebyVκ.

4.1.1. Boundarycontrolthroughforcesandmomenta

Thefistlineof(54)ismultipliedbyvw(scalarmultiplication),thesecondlinebyVκ (tensorcontraction).



v

w

∂

α

w

∂

t d

=  −

v

wdiv

(

Div

(

Eκ

))

d

,

(62)  Vκ:

∂

Aκ

∂

t d

=  Vκ :Grad

(

grad

(

ew

))

d

.

(63)

TherighthandsideofEq.(62)hastobeintegratedbypartstwice:  −

v

wdiv

(

Div

(

Eκ

))

d

=  ∂−n

&

· Div

'(

(

Eκ

)

)

qn

v

wds+  grad

(

v

w

)

· Div

(

Eκ

)

d

(64)

Applyingagaintheintegrationbypartsleadsto:  grad

(

v

w

)

· Div

(

Eκ

)

d

=  ∂grad

(

v

w

)

·

(

n· Eκ

)

ds−  Grad

(

grad

(

v

w

))

:Eκd

(65)

Theusualadditionalmanipulationisperformedontheboundarytermcontainingthemomenta,sothattheproperboundary valuesarise:  ∂grad

(

v

w

)

·

(

n· Eκ

)

ds=  ∂



∂

v

w

∂

n n+

∂

v

w

∂

s s



·

(

n· Eκ

)

ds =  ∂

⎧

⎨

⎩

∂

v

w

∂

n

&

(

n

'(

n

)

:Eκ

)

Mnn +

∂

v

w

∂

s

(

&

n

'(

s

)

:Eκ

)

Mns

⎫

⎬

⎭

ds = " i⊂∂ [Mns

v

w]_∂i+  ∂



∂

v

w

∂

n Mnn−

v

w

∂

Mns

∂

s



ds (66)

CombiningEqs.(64)–(66)thefinal expressionwhichmakes appearthedynamicboundary terms(forcesandmomenta)is

found: 

v

w

∂

α

w

∂

t d

=−  Grad

(

grad

(

v

w

))

:Eκd

+  ∂



∂

v

w

∂

n Mnn+

v

wqn



ds+ " i⊂∂ [Mns

v

w]_∂i. (67)

Iftheboundaryisregular,thefinalexpressionsimplifies: 

v

w

∂

α

w

∂

t d

=−  Grad

(

grad

(

v

w

))

:Eκd

+  ∂



∂

v

w

∂

n Mnn+

v

wqn



ds. (68)

Sothefinalweakformobtainedfromsystem(54)iswrittenas:

⎧

⎪

⎪

⎨

⎪

⎪

⎩



v

w

∂

α

w

∂

t d

=−  Grad

(

grad

(

v

w

))

:Eκd

+  ∂



∂

v

w

∂

n Mnn+

v

wqn



ds,  Vκ:

∂

Aκ

∂

t d

=  Vκ:Grad

(

grad

(

ew

))

d

.

(69)

Thecontrolinputsu_∂ andthecorrespondingconjugateoutputsy_∂ are:

u_∂=



 qn Mnn



∂ , y_∂=



wt

∂

wt

∂

n



∂ .

4.1.2. Boundarycontrolthroughkinematicvariables

Alternatively, the sameprocedure can be performed on thesecond lineof the systemto make appear thekinematic

boundaryconditions,i.e.thevalue ofthe verticalvelocityandits normalderivative alongthe border.Oncethenecessary calculationsarecarriedout,thefollowingresultisfound:

⎧

⎪

⎪

⎨

⎪

⎪

⎩



v

w

∂

α

w

∂

t d

=  −

v

wdiv

(

Div

(

Eκ

))

d

,

 Vκ :

∂

Aκ

∂

t d

=  div

(

Div

(

Vκ

))

ewd

+  ∂



v

Mnn

∂

wt

∂

n +

v

qnwt



ds. (70)

where

v

Mnn=

(

n n

)

:Vκ and

v

qn=−Div

(

Vκ

)

· n−

∂

v

Mns

∂

s with

v

Mns=

(

n s

)

:Eκ.The control inputsu∂ andthe

corre-spondingconjugateoutputsy_∂ are:

u_∂=



wt

∂

wt

∂

n



∂ , y_∂=



qn Mnn



∂ .

4.2.Finite-dimensionalport-Hamiltoniansystem

Inthissection,thediscretizationprocedureisappliedtoformulation(69).Thesameproceduremaybeperformedusing formulation(70).InSection5.2bothstrategieswillbeusedtocomputetheeigenvaluesofasquareplate.

Testandco-energyvariablesarediscretizedusingthesamebasisfunctions(GalerkinMethod):

v

w= Nw " i=1

φ

i w

(

x,y

)

v

iw, Vκ= Nκ " i=1



i κ

(

x,y

)

v

iκ, ew= Nw " i=1

φ

i w

(

x,y

)

eiw

(

t

)

, Eκ = Nκ " i=1



i κ

(

x,y

)

eiκ

(

t

)

, (71)

Thebasisfunctions

φ

i

w,



iκ,havetobechoseninasuitablefunctionspaceVhinthedomainofoperatorJ,i.e.Vh⊂ V∈D

(

J

)

.

ThiswillbediscussedinSection5.Thediscretizedskew-symmetricbilinearformontherightsideof(69)thenyields:

Jd=



0 −DT H DH 0



. (72)

MatrixDHiscomputedinthefollowingway:

DH

(

i,j

)

=





i

κ:Grad

(

grad

(

φ

wj

))

d

,

∈RNκ×Nw, (73)

wherethenotationA(i,j)indicatestheentryinthematrixcorrespondingtotheithrowandjthcolumn.Theenergyvariables arededucedfromtheco-energyvariables:

α

w=

μ

ew, Aκ =D−1Eκ, (74)

where_{Di jkl} is thesymmetric bendingrigidity tensor,the tensorial analogousofmatrixD defined in(15).The symmetric bilinearformontheleftsideof(69)becomes:

M=diag[Mw,Mκ], with Mw

(

i,j

)

= 

μ φ

i w

φ

j wd

,

∈RNw×Nw, M_κ

(

i,j

)

= 



D−1

_

i κ



:



κjd

,

∈RNκ×Nκ. (75)

Theboundaryvariablesarethendiscretizedas: qn= Nqn " i=1

φ

i  qn

(

s

)

q i n, Mnn= N"Mnn i=1

φ

i Mnn

(

s

)

M i nn. (76)

Thevariablesaredefinedonlyovertheboundary

∂

.Consequently,theinputmatrixreads:

B=



Bqn BMnn 0 0



. (77)

Bqn

(

i,j

)

=  ∂

φ

i w

φ

qjnds, ∈R Nw×Nqn, BMnn

(

i,j

)

=  ∂

∂

φ

i w

∂

n

φ

j Mnnds, ∈R Nw×NMnn. (78)

Thefinalport-Hamiltoniansystem,asdefinedin[28]iswrittenas:

Me˙=Jde+Bu∂, y_∂=BTe, (79) wheree₌



e1 w,...,eNκκ



T andu_∂=

1

q1 n,...,M NMnn nn

2

T

aretheconcatenationsofthedegreesoffreedomforthedifferent vari-ables.ThediscreteHamiltonianisthenfoundas:

Hd= 1 2 

{

α

wew+Aκ :Eκ

}

d

= 1 2

.

eT wMwew+eT_κMκeκ

/

= 1 2e T_Me. (80)

UsingEqs.(79)and(80)thetimederivativeoftheHamiltonianisgivenbythescalarproductoftheboundaryflows: ˙

Hd=yT_∂u∂. (81)

TheaboveEquationisequivalenttotheenergybalanceofthecontinuoussystem,expressedby(31).Definition(80),together withsystem(79)arethe finite-dimensionalequivalentof(47)and(54).Thediscretized systemobtainedviaPFEMshares theport-Hamiltonianstructureoftheoriginalinfinite-dimensionalsystem, thediscretizationmethodisthereforestructure preserving.

5. Numericalstudies

InthissectionweillustratenumericallytheconsistencyofdiscretemodelobtainedwithPFEM.Forthispurpose compu-tationoftheeigenvaluesofasquareplateandtime-domainsimulationsforseveralboundaryconditionsarepresented.

5.1. Finiteelementchoice

ThedomainoftheoperatorJin(54)isD

(

J

)

=H2

₍₎

_{× H}divDiv

₍

_,R2×2

sym

)

andboundaryconditions.

Remark8. It hasto beappointedthat,tothebestofauthors’knowledge,thespaceHdivDiv

₍

_,R2×2

sym

)

hasnever addressed

inthemathematicalliterature.ForthisreasonH2₍

_{) conformingfiniteelementswereusedtodealwiththisproblem}

nu-merically.

Asuitablechoiceforthefunctionalspaceisthus:

(

v

w,Vκ

)

∈H2

()

× H2

(,

R2sym×2

)

≡H, (82)

since H ⊂ D

(

J

)

.The H2 _{conformingfinite elements(like the Hermite, Bellor Argyrisfinite elements)do not satisfy the}

properequivalence propertiestogive asimple relationshipbetweenthereferencebasis andnodal basison ageneralcell

[29].TheFiredrakelibrary[11]wasusedtoimplementthenumericalanalysisasitprovidesfunctionalitiestoautomatethe generalizedmappingsfortheseelements.

ThenfortheFiniteElementchoice,denote

Hk

r

(

Pl,

)

=

{

v

∈Hk

()

|

v

|T∈Pl

∀

T∈Tr

}

thefiniteelementspacewhichisasubspaceofHk₍

_{),basedontheshapefunctionspaceofpiecewisepolynomialsofdegree}

l.TheshapefunctionspaceisdefinedoverthemeshTr=-iTi,wherethecellsTiaretriangles.Thesespacescanbe

scalar-valuedorsymmetricmatrixvalued, dependingonthevariables tobediscretized.Theparameterristheaveragesizeofa meshelement.Allthevariables,i.e.thevelocityewandthemomentatensorVκ aswellasthecorrespondingtestfunctions,

arediscretizedbythesamefiniteelementspace,theBellfiniteelementspace[30],denotedH2

r

(

P5,

)

.Forthiselementthe

field iscomputedusingquinticpolynomialswhosedegreesoffreedom arethevaluesofthefunction,its gradientandits Hessianatthevertexofeach triangularelement.TodealwithmixedboundaryconditionsLagrange multipliershavetobe introduced(the readercanrefer to[26],Section4.3foran explanation).Themultipliersarethereforediscretizedbyusing

seconddegreeLagrangepolynomialsdefinedovertheboundaryH1

Table 1

Eigenvalues obtained with 5 Bell element per side for ν= 0 . 3 , considering either the Grad ◦grad formula- tion (69) , either the Div ◦div formulation (70) . For comparison reference [31] is considered. reference,

ε< 0.1%:

Table 2

Eigenvalues obtained with 5 Bell element per side for ν= 0 . 3 , considering either the Grad ◦grad formulation

(69) , either the Div ◦div formulation (70) . For comparison reference [31] is considered: reference, ε< 0.1%,

ε< 1%, ε< 5%, ε< 15%.

5.2.Eigenvaluescomputation

The test case for this analysis is a simple square plate of side L, a benchmark problem which has been studied in

[31,32]for different boundary conditions on each plate side. The possible cases are the following:

• clampedside(C),forwhichwt=0, ∂_∂w_nt=0; • simplysupportedside(S),wt=0,Mnn=0; • freeside(F),qn=0,Mnn=0.

Inordertocompareourresultstheeigenfrequencies

ω

h

narecomputedinthefollowingnon-dimensionalform:

3

ω

h n=4L2

ω

hn



ρ

h D



1/2 , (83)

Theonly parameterwhich influences theresultsis thePoisson’sratio

ν

=0.3.The reportednon-dimensionalfrequencies areindependentoftheremaininggeometricalandphysicalparameters.Theerroriscomputedas:

ε

= abs

(

ω

3hn−

ω

Ln

)

ω

L n

, (84)

where

ω

L

n are the eigenvalues computed in [31]. The results are computed either by using the forces and momenta as

control(69)ortheverticallinearandangularvelocity(70) (columnHessiananddivDivinTables1and2).Theresultsare obtainedusingaregular meshcomposed by5Bellelementoneachside.Hence,thestate vectorhasatotaldimensionof 864.ThedimensionoftheLagrangemultipliervectordependsontheboundaryconditionsuponconsideration.Whenusing

H2

r

(

P2,

∂

)

ontheconsideredmesh,thisnumbercanvaryfrom0to80.Theresultsobtainedbyusing(69)areinperfect

agreementwiththe reference. Thisformulationwas alsoused tocompute the eigenvectorscorresponding to the vertical velocityforthedifferentcasesunderexamination(seeFigs.4–9).Forwhatconcernstheweakformulation(70)theresults deterioratewhenafreecondition(seeTable2)ispresent.

5.3.Time-domainsimulations

Inthisanalysisweconsider asquare plate,subjecteitherto anon nullshearforce ontheboundarieseithertoa dis-tributedforceoverthedomain.ThephysicalparametersandsimulationsettingsarereportedinTable3.Theenergyvariables andLagrangemultipliersarediscretizedusingBellshapefunctions(regularmeshoffiveelementsforeachside)andsecond