Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates



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an author's

https://oatao.univ-toulouse.fr/23900

https://doi.org/10.1016/j.apm.2019.04.035

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

formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. (2019) Applied

Mathematical Modelling, 75. 940-960. ISSN 0307-904X

Port-Hamiltonian

formulation

and

symplectic

discretization

of

plate

models

Part

I:

Mindlin

model

for

thick

plates

Andrea

Brugnoli

∗

,

Daniel

Alazard

,

Valérie

Pommier-Budinger

,

Denis

Matignon

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

a

r

t

i

c

l

e

i

n

f

o

Keywords:

Port-Hamiltonian systems Mindlin–Reissner plate

Partitioned Finite Element Method Geometric spatial discretization Boundary control

a

b

s

t

r

a

c

t

Theport-Hamiltonian formulationisapowerfulmethodformodelingand interconnect-ingsystemsofdifferentnatures.Inthispaper,theport-Hamiltonianformulationin tenso-rialformofathickplatedescribedbytheMindlin–Reissnermodelispresented.Boundary controlandobservationaretakenintoaccount.Thankstotensorialcalculus,itcanbeseen thattheMindlinplatemodelmimicstheinterconnectionstructureofitsone-dimensional counterpart,i.e.theTimoshenkobeam.

ThePartitionedFiniteElementMethod(PFEM)isthenextendedtoboththevectorial and tensorial formulationsinorder toobtainasuitable,i.e.structure-preserving, finite-dimensionalport-Hamiltoniansystem(PHs),whichpreservesthestructureandproperties oftheoriginaldistributedparametersystem.Mixedboundaryconditionsarefinally han-dledbyintroducingsomealgebraicconstraints.

Numericalexamplesarefinallypresentedtovalidatethisapproach.

Introduction

TheHamiltonianformalismarisingfromtheLegendretransformationofEuler–Lagrangesystemhasbeenalreadywidely explored [1].The Legendre transformation gives riseto a system ofequations ruled by a Hamiltonianmatrix ona sym-plecticspace.Port-Hamiltoniansystems(PH)areinsteaddefinedwithrespecttoaHamiltonianoperator [2,Chapter7].For a complete discussion onasthe relation betweenLagrangian andHamiltonianformulation the readermayconsult [3,4]. ThePHframeworkisacquiringmorevisibilityforitscapabilitytorepresentahugeclassofsystemscomingfromdifferent realms of physics in a modular way.Finite-dimensional port-Hamiltonian systems can be easily interconnectedtogether, asshown in [5],allowing the construction ofcomplex multi-physics systems.The interconnection is possiblealso inthe infinite-dimensionalcase [6], eveniftheprocedureisnot asstraightforwardasinthefinite-dimensionalcase. Eventually, itisalsopossibletomergefiniteandinfinitePHsystems [7].Thesefeaturesandcapabilitiesareparticularlyappealingfor controlengineersinordertosimplifythemodelingtaskinpreliminaryanalyses.1,2

Distributed parameter systems are of relevantinterest giventhe increasedcomputational poweravailable for simula-tions.PHdistributedsystemswereinitiallypresentedin [8],byusingthetheory ofdifferentialforms.Linkstowards func-tionalanalysishavebeenmadein [9]andanexhaustivereferenceaboutthesubjectcanbefoundin [10].Thefundamental

∗ _{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 PFEM stands for Partitioned Finite Element Method. 2 PHs stands for port-Hamiltonian systems.

featureofadistributedPHsystemistheunderlyinggeometricinterconnection structure,theso-calledStokes–Dirac struc-ture, that describes the powerflow across the boundary andinside the system, together withan energyfunctional, the Hamiltonian, that determines the nature of the system. Linear/nonlinear, parabolic/hyperbolic systems can be all recast intothisframework, [11].Port-Hamiltonian systemsarebydefinitionopen systems,abletointeractwiththeenvironment through boundaryports. The definitionsof theseboundary variables is ofutmost importance to show that a PHsystem iswellposed(see [12]fortheproofinthe1D case).Applicationscomingfromcontinuummechanics,electrodynamicsand thermodynamicscanbeintegratedinsidethisframework.Academicexamplestypicallyconsideredarethetransmissionline, theshallowwaterequationsandtheMaxwellequations [8].

Numericalsimulations andcontroltechniques requireaspatialdiscretizationthat is meantto preservethe underlying propertiesrelatedtopowercontinuity.ThediscretizationprocedureforsystemsunderPHformalismconsistsoftwosteps:

• Finite-dimensional approximation ofthe Stokes–Dirac structure,i.e.the formally skew symmetric differentialoperator that defines the structure. The duality of the power variables has to be mapped onto the finite approximation. The subspaceofthediscretevariableswillberepresentedbyaDiracstructure.

• TheHamiltonianrequiresaswellasuitablediscretization,whichgivesrisetoadiscreteHamiltonian.

Theresearchcommunityisfocusingonstructure-preservingdiscretizationtechniquessinceseveralyears.In [13],the au-thorsmadeuseofamixedfiniteelementspatialdiscretizationfor1Dhyperbolicsystemofconservationlaws,introducing differentlow-orderbasisfunctionsfortheenergyandco-energyvariables.Pseudo-spectralmethodsrelyingonhigher-order globalpolynomialapproximationswerestudiedin [14].Thismethodwasusedandextendedtotakeintoaccountunbounded controloperatorsin [15].Morerecentlyasimplicialdiscretizationbasedondiscreteexteriorcalculus wasproposedin [16]. Thisapproachcomeswithadditionalcomplexities,sinceaprimalandadualmesheshavetobedefinedbutthe discretiza-tionisstructure-preserving, regardlessofthespatialdimension oftheproblem.WeakformulationswhichleadtoGalerkin numericalapproximationsbegan to beexplored in thelast years. In [17] the prototypicalexample ofhyperbolic systems oftwo conservation lawwasdiscretized by a weak formulation. Inthis approach theboundary is split accordingto the causalityofboundaryports,sothatmixedboundaryconditionsareeasilyhandled,butstilldualmesheshavetobedefined. The symplectic approach based onthe Legendre transformation has beenused to deal withthe Mindlin platemodel in [18].It has beencapable of providinginsights on analytical solutions forthe eigenproblemof rectangular plates.This methodologyisofusewhenevereasyengineeringsolutionsaresoughtafter.Onthecontrary,theport-Hamiltonian frame-work is a powerful tool whenever complex systems have to be modeled in a structured way. The main contribution of this paper is the enrichment of the PH formulation for the Mindlin plate model, by making use of tensor calculus [19, Chapter16].This kindof modelwasalready presentedin [20] andusing the jet bundle formalism in [21] buthere anovelformulationbasedontensorialcalculusisintroduced.Thesecondcontributionofthispaperconcernstheextension ofthePartitionedFiniteElementMethod(PFEM)totheMindlinplatemodel.Inthisapproach,originallypresentedin [22], oncethesystemhasbeenputintoweakform,asubsetoftheequationsisintegratedbyparts,sothatboundaryvariables arenaturallyincludedintotheformulationandappearascontrolinputs,thecollocatedoutputs beingdefinedaccordingly. Then,the discretizationofenergyandco-energy variables(andthe associatedtest functions)leadsdirectlyto afull rank representationforthefinite-dimensionalport-Hamiltoniansystem.Ifmixedboundaryconditionsaretobeconsidered, the finite-dimensionalsystemcontainsconstraints,leadingtoanalgebraicdifferentialsystem(DAE),whichcanbe treatedand analyzedbyreferringto [23,24].Thisapproach,similarlytotheonedetailedin [17],makespossibletheusageofFEM soft-ware,likeFEniCS [25].Thefinaldiscretizedsystemcanbefurtherreducedbyusingappropriatemodelreductiontechniques, asexplainedin [26,27].

Thepaperisdividedintofivemainsections

1.The framework of finite-dimensionalport-Hamiltonian systems(PHs) andthe notionof Diracand Stokes–Dirac struc-tures are recalled. The infinite-dimensionalcaseis thenillustrated by means ofthe Timoshenko beammodel, the1D counterpartoftheMindlinplatemodel.

2.Theconstitutiverelations,equationsofmotionsandenergiesfortheMindlin–Reissnerplatearedetailed.

3.TheMindlinplateport-Hamiltonianformulationishighlightedbydefiningenergyandco-energyvariablesandthe inter-connectionstructure.Theboundaryvariablesarefoundbyintroducingtheenergybalance.ThentheunderlyingStokes– Diracstructureiseasilyobtainedbydefiningtheflowandeffortspaces,togetherwiththespaceofboundaryvariables. 4.ThePFEMdiscretizationfortheMindlinplateisdetailed.Theproblemisputintoweakformfirst,thenthenecessary

in-tegrationsbypartsareperformedandfinallythebasisfunctionsfortheenergy,co-energyandtestfunctionsarechosen. 5.To demonstrate the consistency ofthe approach an eigenvalue studyof a square plateconsidering various boundary

conditionsisperformed,togetherwithsometime-domainsimulation.

1. Reminderonport-Hamiltoniansystems

Inthissection the concepts ofDiracstructure andStokes–Diracstructure are recalled. From theseconcepts finiteand infinitedimensionalport-Hamiltoniansystemarethenderivedbyconsideringtheportbehavior.

1.1. Diracstructures

Considerann-dimensionalspaceF overthefieldRandE≡ Fitsdual,i.e.thespaceoflinearoperatore:F→R.The elementsof_{F are}calledflows,whiletheelementsof_{E are}calledefforts.Thoseare portvariablesandtheir combination givesthepowerflowinginsidethesystem. ThespaceB=F× E iscalledthebondspaceofpowervariables.Thereforethe powerisdefinedas



e,f



=e

(

f

)

,where



e,f



isthedualproductbetweenfande.

Definition 1([28],Definition1.1.1). GiventhespaceF and itsdualE with respecttotheinner product



·,·



:F× E→R,

definethesymmetricbilinearform:





(

f1, e1

)

,

(

f2, e2

)





:=



e1, f2



+



e2, f1



, with

(

fi, ei

)

∈B, i =1, 2 (1)

ADiracstructureon_B:_{=F× E is}asubspace_{D⊂ B,}whichismaximallyisotropicunder



_{· , ·}



. Thisdefinitioncanbeextendedtoconsiderdistributedforcesanddissipation [9].

Proposition 1. Consider the space of power variables F× E and let X denote an n-dimensional space, the space of energy variables.SupposethatF:=

(

Fs, Fe

)

andthatE:=

(

Es, Ee

)

,withdimFs=dimEs=nanddimFe=dimEe=m.Moreover,let

J(x)denoteaskew-symmetricmatrixofdimensionnandbyB(x)amatrixofdimensionn× m.Then,theset

D :=



(

fs, fe, es, ee

)

∈ F × E

|

fs=−J

(

x

)

es− B

(

x

)

fe, ee=B

(

x

)

Tes



(2)

isaDiracstructure.

1.2. Finite-dimensionalPHs

Considerthetime-invariantdynamicalsystem:



˙

x=J

(

x

)

∇

H

(

x

)

+B

(

x

)

u,

y=B

(

x

)

T

∇

_H

₍

_x

₎

_, (3)

where H

(

x

)

:X→R, the Hamiltonian, is a real-valued function bounded from below. Such system is called port-Hamiltonian,asitarisesfromtheHamiltonianmodellingofaphysicalsystemanditinteractswiththeenvironmentthrough theinputuincludedintheformulation.TheconnectionwiththeconceptofDiracstructureisachievedbyconsideringthe followingportbehavior:

fs=− ˙x, es=

∂

H

∂

x,

fe=u, ee=y.

(4)

Withthis choice oftheport variables system (3) defines,by Proposition 1,a Diracstructure. Dissipationanddistributed forcescan beincluded andthecorresponding systemdefinesanextended Diracstructure,once theproperport variables areintroduced.

1.3. Constantmatrixdifferentialoperators

Thetreatmentprovidedin [29]ishererecovered,inordertohavethemostsuitabledefinitionofStokes–Diracstructure forthemechanicalsystemsconsideredinthispaper.Let



denoteacompactsubsetofRd_{representingthespatialdomainof}

thedistributedparametersystem.Then,let_{U andV denote}twosetsofsmoothfunctionsfrom



to_Rqu _andRqv_{respectively.} Definition2. AconstantmatrixdifferentialoperatoroforderNisamapLfromU toV suchthat,givenu=

(

u1,...,uqu

)

∈U

and v =

(

v

1,...,

v

qv

)

∈V:

v

= L u ⇐⇒

v

:= N  |α|=0 P_αD αu, (5)

where

α

:₌

(

α

1,...,

α

d

)

isamulti-index oforder

|

α|

:=di=1

α

i,Pα area setofconstant realqv× qu matricesandDα:=

∂

α1

x1 ...

∂

αd

x_d isadifferentialoperatoroforder|

α

|resultingfromacombinationofspatialderivatives.

Definition3. Considertheconstantmatrixdifferentialoperator (5).ItsformaladjointisthemapL∗fromV toU suchthat:

u= L ∗

v

⇐⇒ u:= N  |α|=0

(

−1

)

|α|_PT αD α

v

. (6)

Definition 4. LetJ denote a constant matrix differential operator. Then, Jis skew-symmetric if andonly if J=−J∗_{. This}

correspondstothecondition:

P_α=

(

−1

)

|α|+1_PT

Animportantrelationbetweenadifferentialoperatoranditsadjointisexpressedbythefollowingtheorem.

Theorem1([30],Chapter9,Theorem9.37). ConsideramatrixdifferentialoperatorLandletL∗denoteitsformaladjoint.Then, foreachfunctionu∈U and v ∈V:

 



v

T_{L u− u}T_L ∗

_v

_d

_

₌ ∂ B L

(

u,

v

)

dA, (8)

whereBLisadifferentialoperatorinducedontheboundary

∂



byL,orequivalently:

v

T_{L u− u}T_L ∗

_v

_{=divB L}

₍

_u,

_v

₎

_{. (9)}

Itisimportanttonote thatBL isa constantdifferentialoperator.ThequantityBL

(

u,

v

)

isaconstant linearcombination

ofthefunctionsuandvtogetherwiththeirspatialderivativesuptoacertainorderanddependingonL.

Corollary1. Consideraskew-symmetricdifferentialoperatorJ.Then,foreachfunctionu∈U and v ∈V withqu=qv=q:

 



v

T_{J u+u}T_J

_v

_d

_

₌  ∂ B J

(

u,

v

)

dA, (10)

whereBJisasymmetricdifferentialoperatoron

∂



dependingonthedifferentialoperatorJ.

1.4.ConstantStokes–Diracstructures

Followingagain [29] let_{F denote}the spaceofflows,i.e.thespaceofsmooth functionsfromthecompactset



_{⊂ R}d

toRq_{. Forsimplicityassumethat the spaceofefforts isE≡ F (generallyspeaking thesespacesare Hilbertspaceslinked}

byduality,asin [9]).Given f=

(

f1,...,fq

)

∈F ande=

(

e1,...,eq

)

∈E.Letz=B_∂

(

e

)

denotetheboundaryterms,whereB_∂

providestherestrictionon

∂



oftheefforteandofitsspatialderivativesofproperorder.Thenitcanbewritten:

 ∂ B J

(

e1, e2

)

dA =  ∂B J

(

z1, z2

)

dA, with B J

(

·,·

)

= B J

(

B ∂

(

·

)

, B ∂

(

·

))

. (11)

Definetheset

Z :=

{

z

|

z= B _∂

(

e

)

}

. (12)

Theorem2. [29]ConsiderthespaceofpowervariablesB=F× E× Z.ThelinearsubspaceD⊂ B

D =

{

(

f, e, z

)

∈ F × E× Z

|

f =−Je, z = B _∂

(

e

)

}

, (13)

isaStokes–DiracstructureonB withrespecttothepairing





(

f1, e1, z1

)

,

(

f2, e2, z2

)





:=  



eT 1f2+eT2f1

d



+  ∂B J

(

z1, z2

)

dA. (14)

Remark1. Theconstant Stokes–Diracstructurehasbeendefinedincaseofsmoothvector valuedfunctionsforsimplicity. InthiscontextthepairinghasbeendefinedastheL2_{innerproductofvector-valuedfunction.Thedefinitionisindeedmore}

generalandencompassesthecaseofmorecomplexfunctionalspaces.TheMindlinplateforexampleisdefinedonamixed functionspaceof scalar-,vector- andtensor-valued functions.The resultpresentedhereremains validprovided that the properpairingisbeingchosen.Furthermore,theconstantdifferentialoperatormaycontainintrinsicoperators(div,grad)as itwillbeshownfortheMindlinplatecase.

1.5.Infinite-dimensionalPHs

Following [10,Chapter3],theprototypicalexampleoftheTimoshenkobeamwillbeusedto illustratetheclassof dis-tributedport-Hamiltoniansystems.ThismodelconsistsoftwocoupledPDEs,describingtheverticaldisplacementand rota-tionscalarfields:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

ρ

(

x

)

∂

2_w

∂

t 2

(

x, t

)

=

∂

∂

x



K

(

x

)



∂

w

∂

x

(

x, t

)

−

φ

(

x, t

)



, x ∈

(

0, L

)

, t ≥ 0 I _ρ

(

x

)

∂

2

φ

∂

t 2

(

x, t

)

=

∂

∂

x



EI

(

x

)

∂φ

∂

x

(

x, t

)



+K

(

x

)



∂

w

∂

x

(

x, t

)

−

φ

(

x, t

)



, (15)

wherew(x,t)isthetransversedisplacementand

φ

(x,t)istherotationangleofafiberofthebeam.Thecoefficients

ρ

(x),

I_ρ(x),E(x),I(x) andK(x) arethemassperunit length,therotary momentofinertia ofa crosssection,Young’smodulus of elasticity,themomentofinertiaofacrosssectionandtheshearmodulus,respectively.Theenergyvariablesarechosenas

follows:

α

w:=

ρ

(

x

)

∂

w

∂

t

(

x, t

)

, LinearMomentum,

α

φ:=I ρ

(

x

)

∂φ

∂

t

(

x, t

)

, AngularMomentum,

α

κ:=

∂φ

_∂

_x

(

x, t

)

, Curvature,

α

γ :=

∂

_∂

w _x

(

x, t

)

−

φ

(

x, t

)

, ShearDeformation. (16)

Thosevariablesare collectedinthevector

α

:_{= (}

α

w,

α

_φ,

α

κ,

α

γ)T,sothattheHamiltoniancanbe writtenasaquadratic

functionalintheenergyvariables:

H =1 2 L 0

α

T_Q

_α

_{dx, where Q =}

⎡

⎢

⎣

1 ρ(x) 0 0 0 0 1 Iρ(x) 0 0 0 0 EI

(

x

)

0 0 0 0 K

(

x

)

⎤

⎥

⎦

. (17)

Theco-energyvariablesarefoundbycomputingthevariationalderivativeoftheHamiltonian(see [31]): e w:=

δ

H

δα

w =

∂

w

∂

t

(

x, t

)

, LinearVelocity, e _φ:=

δ

H

δα

φ =

∂φ

∂

t

(

x, t

)

, AngularVelocity, e _κ :=

δ

H

δα

κ = EI

(

x

)

∂φ

∂

x

(

x, t

)

, FlexuralMomentum, e _γ :=

δ

H

δα

γ = K

(

x

)



∂

w

∂

x

(

x, t

)

−

φ

(

x, t

)



, ShearForce. (18)

These variables are again collected in the vector e=

(

ew,e_φ,eκ,eγ

)

T, so that system (15) can be rewritten in terms of

energyandco-energyvariables:

∂

α

∂

t = Je , where J =

⎡

⎢

⎣

0 0 0 _∂∂_x 0 0 _∂∂_x 1 0 _∂∂_x 0 0 ∂ ∂x −1 0 0

⎤

⎥

⎦

. (19)

ByDefinition (4),theconstantmatrixdifferentialoperatorJisskew-symmetric.InordertounveiltheStokes–Diracstructure forthismodel,boundaryvariableshavetobedefined.Theenergyrateisgivenby(see [10]foradetailedderivation):

˙ H = fT_∂e_∂, (20) where f_∂=



e w

(

0

)

e _φ

(

0

)

e γ

(

L

)

e κ

(

L

)



T , e_∂=



−eγ

(

0

)

−eκ

(

0

)

e w

(

L

)

e φ

(

L

)



T . (21)

Considernowthepowerspace

B :=

{

(

f, e, z

)

∈F × E× Z

}

, (22) whereF≡ E=L2

₍₍

_0,L

₎

_{; R}4

₎

_and Z :=



z

|

z=



f_∂ e_∂



. (23)

ThedualitypairingbetweenelementsofB isthendefinedasfollows:





(

f1, e1, z1

)

,

(

f2, e2, z2

)





:= L 0



eT 1f2+eT2f1

dx +B J

(

z1, z2

)

, (24)

whereBJ

(

z1,z2

)

:=

(

f∂,1

)

Te∂,2+

(

f∂,2

)

Te∂,1.Itisnowpossibletostatethefollowingtheorem:

Theorem3 (Stokes–Dirac structure fortheTimoshenko beam). Consider thespaceof powervariables B defined in (22).By

Theorem2thelinearsubspace_{D⊂ B}

D =



(

f, e, z

)

∈ B

|

f=−

∂

_∂

α

t =−Je, z=



f_∂ e_∂



, (25)

2. Mindlintheoryforthickplates

Inthis section the classical model ofthick platesis first recalledby making useof a tensorial formalism to simplify thediscussiononthe port-Hamiltonianformulation.The Mindlinplatetheory,originallypresentedin [32],ismoresuited forplates having a large thickness.The fibers, i.e.a segment perpendicular to the mid-plane,of the plateare supposed toremainstraight afterthedeformation, butnot necessarilynormaltothemid-plane.Forthisreasontwonewkinematic variableshaveto be added,inorderto representthedeflectionof thecrosssections.Let

θ

x denotethedeflection ofthe

crosssectionwithrespecttotheoppositesideoftheyaxisand

θ

yitsdeflectionalongthexaxis.Thedisplacementfieldis

thereforegivenbythefollowingrelations:

u

(

x, y, z

)

=−z

θ

x

(

x, y

)

, Displacementalongx,

v

(

x, y, z

)

=−z

θ

y

(

x, y

)

, Displacementalongy,

w

(

x, y, z

)

=w

(

x, y

)

, Displacementalongz.

(26)

Bothbending and shear deformations are taken into account. The bending part is described by the second order shear tensor: Eb=





xx

γ

xy

γ

xy



yy



=−zGrad

(

θ

)

=−z

⎡

⎣

∂θ∂xx 1 2



_∂θ y ∂x +∂θ x ∂y



1 2



∂θy ∂x +∂θ x ∂y



∂θy ∂y

⎤

⎦

, (27)

where

θ

=

(

θ

x,

θ

y

)

T.ThetensorGrad(

θ

)standsforthesymmetricgradientofthevector

θ

andcorrespondstothecurvature

tensor: K=



κ

xx

κ

xy

κ

xy

κ

yy



:=Grad

(

θ

)

=1 2



∇



θ

+



∇



θ

T



, (28)

whereu vdenotestheouterproductofvectors,equivalenttothedyadicproductgivenbyuvT_{.Thecorrespondingbending}

stressfieldSbisobtainedbyconsideringtheconstitutiverelationwhich,foranisotropichomogeneousmaterial,reads:

Sb=

E

1−

ν

2

{

(

1−

ν

)

Eb+

ν

I2×2Tr

(

Eb

)

}

, (29)

whereEisYoung’smodulus,

ν

Poisson’sratio,I2× 2theidentityoperatorinR2 andTrthetraceoperator.Byaveragingthe

torquesproducedbystressesalongafiber,itispossibletodefinetheflexuralmomentatensor:

Mi j=  h 2 −h 2 −zSb,i jdz =Di jklKkl, (30)

wherehistheplatethicknessand_Distheafourthordersymmetrictensorandrepresentsthebendingrigiditytensor.The componentsofthemomentatensoraregivenbythefollowingrelations:

M=



M xx M xy M xy M yy



, M xx=D

(

κ

xx+

νκ

yy

)

, M yy=D

(

κ

yy+

νκ

xx

)

, M xy=D

(

1−

ν

)

κ

xy, (31) whereD= Eh3

12(1−ν2₎ isthebendingrigidity.Thesheardeformationsaredescribedusingtheshearstrainvector:



s=



γ

xz

γ

yz



=



_∂w ∂x −

θ

x ∂w ∂y −

θ

y



. (32)

Thecorrespondingstress fieldissimplygivenby

σ

s=G



s,whereGistheshearmodulusG= ₂₍₁₊E_ν).Theshearforcesare

againobtainedbyaveragingtheshearstressalongplatefibers.However,giventheexcessiverigidityoftheshear contribu-tion,acorrectionfactork=5/6(see [32])isintroduced:

q=



q x q y



=  h 2 −h 2 k

σ

sdz = kGh



s= kGh



γ

xz

γ

yz



. (33)

⎧

⎪

⎨

⎪

⎩

ρ

h

∂

2_w

∂

t 2 =div

(

q

)

,

ρ

₁₂h 3

∂

_∂

2_t 2

θ

=q+Div

(

M

)

, (34)

where

ρ

is theplate massdensity. The differentialoperators div()andDiv() denote the divergenceofa vector and ofa tensor,respectively:

a=Div

(

A

)

withai=div

(

Aji

)

= n



j=1

∂

Aji

∂

x j,

whereaisdefinedcolumn-wise.Thekinetic andpotentialenergydensitiesperunitarea(_{K andU),}aregivenrespectively by: K = 1 2

ρ



h



∂

w

∂

t



2 +h 3 12

∂

θ

∂

t ·

∂

θ

∂

t



, U = 1 2

{

M:K+q·



s

}

.

wherethetensorcontractionM:KinCartesiancoordinatesisexpressedas:

M:K=

2



i, j=1

M i j

κ

i j=Tr

(

MTK

)

.

Thetotalenergydensityisthesumofkineticandpotentialenergies

H = K +U (35)

andthecorrespondingenergies H =  H d



, K =  K d



, U =  U d



, (36)

where



isanopenconnectedsetofR2_.

3. PHtensorialformulationoftheMindlinplate

In thissection thenewtensorial formulation fortheMindlinplateis presented.Obviouslythisresult isequivalentto theonefoundinthevectorialcase [20],butthetensorialformulationismoresuitablefromthepointofviewoffunctional analysissince itmakes appearintrinsicoperators(div, Div,grad, Grad)asitwillbe showninthefollowing,regardless of thechoiceofcoordinateframe.TheHamiltonianenergyisfirstconsideredinordertofindtheproperenergyvariables:

H =   1 2



ρ

h



∂

w

∂

t



2 +

ρ

₁₂h 3

∂

_∂

θ

_t ·

∂

_∂

θ

_t +M:K+q·



s



d



, (37)

Thechoiceoftheenergyvariablesisthesameasin [20]butherescalar,vectorandtensorvariablesaregatheredtogether:

α

w=

ρ

h

∂

w

∂

t , Linearmomentum, Aκ=K, CurvatureTensor,

α

θ=

ρ

h 3 12

∂

θ

∂

t , Angularmomentum,

α

γ =



s. ShearDeformation. (38)

Theco-energyvariablesarefoundbycomputingthevariationalderivativeoftheHamiltonian: e w:=

δ

H

δα

w =

∂

w

∂

t , Linearvelocity, Eκ:=

_δ

δ

_AH κ =M, MomentaTensor, e_θ:=

δ

H

δα

θ =

∂

θ

∂

t , Angularvelocity, e_γ :=

δ

_δ

H s = q Shearstress. (39)

Proposition2. ThevariationalderivativeoftheHamiltonianwithrespecttothecurvaturetensoristhemomentatensor δH δAκ =M. Proof. ThecontributionduetothebendingpartinHamiltonianisgivenby:

H b

(

K

)

= 1 2  M:Kd



= 1 2  Tr

(

M T_K

₎

_d

_

_,

wherethemomentatensordepends onthecurvaturestensor_Mi j=Di jklKkl whereD=DT is afourth ordersymmetricpositive

definitetensor.Soavariation

δ

KofthecurvaturestensorwithrespecttoagivenvalueK0 leadsto:

H b

(

K0+

εδ

K

)

= 1 2  Tr

(

M T 0K0

)

d



+

ε

1 2  



Tr

(

MT 0

δ

K

)

+Tr

(

δ

MTK0

)



d



+ O

(

ε

2

₎

_. ThetermTr

(

δ

MT_K

0

)

canbefurthermanipulatedasfollows

Tr

(

δ

MT_K 0

)

=Tr

(

KT0

δ

M

)

=Tr

(

KT0D

δ

K

)

=Tr

(

MT0

δ

K

)

, sothat H b

(

K0+

εδ

K

)

= 1 2  Tr

(

M T 0K0

)

d



+

ε

 



Tr

(

MT 0

δ

K

)



d



+ O

(

ε

2

₎

_.

Bydefinitionofthevariationalderivative(seee.g.[8])itcanbewritten:

H b

(

K0+

εδ

K

)

=H b

(

K0

)

+

ε

δ

H b

δ

K,

δ

K

!

H +O

(

ε

2

₎

_,

whereH isthespaceof thesquare integrablesymmetric tensorsendowedwith theintegralofthetensorcontraction asinner product.Then,byidentification δHb

δK =δδKH =M0. 

Theport-Hamiltoniansystemisexpressedasfollows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

∂α

w

∂

t =div

(

eγ

)

,

∂

α

θ

∂

t =Div

(

Eκ

)

+eγ,

∂

Aκ

∂

t =Grad

(

eθ

)

,

∂

α

γ

∂

t =grad

(

e w

)

− eθ, (40)

Ifthevariablesareconcatenatedtogether,theformallyskew-symmetricoperatorJcanbehighlighted:

∂

∂

t

⎛

⎜

⎝

α

w

α

θ Aκ

α

γ

⎞

⎟

⎠

=

⎡

⎢

⎣

0 0 0 div 0 0 Div I2×2 0 Grad 0 0 grad −I2×2 0 0

⎤

⎥

⎦

(

)*

+

J

⎛

⎜

⎝

e w e_θ Eκ e_γ

⎞

⎟

⎠

, (41)

whereallzerosareintendedasnullifyingoperatorsfromthespaceofinputvariablestothespaceofoutputvariables.

Remark2. It canbe observedthat theinterconnection structuregivenby Jin (41) mimicsthat oftheTimoshenko beam givenin (19).

Theorem4. TheadjointofthetensordivergenceDivis−Grad,theoppositeofthesymmetricgradient.

Proof.LetusconsidertheHilbert spaceofthesquareintegrablesymmetrictensorsofsizen× nover anopenconnectedset



. This space willbe denoted by H1=L2

(

,Rsymn×n

)

.This space is endowed with theintegral of thetensor contraction as scalar

product:



E, F



H1=  E:Fd



=  Tr

(

E T_F

₎

_d

_

_,

_∀

_{E, F∈ L} 2

_(

_{, R}n×n sym

)

.

MoreovertheHilbertspaceofthesquareintegrablevectorfunctionsover thesame openconnectedset



willbedenotedby H2=L2

(

,Rn

)

.Thisspaceisendowedwiththefollowingscalarproduct:



ε

,

φ

H2=  

ε

·

φ

d



=  

ε

T

_φ

_d

_

_,

_∀

_ε

_,

_φ

_{∈ L} 2

_(

_{, R}n

₎

_.

Letusconsiderthetensordivergenceoperatordefinedas:

A : H1→H2,

E→Div

(

E

)

=

ε

, with

ε

i=div

(

Eji

)

=

n  j=1

∂

Eji

∂

x j. WetrytoidentifyA∗ A ∗: H2→H1,

φ

→A ∗

φ

=F,

Fig. 1. Cauchy law for momenta and forces at the boundary. suchthat



A E,

φ

H2=



E, A ∗

_φ

H1,

∀

E∈Domain

(

A

)

⊂H1

∀

φ

∈Domain

(

A ∗

)

⊂H2

Now letustake E∈C1

0

(

,Rnsym×n

)

⊂ Domain

(

A

)

thespace ofdifferentiable symmetrictensorswith compactsupportin



.

Addi-tionally

φ

willbelongtoC1

0

(

,Rn

)

⊂ Domain

(

A∗

)

,thespaceofdifferentiablevectorfunctionswithcompactsupportin



.Then



DivE,

φ

H2=  

ε

·

φ

d



=   n  i=1 n  j=1

∂

Eji

∂

x j

φ

i d



, =−   n  i=1 n  j=1 Eji

∂φ

_∂

_x i j

d



, sincethefunctionsvanishattheboundary, =−   n  i=1 n  j=1 EjiFjid



, ifwefirstchoose Fji=

∂φ

_∂

_x i j .

But inthis latter case, itcouldnot indeedbe statedthatF∈L2

_(

_,Rn×n

sym

)

.Now, since E∈L2

(

,Rsymn×n

)

,Eji=Ei j,thuswe are

allowedtofurtherdecomposethelastequalityas:

 i, j Eji

∂φ

_∂

_x i j = i, j Eji 1 2



∂φ

i

∂

x j +

∂φ

j

∂

x i



= i, j EjiFji, withFji:= 1 2



∂φ

i

∂

x j +

∂φ

j

∂

x i



. ThusF∈L2

_(

_,Rn×n

sym

)

anditcanbestatedthat:



DivE,

φ

H2=−    i, j Eji 1 2



∂φ

i

∂

x j +

∂φ

j

∂

x i



d



=−    i, j EjiFjid



=



E, −Grad

φ

H1.

ItcanbeconcludedthattheformaladjointofDivisDiv∗=−Grad. 

Weshallnowestablishthetotalenergybalanceintermsofboundaryvariables.Thosewillthenbepartoftheunderlying Stokes–Diracstructureofthismodel:

˙ H =  



∂α

w

∂

t e w+

∂

α

θ

∂

t · eθ+

∂

Aκ

∂

t :Eκ+

∂

α

γ

∂

t · eγ



d



=  



div

(

eγ

)

e w+Div

(

Eκ

)

· e_θ+ Grad

(

e_θ

)

:Eκ+grad

(

e w

)

· eγ



d



= 

∂

{

w tq n+

ω

nM nn+

ω

sM ns

}

ds,

(42)

wheresisthecurvilinearabscissa. ThelastintegralisobtainedbyapplyingGreen–Gausstheorem.Theboundaryvariables appearinginthelastlineof (42)andillustratedin Fig.1aredefinedasfollows:

ShearForce q n:=q· n=eγ· n,

Flexuralmomentum M nn:=M:

(

nn

)

=Eκ :

(

nn

)

Torsionalmomentum M ns:=M:

(

sn

)

=Eκ :

(

sn

)

Fig. 2. Reference frames and notations.

Vectorsn andsdesignate thenormalandtangentialunit vectorstothe boundary,asshownin Fig.2.Thecorresponding powerconjugatedvariablesare:

Verticalvelocity w t:=

∂

w

∂

t = e w, Flexuralrotation

ω

n:=

∂

θ

∂

t · n=eθ· n Torsionalrotation

ω

s:=

∂

θ

∂

t · s=eθ· s. , (44)

AnalogouslytotheTimoshenko beamcase, theStokes–DiracstructurefortheMindlinplatecannowbe defined.Consider nowthebondspace:

B :=

{

(

f, e, z

)

∈ F × E× Z

}

, (45)

where F=L2

_()

_:=L2

_()

_{× L}2

_(

_{; R}2

₎

_{× L}2

_(

_,R2×2

sym

)

× L2

(

; R2

)

and E=H1

()

=H1

()

× H1

(

,R2

)

×

HDiv

_(

_,R2×2

sym

)

× Hdiv

(

,R2

)

.Considerthespaceofboundaryportvariables:

Z :=



z

|

z=



f_∂ e_∂



, with f_∂=

,

_w t

ω

n

ω

s

-and e_∂=

,

_q n M nn M ns

-(46)

ThedualitypairingbetweenelementsofB isthendefinedasfollows:





(

f1, e1, z1

)

,

(

f2, e2, z2

)





:=



e1, f2



L2_()+



e2, f1



L2_()+



∂B J

(

z1, z2

)

ds, (47)

wherethepairing



·,·



L2_()istheL2innerproductonspaceL2

()

andBJ

(

z1,z2

)

:=

(

f∂,1

)

Te∂,2+

(

f∂,2

)

Te∂,1.

Theorem5(Stokes–Dirac structurefortheMindlinplateintensorialform). ConsiderthespaceofpowervariablesB defined in(45)andthematrixdifferentialoperatorJin(41).ByTheorem2,thelinearsubspaceD⊂ B:

D =



(

f, e, z

)

∈B

|

f =−

∂

α

∂

t =−Je, z=



f_∂ e_∂



, (48)

isaStokes–Diracstructurewithrespecttothepairing



· , ·



givenby(47).

Remark3. TheHamiltonianformulationandassociatedStokes–DiracstructureinvolveaHamiltonianskew-symmetric dif-ferential operatorJ onthe spaceE of the coenergy variables.This representsa significant difference withrespect to the HamiltoniansystemobtainedbytheLegendretransformationoftheEuler–Lagrangesystem [21].Intheformerformulation thetotaldimensionofthissystemis8,inthelatter6.Furthermore,thelattersystemisruledbyanalgebraicoperator,the differentialpartbeinghiddeninthevariationalderivativeoftheHamiltonian,leadingtothefollowingsystem:

⎛

⎜

⎜

⎜

⎜

⎝

w

θ

x

θ

y p w p _θ,x p _θ,y

⎞

⎟

⎟

⎟

⎟

⎠

=

⎡

⎢

⎢

⎢

⎢

⎣

0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0

⎤

⎥

⎥

⎥

⎥

⎦

⎛

⎜

⎜

⎜

⎜

⎝

δ

wH

δ

θ,xH

δ

θ,yH

δ

pwH

δ

pθ,xH

δ

p_θ,yH

⎞

⎟

⎟

⎟

⎟

⎠

, (49)

wherepwisthelinearmomentumandp_θ,x,pθ,yaretheangularmomenta. 4. DiscretizationoftheMindlinplateusingaPartionedFiniteElementMethod

ThePartionedFiniteElementMethod(PFEM) [22]consistsofputtingthesystemintoweakformfirstandofapplyingthe integrationbypartsonasubsetoftheoverallsystemsecond.FortheMindlinplatetheintegrationbypartscanbeapplied

tofirsttwolinesofsystem (41).Thischoice willmakeappearthemomentaandforcesattheboundaryascontrolinputs. Alternatively,thelasttwoof (41)couldhavebeenselectedtoperformtheintegrationbyparts.Inthislattercasethelinear andangularvelocitiesattheboundarywouldappearascontrolinputs.Keepingthisinmind,themostsuitablechoicewill dependonthephysicalproblemunderconsideration.

4.1. Weakform

Thetestfunctionsareofscalar,vectorialandtensorialnature.Keepingthesamenotationasin Section3,thescalartest functionisdenotedbyvw,thevectorialonebyv_θ,vγ thetensorialonebyVκ.

4.1.1. Boundarycontrolthroughforcesandmomenta

The firstline of (41)is multipliedby vw (multiplication bya scalar), thesecond line andthefourthby v_θ,vγ (scalar

productofR2_{)andthethirdonebyV}

κ (tensorcontraction):  

v

w

∂α

w

∂

t d



=  

v

wdiv

(

eγ

)

d



, (50)  

v

θ·

∂

α

θ

∂

t d



=  

v

θ·

(

Div

(

Eκ

)

+eγ

)

d



, (51)  Vκ:

∂

Aκ

∂

t d



=  Vκ :Grad

(

eθ

)

d



, (52)  

v

γ ·

∂

α

γ

∂

t d



=  

v

γ ·

(

grad

(

e w

)

− eθ

)

d



. (53)

Theright-handsideof Eq.(50)hastobeintegratedbyparts

 

v

wdiv

(

eγ

)

d



=  ∂

v

wn

(

)*

· eγ

+

qn ds −  grad

(

v

w

)

· eγd



, (54)

aswellastheright-handsideof Eq.(51)

 

v

θ·

(

Div

(

Eκ

)

+eγ

)

d



=  ∂

v

θ·

(

n· Eκ

)

ds −  



Grad

(

v

_θ

)

:Eκ−

v

θ· eγ



d



. (55)

The usual additional manipulation is performed on the boundary term containing the momenta, so that the proper boundaryvaluesarise:

 ∂

v

θ·

(

n· Eκ

)

ds =  ∂

{

(

v

θ· n

)

n+

(

v

θ· s

)

s

}

·

(

n· Eκ

)

ds =  ∂

{

v

ωnM nn+

v

ωsM ns

}

ds. (56)

Sodefining

v

ωn:=

v

θ· nand

v

ωs:=

v

θ· sthefinalweakformobtainedfromsystem (41)reads:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

 

v

w

∂α

w

∂

t d



=−  grad

(

v

w

)

· eγd



+  ∂

v

wq nds,  

v

θ·

∂

α

θ

∂

t d



=−  



Grad

(

v

_θ

)

:Eκ−

v

θ· eγ



d



+  ∂

{

v

ωnM nn+

v

ωsM ns

}

ds,  Vκ:

∂

Aκ

∂

t d



=  Vκ :Grad

(

eθ

)

d



,  

v

γ ·

∂

α

γ

∂

t d



=  

v

γ ·

(

grad

(

e w

)

− eθ

)

d



. (57)

Inthisfirstcase,theboundarycontrolsu_∂ andthecorrespondingoutputy_∂ are:

u_∂=

,

_q n M nn M ns

-∂ , y_∂=

,

_w t

ω

n

ω

s

-∂ .

4.1.2. Boundarycontrolthroughkinematicvariables

Alternatively,inthissecond case,thesameprocedurecanbe performedonthetwo lastlinesofthesystemwrittenin weakform(Eqs.(52)and (53)).Oncetheduecalculationsarecarriedout,itisfound:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

 

v

w

∂α

w

∂

t d



=  

v

wdiv

(

eγ

)

d



,  

v

θ·

∂

α

θ

∂

t d



=  

v

θ·

(

Div

(

Eκ

)

+eγ

)

d



,  Vκ :

∂

Aκ

∂

t d



=−  Div

(

Vκ

)

· eθd



+  ∂

{

v

Mnn

ω

n+

v

Mns

ω

s

}

ds,  

v

γ ·

∂

α

γ

∂

t d



=−  



div

(

v

γ

)

e w+

v

γ · eθ



d



+  ∂

v

qnw tds, (58)

where

v

Mnn=Vκ:

(

n n

)

,

v

Mns=Vκ :

(

s n

)

and

v

qn= v γ · n. Inthissecond case, the boundary controlsu∂ and

corre-spondingoutputy_∂ are:

u_∂=

,

_w t

ω

n

ω

s

-∂ , y_∂=

,

_q n M nn M ns

-∂ .

4.2.Finite-dimensionalport-Hamiltoniansystem

Inthissectiontheformulation (57)isusedisordertoexplainthediscretizationprocedure.Testandco-energyvariables arediscretizedusingthesamebasisfunctions(Galerkinmethod):

v

w= Nw  i=1

φ

i w

(

x, y

)

v

iw,

v

θ= N_θ  i=1

φ

i θ

(

x, y

)

v

iθ, Vκ= Nκ  i=1



i κ

(

x, y

)

v

iκ,

v

_γ = Nγ  i=1

φ

i γ

(

x, y

)

v

iγ, e w= Nw  i=1

φ

i w

(

x, y

)

e iw

(

t

)

, e_θ= N_θ  i=1

φ

i θ

(

x, y

)

e iθ

(

t

)

, Eκ = Nκ  i=1



i κ

(

x, y

)

e iκ

(

t

)

, e_γ = Nγ  i=1

φ

i γ

(

x, y

)

e iγ

(

t

)

. (59)

Thebasisfunctions

φ

i w,

φ

i

θ,



iκ,

φ

iγ havetobechoseninasuitablefunctionspaceVhinthedomainofoperatorJ,definedin (41),i.e._Vh_{⊂ V∈D}

₍

_J

₎

_{.Thiswillbediscussedin Section5.Thediscretizedskew-symmetricbilinearformontheright-hand}

sideof (57)thenbecomes:

Jd=

⎡

⎢

⎢

⎣

0 0 0 −DT grad 0 0 −DT Grad −DT0 0 DGrad 0 0 Dgrad D0 0 0

⎤

⎥

⎥

⎦

, (60)

wherethematricesarecomputedinthefollowingway:

DGrad

(

i, j

)

=  



i κ:Grad

(

φ

θj

)

d



, ∈RNκ×Nθ, Dgrad

(

i, j

)

=  

φ

i γ · grad

(

φ

wj

)

d



, ∈RNγ×Nw, D0

(

i, j

)

=−  

φ

i γ ·

φ

θjd



, ∈RNγ×Nθ. (61)

ThenotationD(i,j)indicates theentryinmatrixDcorrespondingtotheithrowandjthcolumn.Theenergyvariablesare deducedfromtheco-energyvariables:

α

w=

ρ

he w,

α

θ=

ρ

h 3 12eθ, Aκ=D−1Eκ,

α

γ =_Ghk 1 e_γ. (62)

Thesymmetricbilinearformontheleft-handsideof (57)isdiscretizedas:

M=diag[Mw, M_θ, Mκ, Mγ], with Mw

(

i, j

)

=  

ρ

h

φ

i w

φ

j wd



, ∈RNw×Nw, M_θ

(

i, j

)

=  

ρ

h 3 12

φ

i θ·

φ

θjd



, ∈RNθ×Nθ, M_κ

(

i, j

)

=  



D−1

_

i κ

:



κjd



, ∈RNκ×Nκ, M_γ

(

i, j

)

=   1 Ghk

φ

i γ ·

φ

γjd



, ∈RNγ×Nγ. (63)

Theboundaryvariablesarethendiscretizedas: q n= Nqn  i=1

φ

i qn

(

s

)

q i n, M nn= NMnn i=1

φ

i Mnn

(

s

)

M i nn, M ns= NMns i=1

φ

i Mns

(

s

)

M i ns. (64)

Thevariablesaredefinedonlyovertheboundary

∂



.Consequently,theinputmatrixreads:

B=

⎡

⎢

⎣

Bqn 0 0 0 BMnn BMns 0 0 0 0 0 0

⎤

⎥

⎦

. (65)

Theinnercomponentsarecomputedas:

Bqn

(

i, j

)

=  ∂

φ

i w

φ

j qnds, ∈R Nw×Nqn, BMnn

(

i, j

)

=  ∂

(

φ

i θ· n

)

φ

Mjnnds, ∈R Nw×NMnn, BMns

(

i, j

)

=  ∂

(

φ

i θ· s

)

φ

Mjnsds, ∈R Nw×NMns. (66)

The final port-Hamiltonian system(as definedin [24],where the presenceof a massmatrix M istakeninto account)is writtenas: Me˙=J_de+Bu_∂, y_∂=BTe, (67) wheree=

(

e1 w,...,e N_γ γ

)

T andu_∂=

(

q1 n,...,M NMns

ns

)

T aretheconcatenationsofthedegreesoffreedomforthedifferent

vari-ables.ThediscreteHamiltonianisthenfoundas: H d= 1 2  



α

we w+

α

_θ· e_θ+Aκ:Eκ+

α

γ· eγ



d



= 1 2



eT wMwew+eT_θMθeθ+eT_κMκeκ+eT_γMγeγ



= 1 2e T_Me. (68)

Thenitnaturallyfollowsthat:

˙

H d=yT_∂u∂. (69)

This is equivalentto the energy balance of thecontinuous system, expressed by Eq.(42). Definition (68),together with system (67)arethefinite-dimensionalequivalentof (37) and (41).Again,thediscretizedsystemobtainedviaPFEMshares thesame propertiesasthose oftheoriginal infinite-dimensionalsystem, thediscretization methodistherefore structure-preserving.

4.3. Mixedboundaryconditions

Thisformulationprovidesascontroltermthedynamicvariablesattheboundary,namelyforcesandmomenta,whereas thekinematicvariablesdonotappear.Inordertohandlemixedboundaryconditions,i.e.aclampedorasimplysupported platewithfreeorloadededges (see Fig.3),theboundarycontroltermhastobesplitintoknownandunknownvariables, i.e.givenboundaryconditionsandreactionsattheboundariesrespectively.Thesetermsmayberearrangedbyintroducing apermutationmatrixP.Thecontroltermmayberewritteninthefollowingway:

u_∂=BP



f

λ



=B



Pf Pλ



f

λ



. (70)