An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations

Contents lists available atScienceDirect

Journal of Computational Physics

www.elsevier.com/locate/jcp

An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations

Igor Tominec

^a,∗

, Pierre-Frédéric Villard

^b

, Elisabeth Larsson

^a

, Víctor Bayona

^c

, Nicola Cacciani

^d,e

aUppsalaUniversity,DepartmentofInformationTechnology,DivisionofScientificComputing,Sweden bUniversitédeLorraine,CNRS,Inria,LORIA,Nancy,France

cDepartamentodeMatemáticas,UC3M,Leganés28911,Spain

dDepartmentofPhysiologyandPharmacology,KarolinskaInstitutet,Stockholm,Sweden

eDepartmentofClinicalNeuroscience,ClinicalNeurophysiology,KarolinskaInstitutet,Stockholm,Sweden

a rt i c l e i nf o a b s t ra c t

Articlehistory:

Received5March2021

Receivedinrevisedform1July2022 Accepted18July2022

Availableonline4August2022

Keywords:

Unfitted RBF-FD Least-squares Elasticity Diaphragm

Mixedboundarycondition

Thethoracicdiaphragmisthemusclethatdrivestherespiratorycycleofahumanbeing.

Using a systemof partial differential equations (PDEs)that models linear elasticitywe computedisplacementsandstressesinatwo-dimensionalcrosssectionofthediaphragm initscontractedstate.Theboundarydataconsistsofamixofdisplacementandtraction conditions. If these are imposed as they are, and the conditions are not compatible, this leadsto reduced smoothness ofthe solution. Therefore, the boundary data is first smoothedusingtheleast-squaresradialbasisfunctiongeneratedfinitedifference(RBF-FD) framework.ThentheboundaryconditionsarereformulatedasaRobinboundarycondition withsmoothcoefficients.Thesameframeworkisalsousedtoapproximatetheboundary curveofthediaphragmcrosssectionbasedondataobtainedfromasliceofacomputed tomography (CT) scan. To solve the PDE weemploy the unfitted least-squares RBF-FD method.Thismakesiteasiertohandlethegeometryofthediaphragm,whichisthinand non-convex.Weshownumericallythatoursolutionconvergeswithhigh-ordertowardsa finiteelementsolutionevaluatedonafinegrid.Throughthissimplifiednumericalmodel wealsogainaninsightintothechallengesassociatedwiththediaphragmgeometryand theboundaryconditionsbeforeapproachingamorecomplexthree-dimensionalmodel.

©^{2022TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticleunderthe} CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Duringthe2020COVID-19pandemicwehaveallbeenmadeawarethatintensivecareunits(ICU)havelimitedcapacity withrespecttothenumberofpatientsthatcanbecaredforatthesametime.TheWHOreportonCOVID-19inChina [1]

indicatesthatCOVID-19patientswithseveresymptomsneed3–6weeksinICU.Patientswithsevererespiratorysymptoms are putundermechanicalventilationtosavetheirlives.Atthesametime, themechanicalventilationhasadverseeffects, suchasventilatorinduceddiaphragmaticdysfunction(VIDD) [2],whichprolongstheICUtime.Hence,improvingmechanical ventilationcanhaveasignificantimpactonICUpatientturnover.

*

Correspondingauthor.

E-mailaddresses:[email protected](I. Tominec),[email protected](P.-F. Villard),[email protected](E. Larsson), [email protected](V. Bayona),[email protected](N. Cacciani).

https://doi.org/10.1016/j.jcp.2022.111496

0021-9991/©^{2022TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).

ThisworkispartoftheINVIVEproject,¹ whereweaimtocreateamechanicallyventilatedvirtualpatient [3] onwhom we can perform tests with different ventilationstrategies andcounter measures against VIDD [4]. With the virtual pa- tient,we canalsovarythegender, physiology,ageandpotential injuriesthataffecttheresponse. Simulationsallowfora controlledcomputer-basedenvironmenttobecreatedinawaythatisnotpossibleinaclinicalsetting.

The diaphragmisthemain respiratorymuscle,andthepresentfocusofourstudy.The diaphragmislocated between thethoracicandabdominalcavities.Ithastwodomes,andisattachedtothelowerribs,thespine,andthesternum.During mechanical ventilation, the normal action of the diaphragm,where inhalation follows the contraction of the muscle, is reversed, i.e., the muscles become passiveandthe airis pumped intothe lungsby the ventilator.As air isentering the lungswithapositivepressure,themusclefibers areinsteadextended.Thissuddenandextrememechanicalperturbationis thetriggerofachainofbiologicaleventscausingtheprogressionofVIDD.

Numerical simulation ofthe biomechanical action ofthe diaphragm during respiration or ventilationis a challenging problem. The shape of the diaphragm isnon-trivial and there are gradual transitions between muscle and tendon with very differentmaterialresponse.Imperfect datacanbe extractedfrommedicalimages,andcan thenbe convertedintoa geometryrepresentation [5]. Thespecificchallengesofconstructinga smoothgeometryrepresentationareaddressedina forthcomingpaper [6].

Inthispaper,wemodelthediaphragmusingalinearelasticPDE.Thisisasimplificationandweplantodevelopamore realistictissuemodelasapartofourfuturework.

TheboundaryconditionsfortheelasticPDEsystemaregivenby acombinationoftractionboundaryconditions,which contains first derivativesof the displacement, and(time-dependent) Dirichletboundary conditions for thedisplacement, wherethediaphragmisattached.Thesearemixedboundaryconditions,ingeneralnotfullycompatible,leadingtoreduced regularityofthesolutionevenwhenweexpect asmoothsolutionfromaphysiologicalperspective.Therefore,wesmooth theboundarydataaswellasthetransitionbetweenthetractionboundarydataandDirichletboundarydatabeforeusingit inthePDEsolver.

Weusethe unfittedradialbasis functiongeneratedfinitedifferencemethodintheleast-squares setting(unfittedRBF- FD-LS) [7] tosolvethe diaphragmproblem.Abenefitofusingtheunfitted RBF-FD-LSmethodisthatthePDE problemis solved onanextendeddomain(see Fig.3).Thissimplifiesnodegenerationasthenodeplacementisthen independentof thegeometry.Anotherbenefitoftheunfittedsettingisasmallerapproximation errornearthe boundaries[7], wherethe stencilsaretypicallyhighlyskewedwhenusingconventionalRBF-FDmethodssuchasthefittedRBF-FD-LSmethod[8] and thecollocationRBF-FDmethod[9].

Asmodelgeometryweuseatwo-dimensionalcrosssectionofthediaphragm.Toinvestigatethepropertiesoftheprob- lemwe usea combinationofdatafrommedicalimagesandknowledge aboutthephysiology ofthediaphragmexpressed intermsofboundaryconditions.Inparticular,weconstructtwobenchmarkproblems:(i)apure Dirichletcase, (ii)acase withmixedboundaryconditions(Dirichlet +traction), rephrasedasa Robinconditionwithsmooth coefficients.Withthe experimentsperformedforthesebenchmarks,weaimtoanswerthefollowingquestions:

•^{CanwesolvethebenchmarksproblemswithunfittedRBF-FD-LS?Aretherespecificnumericalchallengestobenoted?}

•^{Canwe achieve high-order}convergence tothe solutionoftheelastic PDEwhen theimposed boundarydataandthe geometryissmooth?

•^{How isthe} performance and accuracyof the RBF-FD-LSsolver affectedby thetype of boundary conditionsthat are imposed?

•^{HowdoestheRBF-FD-LSsolvercomparewithabasicFEMsolver?Dothesolversgivesimilarsolutions?}

Otherauthorshavealsomodeledthediaphragmnumerically.Themostadvanceddiaphragmsimulationsintheliterature canbefoundinaseriesofpublicationsbyLadjaletal. [10–14].Theapplicationfocusistotrackthemotionoflungtumors duringrespirationforradiotherapypurposes.Thefiniteelementmodelsthatareemployedarehighlyelaborate,takinginto account features such as patient-specific lung compliance in order to define the constitutive law. The simulation times reportedarefarfromreal-timecapability.Inthesestudies,thediaphragmisnotthemaintargetanditisnotwell-resolved inthethicknessdirectionduetothehighaspectratio.Otherrelevant,butlessdetailed,diaphragmsimulationscanbefound in [15],wherethediaphragmisdiscretizedusingshellelementstocomparehealthyandpathologicalsituations,andin [16], wherethewholeregionunderthelungs,includingthediaphragm,formsoneregioninthesimulation.

The outlineofthe paperisasfollows:Section 2 describestheexpectedbehavior ofthediaphragmduring respiration and the relation of the two-dimensional geometry to the real three-dimensional diaphragm geometry. Then the linear elasticitymodelproblemisdefinedinSection3.AnRBF-FDalgorithmforcomputingdifferentiationandevaluationmatrices isdescribedinSection4.InSection5thesematricesarethenusedintheleast-squaresunfittedRBF-FDsettingtodiscretize thelinearelasticityequations.Section6discusseshowtheboundaryofthetwo-dimensionaldiaphragmcross-section,and theboundaryconditions,aresmoothed.InSection7andSection8wecomputethesolutiontothelinearelasticityequations forthetwobenchmarkproblems,andevaluatetheconvergencenumerically.Finally,Section9containstheconclusions.

1 https://www.it.uu.se/research/scientific_computing/project/rbf/biomech.

Fig. 1. Anatomyandphysiologyofthediaphragm(redandgreenparts)fromexpirationtoinspiration.(Forinterpretationofthecolorsinthefigure(s),the readerisreferredtothewebversionofthisarticle.)

Fig. 2. Theexpecteddisplacementindifferentpartsofthesliceofthediaphragm.Theredareascorrespondtotherealphysiologicalbehaviorandthegreen areasaretransitionzones.Inregions3and7,thereisbothathickeningofthemuscleandadownwardmotion,whileinregion5,thedownwardmotion dominates.

2. Theexpectedbehaviorofthethoracicdiaphragm

Thediaphragmisamusculotendinousstructure,approximatelydouble-domeshaped,separatingthethoracicandabdom- inalcavities.Itisthemainmuscleofthephysiologicalrespiration,performing70–80%oftheworkofbreathing,althoughit alsohasnon-ventilatoryfunctions,e.g.,coughing,hiccups,sneezing,vomiting,andposturalfunctions.Itiscomposedofthree main parts,seeFig.1a.Thereare twomuscularparts,one medianandhorizontalinuprightposition,separatingthetho- racicandtheabdominalorgans,andanotherlateralmusclepart,whichendswiththecostalinsertions(attachmenttothe lowerribs),andthereisonecentraltendon,wheretheextremitiesofallthemusclefibersconverge.Whenthediaphragm contractsduring inspiration(inhalation)withapiston-like motion,the musclezonethickens (inspiratorythickening), and thedomesmovecaudally(downward)expandingthethorax.Therefore,theairentersthelungsunderapressuregradient, seeFig.1b.

Thetwo-dimensionalgeometryusedforoursimulationswasextractedfromarealpatientmedicalimage.Weuseda3D CTscanimage(resolution:0.927×⁰.927×⁰.3 mm³)thatwasacquiredformedicalreasons,seeourpreviouswork [5].The diaphragmwasmanuallysegmentedin3D followingvisual cuesasexplainedin[17].Thelabeledvoxelswereexported to atrianglemeshwiththeMarchingCubealgorithmandthemeshwassimplifiedwithadecimationalgorithm[18].

The frontalplaneslice we selectedisinthe middleofthebody,correspondingroughly tothe anatomicillustrationof Fig. 1. The raw data consists of a list of 2D vertices where a topology can easily be extracted. The raw geometry data contains noisefromseveralsources.There issome CTscan deviceincertitude(noise, calibrationetc)[19],thedatais the resultfromadiscretizedprocess,andthelabelingdatacomesfromamanual segmentationthatisprone tohumanerror.

Concerningthislastpoint,thediaphragmisnotentirelyvisible,sometimespartofitsvoxelsalsoincludeotherorgans.The accuracyislinkedtotheimaginationandtheanatomicalknowledgeofthemedicalexpertthatperformedthesegmentation.

The expecteddisplacement ofthe diaphragmbetweenthe relaxedandcontracted statesinthe two-dimensional slice is roughly illustrated inFig. 2. For thenumerical simulations in Sections 7 and8, we construct boundary conditionsto replicatethismotionqualitatively.

3. Equationsoflinearelasticity

Thesimplifiedmodelthatweuseisvalidforstudyingsmalldeformationsofan isotropicandhomogeneous diaphragm witha linearelasticityconstitutivelaw.The deformationofa diaphragm isdescribed byapplyinga displacementfield u= (^u1,u₂)^T,toanobjectlocation y= (^y1,y₂)^T∈ ^suchthat:

y^∗

=

^y

+

^u

,

where y^∗= (^y∗₁,y^∗₂)^T∈ ^{∗ isthenadeformedobject.Afieldderivedfromthe}displacementsisthestresstensor:

σ =

 σ

11

σ

12

σ

21

σ

22

 ,

whichmeasurestheinternalforcesinasaconsequenceofdeformation.Itisrelatedtothedisplacementfieldby:

σ = λ

Tr

( ε ) +

2

με , ε =

¹ 2

 (∇

u

)

^T

+ ∇

u



,

(1)

where

ε

∈ R^{2×2 is thestrain tensorand Tr}(

ε

)= (

ε

11+

ε

22)I is its trace.The scalars λ and

μ

are theLaméparameters, whicharerelatedtotheYoungmodulus E andthePoissonratio

ν

ofthematerialthrough:

λ =

^E

ν

(

1

+ ν ) (

1

−

²

ν ) , μ ₌

^E

2

(

1

+ ν ) .

(2)

ForourcomputationsweuseE=^{105Pa and}

ν

=⁰.3.AspecialstressmeasurethatisalsoofinteresttousistheVonMises stress:



σ

₁₁²

− σ

11

σ

22

+ σ

₂₂²

+

³

σ

₁₂²

,

(3)

whichprovidesascalarmeasureofthetotalstress.

Theequationsoflinearelasticityarederivedfromtheforceequilibrium(Newton’ssecondlaw)imposedon.Wehave:

−∇ · σ =

f on

,

(4)

where f= (^f1(y),f2(y))^T isafieldofinternalforcesinthehorizontalandtheverticaldirection.TheDirichletandtraction boundaryconditionsareprescribedontwodisjointpartsoftheboundary,whichtogether formthewholeboundary∂=

∂0∪ ∂1:

u

=

^{g on}

∂ 

0

,

σ _{· ¯}

n

=

^{h on}

∂ 

1

,

⁽⁵⁾

wheren is¯ an outwardpointingnormalto∂1.Thefirstboundaryconditionwiththerighthandside g= (^g1(y),g2(y))^T isthedisplacementcondition.The secondboundaryconditionwiththerighthandside h= (^h1(y),h₂(y))^T isthetraction condition.Anequivalentformof(5) istheRobinboundarycondition:

u

κ

0

(

y

) + ( σ _{· ¯}

n

) κ

1

(

y

) =

^g

κ

0

(

y

) +

^h

κ

1

(

y

)

on

∂  = ∂

0

∪ ∂

1

,

(6) where

κ

0 and

κ

1 correspondto twospatially dependentcoefficients, inthiscasediscontinuous:

κ

0(y)=^{1 when y}∈ ∂0 andzerootherwise,and

κ

1(y)=^{1 when y}∈ ∂1andzerootherwise.

In Section 7 we first compute solutions using a Dirichlet condition for the whole boundary. (

κ

0(y)=^1,

κ

1(y)=^0, y∈ ∂^{).Then, inSection 8,we computesolutionsusingsmoothedDirichletandtractionconditions(smoothRobincoeffi-} cients),whichapproximatestheimpositionofthesetwoconditionsontwodisjointpartsofthediaphragm.However,since physiologysuggestsasmoothsolution,weseethisasmodelingratherthanasanerror.

For all ofthe computationsin thispaperwe use the displacementformulation of(4), obtained by usingthe relation betweenstressanddisplacementgivenin(1).Theforceequilibrium(4) thenexpandsto:

−∇ · σ = − μ ∇

^2u

− (λ + μ )∇(∇ ·

^u

) =

^f

,

inliteraturealsoreferredtoasNavier-Cauchysystemofequations.Thetractionboundaryconditionexpandsfrom(5) to:

σ · ¯

ⁿ

= 

λ (∇ ·

^u

)

I

+ μ ^ (∇

^u

)

^T

+ ∇

^u



· ¯

ⁿ

=

^h

.

Thedisplacementformulationofthelinearelasticityequationswiththeboundaryconditionfrom(6) isthenwrittenas:

− μ ∇

^2u

− (λ + μ )∇(∇ ·

^u

) =

^{f on}

,

u

κ

0

+ 

λ (∇ ·

^u

)

I

+ μ ^ (∇

^u

)

^T

+ ∇

^u



· ¯

ⁿ



κ

1

=

^g

κ

0

+

^h

κ

1 on

∂.

⁽⁷⁾

Forsimplicitywerewritethesystemaboveas:

Du

(

y

) =

^F

(

y

),

where:

Du

(

y

) =

D2u

(

y

),

y

∈ 

κ

0

(

y

)

D0u

(

y

) + κ

1

(

y

)

D1u

(

y

) · ¯

n

(

y

),

y

∈ ∂

F

(

y

) =

f

(

y

),

y

∈ 

κ

0

(

y

)

g

(

y

) + κ

1

(

y

)

h

(

y

),

y

∈ ∂

(8)

Here D2,D1, D0 aretheexpandedoperators thatin(8) correspondto,∂1 and∂0 respectively.Ifwelet∇i j=_∂y^∂_i∂²_yj and∇i=_∂^∂_yi^{,thentheoperator D}2 is:

D₂

= −

 (λ +

2

μ )∇

11

+ μ ∇

22

(λ + μ )∇

12

(λ + μ )∇

12

μ ∇

11

+ (λ +

²

μ )∇

22



,

(9)

andtheboundaryoperators D1 andD0 are:

D₁

=

 μ

n

¯

2

∇

2

+ (λ +

2

μ )

n

¯

1

∇

1

λ¯

n1

∇

2

+ μ

n

¯

2

∇

1

μ

n

¯

₁

∇

2

+ λ¯

ⁿ2

∇

1

(λ +

²

μ )¯

n₂

∇

2

+ μ

n

¯

₁

∇

1



,

D₀

=



I 0

0 I



.

(10)

4. ApproximationoffunctionsusingtheRBF-FDmethod

Here wedescribe theapproximation framework whichisusedwhen representingtheunknown displacementandthe stress (whicharetheoutputs ofthesimulation),andthe knowndatavectors(the inputdatatothesimulation),asfinite- dimensionalapproximationoffunctions.Thefollowingdiscussionisgeneralinthesensethatthedatavectorcanbeknown orunknown.Concreteexamplesofusingtheframeworkdevelopedinthissectionaregiveninlatersections.

Givenadatavector^u˜(X)= [˜^u(x₁),^u˜(x₂),...,^u˜(x_N)]^T,wherexi∈^{X ,i}=¹,..,N,isonenode,and X∈ R^{N×disanodeset,} we constructa semi-discrete evaluationoperator E=^E(y,X),which, foranypoint y∈ ⊂ R^{d,returns thevalue of the} interpolantofthedatau˜(X)atthatpoint,suchthat:

˜

u

(

y

) =

^E

(

y

,

X

)

u

˜ (

X

).

(11)

Nowweuse(11) foreach y∈^{Y ,whereY}∈ R^{M×d isanevaluationpointset,toobtainasystemofequationsandformthe} matrix E(Y,X)ofsize M×^N,M=^{qN,whereq isan}oversamplingparameter.Notethatinournotationthisisequivalent tosettingy=^{Y .TheRBF-FDprocedureconstructsthematrix E}(Y,X)usingasequenceoflocal,stencil-basedinterpolation problems whichare exact fora cubic orquinticpolyharmonicspline basis (PHS)anda (multivariate) monomial basis of degree p.The matrix E(Y,X)issparse, thatis,it containsn^{N non-zeroelements perrow,wheren is thestencilsize} definedby:

n

=

²



p

+

^d d



.

(12)

DetailsaboutusingPHSplusthemonomialbasisinRBF-FDapproximationscanbefoundin[20–23].Inthesameway,we constructasemi-discreteoperatorfordifferentiationD^L=^DL(Y,X)whichlocallyevaluatesanyderivativeL^ofu˜(Y):

Lu

˜ (

Y

) =

D^L

(

Y

,

X

)

u

˜ (

X

).

(13)

Both matrices, E(Y,X) and D^L(Y,X),can be formedusing theMATLAB codeavailable in[24].For completeness,we belowprovide thesteps tocompute thematrixelements (weights) andtoassemble D^L(Y,X)and E(Y,X)forthepoint setsY and X inanydimensiond.

1. Letx⁽₁^k)=xi∈X (onestencilcenter)andfindn closestneighbors

x⁽_j^k) n

j=¹(stencilpoints)arounditusingtheEuclidean distance.

2. Scaleandshiftthestencilpointstoaunitdomain[−¹,1]^{d.Savethescalingass(k).}

3. FormasquareinterpolationmatrixA^(k),whereA⁽_{i j}^k)=^r3= ||^x(_i^k)−^x(_j^k)||³₂^andx(_i^k),x⁽_j^k),i,j=¹,..,n belongtothestencil.

4. Form a rectangular polynomial matrix P^(k),where P⁽_il^k)= ¯^pl(x⁽_i^k)), i=¹,..,n, l=¹,..,m is a sampled d-dimensional monomial basis withm basisfunctions. When n ischosen asin (12) then m=ⁿ₂ ^{andthe sizeof thematrix P(k) is} n×ⁿ₂^.

5. Using A^(k)andP^(k),formtheaugmentedlocalinterpolationmatrix:

A

˜

^(k)

=



A^(k) P^(k)

(

P^(k)

)

^T 0



(14)

6. Repeatsteps1–5foreveryx∈^{X inordertoformall} A˜^(k).

Thelocalevaluationanddifferentiationweightscanthenbecomputedinthefollowingway:

1. Takeoneevaluationpoint y_l∈^{Y andfindthefirstclosestpointfromthe X pointset.Wedenoteitby x(}₁^k).

2. Shift y_lsuchthatx⁽₁^k)istheorigin,andscaletheshifted vectortoaunitdomain[−¹,1]^{dusingthepreviouslycomputed} scalings^(k).Storetheresultto ¯yl.

3. Form avector b1=L|| ¯yl−^x(j^k)||³2,where{^x(j^k)}ⁿj=¹ isthelocalneighborhood ofthe centerpointx⁽₁^k),whereL=^{I for} constructingevaluationweights,oraderivativeoperatorforconstructingdifferentiationweights.

4. Formavectorb2=L ¯pj(¯yl), j=¹,..,ⁿ₂.

5. Concatenatethetwovectorsintob_L(^y¯l)= [^b1,b₂]^.

6. Use the augmented interpolation matrix that belongs to x⁽₁^k) and compute the local weights by using the relation w_L(¯^yl)= ( ˜^A(k))⁻¹b_L(^y¯l).

7. Storethefirstn weights[w_L(^y¯l)]_1:n tothel-throwofthematrixW_L(Y). 8. Repeatsteps1–7forevery yl∈^{Y inordertoformalllocalweights.}

When using these steps to compute the weights, it is possible to avoid storing the matrix A˜^(k) by combining the two partssuchthat theevaluation/differentiation weightsarecomputedforall yl whichselectx⁽₁^k)asits closeststencilcenter, immediatelyafterstep5ofthefirstlist.Oncethematrixoflocalweights W_L(Y)ofsizeM×^{n iscomputed,theseweights} areassembledintoaglobalrectangularmatrix,E(Y,X)forevaluationandD_L(Y,X)fordifferentiation,bothofsizeM×^N.

Thiscanbedonebyusingamatrix ∈ Z^{N×n,whichisalistofindicesofthelocal}neighborhoodsofpointsaroundevery stencilpoint x⁽₁^k)∈^{X .} Additionally,a matrix

κ

∈ Z^{M×1 isneeded, whichisalist ofindicesofthestencilcenters thatare} closest to each evaluation point y_l. and

κ

are found using the k-nearest neighbor method with k equal to n and 1, respectively.UsingtheMATLABprogramminglanguage,thetwolistsandthesparsematrixcanbeefficientlycomputedby invokingthefollowingthreecommands:

Gamma = knnsearch(X,X,’k’,n) kappa = knnsearch(X,Y,’k’,1)

D_L = sparse(repmat(1:M, 1, n), Gamma(kappa,:), W_L, M, N, N*n)

wherethefirsttwoargumentstothefunctionsparse()havethesameshapeasW_L(Y)andcontaintherowandcolumn indicesforinsertingthe locallycomputedweights intotheglobalmatrix.Wenote thatthealgorithm isnottiedtocubic PHS—itisalsopossibletouseother evenpowersofPHS,i.e.,r⁵,r⁷,andsoon.Inthispaperwe user³ fortwo reasons:

(i) the considered PDE problem contains second derivatives and then the basis functions should be at least two times continuously differentiable,(ii)growthoftheinterpolationmatrixconditionnumberundernode refinementhasa smaller slopecomparedtowhenusinghigherpowersofPHS.

5. Discretizationofthelinearelasticmodelusingtheunfitted RBF-FDmethod

Inthissectionwe employtheunfitted RBF-FDmethod[7] to discretizethelinearelasticityequations(8),(9) and(10) overthediaphragm.ThemethodreliesonconstructingrectangulardifferentiationmatricesD^L(Y,X)asdescribed inSec- tion4.ThematricesthenreplacethedifferentialoperatorsinthePDE,togetherwiththecorrespondingdiscreterighthand sidesF(Y).

Toformthedifferentiationmatrices, wefirstdecidethedegree p ofthemonomial basisthatwe areappendingto the PHSapproximation,andcomputethestencilsizen using(12).Thenweconstructaninterpolationpointset X thatextends overthediaphragm(seeFig.3),andanevaluationpointsetY (seeFig.4)thatconformstothegeometryofthediaphragm.

Thosetwopointsetsareobtainedinfoursimplesteps:

•^{Theinitialpointset X}1isatiltedCartesiannodelayoutwithspacingh inaboxthatenclosesthediaphragm.(thegrey andbluepointsinFig.3).

•^{Then the point set Y}1 is generated by placing q points with average spacing h_y in each Voronoi region inside the diaphragm,definedbythepointsin X1.(theredpointsinFig.4).

Fig. 3. TheinitialpointsetX1(tiltedCartesianpoints)isdistributedoveraboxthatenclosestheboundaryofthediaphragm(blackcurve).Pointsthatare morethanhalfastencilsizeawayfromthegeometry(grey)areremoved.Theremainingpoints(blue)formthepointsetX .

Fig. 4. Theblackcurverepresentsapartoftheboundaryofthediaphragm.NodepointsinX (bluemarkers)andthecorrespondingVoronoiregions(grey lines)areshowntogetherwithinteriorevaluationpoints,Y1,andboundaryevaluationpoints,Yb,(redmarkers).Thesametemplateofinteriorevaluation pointsisusedineachVoronoiregioninsidethediaphragmgeometry.

•^{Inaddition,the pointset Y}b withthesameaveragespacinghy isgenerated byplacingpoints alongthe boundaryof thediaphragm,seeFig.4.

•^{ThefinalevaluationpointsetisgivenbyY}=^Y1∪^Yb,seeFig.4.

•^{Thefinalnodeset X (thebluepointsinFig.3andFig.4)isformedbyreducing X}1 byremovingpoints thatfallmore thanhalfstencilsizeoutsidethediaphragm.InMatlabthiscanbedoneby:

X = X_1(unique(knnsearch(X_1,Y,’k’,ceil(0.5*n)), :);

ThelaststepensuresthatthecolumnsofE(Y,X)andD^L(Y,X)arenon-zero[7].Notethattheevaluationpointsetdoesnot needtobeconstructedbyplacingpreciselyq pointsineveryVoronoiregion.Itispossibletouseanyglobalpointset,which isquasi-uniformbynature(e.g.Haltonpoints),andassuchonaveragesampleseveryVoronoiregionwithapproximatelyq points.

Next, we use the method described in Section 4 to discretize the continuous operator D₂ in (9) using the interior evaluationpoints Y1,andtodiscretizethe continuousoperators D1 andD0 in (10) using theboundaryevaluationpoints Yb.Welet D_k^{i j}(·,X)denotethedifferentiationmatrixthatapproximateselementi,j,oftheoperator Dk,k=⁰,1,2.Ifwe leti,j=¹,2,whilei= ^{j,thenwecanexpressthe}differentiationmatricesas:

Dⁱⁱ₂

(

Y1

,

X

) = (λ +

²

μ )

D^∇ii

(

Y1

,

X

) + μ

D^{∇j j}

(

Y1

,

X

),

D^{i j}₂

(

Y₁

,

X

) = (λ + μ )

D^{∇i j}

(

Y₁

,

X

),

Dⁱⁱ₀

(

Y_b

,

X

) =

^E

(

Y_b

,

X

),

D^{i j}₀

(

Y_b

,

X

) =

⁰

,

Dⁱⁱ₁

(

Y_b

,

X

) = (λ +

²

μ )¯

niD^∇i

(

Y_b

,

X

) + μ

n

¯

jD^∇j

(

Y_b

,

X

),

D^{i j}₁

(

Yb

,

X

) = λ¯

niD^∇j

(

Yb

,

X

) + μ

n

¯

jD^∇i

(

Yb

,

X

).

WeexpressthediscreteRobincoefficientsasKi(Y_b)=^diag(

κ

i(Y_b)),i=⁰,1,andintroduceascalingβi forequationscon- nected withthe operator Di.Finally,we formtherectangular systemofsize 2M×^{2N that}discretizestheNavier-Cauchy systemwithRobinboundaryconditions (8):

⎛

⎜ ⎜

⎝

β2D¹¹₂ β2D¹²₂ β2D²¹₂ β2D²²₂ β0K0D¹¹₀ + β1K1D¹¹₁ β1K1D¹²₁

β1K1D²¹₁ β0K0D²²₀ + β1K1D¹²₁

⎞

⎟ ⎟

⎠



u˜1(X)

˜ u2(X)



=

⎛

⎜ ⎜

⎝

β2f1(Y1) β2f2(Y1) β0K0g1(Y_b)+ β^1K1h1(Y_b) β0K0g2(Yb)+ β1K1h2(Yb)

⎞

⎟ ⎟

⎠ .

⁽¹⁵⁾

Inthenumericalexperiments,weusethefollowingscalefactors:

β

2

=

¹

μ

^hy

^{, β}

¹

⁼

10

μ

1 hh

1

2y

, β

0

=

¹ hh

1

y2

,

(16)

whereh andhy aretheaverageinternodaldistancesinthe X andY pointsets,respectively.Thesearecomputedas:

h

=

¹ N



N j=1

mini=^j

^xi

−

^xj

2

,

hy

=

¹ M



M j=1

mini=^j

^yi

−

^yj

2

.

(17)

Thechoiceofscalefactorsisbasedonthepapers[7,8],wherethescalefactorshyfortheinteriorandh¹_y^/2fortheboundary are used,such thatthenorms ofthediscrete leastsquaresproblemapproximatethecontinuous L2-norm.The additional 1/h scalingincreasestheweightoftheboundaryconditionsandimprovesconvergence.TheLaméparameter

μ

islargeand affectsthescalingbetweendifferentequations.Therefore,wealsoincludethefactor10/

μ

inβ1.

Wesolvetheleast-squaresproblem(15), forthenodalvaluesu˜1(X)andu˜2(X),usingthebackslashcommandinMAT- LAB, which uses a sparse QR decomposition asone of the solution steps in the background. After that the solution is evaluatedattheY pointset,usingtheevaluationmatrix E:

˜

u1

(

Y

) =

E

(

Y

,

X

)

u

˜

1

(

X

),

u

˜

2

(

Y

) =

E

(

Y

,

X

)

u

˜

2

(

X

).

(18) Thestrainsandthestresses(1),(3) arecomputedbyapplyingappropriatedifferentiationmatricestothesolutioncoefficients

˜

u1(X)andu˜2(X).

6. Smoothingofgeometryandboundarydata

Ifweviewthetwo-dimensionaldiaphragmfromthecontinuousperspective,thegeometrycanbedescribedasaclosed curve. We chooseto parametrizethis curve byt∈ [⁰,2

π

_], wherethe starting point t=^{0 isthesame asthefinal point} t=²

π

, i.e., the curve is periodic with period 2

π

. To benefit fromthe potential high-order convergence of the unfitted RBF-FDmethod,weneedtoapproximatetheboundarycurveandtheboundarydataasfunctionsoft withenoughsmooth- nessthat theconvergenceofthePDEproblemisnotadversely affected.There existanumberofalgorithms for(periodic) datarepresentationgivenacertainparameterization.AnRBF-basedrepresentationalgorithmusingRBFswithglobalsupport is givenin[25] wherethe workingcontext aretwo dimensional andthreedimensional platelets.Inthe samepaper, the authors comparetheRBF-basedalgorithm toFourier-based algorithms (usingtrigonometric polynomialsorsphericalhar- monics),andpiecewiselinearapproximations.TheauthorsfoundthatglobalRBF-basedalgorithmsareeasiertoimplement, that their accuracy iscomparable toFourier-based algorithms forsmooth data,andthat globalRBF-basedalgorithms are moreflexiblethanFouriermethodswhenswitchingacrossdifferentparameterizations.Inthispaper,weusealocalizedRBF- basedapproachinstead,i.e.,wereuseleast-squaresunfittedRBF-FDmethodtorepresentthedata.Toenforceperiodicityof thedatarepresentationfunctionsweaddasetofperiodicconditionsintothediscretizedsystemofequations.

Toavoidreducedaccuracydueto boundaryerrorsneartheartificialendpoints oftheinterval,we extendthedomain periodicallytot∈ [−²

π

,4

π

]^fortheapproximation.Wediscretizetheextendeddomainusingtheuniformlyspacednode points T= {^tj}^N_j=^g₁ ^{wherethe setalsoincludespoints 0 and 2}

π

.Given M˜g datapoints, wereplicatetheseperiodicallyto gettheextendeddataset(T^d,G)= {(^td_i,gi)}^M_i=^g₁^,whereMg=³M˜g>Ng.Weuseaone-dimensionalRBF-FDapproximation, basedonaquinticPHSbasisaugmentedwithpolynomialsofdegree p_g=^6,toformanoverdetermined linearsystemfor thenodalvaluesg(T)

E_g

(

T^d

,

T

)

g

(

T

) =

^G

.

(19)

Fig. 5. Thesmoothedboundarygeometrycurve(solidline)isshowninbothsubfigures.ThecurvewascomputedusingNg=133 nodepoints,Mg= 177·³=^{531 datapoints,andstencilsizen}=^{28.Themarkersshowthe}M˜g=^{177 vertexdatapoints(top)anduniformevaluationpoints(bottom).The} normalscomputedfromtheapproximation,aswellasthevaluesoftheparametert alongthecurve,arealsoshowninthebottomsubfigure.

Fig. 6. Thetwosmoothedcoordinatefunctionsapproximatingthegeometry.Thefunction p(t)(left)correspondstothehorizontalcoordinateandq(t) (right)correspondstotheverticalcoordinate.Themarkersshowtheinitialdatalocations.

Toenforcecontinuityatt=^{0,weaddthefollowingequality}constraints

D^∇_g^s

(

0

,

T

) −

^D∇g^s

(

2

π ,

T

) =

⁰

,

s

=

⁰

, . . . ,

pg

−

¹

,

(20)

where D^∇_g^s(0,T) and D^∇_g^s(2

π

,T) are differentiation matricesapproximating the s-th derivative in points 0 and2

π

.We solvetheconstrainedleastsquaresproblem:



2E^T_gEg B^T_g B_g 0

 

_g

(

T

)

¯λ



=



2E^T_gG 0



,

(21)

where B_g containsthe p_gconstraints (20) and ¯λcontainsthecorrespondingLagrangemultipliers.

To find the smooth boundary curve from the initial vertex data x˜^d_i, i=¹,. . . ,M˜g, we first scale the data such that x^d_i = (^pi,q_i)=^s^x˜^d_i^,i=¹,. . . ,M˜_g,wheres_=¹⁵⁶.92⁻¹mm⁻¹.Thescalingwaschosensuchthatalldatapointsfallwithin [−¹,1]^{2.Then we compute an} approximate arclength parametrization using the Euclideandistance between the scaled vertices, such that t^d_i =²

π

^i_j⁻₌¹₁ ^xd_j+1−^xd_j /M^˜g

j=^{1 xd}_j+1−^xd_j ^{,where xd}_M˜

g+1=^xd₁^{.Then we replicatethe data over the} extendeddomain.Newpointcoordinates p(T^e)andq(T^e)areobtainedby:(i)constructinganequidistantpointset T^e on [⁰,2

π

),(ii) solving the system (21) for theunknowns g(T) and ¯λ, (iii)substituting g(T) in(19), where E_g isevaluated in points T^e instead of T^d. The resultingboundary curve is shownin Fig. 5and theindividual coordinate functionsare showninFig.6.Sincethecurve parametrizationhereisintheclockwisedirection, theoutwardnormalsarecomputedas

¯

n(t)= (−^q(t),p(t))/ (−^q(t),p(t)) ^{,wherethefirstderivativeinapointt is}approximatedbyapplyingD^∇1(t,T)tovector g(T)correspondingtoeitherq(t)orp(t).

For the boundary data functions g and h in (7), we manufacture data to mimic the expectedphysiological behavior showninFig.2.Afewdatapointsareplacedintheregionswherewehavesome information(regions1,3,5,7,and9in Fig.2),andthentherestofthedataisgeneratedthroughpiecewiselinearinterpolation.Thisresultsingradualtransitions intheregionswherewelackinformation.ThedatapointsandtheresultingcurvesareshowninFigs.7and15.

Fig. 7. BenchmarkI:Thedisplacementinthehorizontaldirection(left)andtheverticaldirection(right)asafunctionoftheboundaryparametrizationt (seeFig.5,rightimage,foranillustrationoft inrelationtotheboundary).Themarkersshowtheinitiallyplaceddatapoints,whicharethenlinearly interpolatedintoM˜g=^{80 datapoints.Forthe}approximation,Ng=^{120 nodepointsandstencilsizen}=^{28 wereused.Thedashedlinesshowthelocation} oftheendpointsofthediaphragm(regions2and8inFig.2).

Fig. 8. Benchmark I: The diaphragm in its non-deformed state (dashed line) and after displacement of the boundary (solid line).

7. BenchmarkI:deformationofthediaphragmusingthesmoothedDirichletboundaryconditions

Inthissectionwesolvethediscretizedlinearelasticityequations(15).Weareinterestedinwhethertheproblemwith smoothedDirichletboundaryconditionleadstoahigh-orderconvergence.Apointofinterestisalsowhethertheresulting deformationisphysiologicallysensible,andwhethertheVonMisesstressisdistributedasexpected.

TheboundaryconditionthatweuseispurelyDirichlet,whichmeansthatwesettheRobincoefficientsin(8) to

κ

0(y)= 1 and

κ

1(y)=^{0.Thentheonlyboundarydatafunctionspresentinthesystem(15) are g}1(y)andg₂(y),whichcorrespond totheimpositionofdisplacementsinthehorizontalandverticaldirection,respectively.

7.1. Theimpositionofboundarydisplacements

The displacements havebeen synthesized toreproduce the physiologicalbehavior described in Section 2.Particularly, thetranslationofthehorizontalpartofthediaphragm(region5)hasbeendefinedasaconstantverticaldisplacementin thedownwarddirectionandthethickeningoftheappositionalzone(regions3and5)isalsoprescribedasaconstant.We usethisinformationtocompute 2 to5 datapointsineachzone,wherethelargernumberofdatapointsperzoneisused forzoneswherethetransitionindataissharper.Therestofthedataisgeneratedbypiecewiselinearinterpolation.From aphysiologicalperspectivealldisplacementsshouldbesmooth.Wegenerateasmoothfunctionbysolvingtheconstrained leastsquaresproblem (21).TheresultsareshowninFig.7.Imposingsmoothboundarydatamakesitpossibletoobtainhigh- orderconvergenceandprovidesasolutionthat isphysicallyrelevant.InFig.8we displaytheboundaryofthediaphragm beforeandafterapplicationofthedisplacementsfromFig.7.

7.2. SolutionofBenchmarkI

ThesolutiontothebenchmarkproblemisgiveninFig.9,wherewedisplaythespatialdistributionofthedisplacements andtheVonMisesstress.Lookingatthe^u˜1distributioninthefigure,wecanobservethickeningofthediaphragminregions 2,3,7and8accordingtothelabelsfromFig.2.Next,thedistributionof^u˜2 indicatesthattranslationislargestinregions 4,5and6.TheVonMisesstressislargestattheinterfacesbetweenregions2,3andregions7,8.Thismakessense,since thechangeinthethicknessislargestintheseregions.Weconcludethatthebehaviorroughlyfollowsthephysiologicalcues describedinSection2.

7.3. Convergenceundernoderefinement

Whilethe solutionisroughlywhat weexpect froma physiologicalperspective,we areyetto understandwhetherthe simulationgivesacorrectanswerfromanumericalperspective.Weinvestigatetheconvergenceofthenumericalsolution undernoderefinementusingseveraldifferentpolynomialdegreesinthestencil-basedapproximation.Sincewedonotknow

Fig. 9. BenchmarkI:ThecomputeddisplacementsandthecorrespondingvonMisesstressoverthediaphragm.Thissolutionwasobtainedusingtheunfitted RBF-FDdiscretizationwithinternodaldistanceh=⁰.004,oversamplingparameterq=^{5,andanappendedpolynomialbasisofdegreep}=^5.Duetothe scalingappliedtothegeometry(seeSection6),thedisplayedresultsfordisplacementandstressshouldbemultipliedwiths⁻_¹=⁰.15692 mtorecover theresultscorrespondingtotheunscaledproblem.

Table 1

BenchmarkIandII:Therelationbetweenthe inverseinternodaldistance 1/h and(i)theinternodal distanceh and(ii)thenumberofinterpolationpointsN usedfordiscretizingthePDEproblem(8).

1/h 25 50 100 125 166.67 200 250 500

h 0.04 0.02 0.01 0.008 0.006 0.005 0.004 0.002

N 337 853 2500 3617 5904 8126 12129 43841

whatthetruesolutionis,wemeasureconvergenceofthenumericalsolution^u˜(Y)towardsanumericalreferencesolution

˜

u_∗(Y_∗),wherethe node setishighly refined. Wechoose twonumericalreferences: (i) computedusingtheunfitted RBF- FD-LSmethodwithinternodaldistanceh=0.002,leading toN=43841 andpolynomialdegree p=5,(ii)computedusing theGalerkinfiniteelement methodwithlinearelements and236414 interpolationpoints(correspondingtoh=⁰.00085), usingtheGetDPsolver[26].Everynumericalsolution u that˜ weobtainisinterpolated,consistentwiththeapproximation order,tothepointsetofthereferencesolutionY_∗,wherewethencomputetheapproximationerror:

^e

2

= ˜

^u

(

Y_∗

) − ˜

^u_∗

(

Y_∗

)

2

˜

^u∗

(

Y_∗

)

2

.

(22)

Table1showstherelationbetweentheinternodaldistanceh,computedaccordingto(17),andthenumberofinterpolation points N fortheconsideredproblemsizes.

TheresultswhentheunfittedRBF-FDmethodisusedasareferencearedisplayedinFig.10.Weobservethattheerror issmallforallpolynomialdegrees.Theconvergencerateincreasesasp isincreased.Forall p theconvergenceratesofthe displacements arecloseto p−^1.The convergencerateofthe VonMisesstressissimilar totheconvergencerates ofthe displacements.

The resultswhenthefinite elementmethodisused asa referencearegiveninFig. 11.Theconvergenceplots forthe displacementareverysimilartotheresultsinFig.10.Theconvergencerateofthestressforp=^{5 islowercomparedwith} theself-referencetestprovidedinFig.10.Furthermore,whenp=^5,theconvergenceseemstobestallingatthelastpoint ofobservation.Ourspeculativereasoningisthatthederivativesinthefiniteelementspacedonotapproximatethestresses wellenoughthere.ThisimpliesthattheFEMmeshistoocoarsetomatchtheaccuracyoftheunfittedRBF-FDmethodfor derivativeapproximation,whenthepolynomialdegreeisashighasp=^{5,fortherangeofh usedhere.}