Dimension of polynomial splines of mixed smoothness on T-meshes

Delft University of Technology

Dimension of polynomial splines of mixed smoothness on T-meshes

Toshniwal, Deepesh; Villamizar, Nelly

DOI

10.1016/j.cagd.2020.101880

Publication date

2020

Document Version

Final published version

Published in

Computer Aided Geometric Design

Citation (APA)

Toshniwal, D., & Villamizar, N. (2020). Dimension of polynomial splines of mixed smoothness on T-meshes.

Computer Aided Geometric Design, 80, 1-10. [101880]. https://doi.org/10.1016/j.cagd.2020.101880

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Computer Aided Geometric Design 80 (2020) 101880

Contents lists available atScienceDirect

Computer

Aided

Geometric

Design

www.elsevier.com/locate/cagd

Dimension

of

polynomial

splines

of

mixed

smoothness

on

T-meshes

Deepesh Toshniwal

a

,

∗

,

Nelly Villamizar

b

a_{DelftInstituteofAppliedMathematics,DelftUniversityofTechnology,Netherlands}

b_{DepartmentofMathematics,SwanseaUniversity,UnitedKingdomofGreatBritainandNorthernIreland}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline13May2020

Keywords: Splines T-meshes Mixedsmoothness Dimensionformula Homologicalalgebra

Inthispaperwestudythedimensionofsplinesofmixedsmoothnessonaxis-aligned T-meshes.Thisisthesetting whendifferentordersofsmoothness arerequiredacrossthe edgesofthemesh.GivenasplinespacewhosedimensionisindependentofitsT-mesh’s geometricembedding,wepresent constructiveandsufficientconditionsthatensure that the smoothness acrossa subset of the mesh edges can be reduced while maintaining stabilityofthedimension.Theconditionshaveasimplegeometricinterpretation.Examples are presented to show the applicability of the results on both hierarchical and non-hierarchicalT-meshes.ForhierarchicalT-meshesitisshownthatmixedsmoothnessspline spaces that contain the space of PHT-splines (Deng et al., 2008) always have stable dimension.

©2020TheAuthor(s).PublishedbyElsevierB.V.Thisisanopenaccessarticleunderthe CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Polynomialsplinesonpolyhedralpartitionsareubiquitousinapproximationtheory,geometricmodelling,and computa-tionalanalysis. It iscustomaryto asksplines tobe Cr smooth acrossall meshfacetsfora fixed choiceofr

∈ Z

_−1 that dependsontheintendedapplication.However,certainapplicationsalsorequireworkingwithsplinesforwhichsmoothness canbe reducedacrossan arbitrarysubset ofthemeshfacets;e.g.,tomodelnon-smoothorevendiscontinuous geometric features.Suchsplineswillbesaidtohavemixedsmoothness,andtheyconstitutethefocusofthisarticle.

Example(Applicationtofluidflowsaroundthinsolids).Considerthecaseofathinsolid immersedinanincompressiblefluid flow, anda numericalsimulationthat employs a solid-conformingmesh, i.e.,a meshwhere thesolid is modelled asthe union of a subset of the facets. In general, we would like to use smooth splines for approximating the fluid pressure and velocity fields.However, unless the discrete pressure field is allowed to be discontinuous across the thinsolid, the simulationresultswouldbemeaningless.Atthesametime,wewouldliketoretainsmoothnessofthepressurefieldacross theremainingfacets.SeeSauerandLuginsland(2018) foranexampleofsuchanapplication.



An appealingfeature of splines inapplications is theflexibility in the choice ofthe underlyingmeshes. In particular, there isa rich historyofthe use ofsimplicial, quadrilateraland cuboidalmeshes foruniformpolynomial degreesanda fixed order ofglobalsmoothness, see e.g., Cirak etal.(2000) and Hughes etal.(2005). Univariate splinespaces andthe

*

Correspondingauthor.

E-mailaddresses:d.toshniwal@tudelft.nl(D. Toshniwal),n.y.villamizar@swansea.ac.uk(N. Villamizar). https://doi.org/10.1016/j.cagd.2020.101880

0167-8396/©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

construction of a suitable spline basis forthem, calledthe B-spline basis, are well understood, see de Boor (2001) for example. A spline basis for tensor product spline spaces can be easily defined on tensor product quadrilateral meshes by takingtensorproductsofunivariate B-splines;thisprocesscan bedirectlyextendedto higherdimensionsforbuilding multivariatesplinespaces.AcomprehensiveoverviewofsplinesontriangulationscanbefoundinLaiandSchumaker(2007) andthereferencestherein.

When applicationsrequirethe resolutionofthe splinespaceto beincreasedon asubset ofthe meshfaces, themost commonapproach istoemploy localsubdivision.Splineconstructionsonsuch locallysubdividedmesheshavebeen pro-posedinSpeleersetal.(2009);SchumakerandWang(2012b) andKangetal.(2014) fortriangulationsandinSederberget al.(2003);Giannellietal.(2012) andDokkenetal.(2013) forquadrilateralmeshes,amongothers.Wewillfocusonthecase oflocallysubdividedquadrilateralmeshes,theso-calledT-meshes.ExamplesofsuchmesheswillbediscussedinSection5. The studyofmultivariatesplines,andbivariatesplinesonT-meshesinparticular,posesaninteresting challengeasthe splinespacedimensioncandependonthegeometric embeddingofthemesh,seeforinstanceDengetal.(2006);Lietal. (2006);LiandChen(2011);SchumakerandWang(2012a) andLiandWang(2019).Inpractice,identifyingmesheswhere thedimension isstable–i.e.,freefromthisdependence–isusefulforavoidingcaseswheresplinespaceson combinato-rially andtopologicallyequivalent mesheshavedifferentdimensions. Severaltechniqueshave beenusedforstudying the dimension ofmultivariate splines.Wewilldosoforsplinesofmixedsmoothnessusingthehomology-basedapproach in-troducedinBillera(1988),andthereforeinthefollowingwesticktoabriefdiscussionofthesame.Itshouldbenotedthat other approaches suchasBernstein–Bézier methods (AlfeldandSchumaker, 1987) orsmoothingcofactor-conformality (Li andDeng, 2016) areequally suitedtostudy theproblem,andmaybe alternativelyusedtoachieve thesameresultsthat wedo.

Byinterpretingsplines asthetophomologyofachaincomplex,Billera(1988) usedtoolsfromhomologicalalgebrafor studyingthedimensionofsplines. ModificationsofthecomplexesproposedbySchenckandStillman(1997a) andSchenck and Stillman(1997b) havesince been usedby Mourrain andVillamizar(2013) for boundingthespline space dimension on simplicial meshesin two andthree dimensions. Schenck and Sorokina (2018) have recently studied the problem on simplicial mesheswhere onemaximal face hasbeensubdivided. On T-meshes,Mourrain(2014) provided boundson the dimension ofbi-degree

(

m

,

m

)

splines.GeneralizationsoftheboundsfromMourrainandVillamizar(2013) andMourrain (2014) to splines with local polynomial degree adaptivity been recently provided in Toshniwal and Hughes (2019) and Toshniwaletal.(2019).Thetoolsfromhomologyhavealsobeenappliedtostudy ofnon-polynomial splinesonT-meshes where,inparticular,theringstructureofpolynomialscannotbeused;seeBraccoetal.(2016a,b) andBraccoetal.(2019).

Let

R

r_mm denotethespaceofbi-degree

(

m

,

m

)

splinesthat are

r

(

τ

)

smoothacross meshedge

τ

.Asstatedabove,we

willusehomology-basedtechniquessimilartotheonesusedinBillera(1988);SchenckandStillman(1997b) andMourrain (2014) tostudy thedimensionof

R

r

mm.Then,giventhat

R

mmr  hasstabledimension, weprovide sufficientconditionsfor

preservation ofthisstability whenthedesiredordersofsmoothnessaredecreasedacrossa subsetofthemeshedges.Let usdenotethislattersplinespacewith

R

s_mm,with

s

(

τ

)



r

(

τ

)

foralledges

τ

.Notethatingeneraltheresultsproposedin

Mourrain(2014) cannotbeappliedtocomputethedimensionof

R

s_mm.Thisisbecausetheyrequirethesmoothnessacross

allhorizontal(resp.vertical)edgesthatformaconnecteduniontobethesame;wedonotimposethesamerestrictionhere. Insteadofstudying

R

_mms  fromscratch,weuseinformationfrom

R

rmm toconsiderablysimplifytheproblem.Inparticular,in

Section 4we providesufficientconditionsthatensurethatthedimensionof

R

s_mm canbecomputedcombinatoriallyusing

localinformationonly.Theconditionsareconstructiveinnatureandhaveasimplegeometricinterpretation.Applicationof theresultstobothhierarchicalandnon-hierarchicalT-meshesarepresentedinSection5.

2. Preliminaries:splines,meshesandhomology

ThissectionwillintroducetherelevantnotationthatwewilluseforworkingwithpolynomialsplinesonT-meshes.

2.1. SplinesonT-meshes

Definition2.1(T-mesh).AT-mesh

T

of

R

2 isdefinedas:

•

afinitecollection

T

2 ofaxis-alignedrectangles

σ

thatweconsiderasopensetsof

R

2 havingnon-zeromeasure,called 2-cellsorfaces,togetherwith

•

a finiteset

T

1 ofclosed axis-alignedsegments

τ

,called1-cells,which areedges ofthe(closureofthe) faces

σ

∈ T

2, and

•

theset

T

0,ofvertices

γ

,called0-cells,oftheedges

τ

∈ T

1, suchthatthefollowingpropertiesaresatisfied:

•

σ

∈ T

2

⇒

theboundary

∂

σ

of

σ

isafiniteunionofedgesin

T

1,

•

σ

,

σ



∈ T

2

⇒

σ

∩

σ



= ∂

σ

∩ ∂σ

isafiniteunionofedgesin

T

1

∪ T

0,and,

D. Toshniwal, N. Villamizar / Computer Aided Geometric Design 80 (2020) 101880 3

ThedomainoftheT-meshisassumedtobeconnectedandisdefinedas



:= ∪

σ∈T2

σ

⊂ R

2_.

Setsofhorizontalandverticaledgeswillbedenotedbyh

T

1 andv

T

1,respectively.EdgesoftheT-mesharecalledinterior edgesiftheyintersect theinteriorofthedomainoftheT-mesh



◦.Otherwise,theyarecalledboundaryedges.Thesetof interioredgeswillbedenotedby

T

◦1;andthesetsofinteriorhorizontalandverticaledgeswillbedenotedbyh

◦

T

1andv

◦

T

1, respectively.Similarly, ifavertexisin



◦ itwillbe calledaninteriorvertex, andaboundaryvertexotherwise.The setof interiorverticeswillbedenotedby

T

◦0.Wewilldenotethenumberofi-cellswith

t

i

:=

#

T

i.

Assumption2.2.Thedomain



issimplyconnected,and



◦ isconnected.

AT-mesh whichsatisfies Assumption2.2will besaid tobe simplyconnected.We define

P

mm asthe vector spaceof

polynomialsofbi-degreeatmost

(

m

,

m

)

spannedbythemonomialssi_tj_,0

_

_i

_

_{m and0}

_

_j

_

_m_{.Ifeitherofm orm}_are

negative,then

P

mm

:=

0.Thefinalingredientthatweneedfordefiningasplinespaceon

T

isasmoothnessdistributionon

itsedges.

Definition2.3(Smoothnessdistribution).The map r

: T

1

→ Z

−1 is called a smoothnessdistribution if r

(

τ

)

= −

1 for all

τ

∈

/

T

◦1.

Usingthisnotation,wecandefinethesplinespace

R

r_mm thatformstheobjectofourstudy.Fromthefollowingdefinition

andthedefinitionof

r,

itwillbeclearthatweareinterestedinobtaininghighlylocalcontroloverthesmoothnessofsplines in

R

r_mm,afeaturethatismissingfromtheexistingliteraturewhichstudiessplineonT-meshes.

Definition2.4(Splinespace).Givenmesh

T

,bi-degree

(

m

,

m

)

∈ Z

2_0,smoothnessdistribution

r,

wedefinethesplinespace

R

r

_{≡ R}

r mm

(

T)

as

R

r mm

(

T) :=



f

: ∀

σ

∈ T

2 f

|

σ

∈ P

mm

,

and

∀

τ

∈

◦

T

1 f

∈

Cr(τ)smooth across

τ



.

(1)

From theabovedefinition,thepiecesofall splinesin

R

r areconstrainedtomeetwithsmoothness

r

(

τ

)

atan interior edge

τ

;wewillalsodefine

rh

(

γ

)

:=

min_τγ τ∈v_T 1 r(

τ

) ,

rv

(

γ

)

:=

min_τγ τ∈h_T 1 r(

τ

) .

Wewillusethefollowingalgebraiccharacterizationofsmoothnessinthisdocument.Proofsofthischaracterizationcanbe foundinseveraltexts;e.g.,seeChui(1988) andBillera(1988).

Lemma2.5.For

σ

,

σ



∈ T

2,let

σ

∩

σ



=

τ

∈

◦

T

1.Apiecewisepolynomialfunctionequallingp andpon

σ

and

σ

,respectively,isat leastr timescontinuouslydifferentiableacross

τ

ifandonlyif



r_τ+1



p

−

p

,

where



τ isanon-zerolinearpolynomialvanishingon

τ

.

Inlinewiththeabovecharacterizationandforeachinterioredge

τ

,wedefine

I

r_{τ to}bethevectorsubspaceof

P

mm that

containsall polynomialmultiplesof



τr(τ)+1; when

r

(

τ

)

= −

1,

I

rτ issimplydefinedtobe

P

mm.Similarly,foreachinterior

vertex

γ

,wedefine

I

r_γ

:=



_τγ

I

r_{τ .}

Remark2.6.Inthe above, we have suppressedthe dependence ofthe differentvector spaces on

(

m

,

m

)

to simplifythe reading(andwriting)ofthetext.

2.2. Topologicalchaincomplexes

Anyspline f

∈ R

r _{isapiecewise polynomialfunction on}

_T

_{.We canexplicitlyreferto itspiecewise polynomialnature} byequivalently expressingit



_σ

[

σ

]

fσ with fσ

:=

f

|

σ .Thisnotation makesitclearthat thepolynomial fσ isattachedto

theface

σ

of

T

.UsingthisnotationandLemma2.5,thesplinespace

R

r _{canbeequivalentlyexpressedasthekernelofthe} map

∂

,

∂

: ⊕

σ∈T2

[

σ

]P

mm

→ ⊕

τ∈T◦1

[

τ

]P

mm

/I

rτ

,

definedbycomposingtheboundarymap

∂

withthenaturalquotientmap.

As aresultofthisobservation,thesplinespace

R

r _{canbe interpretedasthetop homologyofasuitablydefinedchain} complex

Q

r,

Q

r

_:



σ∈T2

[σ]P

mm



τ∈T◦1

[τ]P

mm

/I

rτ



γ∈T◦0

[γ]P

mm

/I

rγ 0

.

Inotherwords,wehave

R

r

∼

₌

_ker



_∂



₌

_H 2

(

Q

r

) .

As inBillera(1988);SchenckandStillman(1997a) and Mourrain(2014),we willstudy

Q

usingthe followingshortexact sequenceofchaincomplexes,

0 0

I

r

_:

₀



τ∈T◦1

[τ]I

r τ



γ∈T◦0

[γ]I

r γ 0

C :



σ∈T2

[

σ

]P

mm



τ∈T◦1

[

τ

]P

mm



γ∈T◦0

[

γ

]P

mm 0

Q

r

_:



σ∈T2

[

σ

]P

mm



τ∈T◦1

[

τ

]P

mm

/I

rτ



γ∈T◦0

[

γ

]P

mm

/I

rγ 0 0 0 (2)

Thefollowingresultanditsproofcanbefoundin,forinstance,Mourrain(2014).Weincludeithereforcompleteness.

Theorem2.7.ForasimplyconnectedT-mesh

T

2,thedimensionofthesplinespaceofbi-degree

(

m

,

m

)

andsmoothnessdistribution

r is givenby

dim



R

r



=

χ



Q

r



+

dim



H0

(

I

r

)



,

whereH0

(

I

r

)

isthezerothhomologyofthecomplex

I

rand

χ



Q

r



_{istheEulercharacteristicofthecomplex}

_Q

r_,

χ



Q

r



_{= t}

2

(

m

+

1

)(

m

+

1

)

− (

m

+

1

)

τ∈h_T 1

(

min

(r(

τ

),

m

)

+

1

)

− (

m

+

1

)

τ∈v_T1

(

min

(r(

τ

),

m

)

+

1

)

+

γ∈T0

(

min

(r

h

(

γ

),

m

)

+

1

)(

min

(r

v

(

γ

),

m

)

+

1

) .

Proof. Following Assumption2.2,itisclearthat H0

(

C)

=

0

=

H1

(

C)

.Moreover,fromthelongexactsequenceofhomology impliedbytheshortexactsequenceofcomplexesinEquation(2),weobtain

H0

(

Q

r

)

=

0

,

H0

(

I

r

) ∼

=

H1

(

Q

r

) .

Therefore,theclaimfollowsuponrecalling

R

r

∼

₌

_H

2

(

Q

r

)

andthedefinitionoftheEulercharacteristicof

Q

r,

χ



Q

r



=

dim



Q

r₂



−

dim



Q

r₁



+

dim



Q

r₀



,

=

dim



H2

(

Q

r

)



−

dim



H1

(

Q

r

)



+

dim



H0

(

Q

r

)



.



Corollary2.8.Ifdim



H0

(

I

r

)



=

0,thenthedimensionisstableandcanbecomputedusingthefollowing(combinatorial)formula,

D. Toshniwal, N. Villamizar / Computer Aided Geometric Design 80 (2020) 101880 5

3. Splinespace

R

s

⊇ R

rofreducedregularity

Inthisintermediatesection,wewillrelatethedimensionofthesplinespace

R

r _{tothedimensionofasplinespace}

_R

s obtainedby relaxingtheregularityrequirements.Thatis,forallinterioredges

τ

,itwillbe assumedthat

s

(

τ

)



r

(

τ

)

.This relationshipwillbeutilizedinthenextsectiontopresentsufficientconditionsforthedimensionof

R

s tobestable.

Forthesplinespace

R

s_{,letthefirstandlastchaincomplexesinEquation(2) bedenotedby}

_I

s_and

_Q

s_{,respectively.The} splinespacedimensionisthereforegivenasbelow,

dim



R

s



=

dim



H2

(

Q

s

)



=

χ



Q

s



+

dim



H0

(

I

s

)



.

(3)

Then,bydefinitionofthesmoothnessdistributions

r and s,

wehavethefollowinginclusionmapfrom

I

r to

I

s,

I

r

₋

_{→ I}

ι s

_.

Proposition3.1.IfH0

(

I

r

)

=

0,thenH0

(

I

s

)

∼

=

H0

(

I

s

/

I

r

)

.

Proof. The claim follows from the following short exact sequence of chain complexes (and the long exact sequence of homologyimpliedbyit),

0

I

r

I

s

I

s

/

I

r 0

.



Theprevious resultconsiderablysimplifiesthetaskofidentifyingwhen H0

(

I

s

)

willvanishbecause H0

(

I

s

/

I

r

)

canbea simplerobjecttostudy.Let

T

₁s bethesetofedges

τ

forwhich

s

(

τ

)

<

r

(

τ

)

,andlet

T

₀s bethesetofinteriorverticesofthe edges

τ

∈ T

s

1.Thefollowingresultfollows.

Lemma3.2.Thecomplex

I

s

/

I

rissupportedonlyon

T

₁sand

T

s₀.

Proof. The claim follows fromthe definitionof the complexes

I

s _and

_I

r_{. Indeed, if s}

₍

_τ

₎

₌

_r

₍

_τ

₎

_{, then}

_I

s

τ

= I

rτ andthe

cokernelof

ι

iszeroon

τ

;similarlyforthevertices.



4. Dimensionofsplinesofmixedsmoothness

Thissectioncontains ourmainresults.Starting froma splinespacewithstabledimension,wespecifysufficient condi-tions whenthedimension can stillbe computedusingCorollary 2.8afterthesmoothnessrequirements arerelaxedfora subsetoftheinterioredges.Wefirstdefinetheweightofaconnectedunionofhorizontalorverticaledges.

Definition4.1(Segmentanditsweight).Let

ρ

⊆

T

◦1

∪ T

0 be a finitesetofhorizontal (resp.,vertical) edges

τ

∈

◦

T

1 together withtheirvertices

γ

∈

τ

,suchthat

τ∈ρ

τ

isconnectedanditcontainsatleastoneedge.Then

ρ

willbecalledahorizontal (resp.,vertical)segment.Itsweight

ω

r

₍

_ρ

₎

_{willbedefinedas}

ω

r

(

ρ

)

:=

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

γ∈ρ



m

−

rh

(

γ

)



+ if

ρ

is horizontal

,

γ∈ρ



m

−

rv

(

γ

)



+ if

ρ

is vertical

.

Theorem4.2.Let

r be

suchthatH0

(

I

r

)

=

0 andlet

ρ

bea segmentofthemesh. Considerthespace

R

s wherethesmoothness distribution

s is

definedasfollowsforsomer

∈ Z

_−1,

s(

τ

)

=



r(

τ

)

for

τ

∈

/

ρ

∩

T

◦1

,

r



r(

τ

)

for

τ

∈

ρ

∩

T

◦1

.

Ifeitheroneofthefollowingtworequirementsissatisfied,

(a)

ρ

ishorizontaland

ω

s

₍

_ρ

₎

_

_m

₊

_1;otherwise,

_ω

s

₍

_ρ

₎

_

_m

₊

_1;

(b)

ρ



ρ

forsomesegment

ρ

,and

s

(

τ

)



r forall

τ

∈

ρ



thenH0

(

I

s

)

=

0 anddim



Proof. By Lemma 3.2, the study of H0

(

I

s

)

reduces to the study of H0

(

I

s

/

I

r

)

on the segment

ρ

— an essentially one dimensionalproblem.Wewillprovidetheproofforwhen

ρ

isahorizontalsegmentasshownbelow;theproofforvertical segments is analogous. As shown,

ρ

contains the edges

τ

1

,

. . . ,

τ

k

∈

◦

T

1 andthe vertices

γ

0

,

γ

1

,

. . . ,

γ

k. By definition

ρ

containsatleasttwodifferentvertices.

Condition(a).Letusfirstprovetheclaimforthesettingwhen

ρ

satisfiescondition(a)above.Let



ρ beanon-zerolinear

polynomialthatvanisheson

ρ

,andlet



0

,

. . . ,



kbenon-zerolinearpolynomialsthatvanishonverticaledgesthatcontain

thevertices

γ

0

,

. . . ,

γ

k,respectively.Letr0

,

. . . ,

rk besuchthatri

=

rh

(

γ

i

)

=

sh

(

γ

i

)

.

Bydefinition,

I

s_τ

=





r_ρ+1f

:

f

∈ P

m(m−r−1)



.Since

ω

s

₍

_ρ

₎

_

_m

₊

_{1 andthevertices}

_γ

iarealldifferentthen,foranyi

=

j,

therearepolynomials fi,i

=

0

,

. . . ,

k,suchthat1

=



k

i=0



ri+1

i fi(Mourrain,2014,Proposition1.8).Thuswecanwrite

I

s_τ

= 

r_ρ+1 k

i=0



ri+1 i

P

(m−ri−1)(m−r−1)

.

Then, anyelement



r_ρ+1f in

I

s_γi canbewritten asthesumofpolynomials



r_ρ+1



ri+1

i fi

∈ I

s

γi forsome fi ofdegree



m

−

ri

−

1,i

=

0

,

. . . ,

k.But H0

(

I

r

)

=

0 byhypothesis and



rρ+1



ri+1

i fi

∈ I

rγi foralli.Hence,inthecomplex

I

s

_/

_I

r _all

_[

_γ

i

]I

sγi

/I

r

γi areintheimageoftheboundarymap.Therefore,H0

(

I

s

/

I

r

)

=

0 andtheclaimfollowsfromProposition3.1.

Condition(b).Letusnowlookatthecasewhen

ρ

satisfiescondition(b).Withoutlossofgenerality,let

ρ





ρ

beasegment suchthat

γ

0liesintheinteriorof

τ∈ρ

τ

.Then,since

s

(

τ

)



r forall

τ

∈

ρ

,andsince

s and r differ

onlyontheedge

τ

∈

ρ

, thevertex

γ

0 iszeroedoutinthequotient

I

s

/

I

r.Then,forany1



i



k,anyelement



rρ+1f

∈ I

sγi canbeexpressedinthe imageoftheboundarymapas

[γ

i

]

rρ+1f

= ∂

⎛

⎝

i j=1

[τ

j

]

rρ+1f

⎞

⎠ ,

where



r_ρ+1f

∈ I

s

τj for all 1



j



k. This implies that H0

(

I

s

_/

_I

r

₎

₌

_{0 and the claim once again follows from} Proposi-tion3.1.



Remark4.3.Theorem4.2discussesthedimension whenthesmoothnessisreducedacrossasingle segment ofthe mesh. Itssuccessiveapplicationscanhelpuscomputethedimensionofalargeclassofsplineson

T

withmixedsmoothness.

LetuspresentanexampleapplicationofTheorem4.2toaspecialspaceofsplinescalledPHT-splines(Dengetal.,2008). Corollary 4.6helpscompute thedimensionofPHT-splinesofmixedsmoothness;alternatively, Bernstein–Bézier techniques canbeusedtoobtaintheresult.

Definition4.4(

(

m

+

1

,

m

+

1

)

smoothnessdistribution).The smoothness distribution r will be called an

(

m

+

1

,

m

+

1

)

smoothnessdistributionifforalledges

τ

∈

h

_T

1 (resp.,v

T

1),

r

(

τ

)

 (

m

−

1

)/

2 (resp.,

(

m

−

1

)/

2).

Theorem4.5.Let

r be

an

(

m

+

1

,

m

+

1

)

smoothnessdistribution,andlet

s be

anysmoothnessdistributionsuchthat

s

(

τ

)



r

(

τ

)

for alledgesof

T

.IfH0

(

I

r

)

=

0,thenH0

(

I

s

)

=

0 anddim



R

s



₌

_χ



_Q

s



_.

Proof. Sinceeachinterioredgeisintersectedbytwotransversaledgesonitsboundary,bythedefinitionof

r the

weightof each interioredgesatisfiescondition(b)fromTheorem4.2.Therefore,wecanmovefromthesmoothnessdistribution

r to

s one edgeatatime;ateachstage, H0

(

I)

=

0 andTheorem4.2(b)willbeapplicable.



Corollary4.6(PHT-splinesofmixedsmoothness).Let

T

beahierarchicalT-meshandlet

r be

an

(

m

+

1

,

m

+

1

)

smoothness distribu-tionsuchthat

r

(

τ

)

=

r

(

τ



)

foralledges

τ

and

τ

thatbelongtothesamesegment.Then,foranyothersmoothnessdistribution

s as

in

Theorem4.5,wehave

dim



R

s



=

χ



Q

s



.

D. Toshniwal, N. Villamizar / Computer Aided Geometric Design 80 (2020) 101880 7

Fig. 1. TheabovefigurescorrespondtothePHT-splinesettingconsideredinExample5.1.Thesmoothnessrequiredacrosseachedgehasbeenannotated inparenthesesnexttotheedgelabel.Figure(a)showstheinitialsmoothnessdistribution,whileFigure(b)showsthemodifiedinitialdistributions;the modificationsarelimitedtotheedgeslabelledinblue.(Forinterpretationofthecoloursinthefigure(s),thereaderisreferredtothewebversionofthis article.)

5. Examples

ThissectionpresentsexamplesofsettingswhereTheorem4.2applies,andalsowhereitdoesnot.Inparticular,weshow that H0

(

I

s

)

canbenon-trivialwhentheconditionsofTheorem4.2arenotmet.

Example5.1(PHT-splinesofmixedsmoothness).ConsiderthePHT-splinespace

R

r

33showninFig.1(a).FromCorollary4.6,we canreducethesmoothnessacrossanyarbitraryedgeandthedimensionwillstillbegivenbytheEulercharacteristicof

Q

r_. Onesuch modificationis showninFig.1(b)wherethesmoothnessacross edges

τ

27

,

τ

34

,

τ

35 and

τ

36 havebeenreduced. Thedimensionsofthespacescanbeeasilycomputedtobethefollowing,

dim



R

r



=

64

,

dim



R

s



=

66

.



Example5.2(Splinesofmixedsmoothness;hierarchicalT-mesh).Considerthespaceofbi-cubicsplines

R

r

33forthesmoothness distributionshowninFig.2(a).InFigures(b)–(e),wesuccessivelyreducethesmoothnessacrosstheedgeslabelledinblue while ensuring that the conditionsof Theorem 4.2are met.As a result, at each step ofsmoothness reduction,we have

H0

(

I

si

)

=

0,i

=

0

,

. . . ,

3.Thedimensionsofthecorresponding splinespaces canthenbe easilycomputedusingthe Euler characteristicsofcomplexes

Q

si,

dim



R

r



=

56

,

dim



R

s0



=

₅₇

_,

_dim



R

s1



=

₅₈

_,

dim



R

s2



=

₅₈

_,

_dim



R

s3



=

₅₉

_.

Ontheother hand,reducing thesmoothnessfromFigure (a)toFigure (f)doesnotsatisfy theconditionsofTheorem 4.2. Indeed,for

ρ

= {

τ

30

,

τ

37

,

γ

17

,

γ

20

,

γ

25

}

,itcanbeverifiedthat

ω

s4

(

ρ

)

=

3

<

m

+

1

=

4.Inthiscase,itcanalsobecomputed (usingMacaulay2(GraysonandStillman),forinstance)thatdim



H0

(

I

s4

)



=

1.



Example5.3(Splinesofmixedsmoothness;non-hierarchicalT-mesh).Considerthespaceofbi-cubicsplines

R

r₃₃forthe smooth-nessdistribution showninFig.3(a).Notethat inthiscasetheT-mesh cannotbe constructedhierarchically. Nevertheless, it ispossible touse resultsfromLi andWang(2019) to verifythat H0

(

I

r

)

=

0 and dim



R

r



₌

_χ



_Q

r



₌

_{64. Then, using} Theorem4.5,we seethat wecanreducethesmoothnessacrossanysubset ofedges andmaintainH0

(

I

s

)

=

0 forthenew smoothnessdistribution s. Onesuch casehas beenshownin Fig.3(b), andthecorresponding dimension ofthe spaceis givenbydim



R

s



₌

_χ



_Q

s



₌

_71.

_

Remark5.4.Asillustratedintheaboveexamples,ourresultscanbecombinedwithothers,e.g.,Mourrain(2014) andLiand Wang(2019),tocomputethedimensionsofaverygeneralclassofsplinespaces.Whileitisalwayspossibletodirectlyuse theothermethodsforthistask,onecaneasilyfindsimplecaseswhereourapproachgivessuperiorresults.

Fig. 2. Theabovefigurescorrespondtothebi-cubicsplinespaceconsideredinExample5.2.Thesmoothnessrequiredacrosseachedgehasbeenannotated inparenthesesnexttotheedgelabel.ThesmoothnessdistributionsinFigures(b)–(f)differfromtheoneinFigure(a)onlyontheedgeslabelledinblue.

D. Toshniwal, N. Villamizar / Computer Aided Geometric Design 80 (2020) 101880 9

Fig. 3. TheabovefigurescorrespondtothesettingconsideredinExample5.3.Thesmoothnessrequiredacrosseachedgehasbeenannotatedinparentheses nexttotheedgelabel.Figure(a)showstheinitialsmoothnessdistribution,whileFigure(b)showsthemodifiedinitialdistributions;themodificationsare limitedtotheedgeslabelledinblue.

Forinstance,ascommentedearlier,the approachofMourrain(2014) worksbest whenthesmoothnessacrossall con-nected horizontal(resp. vertical) edges isthe same. Using the sameterminology asin Mourrain (2014), define maximal

segments asmaximalconnectedunionsofparalleledgeswiththesamesmoothness.Letusdenotethesemaximalsegments

withthesymbol

λ

.Then,asshowninMourrain(2014),thehomologytermH0

(

I)

canbedescribedsolelyintermsof max-imal segments.Infact,only maximalsegments thatdonointersect themeshboundarymaycontributeto thedimension of H0

(

I)

.LetusapplythisapproachtothesplinespacecorrespondingtoFig.2(c)—thecorresponding interiorhorizontal, interiorverticalandboundarymaximalsegmentshavebeendisplayedasthickred,blueandblacklines,respectively,inthe figureontheright.Thesmoothnessacrosseachmaximalsegmenthasbeenannotatedinparenthesesnexttoitslabel.

Then, only the red and blue maximal segments

λ

i, 1



i



6, will contribute to H0

(

I)

; all black maximal segments labelledas

λ

∂ willnotcontribute asthey intersecttheboundary(Mourrain,2014).Therefore,uponorderingthe

λ

i inthe

followingmanner,

λ

5

 λ

6

 λ

3

 λ

2

 λ

4

 λ

1

,

andfollowingtheproofof(Mourrain,2014,Theorem3.7),theboundsonthedimensionofthesplinespacesimplifyto

0



dim H0

(

I)  (

m

+

1

−

ω

(λ

1

))(

m

−

r(λ1

))

= (

3

+

1

−

1

)(

3

−

0

)

=

9

.

Incontrast,asshowninExample5.2,ourapproachallowedustoshowthatdim H0

(

I) =

0.

6. Conclusions

Smooth polynomial splines are immensely versatile and are routinely utilized forchallenging applications in, for in-stance,geometricmodellingandcomputationalanalysis.However,certaintasksalsorequireworkingwithsplinesofreduced

smoothness, at leastlocally;e.g., geometric objectscontaining C0 featurelines,solutions to physicalproblemsthat show localizeddiscontinuities. Localcontroloverthesmoothnesscanbe verybeneficialinsuchcasesandcan leadtogreat im-provements in thequality of theoutput. Inthis paperwe havestudied thedimension ofbi-degree splines onT-meshes whendifferentordersofsmoothnessarerequiredacrossdifferentmeshedges.Reducingtheproblemtoanessentially uni-variateproblem, we haveprovidedsufficientconditions thatensure thatthedimension canbe combinatoriallycomputed usingonly localinformation.The conditionsareconstructiveinnatureandhavesimplegeometricinterpretation. A forth-comingpaperwillfocusontheconstructionofanormalizedB-spline-likebasisforsuchsplinespaces.

Declarationofcompetinginterest

The authors declare that they have noknown competingfinancial interests orpersonal relationships that could have appearedtoinfluencetheworkreportedinthispaper.

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