Beam-inside-beam contact: Mechanical simulations of slender medical instruments inside the human body

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ContentslistsavailableatScienceDirect

Computer

Methods

and

Programs

in

Biomedicine

journalhomepage:www.elsevier.com/locate/cmpb

Beam-inside-beam

contact:

Mechanical

simulations

of

slender

medical

instruments

inside

the

human

body

Marco

Magliulo

a

_,

_Jakub

_Lengiewicz

a,b

_,

_Andreas

_Zilian

a

_,

_Lars

_A

_.A

_.

_Beex

a,∗

a Institute of Computational Engineering, Faculty of Science, Technology and Communication, University of Luxembourg, Maison du Nombre, 6, Avenue de la

Fonte, 4364 Esch-sur-Alzette, Luxembourg

b Institute of Fundamental Technological Research of the Polish Academy of Sciences (IPPT PAN), ul. Pawinskiego 5B, 02–106 Warsaw, Poland

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 15 January 2020 Revised 6 April 2020 Accepted 29 April 2020 Keywords: Surgical simulation Contact mechanics Beam-inside-beam Artery Cochlea

a

b

s

t

r

a

c

t

BackgroundandObjective: Thiscontributionpresentsarapidcomputationalframeworktomechanically simulatethe insertion ofa slendermedical instrumentin atubular structure such as an artery, the cochleaoranotherslenderinstrument.

Methods: Beamsareemployedtorapidlysimulatethemechanicalbehaviourofthemedicalinstrument and the tubular structure.However, the framework’snovelty isits capabilityto handlethe mechani-calcontactbetweenaninnerbeam(representingthemedicalinstrument)embeddedinahollowouter beam(representing the tubular structure). This“beam-inside-beam” contact framework, whichforces two beamsto remain embedded,is the firstofits kindsince existingcontact frameworks for beams are“beam-to-beam” approaches,i.e.theyrepelbeamsfromeachother.Furthermore,weproposecontact kinematicssuchthatnotonlyinstrumentsandtubeswithcircularcross-sectionscanbeconsidered,but alsothosewithellipticalcross-sections.Thisprovidesflexibilityfortheoptimizationofpatient-specific instruments.

Results: Theresultsdemonstratethattheframework’srobustnessissubstantial,becauseonlyafew incre-mentspersimulationandafewiterationsperincrementarerequired,eventhoughlargedeformations, largerotationsandlargecurvaturechangesofboththeinstrumentandtubularstructureoccur.The sta-bilityoftheframeworkremainshighevenifthemodulusoftheinnertubeisthousandtimeslargerthan thatoftheoutertube.Ameshconvergencestudyfurthermoreexposesthatarelativelysmallnumberof elementsisrequiredtoaccuratelyapproachthereferencesolution.

Conclusions: Theframework’shighsimulationspeedoriginatesfromtheexploitationoftherigidityof thebeams’cross-sectionstoquantifytheexclusionbetweentheinnerandthehollowouterbeam.This rigiditylimitstheaccuracyoftheframeworkatthesametime,butthisisunavoidablesincesimulation accuracyandsimulationspeedaretwocompetinginterests.Hence,theframeworkisparticularly attrac-tiveifsimulationspeedispreferredoveraccuracy.

1. Introduction

Mechanicalsimulationsofsurgicalinterventionscanbeusedto train surgeons, reveal patient-speciﬁccomplicationsthat may oc-cur during surgery and plan interventions patient-speciﬁcally. In the future,mechanical simulations of surgical interventionsmay even be used to optimize medical instruments for each patient (e.g. shape and stiffness) and be exploited to autonomously per-forminterventions.

∗ _{Corresponding author.}

E-mail address: Lars.Beex@uni.lu (L.A .A . Beex).

Numerous frameworks to numerically simulate surgical inter-ventions can be found in the literature. For instance, one set of frameworkssimulatescuttingthroughsofttissuesinreal-time[4– 6,34,42]toprovidehaptic feedbackto thetraineeperforming the “intervention”. Another set of approaches aims to simulate the insertion of needles [2,3,8,10,36]. These frameworks mayalso be usedtoprovidehapticfeedbackand/or tohelp toaccurately steer theneedletothetargetofinterestduringsurgery.

However,theframeworkpresentedinthiscontributionfocuses onmechanicalsimulationsthat involvetheinsertion(or removal) of a slender medical instrument inside a tubular structure such as an artery, the cochlea or another slender instrument such as https://doi.org/10.1016/j.cmpb.2020.105527

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proposed framework and the frameworksof [1,9,15,27,39] is that our framework represents both the slender medical instrument andthe tubular structure as beams, whereas the frameworks of [1,9,15,27,39]only representthe slendermedicalinstrument with beamswhilst conventional 3D ﬁniteelements are usedto repre-sentthetubularstructure.

A wide variety of schemes to handle contact between beams canbedistinguishedintheliterature.Allexisting“beam-to-beam” contact frameworks are formulated to repel penetrating beams. Severalof these beam-to-beam contact frameworks are only ap-plicable if the beams’ cross-sections are circular, shear deforma-tionsareignoredandthecontactarearemainssmall,since unilat-eralcontactconditionsareenforcedattheclosestpairofcentroid points[33,41,43].Thus,iftwobeamscollide,acontactforceis ap-pliedattheclosestpairofcentroidpointstorepelthetwobeams. Ifthebeams’cross-sectionsareelliptical,the considerationofthe centroid-linesalone isinsuﬃcient to determinethe contact loca-tions.Instead,contactforcesmustbeappliedattheclosestpairof surfacepointswherethetangentplanesofthecontactingsurfaces areparallel.This wasdemonstrated byGay Netoetal.[12,13]for frictionlessandfrictionalcases,respectively.

Furthermore, in case of non-localized contact (e.g. for paral-lel beamsin contact),theassumption of point-wisecontactdoes not hold. Meier et al. [30] have therefore proposed a contact frameworktohandlenon-localizedbeam-to-beamcontact,butthe cross-sectionalshapeisrestrictedto(rigidly)circularandshear de-formationisnotaccountedfor.Theserestrictionsenable quantify-ing the penetration solely usingthe centroid-lines of the beams, whichyieldsrapidsimulations.

Magliuloetal.[28,29]presentedothermaster-slaveframeworks forbeam-to-beamcontactapplicabletobothshear-deformableand shear-undeformablebeams,withboth circularaswellaselliptical cross-sections.Bothschemes considerthebeam’s surfaces explic-itly,whichhasresultedinwider applicabilitythanthe schemeof Meieretal.[30],albeitatthe expenseofthesimulation speed.A two-halfpassalgorithmwasfurthermoreformulatedin[29]to re-movethebiasofmaster-slaveapproachesforbeam-to-beam con-tact,butwithlimitedbeneﬁtsfortheresults.

The beam conglomerates of interest to this contribution dif-ferfromthe aforementionedworks [28,29],since the focusis on “beam-inside-beam” contact instead of“beam-to-beam” contact.In otherwords,thebeamsmustremainembeddedforthebeam con-glomeratesofinterestinourcontribution,whereasexisting beam-to-beamcontact frameworks repel penetrating beams. The mea-sureof penetration in our beam-inside-beam framework, on the other hand, shows similarities with the measure of penetration forthe“beam-to-beam” contact frameworkof[28].Penetrationis measuredbetweensectionsdistributedalong theinnerbeamand theinteriorsurfaceoftheouterbeam.Incaseofcontact,unilateral constraintsareregularizedwiththepenaltymethod,whichbrings compliancetotheotherwiserigidcross-sections.

The outline of the remainder of this contribution is as fol-lows. In Section 2, the contact framework is presented in the space-continuous setting along with the associated contact vir-tual work. Also in this section, the spatial discretization applied

The contactkinematics employed inthiscontributionare pre-sented in this section [12,13,28,29]. First, the beam’s surface parametrizationisexplained.Then,theproceduretoquantify pen-etrationisdetailed.Theformulationofthecontactvirtualworkis presentedforapenaltyapproach.

2.1.1. Parametrizationofthesurface

The geometrically exact beam (GEB) Simo-Reissner theory [7,16,17,31,35,37,38] is used in this contribution. This entails that the beamsare shear-deformable andthat rigidcross-sections are considered,whichcannotwarp.

The surface of beam B is parameterized with two convective parametersh=

h1_,_h2

T_._h1_∈_[₀_,_L_]_,_denotes_the_arc-length

param-eterofthebeam’scentroidlinex0c:

(

0,L

)

→R3whileh2∈[0,2

π

]

isa circumferentialparameterofthe perimeterofcross-section _C attachedtox0c

(

h1

)

(see[12]andFig.1).Ldenotestheinitiallength

of the beam. The location ofa surfacepoint in the undeformed conﬁgurationintheglobalcoordinatesystem, x0,canbeobtained

from:

x0

(

h

)

=x0c

(

h1

)

+v0

(

h

)

, (1)

wherev0 denotes a vector containedin C.Here, we assume that

v0alwaysconnectsx0c toasurfacepoint.IncaseC iselliptical,v0

canbeexpressedas:

v0

(

h

)

=acos

(

h2

)

e01

(

h1

)

+bsin

(

h2

)

e02

(

h1

)

, (2)

where a and b denote the dimensions of the elliptical cross-sectionsinitsprincipaldirections.e01ande02 areorthonormal

ba-sisvectorsoftheplanecontaining_C.e03denotesthenormalvector

toC.Thetriad{e01,e02,e03}formsanorthonormalbasis.

Duetothehypothesisofrigidsections,thelocationofthesame materialpoint inthedeformedconﬁgurationcan similarlybe ob-tainedfrom:

x

(

h

)

=

ϕ(

x0

(

h

))

=xc

(

h1

)

+v

(

h

)

, (3)

where:

x_c=x₀_c+u, (4)

denotesthelocationofthecentroidpointinthedeformed conﬁg-uration.u:

(

0_,L

)

_→_R3_denotes_the_{centroid-line’s}_{displacement.}

_ϕ

Fig. 1. Beam kinematics for the current conﬁguration. The beam’s centroid-line is presented with a dashed line. Local basis vector e 3 is not aligned with the vector tangent to the centroid-line ∂xc

∂h1 due to shear deformation. A typical vector v lying

(3)

denotes thedeformation mappingrelatingthecurrentlocationof a point toits location intheundeformed conﬁgurationsuch that

x

(

h

)

=

ϕ

(

X

(

h

))

.visobtainedfrom:

v

(

h

)

=

(

h1

₎

_{· v}

0

(

h

)

=acos

(

h2

)

e1+bsin

(

h2

)

e2, (5)

where

: (0, L)_→SO(3), with SO(3) the rotation group, is a ro-tation tensor that rotates e0i to ei for i∈{1,2, 3}.Because shear

deformationcanbepresent,e3isnotnecessarilyalignedwiththe

tangenttothecentroid-line(seeFig.1).Insuchcases,i.e.:

e3×

∂

xc

∂

h1 =0, (6)

where × denotesthecrossproduct.

For further use, we deﬁne two (covariant) tangent vectors to the surface of_{B at} x

(

h

)

, denoted by

τ

1=_∂∂_hx 1 and

τ

2=_∂∂_hx 2 (see

Fig.1).Ingeneral,

τ

1and

τ

2arenotorthogonaltoeachother.

2.1.2. Contactkinematics

Wefocushereonasystemconsistingoftwobodies:BI_denotes

the thininner beamand_BJ _denotes _the_hollow _outer _beam. _We

assumeherethatbothBI_and_BJ_are_modeled_as_a_GEB_with_plain

andhollowcross-sections,respectively. Wedenoteby

∂

BJ _the

in-teriorsurfaceofBJ_.

ToquantifythepenetrationofBI_with

∂

_BJ_,_and_to_quantify_the

contactareaoverwhichthispenetrationoccurs,we: 1. Seedsectionsalongthecentroid-lineofBI_(see_Fig.₃_),

2. For each seeded section, we solve a projection (local) prob-lemtodetermineifitpenetrates

∂

BJ_and_if_so,_by_how_much.

Thisprojection problemyields twosurfacepoints:one onthe perimeteroftheseededsectionandoneon

∂

_BJ_._These_points

are usedto establisha measure ofpenetration, whichinturn determines theamplitudeofthecontactforces(if penetration ispresent).

AsBI_and_BJ _have_a _different_role,_the_proposed _framework_is

a master-slaveapproach.We call BI _the_slave _and_BJ _the_master

[40]. Next,we discusshow to compute ifa givensection of _BI_,

denoted by C withperimeter

∂

C,penetrates

∂

BJ _and _if_so, _how

theamountofpenetrationiscomputed.

Itmustbenotedthattheproposedcontactalgorithmcanonly be usedifone contactarea occursforeachcross-sections(leftin Fig. 2). The contactframework can thus not handlescenarios as presentedontherightinFig.2.Ifthecross-sectionsofBI _is

per-fectlyalignedwiththecross-sectionsof_BJ_,_only_one_contact_area

occursif:

(

ainner

)

2

binner

<

(

bouter

)

2

aouter

ainner≥ binner aouter≥ bouter, (7)

wheresubscriptinnerreferstotheinnerbeamandsubscriptouter

referstotheinnercross-sectionsoftheouterbeam.

WenowintroducenI_,_an_outward _pointing_unit_vector_normal

to

∂

BI_._It_is_deﬁned_as_follows: nI

_hI

₌

τ

I1

(

h I

₎

_×

_τ

I 2

(

h I

₎

τ

I 1

(

h I

₎

×

τ

I 2

(

h I

₎

. (8)

nJ_on_the_other_hand_is_the_inward_pointing_unit_vector_normal_to

∂

BJ_._It_is_deﬁned_as_follows: nJ

_hJ

₌

τ

J1

(

h J

₎

_×

τ

J 2

(

h J

₎

τ

J 1

(

h J

₎

_×

_τ

J 2

(

h J

₎

. (9)

Localproblemandmeasureofpenetrationbetween

∂

C and

∂

BJ_.

Wewillnowinvestigateif_C,thecross-sectionuniquelydeﬁnedby convectivecoordinatehC,penetrates

∂

BJ _and_if_so,_by_how_much.

Theso-calledgapvectorgconnectsapointontheperimeterofC, xI_∈

∂

_C,_to_a_surface_point_on

∂

_BJ_,_xJ_:

g

(

hC,hI2_,_hJ1_,_hJ2

₎

₌_xJ

(

_hJ1_,_hJ2

₎

_{− x}I

(

_hC_,_hI2

₎

_. ₍₁₀₎

Alocal(or projection)problemmustnowbesolved,whichyields points¯xI_and_¯xJ_,_such_that_an_appropriate_measure_of_penetration

isestablished.Fourconvectivecoordinatesareinvolvedinthelocal problem:hC_,whichisﬁxed,aswellashI2_,_hJ1 _and_hJ2 _that_are_to

bedetermined.

Apossibilitytodeterminetheunknownconvectivecoordinates wouldbe tosolvean optimizationproblembyminimizing an ob-jectivefunction.Anotherpossibility[12–14,21,22,33]istosolvefor a set of equations that does not stem from an objective func-tion,generallyunilateralorbilateralorthogonalityconditions. Pre-viously, we have shown that the latter makes the resolution of thelocal problem20–30% faster to solve forbeam-to-beam con-tact[28].Wethereforeconsiderasimilarapproach forthe beam-inside-beamcontactofthiscontribution.Threeoftheequationswe solveforareexpressedas:

f1

(

¯q

)

=¯xJ− ¯xI− ¯g¯nI=0, (11)

where:

q=

hI2_,_hI1_,_hJ2_,_g

T

_, ₍₁₂₎

denotesthearrayofunknowns.Hereandinthefollowing,an over-headbaroveraquantityindicatesthatitisevaluatedatthe solu-tionoftheprojectionproblem.Thus, ¯qdenotesthearray solution ofEq.11.The independentvariable ¯gquantiﬁespenetration mea-sured in the normal direction ¯nI

(

_hC_,¯hI2

₎

_, _usually _denoted _as_g

N

anddeﬁnedas:

gN= ¯g=

(

¯xJ− ¯xI

)

· ¯nI. (13)

Asfourvariablesarepresentforonlythreeequations,the sys-temof Eq.11 isunder-determined. An additional equation is re-quired.Tothisend, weintroduce plane P spanned bynI _and

τ

I 2

withthefollowingnormalvector:

˜

nI₌

τ

I

2× nI. (14)

Also,wedeﬁnenJp_as_the_normalized_projection_of_nJ_on_P:

nJp₌ nJ−

(

nJ· ˜nI

)

n˜I

nJ−

(

_nJ· ˜_nI

)

_n_˜I

. (15)

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Fig. 3. (a) Two beams in contact. (b) Sections for which penetration has been detected.

Fig. 4. Perpendicular view to the plane P of ∂C. Surface points ¯x I and ¯x J , obtained

after solving Eq. 17 , are presented as red dots. Vectors ¯n I and ¯n Jp are both orthog-

onal to ¯τJ

2 . (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

At thesolution of the local problem, we want nI _and _nJp _to _be

orthogonalto

τ

J

2(see Fig.4).Thisistrueifthefollowingequation

holds:

f2

(

¯q

)

=aC

(

¯nI+¯nJp

)

· ¯

τ

J2=0, (16)

whereaC denotes the dimension ofC along its largestsemi-axis whichisusedtoscalef1such thatf1 andf2 havethesameunits.

Thesetofequationstosolveforisnowabbreviatedasfollows: f

(

¯q

)

=

f1

(

¯q

)

,f2

(

¯q

)

T

=0. (17)

The setof equations to solve forin Eq.17 is nonlinear. Tosolve it,we apply Newton’s method for which we linearise residual f

whichrequiresthefollowingJacobian: H

(

q

)

=

∂

f

∂

q =

⎡

⎣

∂f 1 ∂q

∂f2 ∂q

T

⎤

⎦

. (18)

Tocompute the componentsof H,we need to introduce the fol-lowingquantities:

1.ContravariantcomponentsMKi jofMK,themetrictensorofthe surfaceofbodyK,read:

MK11 MK12 MK21 _MK22

=

MK₁₁ MK₁₂ MK₂₁ MK₂₂

−1 =

τ

K 1·

τ

K1

τ

K1·

τ

K2

τ

K 2·

τ

K1

τ

K2·

τ

K2

−1 . (19)

2. Thesecondordersurfacederivatives:

τ

K

i j=

∂

τ

K

i

∂

hK_j. (20)

3. ThecovariantcomponentsofcurvaturetensorCK: CK_{i j} =

τ

K i j· nK (21) 4. Weingarten’sformula:

∂

nK

∂

hj =−MK jk_CK ki

τ

Kj. (22)

MakinguseofEq.22,thepartialderivativesoff1 withrespect

toqinEq.18yield:

∂

f1

∂

q =

∂

f1

∂

hI2,

∂

f1

∂

hJ1,

∂

f1

∂

hJ2,

∂

f1

∂

g

=

−

τ

I 2+g

(

MIjkCIk2

τ

Ij

)

,

τ

J1,

τ

J2,−nI

. (23)

Thedifferentiationoff2withrespecttoqgives:

∂

f2

∂

q =

∂

f2

∂

hI2,

∂

f2

∂

hJ1,

∂

f2

∂

hJ2,

∂

f2

∂

g

T

. (24)

Thecorrespondingexpressionsaremorecomplicated,inparticular becausenJp_depends_on_hI2_,_hJ1 _and_hJ2_.

WenowrewritenJp_as:

nJp₌Nn Jp ln Jp, (25) where: Nn Jp=nJ_·

(

_I_{− ˜}_nI_n_˜I

)

₌_nJ_{· D}_, ₍₂₆₎ and: ln Jp=

Nn Jp

. (27)

ThepartialderivativeofnJp_with_respect_to_the_kth_-surface

(5)

where

δ

denotes theKronecker symbol (nottobe confused with thevariationsymbol).Theterm ∂n ˜I

∂hlk,where: ˜ nI₌N ˜ n J lN n˜J, (31) with: Nn ˜J =

τ

I 2× nI, (32) and: lN n˜J =

τ

I 2× nI

, (33) reads:

∂

n˜I

∂

hlk = 1 lN n˜J

I−˜nI_n_˜I

_·

∂

N˜ nJ

∂

hlk =F·

∂

Nn˜ J

∂

hlk. (34) Wecanwrite:

∂

Nn ˜ J

∂

hlk =

δ

lI

τ

I 2k× nI+

τ

I2×

−MIi j_CI jk

τ

Ii

=c, (35) suchthat:

∂

n˜I

∂

hlk =F· c. (36)

Finally,thisyields:

∂

nJp

∂

hlk = 1 ln Jp

I−nJp _nJp

_·

∂

Nn Jp

∂

hI2 =E·

δ

lJ

_−MJlm_C mk

τ

K

· D− nJ·

(

F· c

)

˜nI+n˜I

(

F· c

)

=dlk_. ₍₃₇₎

Using Eq. 37, compact expressions forthe components of ∂f2

∂q in

Eq.24canbeobtained:

∂

f2

∂

hI2 =aC

∂

nI

∂

hI2+

∂

nJp

∂

hI2

·

τ

J 2 =aC

−MIjk_CI k2

τ

Ij+dI2

·

τ

J 2, (38)

∂

f2

∂

hJ1 =aC

∂

nJp

∂

hJ1 ·

τ

J 2+nJp·

∂

τ

J 2

∂

hJ1

=aC

dJ1_·

_τ

J 2+nJp·

τ

J21

, (39)

∂

f2

∂

hJ2 =aC

∂

nJp

∂

hJ2 ·

τ

J 2+nJp·

∂

τ

J 2

∂

hI2

=aC

dJ2_·

τ

J 2+nJp·

τ

J22

. (40)

CombiningEqs.23,38,39,40,Hreads:

H= ⎡ ⎣ −τ I 2+g(MIjkCIk2τIj) τJ1 τJ2 −nI aC −MIjk_CI k2τIj+d I2 ·τJ 2aC d J1_·_τJ 2+n Jp·τJ21 aCd J2_·_τJ 2+n Jp·τJ22 0 ⎤ ⎦. (41) First-guessprocedure

As statedabove, Eq.17issolved iteratively.Aninitial guessof the local parameters q must be provided to the solver. We em-ployasimpletwo-stepprocedureto establishan appropriateﬁrst guess:

1. We(approximately)ﬁndthecentroidpointofthemasterbody that is the closest to the centroid point of the slave cross-section,xI

c

(

hC

)

.Asimplewayofachievingthisisby sampling

cross-sectionpointsalong themasterbeam’scentroid-lineand picktheclosestcentroid-pointfromxI

c

(

hC

)

.Theassociated

con-vectiveparameteroftheclosest sampledcentroidpoint is de-notedbyhJ1, f g_.

2.To determine the initial values of circumferential parameters

hI2, f g _and_hJ2, f g_, _we _locate _a _pair _of _perimeter_points _on _the

cross-sectionsattachedtoxI

c

(

hC

)

andxJc

(

hJ1, f g

)

.Thisprocedure

is depictedinFig. 5.We startby samplingfourpointson the perimeter of both cross-sections. The pair of closest points is then chosen. Next,for each cross-section,we seed a point on the middleofeachsub-curveattachedto thepointpreviously selected ((c) inFig. 5). Again, theclosest pair ofpoints is se-lected.Thisprocedureisrepeatedseveraltimes.Inour simula-tions, itwasrepeateduntiltherelative changeofthedistance betweenthepairofclosestpointsfallsbelow10%.

Notethatinpractice,theapproachtoestablishtheinitialguess isonlyperformedforagivenslave cross-sectionifitisnotactive butclose to the master surface. Ifa slave cross-section is active (meaningthat it alreadypenetrated the master surfacein a pre-vious contactdetection),the solutionofthe localproblemof the previouscontactdetection(¯q)isusedastheﬁrstguess.

2.1.3. Contactconstraintsandvirtualworkequationincludingcontact

The impenetrability of

∂

_{C and}

∂

_BJ _is _enforced _via _unilateral

contactconditions:

gN≥ 0 TN<0 gNTN=0, (42)

where TN denotes the nominal contact traction, meaning that it

referstothereferencesurfacearea.

Incaseofcontact,acontactvirtual work,

δ

c,isaddedtothe

virtual work equation for thetwo-body systemand thespace of admissiblevariations_V ismodiﬁed[40].Inthequasi-staticsetting, thevirtualworkreads:

δ(

pIJ_,

δ

_pIJ

)

₌

δ

BI

(

pI,

δ

pI

)

+

δ

_BJ

(

pJ,

δ

pJ

)

+

δ

c

(

pIJ,

δ

pIJ

)

=0,

∀

δ

pIJ∈V, (43)

where

δ

_Bi denotes the internal and external virtual work of

beamBi _(excluding _contact_{interactions).} _Kinematic _variables

as-sociated with Bi _are _denoted _by _pi₌

_ui_,

_θ

i

T_:

₍

₀_,_L

₎

_→_R3_{× R}3

andtheassociatedtest functionsby

δ

pi_._ui _denotes_the

displace-mentﬁeldofthe_Bi_’s_{centroid-line}_and

θ

i_its_ﬁeld_of_rotation

vec-tors parametrizing SO(3) [16]. pi _is _only _admissible _if _pi

₍

_XBi

)

=

pi D

(

XB

i

)

,

∀

XBi∈

∂

Bi

D.

∂

BDi denotesthepartoftheboundaryof

∂

Bi

whereDirichletboundaryconditionsareimposed[40].

pIJ₌

_pI_,_pJ

T _gathers_the _kinematic_variables _of_both _beams.

Similarly,testfunctionsaregatheredin

δ

pIJ₌

δ

_pI_,

δ

_pJ

T_.

InEq.43,thevirtualworkduetocontact,

δ

c,accountsforall

thesectionspenetrated.Theinﬁnitesimalvirtualworkproducedat asinglesection,denotedbyd

δ

c,canbewrittenas[28]:

d

δ

c=TN

δ

gN

∂

xI 0c

∂

hI1

dhI1_, ₍₄₄₎

where

δ

gN denotes the variationof gN that dependson all

kine-matic variables in pIJ_, _and _dhI1 _denotes _the _differential _of _the

slave’scentroid lineparameter associatedto C. dhI1 _is_related _to

thedifferential length of the undeformed centroid-lineaccording to:

dLBI=

∂

xI0c

∂

hI1

dhI1_, ₍₄₅₎

Next,wediscusshowtocalculateTNforthepenaltyapproachused

in this contribution andalso how to compute

δ

gN. The nominal

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Fig. 5. Illustration of the procedure to determine a good initial guess for the local problem. (a) Determination of centroid point x J

c(hJ1, f g) that must be as close as possible

to x I

c(hC) ; (b) Determination of the closest pair of points amongst sampled points on ∂C and the perimeters of the cross-section attached to x Jc(hJ1, f g) ; (c) Determination of

the closest pair of points amongst the closest pair of points from (b) (in orange) and points in the middle of the curve connected to theses points (in red). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

notdepend on pIJ_,_the_{linearization}_of

δ

_yields _shorter

expres-sionsthanifthecurrenttractionvectorwasemployed[26].

Penaltymethod

If a penalty formulationisused, contacttraction TN, actingat

thepairofsurfacepoints¯xJ_and _¯xI_,_is_given_by:

TN=−

N

−gN

, (46)

where

N>0 denotes the penalty stiffness and

•

denote the

Macaulay brackets, representing the positive part of its operand [25]. The fact that

N must be selected can be seen as a

weak-nessofthepenaltymethod.Indeed,other constraintenforcement methods likethe Lagrange multipliersmethod do not need such user-deﬁnedparameters.Ontheotherhand,inthecontextof con-tact frameworks for beams that are characterized by rigid cross sections,the penalty parametercan beinterpreted assome com-plianceofthebeamsinthetransversaldirections[28,33].Inserting Eq.46intoEq.44,thevirtualworkofthecontactforcereads: d

δ

c=−

N

−gN

δ

gN

∂

xI 0c

∂

hI1

dhI1_. ₍₄₇₎

The virtual work of all penetrated sections follows fromthe in-tegration ofthe inﬁnitesimal virtual work of a singlepenetrated section,d

δ

c,alongthecentroid-lineofBIasfollows:

δ

c= h1I U h1I L d

δ

c, (48) whereh1I

L andh1UIarethelowerandupperboundsoftheintegral,

respectively.

Variationofthenormalgap,

δ

gN

gN=¯g· ¯nI,measured fora ﬁxed hI1, dependson pIJ butalso

onthelocalvariablesinq,whichinturnimplicitlydependonpIJ_.

Eventually,

δ

gNmustbesolelyexpressedintermsofthevariations

ofthe primary variables, here

δ

pIJ_. _The _variational _operator

ap-pliedtogNgives:

δ

gN=

δ

¯g· ¯nI+¯g·

δ

¯nI. (49)

Also,wehave:

¯g·

δ

¯nI₌_g

N¯nI·

δ

¯nI=0, (50)

as

δ

¯nI_{· ¯n}I₌₀_._Noting _that _¯g_depends_on _pIJ_and _¯q

(

_pIJ

)

_,_we _get:

δ

¯g=

∂

g

∂

pIJ

q=¯q

T

δ

pIJ₊

∂

g

∂

q

q=¯q

T

δ

q (51) =

∂

g

∂

pIJ

q=¯q

T

δ

pIJ₊

∂

g

∂

q

q=¯q

T

dq dpIJ

q=¯q

δ

pIJ_. ₍₅₂₎

TherelationshipsbetweenqandpIJ_can_be_found_on_the_basis_of

thestationarityof f

(

pIJ_,_¯q

(

_pIJ

))

_with_respect_to_pIJ_:

df dpIJ=

_∂

f

∂

pIJ

q=¯q

δ

pIJ₊

⎛

⎜

⎝

∂

f

∂

q

H

q=¯q

⎞

⎟

⎠

δ

¯q=0, (53)

whichleadsaftersomerearrangementto:

δ

¯q=−

H−1

q=¯q

∂

f

∂

pIJ

q=¯q

δ

pIJ_. ₍₅₄₎

Forfurtheruse,wedeﬁneAasfollows: A=−H−1

q=¯q

∂

f

∂

pIJ

q=¯q, (55) suchthat:

δ

¯q=A

δ

pIJ_. ₍₅₆₎

InsertingthisinEq.52yields:

δ

¯g=

∂

g

∂

pIJ

q=¯q

T

+

∂

g

∂

q

q=¯q

T

A

δ

pIJ_. ₍₅₇₎

Insertingthisinto49yields:

δ

gN=

δ

¯g· ¯nI=

∂

¯g

∂

pIJ

T

δ

pIJ_{· ¯n}I₊

∂

¯g

∂

q

T

A

δ

pIJ_{· ¯n}I ₍₅₈₎ =

(δ

pIJ

)

T

∂

¯g

∂

pIJ· ¯n I₊_AT ∂¯g

∂

q· ¯n I

(59) =

(δ

pIJ

)

T_z_. ₍₆₀₎

Noteforfurtherusethat _∂∂_p¯g IJ¯nIreads:

∂

g

∂

pIJ

q=¯q· ¯n I₌

∂

¯xJ

∂

q T · ¯nI₋

∂

¯xI

∂

q T · ¯nI ₍₆₁₎ =

!

₀

τ

J 1· ¯nI

τ

J 2· ¯nI

"

−

!

_τ

I 1· ¯nI 0 0

"

(62) =

!

₀

τ

J 1· ¯nI

τ

J 2· ¯nI

"

−

!

₀ 0 0

"

, (63)

whereisitobviousfromEq.63that ∂¯xI

∂q T

(7)

Allinall,wecanwrite:

δ

c= h1I U h1I L −

N

−gN

zT

δ

pIJ

∂

xI 0c

∂

hI1

dhI1_. ₍₆₄₎ 2.2. Spatialdiscretization

2.2.1. Interpolationofthebeams’surface

The FiniteElementMethod(FEM)isthe discretisationmethod used in this work. Each beam is now discretized with a set of beamﬁniteelements(BFEs)[16,37,38].Therotationvectorsarethe primary rotational kinematic variables [16] that are interpolated. Fora BFE E,we denote by Xh

c

(

h1

)

:

0,LE

→R3 _the _interpolated

position of its centroid-line, where LE denotes the length of the centroid-line in the undeformed conﬁguration. uh

c

(

h1

)

:

0,LE

→ R3_and

_θ

h

(

h1

₎

_:

₀_,_LE

_→_R3 _denote_the_interpolated_displacement

ﬁeldofthecentroid-lineandtheinterpolatedﬁeldofrotation vec-tors,respectively.Rodrigues’formulaisusedtoobtainrotation ten-sor

fromtheinterpolatedrotationvector1_,_denoted

_θ

h_._The

dis-placement ofnode K isdenoted by uˆE_K andits rotationvector by ˆ

θ

EK.ThekinematicvariablesofelementE aregatheredin:

pE=

uˆE₁,...,uˆE_n_u,

_θ

ˆE 1,...,

θ

ˆ E n_θ

T , (65)

where nu and n_θ denote the number of nodes used to

interpo-late the displacement andthe rotation vector, respectively. Since thenodalvariablesofallBFEsofBI_and_BJ_are_denoted_by _p_ˆ_,_the

associatedvariationsaredenotedby

δ

ˆp.

2.2.2. Contactresidualandstiffness

ThediscretizedformofthevirtualworkinEq.43leadstoaset ofnonlinearequations.Newton’smethodisgenerallyusedto iter-ativelydetermine thesolution ˆpsol of thevirtual work statement. ThisrequiresthelinearizationofEq.43aroundanestimateofpˆsol_,

ˆ

p,whichyields:

δ(

ˆp+pˆ,

δ

pˆ

)

δ(

pˆ,

δ

pˆ

)

+

δ(

ˆp,

δ

ˆp

)

pˆ=

δ

pˆT

(

rg+Kgˆp

)

0,

(66)

where r_g and K

g denote theglobal residual force and the global

stiffness matrix, respectively. pˆ denotes an increment of the nodalvariables.Theglobalforcesread:

rg

(

ˆp

)

=f_int

(

ˆp

)

+rc

(

pˆ

)

− f_ext

(

ˆp

)

, (67)

where fint denotes the internal force vector stemming from the

contributionofallBFEs,and f_ext theexternalforcevector. rc

con-tainsallthecontactcontributionsfromallcontactelements,where acontactelementreferstoaseededsectionattachedatan integra-tion point (see below) along BI _and_its _projection _on _discretized

surface

∂

_BJ_.

Assuming here that f_ext does not depend on pIJ_, _the _global

stiffnessobtainedafterthelinearizationofrg,canbedecomposed

asfollows: K

g=Kint+Kc, (68)

whereK_int,denotesthestiffnessmatrixoftheBFEs,andK_cdenotes thestiffnessmatrixofallcontactelements.

Thecontactvirtualworkisthesumofallcontactcontributions:

δ

c

(

pˆ,

δ

pˆ

)

=

#

e∈S

δ

c,e

(

pˆ_e,

δ

pˆ_e

)

, (69)

1 It is well known that the corresponding ﬁnite element formulation does not share the strain-invariance property of the underlying geometrically exact theory [18] .

whereS denotesthe setofactive contactelements(i.e.thosefor which gN<0), and

δ

c,e denotes the contact virtual work

asso-ciated withcontact element e.If no smoothingprocedure of the surfaceisrequired[26],eachcontactelementsinvolvestwo BFEs, one which is part of the discretization BI_, _and _the _other _which

ispartofthediscretizationofBJ_._However, _if_a_smoothing_of_the

beam’s surface is performed, each contact element directly de-pends on several BFEs of BI _and _of _BJ_. _The _set _of _elements _of

BI_and_BJ_used_to_construct_contact_element_e_are_denoted_by_M

and_N_, respectively.The involvednodalvariables are gatheredin array ˆp_e=

pˆM,pˆN

T. Similarly, theinvolved nodalvariations are denotedby

δ

ˆp_e.

Thelinearizationof

δ

creads:

δ

c

(

ˆp+ˆp,

δ

pˆ

)

= # e∈S

δ

c,e

(

ˆp_e+ˆp_e,

δ

ˆp_e

)

≈# e∈S

δ

c

(

pˆ_e,

δ

ˆp_e

)

+

δ

c

(

ˆp_e,

δ

ˆp_e

)

ˆp_e (70) ≈# e∈S

δ

ˆpT_e

(

rce+K_cepˆ_e

)

. (71)

Next,we discusshow to constructelement contributions r_ce and

K_cetotheglobalresidual(force)vectorandtheglobalstiffness ma-trix,respectively.

Forcevectorandstiffnessofcontactelemente

We numerically integrate Eq. 39 witha quadrature (Gauss or Lobato-type).Tothisend,weplacenM_IP integrationpointsofa sin-glesubdomainalongthecentroid-lineof_M[21,33].Thisyields:

δ

c=−

N 1 −1

−gN

(η)

δ

gN

(η)

J

(η)

d

η

(72) ≈ −

N nM IP # k wk

−gN

(η

k

)

δ

gN

(η

k

)

J

(η

k

)

, (73) ≈ nM IP # k wk

−gNk

δ

gNk

Jk

, (74)

where

η

_∈[₋₁_,1]denotesthecentroidpointcoordinateinthe pa-rameterspaceand_J₌ ∂x I0c

∂η .Theweightandcoordinatesofthekth

integrationpointaredenotedbyw_kand

η

_k,respectively.Giventhe section along M attachedto xc

(

η

)

for whichwe solve the local

problemofEq.17,twosurfacepoints, ¯xMand ¯xN,arecomputed. InEq.73,r_cekisexpressedas:

r_cek=−

N

−gN

dgN

dpˆ_e. (75)

If the Automatic Differentiation (AD) formalism is employed [23,24],the dependency ofthe local variables withrespect to pˆ_e aswell asrelationsofEq.63 canbe directlyincorporated as fol-lows:

δ

gN=

δ

¯g· ¯nI (76) =

δ

¯xJ_{· ¯n}I₋

δ

_¯xI_{· ¯n}I ₍₇₇₎ =

ˆ

∂

xJ ˆ

∂

pˆe

∂q ∂pˆe=A −

∂

ˆxI ˆ

∂

pˆe

∂q ∂pˆe=0 · ¯nI_, ₍₇₈₎ whereoperator ∂ˆ_ˆ

∂w denotesthe AutomaticDifferentiation(AD) of

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Fig. 6. Example 1: (a) Initial configuration; (b) configuration halfway through the loading, and (c) final configuration.

K_cek,stemmingforthecontributionofthekth _integration_point

placedalonghM₁ ,canbeobtainedusingADasfollows: K_cek=

∂

ˆrcek ˆ

∂

ˆp_e

∂¯q ∂ˆp_e=A . (79) r_candK

c,whichcontainthecontributionsofall contactelements

insetS,areobtainedfrom: rc=A # e∈S nM IP # k wkrcek=A # e∈S rce, (80) K_c=A # e∈S nM IP # k wkK_cek=A # e∈S K_ce. (81)

where A $ denotes the ﬁnite-element assembly operator. Note that the exception indifferentiation in Eq.81 allows to properly lineariser_cek suchthattheexception ∂q

∂pˆe =0inEq.78isreplaced

by ∂q

∂pˆe =Aduringlinearization.

3. Results

Inthissection,thebeam-inside-beamcontactschemeisapplied totwo numericalexamples.First,a thinbeamispulledout from anotherbeaminwhichitisinitiallyinserted.Second,athinbeam isinsertedinacurvedbeam.Forthesetwoexamples,asingle in-tegrationpoint isusedto evaluater_ce andK_ce in Eqs.80and81, respectively.

In both examples, large relative displacements ofthe contact-ingsurfacestakeplace.Thisimpliesthatforacontactelement,if projection point ¯xJ _lies _on _the_surface _of_BFE _M_, _this_projection

mightgooff theboundofthesurface

∂

M.Ifthishappens, projec-tionpoint ¯xJ _should_lie _on_an _adjacent_element’s_surface, _namely

thesurfaceofelementBM+1_or_BM−1_._However, _as_two-node

ge-ometricallyexactbeamelementsareemployed,gapsandoverlaps ofthedifferent BFE’ssurfaces arepresentifthe beamisnot ini-tiallystraight.Forthisreason,itmightbediﬃcultorimpossibleto deﬁnethenewlocationof ¯xJ_.

To palliate this problem, a dedicated surfacesmoothing tech-niquewas introduced in[29]. The resultingauxiliary surface has

C1_-continuity_which_is_convenient_for_contact_treatment._This

pro-cedureisusedinthefollowingexamples.

Due to the discretization of the contact kinematics, a sud-denloss of contactnear the inlet andoutlet of the outer beam may result in loss of convergence. The methodology presented inAppendix Ais thereforeadopted inthe following examplesto avoidthesecomplications.

3.1. Example1:Pullout

In the ﬁrst example, two elastic beams with the same initial centroid-lineformapartofahelix (Fig.6a).Theyhavethesame centroid-line as a parameterized helix. The outer beam is hol-low with an elliptical cross-section deﬁned by a=22.3mm and

b=17.8mm.Its wall thicknessis 0.2mm. Its Young’smodulus is settoE₌0_.15MPa[20]and75BFEsareemployedtodiscretizeit. Thethinbeamhasanellipticalsectionwithsemi-axesa=3.9mm is set to andb=3mm. It is stiffer than the hollow beamas its Young’s modulus is set to E₌1_.5 MPa. Its Poisson’s ratio is set to

ν

=0.3 and it is discretized with 100 BFEs. A penalty stiff-nessof1N/misused.Thedisplacementsandrotationsofbothend nodesoftheouterbeamarerestrained.Oneendofthethinbeam is pulledaway fromthe outer beamby 1200mm in300 equally spacedincrements.

Contact interactions substantially deform the thin beam, (Fig. 6b). As the (prescribed) end node ofthe thin beam contin-uestomove awayfromtheouter beam, slidingofthecontacting surfaces occurs until the ﬁnal conﬁguration is reached(Fig 6 c). Astheprescribedbeamdeformsandmovesalongtheouterbeam, the number of penetrated sections changes (Fig. 7). The compo-nentsofthereactionforceandtorqueattheprescribedendofthe thinbeamarereportedinFig.8aandFig.8b,respectively.

3.2. Example2:Insertion

Thesecondexampleinvolvesaninitiallystraightthinbeamthat ispushed in a hollow,largely circularbeam(see Fig. 9). Initially, onlya smallpartofthethinbeamisinsertedinthe hollowone. The kinematic variables of the outer beam’s end node near the thinbeamare restrained.Thez-displacementoftheinnerbeam’s endnodefurthestawayfromtheouterbeamisprescribedtoreach 270mmin300increments,whilsttheotherkinematicvariablesat thisendnodearerestrained.

This inner beamhas a length of54cm, anda Young’s modu-lus of 1.5MPa.The cross sectional shape is given by a=5.4mm andb₌4_.3mm. The outer hollow beamis more compliant with

E=0.15 MPa. Its wall thickness is 1mm and its cross-sectional semi-axes are a=20mm and b=16mm. 100 and 180 BFEs are employedtodiscretizetheinnerandouterbeam,respectively.The Poisson’sratioofbothbeamsissetto0.33.

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Fig. 7. Example 1: top: number of global iterations to reach the convergence criteria f_int_{+ r}c − f ext< 10 −8 ; bottom: number of penetrated sections as a function of the

displacement of the end node of the thin beam. The peak in the number of iterations corresponds to the sliding of a cross-section of the slave beam in contact out of the hollow beam. The peak in the number of iterations corresponds to the sliding of a cross-section of the slave beam in contact out of the hollow beam. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 8. Example 1: (a) components of the reaction force and (b) reaction torque at the prescribed end of the thin beam.

Fig. 9. Example 2: Initial conﬁguration.

determinethecontactlocationaredifficulttoapplyinthese situ-ations.Thereasonisthattheyrelyonthedeterminationofa min-imumofadistancefunctionthatisalmostconstantifsurfacesare closetoeach other onafiniteregion. Thetopdiagram ofFig.11, furthermore,showsthatonlyafewiterationsarerequiredtoreach convergence. This was the same as for the first example, as re-vealed in the top diagram of Fig. 7. The force-displacement and torque-displacementdiagramsofFig.12clearlyshowthatdifferent

regimesarepresent,whereeachregimeisgovernedbyadifferent numberofcontactareas.

Inthecurrentexample,theYoung’smodulusoftheinnerbeam is tentimes larger than that ofthe outer beam. To demonstrate that the framework is also robust for an entirely different ra-tio of Young’s moduli, the example is repeated withexactly the samegeometrical, material and numericalparameters, except for theYoung’smodulusoftheinnerbeam.Insteadof1.5MPa,weset themodulusto150MPasuchthatitisthousandtimeslargerthan thatoftheouterbeam.

The ﬁgures for this additional test case are reported in AppendixB.Theyshowthat, althoughthepredicteddeformations as well as the force- and torque-curves are completely different (cf.Figs.10,11and12),themaximumnumberofiterationsisagain notmorethantwo.

Meshconvergencestudy

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Fig. 10. Example 2: (a) Initial configuration; (b) configuration halfway through the loading, and (c) final configuration.

Fig. 11. Example 2: Top: number of global iterations to reach convergence criteria: f_int_{+ r}c − f ext< 10 −8 ; Bottom: evolution of the number of contact interactions

between sections of the thin beam and the surface of the outer beam.

differentmeshes,thedisplacementfieldoftheinnerbeamis com-paredtoareferencesolution,whichisobtainedwith180elements fortheinnerbeam.Figs.aandbshowthatthedisplacementfields convergetothereferencedisplacementfields.

4. Discussion

The beam-inside-beam contact framework presented in this contributionistheﬁrstapproachtoensurethataninnerbeam re-mainsembeddedinsideanouter,hollowbeambecauseallexisting contactframeworks forbeams aimto achieve the opposite: they repelbeamsfromeachother.

Theadvantageofusingbeamsinsurgicalsimulationsover con-ventional3Dfiniteelementsisthepotentialtoachievefaster sim-ulations (although our particular implementation can surely not compete with frameworks such as SOFA [11]). The disadvantage of using beams over conventional finite elements is a reduction of the simulation accuracy. Hence, the framework proposed in thiscontribution may be perceived to be beneficial if simulation speed is preferred over simulation accuracy. On the other hand, ourframeworkcanhandleslendermedicalinstrumentswithboth circular and elliptical cross-sections, whereas the frameworks of [1,9,15,27,39]haveonlybeendemonstratedtohandlecircularones. The reason that beams, formulated to rapidly simulate the me-chanicalbehaviorofslenderbodies,arefasterformechanical sim-ulationsinvolvingcontactistwofold.First,beamscomewithfewer degreesoffreedom(i.e.kinematicvariables)thanifthetubes’ sur-faces are representedby conventional finite elements, becausein most beam theories the beams’ cross-sections are rigid (i.e. the cross-sections cannot deform). Hence, the entire beam’s

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Fig. 13. Effect of the mesh refinement on the final displacement of the nodes of the thin beam. The displacements in the final configuration are compared to the ones with the finest mesh (180 nodes) (a) Difference of the Y -displacements ; (b) Difference of the Z-displacements.

try can be constructedfrom its centroidlinedescription andthe cross-sections’ orientation. Thisdrasticallyreducesthenumberof degreesoffreedomnecessarytodiscretiseaslenderbody.Second, andseemingly even moreimportantis thefact that the penetra-tion can be quantiﬁed for an entire (rigid)cross-section atonce, whichdrasticallyreducesthenumberoflocalproblemsthatneeds tobesolved.

Onepossibleextensionofthebeam-inside-beamcontact frame-workistheincorporationoffluidflowsinsidethehollowbeamto representblood flowincaseofarteries andveins.Frictional slid-ingbetweentheinnerandouterbeamalsoseemslikeanecessary extensionforthefuture.Surroundingtissueswerefurthermore ne-glectedinthepresentedsimulations,whichdidnotfocusona par-ticulartype ofinterventioninorderto highlightthegeneralityof theframework.

The accuracy of the framework is not as high as that of the frameworks presented in [9,19,27,39]. However, simulation speed andsimulationaccuracyaretwocompetinginterestsandifspeed is preferredover accuracy,our newframeworkis apromising al-ternativetoexistingframeworks.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompeting ﬁnan-cialinterests.

Acknowledgments

MarcoMagliulo,AndreasZilianandLarsBeexwouldliketo ac-knowledge theﬁnancial supportof theUniversity ofLuxembourg forprojectTEXTOOL.JakubLengiewiczgratefullyacknowledgesthe ﬁnancial supportofEUHorizon2020Marie SklodowskaCurie In-dividual Fellowship MOrPhEM (H2020-MSCA-IF-2017, project no. 800150). This work was partially prepared in the framework of the DRIVEN project funded by the European Union’s Horizon 2020research andinnovation programmeunder grantagreement No.811099.

AppendixA. Treatmentofcontactattheendsoftheouter beam.

InthenumericalexamplesofSection3,thesectionatthetipof theinnerbeamslidesalongthewalloftheouterbeamuntilit ex-itstheouterbeam(ﬁrsttwoimagesinFig.14).Theoccurrenceof

thisis monitored,because whenit occurs, the contactconstraint betweenthe section attachedto the last integrationpoint of the innerbeamandthe outer beammustbe deactivated andthe in-crementrepeated.Then, nocontactinteractionsto embedthe in-ner beam inside the outer beam maybe left. Ifthis is thecase, a suddenrelease ofthe inner beammayoccur which makes the simulationdiverge(bottomleftinFig.14).

Toavoidthis,anadditionalconstraintisaddedtotheoutletof thehollowbeam. Itenforces thesection attheedgeoftheouter beamto be incontact withthe surface of the inner beam (bot-tom right in Fig. 14). The local problem to that must be solved to quantifypenetration is againgiven by Eq.17, exceptthat this timeBI_denotes_the_outer_beam_and_BJ_the_inner_one._The_method

ofLagrangemultipliersisusedtoenforcethisconstraint. The rea-sonis thatifthe penalty methodisemployed andthesection at the end of the outer beam is detached from the surface of the

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Fig. 15. Example 2 with a stiffer inner beam: (a) Initial configuration; (b) configu- ration halfway through the loading, and (c) final configuration.

AppendixB.FiguresforExample2withE=150MPaforthe innerbeam.

(a)

(b)

Fig. 16. Example 2 with a stiffer inner beam: Top: number of global iterations to reach convergence criteria: f_int_{+ r}c − f ext< 10 −8 ; Bottom: evolution of the number of

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Fig. 17. Example 2 with a stiffer inner beam: components of the reaction force (a) and torque (b) at the prescribed end of the thin beam.

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