A beam to 3D model switch in transient dynamic analysis

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A beam to 3D model switch in transient dynamic

analysis

Mikhael Tannous, Patrice Cartraud, David Dureisseix, Mohamed Torkhani

To cite this version:

Mikhael Tannous, Patrice Cartraud, David Dureisseix, Mohamed Torkhani. A beam to 3D model

switch in transient dynamic analysis. Finite Elements in Analysis and Design, Elsevier, 2014, 91,

pp.95-107. �10.1016/j.finel.2014.07.003�. �hal-01065975�

A beam to

3

D

model swit h in transient dynami analysis Mikhael Tannous a ,Patri eCartraud a , David Dureisseix b , Mohamed Torkhani a

GéM, E ole Centrale deNantes b

Universitéde Lyon,LaMCos,INSAde Lyon,CNRS UMR 5259

LaMSID UMREDF-CNRS-CEA 2832,EDF R

&

D,F-92141, ClamartCedex,Fran e

Abstra t

Transient stru tural dynami analyses often exhibit dierent phases, whi h

enables to use an adaptive modeling. Thus, a

3D

model is required for

a better understanding of lo al or non-linear ee ts, whereas a simplied

beammodelissu ientfor simulatingthelinear phenomenao urring fora

long periodof time.

This paper proposes a method whi h enables to swit h from a beam to

a 3D model during a transient dynami analysis, and thus, allows toredu e

the omputational ost while preserving a good a ura y.

The method is validated through omparisons with a 3D referen e

solu-tion omputed during allthe simulation.

Keywords: Transient dynami s,nite elements, swit h.

Email address: mikhael.tannouse -nantes.fr(MikhaelTannous)

This is a preprint of the article that appears on its final form as: M. Tannous,

P. Cartraud, D. Dureisseix, M. Torkhani, A beam to 3D model switch in

transient dynamic analysis, Finite Elements in Analysis and Design 91:95-107,

2014. DOI: 10.1016/j.finel.2014.07.003, © 2014, Elsevier. Licensed under

the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0

International http://creativecommons.org/licenses/by-nc-nd/4.0/

Manytransientstru turaldynami problems requirea

3D

modelinorder

to a urately a ount for lo al ee ts, that o ur along a small period of

time. However, a

3D

model for the entire stru ture used during the whole

simulationwillresultinanunaordable omputational ostevenonthebest

nowadays omputational ma hines and softwares. Sin e a

3D

model is

re-quired for a better understanding of lo al or non-linear ee ts, whereas a

simplied beam model is su ient for simulating the linear phenomena

o - urringfor alongperiodof time,anadaptivemodelingte hnique inwhi ha

3D

and a beam modelare used in dierent phases of the transient dynami al ulations an redu e the omputational ost while preserving a good

a - ura y. We, therefore, present a method that an redu e dramati ally the

omputational ost, forproblems where the

3D

non linearitiesare restri ted

in spa e and time.

To solve problems for whi h non linearities are restri ted in time, one

an useatimeintegration s hemeswit hing te hniquesu hasdonebyNoels

et al.[1℄for a blade/ asingintera tion simulation.

For phenomena that are restri ted in spa e, i.e. to a small part of the

omputationaldomain, awide range of methodshas been developed. These

approa hes an bedivided intoexa t (ordire t)methodsand iterativeones.

In the rst group we mention the stati ondensation te hniques and the

exa t stru tural reanalysis methods, su h those used in Hirai et al. [2℄, the

volumepat hes te hniques su hasArlequin (Ben Dhia[3℄)and the beamto

3D

onne tionsorshellto

3D

onne tions,thatenabletoa ounta urately for lo al

3D

phenomena, while the rest of the modelisless omputationally

Theiterativedomainde ompositionmethods anbedividedinto

overlap-pingandnon-overlappingdomainde ompositionmethods. Intherstgroup,

one nds the S hwarz, semi-S hwarz and semi-S hwarz-Lagrange methods

(see Hager et al. [5℄). Multi-s ale methods with pat h, su h as the nite

element pat hes (Glowinski et al. [6℄) and the harmoni pat hes (He et al.

[7℄)enable to have a lo alzoom onthe global domain.

Non-overlappingdomainde ompositionmethods anbe lassedintothree

main ategories (Gosseletand Rey[8℄): the primalapproa hes (Mandel [9℄),

the dual approa hes (FETI method Farhat and Roux [10℄), the hybrid or

mixedapproa hessu hasFETI-DPwhi hisanimprovedversionoftheFETI

method that mixes dual and primal approa hes (Farhat et al. [11℄). FETI

has also a multi-s ale version su h that used in Mobasher Aminiet al. [12℄

for the omputationof ship stru tures where windows are some entimeters

wide,whereasthe stru tureoftheshipishundredofmeterslong. Forsimilar

appli ations we alsond the mi ro-ma ro approa hes (Ladevèze et al.[13℄).

Regarding lo alnon-linear phenomena,FETI was enhan ed todeal with

large number of subdomainsand an take geometri non linearities into

a - ountFarhatetal.[14℄,andwasadapted for onta t problemsinAveryetal.

[15℄,Avery andFarhat[16℄,Dureisseixand Farhat[17℄. In Gendre[18℄,

Gen-dreetal.[19,20℄,theauthorsdevelopedanalgorithmthat enablestorepla e

theglobalmeshbyanelymeshedlo alzone,inordertotakelo alnonlinear

ee ts into onsideration withlow omputationaleort.

Forproblemswhere nonlinearitiesare restri ted both inspa eand time,

tional ost whilepreserving agooda ura yas illustratedin Fig.1. Infa t,

the simulation starts at

t = t

0

with a beam model for a linear simulation, and swit hes at

t = t

s1

to a beam-

3D

mixed modelwhen a non linear phe-nomenon is to take pla e. The simulation swit hes ba k at

t = t

s2

to the beammodel forof the rest of the simulationthat ends at

t

f

, if nomore non linear phenomenon ispresent.

PSfrag repla ements

Beam model

Beam model

Switch one

Switch two

t

s1

t

s2

t

f

t

0

Beam − 3D mixed model

linear simulation

_{non linear simulation}

_{linear simulation}

Figure1: Beamto

3D

swit h

This raises the problem of the swit h from one modelto another. This

paper presents a beam to

3D

model swit h, as well as a beam to a mixed

beam-

3D

modelswit h. The

3D

to beam modelswit h isnot the subje t of

this resear hwork.

Sin e the swit h method enables to swit h from a beam to a

3D

model

whennon-linearorlo alphenomenaaretotakepla e,thentheswit hinstant

hoi e depends on the non-linear problem itself. The main purpose of the

swit h method in this arti le is swit hing from a linear transient dynami

problemwithout largerotationsand with linear materialbehaviorto a

returns onta t=1),the swit hinstantis omputedby

t

s

= t − n × ∆t

,where

t

is the onta t instant,

∆t

the time step value and

n

a safety fa tor (10 is su ient) that is taken in order to prevent the

3D

omputations from

starting with an initial onta t dete ted. However, this arti le is fo used

on the swit h pro ess. Therefore, to demonstrate that the exa titude of the

swit h method is independent from the swit h instant hoi e, this later is

hosenarbitrary in the s ope of our study ases.

2. Mathemati al basi s of the swit h

Abeammodelsimulationthat startedat

t = 0

istobeswit hedfora

3D

modelsimulationat

t = t

s

. Startingwiththe

3D

modelat

t = t

s

requiresthe olle tionofthe beam modelsolutionat

t

s

andtransforming this solutionto have asuitable

3D

modelinitializationatthe same moment.

The fundamental dynami equation of a beam at

t = t

s

an be written as:

M

_b

U

¨

_b

_{+ C}

_b

U

˙

_b

_{+ K}

_b

U

_b

_{= f}

_b

(1)

where,

M

b

,

C

b

,and

K

b

arerespe tively the mass,dampingand stiness

matri es of the beam model.

f

b

is the external loading at

t = t

s

,

U

b

,

U

˙

b

, and

U

¨

b

denote, respe tively, the beam displa ements (in luding rotations), velo ities and a elerations atthe same instant.

The

3D

modelat

t = t

s

an be des ribed by:

where,

M

3

D

,

C

3

D

, and

K

3

D

are respe tively the mass, damping and

stiness matri es of the

3D

model.

f

3

D

is the external loading at

t = t

s

on the

3D

model,

U

3

D

,

U

˙

3

D

, and

U

¨

3

D

denote, respe tively, the

3D

model displa ements,velo ities and a elerationsat the same instant.

Suppose thatwestartwith thebeam modelat

t = 0

andthatwewantto

swit htothe

3D

modelattheswit hmoment(

t = t

s

). Wehaveto onstru t the

3D

solution

U

3

D

fromthe beam solution. This isperformedrst by de- omposingthe

3D

displa ementintoa ross-se tionrigidbodydispla ement

orrespondingtothe lassi alTimoshenkokinemati alassumption

PU

b

,and

a

3D

orre tion

U

3Dc

whi h a ountsfor ross-se tion deformation:

U

_3D

_{= U}

_3Dc

_{+ PU}

_b

(3)

We therefore need to generate

PU

b

and to ompute

U

3Dc

in order to

onstru t the

3D

model displa ementat

t

s

.

2.1. Generating

PU

b

PU

_b

isobtainedthroughaproje tormatrix

P

whi htransformsthebeam

displa ementve tor intoa

3D

rigid bodydispla ement perbeam se tion. It

isnoteworthytosay thatthe

3D

meshandthe beammesh annotbetotally

dis onne ted inorder forthe swit hto bedone. To be ableto onstru t the

displa ement of a node on the

3D

mesh, we should have the displa ements

and rotationsof the beam node that has the same position alongthe beam.

In other words, the beam model should be a proje tion of the

3D

mesh on

itsneutralaxis. However, itisnoteasytobuild

P

be auseitdependsonthe

relationship between the beam mesh and the

3D

mesh, whi h may hange

Let

N

ij

a node that belongs to the

i

th

ross-se tion of the

3D

model,

PU

ij

b

is the displa ement of

N

ij

omputed for a ross-se tion rigid body displa ement. The ross-se tion to whi h belongs

N

ij

has

G

i

on its neutral axis. The

i

th

beam node, whi h has the same oordinates as

G

i

, has a displa ement

U

i

b

and arotationaldispla ement

θ

i

b

. We,then, ompute

PU

ij

b

as follows:

PU

ij

b

= U

i

b

+ N

ij

G

i

∧

θ

i

b

(4) where,

N

ij

G

i

is a ve tor oriented from

N

ij

to

G

i

.

2.2. Computing

U

3Dc

Due to the de omposition of the 3D displa ement a ording to Eq. (3),

the

3D

model initialization will be performed through the

3D

orre tion

U

_3Dc

. Thus, insertingEq. (3) inEq. (2)at (

t = t

s

)

gives:

M

_3D

_(U

¨

3

Dc

+ ¨

PU

b

) + C

3D

( ˙

U

3Dc

+ ˙

PU

b

)

(5)

+ K

3D

(U

3Dc

+ PU

b

) = f

3D

Sin e we have one equation with three unknowns, then the following

as-sumptions are added:

˙

U

_3Dc

_{= 0}

¨

U

_3Dc

_{= 0}

(6)

They result in a displa ement orre tion

U

3

Dc

that orresponds to a stati

omputation for the

3D

model, at

t = t

s

, and that is the solution of the followingequation:

K

₃

_D

U

₃

_Dc

_{= f}

₃

_D

_{− M}

₃

_D

PU

¨

_b

_{− C}

₃

_D

PU

˙

_b

_{− K}

₃

_D

PU

_b

(7)

The omputationsof

P ˙

U

b

and

P ¨

U

b

anbedoneinthe sameway as

PU

b

by derivingEq. (4) with respe t totime.

Now that we have in hand the

3D

displa ements at the swit h instant

orresponding toEq. (3) ,we an initialize the

3D

modelat

t = t

s

by:

U

₃

_D

_{= U}

₃

_Dc

_{+ PU}

_b

˙

U

_3D

_{= P ˙}

U

_b

¨

U

₃

_D

_{= P ¨}

U

_b

(8)

Eq.(6)andEq.(8)are onsistentwithEq.(5),andthusallowtoinitialize

the

3D

model without violating its fundamental equation of motion at the

swit h instant.

However, sin eanintegrations hemeisusedtosolvethefundamental

dy-nami equation,then the initializationdepends alsoonthis time integration

s heme, and that makesthe subje tof Se tion 3.

3. Initializing the

3D

solution

In order to solve a dynami problem,one needs to have in hand the

ini-tial displa ements and velo ities. The initiala elerations are therefore the

solution of the fundamentalequation of motion solved at the initialinstant.

However, when this equation is solved numeri ally via a time integration

s heme, the required initial onditions in that asedepend on the time

is the ase of most softwares) to onsider zero initiala elerations,while for

an impli itintegration s heme to orre tly ompute the initiala elerations

that satisfy the fundamental equation of motion at that instant. Therefore,

for anexpli it integration s heme initializingthe a elerationsis mandatory

to avoid an artifa t transient phenomenon that may lead the integration

s heme to diverge shortly afterswit hing. However, in the examples shown

in this paper, we are using an impli it integration s heme namely, a

New-markintegration s heme that doesnot require initiala elerationssin e the

software, Code_Aster or Abaqus, omputes automati ally the initial

a el-erationshaving inhandthe initialdispla ementsandvelo ities, asindi ated

in Rixen[21℄.

However, itisnoteworthy tomention thatwiththe hoi eof Eq.(8)only

the displa ementsare dierentfromthe ross-se tionrigid-bodyassumption

at the swit h instant. The initial velo ities (as well as the initial

a elera-tions) remain those onstru ted fromthe beam model, and they are around

5%

dierentfromthe

3D

referen evelo itiesanda elerationsformost ases of study shown later in this paper. This dieren eseems quite small, but is

stillstrongenoughfortheproblemswehavesolved andmay auseanartifa t

transient phenomenondepi ted by highfrequen y os illationsinthe

a eler-ations and velo ities values. These high frequen y os illationsmay lead the

solution to diverge. In order to vanish these os illations, one an insert a

numeri aldamping or hange the velo ities and a elerations orre tions as

A numeri aldampingin the integration s heme an lter these high

fre-quen y os illations without any other inuen e on the solution. The HHT

integrations hemehas beenused inthis studytolterthe numeri al

os illa-tions. This numeri aldampingneeds tobemaintainedonseveral time steps

followingthe swit hinorder forthe high frequen y os illationsto vanish,as

shown in the results in the following. However, a more attra tive method

does exist and an redu e the high frequen y os illations that appear after

swit hing onsiderably and is detailedinSe tion 3.2.

3.2. A triple stati swit h pro edure

AsEq.(8)shows, thehigh frequen y os illationsare generatedby apoor

initialization of the velo ities and a elerations, sin e these are later

gen-erated from the beam solution and are not ompletely adapted to the

3D

model. The hypothesis taken in Eq. (6) is too strong and therefore

gener-ates high frequen y os illations. However, assuming that the displa ement

initializationis adapted to the

3D

model, then a strategy enabling a better

initializationof thevelo ities (anda elerationsif needed forthe integration

s heme) based onthe displa ement orre tion an bebuilt with the

integra-tion s heme and thus eliminates the high frequen y transient phenomenon

that o ursafter swit hing.

We therefore rst he k the displa ement orre tion on astati problem

to prove its e ien y and then, a ording to the integration s heme being

The swit h for a stati problem may be seen as a parti ular ase of the

dynami one. It is investigated here to test if the

3D

displa ements after

swit hingare lose toa referen e

3D

stati solution. The beam fundamental

equation for astati problem is:

KU

_b

_{= f}

_b

(9)

The

3D

fundamentalequationis:

K

₃

_D

U

₃

_D

_{= f}

₃

_D

(10)

The

3D

displa ement an be divided asexplained earlierin Eq. (3), and

leads to dene

U

3Dc

as the solutionof:

K

_3D

U

_3Dc

_{= f}

_3D

_{− K}

_3D

PU

_p

(11)

This stati orre tion

U

3

Dc

summed with

PU

b

is ompared to a

refer-en e solution for the same

3D

model mesh, omputed by solving Eq. (10).

This has been performed on several mesh types, ross-se tions shapes and

boundary onditions, and dieren e between the omputed displa ements

and the referen e solution has been found to be negligible, indi ating that

the displa ementsare well orre ted by this swit h method

3.2.2. Basi s of the triple stati swit h pro edure

Sin e astati swit h providesan a urate orre tion of the

3D

displa e-ments,thena orre tionofthe velo itiesanda elerations anbebuiltusing

mationintoa ount. Butthevelo itiesanda elerationsproposedinEq.(8)

areforarigidbody ross-se tionassumption. Sin e,weareusingaNewmark

integration s heme, then there is no need to initialize the a elerations but

we need to improve the velo ities initialization. This an be a hieved if the

stati swit hisappliedonthree onse utivetimesteps,the swit hinstant

t

s

, the pre eding step

t

s−1

and the followingone

t

s+1

. Thenbased on the three su essive displa ements, one an inspirefromthe nite dieren emethoda

better initializationof the velo ities asfollowing:

˙

U

₃

_D

₌

1

2 × ∆T

([PU

b

+ U

3

Dc

]

t

s+1

−

[PU

b

+ U

3

Dc

]

t

s−1

)

(12) This velo ity initialization ombined with the displa ementinitialization

willleadtheNewmarkintegrations hemeto omputetheinitiala elerations

as the solutionof:

M

_3D

U

¨

_3D

_{= (f}

_3D

_{− C}

_3D

U

˙

_3D

_{− K}

_3D

U

_3D

₎

(13)

This initialization te hnique proved to be simple and very e ient, in

the appli ation examples shown in this arti le and on several others. It is

ompletely onsistentwiththeNewmarkintegrations heme,andtherefore,is

proposedas aproperbeamto

3D

modelswit hing te hnique inour resear h

work.

Notethatthe triple stati swit hpro eduredoes not requireanumeri al

damping. Therefore, all the following swit h examples solved by a triple

stati swit hmethodare not damped.

earlier in this arti le, initializing the a elerations is mandatory. For the

entral dieren e integration s heme, the nite dieren e method leads to

the following initiala elerations:

¨

U

_3D

₌

1

∆T

2

([PU

b

+ U

3Dc

]

t

s+1

(14)

−

2 × [PU

b

+ U

3

Dc

]

t

s

+ [PU

b

+ U

3

Dc

]

t

_s−1

)

This a eleration initializationproved to work on several ases of study

not shown inthis resear h work.

4. Energy onsisten y of the swit h

Tovalidate the on ept of the swit h for transientdynami appli ations,

we ompare the

3D

solution after swit hing with a

3D

referen e solution

obtained by performing the same omputation on the whole simulation

pe-riod. Another way to he k the validity of the swit h method on transient

dynami problemsisto he k whether theswit hremovesorinserts parasite

energy in the system at the swit h instant, whi h an lead to non physi al

simulations. Su h solution pre ision analyses are widely used in the

litera-ture su h as in Noels et al.[1, 22, 23℄, where this analysis te hnique served

todemonstrate the stability and onsisten y ofanimpli it andexpli it time

integration s hemes swit h method.

Ifwe have a me hani alsystem subje ted toan external for e

F

, with a

mass

M

, and a stiness

K

, and if the displa ements ata given instant

t

are

denoted by

U

andthevelo itiesatthesameinstantby

U

˙

,the kineti energy

W

c

=

1

2

U

˙

T

_{M ˙}

_U

(15)

The strainenergy reads:

W

d

=

1

2

U

T

_KU

(16)

The work of the external for es

W

f

is omputed by:

W

f

= F

T

U

(17)

Wenote

W

diss

theworkofdissipativefor es(fri tion,damping,et .). The kineti energy theorem gives:

d

dt

W

c

=

d

dt

W

f

+

d

dt

(W

diss

−

W

d

)

(18) In our ases of study, the dissipative for es are negligible,then:

W

c

+ W

d

= W

f

+ cst

(19)

where

cst

is a onstantthat depends on the problembeing solved.

We distinguish three main ases:

ˆ

F

= 0 :

the total energy

W

t

= W

c

+ W

d

is a onstant.

ˆ

F

is a onstant: the total energy is a time dependent fun tion (but

W

c

+ W

d

−

W

f

is a onstant).

ˆ

F

evolves intime (whi h is the ase of all the appli ation examples of

mass

M

ishold by aspringhavinga stiness

k

and subje ted toanexternal

for e

F

. The motion o urs along the

x

-axis. The displa ementsolution is:

x = Acos(ωt) + Bsin(ωt) +

F

k

(20)

where

ω =

r k

M

. The orrespondingkineti energy is:

W

c

=

1

2

M ˙x

2

=

1

2

k(A

2

sin

2

(ωt) + B

2

cos

2

(ωt) − AB sin(2ωt))

(21)

Therefore, the kineti energy involves only the angular frequen y

2ω

,

while the strain energy involvesboth

ω

and

2ω

. In fa t:

W

d

=

1

2

kx

2

=

1

2

k(A

2

cos

2

(ωt) + B

2

sin

2

(ωt

)

+

F

2

k

2

+ 2ABsin(ωt)cos(ωt) + 2A

F

k

cos(ωt) + 2B

F

k

sin(ωt)

(22) Therefore, the strain and kineti energy do not have the same angular

frequen y. If

F = 0

,

W

c

+ W

d

=

1

2

k(A

2

+ B

2

) = cte

. If

F = cst 6= 0

:

W

c

+ W

d

=

1

2

k(A

2

+ B

2

) +

1

2

(

F

2

k

+ 2AF cos(ωt) + 2BF sin(ωt)) = cte + W

f

(23)

If

F = cst

,

W

t

= W

c

+ W

d

= W

f

+ cst

.

Inthisarti le,theenergy onsisten yoftheswit hisveriediftheenergy

losetoits orrespondingvalueforthe

3D

referen esolution. This an prove

that the swit h does not remove nor insert energy in the

3D

solution after

swit hing. A omparison will be set between the evolution of the kineti ,

strain and total energy of the beam model,

3D

referen e modeland the

3D

swit hmodeltoprovethat the swit h method isenergeti ally sound.

5. Appli ation examples

Inthis se tion,wepresentasimplenumeri alexamplethatillustratesthe

e ien y of the beam to

3D

modelswit h for dynami ases.

P

P

PSfrag repla ements

f

_(t)

Figure2: The

3D

modelunder study

In fa t, the method has been validated on more omplex ases, for

dif-ferent ross-se tion shapes, loadings and boundary onditions. In the ase

onsidered here, the beammodelisa Timoshenkobeammodel, witha

re t-angular ross-se tion,havingthefollowingdimensions: width0.012m,height

0.01 m and a length of 0.1m. The beam is made with a steel material with

density

ρ = 7800 kg/m

3

, Young modulus

E = 2.1 × 10

11

N/m

2

and Poisson

oe ient

ν = 0.3

. One side of the beam is xed, the other one issubje ted

to a transverse load equal to

f

(t) = 100 × t

³

×

e

−1.1t

at its surfa e enter.

approximatelyonethousandnodes. Theswit hinstantis

t

s

= 1.5 s

,atwhi h the beam simulationis swit hed to the

3D

model, with the same boundary

onditions and loading. The

3D

solution after swit hing is ompared to a

referen e solution, whi h is a

3D

solution obtained on the same

3D

model

for a simulationthat starts at

t = 0

and last three se onds.

The swit h from the beam model to the

3D

model is performed rst

using the approa h des ribed in se tion Se tion 2.2 (stati orre tion with

numeri al damping) and se ond with the initialization built from the

3D

displa ements omputed at threedierent time steps (see Se tion3.2.2).

0

0.5

1

1.5

2

2.5

3

0

5e-05

0.0001

0.00015

0.0002

PSfrag repla ements

Beam model

3D switch model

3D ref erence model

zoom one

zoom two

zoom three

t(s)

U

(P

)

[m

]

Figure3: Displa ementresults:thenumeri aldampingmethodandthetriplestati swit h

leadtothesameresults

We ompare the displa ements, velo ities and a elerations of node

P

Fig. 3 shows the displa ement results. First we an see a dieren e

be-tween the beam solution and the

3D

referen e solution. This dieren e is

verysmall,but stillnoti eableif wemakeazoom. Immediatelyafter

swit h-ing, the

3D

solution turns out to be very a urate and is very lose to the

referen e one. Both swit h methods exhibit pra ti ally the same pre ision

regarding the displa ements.

However, as shown on Fig. 4, whi h represent a velo ity omparison,

or Fig. 5 whi h represents an a eleration omparison, immediately after

swit hing, highfrequen y os illationswith largeamplitude o ur inthe ase

where there is only the stati orre tion. If anumeri aldamping is used to

lter outthese os illations,then,they willbepresent onlyseveraltime steps

after swit hing. For a HHT integration s heme with

α = 0.25

, in our ase

35

time steps (0.05 s) were su ient for the

3D

solution to onverge to the referen e one. If a triple stati swit h pro edureis performed, the velo ities

donot present anyos illations; however, very smallos illationso ur onthe

a elerations and vanish very shortly afterswit hing.

The results show that both methods work, but the triple stati swit h

appearstobemorea uratewhileeasytoimplement. Thebeamto

3D

model

swit ha elerates the dynami simulationof a

3D

modelwhile preserving a

gooda ura y.

Energy analysis onrms the e ien y ofthe swit h. Infa t, the swit h

does not remove nor insert parasite energy inthe solution

Fig. 6 sets a omparison between the kineti and strain energies of the

0

0.5

1

1.5

2

2.5

3

0

5e-05

0.0001

0.00015

0.0002

PSfrag repla ements

Beam model

3D switch model

3D ref erence model

zoom one

zoom two

zoom three

t(s)

˙ U

(P

)

[m

/s

]

(a)Numeri aldampingmethod

0

0.5

1

1.5

2

2.5

3

0

5e-05

0.0001

0.00015

0.0002

PSfrag repla ements

Beam model

3D switch model

3D ref erence model

zoom one

zoom two

zoom three

t(s)

˙ U

(P

)

[m

/s

]

(b)Triplestati swit hpro edure

0.5

1

1.5

2

2.5

3

-5e-05

0

5e-05

0.0001

0.00015

PSfrag repla ements

Beam model

3D switch model

3D ref erence model

zoom one

zoom two

zoom three

t(s)

¨ U

(P

)

[m

/s

2

]

(a)Numeri aldampingmethod

0

0.5

1

1.5

2

2.5

-5e-05

0

5e-05

0.0001

0.00015

PSfrag repla ements

Beam model

3D switch model

3D ref erence model

zoom one

zoom two

zoom three

t(s)

¨ U

(P

)

[m

/s

2

]

(b)Triplestati swit hpro edure

the same strain energy urve is obtained. However, if the simple swit h is

performedandisstabilizedwithnumeri aldamping,os illationsareobserved

onthekineti energy urveonseveraltimesteps followingtheswit hinstant

beforeit onverges to itsstable value. A small dieren eexists between the

strain energy of the

3D

referen emodeland that of the beam model. That

is due to modelingdieren es, su h as the dieren ein the shapefun tions,

betweenthebeamandthe

3D

models. Afterswit hing,thereremainsasmall

dieren e between the

3D

model strain energy and the

3D

referen e model

strainenergy,but itappears thatthe swit hdoesnot auseadisturban eon

the value of the strain.

0

0

3

5

7

1

1.5

3

0

0

2.5

5

7.5

10

0.5

1

1.5

2

2.5

3

0

1e-10

3e-10

2e-10

1

1.5

2

PSfrag repla ements

3D

Referen e Model BeamModel

3D

Swit hModel

t (s)

t (s)

t (s)

W

c

[×

10

−

1

1

J

]

W

d

[×

10

−

3

J

]

Triplestati swit h

Simpleswit hstabilized

bynumeri aldamping

Zoom

Zoom

Zoom

Figure6: Thekineti (

W

c

)andthestrain(

W

d

)energies

Thissame on lusionisalsoobtainedonthekineti energyon ethislater

te h-perturbation even on the few time steps following the swit h instant.

How-ever, in many industrial ases, the

3D

modelis required for asmall interval

of time, but alsofor a smallarea. It istherefore more appropriateto swit h

from a beam modelto a mixed beam-

3D

model. The

3D

zone is limited to

the zone wherelo alphenomenaare totake pla e as shown inFig. 7. PSfrag repla ements

Beam model

Beam model

Beam to 3D

connexion

3D model

Figure 7: Beam-

3D

mixedmodel

This raises the question of the beam to

3D

onne tion and makes the

subje t of Se tion 6.

6. Beam to

3D

onne tion

As previously mentionedin the introdu tion, when lo alphenomena are

restri ted in spa e and time, a beam to a beam-

3D

mixed model swit h

enables to preserve a good modeling a ura y while de reasing the

om-putational ost. In the following, a beam to

3D

onne tion, available in

Code_Aster (see Pellet [24℄), is presented and will be used in this resear h

work. This beam to

3D

onne tion satises the onsisten y of the beam

and

3D

displa ements(kinemati stability),aswellasasuitableeort

trans-mission from the beam to the

3D

(stati stability) that does not generate

This beam to

3D

onne tion is a non-overlapping one. The onne tion

o urs between a beam node

P

and a

3D

ross-se tion

S

of area

A

at the

gravity enter

G

of

S

.

6.1. Kinemati stability

The

3D

displa ements

U

3D

is the sum of a rigid-body ross-se tion dis-pla ement

U

3

Db

and a ross se tion deformationve tor

U

s

. The beam

dis-pla ementand rotationve torsatpoint

P

are denoted, respe tively,

U

b

and

θ

b

. The kinemati onne tion ondition between

P

and arbitrary node

M

that belongsto se tion

S

reads:

U

3Db

= U

b

+ θ

b

∧ GM

.

Thekinemati stabilityofthe onne tionisfulllediftheorthogonalityof

ve tors

U

3Db

and

U

s

issatised. Thisensures that the

3D

ross-se tion has no inuen e on the displa ementof the beam nodes. This an be expressed

by the following equations:

U

_b

₌

1

A

ˆ

s

U

_3D

_dS

(24)

θ

b

= I

−1

ˆ

s

GM

_{∧ U}

₃

_D

_dS



(25) 6.2. Stati stability

In order to avoid artifa t strains on the onne tion interfa e between

the

3D

model and the beam model, a suitable transmission of the loading

between the beam and the

3D

model is ne essary. It an be a hieved if

the proje tion of se tion

S

stresses onnode

P

result in beam loadingand is

ˆ

s

σ.n.U

3

D

dS = F

p

U

b

+ T

p

θ

b

(26) where

F

p

isaloadingve toronnode

P

and

T

p

isatorqueve toronnode

P

that an be dedu ed from Eq. (26) by solving an optimizationproblem:

F

_p

₌

ˆ

s

σ.ndS

(27)

T

_p

₌

ˆ

s

GM

_∧

_σ.ndS

(28)

The following se tionpresent a beam toa mixed beam-

3D

modelswit h

in transientdynami analysis.

7. A beam to mixed beam-

3D

model swit h example

In this example, we take a beam with a ir ular ross-se tion of radius

0.005 m

and a

0.25 m

length, simply supported from both sides, and that has the following material properties:

ρ = 7800

kg/m³, Poisson oe ient

ν = 0.3

andaYoungmodulus

E = 2.1×10

11

P a

. At

0.12 m

fromonesideitis subje ted toaloadoftheform

f

(t) = −100×sin(ω ×t)

,where

ω = 6.4 rad/s

,

for a

3s

long simulationstartingat

t = 0s

. Animpli itintegration s hemeis

usedwith

2000

timesteps. Theswit hinstantisxed at

t = 2s

. Forabetter

presentation of the results, the displa ements, velo ities and a elerations

are presented in the following illustrations inthe interval

t ∈ [1, 3] s

.

Thedispla ement,velo itiesanda elerationsareregistered withrespe t

to time at a node

D

N

as illustrated in Fig. 8. The later shows the dimen-sions of the model in question. The same physi al model is modeled by a

beam model, a whole

3D

model and a model that ombines beam and

3D

model. The beam to mixed beam-

3D

modelswit h is performed using the PSfrag repla ements

f (t)

f (t)

f (t)

D

N

D

N

D

N

Beam model

3D ref erence model

Beam − 3D mixed model

0.1 m

0.1 m

0.05 m

0.14 m

0.12 m

Figure8: Beammodel,beam-

3D

mixedmodel, and

3D

referen emodel

two initialization methods dis ussed earlier, namely, a numeri al damping

method (HHT integration s heme) with

α = 0.25

and a three stati swit h

pro edure. The displa ements, velo ities and a elerations of the beam-

3D

mixed model after swit hing are ompared with the beam model solution,

the mixedbeam-

3D

modelsolutionanda

3D

modelreferen esolution,three

of them for the same loading, startingat

t = 0 s

and lasting

3 s

.

Ifanumeri aldampingisused tostabilizethe solutionafterswit hing,a

transientstageisinitiatedand anbeseen onthe a elerations,see Fig.10a,

while being less noti eable on the velo ities, see Fig. 9a and absent on the

displa ements,see Fig. 9b.

1

1.5

2

2.5

3

-0.0002

0

0.0002

0.0004

PSfrag repla ements

3D ref erence model

Beam − 3D mixte model

Beam model

Beam − 3D switch model

U

(P

)

[m

]

t(s)

zoom one

zoom two

zoom three

(a)Displa ements: thenumeri aldampingmethod andthetriplestati swit h

leadtothesamedispla ementsresults.

1

1.5

2

2.5

3

-0.002

-0.001

0

0.001

0.002

PSfrag repla ements

˙ U

(P

)

[m

/s

]

t(s)

zoom one

zoom two

zoom three

Numeri aldamping

Triplestati swit h

(b) Velo ities: omparison between the numeri al damping method and the

triplestati swit hmethod

1.5

2

2.5

3

-0.02

-0.01

0

0.01

0.02

1

PSfrag repla ements

3D ref erence model

Beam − 3D mixte model

Beam model

Beam − 3D switch model

¨ U

(P

)

[m

/s

2

]

t(s)

zoom one

zoom two

zoom three

(a)Numeri aldamping

1.5

2

2.5

3

-0.01

0

0.01

1

PSfrag repla ements

¨ U

(P

)

[m

/s

2

]

t(s)

zoom one

zoom two

zoom three

dieren e exists between the displa ements of the

3D

referen e model, the

beamoneandthebeam-

3D

mixedmodelasshownonFig.9a. Thebeam-

3D

mixed model is loser to the beam solution, sin e the

3D

zone is one fth

the lengthofthe beam-

3D

mixed model. This on lusionis thesame forthe

velo ities and a elerations asshown in Fig. 9band Fig. 10respe tively.

Both swit hing te hniques prove to be e ient. The triple stati swit h

is more elegant while easyto implement.

We now he k the energy onsisten y of the swit h for this appli ation

example.

0

0.5

1

_1.5

₂

_2.5

₃

0

0.005

0.01

0.015

PSfrag repla ements

Beam − 3D mixed ref erence model

Beam model

Beam − 3D mixed swicth model

t (s)

W

c

+

W

d

[J

]

Figure11: strainandkineti energysum

Fig.11showsthesumofthekineti andstrainenergyforthebeammodel,

the mixed beam-

3D

swit h model. We avoid to present the energy urves

orresponding tothe

3D

referen emodelsin e they donot provideessential

lues for the analysis of the energy onsisten e of the swit h.

Asmalldieren eisobservedbetweentheenergy urveofthebeammodel

and that of the mixed beam-

3D

model. This dieren e is due to modeling

dieren es (shape fun tions dieren es, et .). After swit hing, the mixed

beam-

3D

model energy urve joins that of the referen e mixed beam-

3D

model. The same on lusion drawn from the previous appli ation example,

inwhi hnobeamto

3D

onne tionisused,ison emoreobtained: theswit h

does not lead to any perturbation in the energy values. The kineti energy

is presented in Fig. 12 in the time interval

t ∈ [1, 3] (s)

, and a zoom on the

kineti energy around the swit hinstant ispresented onthe right hand side

of the this same gure.

1

1.5

2

2.5

0

5e-08

1e-07

1.5e-07

3

PSfrag repla ements

Beam − 3D mixed ref erence model

Beam model

Beam − 3D mixed swicth model

t (s)

W

c

[J

]

Zoom 400%

the swit h.

In this example, at the swit h instant the velo ity is near its maximum

as it an be seen on Fig. 9b, while the displa ements and a elerations are

lowasshown inFig.9aandFig. 10, respe tively. It isinteresting toperform

a swit h at a dierent instant to have a dierent initial onguration su h

as

t

s

= 1.75 (s)

,atwhi hthe velo ities are low, while thedispla ementsand a elerations are high. This an illustrate the e ien y of the swit h and

1

1.5

2

2.5

3

-0.01

-0.005

0

0.005

0.01

0.015

PSfrag repla ements

Beam − 3D mixed ref erence model

3D ref erence model

Beam model

Beam − 3D mixed swicth model

t (s)

¨ U

(m

/s

2

)

Figure13: A elerationresultsfor

t

b

= 1.75 (s)

prove that the swit h instant an be a omplete random in the simulation

interval. Sin e the triple stati swit h is elegant and easy to implement,we

Fig.13showthea elerationresultsa ordingtothe

x

-axisatpoint

D

N

. The same a ura y is obtained onthe displa ementsand velo ities results.

Fig.14shows thekineti andstrain energysum. No energyperturbation

is dete ted. This is also the ase if we he k the strain and kineti energy

urvesseparately. Itisobviousthatwehavethesamee ien yfortheswit h

1

1.5

2

2.5

3

0

0.005

0.01

0.015

PSfrag repla ements

Beam − 3D mixed ref erence model

Beam model

Beam − 3D mixed swicth model

t (s)

W

d

+

W

c

(J

)

Figure 14: Strainandkineti energysumat

t

b

= 1.75 (s)

performedat

t

s

= 2.4 (s)

and

t

s

= 1.75 (s)

.

8. Con lusions

We have proposed a numeri al method that enables to swit h from a

beam to a

3D

model, or from a beam to a mixed beam-

3D

model, when a

a ura y.

Two swit hing te hniques were proposed. One uses a numeri al

damp-ing to lter possible artifa t os illationsin a elerations and velo ities, and

the se ond, the triple stati swit h, is more elegant, do not need numeri al

dampingand donot ause artifa t os illations.

The swit h proved towork on dynami and stati ases. The

3D

swit h

solution ispra ti ally the same asthe

3D

referen e one.

The energy onsisten y of the swit hhas been demonstrated. No energy

is removed nor inserted by the swit h.

In this arti le and as also presented in Tannous et al. [25℄, the swit h

method is developed for transient dynami analyses problems without an

overall rotation. However, the main motivation behind the swit h on ept

proposed inthe PhDthesis of Tannous[26℄, and presented inTannouset al.

[27℄, is its appli ations to turbine a idents involving rotor-stator onta t

intera tions. The swit h method will be extended, in future publi ations,

for appli ation to the slowing down of unbalan ed turbine rotors with lo al

intera tions and fri tions.

A knowledgments

The authors thank the Fren h National Resear h Agen y (ANR) in the

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