on the simulation of deformable bodies using combined discrete and finite element methods

8/10/2019 On the Simulation of Deformable Bodies Using Combined Discrete and Finite Element Methods

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O n the S i mul a t ion o f Deformabl e B odi es Us i ng Com bi ned Di scre te and F i n i te

E l ement Methods

Petros I. Komo dromos and John R. W il l iam s

Introduct ion

In d iscre te e lement methods (DEM, [2] and [4] ) the s imula ted bodies are typ ical ly

assumed to be inf in i te ly r ig id in order to reduce the com putat ional cost . H owever ,

there are m ul t ibod y sys tems where i t i s usefu l to take in to account the deform abi l i ty

of the s imula ted bodies in order to enable the evaluat ion of the i r s t ress and s t ra in

dis tr ibut ions . This paper focuses on the s imula t ion of sys tems o f mul t ip le deformab le

bodies us ing a com binat ion of d iscre te and f in ite e lement methods (FEM), w i th some

s impl i fy ing assumptions that are necessary to make the so lu t ion of the problem

feasible.

In t rad i t ional m ixed FE formulat ions the contact ef fec ts can be taken in to

account us ing Lagrange mul t ip l iers methods and keeping the contact sur faces and

forces as unkno wn s together with the unkn ow n displacements . Th is approach results

in huge sys tems o f h ighly nonl inear coupled equat ions due to geom etr ic as wel l as

boundary nonl inear i t ies . Fur thermore , the par ts of the bodies tha t may come in

contact , typical ly , have to be ident i f ied before per forming the s imula t ion . How ever ,

no p r io r knowledge o f the upcom ing con tac ts i s ava i l ab le in the m ul t ibody systems

under cons idera tion . Cons ider ing the excess ive com putat ional requirements , due to

the hug e nu m ber of degrees -of- f reedom (DOF) an d the h igh non l inear i t ies o f the

coupled sys tems o f equations , i t i s unrealis tic to so lve problems inv olv ing m an y

interactin g bod ies using such c lassical contact FE approaches.

S imulat ions of deformable bodies wi th reasonable computat ional cos t are

enabled by incorporat ing FEM in DE analyses us ing cer ta in assumptions that

uncouple the contact in teract ions f rom the equat ions of dynamic equi l ibr ium. In

par t icu lar , the D EM are em ployed to ident i fy , a t each s im ulat ion s tep , the bod ies in

contact an d determine the contact forces. Then, e i ther a FE or a D E formu lat ion is

used a t the indiv idu al bod y level to descr ibe the equat ions of mot ion , depending

1 Civi l and Env ironm ent Engineer ing Depar tment , M assachusetts Ins t i tu te o f

Tec hnolog y, Cambridge, Massachusetts 02139, U .S .A. Contact e-mai l :

pe t ros@mit . edu

138

Discrete Element Methods

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D I S C R E T E E L E M E N T M E T H O D S 139

whether the bod y unde r cons idera t ion is deformable or rig id , respectively . In case o f

a de formable body, the s t ra ins are assumed to be suf f ic ient ly smal l to perm it a smal l

s t ra ins, a l thoug h large d isp lacements , analys is. The deform abi l i ty o f indiv id ual

bodies i s taken in to account us ing a d isp lacement-based (DB) Updated-Lagrangian

(UL) F in i te Elem ent (FE) formulat ion . F inally , an expl ic i t t ime in tegra t ion me thod,

speci f ica l ly the centra l d i f ference method (CDM), i s used to per form the num er ica l

d i rec t in tegra t ion of the equat ions of m ot ion and d etermine the new disp lacements , as

wel l as the deformat ions and s t resses wherever needed, of each body. Having

com puted the m ot ion of each d iscre te body a t a ne w t ime s tep , the pos i t ions of a l l

d iscre te bodies are updated and a new contact de tec t ion process determines the new

contacts and evaluates the corresponding contact forces , which are then used in the

follow ing t ime s tep.

F i n i t e E l e m e n t F o r m u l a t io n

The UL -FE formulat ion (Bathe [1]) can be der ived f rom the pr incip le of v i r tual wo rk

(PVW ), wh ich is va l id even whe n there are no essent ia l boun dary condi t ions (BCs) as

the mul t ibody unres t ra ined sys tems under cons idera t ion . The la t ter sys tems , in

genera l , have onl y natura l BCs , i .e ., p rescr ibed bound ary forces and mom ents due to

contact ef fec ts. Replacing the unknow n contact forces a t each t ime s tep w i th a D EM

est imat ion a l lows the dec oupl ing of the equat ions of mot ion , which ca n be so lved for

each d iscre te body indep endent ly in order to com pute i t s new pos i t ion .

Acco rding to the U L FE formulat ion , a l l quant it ies and var iables are refer red

to the la tes t computed conf igura t ion . Assum ing that we kno w the so lu t ion up to t ime

t, we can determ ine the s tate, i .e ., the displacem ents t + d t u , strains t + ~c, and stresses

t + at , of a s im ula ted bod y (F igure 1) a t t ime

t + At ,

using the equations:

M . ' ~ I + C . ' I) '+ [ F = t R

(1)

A lumped d iagon al mass m atr ix M and a mass propor t ional dam ping matr ix C are

used to decouple the resul t ing FE equat ions and fur ther reduce the computat ional

cost . Ve ctor tR represents the external , except inert ia and dam ping, forces , and

I F

is

the vec tor of in ternal forces , both com puted us ing the b ody c onf igura t ion a t t ime t .

The vector of in ternal forces [ F = ~ ~V e A B ez

d V

i s computed by in tegra t ing

A e

the p roduct of the s t ra in-d isp lacement matr ix , t B , with the C auc hy s tresses v ector ,

t~ _ t f l 1 t~.22 r33 t r l 2 t r 2 3 t~.13 , o v e r t h e d e f o r m e d v o l u m e o f t h e b o d y a t t i m e

t.

A e

Th e s train-displacem ent ma tr ix, t B , expresses the Alm ansi s trains in terms

of nodal d isp lacements , by proper d i f ferent ia t ion of the d isp lacements .






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1 40 D I S C R E T E E L E M E N T M E T H O D S

X 3

~ j nodek

( ~ ~ k, ~

Element

Original

configuration

(time0)

Configuration

at time

/

C x , x x : )

X2

Fig ure 1 . Conf igura t ion of bod y a t t imes 0 and t

A 6 e = a t B e U

(2)

Th e u sual re la t ions for inf in ites imal s t ra ins cannot be used , because the

displace m ents are large. Instead, the Alm ansi strains , whic h are given b y the

fo l lowing equat ion , should be used as they are the work-conjugate of the Cauchy

stresses.

1 6 t t t ~

A t IJ = - f f : [ t U i , j + t u j , i - t u k , i t U k , j )

3 )

Fo r exam ple, the Alm ansi s train A Cxy n a 3D proble m is equal to:

~tSaY = 2 L O ty Otx Otx O ty OtX Oty Otx

A body is d iscre t ized in to an assemblage of f in i te e lements tha t are

in terconnected a t nodal poin ts on the e lem ent boundar ies . The d isp lacem ents wi th in

a F E , / f , a r e expres sed in t e rms o f in te rpo la t ion funct ions , He , and the noda l po in t

d isp lacements , U, according to a n isoparametr ic FE formulat ion .

e e e T

U e : [ u ( x , y , z ) , v ( x , y , z ~ w ( x , y , z ) ] : H e ( x , y , z ) ' U

(4)

The d isp lacement in terpola t ion matr ix ,/ - /~ , has non zero e lem ents only in the

columns that cor respond to the DOFs associa ted wi th the nodal poin ts of tha t

par t icu lar e lement , s ince the d isp lacements of a poin t ins ide an e leme nt depend o nly

on the d isp lacements of the nodal poin ts o f the par t icu lar dem ent . Therefore ,

assuming d iagonal , ins tead of cons is tent , mass and damping matr ices , a l l the

computa t ions can be done fo r each DOF independen t ly and ve ry e f f i c ien t ly , w i thou t

any need for cons t ruct ing and m anipula t ing matr ices tha t refer to the ent i re

assemblage o f e lements.






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The s t resses can then be computed us ing the mater ia l cons t itu t ive law as i t i s

speci f ied by the e las t i c i ty mat r ix , C ~, which i s the same as the one used for smal l

d i sp lacements w hen the m ater ia l i s i so t ropic an d l inear e las ti c .

r = C e.A 6e + re wh ere r e are any ini t ial s tresses.

(5)

C o n t a c t I n t e r a c t io n s

The contact in teract ion between col l id ing bodies i s an ex t rem ely compl ica ted

pheno me non that involves s t ress and s t ra in d i s tr ibutions wi th in the c o l l id ing bodies,

thermal , acous t ica l, and f r ic t ional d i ss ipation of en ergy due to the con tact, as we l l as

p las t i c deformat ions . I t i s ev ident tha t wi th the current comput ing l imi t s , one can

ant ic ipate to so lve only a very s impl i f i ed vers ion of th i s problem, when i t involves

mu l t ip le unres t ra ined bodies , a f ter mak ing cer ta in assumpt ions and s impl i f ica t ions .

The sof t contact approach i s used , accord ing to wh ich a t an y s imulat ion s tep

that two bodies com e in contact, equivalent normal and tangent ia l spr ings are used to

es t imate the contact forces that are appl ied to the bodies to push them apar t . Some

over lapping of the bodies in contact i s a l lowed, which i s jus t i f i ed by the

deform abi l i ty a t the v ic in i ty of the contact. The mag ni tude o f the contact forces i s

assumed to s tar t f rom zero , when the bodies f i rs t come in contact , and increases as

the bodies in terpenet ra te each o ther up to a ma ximu m value and then s tar t s

decreas ing and ev entual ly becom es equal to zero wh en the bodies detach f rom each

other. Th e in teract ions between bodies m ay involve new contacts , renew ed contacts ,

s l ippages and complete detachments f rom other bodies wi th which they were in

contact.

A contact de tec t ion scheme, which cons i s t s of a spat ia l reasoning and a

pai rwise co ntact reso lu tion phases , i s used to avoid exhaus t ive checks of a l l poss ib le

pai rs , w hich w ould requi re O ( N2 ) checks , where N i s t he number o f bod ies . W henever

two bodies are foun d to be in contact, the i r contact geom et ry i s de termined a nd used

for the calculat ion of the contact forces . In addi t ion, the norma l and tang ent ial contact

p lanes are determ ined in order to ena ble the com putat ion o f the corresponding norm al

and ta nge nt ial relat ive velo ci ty components .

A t each c ontact poin t, pa i rs o f equal and oppos i te contact forces in the norma l

and the tangent ia l d i rec tions are computed f rom the re la tive mot ion o f the bodies in

contact , whi le taking into account any residual contact forces from the previous s tep.

The incremen ts of the ma gni tude of the contact forces are computed us in g the re la tive

veloci t i es of the two bodies in the normal and tangent ia l d i rec t ions , the contact

coeff ic ien t s in these d i rec tions , K, and Ks , respect ively , and the t im e s tep . T he bodies

even tual ly are pushed apar t due to the ac t ion of the contact forces and , W hen there i s

no over lap between the two bodies , the corresponding contact da ta s t ructure i s

removed and no contact forces are appl ied between the two previous ly in-contact

bodies . The normal and tangent ia l contact forces between two bodies in contact are

s tored a t each t ime s tep wi th respect to the associa ted normal and tange nt ia l vectors

to the contact . T his i s necessary in order to be ab le to correct ly evaluate the contact

forces when there i s no change in the in terpenet ra t ion between the two bodies in

contact w hi le the bodies undergo together as a pai r a r ig id bod y ro ta tion . In essence ,






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142 D I S C R E T E E L E M E N T M E T H O D S

the contact forces are computed and s tored in a way that the i r magni tudes are

invar iable to r ig id bod y ro ta t ions o f a pai r o f bodies tha t are in contact .

For example , the increments of the norm al and tangent ia l forces for two

rectangular bodies (F igure 2) in contact are compu ted by the product of the re la t ive

veloc i ty o f the contact poin ts o f the two bodies in the re levant d i rec t ion , i. e ., norm al

or tangen t ia l , t imes the t ime s tep , t imes the corresponding contact coef f ic ient. En ergy

diss ipat ion a t the contact poin ts i s taken in to account us ing dashpots tha t are se t

para l le l to the co ntact spr ings to s imula te v iscous damp ing a t the contact poin ts . This

d iss ipat ion o f energy is in addi t ion to the on e a t the indiv idual bo dy level tha t m ay b e

cons idered us ing the mass propor t ional dam ping matr ix C in the FE formulat ion . The

increment of the contact forces cons is ts of two par ts , the e las t ic and the damping

force increments . Therefore , the n ormal a nd tangent ia l forces are expressed in terms

of elas t ic and damping force components , indicated by the e and d superscripts ,

respectively.

Norm al contact f o r c e : t + A t F f f = t + A t F e c + t + A t F n dc

(6)

Ta ng en tial (i.e. , shear) con tact force: t+AtFC =t+atFsC +t+ atFf* (7)

The mag ni tudes o f the e las t ic forces are accumu lat ing dur ing s imulat ion .

t+ A t F e c = t F e c + v r e l

A t . K n

8 )

t e l A t

K s

+ A t F e c = t F e c + v s

(9)

In contras t, the dam ping forces that lead to energy d iss ipat ion are eq ual to :

t +A t F d C = v r e l C n a n d t + k t F : C = v s e l c s

(io)

Cn and Cs a r e

the dam ping coeff ic ients in the norm al and tang ent ia l d i rec tions .

Norm al forces:

t e l

t + A t l ~ = tF en C + v ~ l . A t . g n + v n C n

Tangential forces:

t + d t F c t c . r e l t e l

= ~ . A t . K s + C s

t - V s V s

C o u l o m b F r i c t i o n : / +

A t F C s I _~ t + A t l; C n . / ~

Figure 2. Con tact forces applied a t a contact poin t of a rec tangular bod y






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Co nside ring tangent ial contact forces , i .e . , perpe ndicu lar to the norm al contac t

d i rec t ion , Coulomb f r ic t ion i s used to l imi t these shear forces below a cer ta in

mag ni tude tak ing in to account the magni tude o f the normal contact force and the

coeff ic ien t s of f r i c t ion of the two bodies in contact.

The computed normal and tangent ia l contact forces a t each contact poin t are

taken in to account wh en forming the equations of mo t ion in order to determ ine the

response of the body. In the case o f inf in i te ly r ig id bodies the contact forces f rom

each contact poin t are transformed to the cen t ro id o f each inf in i te ly r ig id bo dy and

added to al l other extern al force t ract ions and body forces . In contrast , in the case o f

deformable bodies , equivalent nodal forces to the computed contact forces are

calcu la ted and u sed together wi th any o ther nodal forces in the UL FE formulat ion . In

both cases , the equat ions of mot ion of each body are so lved us ing the CD M as br ief ly

descr ibed in the next paragraph.

N umerical Integrat ion of Equat ions of Mot ion

The mot ion of a deformable d i scre te body for a t t ime

t+At

i s de termined f rom the

dynam ic equi l ibr ium equat ions (1) a t t ime t , which are nu me rica l ly in tegra ted us ing

the CDM. Subs t i tu t ing f in i te d i f ference approximat ions for the veloci t i es and

accelera t ions in to the govern ing equat ions we obta in an express ion wi th the

disp lacements a t t ime

t+At

the only unknow ns for wh ich we can so lve:

AT 'M + 2.At XT ' M - - - '-~u

- At

From the nodal d i sp lacements , the bod y deformat ions and the in tem al s t resses

are computed , as wel l as the new p os i t ions of the s imula ted bodies , wh ich are used

for the next s imulat ion s tep. Similar , but much s impler, equat ion and solut ion are

used for inf in i te ly r ig id bodies w i th only three , or s ix , DO F per b od y in 2D or 3D,

respect ively , expressed w i th respect to the cent ro id o f the body under cons idera tion .

The combina t i on o f t he UL-F E fo rmula t i on wi th C DM enab l es s i gn i f i can t

s impl i f i ca t ions of the problem, because i t a l lows the d i rec t in tegra t ion of the

equat ions of mot ion wi thout the need for any i t era t ive procedure for convergence,

regard less o f the l arge d isp lacements and ro ta tions . Selec t ing d iagonal m ass an d

mass-propor t ional dam ping mat r ices , the d i sp lacement in each D OF i s ca lcu la ted as a

f rac t ion of two num bers . T he shor tcoming of an e xpl ic i t me thod i s tha t the method i s

cond i t ional ly s tab le and a suff ic ien t ly smal l t ime s tep i s requi red to ensure s tab i l ity .

How ever , regard less of th is requirement we n eed to use sm al l t ime s teps anyw ay, in

order to avoid er rors dur ing the contact de tec t ion par t and the ca lcu la t ion of the

contact forces . In addi t ion, decoupl ing the system of equat ions faci l i tates the use of

para l le l comput ing , w hich i s usefu l wh en s im ulat ing ver y large num bers o f bodies .






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144 D I S C R E T E E L E M E N T M E T H O D S

A p p l ic a ti on s a n d F u tu r e W o r k

A so t tware app l ica t ion tha t imp lemen ts the com bina t ion o f DE M and FEM , as

descr ibed here , i s presented in the pape r [3] a long wi th a ver i f ica t ion exam ple o f th is

approach. Th at paper d iscusses how Java and database technologies can be u t i l ized to

develop a por table , robus t and extendable DE M appl icat ion . The com binat ion o f

FEM and DEM can be used to s imula te sys tems , such as masonry s t ructures , tha t

have been typ ica l ly ana lyzed using con t inuum m ode l s, wh ich canno t cap tu re dyn amic

response phen om ena that involve rocking , impact and deform at ion of the indiv idu al

b locks . Us ing D EM , a m ason ry s tructure can be mode led as a sys tem o f m any d is t inc t

bodies pu t together in the same w ay that the s tructure i s phys ica l ly cons t ructed and

al lowed to in teract through contact s t resses wi th thei r ad jacent bodies , whi le the

s t ress and s t ra in d is t ribut ions are comp uted a t the ind iv idual bod y leve l us ing the

emb edded FE M formulat ion and analys is .

Tw o natura l ex tens ions of th is work are the co ns idera t ion of large s t ra ins and

non l inear m ater ia l by u s ing the proper FE form ulat ion , such as the to ta l Lagrangian

formulat ion . The la t ter requires use of the second P io la -Kirchh off s tresses and the

Green-Lagrange s trains , which ra ise the com putat ional requirements s ignif icant ly

higher.

References

[1] Bathe

Klaus-Jurgen, F in i te Elem ent Procedures , Prent ice-Hal l , Inc . ,

En glew ood Cl i f fs , New Jersey , 1996.

[2] Cundal l , P . and S track O. , A d is t inc t e lement m odel for granular assembl ies ,

Geotechnique, 29:47,6, 1979.

[3] Kom odrom os , Pet ros , Ut i l iza t ion o f Java and Database Te chn ology in the

Deve lopmen t o f a Combined D isc re te and F in i t e E lemen t Mul t ibody Dynamics

Simulator , 3rd Intern. Confer. on Discrete Element Methods, Santa Fe , New

M exico , 2002.

[4] Will iams, John, and Mustoe, Graham, editors . Proceedings of the 2nd

International Conference on Discrete Element Methods DEM), Dept. o f C iv i l and

En vironm enta l Engineer ing , Massachusetts Ins t itu te of Techn ology, 1993. IES L

Publ ica t ions .





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on the simulation of deformable bodies using combined discrete and finite element methods

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