A modular approach to 3-D impact computation withfrictional contact

P. Ladeveze *, H. Lemoussu, P.A. Boucard

Laboratoire de Mechanique et Technologie Cachan, ENS de Cachan, CNRS, Univ. Paris VI 61, Avenue du President Wilson, 94235

Cachan Cedex, France

Received 30 November 1998; accepted 17 July 1999

Abstract

An approach suitable for 3-D impact computation with a large number of unilateral friction contact surfaces is

presented. Applications concern connection re®ned models for which the structure components are assumed to be

elastic. It is a modular approach based on a mixed domain decomposition method and the LATIN method. The

resolution process is iterative and operates over the entire time-space domain. Several examples are ®nally analyzed to

show the method's capability of describing shocks. Ó 2000 Civil-Comp Ltd. and Elsevier Science Ltd. All rights re-

served.

Keywords: Impact; Frictional contact; Computation

1. Introduction

The analysis of complex assemblies, subjected to

shock leads to serious diculties especially when the

global behaviour of the structure is very non-linear be-

cause of a large number of unilateral friction contact

surfaces, on which the contact ¯uctuates. Even within a

static framework, industrial computational codes imply

prohibitive computation times in such cases.

The aim of the present study is to propose a new

approach that provides a satisfactory analysis of the

shock response of assemblies, or shock attenuator. Such

an approach would focus, for example, on the major

phenomena occurring during the two-stage separation

of a launcher, when the shock, initiated by the explosion

of a cord cutter, in¯uences the assemblies that serve to

install the on-board devices. Hence, this study has been

part of the pyrotechnic shock investigation group con-

ducted by the National Center for Space Studies

(CNES), based in Evry (France).

The approach involved consists of the continuation

of previous works conducted in statics for which the

computational time has been divided by 50 for several

industrial 3-D connection problems presenting many

contact surfaces, regarding industrial ®nite element

codes [1,3,5].

The ®rst step, already used for static problem [1,5], is

decomposition of a structure into elastic substructures

linked together by interfaces exhibiting a distinct be-

haviour adapted with the phenomena encountered.

Moreover, the dialogue existing between the two entities

is mixed and performed with a velocity±force duality,

which is very well suited to the dynamic framework. The

interfaces contain all non-linear information, such as

unilateral contact with friction [2]. This represents a

meso-modellisation of the structure which, adapted to

parallel computers, provides a high level of modularity

and ¯exibility in the problem's description.

The second step is to use a non-incremental compu-

tational strategy and called the Latin method, where the

non-linearities are concentrated at the interfaces. The

relevant principles are described in Ref. [5]. This method

is an iterative one and operates over the entire space

domain and the entire studied time interval. The added

Computers and Structures 78 (2000) 45±51

www.elsevier.com/locate/compstruc

* Corresponding author.

0045-7949/00/$ - see front matter Ó 2000 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.

PII: S0 04 5 -7 94 9 (00 )0 0 09 4 -8

step herein concern the resolution of the global prob-

lems, whereby the constancy of the matrices present

throughout the iterations is used. Consequently, these

matrices have been pre-calculated during the method's

initialization.

The basic principles of the Latin method will be re-

called ®rst; a full description of the method will be

provided within a dynamic 3-D framework introducing

a speci®c strategy for solving the global problem, on the

time-space domain, that we get at each interation.This

approach has been introduced into the ®nite element

code Castem 2000. A few simple examples will subse-

quently be used to highlight the feasibility of this

method, and its capability of describing shock and

frictional unilateral contact problems with good con-

vergence results. A full comparison with industrial codes

will be developed afterwards.

2. Principles of the approach

Two major components constitute this approach

which is the association of a mixed domain decompo-

sition method and a non-incremental resolution tech-

nique.

2.1. Structural decomposition

The studied assembly X is decomposed into two

mechanical quantities: the substructures XE and the in-

terfaces cEE0 . A substructure can only have dialogue with

adjacent interfaces. This exchange is described in Fig. 1.

This ®gure shows the two quantities _WE

and F E which

are the velocity and surface traction ®elds, respectively.

This velocity±force duality, associated with the power

quantity, is very well suited to the dynamic framework.

2.2. Mechanical problem

A linear behaviour of the material within the small

perturbation hypothesis has been considered herein. The

external loadings are time dependent.

Each entity (substructure and interface) has its own

set of equations. The problem can be written as follows:

For a substructure XE

Find a velocity and a stress ®eld (V EM ; t and rEM ; t),de®ned on XE 0; T , that satisfy

· kinematic admissibility

V E joXE _WE; V E 2 V0;T ; 1

· initial conditions

U Et0 U E

0 ; V Et0 V E

0 ; 2· equilibrium equations

divrEM ; t fd q _V EM ; t; 3

rEoXE M ; tn F E; 4

· constitutive law

rEM ; t KU E; 5

where oXE is the boundary of the substructure XE, fd is

the prescribed body force and U E is the displacement

®eld on XE.

For an interface cEE0

Find _WE; _W

E0; F E; F E0

, de®ned on cEE0 0; T , that

satis®es

· constitutive law

R _WE; _W

E0; F E; F E0

0: 6

The interface equations depend on the interface's be-

haviour. In the case of perfectly connected interfaces,

these relations become

8t 2 0; T ; _WE _W

E0; F E F E0 0: 7

2.3. Solution process

In order to solve the problem associated with the

above decomposition, a non-incremental approach,

called the Latin method was proposed by P. Ladeveze in

1985 (more details can be found in Ref. [5]). This

method has yielded some excellent results for quasi-

static loadings [6]. Previous works have shown compu-

tation times divided by 50 for 3-D connections problems

with many contact surfaces [1]. The present study con-

sists of its development in dynamics.

The Latin method is based on three principles. The

®rst principle is to separate diculties in order to avoid

the simultaneity of global and non-linear problems. By

taking into account the mechanical properties of the

equations, two groups can be distinguished: the local in-

space variable and eventually non-linear equations on

the one hand (the associated space will be called C), and

the linear and global in-space variable equations on the

other hand (the associated space will be called Ad).

The chosen separation of the equation de®nes Ad and

C as

WE

γEE'

ΩE

WE'

FE FE'

Fig. 1. Exchange between substructures and interfaces.

46 P. Ladeveze et al. / Computers and Structures 78 (2000) 45±51

The second principle of the method is a two-stage

iteration scheme which alternatively solves each set of

equations. The local stage solves the problem associated

with C and leads to a non-unique solution; it is therefore

necessary to add other equations, called search direction

equations, to be written E. The linear global stage

solves the problem associated with Ad; it becomes nec-

essary once again to add search direction equations to be

written Eÿ.The third principle of the method lies in the resolu-

tion of the global problem. One characteristic of the

method is that the global operators involved in this stage

are constant with respect to the iterations. Thus, they are

all treated during the method's initialization. More de-

tails about the Latin method can be found in Ref. [5].

Let s be the entire set of unknowns, associated with

the functional space S. s is de®ned on XE [ cEE0 0; T as

s ; r; _W ; F [E sEn E; rE; _W

E; F E

o: 8

sn is the solution to a linear global stage and sn1=2 is

the solution to the local stage at iteration n. This itera-

tive scheme is described in Fig. 2.

2.4. Global linear stage at iteration n

This stage begins with a known element sn1=2

_WE

n1=2;_W

E0

n1=2; FE

n1=2; FE0

n1=2

that belongs to C and

is de®ned on cEE0 0; T .One must then ®nd an element sn1 (V E

n1; rEn1, de-

®ned on XE 0; T , that satis®es

· motion equation

divrEn1 fd q _V E

n1; 9

rEn1oXE n F E; 10

· constitutive law

rEn1 KU E

n1; 11· search direction (SD)

F En1 ÿ F

E

n1=2 kÿ _WE

n1

ÿ _W

E

n1=2

; 12

F E0n1 ÿ F

E0

n1=2 kÿ _WE0

n1

ÿ _W

E0

n1=2

; 13

where kÿ is a positive constant. The used value is

GTc=Lc, where G is the shear modulus, Lc a characteristic

length and Tc a characteristic time.

The linear global stage leads to a global problem on

each substructure, which serves to considerably reduce

the size of the problem. Moreover, these small problems

are independent and as such can be parallelized. All of

these problems are non-classical in nature. Over the

entire boundary oXE, both density of surface traction

and velocity are applied. The problem associated with

the substructure XE can then be written asZXE

KU En1V dXE

ZoXE

kÿV En1V dSE

Z

XE

q _VE

n1V dXE

Z

oXE

FE

n1=2

kÿ _W

E

n1=2

V dSE 8V 2V

0;T 0 :

14This is a classical formulation of a problem in which

the density of the surface traction is applied: ~F F

E

n1=2 kÿ _WE

n1=2. It has a unique solution which can be

obtained using various techniques [4,8±11].

The space domain is discretized using ®nite elements.

The resolution of the dierential system is performed

with an implicit Newmark scheme [7] over the complete

time interval 0; T .

2.5. Local stage at iteration n

This stage begins with a known element sn (V En ; r

En

that belongs to Ad and is de®ned on XE 0; T .

Γ

Ads ex

s

s

s0s1 s2

( )-

directionsearch of global stage

of local stage

E

( + )E

1 2

3 2

directionsearch

Fig. 2. Presentation of the iterative process.

Ad

kinematic admissibility;initial conditions;equilibrium equations;

8<: C constitutive law on cEE0 :n

P. Ladeveze et al. / Computers and Structures 78 (2000) 45±51 47

One then must ®nd an element sn1=2

_WE

n1=2;_W

E0

n1=2; FE

n1=2; FE0

n1=2

, de®ned on cEE0

0; T , that satis®es

· interface behaviour

R _WE; _W

E0; F E; F E0

0; 15

· search direction (SD)

FE

n1=2 ÿ F En k _W

E

n1=2

ÿ _W

E

n

; 16

FE0

n1=2 ÿ F E0n k _W

E0

n1=2

ÿ _W

E0

n

; 17

where k is a positive constant, equal to kÿ.

2.5.1. Interface with unilateral friction contact

For a unilateral friction contact between XE and XE0 ,

the largest contact zone is supposed to be known.

The velocity ®eld _WE

n1=2 and the surface traction

density FE

n1=2 are written as follows:

_WE

n1=2 NEE0 : _WE

n1=2

N EE0 p _W

E

n1=2; 18

FE

n1=2 N EE0 :FE

n1=2

N EE0 pF

E

n1=2; 19

where N EE0 is the exterior unit vector, oriented from E to

E0, and p denotes the tangential coecient operator

(Fig. 3). Contact conditions are

if WE

n1=2

ÿ W

E0

n1=2

NEE0 > 0 ) separation

) FE

n1=2 FE0

n1=2 0;

Two solutions are possible in the contact case:

if pFE

n1=2

< l FE

n1=2 NEE0

) sticking ) p _W

E

n1=2 p _WE0

n1=2;

if pFE

n1=2

l FE

n1=2 NEE0

) sliding ) 9 k P 0;

such as p _WE

n1=2

ÿ _W

E0

n1=2

ÿkF

E

n1=2:

The resolution of the local stage is explicit.

In the normal direction, a scalar indicator Cn gives

the solution:

Cn 1

2_W

E0

n1=2

ÿ _W

E

n1=2

ÿ 1

2kF

E0

n1=2

ÿ F

E

n1=2

N EE0 ; 20

if Cn > 0 ) separation, and if Cn 6 0 ) contact.

In the tangential direction, a vectorial indicator Gn

gives the solution:

Gn 12kp _W

E0

n1=2

ÿ _W

E

n1=2

ÿ 1

2p F

E0

n1=2

ÿ F

E

n1=2

:

21

The sliding conditions can then be written, using the

sliding limit g l FE

n1=2 NEE0

, as follows:

sticking if kGnk 6 g

)p F

E

n1=2

ÿp F

E0

n1=2

Gn

p _WE

n1=2

and p _W

E0

n1=2

SD;

22

M

ΩE

ΩE'

γEE'

NEE'

Fig. 3. Contact.

if WE

n1=2

ÿ W

E0

n1=2

NEE0 0 ) contact )

_WE

n1=2 NEE0 _WE0

n1=2 NEE0 ;

FE

n1=2 FE0

n1=2 0;

FE

n1=2 N EE0 6 0:

48 P. Ladeveze et al. / Computers and Structures 78 (2000) 45±51

sliding if kGnk > g

)p F

E

n1=2

ÿp F

E0

n1=2

g Gn

kGnk

p _WE

n1=2

and p _W

E0

n1=2

SD:

23

In this manner, contact with friction problems are no

more dicult to solve than are perfect connection

problems. Moreover, the convergence characteristics of

the method are not aected by the presence of contact

interfaces.

2.6. Control of the iterative process

An error estimator can be based upon the degree of

non-satisfaction of the behaviour of the interfaces for an

approximate solution sn, an element of Ad. In order to

stop the algorithm, we use an energy indicator, which is

computed after each local stage as follows:

g Rksn ÿ sn1=2k2E

Rksnk2E ksn1=2k2

E

;

where

ksk2E

ZoXE

F Tkÿ1F dSE Z

oXE

_WTk _W dSE: 24

3. Examples

3.1. Beam in traction

This example demonstrates the capacity of the

method in treating shock-related problems. An example

problem is presented in Fig. 4. Two beams, composed of

the same material, are separated by a gap j and then

shocked by a compression load.

The meso-model employed is composed of three

substructures each of them containing ®fty elements

(eight-node cubes). The time step for the studied interval

consists of 250 discretized time values to represent the

loading history.

The solution is given in Fig. 5 with a plot of the

uniaxial stress along the beams. Once the wave has

moved suciently, with respect to the ®rst beam, the gap

vanishes. This phenomenon creates one wave in each

beam; both of these waves have the same speed and the

same level.

In Fig. 6, the state of the uniaxial stress is repeated

and associated with the time history of this stress for a

single point. This procedure reveals, by means of the

iterations, how the solution is obtained. At the ®rst it-

eration, only one sub-structure can perceive the loading.

Therefore, the entire structure requires several iterations

in order to be in¯uenced by the loading. In any event, at

each iteration, the state of all of the unknowns is ob-

tained throughout the entire structure and over the en-

tire studied time interval.

Fig. 4. Studied problem.

Fig. 5. Problem solution.

Fig. 6. Method behaviour.

P. Ladeveze et al. / Computers and Structures 78 (2000) 45±51 49

At this stage, the equilibrium equations are solved

globally. Thus, the contact conditions can be trans-

gressed and get veri®ed only at convergence. It is im-

portant to note that corrections are done not only in

space but in time as well. Both the level and the moment

of the shock are therefore modi®ed throughout the it-

erations. At convergence (six iterations), the theoretical

solution is correctly obtained.

3.2. Two-specimen crushing

This example demonstrates the capacity of the

method in treating friction problems. Two specimens,

whose mechanical characteristics vary greatly, are in

contact on a wall. The only non-zero sliding coecient is

between the two sub-structures. By virtue of symmetry,

only a quarter of the structure needs to be studied. This

problem and its corresponding meso-model are pre-

sented in Fig. 7.

Since this problem has no analytical solution, the

reference then becomes our solution based on a large

number of iterations (30 iterations). This solution has

been presented in Fig. 8. The central displacement of the

interface is plotted for both specimen. The solid lines

represent the hard specimen and the dashed lines rep-

resent the soft specimen. The evolution of these curves

throughout the method's iterations is also plotted. Once

again, the method's classical behaviour appears. The

plotted quantities are relevant to the substructures; thus,

they satisfy the contact condition only at convergence.

The solution in terms of tangential displacement is

plotted in Fig. 9. The solid lines represent the hard

specimen and the dashed lines represent the soft speci-

men. The studied point is located at the center of the two

specimen's interface. Four dierent friction coecients lhas been studied.

Fig. 7. Studied problem.

Fig. 8. Evolution in contact during the iterations.

Fig. 9. Evolution in sliding.

Fig. 10. Convergence indicator.

50 P. Ladeveze et al. / Computers and Structures 78 (2000) 45±51

All these cases yield dierent behaviour for the

structure even when the moment of the separation is the

same. No sticking occurs when the sliding coecient is

zero, and no sliding occurs when l is suciently large.

For the other two cases, three phases are encountered:

sticking, sliding and separation.

In Fig. 10, the method's convergence behaviour is

shown. It can be easily observed both that the value of

the sliding coecient does not aect this convergence

behaviour, and that sliding problems can be treated with

the same facility as frictionless problems.

4. Conclusion

A new approach for solving three-dimensional dy-

namic assemblage problems has been presented here. It

is based on two components: a mixed decomposition

of the structure which provides signi®cant modularity

to the problem description, and an iterative solution

scheme that is well-adapted to the problem.

The initial numerical results in the case of elastic

problems have shown the capacity and behaviour of this

approach, in addition to demonstrating its capability in

solving friction and frictionless problems with the same

facility. Moreover, the special treatment of contact con-

ditions and the resolution technique used here involving

constant operators through out the iterations make this

approach suitable for 3-D computations. Other com-

parisons with industrial codes will be conducted subse-

quently using problems with large numbers of d.o.f.

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