Frictionless contact

Over the last years, an increasing interest has been devoted to energy conserving time integration schemes for contact mechanics. In particular, in the framework of frictionless contact, both Laursen and Chawla [LC97] and Armero and Pet¨ocz [AP98] have shown the interest of the persistency condition to obtain energy conservation in the discrete framework. Nevertheless, as underlined in [LL02], both contributions encounter a difficulty in enforcing standard Kuhn-Tucker conditions associated to frictionless contact, so that they concede an interpenetration of the structures in interaction, vanishing as the time step goes to zero. This drawback is resolved by Laursen and Love in [LL02], by introducing a discrete jump in velocities during impact, making possible the enforcement of contact conditions at each time step, at the computational price of resolving a problem on the jump in velocities. In the framework of a penalized enforcement of the contact condition, and by adapting the correction technique from [Gon00], we propose an energy conserving scheme while enforcing the standard Kuhn-Tucker contact conditions at entire time steps.

Let Ω⁽¹⁾ and Ω⁽²⁾, two open sets inR³ representing the interior of the reference confi- gurations of two solids potentially in contact on the parts Γ⁽ⁱ⁾c ⊂∂Ω⁽ⁱ⁾ (i∈ {1,2}) of their boundaries. For each i∈ {1,2}, ⁽ⁱ⁾ will denote the quantityrelative to Ω⁽ⁱ⁾, and with the notation introduced above, we assume that Γ⁽ⁱ⁾_D, Γ⁽ⁱ⁾_N, and Γ⁽ⁱ⁾_c constitute a partition of the boundary∂Ω⁽ⁱ⁾. In this presentation, Γ⁽²⁾c will be considered as the master surface.

Let us introduce for allx∈Γ⁽¹⁾c , the closest-point projection : y(t, x) = arg min

y∈Γ⁽²⁾_c kϕ⁽¹⁾(t, x)−ϕ⁽²⁾(t, y)k2.

Assuming that the manifold Γ⁽²⁾c is continuously differentiable, it follows that there exists a functiong: [0, T]×Γ⁽¹⁾c →R continuous with respect to the space variables, such that :

ϕ⁽¹⁾(t, x)−ϕ⁽²⁾(t, y(t, x)) =−g(t, x)ν(t, y(t, x)),

whereν(t, y) is the normal outward unit vector toϕ⁽²⁾(t,Γ⁽²⁾_c ) at timet∈[0, T] and point y∈Γ⁽²⁾c . As a consequence, we get :

g(t, x) =−

ϕ⁽¹⁾(t, x)−ϕ⁽²⁾(t, y(t, x))

·ν(t, y(t, x)), and the non-penetration condition expresses as :

g(t, x)≤0,

for all displacements fields ϕ⁽¹⁾ and ϕ⁽²⁾. Then, the weak form of the balance of linear momentum reads :

X2 i=1

Z

Ω⁽ⁱ⁾

ρ⁽ⁱ⁾ϕ¨⁽ⁱ⁾·v⁽ⁱ⁾+ Z

Ω⁽ⁱ⁾

Π⁽ⁱ⁾:∇v⁽ⁱ⁾ = X2

i=1

Z

Ω⁽ⁱ⁾

f⁽ⁱ⁾·v⁽ⁱ⁾

− Z

Γ⁽¹⁾c

λ(t, x) ∂g

∂ϕ⁽¹⁾ ·v⁽¹⁾+ ∂g

∂ϕ⁽²⁾ ·v⁽²⁾

| {z }

G(v⁽¹⁾,v⁽²⁾)

, (3.65)

for all admissible virtual displacements v⁽ⁱ⁾ ∈ U0(Ω⁽ⁱ⁾) , i ∈ {1,2}, such that the Kuhn- Tucker conditions are satisfied :







λ(t, x)≥0, g(t, x)≤0, λ(t, x)g(t, x) = 0,

(3.66)

for almost all (t, x)∈[0, T]×Ω (see [Lau02]). Moreover, it is well known that the frictionless contact reaction is then normal to Γ⁽¹⁾c , and as proved in [Lau02] for example, the following relation holds :

G(v⁽¹⁾, v⁽²⁾) =− Z

Γ⁽¹⁾c

λ(t, x)ν(t, y(t, x))·h

v⁽¹⁾(x)−v⁽²⁾(y(t, x))i .

For energy conservation purpose, the following persistency condition (see [SL92]) has to be added :

λ(t, x) ˙g(ϕ(t, x)) = 0. (3.67) The condition (3.67) means that normal contact reactions can only appear during per- sistent contact on the rigid surface.

Conservation properties in the continuous framework

The two-body system, assumed here to be compressible for simplicity, achieves usual conservation properties in the absence of external forces, as proved in [AP98] for example.

Indeed, the work of normal contact reactions at time t, obtained as G( ˙ϕ⁽¹⁾(t),ϕ˙⁽²⁾(t)) vanishes :

G( ˙ϕ⁽¹⁾(t),ϕ˙⁽²⁾(t)) = − Z

Γc

λ(t, x)ν(t, y(t, x))·

˙

ϕ⁽¹⁾(t, x)−ϕ˙⁽²⁾(t, y(t, x))

= Z

Γc

λ(t, x) ∂g

∂ϕ⁽¹⁾ ·ϕ˙⁽¹⁾(t, x) + ∂g

∂ϕ⁽²⁾ ·ϕ˙⁽²⁾(t, x)

= Z

Γc

λ(t, x) ˙g(t, x)

= 0.

As a consequence, when the persistency condition (3.67) is enforced, the total energy of the two-body-system :

E(t) = X2

i=1

E⁽ⁱ⁾(t), with :

E⁽ⁱ⁾(t) = 1 2

Z

Ω⁽ⁱ⁾

ρ⁽ⁱ⁾ϕ¨⁽ⁱ⁾(t) + Z

Ω⁽ⁱ⁾

Wˆ⁽ⁱ⁾(∇ϕ⁽ⁱ⁾(t)),

is conserved. The same statement can be established for angular and linear momentum.

Indeed, the resultant moment of the contact forces with respect to any axis a ∈ R³ va- nishes :

G(a×ϕ⁽¹⁾(t), a×ϕ⁽²⁾(t)) = −a· Z

Γc

λ(t, x)

ϕ⁽¹⁾(t, x)−ϕ⁽²⁾(t, y(t, x))

×ν(t, y(t, x))

= a· Z

Γc

λ(t, x)g(t, x)ν(t, y(t, x))×ν(t, y(t, x))

= 0,

which entails, in the absence of external efforts, the conservation of the two-body system angular momentum :

M(t) = X2

i=1

M⁽ⁱ⁾(t), with :

M⁽ⁱ⁾(t) = Z

Ω

ρ⁽ⁱ⁾ϕ⁽ⁱ⁾(t)×ϕ˙⁽ⁱ⁾(t).

Finally, for any translation a ∈ R³ of the two-body system, it is straightforward that G(a, a) = 0, which entails that the sum of resultant forces between the two bodies vanishes,

and in the absence of external efforts, the conservation of the two-body system linear momentum :

I(t) = X2

i=1

I⁽ⁱ⁾(t), with :

I⁽ⁱ⁾(t) = Z

Ω

ρ⁽ⁱ⁾ϕ˙⁽ⁱ⁾(t).

When the conditions (3.66) are enforced through a penalized formulation, the Lagrange multiplier λis defined as :

λ= 1

ηg⁺, [0, T]×Ω,

with g⁺ = g ifg ≥0 and g⁺ = 0 otherwise. Then, the persistency condition (3.67) is no more necessary to achieve energy conservation. The work of contact forces is given by :

Z

Γc

λ(t, x) ˙g(t, x) = d dt

1 2η

Z

Γc

g⁺2 ,

resulting in the absence of external forces, in the conservation of a penalized total energy of the two-body system :

E(t) = 1 2η

Z

Γc

g⁺(t)2

+ X2 i=1

E⁽ⁱ⁾(t).

A conserving time integration approach for frictionless contact

To reproduce in the discrete framework the previous conservation properties, we adapt the energy correction approach of [Gon00] and propose the following midtime approxima- tion of the normal contact reaction :

G_n+1/2(v⁽¹⁾, v⁽²⁾) = Z

Γ⁽¹⁾c

Λ_n+1/2∆g_n+1/2·h

v⁽¹⁾(x)−v⁽²⁾(y_n+1/2(x))i

, (3.68)

where y_n+1/2(x) is the projection of ϕ⁽¹⁾_n+1/2(x) over ϕ⁽²⁾_n+1/2(Γ_c) with the notation : ϕ⁽ⁱ⁾_n+1/2 = 1

2

ϕ⁽ⁱ⁾_n +ϕ⁽ⁱ⁾_n+1 . Moreover, we propose to adopt :

∆g_n+1/2=−ν_n+1/2+

g_n+1−g_n+ν_n+1/2·δϕ_n δϕn

δϕ_n·δϕ_n,

where ν_n+1/2(x) is the normal outward unit vector to ϕ⁽²⁾_n+1/2(Γ⁽²⁾c ) at point y_n+1/2(x) ∈ Γ⁽²⁾c , and :





gn(x) =−

ϕ⁽¹⁾n (x)−ϕ⁽²⁾n (y_n(x))

·νn(x), δϕ_n(x) =h

ϕ⁽¹⁾_n+1(x)−ϕ⁽²⁾_n+1(y_n+1/2(x))i

−h

ϕ⁽¹⁾n (x)−ϕ⁽²⁾n (y_n+1/2(x))i .

With rather obvious notation, we have denoted by y_n(x) the projection of ϕ⁽¹⁾_n (x) over ϕ⁽²⁾_n (Γ_c) and byν_n(x) the outward normal unit vector to ϕ⁽²⁾_n (Γ_c) at pointy_n(x). Finally, we propose :

Λ_n+1/2 =λ_n+1/2+

λ_n+1g_n+1−λ_ng_n

2 −λ_n+1/2δg_n δg_n

(δgn)²,

whereλ_n+1/2=g_n+1/2(x)⁺/η in whichg_n+1/2(x) is the gap functiong evaluated at point x∈Γ⁽¹⁾_c for the displacements fieldsϕ⁽¹⁾_n+1/2 and ϕ⁽²⁾_n+1/2. Moreover, the following notation

has been used : (

λ_n(x) =g⁺_n(x)/η,

δg_n(x) =g_n+1(x)−g_n(x).

With this construction, the following properties hold :

Proposition 3.7. The discrete work of frictionless contact forces we have defined in (3.68), achieves :

1. exact discrete work, that is :

G_n+1/2(ϕ⁽¹⁾_n+1−ϕ⁽¹⁾_n , ϕ⁽²⁾_n+1−ϕ⁽²⁾_n ) = 1 2

Z

Γ⁽¹⁾c

λ_n+1g_n+1−λ_ng_n

= 1

2η Z

Γc

g⁺_n+12

− g_n⁺2

2. zero resultant force, that is :

G_n+1/2(a, a) = 0, ∀a∈R³.

Proof :The zero resultant force is readily obtained from (3.68). Concerning discrete work, we have by construction :

G_n+1/2(ϕ⁽¹⁾_n+1−ϕ⁽¹⁾_n , ϕ⁽²⁾_n+1−ϕ⁽²⁾_n ) = Z

Γ⁽¹⁾c

Λ_n+1/2∆g_n+1/2·δϕ_n

= Z

Γ⁽¹⁾c

Λ_n+1/2 δg_n

= 1

2 Z

Γ⁽¹⁾c

λ_n+1g_n+1−λ_ng_n,

hence the proof.

Remark 3.13. A more simple energy conserving formulation is given by : G_n+1/2(v⁽¹⁾, v⁽²⁾) =

Z

Γ⁽¹⁾c

N_n+1/2·h

v⁽¹⁾(x)−v⁽²⁾(y_n+1/2(x))i

, (3.69)

in which :

N_n+1/2 =λ_n+1/2ν_n+1/2−

λ_n+1g_n+1−λ_ng_n

2 −λ_n+1/2ν_n+1/2·δϕ_n

δϕ_n δϕn·δϕn

, with the above notation. Nevertheless, the section dealing with unilateral frictionless contact against a plane wall will illustrate a major difference between the two approaches (3.68) and (3.69).

Remark 3.14. In order to preserve symmetric tangent operators in a Newton’s method, we propose not to differentiate correction terms, that is in the expressions of ∆g_n+1/2 and Λ_n+1/2, the terms marked with a (†) :

∆g_n+1/2=−ν_n+1/2+

g_n+1−g_n+ν_n+1/2·δϕ_n δϕ_n δϕ_n·δϕ_n

| {z }

(†)

,

Λ_n+1/2 =λ_n+1/2+

λ_n+1 g_n+1−λ_ng_n

2 −λ_n+1/2δg_n δg_n

(δg_n)²

| {z }

(†)

.

This proposal enables to solve the problem in the midtime displacements field ϕ_n+1/2, as in the proposed implementation of Gonzalez time integration scheme.

Unilateral frictionless contact against a plane wall

We analyze here the case of unilateral frictionless contact against a plane wall. Then, we assume that the infinite half space Ω⁽²⁾=R²×R₊is fixed and perfectly rigid, and that the deformable body of reference configuration Ω⁽¹⁾is submitted to a unilateral frictionless contact against the boundary Γ⁽²⁾_c =R²× {0} of Ω⁽²⁾. This assumption impose that the displacement fieldsϕ⁽²⁾ =idand its variationsv⁽²⁾vanishes. Moreover, the outward normal unit vector ν is constant over Γ⁽²⁾c . Then, by using the above definitions, we get :







y_n(x) =ϕn(x)−(ϕn(x)·ν)ν, g_n(x) =−ϕ⁽¹⁾_n (x)·ν,

δϕ_n(x) =ϕ⁽¹⁾_n+1(x)−ϕ⁽¹⁾_n (x) and deduce that ∆g_n+1/2(x) =−ν, so that :

G_n+1/2(v⁽¹⁾,0) =− Z

Γ⁽¹⁾c

Λ_n+1/2 ν·v⁽¹⁾,

for allv⁽¹⁾ ∈ U0(Ω⁽¹⁾). The fact, that the contact force remains normal to the wall explains the superiority of the proposed discrete formulation, when compared to the apparently sim- pler formulation (3.69). Indeed, the latter induces a non-physical variation of the contact force direction to achieve energy conservation, while (3.68) only plays on the intensity of the contact force. Moreover, we have noticed numerically the superiority of (3.68) in terms of number of iterations in Newton’s algorithm in the framework of the previous imple- mentation, since the formulation (3.69) implemented with the same strategy experiment difficulty in converging.

No documento Méthodes numériques pour la dynamique des structures non-linéaires incompressibles à deux échelles. (páginas 82-88)