Computational contact mechanics

C. Hesch and P. Betsch

Introduction

Large deformation contact problems [1]

Geometrical and material nonlinearities

Inequality contact constraints

Active set strategy to resolve the Karush-Kuhn-Tucker conditions

Consitent formulation [2]

Mortar concept for spatial discretisation

Complex segmentation procedure

Energy-momentum scheme for temporal discretisation

Problem setting

Decompsition of boundaries

γ(i)σ ∪ γ(i)

u ∪ γ(i)c = ∂Ω(i)

Non-penetration condition

g = n · (ϕ(1) −ϕ(2)) ≥ 0

Karush-Kuhn-Tucker

inequality conditions

t ≥ 0; g ≥ 0; tg = 0

γ(1)σ

γ(1)c ϕ(1)(ϕ(2))

γ(2)σ

γ(1)u

Ω(1)

Ω(2)

γ(2)c

n

γ(2)u

ϕ(2)

Rnsd

Discretization in space

ϕh,(i)(X , t) =∑

I

NI(X)q(i)I (t)

Augmented Hamiltonian

Hλ(p, q) =1

2p ·M−1p + V int(q) + V ext(q) + λ ·Φ(q)

Equations of motion

qn+1 − qn =∆t

2M−1(pn+1 + pn)

(pn+1 − pn) = −∆t∇qV (qn+12)−∆tλ∇qΦ(qn+1

2)

0 = Φ(qn+1)

Mortar method

Weak formulation of the contact virtual work

Gc =

∫

γ(1)c

th,(1) ·[δϕh,(1) − δϕh,(2)

]dγ

Dual field

th,(1)(X, t) =∑

I∈ω(1)

NI(X)λ(i)I (t)

Segment contribution of the contact constraints

Φκe1,seg

(qseg) = n ·[nκβq

(1)β − nκζq

(2)ζ

]

Assembly of the segment contributions

Φmortar(q) = Ae1∈ǫ(1)

⋃

seg

Φe1,seg(qseg) = Ae1∈ǫ(1)

⋃

seg

Φκ=1e1,seg

(qseg)...

Φκ=4e1,seg

(qseg)

Concept of a discrete gradient

Reparametrization of the augmented Hamiltonian using at most

quadratic invariants

Hλ(p, q) = Hλ(π(p, q))

Application of the chain rule

∇qHλ = ∇qπ(pn+12, qn+1

2)T∇πHλ(π(zn),π(zn+1))

G-equivariant discrete gradient

∇πHλ(πn,πn+1) = ∇πHλ(πn+12)+

Hλ(πn+1)− Hλ(πn)−∇πHλ(πn+12) · (πn+1 − πn)

‖(πn+1 − πn)‖2(πn+1 − πn)

Numerical example

Impact problem of two hollow tori. The inner and the outer radius of

the tori is 52 and 100, respectively. The wall thickness of each hollow

torus is 4.5. Both tori are subdivided into 3120 elements, using a

Neo-Hookean hyperelastic material with E = 2250 and ν = 0.3. The

initial densities are ρ = 0.1 and the homogeneous, initial velocity of

the left torus is given by v = [30, 0, 23].

Several configurations and the stress distributions are displayed. The

segmentation after 2s as well as the three components of the angular

momentum are shown below.

0 1 2 3 4 5−3

−2

−1

0

0.5x 10

6

t

J

J1

J2

J3

References

P. Betsch and C. Hesch.

Energy-momentum conserving schemes for frictionless dynamic contact problems.

Part I: NTS method.

IUTAM Bookseries, 3:77–96. Springer-Verlag, 2007.

C. Hesch and P. Betsch.

A mortar method for energy-momentum conserving schemes in frictionless

dynamic contact problems.

Int. J. Numer. Meth. Engng, 77:1468–1500, 2009.

University of SiegenChair of Computational Mechanics Prof. Dr.-Ing. habil. C. Hesch