3.7 Numerical experiment
3.7.2 Ball impact
time(s)
vertical displacement (m)
0 0.5 1 1.5 2
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1
midpoint trapezoidal
time (s)
vertical displacement (m)
0 5 10
-0.4 -0.3 -0.2 -0.1 0
HHT(alpha=0.05) Gonzalez
Fig.3.4 – Vertical displacement of the tip of the cantilever beam. Simulation stops when the time step calculation exceeds 20 Newton’s iterations.
Remark 3.15. The present numerical analysis holds for quasi-incompressible nonlinear elastodynamics, but the same phenomena can be observed in large displacements for com- pressible systems.
time (s)
H
0 0.5 1 1.5 2
-100 0
-150 -50
Midpoint
time (s)
H
1 2 3 4 5 6 7
-8 -7 -6 -5 -4 -3 -2 -1 0
HHT(alpha=0.05)
time (s)
H
0 0.5 1 1.5 2
-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
Trapezoidal
time (s)
H
0 5 10
-4. 10-7 -3. 10-7 -2. 10-7 -1. 10-7 0
Gonzalez
Fig. 3.5 – Evolution of the discrete total potential H (in Joules) as a function of time.
As an indication, the maximal value of the deformation potential R
ΩW(C) is about 0.5 Joules.
3.12, the evolution of discrete energy in the ball during the dynamics is very sensitive to the time integration strategy.
In particular, the discrete energy explodes when using a midpoint scheme (or a trapezoidal scheme), and the deformation of the ball just before energy explosion is shown on figure 3.13. In particular, non-physical irregular displacements can be noticed around the hole.
The conservative Gonzalez scheme enriched with our energy conserving impact formulation keeps its promise and the relative loss of energy through the impact is 1.8 E-4, only depending on the required accuracy in Newton’s algorithm. The interest of our energy dissipative formulation is also confirmed, showing here the control of the mechanical energy in the ball. To complete this discussion, let us mention that when considering practical industrial use of time integration schemes for non-smooth dynamics, first order implicit schemes are sometimes prefered for their robustness. The best proof is the frequent use of implicit Euler strategies in coupled systems [TM01, GLB03]. In order to compare energy evolution when using first order strategy, we introduce the following time integration approach, obtained by a trapezoidal integration of the inertial term, and an implicit Euler
time (s)
speed (m/s)
0 0.5 1 1.5 2
-10 0
-5 5
Midpoint Trapezoidal
Fig.3.6 – Instability of the vertical velocity at the tip of the cantilever beam, for midpoint and trapezoidal schemes.
time (s)
H
1 1.5 2
0 0.01 0.02 0.03
0.005 0.015 0.025 0.035
HHT (alpha = 0.2) Energy conserving
time (s)
vertical displacement (m)
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
-0.4 -0.3 -0.2 -0.1 0
-0.45 -0.35 -0.25 -0.15 -0.05
HHT(alpha=0.2)
Fig.3.7 – Zoom on total discrete potential and displacement for HHT scheme (α= 0.2).
strategy for the stress part :
Z
Ω
ρϕ˙n+1−ϕ˙n
∆tn ·v+ Z
Ω
∂Wˆ
∂F (∇ϕn+1) :∇v= Z
Ω
fn+fn+1
2 ·v, ∀v∈ U0, ϕn+1−ϕn
∆tn = ϕ˙n−ϕ˙n+1
2 ,
(3.73)
written here in the compressible framework. The kinematical constraint will be naturally satisfied by the displacements fieldϕn+1at timetn+1. It is a Euler-like degraded first order version of the trapezoidal second order time integration scheme. It can be readily checked with the analysis of section 4, that the Euler-Newmark scheme (3.73) is energy dissipating
time (s)
vertical displacement (m)
0 5 10
-0.4 -0.3 -0.2 -0.1
0 HHT(alpha=0.2)
Fig.3.8 – Overdissipation for vertical displacement at the tip of the beam for HHT scheme (α= 0.2).
radius 0.1 m
density 1200 kg/m3
Young’s modulus 0.2 M Pa
Poisson’s ratio 0.33
initial distance of the center
of the ball to the wall 0.12 m
initial velocity 0.4 m/s
η 1.E-4
time step 0.002 s
T 1.0 s
# nodes in the mesh 11.160
Fig.3.9 – Data for ball impact, made of a Saint-Venant Kirchhoff material.
whenever the stored energy ˆW(F) is locally convex. The ball impact simulation performed with this scheme proves to achieve global energy dissipation, with a 9 % relative loss of energy through the impact. To illustrate the better accuracy of our second order energy conserving/dissipating schemes, the figure 3.14 illustrates the evolution of elastic energy after impact.
A noticeable statement is that the computation of contact pressures is relatively inde- pendent of the considered scheme for the simulation, as illustrated on figure 3.15.
Finally, we use the present example to illustrate the well known sensitivity of contact pressures in the penalization coefficientη. Indeed, it is shown on figure 3.16 that whenηis divided by 10, oscillations on the contact force appear. This phenomenon can be explained by the absence of strong limit for the linearized dynamics asηgoes to zero (see for example [Sca04]) in the absence of viscosity. Such oscillations are typical of weak limits in two-scale
Fig. 3.10 – Mesh of the ball.
Fig.3.11 – Snapshots of the impact simulation.
problems, as in [BLP78, All97]. The phenomenon is even much more visible in the case of a cube impact, described on figure (3.17).
time (s)
mechanical energy (J)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 1. 109
5. 108
midpoint
time (s)
mechanical energy (J)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
31 32 33 34 35
Euler-Newmark Energy conserving Energy dissipating (alpha = 0.5)
Fig. 3.12 – Evolution of the ball mechanical energy through impact for midpoint, Euler- Newmark, energy conserving, and dissipating (α = 0.5) schemes.
From a mathematical point of view, the presence of viscosity enables to obtain com- pacity, enabling the possibility to build a converging sequence of solutions asη→0. From a physical point of view, viscosity would enable the dissipation of high frequency vibra- tions. In [AP98], the authors propose the enforcement of the persistency condition (3.67) at entire time steps in their formulation, then adding an energy term associated with this constraint. An alternative could consist in introducing a real internal viscoelastic behavior of the material in the structures. In this framework, the “conserving” scheme proposed in the viscoelastic framework could be exploited.
Fig.3.13 – Snapshots at times t = 0.2 and 0.21 s, illustrating the deformation of the ball before energy explosion when using a midpoint time integration scheme.
time (s)
elastic energy (J)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1
0.5
Energy conserving Euler-Newmark Energy dissipating (alpha = 0.5)
Fig.3.14 – Evolution of elastic energy after impact for first order accuracy Euler-Newmark scheme, and second order accuracy energy conserving and dissipating (α= 0.5) schemes.