Conservation properties – falling coin

The next example demonstrates the conservation properties of the Lagrange multiplier contact al-gorithms developed in this thesis, which were previously discussed in Section 4.8.3. In addition, the convergence of the contact algorithms within the semi-smooth Newton scheme is investi-gated. The example consists of an elastic coin (flat cylinder) and an elastic foundation, see Fig-ures 4.30 and 4.31. The employed material model for both bodies is of Saint-Venant-Kirchhoff

Figure 4.30: Initial setting for the falling coin example. The figure is taken from Farah et al. [71].

type. The material properties of the coin are defined with Young’s modulus being E = 1·10⁵, Poisson’s ratio beingν = 0.0and the density beingρ₀ = 0.3. The properties of the foundation are the same except for Young’s modulus, which is defined asE = 4·10³. The dimensions of the bodies can be seen in Figure 4.30. During the entire simulation, the coin is subjected to a constant body forceˆb = −700in negative Z-direction. The edges of the lower surface of the foundation are completely fixed during the simulation. The employed finite element discretization can also be seen in the Figures 4.30 and 4.31. All in all,14248tri-linear hexahedral elements (hex8) with

Figure 4.31: 3D view on the initial setting of the falling coin example. The figure is taken from Farah et al. [71].

EAS element technology (cf. Simo and Rifai [255]) are employed. For the simulation, inertia

effects are considered and implicit time integration is done with a generalized-α scheme intro-duced in Section 2.2.3. The time integration parameter is chosen asρ_∞= 0.95, which introduces slight numerical dissipation. The overall simulation time is T = 0.055and the time step is de-fined to be∆t = 5·10⁻⁴. The contact scenario is defined with the coin being the slave body and the foundation being the master body, respectively. Contact without any frictional effects is as-sumed, and line and surface Lagrange multipliers are introduced at the edges and the surfaces of the coin. A Petrov-Galerkin scheme is utilized for the Lagrange multiplier interpolation, which changes the definition of the weighted gaps according to (3.42), see also Popp et al. [214].

The resulting deformation of the coin and the foundation is visualized for characteristic time steps in Figure 4.32. Herein, the deformation corresponding to the first impact is shown in the

Figure 4.32: Falling coin example: characteristic stages of deformation. The figure is taken from Farah et al. [71].

top left part. This impact is resolved entirely by the line Lagrange multipliers. It introduces a rotation of the coin, which then leads to the next contact situation being dominated by the surface Lagrange multipliers, see top right part of Figure 4.32. The bottom left part of Figure 4.32 illustrates nicely the elastic wave traveling through the foundation after the first impact. The final deformation at the end of the simulation is shown in the bottom right part of Figure 4.32.

4.9. Numerical examples

0.02 0.025 0.04 0.045 0.05 0.055

active nodes

time

sum of edge and surface nodes

Figure 4.33: Falling coin example: active edge and surface nodes (left) and sum of edge and surface nodes (right). The figure is taken from Farah et al. [71].

The mentioned activation and deactivation of the line- and surface-based Lagrange multipli-ers is additionally shown in Figure 4.33. Therein, active edge nodes correspond to discrete line Lagrange multipliers. It can be seen that at most points in time contact interaction is actually dominated by the line Lagrange multipliers, while only very few situations, such as the one il-lustrated in the top right and bottom left corners of Figure 4.32, are characterized by surface contact. Thus, it can be stated that the overall robustness of the simulation is strongly affected by the newly developed segment-based integration scheme for line contact presented in Sec-tion 4.6.2.

Again, investigations concerning conservation properties are based on the explanations in Sec-tion 4.8.3. Thus, conservaSec-tion of linear momentum is achieved, when the sum of contact forces that act on slave and master side vanishes. In analogy, conservation of angular momentum is achieved, when the sum of interface moments due to contact forces that act on slave and master side vanishes. For this investigation, the interface forces and moments of slave and master sides are visualized in Figure 4.34. It can be seen that the absolute values of force and moment of slave and master side behave identically at first sight and correspond to the impact situations characterized by the number of active nodes in Figure 4.33. However, an in-depth investigation of the conservation properties requires a closer look at the relative error of the sums with respect to the slave quantity. The corresponding results are plotted in Figure 4.35. As described in Sec-tion 4.8.3, conservaSec-tion of linear momentum isguaranteedfor the developed contact algorithms, since slave and master forces balance perfectly. This conservation is achieved up to machine pre-cision. Conservation of angular momentum is not guaranteed, since the interface moments do not balance. However, the obtained error is very small (max. 0.025%) and thus can be considered negligible from an engineering point of view. Note that conservation of energy can not be guar-anteed with the presented algorithm, since adequate time integration schemes that resolve the discontinuities of the interface velocities in the event of an impact are required for this purpose, see Laursen and Chawla [152] and Laursen and Love [153]. These time integration schemes are not employed and not in the focus of this thesis.

Finally, the convergence rate of the contact algorithms within the semi-smooth Newton scheme is investigated in the following. Therefore, a characteristic time step with line and surface

La--1500

Figure 4.34: Falling coin example: absolute values of interface forces (left) and moments (right) for investigation of conservation of linear momentum and angular momentum. Mas-ter quantities are defined to be negative for betMas-ter visualization. The figure is taken from Farah et al. [71].

Figure 4.35: Relative errors for balance of interface forces (left) and moments (right) for the falling coin example. The figure is taken from Farah et al. [71].

grange multipliers being active is considered. The convergence behavior is shown in Table 4.1.

There, the L²-norm of the displacement residual, the displacement increment, the constraint residual and the Lagrange multiplier increment are given for the required Newton steps. In ad-dition, the number of active nodes is provided and the nonlinear solution steps with changing active set are highlighted. At the beginning of the nonlinear solution procedure,5changes in the active set are carried out. Within these steps, the considered norms only slowly decrease, which is expected for semi-smooth Newton methods, see Popp [210]. After the correct active set is found, all norms converge approximately quadratic to zero, which is the expected convergence rate for Newton methods when being close enough to the solution, see Section 2.2.4. However, the Lagrange multiplier increment norm approaches latest to zero, which is also expected ac-cording to the classic textbook about constrained optimization Bertsekas [27]. For the sake of completeness, it should be noted, that the final number of 88active nodes consists of 24edge nodes and64surface nodes

4.9. Numerical examples step displ. residual displ. incr. constr. residual LM incr. active nodes

1 2.86708e+ 01 2.49036e−01 1.78943e+ 05 2.86007e+ 05 207 (∗) 2 1.74212e+ 01 2.57951e−02 1.77194e+ 05 3.21509e+ 05 130 (∗) 3 8.92433e+ 00 1.76936e−02 1.11207e+ 05 2.47264e+ 05 96 (∗) 4 2.92589e+ 00 7.29371e−03 3.16991e+ 04 1.30408e+ 05 89 (∗) 5 3.44270e−01 1.72936e−03 2.60689e+ 03 3.59133e+ 04 88 (∗) 6 3.81247e−03 1.42311e−04 3.00870e−01 2.92161e+ 03 88 7 5.09491e−08 3.46040e−07 4.57391e−06 1.38196e+ 00 88 8 8.51222e−11 5.54522e−12 2.81013e−11 1.68548e−05 88 9 8.22572e−11 6.50431e−15 2.07129e−11 4.33384e−09 88

(∗) = change in active set

Table 4.1.: Convergence behavior of the all entity contact algorithm in terms of the displace-ment residual norm, the displacedisplace-ment incredisplace-ment norm, the constraint norm and the Lagrange multiplier increment norm for a characteristic time step. In addition, the change in active set and the number of active nodes are given.

Figure 4.36: Initial setting for the frictional plate on plate example. The figure is taken from Farah et al. [71].

The convergence results demonstrate that the all entity contact formulation developed in this thesis is perfectly incorporated into the semi-smooth Newton methods from Section 3.5.2. Al-though the Lagrange multipliers for line and surface contact have different physical interpreta-tion and are of different units, an excellent performance of the nonlinear soluinterpreta-tion scheme could be achieved. One basic aspect of this result is the consistent linearization of the line contact algorithm (cf. Appendix B) with all geometrical operations being considered.