Cardboard box drop test

This example models the drop test of a simplied cardboard box on its corner and edge. The interest of this example is that it combines in the same model several contact interfaces and a hybrid discretization with FE elements and B-Splines. The cardboard is composed of several parts with dierent materials. The box inners, colored in pink on Figure 4.20, are represented by aluminium blocks and are con-nected at their sides by rubber pads (in blue) to a steel housing (in light red). This housing rests at its corners on styrofoam cushions (in green). Finally, a cardboard (in orange) packs all the parts. The purpose of this example is to check that the packing of the inners is ecient: energy absorption...

Inners

Rubber pads Housing Styrofoam

cushions Cardboard

Fig. 4.20: The cardboard box and its internal parts. Some parts of the cardboard were removed for a better visualization.

The cardboard box and the styrofoam cushions are discretized withC¹ quadratic B-Splines. The housing is discretized with piecewize linear shell elements. For all the other parts, piecewize linear FE bricks are used. Type 7 contact interfaces are dened between the cardboard box, the styrofoam cushions and the housing.

All other part connections (e.g. between inners and rubber pads) are imposed by merging nodes with compatible meshes. An initial vertical velocity of 5 m.s⁻¹ is set as well as a gravity eld, which represents the drop. A rigid wall is xed in front of the corner or the edge of the cardboard, depending on the case considered.

Material models used in this example are elastic for inners, elastic perfectly plastic for the cardboard and for the housing, foam-plastic for all the styrofoams, and hyperelastic for the rubber. The cardboard is considered as isotropic for simplicity.

Results and observations

Simulation results for the corner and the egde drop tests are given on Figure 4.21 and Figure 4.22. As said previously, the analysis of those results is focused on the energy balance of the inner parts. Energy balances for each test can be plotted to compare the total kinetic energy, the total internal energy, the contact energy and the inners internal energy, in a logarithmic scale. Results show a conversion between kinetic and internal energy during the drop test. Most of the energy is absorbed by the styrofoams and the cardboard and let the housing and the inners safe. Left gures show the internal force magnitude on the inners. We observe that the highest value is at least six times lower than that of the other parts of the model.

Fig. 4.21: Deformation and energy balance for a corner drop test.

Fig. 4.22: Deformation and energy balance for an edge drop test.

Conclusion

This thesis was devoted to the integration of isogeometric analysis in an existing industrial explicit nite element software. All the needed ingredients for a complete simulation of dynamic phenoma such as stamping or crash have been identied, studied and modied according to the specicities of the IGA.

We were able to dene heuristic estimates for the critical time increment asso-ciated to these new elements. These estimates are initially given for conventional nite elements and have been adapted to suit our basis functions. Simulations have shown that the time increment of simulations obtained with isogeometric elements increases signicantly. This gain is a very strong argument to put forward IGA in solvers, especially explicit ones.

An important part this work was dedicated to the modication of the contact interface. We have been able to integrate IGA, only addressing the representation and discretization of Spline surfaces. Spline surface discretization was simply done by a set of linear facets. Nevertheless, it allows a better representation of surfaces compared to what is obtained with conventional nite elements with similar initial element densities. We showed the benets given by this simple contact formulation in some industrial cases.

We highlighted and analyzed several Spline basis functions allowing local rene-ment with their characteristics, taking into account the implerene-mentation constraints of each of them. We implemented the LR B-Spline basis functions to dene locally rened tridimensional models. We showed that the previously given renement schemes were not sucient to guarantee a set of non-overloaded elements, especially for quadratic cases. The renement complexity that can be found in industrial models is such that it is beyond the scope of the study established until then. We introduced an improved full span scheme, based on the full span scheme. More complex renement shapes such as L-shaped renement, or more general shapes with an inward corner, have been taken into account. The improved full span scheme thus appears to be the appropriate method for dealing with rened models.

It appeared in the LR B-Splines implementation that it was indeed possible to have a native integration within Radioss keeping the engineers usage.

The IGA in Radioss has improved the quality of the results obtained compared to traditional FEA. B-Splines also provide better accuracy and will quickly give a solution without compromising the quality of the result. However, the LR B-Splines computational costs are for the moment not satisfying but improve the quality of solution.

A fair comparison of IGA and FEA is dicult when having FEA engineers habits. We can state without compromises that IGA can show better accuracy or better time step size and on some cases both at the same time. Improving its computational cost to reach industrial solvers standard beyond the work shown in this thesis will make it a very good competitive method compared to FEA.