
Figure 2: Driving backwards into a fixed barrier at 15 km/h, i.e. the Allianz test, without damaging the car body is one of the toughest requirements. The figure shows the CAE model built in ANSA. This model of a full vehicle was used for verification.

Performance varies due to tolerances in production
Using modern crash simulation software such as LSDYNA, it is now possible to predict the behavior in a crash with good accuracy. However, everything that is manufactured has its tolerances on geometry, material properties etc which means that in practice a certain range of variation on the performance parameters always exists. Any small deviation, even a random noise, could influence the real crash, but may not be visible in the CAE analysis when nominal values are used for simulation.

Figure 3: modeFRONTIER was used to automate the robustness study using LSDYNA and METApost. In order to save computational cost, a submodel instead of a full vehiclemodel was used for the robustness and metamodel evaluations.

A robustness study looks into groups of simulations with different combinations of input parameters, to see if they give similar responses or not. Just as with the input parameters, it is important to identify the relevant and interesting output parameters which are then traced in the robustness study. The analysis will show how the performance varies due to scatter in the input parameters.
Evaluation of robustness
Performing a robustness study is both complex and expensive. Complex, since the crashworthiness is determined by variations in a large number of parameters, such as material properties of different parts, friction, impact angle and speed. Complexity includes both choosing the most influential parameters and implementing them for automatic evaluation. Expensive, since a single simulation takes about 2 hours using parallel execution on 24 CPUs and a robustness study may need more than 100 evaluations.
The selected input parameters in this study are:
 Material properties of the bumper beam
 Thickness of the bumper beam
 Material properties of the parts behind the bumper beam
 Barrier impact and tilt angle
 Friction
The selected output parameters are:
 Maximum plastic strain in all parts except bumper and barrier
 Mean plastic strain in all parts except bumper and barrier
 Number of high plastic strain nodes in all parts except bumper and barrier
 Maximum deformation of the bumper beam
 Kinetic and internal energy of the model
 Maximum bumper beam internal energy
 Section forces of the side member
 Latch displacements
The preferred sampling method for this type of robustness study is Latin Hypercube. A central question is how many samples are needed for the chosen 10 variables in the study. A possible answer is to study the correlations between the input variables as shown in figure 4. Figure 5 shows the absolute max and arithmetic mean of the correlation versus the number of designs. It can be seen that both values approach the ideal correlation of 0 as the number of designs grow. A correlation of 0.1 is regarded as acceptable which corresponds to about 75 to 100 samples. In the crashworthiness study, the complexity of the evaluated results as well as the number and complexity of significant interactions among the input variables may require even more samples to be evaluated in order to reach converged stochastic results.

Figure 4: Linear correlations between the 10 input variables for the Latin Hypercube sampling. 

Figure 5: Correlation between input variables approach the ideal value of zero as the number of designs grows. A maximum correlation of 0.1 between two inputs is regarded as acceptable which corresponds to a requirement of approximately 75 to 100 samples. 
In this study, convergence of the stochastic results of the initial sampling of 200 design points is verified by an additional 100 design points. The additional 100 designs are also generated from Latin Hypercube, but from a different random seed. This means that the additional 100 designs differ from the original 200 designs and the 300 designs as a whole still follow the Latin Hypercube space filler distribution. It is observed that there was not a big difference between the output correlations or the output distributions gained from the 200 and 300 design sets.

Table 1: Comparison of main and interaction effects of the inputs on maximum level of the bumper beam internal energy. 
Results of the robustness study
One result of the robustness study is a list of the main effects for each results quantity. Figure 6 shows the effect of input parameters on the maximum internal energy of the bumper beam, ranked from most to least influential. It can be seen that the maximum internal energy of the bumper beam is critically influenced by changes to the tilt and impact angle of the barrier. In addition, an increase in the friction and a decrease in the bumper beam material strength could give higher energy absorption.
Besides, the effect of each individual input parameter interactions of several inputs can be significant. As it can be seen in table 1, the combination of material properties of the rear side members and the impact angle have more effect on the results than the single factors friction or material properties of the bumper beam.

Figure 3: modeFRONTIER was used to automate the robustness study using LSDYNA and METApost. In order to save computational cost, a submodel instead of a full vehiclemodel was used for the robustness and metamodel evaluations. 
The robustness study also uncovered a set of designs giving extreme results. A separate study on these outliers revealed that they all had low values of friction. The root cause of the outliers is related to the way LSDYNA deals with friction. As a result, 200 new FE simulations were performed with the friction fixed at the nominal value. The ranking of main and interaction effects was not affected while the output values and their distributions had to be updated. Table 2 shows how the most important stochastic data changes when friction is removed as a stochastic input variable. The table also shows that the standard deviation of the internal energy is in the order of 510% of the nominal value. By comparison, the number of deformed elements, i.e. elements exceeding a specified plastic strain, has a standard deviation exceeding 50% of the nominal value.
The correlation chart is a versatile tool and figure 7 shows the original 10 input variables versus 4 outputs. Marked boxes are regarded to have high values of correlation. Since the variables Tilt, Thickness, Impact Angle and Friction have many marked boxes but only one box is marked for the material properties, it is concluded that variations in material properties are of less importance than variations in the loading case.
Another important result is the correlation between the outputs. Figure 8 shows that an increase in the maximum internal energy of the bumper beam leads to a decrease in the number of deformed elements on the ring frame.

Table 2: Variation of friction has a significant effect on some of the stochastic results. It is also clear that the robustness properties can hardly be ignored when the maximum value in the study exceed the nominal value by more than 5 times. 
The necessity of metamodels
As seen in the robustness study, the scatter of the results cannot be neglected in an optimization. Furthermore, the computational expense makes it most desirable to find a fast replacement for the FE simulation during the optimization. In modeFRONTIER there are 7 types of metamodels which aim to replace the underlying simulation model with a very fast but approximate function. The evaluation time is in the order of 0.05 seconds, making it possible to evaluate thousands of design candidates in order to solve the robust design optimization task.

Figure 7: Correlation between input and output variables. The variation in crashworthiness due to scatter in material properties is small if compared to the scatter in the load case variables. 
The process of using metamodels is divided into 3 steps:
 Training the metamodel
 Evaluating the quality of the fit
 Using of the metamodel
It was not obvious which metamodel would deliver the best fit so Kriging, Radial Basis Function and Neural Networks were included and evaluated.
Besides the previously mentioned robustness parameters, 3 new geometry parameters, implemented through mesh morphing in ANSA, were introduced.

Figure 8: Correlation between output and output variables. An increase in the internal energy is strongly correlated to fewer nodes with high strain in the ring frame. 
The training set consisted of 1000 FE simulations and another 170 FE simulations were used to check the quality of the metamodels. Figure 9 shows the difference between the Radial Basis Function and the evaluation set for one of the results. The mean residual values between the three methods were close and the response looked similar to the same design IDs. As such, all three methods in this study are considered to give equally good results. In the end, the parameters given by the Neural Networks were chosen for final verification.

Figure 9: The residual chart shows the difference between the forecasted value by the Radial Basis Function and the FE simulations for the evaluation set. 
Robust Design Optimization
The metamodels were used to run a multiobjective robust design optimization. A design found through optimization on the metamodels was then selected and verified using real FE simulations. Table 3 shows results for highly strained elements and it is clear that the optimized bumper beam is a big improvement over the original. Both the mean value and standard deviation have decreased. The comparison is also done for the full car model, to confirm that results calculated from the submodel can be applied to the full car, cf. figures 10 and 11.

Figure 10: a) shows the plastic strain on the ring frame (i.e. a rear part of car body) in the submodel with original bumper beam. b) shows the plastic strain on the ring frame in the submodel with optimized bumper beam. 

Figure 11: a) shows the plastic strain on the ring frame in the full car model with original bumper beam. b) shows the plastic strain on the ring frame in the full car model with optimized bumper beam. 
The bumper which was optimized according to the Allianz load case was also tested in other low and high speed crashes. The results highlighted the necessity to consider multiple load cases at the same time during the optimization.

Table 3: The optimized bumper has been improved in all studied outputs. 
Summary
Overall the results were very promising, proving the potential of running robust design optimization on metamodels for crash simulations. The initial robustness study also provided great value and insight into the dominant parameters and considerations regarding the FE simulations. The arithmetic mean and standard deviation for the stochastics simulations were improved for all studied outputs, e.g. for the ringframe the results were improved by about 50% and 20% respectively.
Reference
[1] Xin Li and Tolga Olpak, "Robustness and Optimization Study of a Rear Bumper Beam During a Low Speed Impact", M.Sc. Thesis at Volvo Car Corporation, Göteborg, Sweden, Department of Solid Mechanics at the Royal Institute of Technology (KTH), Stockholm, 2009