Fig. 3 - Sketch of the modeFRONTIER – LS-DYNA workflow
In order to better understand the influence of such parameters on the simulation results, a sensitivity analysis was performed by coupling the LS-DYNA FE model with modeFRONTIER, a process integration and design optimization tool. modeFRONTIER is able to explore the design space (i.e. the permitted values of free parameters) and find configurations which satisfy several objective functions. The integration of the LS-DYNA FE model described above into the modeFRONTIER environment is roughly described by the workflow in Figure 3. The blocks on the top define the input variables for which a suitable range of variations was set. These input variables included: the damping constant (variable sf in the DAMPING_PART_MASS card), the shear strength for tiebreak contact (variable sfls in the tiebreak CONTACT card), the degradation factor for tensile failures (variable slimt in the MAT54 card) and the degradation factor for compression failure (variable slimc in the MAT54 card). Each time a new combination of their values was proposed by modeFRONTIER, the LS-DYNA input file was updated and a new LS-DYNA analysis performed in batch mode. The output of each simulation was then post-processed and the results of the analysis evaluated. The outputs used in this study were the contact force time history, the plate deflection time history, the absorbed energy and the damaged area size. Three of these output were evaluated directly in LS-DYNA (contact force, plate deflection, absorbed energy), while the damaged area was evaluated using ANSYS FE by means of an APDL macro. These numerical results were compared to experimental data during the post-processing phase and the relative errors were computed. Such errors, which will be indicated respectively as “err_f_min”, “err_d_min”, “delta_energy” and “min_del_area” were thus the objective functions to be minimized. In the block labelled “DOE” (which stands for “Design of Experiments”) the user can generate an initial population of designs, each possessing a different combination of input variables. Starting with the results obtained from these initial designs, the “Scheduler” block iteratively generates completely new designs with the aim of achieving the defined goals using various optimization algorithms.
Fig. 4 - a) Scatter matrix chart; b) 4D Bubble Chart
In order to study the interaction between the input variables and the four chosen objectives a statistical analysis was performed by evaluating an initial population of 81 designs generated by using the Full-Factorial method with 3 levels for each variables. The scatter matrix chart, which is a very useful tool to analyze the data of a statistical analysis, is shown in Figure 4a).
It was found that the variable slimt is the more significant input variable (high correlation with the 4 objectives). All parameters were found to affect significantly the damaged area objective. All objectives are positively correlated, indicating that the objectives were not conflicting.
A multi-objective optimization analysis with the algorithm MOGA-II was then performed. The optimization strategy evaluated 137 designs (the initial 49 Full Factorial designs followed by 88 designs specified by the MOGA-II algorithm), leading to several candidate optimal solutions. These can be easily detected in the 4D bubble chart of Figure 4b, where each solution is represented by a coloured bubble of a particular size. A good configuration which minimizes all four objectives should therefore be blue, have a small diameter and lie towards the bottom left of the chart. Design 189 (indicated by the red arrow) was considered to be a good compromise in achieving these goals.
The correlation between the numerical results obtained with this configuration and the experimental data, in terms of damaged area size, contact force, deflection, absorbed energy time histories and force versus displacement trend, are shown in Figures 5, 6a, 6b, 6c and 6d, respectively. The comparison shows that the fitted simulation results and experimental data to be well-correlated.