Fig. 2 – modeFRONTIER – LS-DYNA workflow

The set-up of the procedure firstly required the selection of a set of simple experimental tests to obtain proper data fitting, and also to integrate the finite element LS-DYNA models into the modeFRONTIER environment.

2.1 Selection of experimental tests

The experimental tests should describe in an accurate way the different material behaviors under different loading conditions, avoiding at the same time information redundancy. By using the same ASTM D 790 specimen geometry (rectangular cross section bar), the following four destructive tests have been selected and performed:

• 3 Points Bending Test (16mm width specimen);
• 3 Points Bending Test (25mm width specimen);
• 4 Points Bending Test;
• Charpy Pendulum Test.

The Global Stiffness and the Total Absorbed Energy quantities have been measured for 3 and 4-Points Bending tests, while Absorbed Energy and Local Failure Mode were selected for Charpy Pendulum. For these quantities, 9 objective functions to be minimized (nine relative errors) were considered in order to investigate the material behaviors.

2.2 Process Integration: modeFRONTIER – LS-DYNA workflow

As the LS-DYNA material model calibration process is characterized by a large number of goals (i.e. the 9 objective functions) and design parameters (MAT 58 – MAT_LAMINATED_COMPOSITE_FABRIC – selected as the most suitable to describe the composite sandwich which requires 23 input variables), it cannot be solved by using a simple trial-and-error procedure. The efficient method we propose here, instead, takes advantage of the modeFRONTIER Process Integration and Design Optimization capabilities.

The process integration of the numerical models related to the experimental tests can be implemented and described within modeFRONTIER by means of the workflow illustrated in Fig. 2.

From top to bottom, following the blue links, the so-called “Data Flow” can be seen. The green block at the top defines the input variables (the constitutive parameters) for which a suitable range of variations can be set. Each time a new combination of their values is proposed by the modeFRONTIER optimization strategy, the MAT 58 card file is updated (node “mat_inp”) and transferred to the four LS-DYNA models. These computations deliver outputs that are post-processed and finally provide numerical forecasts of the nine physical responses, whose values are re-arranged in the nine relative errors with the aim to minimize the discrepancy (red blocks).

From left to right, following the dotted link, the so-called “Logic Flow” is shown: it represents the sequence of operations that modeFRONTIER will automate, and the logic behind them. The “DOE” node, the first block on the left side, means “Design Of Experiments”. This node allows the user to design a suitable initial population (combinations of input variable values) in respect of an efficient exploration of the design phase. Looking at the performances provided by these configurations, the “Scheduler” node realizes how the investigated phenomenons behave and based on its internal search strategy, starts to generate completely new designs. The new configurations flow sequentially into the four LS-DYNA models, whose numerical simulation is run in batch modality by modeFRONTIER, so that eventually a new evaluation of the objective functions is performed.
In this way, the workflow describes in a graphical and a very intuitive way, how the whole process is carried out.

Fig. 3 - objectives’ Correlation Chart

The whole procedure is composed of 3 fundamental steps:
1) Check outputs’ correlations to reduce number of experiments;
2) Check for the least-influential inputs to reduce their number;
3) Calibration and validation procedure.

In the first two steps, a numerical-experimental data comparison is performed in respect of all experimental tests. No optimization (and consequently no calibration) is carried out, as only the designs that belong to the initial DOE are evaluated. By selecting a suitable DOE (full or reduced factorial, latin square, etc.), it is indeed possible to point out both, the correlations between the different objective functions and the correlations between the constitutive parameters as well as the objective functions.

Getting the outputs-outputs relationships (1st step), allows reducing the calibration challenge size because the correlated outputs (i.e. the redundant ones) can be omitted, as well as a whole experimental test. Fig. 3 – objectives’ Correlation Chart – highlights the first-order correlation between two entities by means of a normalized index spanning from –1 to +1: a value equal to +1 (-1) denotes a full direct (inverse) correlation, while a low absolute value means low correlation. In such cases, the outputs of the two 3-Points Bending tests are strongly directly correlated between themselves, hence only a single Point Bending test can be taken into account. Thus the number of objective functions can be reduced (currently from nine to four objectives).

Fig. 4 - Overall Student Chart

At the same time, highlighting the correlations input-output (2nd step) enables to freeze, at a constant value, the least significant input variables. Thus the number of input variables can be reduced. For this purpose, a Correlation Chart sketching both input and output variables could be used. Another efficient tool is the so-called Overall Student Chart; based on Student’s t-test, it is extremely useful to easily detect simultaneously which input variables are the least sensitive ones for the set of objectives functions.

In particular, as shown in Fig. 4 – Overall Student Chart – every objective function is given by means of a pie chart. In the pie chart, those input variables that affect more than a given threshold are highlighted while the least sensitive ones are grey-shaded and hidden from the table at the right side of the picture after moving the “threshold filter” at the bottom. The input variables reduction (currently from 23 to 15) allows to get a faster model calibration.

In the 3rd step, the modeFRONTIER workflow has been simplified according to the above findings, and the multi-objective optimization campaign started with the Multi-Objective-Optimization-Algorithm MOGA-II. A calibration using only the most sensitive input variables and non-correlated numerical models has been carried out. The optimization strategy provided, in less than 4000 evaluations, three candidate optimal solutions: they are red-framed in Fig. 5 – 4-Dimensional Bubble Chart – where each solution is represented by a bubble in the 4-D plane of the objectives. A good configuration should stay bottom left in the chart (low “c_d_Energy” and “p3_d_MaxForce” deviations from experimental data), should be blue (cold)-colored (low “p3_d_E_val”) and look like a small-sized bubble (low “c_d_BreakTime”). The three selected configurations are good compromises between the simultaneous reduction of the four uncorrelated percentage deviation indexes, selected from the Charpy and 4-Bending Points tests.

The full validation has been eventually assessed by inserting the optimum candidates coming from the previous calibration into the complete workflow: one-digit % deviations from experimental values have been confirmed also by the remaining five objectives.