With simulation methods replacing more and more hardware tests, major decisions in the development process are based on simulation results. It is important for the CAE engineer to be able to assess the degree of reliability of the models. Stochastic simulations take into account uncertainties in the input parameters, such as boundary conditions, material properties, geometric tolerances, etc.

Due to non-linear behaviour under crash conditions, even small changes in model parameters may cause significant changes in the results (fig. 1). In simulations, however, certain modelling techniques can add computational uncertainties, causing additional scatter in results that is unexplained by physical phenomena. Such computational uncertainties should be detected and removed from simulation models wherever possible.

fig.1:  Concept of stochatic simulations

The engineer needs to have confidence in simulation models if major decisions are based on simulation results. Monte Carlo (MC) methods are one way to evaluate the behaviour of simulation models under non-deterministic conditions. The MC-based stochastic simulations method is rather simple. Certain parameters in the deterministic model that are known to scatter are selected. A statistical distribution is defined for each parameter.

Then a sufficient number of simulations are conducted, where the nominal values of the parameters are replaced with random ones. The MC method generates random values that reflect the predefined distribution. The result variables of interest are extracted from the results file (e.g. the PEAK file in MADYMO) and stored. Then the statistical moments, such as mean, standard deviation, 95% confidence interval for the mean, etc., can be computed from the results. Typically, the statistical moments converge after 50 - 100 runs, which means that this number of runs is required in order to properly assess the results statistically. For this reason, MADYMO simulations, which are relatively fast, are well-suited for stochastic simulations, whereas stochastic simulations with large FE models require significant CPU resources. EASi Engineering currently has more than one year of experience in stochastic simulations. We have established a multi-step approach to evaluate models stochastically (fig. 2). The first step is to evaluate model robustness by applying minor scatter to a set of parameters. If the results show unexpected scatter, an in-depth analysis of the model is required to determine possible sources.

fig. 2: EASi's multi-step stochastic approach

Using this approach, we were able to identify certain modelling techniques as sources of "computational" scatter. The second step is to use stochastic simulations for optimization. The MC method generates response surfaces on which local and absolute optima can be found. The quality of the response surface is, of course, highly dependent on the number of runs. The main advantage of this method is that it does not require any functional relationships between input and output variables, as most other optimization methods do. The last step is to evaluate the scatter of the final results based on 'real-world' scatter. Here the distributions of input parameters are typically based on tolerances known for materials (e.g. Young's modulus, density), geometry (e.g. production tolerances, dummy positioning procedure) and boundary conditions (e.g. impact velocity). The result of this step is a cluster of results rather than just one deterministic value. From this cluster, we can compute a confidence interval for any result variable (fig. 3).

fig. 3: Probalistic vs. deterministic results

The confidence interval is the range of values within which a result with a given probability is situated. For example, we might find that the HIC (Head Injury Criterion) is between 950 and 1073 with a probability of 95%. In this case, the deterministic result might be 961. Assuming that we have set a limit of 1000, the deterministic result is below it, but the probabilistic result shows that this is uncertain. This indicates that we have to further optimize the system in order to get the entire confidence interval below the limit. The stochastic approach described above adds significant value to MADYMO simulations, as it increases their degree of realism and provides much deeper insight into the physics of a simulation model.

Last updated: 10. Dez. 1998

© EASi Engineering GmbH 1998