## Uncertainty propagation

The uncertainty propagation in computer simulation models is the core business of PHIMECA Engineering. With the solid expertize of its collaborators in high-performance computing (HPC), PHIMECA Engineering is able to automate complicated algebraic calculation chains to urge them on intensively (that is, for a large number of input parameters) to explore them and provide the expected answer in the time limits.

Simulation method and planning of experiences

The simulation and planning of experience methods bring together all the techniques which permit to manage the calculations. Their choice depends on the purpose of the calculation.

• The parametric analysis is to examine the model response on a grid more or less dense of input parameters to produce graphs of tendency ;
• The direct Monte Carlo simulation uses pseudo-random number generator to reproduce the hazard characterized by the input data probability model to study the probability model response. Methods vary depending on if you want to calculate averages, variances or probabilities of rare.
• The reverse Monte Carlo simulation use the Bayes’s formula to study the probability distribution of the input parameter model when distribution of the response is specified.The applications of these techniques are the resetting of model and the indirect parametric identification (for instance, in non destructive testing where sometimes we have to measure a quantity which is not an interest quantity but which it is linked to it by a model).
• Finally, the experience planning is art of selecting at best the parameters in which the model will be calculated to maximize the information about the relation between input and output to built an approximation (less expensive in calculation time) : the response surface.

It has to be noted that the optimization is also an model exploration technique, controlled in PHIMECA Engineering, which focused on the research of a particular value of the input parameters (for instance : the value which minimize or maximize a criterion established on the model response).

### Response surfaces

The total cost of our customers’ models, according to the time or financial (licenses, calculation resources mobilization), and the large number of simulations that it have to be realized leads to the construction of approximations of these models to accelerate processing times. Often, these approximations are called response surfaces, but PHIMECA Engineering also uses substitution model (British word) which is more suggestive.

These substitution models are simplified analytic representations based on input/output data sample of the original model. PHIMECA Engineering controls generalized linear models (for instance : the polynomial response surface), recession and/or support vector classification models (support vector machines) or even the Kriging (Gaussian processes for machine learning, GPML).
PHIMECA Engineering attaches great importance to the validation of its response surface and has pertinent tools for this task : the metric adjustment and their calculation by cross validation methods.

### PHIMECA Cournon

34 rue de Sarliève
F-63800 Cournon d'Auvergne
Tél. : (00 33) 4 73 28 93 66
Fax : (00 33) 4 73 28 95 78

### PHIMECA Paris

18/20 boulevard de Reuilly
F-75012 Paris
FRANCE
Tél. : (00 33) 1 58 51 18 02

PHIMECA engineering is an engineering company specialized in the optimization, the reliability and the robustness of products and structures.
Its goal is to provide solutions for robust engineering based on solid scientific foundations, entrepreneurial spirit and moral values shared by all its collaborators.