Metamodels and industrial applications

Finite element simulation is a valuable tool for estimating the behavior of mechanical systems. However, the uncertainties associated with these systems (materials, geometry, conditions, etc.) make predictions considerably more complex. To improve the reliability of simulations, researchers are increasingly turning to probabilistic modeling, which can take into account the variability inherent in these systems.

Numerical accuracy and error control have been used in simulations for dynamic responses of structures. Among these probabilistic methods, the most frequently used is the statistical approach or sampling technique, such as Monte-Carlo. In this method, a large number of input variable samples are required for reasonable accuracy. The problem is then solved for each realization. This technique is widely used, as it is the easiest to implement and very robust. However, the number of realizations must be sufficient, i.e. deterministic FESs of 10 ⁵ or 10 ⁶ must be run in order to obtain accurate results.

As a general rule, when using computationally-intensive simulation codes in complex engineering problems, it becomes impractical to run a large number of simulations for uncertainty quantification or design optimization. A better alternative is to use approximations to the original models, often called metamodels (or surrogate models). These metamodels aim to build mathematical models to define the relationship between the inputs and outputs of specific systems. Substitution models were mainly developed to approximate deterministic simulations. Recent developments have explored their use in probabilistic analysis and design optimization. The most popular metamodeling methods are polynomial response surface (PRS) methodology, kriging, radial basis function (RBF) and support vector machines (SVM).

In this article, we present methods for eliminating from the model those factors that have no influence on the model. Once this selection has been made, experimental designs for numerical response surfaces, Doehlert designs and Latin Hypercube designs are presented. Once the experimental design has been run, a response surface is fitted to the test runs using PLS regression or kriging. A sensitivity analysis is performed to identify the factors that contribute most variability to the response. A robust design method using factor compensation is then applied, and illustrated in a mechatronic system.

Three applications of metamodels are presented: sensitivity analysis, study of a mechatronic system and study of a shape memory alloy actuator.

A glossary of terms is provided at the end of the article....