Response Surfaces
The progresses in finite elements methods (FEM) and high performance computing offer to engineers accurate and reliable virtual environments to explore several possible configurations. On the other hand, users' requests had grown up going even beyond computational exhaustiveness.
In real case applications, it is not always possible to reduce the complexity of the problem and obtain a model that can be solved quickly. Usually every single simulation can take hours or even days. In these cases, the time to run a single analysis makes running more than a few simulations prohibitive and some other smart approaches are needed. The engineers can turn to a Design of Experiment (DOE) technique to perform a reduced number of calculations. After that, engineers can use these well-distributed results to create an interpolating surface. This surface represents a meta-model of the original problem and can be used to perform the optimization without computing any further analyses.
The use of mathematical and statistical tools to approximate, analyze and simulate complex real world systems is widely applied in many scientific domains. These kinds of interpolation and regression methodologies are now becoming common even in engineering where they are also known as Response Surface Methods (RSMs). RSMs are becoming very popular offering a surrogated model with a second generation of improvements in speed and accuracy in computer aided engineering.
Available methods
In modeFRONTIER there are available all the tools for measuring the quality of a meta-models in term of statistical reliability. Moreover, modeFRONTIER gives a set of reasonable meta-modeling methods to interpolate different kind of data. These methods include:
- Multivariate Polynomial Interpolation based on the Singular Value Decomposition (SVD).
- K-Nearest, a statistical interpolator which works averaging the known values of the target function. The weights are assigned according to the reciprocal of the mutual distances between the target point and the training dataset points.
- Kriging: a regression methodology originated from the extensive work of Professor Daniel Krige, from the Witwatersrand University of South Africa, especially from problems of gold extraction. The formalization and dissemination of this methodology, now universally employed in all branches of geostatistics, as oil extraction and idrology among others, is due to Professor Georges Matheron, who indicated the Krige's regression technique as krigeage.
- Parametric Surfaces: useful when the mathematical expression of the response is known, except for some unknown parameters. The training algorithm calculates the values of the unknown parameters that yield the best fit
- Gaussian Processes implement the Bayesian approach to regression problems; the knowledge of the response is expressed in terms of probability distributions. This algorithm is best suited for non polynomial responses.
- Artificial Neural Networks: is a machine designed to model the way in which the brain performs a particular task or function of interest. To achieve their aims, neural networks massively employ mutual interconnections between simple computing cells usually called neurons. Networks simulate the brain in two aspects: the knowledge is acquired through a learning process and the information is stored in the synaptic weights. The class of Neural Networks included in modeFrontier with a single hidden layer is shown to be capable to interpolate any functions with minimum request of regularity.
Considering that there are several methods for interpolation available both in modeFRONTIER and in the literature, an engineer may ask which would be the best model to be use. There is an obvious notion for which simpler functions can be approximated better and more complex functions are in general more difficult to approximate regardless of the meta-modeling type, design type and design size. A general recommendation is to use simple meta-models first (such as on low order polynomials). Kriging, Gaussian and Neural Network should be used for more complex responses. In general, regardless of the meta-model type, design type, or the complexity of the response, the performance tends to improve with the size of the design, especially for Kriging and Artificial Neural Networks.