Validity Domains of Beams Behavioural Models: Efficiency and Reduction with Artificial Neural Networks Keny Ordaz-Hernandez, Xavier Fischer, and Fouad Bennis Abstract— In a particular case of behavioural model reduction by ANNs, a validity domain shortening has been found. In me- chanics, as in other domains, the notion of validity domain allows the engineer to choose a valid model for a particular analysis or simulation. In the study of mechanical behaviour for a cantilever beam (using linear and non-linear models), Multi-Layer Perceptron (MLP) Backpropagation (BP) networks have been applied as model reduction technique. This reduced model is constructed to be more efficient than the non-reduced model. Within a less extended domain, the ANN reduced model estimates correctly the non-linear response, with a lower computational cost. It has been found that the neural network model is not able to approximate the linear behaviour while it does approximate the non-linear behaviour very well. The details of the case are provided with an example of the cantilever beam behaviour modelling. Keywords— artificial neural network; validity domain; cantilever beam; non-linear behaviour; model reduction. I. I NTRODUCTION A MODEL is an abstraction of reality and no model represents it perfectly [16]. In mechanical engineering, as in other areas, it is possible to find a number of models to simulate the same phenomenon. These variety is created for many reasons. Mainly, because a model represents only a limited view of reality: some aspects of reality are incorpo- rated, others are leaved out. Also, because disposing of several models permits the engineer to select an more efficient or more adequate model. While that is true, another reason is that the same phenomenon may be modelled differently according the activity where it is to be employed; e.g. within an off-line analysis, models are essentially accurate and precise; while within an on-line simulation, models are primarily fast. Also, specific models are usually developed to be more efficient than their generic counterparts; but not without a penalty —typically the diminution of the applicability domain. The domain where a model can be applied limits its validity domain, since only there its validity can be assessed. In many Manuscript received March 28, 2007. This work was supported in part by the Communauté d’agglomération de Bayonne-Anglet-Biarritz (CABAB). K. Ordaz-Hernandez is with the ECN IRCCyN Institute; 1, rue de la Noë - BP 92101; 44321 Nantes Cedex 03, France and with the ESTIA LIPSI laboratory ; Technopôle Izarbel, 64210 Bidart, France (corresponding author: +33 559-438-512; fax: +33 559-438-401; e-mail: Keny.Ordaz-Hernandez@ irccyn.ec-nantes.fr). X. Fischer is with the ESTIA LIPSI laboratory ; Technopôle Izarbel, 64210 Bidart, France and with the laboratory TREFLE - UMR CNRS 8508 ENSAM; Esplanade des Arts et Métiers, 33405 Talence Cedex, France (e- mail: x.fischer@estia.fr). F. Bennis is with the ECN IRCCyN Institute; 1, rue de la Noë - BP 92101; 44321 Nantes Cedex 03, France (e-mail: Fouad.Bennis@irccyn.ec-nantes.fr). situations (as in the reuse of mechanical models, see [40]), a model should always be accompanied of its validity domain in order to be used. Moreover, the validity domain must be verified if the model is modified: during the application of a model reduction to beam behavioural models, a modification of the resulting validity domain was noticed. While using model reduction techniques to create a more efficient model (lower time of response with a negligible loss of accuracy), changes in the validity domain must be expected. In this paper, the case of Artificial Neural Networks (ANN) employment as model reduction technique for beam behavioural models is considered. The efficiency and validity domain of the reduced model are studied to a means to support decision making in the successful use of models: • A shortening of validity domain after reduction of the original model is reported. • An improvement of the efficiency of the model after reduction is reported. The next section (sect. III) provides a background of the utilisation of ANN in mechanics and engineering, and as model reduction techniques. Section IV presents some behavioural models for beams and discusses their validity domain. Section V illustrates the application of ANN-based model reduction to a cantilever beam case. Section VI dis- cusses the resulting efficiency and validity domain of the cantilever beam case. II. PROBLEM STATEMENT A recent trend in the creation of virtual prototypes for product design is the inclusion of interactivity. Virtual proto- types are digital representations or simulations of the product concept. Simulation of interactive prototypes (or interactive simulations) can be used to explore and experiment prod- uct concepts according to the expertise and intuition of the designer[9] and the future user. Similarly, it has been sug- gested that the use of interactive simulation shall speed up the findings and reviewing of concept design in the early stages of the development process [5]. Interactivity in the virtual prototype is its capability to simulate the human’s interaction with the design. In the past, the effectiveness of an interactive virtual prototype was limited to the following features: realistic visualisation, geometry-related constrains, and realistic simulation of physical behaviour [39]. However, human-product interaction should be included [36] as well as real-time processing and rendering [29], [5] to maintain the World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:2, No:6, 2008 2080 International Scholarly and Scientific Research & Innovation 2(6) 2008 scholar.waset.org/1307-6892/4871 International Science Index, Computer and Information Engineering Vol:2, No:6, 2008 waset.org/Publication/4871