Draft: Empirical Analysis Using Advanced Similarity Methods

Traditional dimensional analysis techniques for predicting the performance characteristics of a product can be greatly improved in both accuracy and domain of applicability by the infusion of empirical data, derived from material tests, into the equations that characterize the system parameters of interest. Advanced similarity methods are investigated which overcome the constraints associated with the traditional methods and provide increased analysis capability and improved insight into the phenomenon governing the problem. Such capability greatly increases the design toolbox available to product developers, across a large range of scale and application. It also significantly enhances a developer’s choices for prototype portioning during a development cycle. Solid mechanics and heat transfer applications are used to illustrate the basic utility of the methods. INTRODUCTION The Traditional Similarity Method (TSM) is well understood as a methodology that uses the concept of dimensional analysis, based upon the Buckingham  theorem, to correlate the respective states of a model and a product. The primary goal of the technique is to allow the prediction of the performance of a product from the measured states of a representative model. This process has been used to save considerable time and expense in contrast to testing complex and expensive full-scale products. Dimensional analysis, theory of dimensions, theory of models, similitude, theory of similarity; all of these terms have been used interchangeably over the past century to describe the process of using the exponents of variables as a starting point for a more detailed dimension-based analysis [1]. Dimensional analysis has long been recognized as a valuable design and analysis tool but has suffered from severe constraints and limits of applicability. Advanced similarity analysis methods, in tandem with recent advances in rapid prototyping technology, have afforded the development of functional testing for geometrically complex physical models [2], free of many of the constraints that limit the application of traditional methods. Until recently, functional tests on rapid prototyped products was rare with the key factor the severe material properties limitations [3]. Advanced similarity methods represent an opportunity to overcome these limitations and those imposed by traditional methods by using scale testing on rapid-prototyped models in concert with advanced similarity prediction techniques. Advanced similarity methods represent a powerful class of analysis tools in development, derived from the basic principles of dimensional analysis. In this paper, the theoretical foundation for traditional dimensional analysis is reviewed to gain a full understanding of the method including constraints and limits of applicability. This review is followed by an introduction to an advanced method, referred to as the Empirical Similarity Method (ESM), to include the derivation of the method and evaluation using both a numerical example and an empirical example to demonstrate the potential of the new methods for empirical engineering analysis.