New trends in asymptotic approaches: Summation and interpolation methods Igor V Andrianov Pridneprovye State Academy of Civil Engineering and Architecture, 24a Chernyshevskogo St., Dnepropetrovsk 49000, Ukraine; igor_andrianov@hotmail.com Jan Awrejcewicz Technical University of Lodz, Division ofAutomatics and Biomechanics, 1/15 Stefanowskiego St, 90-924 Lodz, Poland; awrejcew@ck-sg.p.lodz.pl In this review article, we present in some detail new trends in application of asymptotic techniques to me- chanical problems. First we consider the various methods which allows for the possibility of extending the perturbation series application space and hence omiting their local character. While applying the asymptotic methods very often the following situation appears: an existence of the asymptotics £ —» 0 implies an exis- tence of the asymptotics e —> °° (or, in a more general sense, e —> a and e —> b). Therefore, an idea of constructing a single solution valid for a whole interval of parameter e changes is very attractive. In other words, we discuss a problem of asymptotically equivalent function constructions possessing for e —> a and E —> b a known asymptotic behavior. The defined problems are very important from the point of view of both theoretical and applied sciences. In this work, we review the state-of-the-art, by presenting the existing methods and by pointing out their advantages and disadvantages, as well as the fields of their applications. In addition, some new methods are also proposed. The methods are demonstrated on a wide variety of static and dynamic solid mechanics problems and some others involving fluid mechanics. This review article contains 340 references. 1 INTRODUCTION: ASYMPTOTIC APPROACHES IN THE INFORMATION AGE The end of the century and the birth of a new millennium focus the attention of many researchers on the future of and relations between analytical and numerical strategies applied in science and engineering (Elishakoff, 1998; Gromov, 1998; Gucken- heimer, 1998; Morgan, 1998). Those problems had previously attracted the attention of researchers (Andrianov and Manevitch, 1992, 1994; Awrejcewicz et al, 1998; Barantsev, 1989; Ham- ming, 1973; Nayfeh, 1973, 1981; Obraztsov et al, 1991; Van Dyke, 1975a, 1991; Wilcox, 1995). The fundamental impressions can be summarized in the fol- lowing way. The possibility of obtaining required information has significantly increased in recent years. This implies high quality tools for its safe keeping and a transformation in a man- ner to be understandable by a human being. The last require- ment is related to: a) construction of low dimensional models; b) extraction of high dimensional information; c) extraction of the most important singularities in a system's behavior (for in- stance, the bifurcation points), and so on. The most suitable tools to realize these requirements are related to analytical methods and, in particular, to asymptotic approaches. On the other hand, it has been observed that in recent years asymptotic approaches have been highly developed from both qualitative (an increase of number of possible ap- plications) and quantitative points of view. The most inter- esting directions of development of the analytical approaches refer to detection of new non-trivial small (perturbation) pa- rameters (even in classical problems) and an application of various methods of summations and interpolations. The last problems are dealt with in this review paper, whereas a problem related to detection of new small parameters will be reconsidered later. Last, but certainly not least, we would like to briefly de- scribe the role of asymptotic approaches in investigations of chaotic dynamics, as well as in quasi-periodic solutions. In this work, we also are going to emphasize the role of Pade approximants, which seems to play a more important role in today's nonlinear mechanics. For instance, they have been widely used during analysis of nonlinear oscillations through the concept of normal modes of nonlinear oscillations (Mikhlin, 1985, 1995; Manevitch and Mikhlin, 1989; Manevitch et al, 1989; Vakakis et al, 1996; Salenger et al, 1999) and normal form (Robnik, 1993), padeons (Lambert and Musette, 1984, 1986), and even in the theory of chaos and fractals (Barnsley et al, 1983; Barnsley, 1988; Barnsley and Demko, 1984,1985; Karlsson and Wallin, 1994). 2 SUMMATION METHODS 2.1 General remarks The principal shortcoming of perturbation methods is the lo- cal nature of solutions based on them. Besides that, the fol- Transmitted by Associate Editor J Simmonds ASME Reprint No AMR302 $20 Appl Mech Rev vol 54, no 1, January 2001 69 © 2001 American Society of Mechanical Engineers Downloaded 25 Nov 2010 to 130.161.210.163. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm