Italiano
Ettore Antona
22 June 1931 13 January 2009



Picture of Professor Ettore Antona

Prof. Ettore Antona

Outline of Scientific Activities

Analyisis of objectives:
Towards a conceptual unification of exising approaches to structural analysis, including aspects of application in engineering and mathematical theory aspects

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Asymptotic approaches

An approach where each theory axiomatically derived is contained as a particular case in a progressive approximation process is the "asymptotic" approach proposed by Cicala, and targeted in particular to shells and beams. Such an approach, in the domains where it can be applied, allows generating processes for the identification of progressive approximations without the need of comparing the results of each step with empirical data, and can therfore be considered as even more general than the approach proposed here.

Asymptotic approaches can be applied to all those problems where approximate solutions are sought, and among the quantities which define the problem some can be considered as smaller with respect to others (e.g., the thickness of a shell structure), and can be adopted as principal infinitesimals. Such approaches, which can be applied to obtain general theoris both in engineering and in mathemtical physics problems, are the opposite of axiomatic approaches, where the adoption of specific (sometimes really ingenious) assumptions poses constraints on the very nature of the solutions.

The rigour and depth of asymptotic appraoches allows obtaining the following characteristics:

  • the infinitesimal order of the unknowns (be they quantities or functions) is evaluated beforehand the solutions

  • the effect of differentiation is taken into account in the infimitesimal order, with reference to the nature of the unknown functions and its asymptotic behaviour

  • on these bases it is possible to determine the infinitesimal orders of each term in the equations to be solved

  • the asymptotic approach can be applied also to boundary conditions, in particular for determining the constants of integration, both in homogeneous and non homogeneous equations.

A very important contribution on asymptotic approaches is due to Cicala, who has proposed a theory where asymptotic approaches are considered as particular cases, and can be framed in their limitations and implications.

The results obtained by de St. Venant, Timoshenko, Kirchhoff, Mindlin, and others are surpassed, in a logico-mathematic perspective, by a vision based on the following:

  1. each problem is part of a family, obtained by having one (or more) parameters tending towards zero,

  2. considering the solutions and each addend of the equations, including the contributions of the unknowns, as extensions of function series, constituing a complete basis for such extensions in the space of the coordinates, in which some of the dimensions are small with respect ot the others, and considering the infinitesimal orders of each component with respect ot the fundamental parameters (it should be noted that tridimensional continuum equations become infinite in number and with infinite addends in bidimensional and monodimensional spaces for shells and beams respectively),

  3. decomposing the system of infinite equations into reduce systems, each of which has a finite number of equations with a finite number of addends, down to a specific infinitesimal order and ignoring the rest.

For each of the above steps Cicala has created original procedures, in which very novel elements can be observed (e.g., the possibility of evaluating the infinitesimal order before solving the equations, the semi-convergence of the series obtained during the solution processes by successive approximations, etc.). The asymptotic approach can be used in all those problems where the analytical formulation preserve its meaning when one of the parameters tends towards zero. Compared to an axiomatic approach, where the simplifications which can be obtained by the small dimension of one parameter are introduced heuristically as  "a priori" assumptions, the foolowing advantages can be observed:

  • the approximation level is not imposed by axioms regarding the solutions

  • a sequence of steps of increasing approximations is determined, where the results of axiomatic approaches can be recognised as intermediate steps.