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A counterexample to FORM and SORM

Claudio Roberto Ávila da Silva Júnior (Nucleus of Applied and Theoretical Mechanics (NuMAT), Post Graduated Program at Mechanical and Material Engineering (PPGEM), Federal Technology University of Parana (UTFPR), Curitiba, Brazil)
Pedro Danizete Damazio (Department of Mathematics, Federal University of Parana (UFPR), Curitiba, Brazil)
Luiz Carlos Matioli (Department of Mathematics, Federal University of Parana (UFPR), Curitiba, Brazil)
João Lucas Cavichiolo (Nucleus of Applied and Theoretical Mechanics (NuMAT), Post Graduated Program at Mechanical and Material Engineering (PPGEM), Federal Technology University of Parana (UTFPR), Curitiba, Brazil)

Engineering Computations

ISSN: 0264-4401

Article publication date: 10 February 2020

Issue publication date: 21 May 2020

91

Abstract

Purpose

This paper aims to presents a counterexample that points to an inconsistency generated by the first- and second-order approximation methods, FORM and SORM, respectively, procedures for elliptical problems.

Design/methodology/approach

The classical results of theory measure and functional analysis were used.

Findings

The FORM and SORM are known to find solutions in a Gaussian space. This procedure does not satisfy the conditions of the Lax–Milgram theorem and does not assure the existence and uniqueness of the solution.

Research limitations/implications

This paper alerts the engineering research community that uses these methods, initiating discussion and improvement of FORM and SORM procedures.

Practical implications

This paper puts in check the feasibility of using FORM and SORM in engineering problems.

Originality/value

From the moment they were introduced to the engineering and scientific communities, the FORM and SORM were taken as the bases for solving various problems found in the literature and indifferent documents scattered throughout the world over the past 50 years, for FORM, and 40 years, for SORM. Even though it was a very serious fault, at least for elliptical problems, pointed out in this work, it went unnoticed all those years by the research community. Therefore, the contribution of this paper is to present the engineering community that uses FORM and SORM in elliptical problems an unnoticed failure since the introduction of these methods.

Keywords

Acknowledgements

This article is part of a research project of process numbers 420615/2016-4 and 5071576014987215 developed in the PPGEM/UTFPR and supported by the National Research Council, CNPq, Brazil.

Citation

Ávila da Silva Júnior, C.R., Damazio, P.D., Matioli, L.C. and Cavichiolo, J.L. (2020), "A counterexample to FORM and SORM", Engineering Computations, Vol. 37 No. 6, pp. 2127-2135. https://doi.org/10.1108/EC-06-2019-0286

Publisher

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Emerald Publishing Limited

Copyright © 2020, Emerald Publishing Limited

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