Vol. 5, No. 3-4, 2017

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ISSN: 2325-3444 (e-only)
ISSN: 2326-7186 (print)
A variational formulation for fuzzy analysis in continuum mechanics

Rolf Mahnken

Vol. 5 (2017), No. 3-4, 261–298
DOI: 10.2140/memocs.2017.5.261
Abstract

In order to improve the credibility of modern simulation tools, uncertainties of different kinds have to be considered. This work is focused on epistemic uncertainties in the framework of continuum mechanics, which are taken into account by fuzzy analysis. The underlying min-max optimization problem of the extension principle is approximated by α-discretization, resulting in a separation of minimum and maximum problems. To become more universal, so-called quantities of interest are employed, which allow a general formulation for the target problem of interest. In this way, the relation to parameter identification problems based on least-squares functions is highlighted. The solutions of the related optimization problems with simple constraints are obtained with a gradient-based scheme, which is derived from a sensitvity analysis for the target problem by means of a variational formulation. Two numerical examples for the fuzzy analysis of material parameters are concerned with a necking problem at large strain elastoplasticity and a perforated strip at large strain hyperelasticity to demonstrate the versatility of the proposed variational formulation.

Keywords
fuzzy analysis, $\alpha$-level optimization, quantities of interest, optimization with simple constraints, large strain elasticity, large strain elastoplasticity
Mathematical Subject Classification 2010
Primary: 74-XX
Milestones
Received: 17 December 2016
Accepted: 27 June 2017
Published: 30 December 2017

Communicated by Paul Steinmann
Authors
Rolf Mahnken
Paderborn
Germany