Three-term Method and Dual Estimate on Static Problems of Continuum Bodies

TL;DR

This paper introduces the three-term method and dual estimate for direct minimization in static continuum mechanics problems, demonstrating improved convergence and applicability to tension structures and large deformation analyses.

Contribution

It presents a novel three-term iterative method and dual estimate approach, enhancing convergence efficiency in static continuum mechanics problems.

Findings

Three-term method shows consistent global convergence

Dual estimate effectively incorporates constraints

Applicable to tension structures and large deformation problems

Abstract

This work aims to provide standard formulations for direct minimization approaches on various types of static problems of continuum mechanics. Particularly, form-finding problems of tension structures are discussed in the first half and the large deformation problems of continuum bodies are discussed in the last half. In the first half, as the standards of iterative direct minimization strategies, two types of simple recursive methods are presented, namely the two-term method and the three-term method. The dual estimate is also introduced as a powerful means of involving equally constraint conditions into minimization problems. As examples of direct minimization approaches on usual engineering issues, some form finding problems of tension structures which can be solved by the presented strategies are illustrated. Additionally, it is pointed out that while the two-term method sometimes becomes useless, the three-term method always provides remarkable rate of global convergence efficiency. Then, to show the potential ability of the three-term method, in the last part of this work, some principle of virtual works which usually appear in the continuum mechanics are approximated and discretized in a common manner, which are suitable to be solved by the three-term method. Finally, some large deformation analyses of continuum bodies which can be solved by the three-term method are presented.