Uncertainty in Mechanics Problems—Interval–Based Approach

TLDR

Guaranteed enclosures for interval linear systems exist but are often conservative, and exact enclosures are generally unknown. The study introduces a nontraditional interval‑based uncertainty treatment for mechanics problems by developing an interval linear system of equations to handle various uncertainties in linear static analyses. The authors employ interval finite‑element methods, constructing element matrices from material physics and using a Lagrange multiplier technique to enforce compatibility and equilibrium in an element‑by‑element formulation. The approach yields a very sharp enclosure of the solution set for solid mechanics problems, eliminating most overestimation, and is validated by several numerical examples.

Abstract

This paper presents a nontraditional uncertainty treatment for mechanics problems. Uncertainties are introduced as bounded possible values (intervals). Interval finite-element methods, developed by the authors, are used in the present formulation. To account for different types of uncertainties in linear static problems an interval linear system of equations is developed. A guaranteed enclosure for the solution of interval linear system of equations is achievable and usually is not sharp and very conservative; however, an exact enclosure is not known to be obtained in the general case of such systems. In this work, a very sharp enclosure for the solution set, due to loading, material and geometric uncertainty in solid mechanics problems, is obtained. The new formulation is based on an element-by-element technique. Element matrices are formulated, based on the physics of materials, and the Lagrange multiplier method is applied to impose the necessary constraints for compatibility and equilibrium. Most sources of overestimation were eliminated, and a very sharp solution is obtained. A number of numerical examples are introduced.