Application of interval arithmetic to structural mechanics problems with uncertainty

Abstract

Many problems in structural mechanics can be solved using the finite element method (FEM), wherein a model for a mechanical system is constructed by discretising the structure into a finite set of structural elements, connected at nodes, leading to a system of equations to be solved. In the case of linearised geometric displacement equations and linear elastic material behaviour, a system of linear equations is obtained. Uncertainty may be present in some or all of the physical model parameters, caused for example by measurement and fabrication imprecision or rounding errors. A mechanical frame or truss structure will typically be constructed so that the node positions, before loading, are only known to a tolerance of several millimetres. We therefore consider not only uncertain material parameters but also uncertain node locations and correspondingly uncertain element lengths, as well as uncertain loading forces. Intervals can be used to model such parameters when their values are known to lie within certain bounds. In this case, we obtain a system of equations involving interval parameters. However, a naive solution of this system, using interval arithmetic, will typically lead to a solution with result intervals that are hopelessly wide. In our approach, firstly guaranteed starting interval enclosures for the node displacements which are relatively wide are computed. These solution intervals are then iteratively tightenend by performing a monotonicity analysis of all the parameters coupled with a solver for interval systems of linear equations, providing tight guaranteed enclosures for the node displacements of the model.