Interval Finite Element as a Basis for Generalized Models of Uncertainty in Engineering Mechanics

Latest scientific and engineering advances have started to recognize the need of defining multiple types of uncertainty. Probabilistic modeling cannot handle situations with incomplete or little information on which to evaluate a probability, or when that information is nonspecific, ambiguous, or conflicting [46, 11, 43]. Many interval-based models of uncertainty have been developed to treat such situations. This paper presents an interval approach for the treatment of parameter uncertainty for linear static problems of mechanics. Uncertain parameters are introduced in the form of unknown but bounded quantities (intervals). Interval analysis is applied to Finite Element Method to analyze the system response due to uncertain stiffness and loading. To avoid overestimation, the formulation is based on an element-by-element (EBE) 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. Earlier EBE formulation provided sharp bounds only on displacements [29]. Based on the developed formulation, the bounds on the system’s displacement and forces are obtained simultaneously and have the same level of accuracy. Very sharp enclosures for the exact system responses are obtained. A number of numerical examples are introduced and scalability is illustrated.