A novel reduced-basis method with upper and lower bounds for real-time computation of linear elasticity problems

This paper presents a novel reduced-basis method for analyzing problems of linear elasticity in a systematical, rapid and reliable fashion for solutions with both upper and lower bounds to the exact solution in the form of energy norm or compliance output. The lower bound of the solution output is obtained form the well-known reduced-basis method based on the Galerkin projection used in the finite element method, which is termed as GP_RBM. For the upper bound, a new reduced-basis approach is developed by the combination of the reduced-basis method and a smoothed Galerkin projection used in the linearly conforming point interpolation method, and it is thus termed as SGP_RBM. To examine the present SGP_RBM, we first conduct a theoretical study on the very important upper bound property. Reduced-basis models for both GP_RBM and SGP_RBM are constructed with the aid of an asymptotic error estimation and greedy adaptive procedure. The GP_RBM and the newly proposed SGP_RBM are applied to analyze a cantilever beam with an oblique crack to verify the proposed RBM technique in terms of accuracy, convergence, bound properties and computational savings. Both theoretical analysis and numerical results have demonstrated that the present method is a very efficient method for real-time solutions providing exact output bounds.