Design of de-noising FEM-FIR filters for the evaluation of temporal and spatial derivatives of measured displacement in elastic solids

摘要

This paper presents an FEM-FIR filter de-noising measured displacement to estimate temporal and spatial derivatives of displacement in an elastic solid. The filter is formulated by two independent inverse problems, which are defined as the minimization of the difference between measured and de-noised displacement in the temporal and spatial domains. The regularization functions are introduced in the minimization problems to eliminate noise in measured displacement. The L-2-norm of the acceleration and the strain energy of an elastic solid are utilized as the regularization functions in the temporal and spatial domains, respectively, to impose the integrability of the temporal and spatial derivatives. The moving time-window technique is adopted in the temporal filter, and a usual finite element procedure is applied to discretize the minimization problems. The temporal and spatial filters are combined together to form a single, unified filter, which not only denoises measured displacement in both domains but also reconstructs the velocity field. The acceleration field is obtained by the 1st-order central finite difference of the reconstructed velocity field. The strain field is evaluated at the Gauss points of each finite element using the de-noised nodal displacement. The regularization factors, which control the degree of de-noising strength, are determined by the desired accuracy of the filters at the target frequency. It is shown through a numerical simulation study that the proposed filters are capable of de-noising measured displacement effectively and yielding accurate temporal and spatial derivatives. (C) 2018 Elsevier Ltd. All rights reserved.