Non-negative constrained inverse eigenvalue problems – Application to damage identification. (15th August 2019)

• Record Type: Journal Article
• Authors: Smith, Chandler B.; Hernandez, Eric M.
• Is Part Of: Mechanical systems and signal processing. Volume 129(2019)
• Page Start: 629
• Page End: 644
• Publication Date: 2019-08-15
• Subjects: Sparsity -- Non-negative constraint -- Ill-posed inverse -- Damage identification -- Finite element model updating -- Constrained nonlinear least squares; Structural dynamics -- Periodicals; Vibration -- Periodicals; Constructions -- Dynamique -- Périodiques; Vibration -- Périodiques; Structural dynamics; Vibration; Periodicals; 621
• DOI: 10.1016/j.ymssp.2019.04.052
• Languages: English
• ISSNs: 0888-3270

Abstract

Highlights: Local damage identification using non-negative least squares is investigated. Unique non-negative and sparse solutions to ill-posed linearized inverse problem. Theory and numerical simulations on shear beam and truss support findings. Iterative method for non-negative constrained nonlinear least squares is proposed. Local damage identified from underdetermined nonlinear system using proposed method. Abstract: Damage identification using eigenvalue shifts is ill-posed because the number of identifiable eigenvalues is typically far less than the number of potential damage locations. This paper shows that if damage is defined by sparse and non-negative vectors, such as the case for local stiffness reductions, then the non-negative solution to the linearized inverse eigenvalue problem can be made unique with respect to a subset of eigenvalues significantly smaller than the number of potential damage locations. Theoretical evidence, numerical simulations, and performance comparisons to sparse vector recovery methods based on l 1 -norm optimization are used to validate the findings. These results are then extrapolated to the ill-posed nonlinear inverse eigenvalue problem in cases where damage is large, and linearization induces non-negligible truncation errors. In order to approximate the solution to the non-negative nonlinear least squares, a constrained finite element model updating approach is presented. The proposed method is verified using three simulated structures of increasing complexity: a one-dimensional shear beam, a planar truss, and a three-dimensional space structure. For multiple structures, this paper demonstrates that the proposed method finds sparse solutions in the presence of measurement noise.