Non-negative constrained inverse eigenvalue problems – Application to damage identification

• Published in: Mechanical systems and signal processing Vol. 129; pp. 629 - 644
• Main Authors: Smith, Chandler B., Hernandez, Eric M.
• Format: Journal Article
• Language: English
• Published: Berlin Elsevier Ltd 15.08.2019; Elsevier BV
• Subjects: Computer simulation; Constrained nonlinear least squares; Damage detection; Damage identification; Eigenvalues; Finite element method; Finite element model updating; Ill posed problems; Ill-posed inverse; Linearization; Model updating; Noise measurement; Non-negative constraint; Optimization; Performance assessment; Sparsity; Stiffness; Truncation errors; Non-negative constraint; Damage identification; Sparsity; Ill-posed inverse; Constrained nonlinear least squares; Finite element model updating
• ISSN: 0888-3270; 1096-1216
• DOI: 10.1016/j.ymssp.2019.04.052

•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. 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 l1-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.