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

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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.