Non-negative and sparsity constrained inverse problems in damage identification – Application to a full-scale 3D truss

•Multiple damages identified on full-scale experiment using natural frequencies.•Unique non-negative and sparse solutions to underdetermined inverse problems.•Bounded non-negative solution space investigated.•Non-negative constrained L1-norm optimization investigated. Non-negative constrained least squares and l1-norm optimization are sometimes viable inverse-based methods used to quantify and locate damage described by local stiffness reductions using measured changes in natural frequencies. Although the two methods provide meaningful solutions to the associated underdetermined inverse problem when the physically correct solution is sufficiently sparse, each method is disadvantaged in terms of either solution uniqueness, regularization, or forced sparsity. This paper addresses these challenges and argues that combining the non-negative constraint and l1-norm optimization improves performance and abates or improves upon the deficiencies of the standalone methods. This paper demonstrates that the optimal set of solutions satisfying the non-negative least squares is bounded and that estimating these bounds provides a novel measure for interpreting the validity of the sparse solution recovered from the proposed non-negative constrained l1-norm optimization method. The proposed method is numerically verified and experimentally tested on vibration data taken from a 17.24 m × 1.98 m × 1.83 m full-scale three-dimensional truss subjected to three progressive local damage cases.