Finite element model updating through sum-of-squares (SOS) optimization and constrained Kalman filters
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Finite element (FE) modeling techniques optimize model parameter values for improving the predication accuracy of a numerical model. This research investigates algorithms utilizing the frequency-domain modal properties and the time-domain dynamic responses for FE model updating. In terms of frequency-domain model updating, a global optimization algorithm, the sum-of-squares (SOS) method, is proposed to solve the modal dynamic residual formulation for finding optimal model parameters. The SOS method can reformulate a nonconvex polynomial optimization problem into a convex semidefinite programming (SDP) problem, the global optimal solution of which can be reliably solved. In order to improve the computational efficiency, the sparsity of the optimization problem and facial reduction technique are investigated for reducing the size of the reformulated SDP problem. The proposed SOS method and efficiency improvement techniques are validated through numerical studies of a four-story shear frame structure and a plane truss structure. In terms of time-domain model updating, the constrained extended Kalman filter (CEKF) and the constrained unscented Kalman filter (CUKF) are proposed to recursively update model parameters for both linear and nonlinear structures. Incorporating constraints during the model updating process can effectively prevent parameter estimates from being unrealistic. Analytical solution of the Kalman gain is derived when there are inequality constraints. With the explicit expression of the Kalman gain, the estimation process can be significantly accelerated. The proposed CEKF and CUKF are validated through numerical studies of a linear four-story shear frame structure and a single degree of freedom (SDOF) Bouc-Wen hysteretic system. Besides numerical studies, experimental measurements are also used to evaluate the model updating performance of the proposed methods. The first example is a four-story shear frame structure in laboratory. The second example is a full-scale reinforced concrete frame structure in field. Model updating performance of proposed SOS optimization method and constrained Kalman filters are investigated through comparison between simulated responses and experimental measurements.