23 Optimal Sensor Placement for Response Reconstruction in Structural Dynamics 209 The optimal sensor configuration δopt is obtained by maximizing the U δ with respect to the design variables δ δopt =argmax δ U δ (23.13) The optimal number of sensors in the sensor configuration can be estimated by monitoring the gain in information as additional sensors are placed in the structure. Usually, after sufficiently number of sensors placed in the structure, the information gain using additional sensors is relatively small. The optimal number of sensors is a trade-off between information gain and cost of sensors. The optimization in Eq. (23.13) may result in multiple local/global solutions. The optimization problem is solved in the continuous physical domain of variation of the sensor locations. Stochastic optimization algorithms, such as CMA-ES [13], can be employed in order to avoid premature convergence to a local optimum. Alternative heuristic forward and backward sequential sensor placement algorithms [14, 15] are effective in solving the optimization problem. The heuristic algorithm bypasses the problem of multiple local/global optima manifested in optimal experimental designs, providing near optima solutions in a fraction of the computational effort required in expensive stochastic optimization algorithms. 23.4 Conclusion Using information theory and utility function, the optimal sensor placement problem for response reconstruction is formulated as a problem of maximizing a multi-dimensional integral of the minus the information entropy in the parameter space. The framework provides optimal sensor configurations that are robust to uncertainties in the model parameters as well as uncertainties in the state and measurement errors. Such uncertainties are not known in the initial optimal experimental design phase and thus need to be postulated or partly learned from the data using Bayesian inference techniques. Monte Carlo or sparse grid techniques can be used to estimate the multidimensional integral. Computationally efficient heuristic sequential sensor placement strategies can be employed to estimate the near optimal sensor locations. The proposed framework is applicable to complex linear systems involving uncertainties in their parameters. It is appropriate to use for reliably reconstructing responses that are important for providing data-driven reliability and safety estimates of systems, as well as reconstruct stress responses that are needed in fatigue damage accumulation theories [1, 7]. The proposed framework can be implemented with response reconstruction techniques based on modal expansion methods, as well as filter-based methods for joint input-state estimation. Moreover, it can be used to explore the number and type of sensors that are needed to provide reliable estimates of output QoI. Acknowledgements The author gratefully acknowledges the European Commission for its support of the Marie Sklodowska Curie program through the ETN DyVirt project (GA 764547). References 1. Papadimitriou, C., Fritzen, C., Kraemer, P., Ntotsios, E.: Fatigue predictions in entire body of metallic structures from a limited number of vibration sensors using Kalman filtering. Struct. Control. Health Monit. 18, 554–573 (2011) 2. Lourens, E., Reynders, E., De Roeck, G., Degrande, G., Lombaert, G.: An augmented Kalman filter for force identification in structural dynamics. Mech. Syst. Signal Process. 27, 446–460 (2012) 3. Lourens, E., Papadimitriou, C., Gillijns, S., Reynders, E., De Roeck, G., Lombaert, G.: Joint input-response estimation for structural systems based on reduced-order models and vibration data from a limited number of sensors. Mech. Syst. Signal Process. 29, 310–327 (2012) 4. Azam, S.E., Chatzi, E., Papadimitriou, C.: A dual Kalman filter approach for state estimation via output-only acceleration measurements. Mech. Syst. Signal Process. 60, 866–886 (2015) 5. Lourens, E., Fallais, D.J.M.: Full-field response monitoring in structural systems driven by a set of identified equivalent forces. Mech. Syst. Signal Process. 114, 106–119 (2019) 6. Maes, K., Smyth, A.W., De Roeck, G., Lombaert, G.: Joint input-state estimation in structural dynamics. Mech. Syst. Signal Process. 70–71, 445–466 (2016) 7. Papadimitriou, C., Chatzi, E.N., Azam, S.E., Dertimanis, V.K.: Fatigue monitoring and remaining lifetime prognosis using operational vibration measurements. In: Model Validation and Uncertainty Quantification, vol. 3, pp. 133–136. Springer, Berlin (2019) 8. Maes, K., Lourens, E., Van Nimmen, K., Reynders, E., De Roeck, G., Lombaert, G.: Design of sensor networks for instantaneous inversion of modally reduced order models in structural dynamics. Mech. Syst. Signal Process. 52–53, 628–644 (2015) 9. Zhang, C.D., Xu, Y.L.: Optimal multi-type sensor placement for response and excitation reconstruction. J. Sound Vib. 360, 112–128 (2016)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==