noise and number of sensors Giovanni Capellari Eleni Chatzi - - PowerPoint PPT Presentation

Optimal sensor placement through Bayesian experimental design: effect of measurement noise and number of sensors

3rd International Electronic Conference on Sensors and Aplications, 10-15 November 2016

Giovanni Capellari Eleni Chatzi Stefano Mariani

Motivation

Data collection 𝒛 Parameters estimation 𝜾 SHM system design 𝒆 Decision making

Structural Health Monitoring can be conceptually divided in three stages: in our work, we will focus on the design of the sensor network

Motivation

Identifiability Estimates Uncertainty SHM system cost # sensors measurement error Optimal SHM system design configuration

The usefulness of the sensor network depends on the number, type and location of the sensors. Therefore, we need a method to quantify the information obtained by the acquisition system.

Optimal sensor placement: deterministic methods

EFI KE EVP

• M. Meo, G. Zumpano, (2005), M. Bruggi, S. Mariani, (2013), Leyder, C., Ntertimanis, V., Chatzi, E., Frangi, A. (2015).

Sensitivity to damage

The existing approaches does not take into account the measurement noise, i.e. the sensors accuracy.

Optimal sensor placement: Bayesian framework

• X. Huan, Y. M. Marzouk, (2013).

Expected gain in Shannon information 𝑉 𝒆 = න

𝒁

න

Θ

𝑣 𝒆, 𝒛, 𝜾 𝑞 𝜾, 𝒛 𝒆 𝑒𝜾𝑒𝒛 𝒆∗ = arg max

𝒆∈𝑬 𝑉(𝒆)

Monte Carlo sampling Prior: 𝜾~𝑞 𝜾 Likelihood: 𝒛~𝑞(𝒛|𝜾, 𝒆) 𝑉 𝒆 ≈ 1 𝑜𝑝𝑣𝑢 ෍

𝑗=1 𝑜𝑝𝑣𝑢

ln 𝑞 𝒛𝑗 𝜾𝑗, 𝒆 − ln 1 𝑜𝑗𝑜 ෍

𝑘=1 𝑜𝑗𝑜

𝑞 𝒛𝑗 𝜾𝑘, 𝒆

In a Bayesian sense, the optimal spatial configuration 𝒆∗ of the sensor network can be found by maximizing the Shannon information gain. In

• rder to compute it, we use a Monte Carlo approximation.

Model evaluation

• Evaluation of the likelihood

𝑞 𝒛𝑗 𝜾𝑘, 𝒆 = 𝑞𝛝 𝒛𝑗 − 𝑯 𝜾𝑘, 𝒆 𝑯 𝜾, 𝒆 = 𝑴 𝒆 𝑳(𝜾)−1𝑮

Observation matrix 𝑳 𝜾 = ෍

𝑗=1 𝑜𝜾

𝐹𝑗 𝐹 − 1 𝑳𝑣𝑜𝑒 − 𝑳𝑗

• Forward model

𝒛 = 𝑯 𝜾, 𝒆 + 𝛝

Measurement noise

The measurements are related to the mechanical parameters to be estimated through a FEM-based forward model. The sensor accuracy is taken into account through a fictitious measurement noise.

Optimization

𝜾𝑗~𝑞 𝜾 , 𝒆𝑗~𝒱 𝓔 𝒀𝑗= 𝜾𝑗

𝑈 𝒆𝑗 𝑈

𝒛𝑄𝐷𝐹 = 𝑁 𝒀 = σ𝜷∈ℕ𝑁 𝑧𝛽𝜔𝛽 𝒀 𝒛𝑗𝐺𝐹 = 𝑯 𝜾𝑗, 𝒆𝑗

• Surrogate model: polynomial chaos expansion
• Optimization: Covariance Matrix Adaptation Evolution Strategy

(CMA-ES)

• 1. 𝒆𝑗~𝒏 + 𝜏𝒪

𝑗 𝟏, 𝑫

𝒏 ∈ ℝ𝑜𝒆, 𝑫 ∈ ℝ𝑜𝒆×𝑜𝒆

• 2. 𝒏 and 𝑫 are updated through cumulation
• 3. Check the tolerance on 𝑉 𝒆
• N. Hansen, S.D. Müller, P. Koumoutsakos, (2003).

In order to reduce the computational cost of the forward model, a cheaper surrogate model is built.

Bayesian OSP framework

Sample input variables 𝜾𝑗~𝑞 𝜾 , 𝒆𝑗~𝒱 𝓔 𝒀𝑗= 𝜾𝑗

𝑈 𝒆𝑗 𝑈

System response 𝑯𝐺𝐹 𝜾𝑗, 𝒆𝑗 PCE surrogate 𝑯𝐺𝐹 𝜾𝑗, 𝒆𝑗 ≅ 𝑯𝑄𝐷𝐹 𝜾𝑗, 𝒆𝑗 Maximizing information Sample design variable 𝒆𝑚 MC approximation 𝑉(𝒆𝑚) Update 𝒆𝑚→ 𝒆𝑚+1 (CMA-ES) Check tolerance on 𝑉 𝒆𝑚 − 𝑉 𝒆𝑚+1 Optimal configuration 𝒆∗ Training surrogate model

Application: simply supported plate

10x10 mesh: 726 d.o.f. Displacement measurements 4 zones: 𝜾 = 𝐹1, 𝐹2, 𝐹3, 𝐹4

Application: simply supported plate Choice of prior distribution 𝑞 𝜾

𝑞 𝜾 ~𝒱 0, 𝐹 𝑞 𝜾 ~𝒱 2 𝐹 3 , 𝐹 𝑂𝑡: # sensors 𝑂𝑄𝐷𝐹: # PCE samples 𝑞: PCE polynomial degree 𝑂𝑁𝐷: # MC samples 𝜾 = 𝐹1 𝐹2 𝐹3 𝐹4 𝑂𝑡 = 4, 𝑂𝑄𝐷𝐹 = 104, 𝑞 = 10, 𝑂𝑁𝐷 = 5 · 103

Optimal position of 𝑜𝑡 = 4 sensors, results of 10 algorithm runs

Application: simply supported plate Effect of σ𝛝

2.4 2.5 2.6 2.7 2.8 2.9

𝜗~𝒪 0, 𝜏𝜗

2

𝜾 = 𝐹2 𝑂𝑡 = 1, 𝑂𝑄𝐷𝐹 = 104, 𝑞 = 10, 𝑂𝑁𝐷 = 5 · 103

0.693 0.6935 0.694 0.6945 0.695 0.6955 0.696 0.6965 0.697 0.6926 0.6927 0.6928 0.6929 0.693 0.6931 0.6932 0.6933 0.6934 0.6935 0.6936

σ𝜗 = 10−3 m σ𝜗 = 10−4 m σ𝜗 = 10−5 m 𝑂𝑡: # sensors 𝑂𝑄𝐷𝐹: # PCE samples 𝑞: PCE polynomial degree 𝑂𝑁𝐷: # MC samples

Contour of the objective function with one sensor for each possible location

• n the plate with different standard deviations of the measurement noise.

Application: simply supported plate Effect of σ𝛝 and number of sensors

𝜗~𝒪 0, 𝜏𝜗

2

𝜾 = 𝐹2 𝑂𝑄𝐷𝐹 = 104, 𝑞 = 10, 𝑂𝑁𝐷 = 5 · 103 𝑂𝑡: # sensors 𝑂𝑄𝐷𝐹: # PCE samples 𝑞: PCE polynomial degree 𝑂𝑁𝐷: # MC samples

Contour of the objective function with one sensor for different standard deviations and number of sensors.

Conclusions

• Optimal sensor placement and SHM system design
• Take into account:
• Measurements uncertainties
• Number of sensors
• Maximization of expected information gain between prior and

posterior

• Use of surrogate model (PCE) for MC approximation and stochastic
• ptimization (CMA-ES) methods for computational speed-up
• Future developments: larger number of sensors, larger number of

parameters, application to complex cases

Bayesian optimal experimental design

References

Bruggi, M., and Mariani, S. (2013). “Optimization of sensor placement to detect damage in flexible plates.” Engineering Optimization, 45(6), 659–676. Capellari, G., Eftekhar Azam, S., Mariani, S. (2016). “Towards real-time health monitoring of structural systems via recursive Bayesian filtering and reduced order modelling.” International Journal of Sustainable Materials and Structural Systems, In Press. Hansen, N., Müller, S. D., Koumoutsakos, P. (2003). “Reducing the time complexity of the derandomized evolution strategy with Covariance Matrix Adaptation (CMA-ES).” Evolutionary Computation, 11(1), 1-18. Huan, X., and Marzouk, Y. M. (2013). “Simulation-based optimal Bayesian experimental design for nonlinear systems.” Journal of Computational Physics, 232(1), 288–317. Leyder, C., Ntertimanis, V., Chatzi, E., Frangi, A. (2015). “Optimal sensors placement for the modal identification of an innovative timber structure.” Proceedings of the 1st International Conference on Uncertainty Quantification in Computational Sciences and Engineering, 467-476. Lindley, D. V. (1972). Bayesian Statistics, A Review, Society for Industrial and Applied Mathematics, SIAM. Marelli, S., and Sudret, B. (2015). UQLab User Manual, Chair of Risk, Safety & Uncertainty Quantification, ETH Zürich. Capellari, G., Chatzi, C., Mariani, S. (2016). An optimal sensor placement method for SHM based on Bayesian experimental design and Polynomial Chaos Expansion Proceedings of the VII European Congress

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