Abstract
The Inverse Finite Element method (iFEM), employing a network of strain sensors installed on a structure reconstructs the displacement field independently of the structural loading conditions and material properties. However, the solution is compromised when the sensor network, due to logistic or cost constraints, is sparse and measureless finite elements are present. To overcome this issue the iFEM minimizes a weighted functional, assigning smaller weights to the elements missing experimental measures. Strain field pre-extrapolation techniques have been proposed to improve the iFEM performance, although still assigning arbitrarily small weights to the extrapolated strains. The current paper proposes a Gaussian Process as the pre-extrapolation technique for the strain field, which natively incorporates measurement uncertainty, therefore providing a metric to assign the functional weights, as well as confidence intervals for the displacement field computed through the iFEM. The proposed technique is validated with a virtual experiment; advantages and limitations of the proposed approach are also discussed.
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Notes
- 1.
It should be noted that since the solution is derived from a least square minimization of a weighted functional, the result does not depend on the absolute values of weights, rather on their ratio.
- 2.
In most shells, it is physically impossible to install strain sensors measuring the out-of-plane strain components g. Moreover, their contribution to the displacements in thin shells is at least one order of magnitude lower with respect to the contributions of the membrane and of the bending strains. Hence, in this article they are neglected and set to zero, without hindering the generality of the proposed method.
- 3.
The explicit form of the term \(\xi ^{i}\) has been omitted since it is a constant not multiplied by \({\mathbf {u}}^{i}\).
- 4.
A more rigorous approach for a proper hyperparameter inference would entail using a Markov-Chain Monte Carlo method to integrate out the hyperparameters from the posterior distribution. This approach has not been followed in this paper considering that in some applications the differences in the predictions are limited [3], but it is left for future research.
- 5.
It should be noted that only one sample from the sensors was needed to compute the uncertainty. Including more samples from the same sensors would shrink the CI.
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Poloni, D., Oboe, D., Sbarufatti, C., Giglio, M. (2023). Gaussian Process Strain Pre-extrapolation and Uncertainty Estimation for Inverse Finite Elements. In: Rizzo, P., Milazzo, A. (eds) European Workshop on Structural Health Monitoring. EWSHM 2022. Lecture Notes in Civil Engineering, vol 254. Springer, Cham. https://doi.org/10.1007/978-3-031-07258-1_32
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