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# Inferring Displacement Fields from Sparse Measurements Using the Statistical Finite Element Method

Dec 2022

A well-established approach for inferring full displacement and stress fieldsfrom possibly sparse data is to calibrate the parameter of a given constitutivemodel using a Bayesian update. After calibration, a (stochastic) forwardsimulation is conducted with the identified model parameters to resolvephysical fields in regions that were not accessible to the measurement device.A shortcoming of model calibration is that the model is deemed to bestrepresent reality, which is only sometimes the case, especially in the contextof the aging of structures and materials. While this issue is often addressedwith repeated model calibration, a different approach is followed in therecently proposed statistical Finite Element Method (statFEM). Instead of usingBayes' theorem to update model parameters, the displacement is chosen as thestochastic prior and updated to fit the measurement data more closely. For thispurpose, the statFEM framework introduces a so-called model-reality mismatch,parametrized by only three hyperparameters. This makes the inference offull-field data computationally efficient in an online stage: If the stochasticprior can be computed offline, solving the underlying partial differentialequation (PDE) online is unnecessary. Compared to solving a PDE, identifyingonly three hyperparameters and conditioning the state on the sensor datarequires much fewer computational resources. This paper presents two contributions to the existing statFEM approach:First, we use a non-intrusive polynomial chaos method to compute the prior,enabling the use of complex mechanical models in deterministic formulations.Second, we examine the influence of prior material models (linear elastic andSt.Venant Kirchhoff material with uncertain Young's modulus) on the updatedsolution. We present statFEM results for 1D and 2D examples, while an extensionto 3D is straightforward.

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