A well-established approach for inferring full displacement and stress fields from possibly sparse data is to calibrate the parameter of a given constitutive model using a Bayesian update. After calibration, a (stochastic) forward simulation is conducted with the identified model parameters to resolve physical fields in regions that were not accessible to the measurement device. A shortcoming of model calibration is that the model is deemed to best represent reality, which is only sometimes the case, especially in the context of the aging of structures and materials. While this issue is often addressed with repeated model calibration, a different approach is followed in the recently proposed statistical Finite Element Method (statFEM). Instead of using Bayes' theorem to update model parameters, the displacement is chosen as the stochastic prior and updated to fit the measurement data more closely. For this purpose, the statFEM framework introduces a so-called model-reality mismatch, parametrized by only three hyperparameters. This makes the inference of full-field data computationally efficient in an online stage: If the stochastic prior can be computed offline, solving the underlying partial differential equation (PDE) online is unnecessary. Compared to solving a PDE, identifying only three hyperparameters and conditioning the state on the sensor data requires 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 and St.Venant Kirchhoff material with uncertain Young's modulus) on the updated solution. We present statFEM results for 1D and 2D examples, while an extension to 3D is straightforward.

翻译：一种从可能稀疏的数据中推断完整位移场和应力场的成熟方法是通过贝叶斯更新对给定本构模型的参数进行标定。标定后，使用已识别的模型参数执行（随机）正向模拟，以解析测量设备无法触及区域的物理场。模型标定的一个缺点在于该模型被视为最能代表真实情况，而这种情况仅在少数情况下成立，尤其是在结构和材料老化的背景下。虽然此问题常通过重复模型标定来解决，但近期提出的统计有限元方法（statFEM）采用了不同的思路。该方法并非利用贝叶斯定理更新模型参数，而是将位移场选为随机先验，并通过更新使其更贴合测量数据。为此，statFEM框架引入了所谓的"模型-现实失配"，仅由三个超参数参数化。这使得全场数据的推断在在线阶段具有计算高效性：若随机先验可离线计算，则无需在线求解底层偏微分方程（PDE）。与求解PDE相比，仅识别三个超参数并根据传感器数据对状态进行条件化所需的计算资源少得多。本文对现有statFEM方法作出两项贡献：首先，我们采用非侵入式多项式混沌方法计算先验，从而能在确定性公式中应用复杂力学模型；其次，我们考察了先验材料模型（线弹性材料与具有不确定性杨氏模量的圣维南-基尔霍夫材料）对更新解的影响。我们展示了1D和2D示例的statFEM结果，而扩展到3D的方法则具有直接性。