论文标题

使用统计有限元方法从稀疏测量中推断位移字段

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

论文作者

Narouie, Vahab B., Wessels, Henning, Römer, Ulrich

论文摘要

从稀疏数据中推断出完全位移和应力场的一种完善的方法是使用贝叶斯更新校准给定本构模型的参数。校准后,使用已确定的模型参数进行A(随机)正向模拟,以解决测量设备无法访问的区域中的物理场。模型校准的缺点是,该模型被视为最能代表现实,这是有时的情况,尤其是在结构和材料的老化的背景下。虽然通常通过重复的模型校准解决了此问题,但最近提出的统计有限元方法(StatFem)遵循了另一种方法。与其使用贝叶斯定理更新模型参数,而是选择位移作为随机先验,并更新以更接近测量数据。为此,STATFEM框架引入了所谓的模型不匹配,仅通过三个超参数进行了参数。这使得在线阶段的全场数据计算上有效地推断了:如果可以离线计算随机先验,则不需要在线求解基本的部分微分方程(PDE)。与解决PDE相比,仅识别三个超参数并在传感器数据上调节状态需要更少的计算资源。 本文对现有的STATFEM方法提出了两种贡献:首先,我们使用非侵入性多项式混乱方法来计算先验,从而在确定性公式中使用复杂的机械模型。其次,我们研究了先前的材料模型(具有不确定杨氏模量的线性弹性和圣文族基尔奇霍夫材料)对更新解决方案的影响。我们提供了1D和2D示例的STATFEM结果,而3D的扩展名很简单。

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.

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